3.88 \(\int \frac {x}{\sqrt {-1+x^3} (-10-6 \sqrt {3}+x^3)} \, dx\)
Optimal. Leaf size=222 \[ \frac {\left (2-\sqrt {3}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) (1-x)}{\sqrt {2} \sqrt {x^3-1}}\right )}{6 \sqrt {2} \sqrt [4]{3}}+\frac {\left (2-\sqrt {3}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{3} \left (2 x+\sqrt {3}+1\right )}{\sqrt {2} \sqrt {x^3-1}}\right )}{3 \sqrt {2} \sqrt [4]{3}}+\frac {\left (2-\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) (1-x)}{\sqrt {2} \sqrt {x^3-1}}\right )}{2 \sqrt {2} 3^{3/4}}-\frac {\left (2-\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt {x^3-1}}{\sqrt {2} 3^{3/4}}\right )}{3 \sqrt {2} 3^{3/4}} \]
[Out]
1/36*arctan(1/2*3^(1/4)*(1-x)*(1-3^(1/2))*2^(1/2)/(x^3-1)^(1/2))*(2-3^(1/2))*3^(3/4)*2^(1/2)+1/18*arctan(1/2*3
^(1/4)*(1+2*x+3^(1/2))*2^(1/2)/(x^3-1)^(1/2))*(2-3^(1/2))*3^(3/4)*2^(1/2)+1/12*arctanh(1/2*3^(1/4)*(1-x)*(1+3^
(1/2))*2^(1/2)/(x^3-1)^(1/2))*(2-3^(1/2))*3^(1/4)*2^(1/2)-1/18*arctanh(1/6*(1-3^(1/2))*(x^3-1)^(1/2)*3^(1/4)*2
^(1/2))*(2-3^(1/2))*3^(1/4)*2^(1/2)
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 222, normalized size of antiderivative = 1.00,
number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used =
{488} \[ \frac {\left (2-\sqrt {3}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) (1-x)}{\sqrt {2} \sqrt {x^3-1}}\right )}{6 \sqrt {2} \sqrt [4]{3}}+\frac {\left (2-\sqrt {3}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{3} \left (2 x+\sqrt {3}+1\right )}{\sqrt {2} \sqrt {x^3-1}}\right )}{3 \sqrt {2} \sqrt [4]{3}}+\frac {\left (2-\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) (1-x)}{\sqrt {2} \sqrt {x^3-1}}\right )}{2 \sqrt {2} 3^{3/4}}-\frac {\left (2-\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt {x^3-1}}{\sqrt {2} 3^{3/4}}\right )}{3 \sqrt {2} 3^{3/4}} \]
Antiderivative was successfully verified.
[In]
Int[x/(Sqrt[-1 + x^3]*(-10 - 6*Sqrt[3] + x^3)),x]
[Out]
((2 - Sqrt[3])*ArcTan[(3^(1/4)*(1 - Sqrt[3])*(1 - x))/(Sqrt[2]*Sqrt[-1 + x^3])])/(6*Sqrt[2]*3^(1/4)) + ((2 - S
qrt[3])*ArcTan[(3^(1/4)*(1 + Sqrt[3] + 2*x))/(Sqrt[2]*Sqrt[-1 + x^3])])/(3*Sqrt[2]*3^(1/4)) + ((2 - Sqrt[3])*A
rcTanh[(3^(1/4)*(1 + Sqrt[3])*(1 - x))/(Sqrt[2]*Sqrt[-1 + x^3])])/(2*Sqrt[2]*3^(3/4)) - ((2 - Sqrt[3])*ArcTanh
[((1 - Sqrt[3])*Sqrt[-1 + x^3])/(Sqrt[2]*3^(3/4))])/(3*Sqrt[2]*3^(3/4))
Rule 488
Int[(x_)/(Sqrt[(a_) + (b_.)*(x_)^3]*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[b/a, 3], r = Simplify[(b
*c - 10*a*d)/(6*a*d)]}, Simp[(q*(2 - r)*ArcTanh[((1 - r)*Sqrt[a + b*x^3])/(Sqrt[2]*Rt[-a, 2]*r^(3/2))])/(3*Sqr
t[2]*Rt[-a, 2]*d*r^(3/2)), x] + (-Simp[(q*(2 - r)*ArcTanh[(Rt[-a, 2]*Sqrt[r]*(1 + r)*(1 + q*x))/(Sqrt[2]*Sqrt[
a + b*x^3])])/(2*Sqrt[2]*Rt[-a, 2]*d*r^(3/2)), x] - Simp[(q*(2 - r)*ArcTan[(Rt[-a, 2]*Sqrt[r]*(1 + r - 2*q*x))
/(Sqrt[2]*Sqrt[a + b*x^3])])/(3*Sqrt[2]*Rt[-a, 2]*d*Sqrt[r]), x] - Simp[(q*(2 - r)*ArcTan[(Rt[-a, 2]*(1 - r)*S
qrt[r]*(1 + q*x))/(Sqrt[2]*Sqrt[a + b*x^3])])/(6*Sqrt[2]*Rt[-a, 2]*d*Sqrt[r]), x])] /; FreeQ[{a, b, c, d}, x]
&& NeQ[b*c - a*d, 0] && EqQ[b^2*c^2 - 20*a*b*c*d - 8*a^2*d^2, 0] && NegQ[a]
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {-1+x^3} \left (-10-6 \sqrt {3}+x^3\right )} \, dx &=\frac {\left (2-\sqrt {3}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) (1-x)}{\sqrt {2} \sqrt {-1+x^3}}\right )}{6 \sqrt {2} \sqrt [4]{3}}+\frac {\left (2-\sqrt {3}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}+2 x\right )}{\sqrt {2} \sqrt {-1+x^3}}\right )}{3 \sqrt {2} \sqrt [4]{3}}+\frac {\left (2-\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) (1-x)}{\sqrt {2} \sqrt {-1+x^3}}\right )}{2 \sqrt {2} 3^{3/4}}-\frac {\left (2-\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt {-1+x^3}}{\sqrt {2} 3^{3/4}}\right )}{3 \sqrt {2} 3^{3/4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.07, size = 65, normalized size = 0.29 \[ -\frac {x^2 \sqrt {1-x^3} F_1\left (\frac {2}{3};\frac {1}{2},1;\frac {5}{3};x^3,\frac {x^3}{10+6 \sqrt {3}}\right )}{\left (20+12 \sqrt {3}\right ) \sqrt {x^3-1}} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[x/(Sqrt[-1 + x^3]*(-10 - 6*Sqrt[3] + x^3)),x]
[Out]
-((x^2*Sqrt[1 - x^3]*AppellF1[2/3, 1/2, 1, 5/3, x^3, x^3/(10 + 6*Sqrt[3])])/((20 + 12*Sqrt[3])*Sqrt[-1 + x^3])
)
________________________________________________________________________________________
fricas [B] time = 9.99, size = 7910, normalized size = 35.63 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x/(-10+x^3-6*3^(1/2))/(x^3-1)^(1/2),x, algorithm="fricas")
[Out]
1/432*sqrt(2*(7*sqrt(3) + 12)*sqrt(-672*sqrt(3) + 1164) + 24)*(56*sqrt(3) + 97)*sqrt(-56*sqrt(3) + 97)*(-672*s
qrt(3) + 1164)^(3/4)*arctan(1/1296*(6*sqrt(x^3 - 1)*((459*x^16 + 13425*x^15 - 33201*x^14 - 950652*x^13 - 99730
2*x^12 + 14760972*x^11 + 47069892*x^10 + 49762248*x^9 - 8212536*x^8 - 84377808*x^7 - 88427328*x^6 - 25613856*x
^5 + 27458496*x^4 + 36433344*x^3 + 12609792*x^2 + sqrt(3)*(265*x^16 + 7751*x^15 - 19167*x^14 - 548864*x^13 - 5
75818*x^12 + 8522268*x^11 + 27175852*x^10 + 28730312*x^9 - 4741560*x^8 - 48715600*x^7 - 51053600*x^6 - 1478812
8*x^5 + 15853184*x^4 + 21034816*x^3 + 7280256*x^2 + 2488832*x - 1889792) - (3691*x^16 + 6128*x^15 - 537864*x^1
4 - 1586477*x^13 + 16210952*x^12 + 77181756*x^11 + 84218362*x^10 - 71018320*x^9 - 254455812*x^8 - 196076008*x^
7 + 120105208*x^6 + 256326864*x^5 + 134645168*x^4 - 78464672*x^3 - 78514944*x^2 + sqrt(3)*(2131*x^16 + 3538*x^
15 - 310536*x^14 - 915953*x^13 + 9359398*x^12 + 44560908*x^11 + 48623494*x^10 - 41002448*x^9 - 146910132*x^8 -
113204536*x^7 + 69342776*x^6 + 147990384*x^5 + 77737424*x^4 - 45301600*x^3 - 45330624*x^2 - 12242560*x + 7598
336) - 21204736*x + 13160704)*sqrt(-672*sqrt(3) + 1164) + 4310784*x - 3273216)*(-672*sqrt(3) + 1164)^(3/4) + 3
*(984*x^15 + 30612*x^14 + 164676*x^13 + 205368*x^12 - 289200*x^11 - 183720*x^10 + 886752*x^9 + 71568*x^8 - 196
0992*x^7 - 1849440*x^6 + 1558464*x^5 + 2478912*x^4 + 66432*x^3 - 750336*x^2 + 4*sqrt(3)*(142*x^15 + 4419*x^14
+ 23781*x^13 + 29608*x^12 - 41940*x^11 - 26454*x^10 + 128152*x^9 + 10692*x^8 - 283320*x^7 - 267064*x^6 + 22478
4*x^5 + 357936*x^4 + 9632*x^3 - 108288*x^2 - 96000*x + 33920) - (4945*x^15 + 88617*x^14 + 738528*x^13 + 186004
6*x^12 - 784596*x^11 - 7668708*x^10 - 6570680*x^9 + 6903864*x^8 + 15444144*x^7 + 4312832*x^6 - 9559200*x^5 - 9
359808*x^4 - 155968*x^3 + 3016704*x^2 + sqrt(3)*(2855*x^15 + 51163*x^14 + 426388*x^13 + 1073898*x^12 - 452980*
x^11 - 4427548*x^10 - 3793592*x^9 + 3985944*x^8 + 8916720*x^7 + 2490016*x^6 - 5519008*x^5 - 5403904*x^4 - 9004
8*x^3 + 1741696*x^2 + 1543936*x - 545536) + 2674176*x - 944896)*sqrt(-672*sqrt(3) + 1164) - 665088*x + 235008)
*(-672*sqrt(3) + 1164)^(1/4))*sqrt(2*(7*sqrt(3) + 12)*sqrt(-672*sqrt(3) + 1164) + 24)*sqrt(-56*sqrt(3) + 97) +
36*(144*x^17 + 5976*x^16 + 5544*x^15 - 299664*x^14 - 1062360*x^13 - 116712*x^12 + 3600000*x^11 + 4761216*x^10
- 1046592*x^9 - 8676864*x^8 - 6592896*x^7 + 2641536*x^6 + 7016832*x^5 + 3699072*x^4 - 1861632*x^3 - 1640448*x
^2 + 12*sqrt(3)*(7*x^17 + 286*x^16 + 238*x^15 - 14255*x^14 - 50390*x^13 - 5942*x^12 + 171808*x^11 + 226888*x^1
0 - 48920*x^9 - 415384*x^8 - 315088*x^7 + 125600*x^6 + 336608*x^5 + 177344*x^4 - 89152*x^3 - 78784*x^2 - 39040
*x + 18176) + (1164*x^17 + 6276*x^16 - 26052*x^15 - 332844*x^14 - 1632156*x^13 - 4149132*x^12 - 5805024*x^11 -
318696*x^10 + 12621072*x^9 + 19878720*x^8 + 9619008*x^7 - 13361088*x^6 - 20168256*x^5 - 10936128*x^4 + 643430
4*x^3 + 6426240*x^2 + 24*sqrt(3)*(28*x^17 + 151*x^16 - 626*x^15 - 8006*x^14 - 39266*x^13 - 99812*x^12 - 139652
*x^11 - 7661*x^10 + 303610*x^9 + 478214*x^8 + 231392*x^7 - 321412*x^6 - 485176*x^5 - 263080*x^4 + 154784*x^3 +
154592*x^2 + 78464*x - 36544) - (2340*x^17 + 96354*x^16 + 84798*x^15 - 4817124*x^14 - 17052930*x^13 - 1941678
*x^12 + 57963744*x^11 + 76603680*x^10 - 16678512*x^9 - 139922496*x^8 - 106227360*x^7 + 42453216*x^6 + 11326953
6*x^5 + 59694624*x^4 - 30025728*x^3 - 26496000*x^2 + sqrt(3)*(1351*x^17 + 55630*x^16 + 48958*x^15 - 2781167*x^
14 - 9845510*x^13 - 1121030*x^12 + 33465376*x^11 + 44227144*x^10 - 9629336*x^9 - 80784280*x^8 - 61330384*x^7 +
24510368*x^6 + 65396192*x^5 + 34464704*x^4 - 17335360*x^3 - 15297472*x^2 - 7571584*x + 3526400) - 13114368*x
+ 6107904)*sqrt(-672*sqrt(3) + 1164) + 3261696*x - 1519104)*sqrt(-672*sqrt(3) + 1164) - 12*(97*x^17 + 523*x^16
- 2171*x^15 - 27737*x^14 - 136013*x^13 - 345761*x^12 - 483752*x^11 - 26558*x^10 + 1051756*x^9 + 1656560*x^8 +
801584*x^7 - 1113424*x^6 - 1680688*x^5 - 911344*x^4 + 536192*x^3 + 535520*x^2 + 2*sqrt(3)*(28*x^17 + 151*x^16
- 626*x^15 - 8006*x^14 - 39266*x^13 - 99812*x^12 - 139652*x^11 - 7661*x^10 + 303610*x^9 + 478214*x^8 + 231392
*x^7 - 321412*x^6 - 485176*x^5 - 263080*x^4 + 154784*x^3 + 154592*x^2 + 78464*x - 36544) + 271808*x - 126592)*
sqrt(-672*sqrt(3) + 1164) - 811008*x + 377856)*sqrt(-56*sqrt(3) + 97) - (sqrt(x^3 - 1)*((459*x^16 + 1557*x^15
- 26415*x^14 + 1449954*x^13 + 4677912*x^12 - 12651948*x^11 - 55684800*x^10 - 62834256*x^9 + 8526168*x^8 + 1053
13392*x^7 + 99605088*x^6 + 18897984*x^5 - 42499296*x^4 - 37357632*x^3 - 8256960*x^2 + sqrt(3)*(265*x^16 + 899*
x^15 - 15249*x^14 + 837130*x^13 + 2700776*x^12 - 7304604*x^11 - 32149640*x^10 - 36277360*x^9 + 4922568*x^8 + 6
0802736*x^7 + 57507040*x^6 + 10910784*x^5 - 24536992*x^4 - 21568448*x^3 - 4767168*x^2 - 1207168*x + 1383424) -
(3691*x^16 - 17731*x^15 - 951114*x^14 - 450359*x^13 + 4370159*x^12 - 30318522*x^11 - 78096668*x^10 - 9429316*
x^9 + 146877876*x^8 + 197107784*x^7 - 30834152*x^6 - 185125776*x^5 - 132260896*x^4 + 45545344*x^3 + 69517536*x
^2 + sqrt(3)*(2131*x^16 - 10237*x^15 - 549126*x^14 - 260015*x^13 + 2523113*x^12 - 17504406*x^11 - 45089132*x^1
0 - 5444020*x^9 + 84799980*x^8 + 113800232*x^7 - 17802104*x^6 - 106882416*x^5 - 76360864*x^4 + 26295616*x^3 +
40135968*x^2 + 7907648*x - 5562368) + 13696448*x - 9634304)*sqrt(-672*sqrt(3) + 1164) - 2090880*x + 2396160)*(
-672*sqrt(3) + 1164)^(3/4) + 3*(984*x^15 + 14712*x^14 - 53940*x^13 - 411732*x^12 - 280248*x^11 + 324624*x^10 +
180816*x^9 + 518544*x^8 + 974304*x^7 + 887136*x^6 - 1404096*x^5 - 1843584*x^4 + 135936*x^3 + 696192*x^2 + 4*s
qrt(3)*(142*x^15 + 2124*x^14 - 7773*x^13 - 59447*x^12 - 40626*x^11 + 46860*x^10 + 26308*x^9 + 75276*x^8 + 1404
72*x^7 + 127784*x^6 - 202896*x^5 - 266016*x^4 + 19712*x^3 + 100512*x^2 + 62400*x - 24832) - (4945*x^15 + 37473
*x^14 - 490698*x^13 - 2249468*x^12 + 474132*x^11 + 8423784*x^10 + 5853520*x^9 - 8451720*x^8 - 15320016*x^7 - 7
68064*x^6 + 10405056*x^5 + 6627744*x^4 - 700480*x^3 - 2799552*x^2 + sqrt(3)*(2855*x^15 + 21635*x^14 - 283306*x
^13 - 1298732*x^12 + 273748*x^11 + 4863472*x^10 + 3379536*x^9 - 4879608*x^8 - 8845008*x^7 - 443456*x^6 + 60073
60*x^5 + 3826528*x^4 - 404416*x^3 - 1616320*x^2 - 1003648*x + 399360) - 1738368*x + 691712)*sqrt(-672*sqrt(3)
+ 1164) + 432384*x - 172032)*(-672*sqrt(3) + 1164)^(1/4))*sqrt(2*(7*sqrt(3) + 12)*sqrt(-672*sqrt(3) + 1164) +
24)*sqrt(-56*sqrt(3) + 97) + 6*(4680*x^16 + 60552*x^15 + 89856*x^14 - 278280*x^13 + 64440*x^12 + 1285200*x^11
- 255600*x^10 - 3098880*x^9 - 1770336*x^8 + 3614400*x^7 + 3895488*x^6 - 1199232*x^5 - 2905344*x^4 - 681984*x^3
+ 649728*x^2 + 108*sqrt(3)*(25*x^16 + 324*x^15 + 489*x^14 - 1482*x^13 + 316*x^12 + 6984*x^11 - 1312*x^10 - 16
624*x^9 - 9792*x^8 + 19328*x^7 + 20976*x^6 - 6240*x^5 - 15552*x^4 - 3712*x^3 + 3456*x^2 + 4096*x - 1280) + (11
64*x^17 - 1248*x^16 - 246120*x^15 - 518172*x^14 + 2607528*x^13 + 8301144*x^12 + 7017600*x^11 - 6258120*x^10 -
21360336*x^9 - 16998960*x^8 + 966336*x^7 + 18216672*x^6 + 15860544*x^5 + 4720704*x^4 - 6023424*x^3 - 5362176*x
^2 + 48*sqrt(3)*(14*x^17 - 15*x^16 - 2960*x^15 - 6232*x^14 + 31362*x^13 + 99844*x^12 + 84404*x^11 - 75267*x^10
- 256916*x^9 - 204458*x^8 + 11616*x^7 + 219104*x^6 + 190768*x^5 + 56784*x^4 - 72448*x^3 - 64496*x^2 - 24480*x
+ 13376) - (2340*x^17 + 35850*x^16 - 106410*x^15 + 2064744*x^14 + 11945946*x^13 + 1710042*x^12 - 46293732*x^1
1 - 59161524*x^10 + 18480192*x^9 + 122366520*x^8 + 81203856*x^7 - 45222000*x^6 - 100598112*x^5 - 42207168*x^4
+ 29609472*x^3 + 22458240*x^2 + sqrt(3)*(1351*x^17 + 20698*x^16 - 61436*x^15 + 1192081*x^14 + 6896998*x^13 + 9
87292*x^12 - 26727704*x^11 - 34156928*x^10 + 10669552*x^9 + 70648352*x^8 + 46883072*x^7 - 26108944*x^6 - 58080
352*x^5 - 24368320*x^4 + 17095040*x^3 + 12966272*x^2 + 4724480*x - 2581504) + 8183040*x - 4471296)*sqrt(-672*s
qrt(3) + 1164) - 2035200*x + 1112064)*sqrt(-672*sqrt(3) + 1164) - 24*(627*x^16 + 14286*x^15 + 39762*x^14 - 501
42*x^13 - 216816*x^12 - 112284*x^11 + 325707*x^10 + 586326*x^9 - 3294*x^8 - 631752*x^7 - 539220*x^6 + 184392*x
^5 + 483816*x^4 + 115296*x^3 - 108576*x^2 + 2*sqrt(3)*(181*x^16 + 4124*x^15 + 11478*x^14 - 14474*x^13 - 62584*
x^12 - 32412*x^11 + 94021*x^10 + 169244*x^9 - 954*x^8 - 182368*x^7 - 155648*x^6 + 53232*x^5 + 139664*x^4 + 332
80*x^3 - 31344*x^2 - 37024*x + 11584) - 128256*x + 40128)*sqrt(-672*sqrt(3) + 1164) + 764928*x - 239616)*sqrt(
-56*sqrt(3) + 97))*sqrt((36*x^8 - 72*x^7 + 1656*x^6 - 720*x^5 + 1440*x^4 - 2016*x^3 + (60*x^6 - 324*x^5 + 576*
x^4 - 696*x^3 + 432*x^2 + 36*sqrt(3)*(x^6 - 5*x^5 + 10*x^4 - 10*x^3 + 8*x^2 - 4*x) - (123*x^6 - 2016*x^5 + 221
4*x^4 - 2064*x^3 + 396*x^2 + sqrt(3)*(71*x^6 - 1164*x^5 + 1278*x^4 - 1192*x^3 + 228*x^2 - 112) - 192)*sqrt(-67
2*sqrt(3) + 1164) - 144*x + 96)*sqrt(x^3 - 1)*sqrt(2*(7*sqrt(3) + 12)*sqrt(-672*sqrt(3) + 1164) + 24)*(-672*sq
rt(3) + 1164)^(1/4) - 288*x^2 - 144*sqrt(3)*(x^7 - 4*x^6 + 6*x^5 - 5*x^4 - 4*x^3 - 6*x^2 + 4*x + 8) + 72*(26*x
^7 - 38*x^6 + 42*x^5 - 46*x^4 + 46*x^3 - 42*x^2 + sqrt(3)*(15*x^7 - 22*x^6 + 24*x^5 - 27*x^4 + 26*x^3 - 24*x^2
+ 12*x - 4) + 20*x - 8)*sqrt(-672*sqrt(3) + 1164) + 576*x + 2304)/(x^8 + 4*x^7 + 16*x^6 + 16*x^5 + 28*x^4 - 3
2*x^3 + 64*x^2 - 32*x + 16)))/(x^17 - 13*x^16 - 522*x^15 - 1742*x^14 + 3008*x^13 + 16884*x^12 + 11656*x^11 - 2
3944*x^10 - 42336*x^9 - 9136*x^8 + 36256*x^7 + 27360*x^6 - 256*x^5 - 13376*x^4 - 5760*x^3 + 1664*x^2 + 256*x))
+ 1/432*sqrt(2*(7*sqrt(3) + 12)*sqrt(-672*sqrt(3) + 1164) + 24)*(56*sqrt(3) + 97)*sqrt(-56*sqrt(3) + 97)*(-67
2*sqrt(3) + 1164)^(3/4)*arctan(1/1296*(6*sqrt(x^3 - 1)*((459*x^16 + 13425*x^15 - 33201*x^14 - 950652*x^13 - 99
7302*x^12 + 14760972*x^11 + 47069892*x^10 + 49762248*x^9 - 8212536*x^8 - 84377808*x^7 - 88427328*x^6 - 2561385
6*x^5 + 27458496*x^4 + 36433344*x^3 + 12609792*x^2 + sqrt(3)*(265*x^16 + 7751*x^15 - 19167*x^14 - 548864*x^13
- 575818*x^12 + 8522268*x^11 + 27175852*x^10 + 28730312*x^9 - 4741560*x^8 - 48715600*x^7 - 51053600*x^6 - 1478
8128*x^5 + 15853184*x^4 + 21034816*x^3 + 7280256*x^2 + 2488832*x - 1889792) - (3691*x^16 + 6128*x^15 - 537864*
x^14 - 1586477*x^13 + 16210952*x^12 + 77181756*x^11 + 84218362*x^10 - 71018320*x^9 - 254455812*x^8 - 196076008
*x^7 + 120105208*x^6 + 256326864*x^5 + 134645168*x^4 - 78464672*x^3 - 78514944*x^2 + sqrt(3)*(2131*x^16 + 3538
*x^15 - 310536*x^14 - 915953*x^13 + 9359398*x^12 + 44560908*x^11 + 48623494*x^10 - 41002448*x^9 - 146910132*x^
8 - 113204536*x^7 + 69342776*x^6 + 147990384*x^5 + 77737424*x^4 - 45301600*x^3 - 45330624*x^2 - 12242560*x + 7
598336) - 21204736*x + 13160704)*sqrt(-672*sqrt(3) + 1164) + 4310784*x - 3273216)*(-672*sqrt(3) + 1164)^(3/4)
+ 3*(984*x^15 + 30612*x^14 + 164676*x^13 + 205368*x^12 - 289200*x^11 - 183720*x^10 + 886752*x^9 + 71568*x^8 -
1960992*x^7 - 1849440*x^6 + 1558464*x^5 + 2478912*x^4 + 66432*x^3 - 750336*x^2 + 4*sqrt(3)*(142*x^15 + 4419*x^
14 + 23781*x^13 + 29608*x^12 - 41940*x^11 - 26454*x^10 + 128152*x^9 + 10692*x^8 - 283320*x^7 - 267064*x^6 + 22
4784*x^5 + 357936*x^4 + 9632*x^3 - 108288*x^2 - 96000*x + 33920) - (4945*x^15 + 88617*x^14 + 738528*x^13 + 186
0046*x^12 - 784596*x^11 - 7668708*x^10 - 6570680*x^9 + 6903864*x^8 + 15444144*x^7 + 4312832*x^6 - 9559200*x^5
- 9359808*x^4 - 155968*x^3 + 3016704*x^2 + sqrt(3)*(2855*x^15 + 51163*x^14 + 426388*x^13 + 1073898*x^12 - 4529
80*x^11 - 4427548*x^10 - 3793592*x^9 + 3985944*x^8 + 8916720*x^7 + 2490016*x^6 - 5519008*x^5 - 5403904*x^4 - 9
0048*x^3 + 1741696*x^2 + 1543936*x - 545536) + 2674176*x - 944896)*sqrt(-672*sqrt(3) + 1164) - 665088*x + 2350
08)*(-672*sqrt(3) + 1164)^(1/4))*sqrt(2*(7*sqrt(3) + 12)*sqrt(-672*sqrt(3) + 1164) + 24)*sqrt(-56*sqrt(3) + 97
) - 36*(144*x^17 + 5976*x^16 + 5544*x^15 - 299664*x^14 - 1062360*x^13 - 116712*x^12 + 3600000*x^11 + 4761216*x
^10 - 1046592*x^9 - 8676864*x^8 - 6592896*x^7 + 2641536*x^6 + 7016832*x^5 + 3699072*x^4 - 1861632*x^3 - 164044
8*x^2 + 12*sqrt(3)*(7*x^17 + 286*x^16 + 238*x^15 - 14255*x^14 - 50390*x^13 - 5942*x^12 + 171808*x^11 + 226888*
x^10 - 48920*x^9 - 415384*x^8 - 315088*x^7 + 125600*x^6 + 336608*x^5 + 177344*x^4 - 89152*x^3 - 78784*x^2 - 39
040*x + 18176) + (1164*x^17 + 6276*x^16 - 26052*x^15 - 332844*x^14 - 1632156*x^13 - 4149132*x^12 - 5805024*x^1
1 - 318696*x^10 + 12621072*x^9 + 19878720*x^8 + 9619008*x^7 - 13361088*x^6 - 20168256*x^5 - 10936128*x^4 + 643
4304*x^3 + 6426240*x^2 + 24*sqrt(3)*(28*x^17 + 151*x^16 - 626*x^15 - 8006*x^14 - 39266*x^13 - 99812*x^12 - 139
652*x^11 - 7661*x^10 + 303610*x^9 + 478214*x^8 + 231392*x^7 - 321412*x^6 - 485176*x^5 - 263080*x^4 + 154784*x^
3 + 154592*x^2 + 78464*x - 36544) - (2340*x^17 + 96354*x^16 + 84798*x^15 - 4817124*x^14 - 17052930*x^13 - 1941
678*x^12 + 57963744*x^11 + 76603680*x^10 - 16678512*x^9 - 139922496*x^8 - 106227360*x^7 + 42453216*x^6 + 11326
9536*x^5 + 59694624*x^4 - 30025728*x^3 - 26496000*x^2 + sqrt(3)*(1351*x^17 + 55630*x^16 + 48958*x^15 - 2781167
*x^14 - 9845510*x^13 - 1121030*x^12 + 33465376*x^11 + 44227144*x^10 - 9629336*x^9 - 80784280*x^8 - 61330384*x^
7 + 24510368*x^6 + 65396192*x^5 + 34464704*x^4 - 17335360*x^3 - 15297472*x^2 - 7571584*x + 3526400) - 13114368
*x + 6107904)*sqrt(-672*sqrt(3) + 1164) + 3261696*x - 1519104)*sqrt(-672*sqrt(3) + 1164) - 12*(97*x^17 + 523*x
^16 - 2171*x^15 - 27737*x^14 - 136013*x^13 - 345761*x^12 - 483752*x^11 - 26558*x^10 + 1051756*x^9 + 1656560*x^
8 + 801584*x^7 - 1113424*x^6 - 1680688*x^5 - 911344*x^4 + 536192*x^3 + 535520*x^2 + 2*sqrt(3)*(28*x^17 + 151*x
^16 - 626*x^15 - 8006*x^14 - 39266*x^13 - 99812*x^12 - 139652*x^11 - 7661*x^10 + 303610*x^9 + 478214*x^8 + 231
392*x^7 - 321412*x^6 - 485176*x^5 - 263080*x^4 + 154784*x^3 + 154592*x^2 + 78464*x - 36544) + 271808*x - 12659
2)*sqrt(-672*sqrt(3) + 1164) - 811008*x + 377856)*sqrt(-56*sqrt(3) + 97) - (sqrt(x^3 - 1)*((459*x^16 + 1557*x^
15 - 26415*x^14 + 1449954*x^13 + 4677912*x^12 - 12651948*x^11 - 55684800*x^10 - 62834256*x^9 + 8526168*x^8 + 1
05313392*x^7 + 99605088*x^6 + 18897984*x^5 - 42499296*x^4 - 37357632*x^3 - 8256960*x^2 + sqrt(3)*(265*x^16 + 8
99*x^15 - 15249*x^14 + 837130*x^13 + 2700776*x^12 - 7304604*x^11 - 32149640*x^10 - 36277360*x^9 + 4922568*x^8
+ 60802736*x^7 + 57507040*x^6 + 10910784*x^5 - 24536992*x^4 - 21568448*x^3 - 4767168*x^2 - 1207168*x + 1383424
) - (3691*x^16 - 17731*x^15 - 951114*x^14 - 450359*x^13 + 4370159*x^12 - 30318522*x^11 - 78096668*x^10 - 94293
16*x^9 + 146877876*x^8 + 197107784*x^7 - 30834152*x^6 - 185125776*x^5 - 132260896*x^4 + 45545344*x^3 + 6951753
6*x^2 + sqrt(3)*(2131*x^16 - 10237*x^15 - 549126*x^14 - 260015*x^13 + 2523113*x^12 - 17504406*x^11 - 45089132*
x^10 - 5444020*x^9 + 84799980*x^8 + 113800232*x^7 - 17802104*x^6 - 106882416*x^5 - 76360864*x^4 + 26295616*x^3
+ 40135968*x^2 + 7907648*x - 5562368) + 13696448*x - 9634304)*sqrt(-672*sqrt(3) + 1164) - 2090880*x + 2396160
)*(-672*sqrt(3) + 1164)^(3/4) + 3*(984*x^15 + 14712*x^14 - 53940*x^13 - 411732*x^12 - 280248*x^11 + 324624*x^1
0 + 180816*x^9 + 518544*x^8 + 974304*x^7 + 887136*x^6 - 1404096*x^5 - 1843584*x^4 + 135936*x^3 + 696192*x^2 +
4*sqrt(3)*(142*x^15 + 2124*x^14 - 7773*x^13 - 59447*x^12 - 40626*x^11 + 46860*x^10 + 26308*x^9 + 75276*x^8 + 1
40472*x^7 + 127784*x^6 - 202896*x^5 - 266016*x^4 + 19712*x^3 + 100512*x^2 + 62400*x - 24832) - (4945*x^15 + 37
473*x^14 - 490698*x^13 - 2249468*x^12 + 474132*x^11 + 8423784*x^10 + 5853520*x^9 - 8451720*x^8 - 15320016*x^7
- 768064*x^6 + 10405056*x^5 + 6627744*x^4 - 700480*x^3 - 2799552*x^2 + sqrt(3)*(2855*x^15 + 21635*x^14 - 28330
6*x^13 - 1298732*x^12 + 273748*x^11 + 4863472*x^10 + 3379536*x^9 - 4879608*x^8 - 8845008*x^7 - 443456*x^6 + 60
07360*x^5 + 3826528*x^4 - 404416*x^3 - 1616320*x^2 - 1003648*x + 399360) - 1738368*x + 691712)*sqrt(-672*sqrt(
3) + 1164) + 432384*x - 172032)*(-672*sqrt(3) + 1164)^(1/4))*sqrt(2*(7*sqrt(3) + 12)*sqrt(-672*sqrt(3) + 1164)
+ 24)*sqrt(-56*sqrt(3) + 97) - 6*(4680*x^16 + 60552*x^15 + 89856*x^14 - 278280*x^13 + 64440*x^12 + 1285200*x^
11 - 255600*x^10 - 3098880*x^9 - 1770336*x^8 + 3614400*x^7 + 3895488*x^6 - 1199232*x^5 - 2905344*x^4 - 681984*
x^3 + 649728*x^2 + 108*sqrt(3)*(25*x^16 + 324*x^15 + 489*x^14 - 1482*x^13 + 316*x^12 + 6984*x^11 - 1312*x^10 -
16624*x^9 - 9792*x^8 + 19328*x^7 + 20976*x^6 - 6240*x^5 - 15552*x^4 - 3712*x^3 + 3456*x^2 + 4096*x - 1280) +
(1164*x^17 - 1248*x^16 - 246120*x^15 - 518172*x^14 + 2607528*x^13 + 8301144*x^12 + 7017600*x^11 - 6258120*x^10
- 21360336*x^9 - 16998960*x^8 + 966336*x^7 + 18216672*x^6 + 15860544*x^5 + 4720704*x^4 - 6023424*x^3 - 536217
6*x^2 + 48*sqrt(3)*(14*x^17 - 15*x^16 - 2960*x^15 - 6232*x^14 + 31362*x^13 + 99844*x^12 + 84404*x^11 - 75267*x
^10 - 256916*x^9 - 204458*x^8 + 11616*x^7 + 219104*x^6 + 190768*x^5 + 56784*x^4 - 72448*x^3 - 64496*x^2 - 2448
0*x + 13376) - (2340*x^17 + 35850*x^16 - 106410*x^15 + 2064744*x^14 + 11945946*x^13 + 1710042*x^12 - 46293732*
x^11 - 59161524*x^10 + 18480192*x^9 + 122366520*x^8 + 81203856*x^7 - 45222000*x^6 - 100598112*x^5 - 42207168*x
^4 + 29609472*x^3 + 22458240*x^2 + sqrt(3)*(1351*x^17 + 20698*x^16 - 61436*x^15 + 1192081*x^14 + 6896998*x^13
+ 987292*x^12 - 26727704*x^11 - 34156928*x^10 + 10669552*x^9 + 70648352*x^8 + 46883072*x^7 - 26108944*x^6 - 58
080352*x^5 - 24368320*x^4 + 17095040*x^3 + 12966272*x^2 + 4724480*x - 2581504) + 8183040*x - 4471296)*sqrt(-67
2*sqrt(3) + 1164) - 2035200*x + 1112064)*sqrt(-672*sqrt(3) + 1164) - 24*(627*x^16 + 14286*x^15 + 39762*x^14 -
50142*x^13 - 216816*x^12 - 112284*x^11 + 325707*x^10 + 586326*x^9 - 3294*x^8 - 631752*x^7 - 539220*x^6 + 18439
2*x^5 + 483816*x^4 + 115296*x^3 - 108576*x^2 + 2*sqrt(3)*(181*x^16 + 4124*x^15 + 11478*x^14 - 14474*x^13 - 625
84*x^12 - 32412*x^11 + 94021*x^10 + 169244*x^9 - 954*x^8 - 182368*x^7 - 155648*x^6 + 53232*x^5 + 139664*x^4 +
33280*x^3 - 31344*x^2 - 37024*x + 11584) - 128256*x + 40128)*sqrt(-672*sqrt(3) + 1164) + 764928*x - 239616)*sq
rt(-56*sqrt(3) + 97))*sqrt((36*x^8 - 72*x^7 + 1656*x^6 - 720*x^5 + 1440*x^4 - 2016*x^3 - (60*x^6 - 324*x^5 + 5
76*x^4 - 696*x^3 + 432*x^2 + 36*sqrt(3)*(x^6 - 5*x^5 + 10*x^4 - 10*x^3 + 8*x^2 - 4*x) - (123*x^6 - 2016*x^5 +
2214*x^4 - 2064*x^3 + 396*x^2 + sqrt(3)*(71*x^6 - 1164*x^5 + 1278*x^4 - 1192*x^3 + 228*x^2 - 112) - 192)*sqrt(
-672*sqrt(3) + 1164) - 144*x + 96)*sqrt(x^3 - 1)*sqrt(2*(7*sqrt(3) + 12)*sqrt(-672*sqrt(3) + 1164) + 24)*(-672
*sqrt(3) + 1164)^(1/4) - 288*x^2 - 144*sqrt(3)*(x^7 - 4*x^6 + 6*x^5 - 5*x^4 - 4*x^3 - 6*x^2 + 4*x + 8) + 72*(2
6*x^7 - 38*x^6 + 42*x^5 - 46*x^4 + 46*x^3 - 42*x^2 + sqrt(3)*(15*x^7 - 22*x^6 + 24*x^5 - 27*x^4 + 26*x^3 - 24*
x^2 + 12*x - 4) + 20*x - 8)*sqrt(-672*sqrt(3) + 1164) + 576*x + 2304)/(x^8 + 4*x^7 + 16*x^6 + 16*x^5 + 28*x^4
- 32*x^3 + 64*x^2 - 32*x + 16)))/(x^17 - 13*x^16 - 522*x^15 - 1742*x^14 + 3008*x^13 + 16884*x^12 + 11656*x^11
- 23944*x^10 - 42336*x^9 - 9136*x^8 + 36256*x^7 + 27360*x^6 - 256*x^5 - 13376*x^4 - 5760*x^3 + 1664*x^2 + 256*
x)) + 1/5184*sqrt(2*(7*sqrt(3) + 12)*sqrt(-672*sqrt(3) + 1164) + 24)*((7*sqrt(3) + 12)*sqrt(-672*sqrt(3) + 116
4) - 12)*(-672*sqrt(3) + 1164)^(1/4)*log(1/36*(36*x^8 - 72*x^7 + 1656*x^6 - 720*x^5 + 1440*x^4 - 2016*x^3 + (6
0*x^6 - 324*x^5 + 576*x^4 - 696*x^3 + 432*x^2 + 36*sqrt(3)*(x^6 - 5*x^5 + 10*x^4 - 10*x^3 + 8*x^2 - 4*x) - (12
3*x^6 - 2016*x^5 + 2214*x^4 - 2064*x^3 + 396*x^2 + sqrt(3)*(71*x^6 - 1164*x^5 + 1278*x^4 - 1192*x^3 + 228*x^2
- 112) - 192)*sqrt(-672*sqrt(3) + 1164) - 144*x + 96)*sqrt(x^3 - 1)*sqrt(2*(7*sqrt(3) + 12)*sqrt(-672*sqrt(3)
+ 1164) + 24)*(-672*sqrt(3) + 1164)^(1/4) - 288*x^2 - 144*sqrt(3)*(x^7 - 4*x^6 + 6*x^5 - 5*x^4 - 4*x^3 - 6*x^2
+ 4*x + 8) + 72*(26*x^7 - 38*x^6 + 42*x^5 - 46*x^4 + 46*x^3 - 42*x^2 + sqrt(3)*(15*x^7 - 22*x^6 + 24*x^5 - 27
*x^4 + 26*x^3 - 24*x^2 + 12*x - 4) + 20*x - 8)*sqrt(-672*sqrt(3) + 1164) + 576*x + 2304)/(x^8 + 4*x^7 + 16*x^6
+ 16*x^5 + 28*x^4 - 32*x^3 + 64*x^2 - 32*x + 16)) - 1/5184*sqrt(2*(7*sqrt(3) + 12)*sqrt(-672*sqrt(3) + 1164)
+ 24)*((7*sqrt(3) + 12)*sqrt(-672*sqrt(3) + 1164) - 12)*(-672*sqrt(3) + 1164)^(1/4)*log(1/36*(36*x^8 - 72*x^7
+ 1656*x^6 - 720*x^5 + 1440*x^4 - 2016*x^3 - (60*x^6 - 324*x^5 + 576*x^4 - 696*x^3 + 432*x^2 + 36*sqrt(3)*(x^6
- 5*x^5 + 10*x^4 - 10*x^3 + 8*x^2 - 4*x) - (123*x^6 - 2016*x^5 + 2214*x^4 - 2064*x^3 + 396*x^2 + sqrt(3)*(71*
x^6 - 1164*x^5 + 1278*x^4 - 1192*x^3 + 228*x^2 - 112) - 192)*sqrt(-672*sqrt(3) + 1164) - 144*x + 96)*sqrt(x^3
- 1)*sqrt(2*(7*sqrt(3) + 12)*sqrt(-672*sqrt(3) + 1164) + 24)*(-672*sqrt(3) + 1164)^(1/4) - 288*x^2 - 144*sqrt(
3)*(x^7 - 4*x^6 + 6*x^5 - 5*x^4 - 4*x^3 - 6*x^2 + 4*x + 8) + 72*(26*x^7 - 38*x^6 + 42*x^5 - 46*x^4 + 46*x^3 -
42*x^2 + sqrt(3)*(15*x^7 - 22*x^6 + 24*x^5 - 27*x^4 + 26*x^3 - 24*x^2 + 12*x - 4) + 20*x - 8)*sqrt(-672*sqrt(3
) + 1164) + 576*x + 2304)/(x^8 + 4*x^7 + 16*x^6 + 16*x^5 + 28*x^4 - 32*x^3 + 64*x^2 - 32*x + 16)) + 1/72*sqrt(
14*sqrt(3) - 24)*log((x^8 + 16*x^7 + 112*x^6 + 16*x^5 + 112*x^4 - 224*x^3 + 64*x^2 - 2*(5*x^6 + 54*x^5 + 96*x^
4 + 56*x^3 - 36*x^2 + 3*sqrt(3)*(x^6 + 10*x^5 + 20*x^4 + 8*x^3 - 4*x^2 - 8*x) - 24*x - 16)*sqrt(x^3 - 1)*sqrt(
14*sqrt(3) - 24) + 16*sqrt(3)*(x^7 + 2*x^6 + 6*x^5 - 5*x^4 + 2*x^3 - 6*x^2 + 4*x - 4) - 128*x + 112)/(x^8 - 8*
x^7 + 16*x^6 + 16*x^5 - 56*x^4 - 32*x^3 + 64*x^2 + 64*x + 16))
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{{\left (x^{3} - 6 \, \sqrt {3} - 10\right )} \sqrt {x^{3} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x/(-10+x^3-6*3^(1/2))/(x^3-1)^(1/2),x, algorithm="giac")
[Out]
integrate(x/((x^3 - 6*sqrt(3) - 10)*sqrt(x^3 - 1)), x)
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maple [C] time = 0.27, size = 349, normalized size = 1.57 \[ \frac {\left (-1-\sqrt {3}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \EllipticPi \left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{9 \left (2+\sqrt {3}\right ) \sqrt {x^{3}-1}}-\frac {\sqrt {2}\, \left (-\sqrt {3}\, \RootOf \left (\textit {\_Z}^{2}+\left (1+\sqrt {3}\right ) \textit {\_Z} +2 \sqrt {3}+4\right )+\RootOf \left (\textit {\_Z}^{2}+\left (1+\sqrt {3}\right ) \textit {\_Z} +2 \sqrt {3}+4\right )+2\right ) \left (-i \sqrt {3}-3\right ) \sqrt {\frac {x -1}{-i \sqrt {3}-3}}\, \sqrt {\frac {2 x +1-i \sqrt {3}}{-i \sqrt {3}+3}}\, \sqrt {\frac {2 x +1+i \sqrt {3}}{i \sqrt {3}+3}}\, \left (2 \RootOf \left (\textit {\_Z}^{2}+\left (1+\sqrt {3}\right ) \textit {\_Z} +2 \sqrt {3}+4\right )-\sqrt {3}\, \RootOf \left (\textit {\_Z}^{2}+\left (1+\sqrt {3}\right ) \textit {\_Z} +2 \sqrt {3}+4\right )+1\right ) \EllipticPi \left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \sqrt {3}\, \RootOf \left (\textit {\_Z}^{2}+\left (1+\sqrt {3}\right ) \textit {\_Z} +2 \sqrt {3}+4\right )}{3}-\frac {\sqrt {3}\, \RootOf \left (\textit {\_Z}^{2}+\left (1+\sqrt {3}\right ) \textit {\_Z} +2 \sqrt {3}+4\right )}{2}-\frac {i \RootOf \left (\textit {\_Z}^{2}+\left (1+\sqrt {3}\right ) \textit {\_Z} +2 \sqrt {3}+4\right )}{2}+\RootOf \left (\textit {\_Z}^{2}+\left (1+\sqrt {3}\right ) \textit {\_Z} +2 \sqrt {3}+4\right )+\frac {i \sqrt {3}}{6}+\frac {1}{2}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{18 \left (-2 \RootOf \left (\textit {\_Z}^{2}+\left (1+\sqrt {3}\right ) \textit {\_Z} +2 \sqrt {3}+4\right )-\sqrt {3}-1\right ) \sqrt {x^{3}-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x/(-10+x^3-6*3^(1/2))/(x^3-1)^(1/2),x)
[Out]
1/9*(-1-3^(1/2))/(2+3^(1/2))*(-3/2-1/2*I*3^(1/2))*((x-1)/(-3/2-1/2*I*3^(1/2)))^(1/2)*((x+1/2-1/2*I*3^(1/2))/(3
/2-1/2*I*3^(1/2)))^(1/2)*((x+1/2+1/2*I*3^(1/2))/(3/2+1/2*I*3^(1/2)))^(1/2)/(x^3-1)^(1/2)*3^(1/2)*EllipticPi(((
x-1)/(-3/2-1/2*I*3^(1/2)))^(1/2),-1/3*(3/2+1/2*I*3^(1/2))*3^(1/2),((3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1
/2))-1/18*2^(1/2)*sum((-3^(1/2)*_alpha+_alpha+2)/(-3^(1/2)-2*_alpha-1)*(-I*3^(1/2)-3)*((x-1)/(-I*3^(1/2)-3))^(
1/2)*((2*x+1-I*3^(1/2))/(-I*3^(1/2)+3))^(1/2)*((2*x+1+I*3^(1/2))/(I*3^(1/2)+3))^(1/2)/(x^3-1)^(1/2)*(1+2*_alph
a-3^(1/2)*_alpha)*EllipticPi(((x-1)/(-3/2-1/2*I*3^(1/2)))^(1/2),1/3*I*3^(1/2)*_alpha-1/2*3^(1/2)*_alpha-1/2*I*
_alpha+_alpha+1/6*I*3^(1/2)+1/2,((3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2)),_alpha=RootOf(_Z^2+(1+3^(1/2)
)*_Z+2*3^(1/2)+4))
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{{\left (x^{3} - 6 \, \sqrt {3} - 10\right )} \sqrt {x^{3} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x/(-10+x^3-6*3^(1/2))/(x^3-1)^(1/2),x, algorithm="maxima")
[Out]
integrate(x/((x^3 - 6*sqrt(3) - 10)*sqrt(x^3 - 1)), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int -\frac {x}{\sqrt {x^3-1}\,\left (-x^3+6\,\sqrt {3}+10\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-x/((x^3 - 1)^(1/2)*(6*3^(1/2) - x^3 + 10)),x)
[Out]
int(-x/((x^3 - 1)^(1/2)*(6*3^(1/2) - x^3 + 10)), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {\left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x^{3} - 6 \sqrt {3} - 10\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x/(-10+x**3-6*3**(1/2))/(x**3-1)**(1/2),x)
[Out]
Integral(x/(sqrt((x - 1)*(x**2 + x + 1))*(x**3 - 6*sqrt(3) - 10)), x)
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