I have been playing around with some old algebraic integrands in the
new version 1.3.10 of FriCAS on the web interface.

Sam Blake's pseudo-elliptic of April 2020 still gives:

integrate((x^4 - 1)*(x^4 + x^2 + 1)*sqrt(-x^4 + x^2 - 1)
/(x^4 + 1)^3, x)

Error detected within library code:

catdef: division by zero

perhaps because the radicand is negative everywhere on the real axis.

And an older and presumably truly elliptic case still fails:

integrate((5*x - 9*sqrt(6) + 26)
/((x^2 - 4*x - 50)*sqrt(x^3 - 30*x - 56)), x)

Error detected within library code:

catdef: division by zero

in the same manner, although the radicand is cubic here.

The following integrand by Legendre is still evaluated to six complex logarithms:

integrate(x/((4 - x^3)*sqrt(1 - x^3)), x)

(((-3)^(1/2)+(-1))*((-1)/432)^(1/6)*log((((3888*x^5*(-3)^(1/2)+3888*x^5)*(((-1)/432)^(1/6))^5+(360*x^6+1440*x^3+(-1152))*(((-1)/432)^(1/6))^3+(((-6)*x^7+(-96)*x^4+48*x)*(-3)^(1/2)+(6*x^7+96*x^4+(-48)*x))*((-1)/432)^(1/6))*((-1)*x^3+1)^(1/2)+((((-864)*x^7+
(-864)*x^4+1728*x)*(-3)^(1/2)+(864*x^7+864*x^4+(-1728)*x))*(((-1)/432)^(1/6))^4+(((-36)*x^8+(-252)*x^5+288*x^2)*(-3)^(1/2)+((-36)*x^8+(-252)*x^5+288*x^2))*(((-1)/432)^(1/6))^2+((-1)*x^9+(-66)*x^6+72*x^3+(-32))))/(x^9+(-12)*x^6+48*x^3+(-64)))+(((-3)^(1/2)+
1)*((-1)/432)^(1/6)*log((((3888*x^5*(-3)^(1/2)+(-3888)*x^5)*(((-1)/432)^(1/6))^5+((-360)*x^6+(-1440)*x^3+1152)*(((-1)/432)^(1/6))^3+(((-6)*x^7+(-96)*x^4+48*x)*(-3)^(1/2)+((-6)*x^7+(-96)*x^4+48*x))*((-1)/432)^(1/6))*((-1)*x^3+1)^(1/2)+(((864*x^7+864*x^4+(-
1728)*x)*(-3)^(1/2)+(864*x^7+864*x^4+(-1728)*x))*(((-1)/432)^(1/6))^4+((36*x^8+252*x^5+(-288)*x^2)*(-3)^(1/2)+((-36)*x^8+(-252)*x^5+288*x^2))*(((-1)/432)^(1/6))^2+((-1)*x^9+(-66)*x^6+72*x^3+(-32))))/(x^9+(-12)*x^6+48*x^3+(-64)))+((-2)*((-1)/432)^(1/6)*
log(((7776*x^5*(((-1)/432)^(1/6))^5+((-360)*x^6+(-1440)*x^3+1152)*(((-1)/432)^(1/6))^3+(12*x^7+192*x^4+(-96)*x)*((-1)/432)^(1/6))*((-1)*x^3+1)^(1/2)+(((-1728)*x^7+(-1728)*x^4+3456*x)*(((-1)/432)^(1/6))^4+(72*x^8+504*x^5+(-576)*x^2)*(((-1)/432)^(1/6))^2+((
-1)*x^9+(-66)*x^6+72*x^3+(-32))))/(x^9+(-12)*x^6+48*x^3+(-64)))+(2*((-1)/432)^(1/6)*log((((-7776)*x^5*(((-1)/432)^(1/6))^5+(360*x^6+1440*x^3+(-1152))*(((-1)/432)^(1/6))^3+((-12)*x^7+(-192)*x^4+96*x)*((-1)/432)^(1/6))*((-1)*x^3+1)^(1/2)+(((-1728)*x^7+(-
1728)*x^4+3456*x)*(((-1)/432)^(1/6))^4+(72*x^8+504*x^5+(-576)*x^2)*(((-1)/432)^(1/6))^2+((-1)*x^9+(-66)*x^6+72*x^3+(-32))))/(x^9+(-12)*x^6+48*x^3+(-64)))+(((-1)*(-3)^(1/2)+(-1))*((-1)/432)^(1/6)*log(((((-3888)*x^5*(-3)^(1/2)+3888*x^5)*(((-1)/432)^(1/6))^
5+(360*x^6+1440*x^3+(-1152))*(((-1)/432)^(1/6))^3+((6*x^7+96*x^4+(-48)*x)*(-3)^(1/2)+(6*x^7+96*x^4+(-48)*x))*((-1)/432)^(1/6))*((-1)*x^3+1)^(1/2)+(((864*x^7+864*x^4+(-1728)*x)*(-3)^(1/2)+(864*x^7+864*x^4+(-1728)*x))*(((-1)/432)^(1/6))^4+((36*x^8+252*x^5+(
-288)*x^2)*(-3)^(1/2)+((-36)*x^8+(-252)*x^5+288*x^2))*(((-1)/432)^(1/6))^2+((-1)*x^9+(-66)*x^6+72*x^3+(-32))))/(x^9+(-12)*x^6+48*x^3+(-64)))+((-1)*(-3)^(1/2)+1)*((-1)/432)^(1/6)*log(((((-3888)*x^5*(-3)^(1/2)+(-3888)*x^5)*(((-1)/432)^(1/6))^5+((-360)*x^6+(
-1440)*x^3+1152)*(((-1)/432)^(1/6))^3+((6*x^7+96*x^4+(-48)*x)*(-3)^(1/2)+((-6)*x^7+(-96)*x^4+48*x))*((-1)/432)^(1/6))*((-1)*x^3+1)^(1/2)+((((-864)*x^7+(-864)*x^4+1728*x)*(-3)^(1/2)+(864*x^7+864*x^4+(-1728)*x))*(((-1)/432)^(1/6))^4+(((-36)*x^8+(-252)*x^5+
288*x^2)*(-3)^(1/2)+((-36)*x^8+(-252)*x^5+288*x^2))*(((-1)/432)^(1/6))^2+((-1)*x^9+(-66)*x^6+72*x^3+(-32))))/(x^9+(-12)*x^6+48*x^3+(-64))))))))/36

... even though a real expression for the antiderivative exists:

INT(x/((4 - x^3)*SQRT(1 - x^3)), x) =
2^(1/3)/18*(ATANH(SQRT(1 - x^3))
- 3*ATANH((1 + 2^(1/3)*x)/SQRT(1 - x^3))
- SQRT(3)*ATAN((2^(1/3) - 2^(2/3)*x - x^2)
/(SQRT(3)*2^(1/3)*SQRT(1 - x^3))))

Can't these complex logarithms be broken down similar to those for
integral 5.66 (#401) from the Timofeev suite?

And for the next integrand, FriCAS still produces unreasonable integers
in an arc tangent's argument:

integrate(1/((x + 1)*(x^3 + 2)^(1/3)), x)

(log(((21*x^4+(-6)*x^3+(-96)*x^2+(-60)*x+12)*((x^3+2)^(1/3))^2+(21*x^5+(-48)*x^3+102*x^2+228*x+96)*(x^3+2)^(1/3)+(22*x^6+6*x^5+(-48)*x^4+44*x^3+24*x^2+(-192)*x+(-140)))/(x^6+6*x^5+15*x^4+20*x^3+15*x^2+6*x+1))+2*3^(1/2)*atan(((
98966744593197647869364591874*x^4+190053406517364372745124029472*x^3+(-642339750020464731448133545632)*x^2+(-1764382450892402509391037276448)*x+(-1072244631963565627440642667696))*3^(1/2)*((x^3+2)^(1/3))^2+((-45228634350310035870300951616)*x^5+(-
453545129950193664973324584892)*x^4+(-726175722499147186465445363320)*x^3+735314591615271415729365586328*x^2+2230842809300000322439227290544*x+1190118508012558386973005239952)*3^(1/2)*(x^3+2)^(1/3)+(93292570833559435663132301885*x^6+
382151535711085278859235047618*x^5+673924074224408772959625384792*x^4+889426563183087468015580290048*x^3+888876515195959220955879945824*x^2+351260598258508240019971964880*x+(-47674000995597211057816884304))*3^(1/2))/(236716304443694165237125394649*x^6+
1013240117509374668590043803350*x^5+46796858328175763683008212928*x^4+(-2686291575945300326054363894472)*x^3+1085003586721431086608600126056*x^2+7625406903034897531937916271008*x+4664445860470002276943457906640)))/12

... while the antiderivative can in fact be compactly stated as:

INT(1/((x + 1)*(x^3 + 2)^(1/3)), x) =
1/12*(- 3*LN((x^3 + 2)^(1/3) - x)
+ 2*SQRT(3)*ATAN(1/SQRT(3)*(1 + 2*x/(x^3 + 2)^(1/3))))
- 1/4*(LN((x + 2)^3 - (x^3 + 2))
- 3*LN((x + 2) - (x^3 + 2)^(1/3))
+ 2*SQRT(3)*ATAN(1/SQRT(3)*(1 + 2*((x + 2)/(x^3 + 2)^(1/3)))))

If the unreasonable numbers cannot be avoided earlier, they could at
least be removed by subtracting an arc tangent for a suitably chosen
value of x; both x = infinity and x = -2^(1/3) turn out to work well.

Finally I find that FriCAS version 1.3.10 still cannot solve:

integrate((3*x + 2)/((x + 6)*(9*x - 2)*(3*x^2 + 4)^(1/3)), x)

Error detected within library code:

integrate: implementation incomplete (residue poly has multiple
non-linear factors)

as first presented in the thread "Risch integrator troubles" of
Autumn/Winter 2019/2020. Why does this one remain too hard for an
algebraic Risch integrator, unlike the many cube-root integrands of
Goursat type now mastered by FriCAS?

Martin.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

T24gMi85LzIwMjQgMTE6MjMgQU0sIGNsaWNsaWNsaWNAZnJlZW5ldC5kZSB3cm90ZToNCj4g DQo+IEkgaGF2ZSBiZWVuIHBsYXlpbmcgYXJvdW5kIHdpdGggc29tZSBvbGQgYWxnZWJyYWlj IGludGVncmFuZHMgaW4gdGhlDQo+IG5ldyB2ZXJzaW9uIDEuMy4xMCBvZiBGcmlDQVMgb24g dGhlIHdlYiBpbnRlcmZhY2UuDQo+IA0KPiBTYW0gQmxha2UncyBwc2V1ZG8tZWxsaXB0aWMg b2YgQXByaWwgMjAyMCBzdGlsbCBnaXZlczoNCj4gDQo+IGludGVncmF0ZSgoeF40IC0gMSkq KHheNCArIHheMiArIDEpKnNxcnQoLXheNCArIHheMiAtIDEpDQo+IC8oeF40ICsgMSleMywg eCkNCj4gDQo+Pj4gRXJyb3IgZGV0ZWN0ZWQgd2l0aGluIGxpYnJhcnkgY29kZToNCj4gY2F0 ZGVmOiBkaXZpc2lvbiBieSB6ZXJvDQo+IA0KPiBwZXJoYXBzIGJlY2F1c2UgdGhlIHJhZGlj YW5kIGlzIG5lZ2F0aXZlIGV2ZXJ5d2hlcmUgb24gdGhlIHJlYWwgYXhpcy4NCj4gDQoNCkZ5 aTsNCg0KSSd2ZSByZXBvcnRlZCBkaXZpc2lvbiBieSB6ZXJvIHRvIEZyaWNhcyBuZXdzZ3Jv dXANCg0KaHR0cHM6Ly9ncm91cHMuZ29vZ2xlLmNvbS9nL2ZyaWNhcy1kZXZlbC9jLzZnMEI1 M3FYMlRVDQoNCkJ0dywgSSBkbyBub3QgdGhpbmsgbWFueSBGcmljYXMgZGV2ZWxvcGVycyBy ZWFkIHNjaS5tYXRoLnN5bWJvbGljDQoNCk1heSBiZSB5b3UgY291bGQgQ0MNCg0KICAgICAg ICAgICBmcmljYXMtZGV2ZWxAZ29vZ2xlZ3JvdXBzLmNvbQ0KDQphbHNvLiBJIGRvIG5vdCBr bm93IGlmIGl0IHdpbGwgd29yayBvciBub3QgZnJvbSB5b3VyIGVuZA0Kb3IgaWYgcmVnaXN0 cmF0aW9uIGlzIG5lZWRlZCBvciBub3QuIFNvbWV0aW1lcyBJIGdldCBkaXJlY3QgZW1haWwN CmZyb20gdGhlIGFib3ZlIG15c2VsZi4NCg0KPiBBbmQgYW4gb2xkZXIgYW5kIHByZXN1bWFi bHkgdHJ1bHkgZWxsaXB0aWMgY2FzZSBzdGlsbCBmYWlsczoNCj4gDQo+IGludGVncmF0ZSgo NSp4IC0gOSpzcXJ0KDYpICsgMjYpDQo+IC8oKHheMiAtIDQqeCAtIDUwKSpzcXJ0KHheMyAt IDMwKnggLSA1NikpLCB4KQ0KPiANCj4+PiBFcnJvciBkZXRlY3RlZCB3aXRoaW4gbGlicmFy eSBjb2RlOg0KPiBjYXRkZWY6IGRpdmlzaW9uIGJ5IHplcm8NCj4gDQo+IGluIHRoZSBzYW1l IG1hbm5lciwgYWx0aG91Z2ggdGhlIHJhZGljYW5kIGlzIGN1YmljIGhlcmUuDQo+IA0KPiBU aGUgZm9sbG93aW5nIGludGVncmFuZCBieSBMZWdlbmRyZSBpcyBzdGlsbCBldmFsdWF0ZWQg dG8gc2l4IGNvbXBsZXgNCj4gbG9nYXJpdGhtczoNCj4gDQo+IGludGVncmF0ZSh4LygoNCAt IHheMykqc3FydCgxIC0geF4zKSksIHgpDQo+IA0KPiAoKCgtMyleKDEvMikrKC0xKSkqKCgt MSkvNDMyKV4oMS82KSpsb2coKCgoMzg4OCp4XjUqKC0zKV4oMS8yKSszODg4KnheNSkqKCgo LTEpLzQzMileKDEvNikpXjUrKDM2MCp4XjYrMTQ0MCp4XjMrKC0xMTUyKSkqKCgoLTEpLzQz MileKDEvNikpXjMrKCgoLTYpKnheNysoLTk2KSp4XjQrNDgqeCkqKC0zKV4oMS8yKSsoNip4 XjcrOTYqeF40KygtNDgpKngpKSooKC0xKS80MzIpXigxLzYpKSooKC0xKSp4XjMrMSleKDEv MikrKCgoKC04NjQpKnheNysoLTg2NCkqeF40KzE3MjgqeCkqKC0zKV4oMS8yKSsoODY0Knhe Nys4NjQqeF40KygtMTcyOCkqeCkpKigoKC0xKS80MzIpXigxLzYpKV40KygoKC0zNikqeF44 KygtMjUyKSp4XjUrMjg4KnheMikqKC0zKV4oMS8yKSsoKC0zNikqeF44KygtMjUyKSp4XjUr Mjg4KnheMikpKigoKC0xKS80MzIpXigxLzYpKV4yKygoLTEpKnheOSsoLTY2KSp4XjYrNzIq eF4zKygtMzIpKSkpLyh4XjkrKC0xMikqeF42KzQ4KnheMysoLTY0KSkpKygoKC0zKV4oMS8y KSsxKSooKC0xKS80MzIpXigxLzYpKmxvZygoKCgzODg4KnheNSooLTMpXigxLzIpKygtMzg4 OCkqeF41KSooKCgtMSkvNDMyKV4oMS82KSleNSsoKC0zNjApKnheNisoLTE0NDApKnheMysx MTUyKSooKCgtMSkvNDMyKV4oMS82KSleMysoKCgtNikqeF43KygtOTYpKnheNCs0OCp4KSoo LTMpXigxLzIpKygoLTYpKnheNysoLTk2KSp4XjQrNDgqeCkpKigoLTEpLzQzMileKDEvNikp KigoLTEpKnheMysxKV4oMS8yKSsoKCg4NjQqeF43Kzg2NCp4XjQrKC0xNzI4KSp4KSooLTMp XigxLzIpKyg4NjQqeF43Kzg2NCp4XjQrKC0xNzI4KSp4KSkqKCgoLTEpLzQzMileKDEvNikp XjQrKCgzNip4XjgrMjUyKnheNSsoLTI4OCkqeF4yKSooLTMpXigxLzIpKygoLTM2KSp4Xjgr KC0yNTIpKnheNSsyODgqeF4yKSkqKCgoLTEpLzQzMileKDEvNikpXjIrKCgtMSkqeF45Kygt NjYpKnheNis3Mip4XjMrKC0zMikpKSkvKHheOSsoLTEyKSp4XjYrNDgqeF4zKygtNjQpKSkr KCgtMikqKCgtMSkvNDMyKV4oMS82KSpsb2coKCg3Nzc2KnheNSooKCgtMSkvNDMyKV4oMS82 KSleNSsoKC0zNjApKnheNisoLTE0NDApKnheMysxMTUyKSooKCgtMSkvNDMyKV4oMS82KSle MysoMTIqeF43KzE5Mip4XjQrKC05NikqeCkqKCgtMSkvNDMyKV4oMS82KSkqKCgtMSkqeF4z KzEpXigxLzIpKygoKC0xNzI4KSp4XjcrKC0xNzI4KSp4XjQrMzQ1Nip4KSooKCgtMSkvNDMy KV4oMS82KSleNCsoNzIqeF44KzUwNCp4XjUrKC01NzYpKnheMikqKCgoLTEpLzQzMileKDEv NikpXjIrKCgtMSkqeF45KygtNjYpKnheNis3Mip4XjMrKC0zMikpKSkvKHheOSsoLTEyKSp4 XjYrNDgqeF4zKygtNjQpKSkrKDIqKCgtMSkvNDMyKV4oMS82KSpsb2coKCgoLTc3NzYpKnhe NSooKCgtMSkvNDMyKV4oMS82KSleNSsoMzYwKnheNisxNDQwKnheMysoLTExNTIpKSooKCgt MSkvNDMyKV4oMS82KSleMysoKC0xMikqeF43KygtMTkyKSp4XjQrOTYqeCkqKCgtMSkvNDMy KV4oMS82KSkqKCgtMSkqeF4zKzEpXigxLzIpKygoKC0xNzI4KSp4XjcrKC0xNzI4KSp4XjQr MzQ1Nip4KSooKCgtMSkvNDMyKV4oMS82KSleNCsoNzIqeF44KzUwNCp4XjUrKC01NzYpKnhe MikqKCgoLTEpLzQzMileKDEvNikpXjIrKCgtMSkqeF45KygtNjYpKnheNis3Mip4XjMrKC0z MikpKSkvKHheOSsoLTEyKSp4XjYrNDgqeF4zKygtNjQpKSkrKCgoLTEpKigtMyleKDEvMikr KC0xKSkqKCgtMSkvNDMyKV4oMS82KSpsb2coKCgoKC0zODg4KSp4XjUqKC0zKV4oMS8yKSsz ODg4KnheNSkqKCgoLTEpLzQzMileKDEvNikpXjUrKDM2MCp4XjYrMTQ0MCp4XjMrKC0xMTUy KSkqKCgoLTEpLzQzMileKDEvNikpXjMrKCg2KnheNys5Nip4XjQrKC00OCkqeCkqKC0zKV4o MS8yKSsoNip4XjcrOTYqeF40KygtNDgpKngpKSooKC0xKS80MzIpXigxLzYpKSooKC0xKSp4 XjMrMSleKDEvMikrKCgoODY0KnheNys4NjQqeF40KygtMTcyOCkqeCkqKC0zKV4oMS8yKSso ODY0KnheNys4NjQqeF40KygtMTcyOCkqeCkpKigoKC0xKS80MzIpXigxLzYpKV40KygoMzYq eF44KzI1Mip4XjUrKC0yODgpKnheMikqKC0zKV4oMS8yKSsoKC0zNikqeF44KygtMjUyKSp4 XjUrMjg4KnheMikpKigoKC0xKS80MzIpXigxLzYpKV4yKygoLTEpKnheOSsoLTY2KSp4XjYr NzIqeF4zKygtMzIpKSkpLyh4XjkrKC0xMikqeF42KzQ4KnheMysoLTY0KSkpKygoLTEpKigt MyleKDEvMikrMSkqKCgtMSkvNDMyKV4oMS82KSpsb2coKCgoKC0zODg4KSp4XjUqKC0zKV4o MS8yKSsoLTM4ODgpKnheNSkqKCgoLTEpLzQzMileKDEvNikpXjUrKCgtMzYwKSp4XjYrKC0x NDQwKSp4XjMrMTE1MikqKCgoLTEpLzQzMileKDEvNikpXjMrKCg2KnheNys5Nip4XjQrKC00 OCkqeCkqKC0zKV4oMS8yKSsoKC02KSp4XjcrKC05NikqeF40KzQ4KngpKSooKC0xKS80MzIp XigxLzYpKSooKC0xKSp4XjMrMSleKDEvMikrKCgoKC04NjQpKnheNysoLTg2NCkqeF40KzE3 MjgqeCkqKC0zKV4oMS8yKSsoODY0KnheNys4NjQqeF40KygtMTcyOCkqeCkpKigoKC0xKS80 MzIpXigxLzYpKV40KygoKC0zNikqeF44KygtMjUyKSp4XjUrMjg4KnheMikqKC0zKV4oMS8y KSsoKC0zNikqeF44KygtMjUyKSp4XjUrMjg4KnheMikpKigoKC0xKS80MzIpXigxLzYpKV4y KygoLTEpKnheOSsoLTY2KSp4XjYrNzIqeF4zKygtMzIpKSkpLyh4XjkrKC0xMikqeF42KzQ4 KnheMysoLTY0KSkpKSkpKSkvMzYNCj4gDQo+IC4uLiBldmVuIHRob3VnaCBhIHJlYWwgZXhw cmVzc2lvbiBmb3IgdGhlIGFudGlkZXJpdmF0aXZlIGV4aXN0czoNCj4gDQo+IElOVCh4Lygo NCAtIHheMykqU1FSVCgxIC0geF4zKSksIHgpID0NCj4gMl4oMS8zKS8xOCooQVRBTkgoU1FS VCgxIC0geF4zKSkNCj4gICAtIDMqQVRBTkgoKDEgKyAyXigxLzMpKngpL1NRUlQoMSAtIHhe MykpDQo+ICAgLSBTUVJUKDMpKkFUQU4oKDJeKDEvMykgLSAyXigyLzMpKnggLSB4XjIpDQo+ IC8oU1FSVCgzKSoyXigxLzMpKlNRUlQoMSAtIHheMykpKSkNCj4gDQo+IENhbid0IHRoZXNl IGNvbXBsZXggbG9nYXJpdGhtcyBiZSBicm9rZW4gZG93biBzaW1pbGFyIHRvIHRob3NlIGZv cg0KPiBpbnRlZ3JhbCA1LjY2ICgjNDAxKSBmcm9tIHRoZSBUaW1vZmVldiBzdWl0ZT8NCj4g DQo+IEFuZCBmb3IgdGhlIG5leHQgaW50ZWdyYW5kLCBGcmlDQVMgc3RpbGwgcHJvZHVjZXMg dW5yZWFzb25hYmxlIGludGVnZXJzDQo+IGluIGFuIGFyYyB0YW5nZW50J3MgYXJndW1lbnQ6 DQo+IA0KPiBpbnRlZ3JhdGUoMS8oKHggKyAxKSooeF4zICsgMileKDEvMykpLCB4KQ0KPiAN Cj4gKGxvZygoKDIxKnheNCsoLTYpKnheMysoLTk2KSp4XjIrKC02MCkqeCsxMikqKCh4XjMr MileKDEvMykpXjIrKDIxKnheNSsoLTQ4KSp4XjMrMTAyKnheMisyMjgqeCs5NikqKHheMysy KV4oMS8zKSsoMjIqeF42KzYqeF41KygtNDgpKnheNCs0NCp4XjMrMjQqeF4yKygtMTkyKSp4 KygtMTQwKSkpLyh4XjYrNip4XjUrMTUqeF40KzIwKnheMysxNSp4XjIrNip4KzEpKSsyKjNe KDEvMikqYXRhbigoKDk4OTY2NzQ0NTkzMTk3NjQ3ODY5MzY0NTkxODc0KnheNCsxOTAwNTM0 MDY1MTczNjQzNzI3NDUxMjQwMjk0NzIqeF4zKygtNjQyMzM5NzUwMDIwNDY0NzMxNDQ4MTMz NTQ1NjMyKSp4XjIrKC0xNzY0MzgyNDUwODkyNDAyNTA5MzkxMDM3Mjc2NDQ4KSp4KygtMTA3 MjI0NDYzMTk2MzU2NTYyNzQ0MDY0MjY2NzY5NikpKjNeKDEvMikqKCh4XjMrMileKDEvMykp XjIrKCgtNDUyMjg2MzQzNTAzMTAwMzU4NzAzMDA5NTE2MTYpKnheNSsoLTQ1MzU0NTEyOTk1 MDE5MzY2NDk3MzMyNDU4NDg5MikqeF40KygtNzI2MTc1NzIyNDk5MTQ3MTg2NDY1NDQ1MzYz MzIwKSp4XjMrNzM1MzE0NTkxNjE1MjcxNDE1NzI5MzY1NTg2MzI4KnheMisyMjMwODQyODA5 MzAwMDAwMzIyNDM5MjI3MjkwNTQ0KngrMTE5MDExODUwODAxMjU1ODM4Njk3MzAwNTIzOTk1 MikqM14oMS8yKSooeF4zKzIpXigxLzMpKyg5MzI5MjU3MDgzMzU1OTQzNTY2MzEzMjMwMTg4 NSp4XjYrMzgyMTUxNTM1NzExMDg1Mjc4ODU5MjM1MDQ3NjE4KnheNSs2NzM5MjQwNzQyMjQ0 MDg3NzI5NTk2MjUzODQ3OTIqeF40Kzg4OTQyNjU2MzE4MzA4NzQ2ODAxNTU4MDI5MDA0OCp4 XjMrODg4ODc2NTE1MTk1OTU5MjIwOTU1ODc5OTQ1ODI0KnheMiszNTEyNjA1OTgyNTg1MDgy NDAwMTk5NzE5NjQ4ODAqeCsoLTQ3Njc0MDAwOTk1NTk3MjExMDU3ODE2ODg0MzA0KSkqM14o MS8yKSkvKDIzNjcxNjMwNDQ0MzY5NDE2NTIzNzEyNTM5NDY0OSp4XjYrMTAxMzI0MDExNzUw OTM3NDY2ODU5MDA0MzgwMzM1MCp4XjUrNDY3OTY4NTgzMjgxNzU3NjM2ODMwMDgyMTI5Mjgq eF40KygtMjY4NjI5MTU3NTk0NTMwMDMyNjA1NDM2Mzg5NDQ3MikqeF4zKzEwODUwMDM1ODY3 MjE0MzEwODY2MDg2MDAxMjYwNTYqeF4yKzc2MjU0MDY5MDMwMzQ4OTc1MzE5Mzc5MTYyNzEw MDgqeCs0NjY0NDQ1ODYwNDcwMDAyMjc2OTQzNDU3OTA2NjQwKSkpLzEyDQo+IA0KPiAuLi4g d2hpbGUgdGhlIGFudGlkZXJpdmF0aXZlIGNhbiBpbiBmYWN0IGJlIGNvbXBhY3RseSBzdGF0 ZWQgYXM6DQo+IA0KPiBJTlQoMS8oKHggKyAxKSooeF4zICsgMileKDEvMykpLCB4KSA9DQo+ IDEvMTIqKC0gMypMTigoeF4zICsgMileKDEvMykgLSB4KQ0KPiAgICsgMipTUVJUKDMpKkFU QU4oMS9TUVJUKDMpKigxICsgMip4Lyh4XjMgKyAyKV4oMS8zKSkpKQ0KPiAgIC0gMS80KihM TigoeCArIDIpXjMgLSAoeF4zICsgMikpDQo+ICAgLSAzKkxOKCh4ICsgMikgLSAoeF4zICsg MileKDEvMykpDQo+ICAgKyAyKlNRUlQoMykqQVRBTigxL1NRUlQoMykqKDEgKyAyKigoeCAr IDIpLyh4XjMgKyAyKV4oMS8zKSkpKSkNCj4gDQo+IElmIHRoZSB1bnJlYXNvbmFibGUgbnVt YmVycyBjYW5ub3QgYmUgYXZvaWRlZCBlYXJsaWVyLCB0aGV5IGNvdWxkIGF0DQo+IGxlYXN0 IGJlIHJlbW92ZWQgYnkgc3VidHJhY3RpbmcgYW4gYXJjIHRhbmdlbnQgZm9yIGEgc3VpdGFi bHkgY2hvc2VuDQo+IHZhbHVlIG9mIHg7IGJvdGggeCA9IGluZmluaXR5IGFuZCB4ID0gLTJe KDEvMykgdHVybiBvdXQgdG8gd29yayB3ZWxsLg0KPiANCj4gRmluYWxseSBJIGZpbmQgdGhh dCBGcmlDQVMgdmVyc2lvbiAxLjMuMTAgc3RpbGwgY2Fubm90IHNvbHZlOg0KPiANCj4gaW50 ZWdyYXRlKCgzKnggKyAyKS8oKHggKyA2KSooOSp4IC0gMikqKDMqeF4yICsgNCleKDEvMykp LCB4KQ0KPiANCj4+PiBFcnJvciBkZXRlY3RlZCB3aXRoaW4gbGlicmFyeSBjb2RlOg0KPiBp bnRlZ3JhdGU6IGltcGxlbWVudGF0aW9uIGluY29tcGxldGUgKHJlc2lkdWUgcG9seSBoYXMg bXVsdGlwbGUNCj4gbm9uLWxpbmVhciBmYWN0b3JzKQ0KPiANCj4gYXMgZmlyc3QgcHJlc2Vu dGVkIGluIHRoZSB0aHJlYWQgIlJpc2NoIGludGVncmF0b3IgdHJvdWJsZXMiIG9mDQo+IEF1 dHVtbi9XaW50ZXIgMjAxOS8yMDIwLiBXaHkgZG9lcyB0aGlzIG9uZSByZW1haW4gdG9vIGhh cmQgZm9yIGFuDQo+IGFsZ2VicmFpYyBSaXNjaCBpbnRlZ3JhdG9yLCB1bmxpa2UgdGhlIG1h bnkgY3ViZS1yb290IGludGVncmFuZHMgb2YNCj4gR291cnNhdCB0eXBlIG5vdyBtYXN0ZXJl ZCBieSBGcmlDQVM/DQo+IA0KPiBNYXJ0aW4uDQoNCg0KLS1OYXNzZXINCg==

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

"Nasser M. Abbasi" schrieb:

On 2/9/2024 11:23 AM, [email protected] wrote:

I have been playing around with some old algebraic integrands in the
new version 1.3.10 of FriCAS on the web interface.

Sam Blake's pseudo-elliptic of April 2020 still gives:

integrate((x^4 - 1)*(x^4 + x^2 + 1)*sqrt(-x^4 + x^2 - 1)
/(x^4 + 1)^3, x)

Error detected within library code:

catdef: division by zero

perhaps because the radicand is negative everywhere on the real
axis.

Fyi;

I've reported division by zero to Fricas newsgroup

https://groups.google.com/g/fricas-devel/c/6g0B53qX2TU

Btw, I do not think many Fricas developers read sci.math.symbolic

May be you could CC

[email protected]

also. I do not know if it will work or not from your end
or if registration is needed or not. Sometimes I get direct email
from the above myself.

Thanks, will try this next time and see what happpens. In fact, I more
or less regularly follow the posts at <https://www.mail-archive.com/ [email protected]>, and thus see Waldek occasionally
responding to <sci.math.symbolic> messages over there - so he's still
reading them. Dunno why he doesn't register at Eternal-September - he
even posted via Telekomunikacja Polska in April last year (I suppose
they offered a free trial which ran out).

I have verified that registration at <www.solani.org> also works, but
one may have to remind the operators via e-mail and wait for a week
until one receives a password.

And as stated before, I can e-mail a password for <news.killfile.org>
which I was able to guess to any one of the serious <sci.math.symbolic> posters.

And an older and presumably truly elliptic case still fails:

integrate((5*x - 9*sqrt(6) + 26)
/((x^2 - 4*x - 50)*sqrt(x^3 - 30*x - 56)), x)

Error detected within library code:

catdef: division by zero

in the same manner, although the radicand is cubic here.

[...]

And for the next integrand, FriCAS still produces unreasonable
integers in an arc tangent's argument:

integrate(1/((x + 1)*(x^3 + 2)^(1/3)), x)

(log(((21*x^4+(-6)*x^3+(-96)*x^2+(-60)*x+12)*((x^3+2)^(1/3))^2+(21*x^5+(-48)*x^3+102*x^2+228*x+96)*(x^3+2)^(1/3)+(22*x^6+6*x^5+(-48)*x^4+44*x^3+24*x^2+(-192)*x+(-140)))/(x^6+6*x^5+15*x^4+20*x^3+15*x^2+6*x+1))+2*3^(1/2)*atan(((
98966744593197647869364591874*x^4+190053406517364372745124029472*x^3+(-642339750020464731448133545632)*x^2+(-1764382450892402509391037276448)*x+(-1072244631963565627440642667696))*3^(1/2)*((x^3+2)^(1/3))^2+((-45228634350310035870300951616)*x^5+(-
453545129950193664973324584892)*x^4+(-726175722499147186465445363320)*x^3+735314591615271415729365586328*x^2+2230842809300000322439227290544*x+1190118508012558386973005239952)*3^(1/2)*(x^3+2)^(1/3)+(93292570833559435663132301885*x^6+
382151535711085278859235047618*x^5+673924074224408772959625384792*x^4+889426563183087468015580290048*x^3+888876515195959220955879945824*x^2+351260598258508240019971964880*x+(-47674000995597211057816884304))*3^(1/2))/(236716304443694165237125394649*x^6+
1013240117509374668590043803350*x^5+46796858328175763683008212928*x^4+(-2686291575945300326054363894472)*x^3+1085003586721431086608600126056*x^2+7625406903034897531937916271008*x+4664445860470002276943457906640)))/12

... while the antiderivative can in fact be compactly stated as:

INT(1/((x + 1)*(x^3 + 2)^(1/3)), x) =
1/12*(- 3*LN((x^3 + 2)^(1/3) - x)
+ 2*SQRT(3)*ATAN(1/SQRT(3)*(1 + 2*x/(x^3 + 2)^(1/3))))
- 1/4*(LN((x + 2)^3 - (x^3 + 2))
- 3*LN((x + 2) - (x^3 + 2)^(1/3))
+ 2*SQRT(3)*ATAN(1/SQRT(3)*(1 + 2*((x + 2)/(x^3 + 2)^(1/3)))))

If the unreasonable numbers cannot be avoided earlier, they could at
least be removed by subtracting an arc tangent for a suitably chosen
value of x; both x = infinity and x = -2^(1/3) turn out to work
well.

I also find that x = -1 works less well; perhaps one should simply try
x = infinity in all cases of algebraic antiderivatives with
unreasonable arc tangent arguments (but only if the radical stays
real?), and perhaps for reasonable arguments as well to avoid deciding
what's unreasonable.

[...]

Martin.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

"[email protected]" schrieb:

"Nasser M. Abbasi" schrieb:

On 2/9/2024 11:23 AM, [email protected] wrote:

And for the next integrand, FriCAS still produces unreasonable
integers in an arc tangent's argument:

integrate(1/((x + 1)*(x^3 + 2)^(1/3)), x)

(log(((21*x^4+(-6)*x^3+(-96)*x^2+(-60)*x+12)*((x^3+2)^(1/3))^2+(21*x^5+(-48)*x^3+102*x^2+228*x+96)*(x^3+2)^(1/3)+(22*x^6+6*x^5+(-48)*x^4+44*x^3+24*x^2+(-192)*x+(-140)))/(x^6+6*x^5+15*x^4+20*x^3+15*x^2+6*x+1))+2*3^(1/2)*atan(((

98966744593197647869364591874*x^4+190053406517364372745124029472*x^3+(-642339750020464731448133545632)*x^2+(-1764382450892402509391037276448)*x+(-1072244631963565627440642667696))*3^(1/2)*((x^3+2)^(1/3))^2+((-45228634350310035870300951616)*x^5+(-
453545129950193664973324584892)*x^4+(-726175722499147186465445363320)*x^3+735314591615271415729365586328*x^2+2230842809300000322439227290544*x+1190118508012558386973005239952)*3^(1/2)*(x^3+2)^(1/3)+(93292570833559435663132301885*x^6+
382151535711085278859235047618*x^5+673924074224408772959625384792*x^4+889426563183087468015580290048*x^3+888876515195959220955879945824*x^2+351260598258508240019971964880*x+(-47674000995597211057816884304))*3^(1/2))/(236716304443694165237125394649*x^6+
1013240117509374668590043803350*x^5+46796858328175763683008212928*x^4+(-2686291575945300326054363894472)*x^3+1085003586721431086608600126056*x^2+7625406903034897531937916271008*x+4664445860470002276943457906640)))/12

... while the antiderivative can in fact be compactly stated as:

INT(1/((x + 1)*(x^3 + 2)^(1/3)), x) =
1/12*(- 3*LN((x^3 + 2)^(1/3) - x)
+ 2*SQRT(3)*ATAN(1/SQRT(3)*(1 + 2*x/(x^3 + 2)^(1/3))))
- 1/4*(LN((x + 2)^3 - (x^3 + 2))
- 3*LN((x + 2) - (x^3 + 2)^(1/3))
+ 2*SQRT(3)*ATAN(1/SQRT(3)*(1 + 2*((x + 2)/(x^3 + 2)^(1/3)))))

If the unreasonable numbers cannot be avoided earlier, they could
at least be removed by subtracting an arc tangent for a suitably
chosen value of x; both x = infinity and x = -2^(1/3) turn out to
work well.

I also find that x = -1 works less well; perhaps one should simply try
x = infinity in all cases of algebraic antiderivatives with
unreasonable arc tangent arguments (but only if the radical stays
real?), and perhaps for reasonable arguments as well to avoid deciding
what's unreasonable.

I was wondering if it could be possible to determine automatically how
many FriCAS antiderivatives among the test results presented at

<https://www.12000.org/my_notes/CAS_integration_tests/index.htm>

involve integers with more than 5 or 10 decimal digits? Can a list of
these integrals be extracted?

Martin.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)