3.217 \(\int \frac{1}{(-1+x)^{2/3} x^5} \, dx\)
Optimal. Leaf size=104 \[ \frac{\sqrt [3]{x-1}}{4 x^4}+\frac{11 \sqrt [3]{x-1}}{36 x^3}+\frac{11 \sqrt [3]{x-1}}{27 x^2}+\frac{55 \sqrt [3]{x-1}}{81 x}+\frac{55}{81} \log \left (\sqrt [3]{x-1}+1\right )-\frac{55 \log (x)}{243}-\frac{110 \tan ^{-1}\left (\frac{1-2 \sqrt [3]{x-1}}{\sqrt{3}}\right )}{81 \sqrt{3}} \]
[Out]
(-1 + x)^(1/3)/(4*x^4) + (11*(-1 + x)^(1/3))/(36*x^3) + (11*(-1 + x)^(1/3))/(27*
x^2) + (55*(-1 + x)^(1/3))/(81*x) - (110*ArcTan[(1 - 2*(-1 + x)^(1/3))/Sqrt[3]])
/(81*Sqrt[3]) + (55*Log[1 + (-1 + x)^(1/3)])/81 - (55*Log[x])/243
_______________________________________________________________________________________
Rubi [A] time = 0.0933074, antiderivative size = 104, normalized size of antiderivative =
1., number of steps used = 8, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454
\[ \frac{\sqrt [3]{x-1}}{4 x^4}+\frac{11 \sqrt [3]{x-1}}{36 x^3}+\frac{11 \sqrt [3]{x-1}}{27 x^2}+\frac{55 \sqrt [3]{x-1}}{81 x}+\frac{55}{81} \log \left (\sqrt [3]{x-1}+1\right )-\frac{55 \log (x)}{243}-\frac{110 \tan ^{-1}\left (\frac{1-2 \sqrt [3]{x-1}}{\sqrt{3}}\right )}{81 \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[1/((-1 + x)^(2/3)*x^5),x]
[Out]
(-1 + x)^(1/3)/(4*x^4) + (11*(-1 + x)^(1/3))/(36*x^3) + (11*(-1 + x)^(1/3))/(27*
x^2) + (55*(-1 + x)^(1/3))/(81*x) - (110*ArcTan[(1 - 2*(-1 + x)^(1/3))/Sqrt[3]])
/(81*Sqrt[3]) + (55*Log[1 + (-1 + x)^(1/3)])/81 - (55*Log[x])/243
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Rubi in Sympy [A] time = 3.23367, size = 99, normalized size = 0.95 \[ - \frac{55 \log{\left (x \right )}}{243} + \frac{55 \log{\left (\sqrt [3]{x - 1} + 1 \right )}}{81} + \frac{110 \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{x - 1}}{3} - \frac{1}{3}\right ) \right )}}{243} + \frac{55 \sqrt [3]{x - 1}}{81 x} + \frac{11 \sqrt [3]{x - 1}}{27 x^{2}} + \frac{11 \sqrt [3]{x - 1}}{36 x^{3}} + \frac{\sqrt [3]{x - 1}}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(-1+x)**(2/3)/x**5,x)
[Out]
-55*log(x)/243 + 55*log((x - 1)**(1/3) + 1)/81 + 110*sqrt(3)*atan(sqrt(3)*(2*(x
- 1)**(1/3)/3 - 1/3))/243 + 55*(x - 1)**(1/3)/(81*x) + 11*(x - 1)**(1/3)/(27*x**
2) + 11*(x - 1)**(1/3)/(36*x**3) + (x - 1)**(1/3)/(4*x**4)
_______________________________________________________________________________________
Mathematica [C] time = 0.0247523, size = 63, normalized size = 0.61 \[ \frac{-220 \left (\frac{x-1}{x}\right )^{2/3} x^4 \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{1}{x}\right )+220 x^4-88 x^3-33 x^2-18 x-81}{324 (x-1)^{2/3} x^4} \]
Antiderivative was successfully verified.
[In] Integrate[1/((-1 + x)^(2/3)*x^5),x]
[Out]
(-81 - 18*x - 33*x^2 - 88*x^3 + 220*x^4 - 220*((-1 + x)/x)^(2/3)*x^4*Hypergeomet
ric2F1[2/3, 2/3, 5/3, x^(-1)])/(324*(-1 + x)^(2/3)*x^4)
_______________________________________________________________________________________
Maple [B] time = 0.026, size = 158, normalized size = 1.5 \[ -{\frac{1}{324} \left ( 1+\sqrt [3]{-1+x} \right ) ^{-4}}-{\frac{5}{243} \left ( 1+\sqrt [3]{-1+x} \right ) ^{-3}}-{\frac{20}{243} \left ( 1+\sqrt [3]{-1+x} \right ) ^{-2}}-{\frac{25}{81} \left ( 1+\sqrt [3]{-1+x} \right ) ^{-1}}+{\frac{110}{243}\ln \left ( 1+\sqrt [3]{-1+x} \right ) }-{\frac{1}{243} \left ( -75\, \left ( -1+x \right ) ^{7/3}+190\, \left ( -1+x \right ) ^{2}-350\, \left ( -1+x \right ) ^{5/3}+{\frac{1157}{4} \left ( -1+x \right ) ^{{\frac{4}{3}}}}+{\frac{149}{4}}-138\,x-116\, \left ( -1+x \right ) ^{2/3}+137\,\sqrt [3]{-1+x} \right ) \left ( \left ( -1+x \right ) ^{{\frac{2}{3}}}-\sqrt [3]{-1+x}+1 \right ) ^{-4}}-{\frac{55}{243}\ln \left ( \left ( -1+x \right ) ^{{\frac{2}{3}}}-\sqrt [3]{-1+x}+1 \right ) }+{\frac{110\,\sqrt{3}}{243}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,\sqrt [3]{-1+x}-1 \right ) } \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(-1+x)^(2/3)/x^5,x)
[Out]
-1/324/(1+(-1+x)^(1/3))^4-5/243/(1+(-1+x)^(1/3))^3-20/243/(1+(-1+x)^(1/3))^2-25/
81/(1+(-1+x)^(1/3))+110/243*ln(1+(-1+x)^(1/3))-1/243*(-75*(-1+x)^(7/3)+190*(-1+x
)^2-350*(-1+x)^(5/3)+1157/4*(-1+x)^(4/3)+149/4-138*x-116*(-1+x)^(2/3)+137*(-1+x)
^(1/3))/((-1+x)^(2/3)-(-1+x)^(1/3)+1)^4-55/243*ln((-1+x)^(2/3)-(-1+x)^(1/3)+1)+1
10/243*3^(1/2)*arctan(1/3*(2*(-1+x)^(1/3)-1)*3^(1/2))
_______________________________________________________________________________________
Maxima [A] time = 1.49667, size = 142, normalized size = 1.37 \[ \frac{110}{243} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \,{\left (x - 1\right )}^{\frac{1}{3}} - 1\right )}\right ) + \frac{220 \,{\left (x - 1\right )}^{\frac{10}{3}} + 792 \,{\left (x - 1\right )}^{\frac{7}{3}} + 1023 \,{\left (x - 1\right )}^{\frac{4}{3}} + 532 \,{\left (x - 1\right )}^{\frac{1}{3}}}{324 \,{\left ({\left (x - 1\right )}^{4} + 4 \,{\left (x - 1\right )}^{3} + 6 \,{\left (x - 1\right )}^{2} + 4 \, x - 3\right )}} - \frac{55}{243} \, \log \left ({\left (x - 1\right )}^{\frac{2}{3}} -{\left (x - 1\right )}^{\frac{1}{3}} + 1\right ) + \frac{110}{243} \, \log \left ({\left (x - 1\right )}^{\frac{1}{3}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x - 1)^(2/3)*x^5),x, algorithm="maxima")
[Out]
110/243*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x - 1)^(1/3) - 1)) + 1/324*(220*(x - 1)^(
10/3) + 792*(x - 1)^(7/3) + 1023*(x - 1)^(4/3) + 532*(x - 1)^(1/3))/((x - 1)^4 +
4*(x - 1)^3 + 6*(x - 1)^2 + 4*x - 3) - 55/243*log((x - 1)^(2/3) - (x - 1)^(1/3)
+ 1) + 110/243*log((x - 1)^(1/3) + 1)
_______________________________________________________________________________________
Fricas [A] time = 0.215965, size = 128, normalized size = 1.23 \[ -\frac{\sqrt{3}{\left (220 \, \sqrt{3} x^{4} \log \left ({\left (x - 1\right )}^{\frac{2}{3}} -{\left (x - 1\right )}^{\frac{1}{3}} + 1\right ) - 440 \, \sqrt{3} x^{4} \log \left ({\left (x - 1\right )}^{\frac{1}{3}} + 1\right ) - 1320 \, x^{4} \arctan \left (\frac{2}{3} \, \sqrt{3}{\left (x - 1\right )}^{\frac{1}{3}} - \frac{1}{3} \, \sqrt{3}\right ) - 3 \, \sqrt{3}{\left (220 \, x^{3} + 132 \, x^{2} + 99 \, x + 81\right )}{\left (x - 1\right )}^{\frac{1}{3}}\right )}}{2916 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x - 1)^(2/3)*x^5),x, algorithm="fricas")
[Out]
-1/2916*sqrt(3)*(220*sqrt(3)*x^4*log((x - 1)^(2/3) - (x - 1)^(1/3) + 1) - 440*sq
rt(3)*x^4*log((x - 1)^(1/3) + 1) - 1320*x^4*arctan(2/3*sqrt(3)*(x - 1)^(1/3) - 1
/3*sqrt(3)) - 3*sqrt(3)*(220*x^3 + 132*x^2 + 99*x + 81)*(x - 1)^(1/3))/x^4
_______________________________________________________________________________________
Sympy [A] time = 4.89729, size = 7998, normalized size = 76.9 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(-1+x)**(2/3)/x**5,x)
[Out]
660*(x - 1)**(34/3)*gamma(1/3)/(2916*(x - 1)**12*gamma(4/3) + 32076*(x - 1)**11*
gamma(4/3) + 160380*(x - 1)**10*gamma(4/3) + 481140*(x - 1)**9*gamma(4/3) + 9622
80*(x - 1)**8*gamma(4/3) + 1347192*(x - 1)**7*gamma(4/3) + 1347192*(x - 1)**6*ga
mma(4/3) + 962280*(x - 1)**5*gamma(4/3) + 481140*(x - 1)**4*gamma(4/3) + 160380*
(x - 1)**3*gamma(4/3) + 32076*(x - 1)**2*gamma(4/3) + 2916*(x - 1)*gamma(4/3)) +
6996*(x - 1)**(31/3)*gamma(1/3)/(2916*(x - 1)**12*gamma(4/3) + 32076*(x - 1)**1
1*gamma(4/3) + 160380*(x - 1)**10*gamma(4/3) + 481140*(x - 1)**9*gamma(4/3) + 96
2280*(x - 1)**8*gamma(4/3) + 1347192*(x - 1)**7*gamma(4/3) + 1347192*(x - 1)**6*
gamma(4/3) + 962280*(x - 1)**5*gamma(4/3) + 481140*(x - 1)**4*gamma(4/3) + 16038
0*(x - 1)**3*gamma(4/3) + 32076*(x - 1)**2*gamma(4/3) + 2916*(x - 1)*gamma(4/3))
+ 33561*(x - 1)**(28/3)*gamma(1/3)/(2916*(x - 1)**12*gamma(4/3) + 32076*(x - 1)
**11*gamma(4/3) + 160380*(x - 1)**10*gamma(4/3) + 481140*(x - 1)**9*gamma(4/3) +
962280*(x - 1)**8*gamma(4/3) + 1347192*(x - 1)**7*gamma(4/3) + 1347192*(x - 1)*
*6*gamma(4/3) + 962280*(x - 1)**5*gamma(4/3) + 481140*(x - 1)**4*gamma(4/3) + 16
0380*(x - 1)**3*gamma(4/3) + 32076*(x - 1)**2*gamma(4/3) + 2916*(x - 1)*gamma(4/
3)) + 96075*(x - 1)**(25/3)*gamma(1/3)/(2916*(x - 1)**12*gamma(4/3) + 32076*(x -
1)**11*gamma(4/3) + 160380*(x - 1)**10*gamma(4/3) + 481140*(x - 1)**9*gamma(4/3
) + 962280*(x - 1)**8*gamma(4/3) + 1347192*(x - 1)**7*gamma(4/3) + 1347192*(x -
1)**6*gamma(4/3) + 962280*(x - 1)**5*gamma(4/3) + 481140*(x - 1)**4*gamma(4/3) +
160380*(x - 1)**3*gamma(4/3) + 32076*(x - 1)**2*gamma(4/3) + 2916*(x - 1)*gamma
(4/3)) + 181881*(x - 1)**(22/3)*gamma(1/3)/(2916*(x - 1)**12*gamma(4/3) + 32076*
(x - 1)**11*gamma(4/3) + 160380*(x - 1)**10*gamma(4/3) + 481140*(x - 1)**9*gamma
(4/3) + 962280*(x - 1)**8*gamma(4/3) + 1347192*(x - 1)**7*gamma(4/3) + 1347192*(
x - 1)**6*gamma(4/3) + 962280*(x - 1)**5*gamma(4/3) + 481140*(x - 1)**4*gamma(4/
3) + 160380*(x - 1)**3*gamma(4/3) + 32076*(x - 1)**2*gamma(4/3) + 2916*(x - 1)*g
amma(4/3)) + 237951*(x - 1)**(19/3)*gamma(1/3)/(2916*(x - 1)**12*gamma(4/3) + 32
076*(x - 1)**11*gamma(4/3) + 160380*(x - 1)**10*gamma(4/3) + 481140*(x - 1)**9*g
amma(4/3) + 962280*(x - 1)**8*gamma(4/3) + 1347192*(x - 1)**7*gamma(4/3) + 13471
92*(x - 1)**6*gamma(4/3) + 962280*(x - 1)**5*gamma(4/3) + 481140*(x - 1)**4*gamm
a(4/3) + 160380*(x - 1)**3*gamma(4/3) + 32076*(x - 1)**2*gamma(4/3) + 2916*(x -
1)*gamma(4/3)) + 217791*(x - 1)**(16/3)*gamma(1/3)/(2916*(x - 1)**12*gamma(4/3)
+ 32076*(x - 1)**11*gamma(4/3) + 160380*(x - 1)**10*gamma(4/3) + 481140*(x - 1)*
*9*gamma(4/3) + 962280*(x - 1)**8*gamma(4/3) + 1347192*(x - 1)**7*gamma(4/3) + 1
347192*(x - 1)**6*gamma(4/3) + 962280*(x - 1)**5*gamma(4/3) + 481140*(x - 1)**4*
gamma(4/3) + 160380*(x - 1)**3*gamma(4/3) + 32076*(x - 1)**2*gamma(4/3) + 2916*(
x - 1)*gamma(4/3)) + 137601*(x - 1)**(13/3)*gamma(1/3)/(2916*(x - 1)**12*gamma(4
/3) + 32076*(x - 1)**11*gamma(4/3) + 160380*(x - 1)**10*gamma(4/3) + 481140*(x -
1)**9*gamma(4/3) + 962280*(x - 1)**8*gamma(4/3) + 1347192*(x - 1)**7*gamma(4/3)
+ 1347192*(x - 1)**6*gamma(4/3) + 962280*(x - 1)**5*gamma(4/3) + 481140*(x - 1)
**4*gamma(4/3) + 160380*(x - 1)**3*gamma(4/3) + 32076*(x - 1)**2*gamma(4/3) + 29
16*(x - 1)*gamma(4/3)) + 57375*(x - 1)**(10/3)*gamma(1/3)/(2916*(x - 1)**12*gamm
a(4/3) + 32076*(x - 1)**11*gamma(4/3) + 160380*(x - 1)**10*gamma(4/3) + 481140*(
x - 1)**9*gamma(4/3) + 962280*(x - 1)**8*gamma(4/3) + 1347192*(x - 1)**7*gamma(4
/3) + 1347192*(x - 1)**6*gamma(4/3) + 962280*(x - 1)**5*gamma(4/3) + 481140*(x -
1)**4*gamma(4/3) + 160380*(x - 1)**3*gamma(4/3) + 32076*(x - 1)**2*gamma(4/3) +
2916*(x - 1)*gamma(4/3)) + 14241*(x - 1)**(7/3)*gamma(1/3)/(2916*(x - 1)**12*ga
mma(4/3) + 32076*(x - 1)**11*gamma(4/3) + 160380*(x - 1)**10*gamma(4/3) + 481140
*(x - 1)**9*gamma(4/3) + 962280*(x - 1)**8*gamma(4/3) + 1347192*(x - 1)**7*gamma
(4/3) + 1347192*(x - 1)**6*gamma(4/3) + 962280*(x - 1)**5*gamma(4/3) + 481140*(x
- 1)**4*gamma(4/3) + 160380*(x - 1)**3*gamma(4/3) + 32076*(x - 1)**2*gamma(4/3)
+ 2916*(x - 1)*gamma(4/3)) + 1596*(x - 1)**(4/3)*gamma(1/3)/(2916*(x - 1)**12*g
amma(4/3) + 32076*(x - 1)**11*gamma(4/3) + 160380*(x - 1)**10*gamma(4/3) + 48114
0*(x - 1)**9*gamma(4/3) + 962280*(x - 1)**8*gamma(4/3) + 1347192*(x - 1)**7*gamm
a(4/3) + 1347192*(x - 1)**6*gamma(4/3) + 962280*(x - 1)**5*gamma(4/3) + 481140*(
x - 1)**4*gamma(4/3) + 160380*(x - 1)**3*gamma(4/3) + 32076*(x - 1)**2*gamma(4/3
) + 2916*(x - 1)*gamma(4/3)) - 440*(x - 1)**12*exp(5*I*pi/3)*log(-(x - 1)**(1/3)
*exp_polar(I*pi/3) + 1)*gamma(1/3)/(2916*(x - 1)**12*gamma(4/3) + 32076*(x - 1)*
*11*gamma(4/3) + 160380*(x - 1)**10*gamma(4/3) + 481140*(x - 1)**9*gamma(4/3) +
962280*(x - 1)**8*gamma(4/3) + 1347192*(x - 1)**7*gamma(4/3) + 1347192*(x - 1)**
6*gamma(4/3) + 962280*(x - 1)**5*gamma(4/3) + 481140*(x - 1)**4*gamma(4/3) + 160
380*(x - 1)**3*gamma(4/3) + 32076*(x - 1)**2*gamma(4/3) + 2916*(x - 1)*gamma(4/3
)) + 440*(x - 1)**12*log(-(x - 1)**(1/3)*exp_polar(I*pi) + 1)*gamma(1/3)/(2916*(
x - 1)**12*gamma(4/3) + 32076*(x - 1)**11*gamma(4/3) + 160380*(x - 1)**10*gamma(
4/3) + 481140*(x - 1)**9*gamma(4/3) + 962280*(x - 1)**8*gamma(4/3) + 1347192*(x
- 1)**7*gamma(4/3) + 1347192*(x - 1)**6*gamma(4/3) + 962280*(x - 1)**5*gamma(4/3
) + 481140*(x - 1)**4*gamma(4/3) + 160380*(x - 1)**3*gamma(4/3) + 32076*(x - 1)*
*2*gamma(4/3) + 2916*(x - 1)*gamma(4/3)) - 440*(x - 1)**12*exp(I*pi/3)*log(-(x -
1)**(1/3)*exp_polar(5*I*pi/3) + 1)*gamma(1/3)/(2916*(x - 1)**12*gamma(4/3) + 32
076*(x - 1)**11*gamma(4/3) + 160380*(x - 1)**10*gamma(4/3) + 481140*(x - 1)**9*g
amma(4/3) + 962280*(x - 1)**8*gamma(4/3) + 1347192*(x - 1)**7*gamma(4/3) + 13471
92*(x - 1)**6*gamma(4/3) + 962280*(x - 1)**5*gamma(4/3) + 481140*(x - 1)**4*gamm
a(4/3) + 160380*(x - 1)**3*gamma(4/3) + 32076*(x - 1)**2*gamma(4/3) + 2916*(x -
1)*gamma(4/3)) - 4840*(x - 1)**11*exp(5*I*pi/3)*log(-(x - 1)**(1/3)*exp_polar(I*
pi/3) + 1)*gamma(1/3)/(2916*(x - 1)**12*gamma(4/3) + 32076*(x - 1)**11*gamma(4/3
) + 160380*(x - 1)**10*gamma(4/3) + 481140*(x - 1)**9*gamma(4/3) + 962280*(x - 1
)**8*gamma(4/3) + 1347192*(x - 1)**7*gamma(4/3) + 1347192*(x - 1)**6*gamma(4/3)
+ 962280*(x - 1)**5*gamma(4/3) + 481140*(x - 1)**4*gamma(4/3) + 160380*(x - 1)**
3*gamma(4/3) + 32076*(x - 1)**2*gamma(4/3) + 2916*(x - 1)*gamma(4/3)) + 4840*(x
- 1)**11*log(-(x - 1)**(1/3)*exp_polar(I*pi) + 1)*gamma(1/3)/(2916*(x - 1)**12*g
amma(4/3) + 32076*(x - 1)**11*gamma(4/3) + 160380*(x - 1)**10*gamma(4/3) + 48114
0*(x - 1)**9*gamma(4/3) + 962280*(x - 1)**8*gamma(4/3) + 1347192*(x - 1)**7*gamm
a(4/3) + 1347192*(x - 1)**6*gamma(4/3) + 962280*(x - 1)**5*gamma(4/3) + 481140*(
x - 1)**4*gamma(4/3) + 160380*(x - 1)**3*gamma(4/3) + 32076*(x - 1)**2*gamma(4/3
) + 2916*(x - 1)*gamma(4/3)) - 4840*(x - 1)**11*exp(I*pi/3)*log(-(x - 1)**(1/3)*
exp_polar(5*I*pi/3) + 1)*gamma(1/3)/(2916*(x - 1)**12*gamma(4/3) + 32076*(x - 1)
**11*gamma(4/3) + 160380*(x - 1)**10*gamma(4/3) + 481140*(x - 1)**9*gamma(4/3) +
962280*(x - 1)**8*gamma(4/3) + 1347192*(x - 1)**7*gamma(4/3) + 1347192*(x - 1)*
*6*gamma(4/3) + 962280*(x - 1)**5*gamma(4/3) + 481140*(x - 1)**4*gamma(4/3) + 16
0380*(x - 1)**3*gamma(4/3) + 32076*(x - 1)**2*gamma(4/3) + 2916*(x - 1)*gamma(4/
3)) - 24200*(x - 1)**10*exp(5*I*pi/3)*log(-(x - 1)**(1/3)*exp_polar(I*pi/3) + 1)
*gamma(1/3)/(2916*(x - 1)**12*gamma(4/3) + 32076*(x - 1)**11*gamma(4/3) + 160380
*(x - 1)**10*gamma(4/3) + 481140*(x - 1)**9*gamma(4/3) + 962280*(x - 1)**8*gamma
(4/3) + 1347192*(x - 1)**7*gamma(4/3) + 1347192*(x - 1)**6*gamma(4/3) + 962280*(
x - 1)**5*gamma(4/3) + 481140*(x - 1)**4*gamma(4/3) + 160380*(x - 1)**3*gamma(4/
3) + 32076*(x - 1)**2*gamma(4/3) + 2916*(x - 1)*gamma(4/3)) + 24200*(x - 1)**10*
log(-(x - 1)**(1/3)*exp_polar(I*pi) + 1)*gamma(1/3)/(2916*(x - 1)**12*gamma(4/3)
+ 32076*(x - 1)**11*gamma(4/3) + 160380*(x - 1)**10*gamma(4/3) + 481140*(x - 1)
**9*gamma(4/3) + 962280*(x - 1)**8*gamma(4/3) + 1347192*(x - 1)**7*gamma(4/3) +
1347192*(x - 1)**6*gamma(4/3) + 962280*(x - 1)**5*gamma(4/3) + 481140*(x - 1)**4
*gamma(4/3) + 160380*(x - 1)**3*gamma(4/3) + 32076*(x - 1)**2*gamma(4/3) + 2916*
(x - 1)*gamma(4/3)) - 24200*(x - 1)**10*exp(I*pi/3)*log(-(x - 1)**(1/3)*exp_pola
r(5*I*pi/3) + 1)*gamma(1/3)/(2916*(x - 1)**12*gamma(4/3) + 32076*(x - 1)**11*gam
ma(4/3) + 160380*(x - 1)**10*gamma(4/3) + 481140*(x - 1)**9*gamma(4/3) + 962280*
(x - 1)**8*gamma(4/3) + 1347192*(x - 1)**7*gamma(4/3) + 1347192*(x - 1)**6*gamma
(4/3) + 962280*(x - 1)**5*gamma(4/3) + 481140*(x - 1)**4*gamma(4/3) + 160380*(x
- 1)**3*gamma(4/3) + 32076*(x - 1)**2*gamma(4/3) + 2916*(x - 1)*gamma(4/3)) - 72
600*(x - 1)**9*exp(5*I*pi/3)*log(-(x - 1)**(1/3)*exp_polar(I*pi/3) + 1)*gamma(1/
3)/(2916*(x - 1)**12*gamma(4/3) + 32076*(x - 1)**11*gamma(4/3) + 160380*(x - 1)*
*10*gamma(4/3) + 481140*(x - 1)**9*gamma(4/3) + 962280*(x - 1)**8*gamma(4/3) + 1
347192*(x - 1)**7*gamma(4/3) + 1347192*(x - 1)**6*gamma(4/3) + 962280*(x - 1)**5
*gamma(4/3) + 481140*(x - 1)**4*gamma(4/3) + 160380*(x - 1)**3*gamma(4/3) + 3207
6*(x - 1)**2*gamma(4/3) + 2916*(x - 1)*gamma(4/3)) + 72600*(x - 1)**9*log(-(x -
1)**(1/3)*exp_polar(I*pi) + 1)*gamma(1/3)/(2916*(x - 1)**12*gamma(4/3) + 32076*(
x - 1)**11*gamma(4/3) + 160380*(x - 1)**10*gamma(4/3) + 481140*(x - 1)**9*gamma(
4/3) + 962280*(x - 1)**8*gamma(4/3) + 1347192*(x - 1)**7*gamma(4/3) + 1347192*(x
- 1)**6*gamma(4/3) + 962280*(x - 1)**5*gamma(4/3) + 481140*(x - 1)**4*gamma(4/3
) + 160380*(x - 1)**3*gamma(4/3) + 32076*(x - 1)**2*gamma(4/3) + 2916*(x - 1)*ga
mma(4/3)) - 72600*(x - 1)**9*exp(I*pi/3)*log(-(x - 1)**(1/3)*exp_polar(5*I*pi/3)
+ 1)*gamma(1/3)/(2916*(x - 1)**12*gamma(4/3) + 32076*(x - 1)**11*gamma(4/3) + 1
60380*(x - 1)**10*gamma(4/3) + 481140*(x - 1)**9*gamma(4/3) + 962280*(x - 1)**8*
gamma(4/3) + 1347192*(x - 1)**7*gamma(4/3) + 1347192*(x - 1)**6*gamma(4/3) + 962
280*(x - 1)**5*gamma(4/3) + 481140*(x - 1)**4*gamma(4/3) + 160380*(x - 1)**3*gam
ma(4/3) + 32076*(x - 1)**2*gamma(4/3) + 2916*(x - 1)*gamma(4/3)) - 145200*(x - 1
)**8*exp(5*I*pi/3)*log(-(x - 1)**(1/3)*exp_polar(I*pi/3) + 1)*gamma(1/3)/(2916*(
x - 1)**12*gamma(4/3) + 32076*(x - 1)**11*gamma(4/3) + 160380*(x - 1)**10*gamma(
4/3) + 481140*(x - 1)**9*gamma(4/3) + 962280*(x - 1)**8*gamma(4/3) + 1347192*(x
- 1)**7*gamma(4/3) + 1347192*(x - 1)**6*gamma(4/3) + 962280*(x - 1)**5*gamma(4/3
) + 481140*(x - 1)**4*gamma(4/3) + 160380*(x - 1)**3*gamma(4/3) + 32076*(x - 1)*
*2*gamma(4/3) + 2916*(x - 1)*gamma(4/3)) + 145200*(x - 1)**8*log(-(x - 1)**(1/3)
*exp_polar(I*pi) + 1)*gamma(1/3)/(2916*(x - 1)**12*gamma(4/3) + 32076*(x - 1)**1
1*gamma(4/3) + 160380*(x - 1)**10*gamma(4/3) + 481140*(x - 1)**9*gamma(4/3) + 96
2280*(x - 1)**8*gamma(4/3) + 1347192*(x - 1)**7*gamma(4/3) + 1347192*(x - 1)**6*
gamma(4/3) + 962280*(x - 1)**5*gamma(4/3) + 481140*(x - 1)**4*gamma(4/3) + 16038
0*(x - 1)**3*gamma(4/3) + 32076*(x - 1)**2*gamma(4/3) + 2916*(x - 1)*gamma(4/3))
- 145200*(x - 1)**8*exp(I*pi/3)*log(-(x - 1)**(1/3)*exp_polar(5*I*pi/3) + 1)*ga
mma(1/3)/(2916*(x - 1)**12*gamma(4/3) + 32076*(x - 1)**11*gamma(4/3) + 160380*(x
- 1)**10*gamma(4/3) + 481140*(x - 1)**9*gamma(4/3) + 962280*(x - 1)**8*gamma(4/
3) + 1347192*(x - 1)**7*gamma(4/3) + 1347192*(x - 1)**6*gamma(4/3) + 962280*(x -
1)**5*gamma(4/3) + 481140*(x - 1)**4*gamma(4/3) + 160380*(x - 1)**3*gamma(4/3)
+ 32076*(x - 1)**2*gamma(4/3) + 2916*(x - 1)*gamma(4/3)) - 203280*(x - 1)**7*exp
(5*I*pi/3)*log(-(x - 1)**(1/3)*exp_polar(I*pi/3) + 1)*gamma(1/3)/(2916*(x - 1)**
12*gamma(4/3) + 32076*(x - 1)**11*gamma(4/3) + 160380*(x - 1)**10*gamma(4/3) + 4
81140*(x - 1)**9*gamma(4/3) + 962280*(x - 1)**8*gamma(4/3) + 1347192*(x - 1)**7*
gamma(4/3) + 1347192*(x - 1)**6*gamma(4/3) + 962280*(x - 1)**5*gamma(4/3) + 4811
40*(x - 1)**4*gamma(4/3) + 160380*(x - 1)**3*gamma(4/3) + 32076*(x - 1)**2*gamma
(4/3) + 2916*(x - 1)*gamma(4/3)) + 203280*(x - 1)**7*log(-(x - 1)**(1/3)*exp_pol
ar(I*pi) + 1)*gamma(1/3)/(2916*(x - 1)**12*gamma(4/3) + 32076*(x - 1)**11*gamma(
4/3) + 160380*(x - 1)**10*gamma(4/3) + 481140*(x - 1)**9*gamma(4/3) + 962280*(x
- 1)**8*gamma(4/3) + 1347192*(x - 1)**7*gamma(4/3) + 1347192*(x - 1)**6*gamma(4/
3) + 962280*(x - 1)**5*gamma(4/3) + 481140*(x - 1)**4*gamma(4/3) + 160380*(x - 1
)**3*gamma(4/3) + 32076*(x - 1)**2*gamma(4/3) + 2916*(x - 1)*gamma(4/3)) - 20328
0*(x - 1)**7*exp(I*pi/3)*log(-(x - 1)**(1/3)*exp_polar(5*I*pi/3) + 1)*gamma(1/3)
/(2916*(x - 1)**12*gamma(4/3) + 32076*(x - 1)**11*gamma(4/3) + 160380*(x - 1)**1
0*gamma(4/3) + 481140*(x - 1)**9*gamma(4/3) + 962280*(x - 1)**8*gamma(4/3) + 134
7192*(x - 1)**7*gamma(4/3) + 1347192*(x - 1)**6*gamma(4/3) + 962280*(x - 1)**5*g
amma(4/3) + 481140*(x - 1)**4*gamma(4/3) + 160380*(x - 1)**3*gamma(4/3) + 32076*
(x - 1)**2*gamma(4/3) + 2916*(x - 1)*gamma(4/3)) - 203280*(x - 1)**6*exp(5*I*pi/
3)*log(-(x - 1)**(1/3)*exp_polar(I*pi/3) + 1)*gamma(1/3)/(2916*(x - 1)**12*gamma
(4/3) + 32076*(x - 1)**11*gamma(4/3) + 160380*(x - 1)**10*gamma(4/3) + 481140*(x
- 1)**9*gamma(4/3) + 962280*(x - 1)**8*gamma(4/3) + 1347192*(x - 1)**7*gamma(4/
3) + 1347192*(x - 1)**6*gamma(4/3) + 962280*(x - 1)**5*gamma(4/3) + 481140*(x -
1)**4*gamma(4/3) + 160380*(x - 1)**3*gamma(4/3) + 32076*(x - 1)**2*gamma(4/3) +
2916*(x - 1)*gamma(4/3)) + 203280*(x - 1)**6*log(-(x - 1)**(1/3)*exp_polar(I*pi)
+ 1)*gamma(1/3)/(2916*(x - 1)**12*gamma(4/3) + 32076*(x - 1)**11*gamma(4/3) + 1
60380*(x - 1)**10*gamma(4/3) + 481140*(x - 1)**9*gamma(4/3) + 962280*(x - 1)**8*
gamma(4/3) + 1347192*(x - 1)**7*gamma(4/3) + 1347192*(x - 1)**6*gamma(4/3) + 962
280*(x - 1)**5*gamma(4/3) + 481140*(x - 1)**4*gamma(4/3) + 160380*(x - 1)**3*gam
ma(4/3) + 32076*(x - 1)**2*gamma(4/3) + 2916*(x - 1)*gamma(4/3)) - 203280*(x - 1
)**6*exp(I*pi/3)*log(-(x - 1)**(1/3)*exp_polar(5*I*pi/3) + 1)*gamma(1/3)/(2916*(
x - 1)**12*gamma(4/3) + 32076*(x - 1)**11*gamma(4/3) + 160380*(x - 1)**10*gamma(
4/3) + 481140*(x - 1)**9*gamma(4/3) + 962280*(x - 1)**8*gamma(4/3) + 1347192*(x
- 1)**7*gamma(4/3) + 1347192*(x - 1)**6*gamma(4/3) + 962280*(x - 1)**5*gamma(4/3
) + 481140*(x - 1)**4*gamma(4/3) + 160380*(x - 1)**3*gamma(4/3) + 32076*(x - 1)*
*2*gamma(4/3) + 2916*(x - 1)*gamma(4/3)) - 145200*(x - 1)**5*exp(5*I*pi/3)*log(-
(x - 1)**(1/3)*exp_polar(I*pi/3) + 1)*gamma(1/3)/(2916*(x - 1)**12*gamma(4/3) +
32076*(x - 1)**11*gamma(4/3) + 160380*(x - 1)**10*gamma(4/3) + 481140*(x - 1)**9
*gamma(4/3) + 962280*(x - 1)**8*gamma(4/3) + 1347192*(x - 1)**7*gamma(4/3) + 134
7192*(x - 1)**6*gamma(4/3) + 962280*(x - 1)**5*gamma(4/3) + 481140*(x - 1)**4*ga
mma(4/3) + 160380*(x - 1)**3*gamma(4/3) + 32076*(x - 1)**2*gamma(4/3) + 2916*(x
- 1)*gamma(4/3)) + 145200*(x - 1)**5*log(-(x - 1)**(1/3)*exp_polar(I*pi) + 1)*ga
mma(1/3)/(2916*(x - 1)**12*gamma(4/3) + 32076*(x - 1)**11*gamma(4/3) + 160380*(x
- 1)**10*gamma(4/3) + 481140*(x - 1)**9*gamma(4/3) + 962280*(x - 1)**8*gamma(4/
3) + 1347192*(x - 1)**7*gamma(4/3) + 1347192*(x - 1)**6*gamma(4/3) + 962280*(x -
1)**5*gamma(4/3) + 481140*(x - 1)**4*gamma(4/3) + 160380*(x - 1)**3*gamma(4/3)
+ 32076*(x - 1)**2*gamma(4/3) + 2916*(x - 1)*gamma(4/3)) - 145200*(x - 1)**5*exp
(I*pi/3)*log(-(x - 1)**(1/3)*exp_polar(5*I*pi/3) + 1)*gamma(1/3)/(2916*(x - 1)**
12*gamma(4/3) + 32076*(x - 1)**11*gamma(4/3) + 160380*(x - 1)**10*gamma(4/3) + 4
81140*(x - 1)**9*gamma(4/3) + 962280*(x - 1)**8*gamma(4/3) + 1347192*(x - 1)**7*
gamma(4/3) + 1347192*(x - 1)**6*gamma(4/3) + 962280*(x - 1)**5*gamma(4/3) + 4811
40*(x - 1)**4*gamma(4/3) + 160380*(x - 1)**3*gamma(4/3) + 32076*(x - 1)**2*gamma
(4/3) + 2916*(x - 1)*gamma(4/3)) - 72600*(x - 1)**4*exp(5*I*pi/3)*log(-(x - 1)**
(1/3)*exp_polar(I*pi/3) + 1)*gamma(1/3)/(2916*(x - 1)**12*gamma(4/3) + 32076*(x
- 1)**11*gamma(4/3) + 160380*(x - 1)**10*gamma(4/3) + 481140*(x - 1)**9*gamma(4/
3) + 962280*(x - 1)**8*gamma(4/3) + 1347192*(x - 1)**7*gamma(4/3) + 1347192*(x -
1)**6*gamma(4/3) + 962280*(x - 1)**5*gamma(4/3) + 481140*(x - 1)**4*gamma(4/3)
+ 160380*(x - 1)**3*gamma(4/3) + 32076*(x - 1)**2*gamma(4/3) + 2916*(x - 1)*gamm
a(4/3)) + 72600*(x - 1)**4*log(-(x - 1)**(1/3)*exp_polar(I*pi) + 1)*gamma(1/3)/(
2916*(x - 1)**12*gamma(4/3) + 32076*(x - 1)**11*gamma(4/3) + 160380*(x - 1)**10*
gamma(4/3) + 481140*(x - 1)**9*gamma(4/3) + 962280*(x - 1)**8*gamma(4/3) + 13471
92*(x - 1)**7*gamma(4/3) + 1347192*(x - 1)**6*gamma(4/3) + 962280*(x - 1)**5*gam
ma(4/3) + 481140*(x - 1)**4*gamma(4/3) + 160380*(x - 1)**3*gamma(4/3) + 32076*(x
- 1)**2*gamma(4/3) + 2916*(x - 1)*gamma(4/3)) - 72600*(x - 1)**4*exp(I*pi/3)*lo
g(-(x - 1)**(1/3)*exp_polar(5*I*pi/3) + 1)*gamma(1/3)/(2916*(x - 1)**12*gamma(4/
3) + 32076*(x - 1)**11*gamma(4/3) + 160380*(x - 1)**10*gamma(4/3) + 481140*(x -
1)**9*gamma(4/3) + 962280*(x - 1)**8*gamma(4/3) + 1347192*(x - 1)**7*gamma(4/3)
+ 1347192*(x - 1)**6*gamma(4/3) + 962280*(x - 1)**5*gamma(4/3) + 481140*(x - 1)*
*4*gamma(4/3) + 160380*(x - 1)**3*gamma(4/3) + 32076*(x - 1)**2*gamma(4/3) + 291
6*(x - 1)*gamma(4/3)) - 24200*(x - 1)**3*exp(5*I*pi/3)*log(-(x - 1)**(1/3)*exp_p
olar(I*pi/3) + 1)*gamma(1/3)/(2916*(x - 1)**12*gamma(4/3) + 32076*(x - 1)**11*ga
mma(4/3) + 160380*(x - 1)**10*gamma(4/3) + 481140*(x - 1)**9*gamma(4/3) + 962280
*(x - 1)**8*gamma(4/3) + 1347192*(x - 1)**7*gamma(4/3) + 1347192*(x - 1)**6*gamm
a(4/3) + 962280*(x - 1)**5*gamma(4/3) + 481140*(x - 1)**4*gamma(4/3) + 160380*(x
- 1)**3*gamma(4/3) + 32076*(x - 1)**2*gamma(4/3) + 2916*(x - 1)*gamma(4/3)) + 2
4200*(x - 1)**3*log(-(x - 1)**(1/3)*exp_polar(I*pi) + 1)*gamma(1/3)/(2916*(x - 1
)**12*gamma(4/3) + 32076*(x - 1)**11*gamma(4/3) + 160380*(x - 1)**10*gamma(4/3)
+ 481140*(x - 1)**9*gamma(4/3) + 962280*(x - 1)**8*gamma(4/3) + 1347192*(x - 1)*
*7*gamma(4/3) + 1347192*(x - 1)**6*gamma(4/3) + 962280*(x - 1)**5*gamma(4/3) + 4
81140*(x - 1)**4*gamma(4/3) + 160380*(x - 1)**3*gamma(4/3) + 32076*(x - 1)**2*ga
mma(4/3) + 2916*(x - 1)*gamma(4/3)) - 24200*(x - 1)**3*exp(I*pi/3)*log(-(x - 1)*
*(1/3)*exp_polar(5*I*pi/3) + 1)*gamma(1/3)/(2916*(x - 1)**12*gamma(4/3) + 32076*
(x - 1)**11*gamma(4/3) + 160380*(x - 1)**10*gamma(4/3) + 481140*(x - 1)**9*gamma
(4/3) + 962280*(x - 1)**8*gamma(4/3) + 1347192*(x - 1)**7*gamma(4/3) + 1347192*(
x - 1)**6*gamma(4/3) + 962280*(x - 1)**5*gamma(4/3) + 481140*(x - 1)**4*gamma(4/
3) + 160380*(x - 1)**3*gamma(4/3) + 32076*(x - 1)**2*gamma(4/3) + 2916*(x - 1)*g
amma(4/3)) - 4840*(x - 1)**2*exp(5*I*pi/3)*log(-(x - 1)**(1/3)*exp_polar(I*pi/3)
+ 1)*gamma(1/3)/(2916*(x - 1)**12*gamma(4/3) + 32076*(x - 1)**11*gamma(4/3) + 1
60380*(x - 1)**10*gamma(4/3) + 481140*(x - 1)**9*gamma(4/3) + 962280*(x - 1)**8*
gamma(4/3) + 1347192*(x - 1)**7*gamma(4/3) + 1347192*(x - 1)**6*gamma(4/3) + 962
280*(x - 1)**5*gamma(4/3) + 481140*(x - 1)**4*gamma(4/3) + 160380*(x - 1)**3*gam
ma(4/3) + 32076*(x - 1)**2*gamma(4/3) + 2916*(x - 1)*gamma(4/3)) + 4840*(x - 1)*
*2*log(-(x - 1)**(1/3)*exp_polar(I*pi) + 1)*gamma(1/3)/(2916*(x - 1)**12*gamma(4
/3) + 32076*(x - 1)**11*gamma(4/3) + 160380*(x - 1)**10*gamma(4/3) + 481140*(x -
1)**9*gamma(4/3) + 962280*(x - 1)**8*gamma(4/3) + 1347192*(x - 1)**7*gamma(4/3)
+ 1347192*(x - 1)**6*gamma(4/3) + 962280*(x - 1)**5*gamma(4/3) + 481140*(x - 1)
**4*gamma(4/3) + 160380*(x - 1)**3*gamma(4/3) + 32076*(x - 1)**2*gamma(4/3) + 29
16*(x - 1)*gamma(4/3)) - 4840*(x - 1)**2*exp(I*pi/3)*log(-(x - 1)**(1/3)*exp_pol
ar(5*I*pi/3) + 1)*gamma(1/3)/(2916*(x - 1)**12*gamma(4/3) + 32076*(x - 1)**11*ga
mma(4/3) + 160380*(x - 1)**10*gamma(4/3) + 481140*(x - 1)**9*gamma(4/3) + 962280
*(x - 1)**8*gamma(4/3) + 1347192*(x - 1)**7*gamma(4/3) + 1347192*(x - 1)**6*gamm
a(4/3) + 962280*(x - 1)**5*gamma(4/3) + 481140*(x - 1)**4*gamma(4/3) + 160380*(x
- 1)**3*gamma(4/3) + 32076*(x - 1)**2*gamma(4/3) + 2916*(x - 1)*gamma(4/3)) - 4
40*(x - 1)*exp(5*I*pi/3)*log(-(x - 1)**(1/3)*exp_polar(I*pi/3) + 1)*gamma(1/3)/(
2916*(x - 1)**12*gamma(4/3) + 32076*(x - 1)**11*gamma(4/3) + 160380*(x - 1)**10*
gamma(4/3) + 481140*(x - 1)**9*gamma(4/3) + 962280*(x - 1)**8*gamma(4/3) + 13471
92*(x - 1)**7*gamma(4/3) + 1347192*(x - 1)**6*gamma(4/3) + 962280*(x - 1)**5*gam
ma(4/3) + 481140*(x - 1)**4*gamma(4/3) + 160380*(x - 1)**3*gamma(4/3) + 32076*(x
- 1)**2*gamma(4/3) + 2916*(x - 1)*gamma(4/3)) + 440*(x - 1)*log(-(x - 1)**(1/3)
*exp_polar(I*pi) + 1)*gamma(1/3)/(2916*(x - 1)**12*gamma(4/3) + 32076*(x - 1)**1
1*gamma(4/3) + 160380*(x - 1)**10*gamma(4/3) + 481140*(x - 1)**9*gamma(4/3) + 96
2280*(x - 1)**8*gamma(4/3) + 1347192*(x - 1)**7*gamma(4/3) + 1347192*(x - 1)**6*
gamma(4/3) + 962280*(x - 1)**5*gamma(4/3) + 481140*(x - 1)**4*gamma(4/3) + 16038
0*(x - 1)**3*gamma(4/3) + 32076*(x - 1)**2*gamma(4/3) + 2916*(x - 1)*gamma(4/3))
- 440*(x - 1)*exp(I*pi/3)*log(-(x - 1)**(1/3)*exp_polar(5*I*pi/3) + 1)*gamma(1/
3)/(2916*(x - 1)**12*gamma(4/3) + 32076*(x - 1)**11*gamma(4/3) + 160380*(x - 1)*
*10*gamma(4/3) + 481140*(x - 1)**9*gamma(4/3) + 962280*(x - 1)**8*gamma(4/3) + 1
347192*(x - 1)**7*gamma(4/3) + 1347192*(x - 1)**6*gamma(4/3) + 962280*(x - 1)**5
*gamma(4/3) + 481140*(x - 1)**4*gamma(4/3) + 160380*(x - 1)**3*gamma(4/3) + 3207
6*(x - 1)**2*gamma(4/3) + 2916*(x - 1)*gamma(4/3))
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.203377, size = 111, normalized size = 1.07 \[ \frac{110}{243} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \,{\left (x - 1\right )}^{\frac{1}{3}} - 1\right )}\right ) + \frac{220 \,{\left (x - 1\right )}^{\frac{10}{3}} + 792 \,{\left (x - 1\right )}^{\frac{7}{3}} + 1023 \,{\left (x - 1\right )}^{\frac{4}{3}} + 532 \,{\left (x - 1\right )}^{\frac{1}{3}}}{324 \, x^{4}} - \frac{55}{243} \,{\rm ln}\left ({\left (x - 1\right )}^{\frac{2}{3}} -{\left (x - 1\right )}^{\frac{1}{3}} + 1\right ) + \frac{110}{243} \,{\rm ln}\left ({\left (x - 1\right )}^{\frac{1}{3}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x - 1)^(2/3)*x^5),x, algorithm="giac")
[Out]
110/243*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x - 1)^(1/3) - 1)) + 1/324*(220*(x - 1)^(
10/3) + 792*(x - 1)^(7/3) + 1023*(x - 1)^(4/3) + 532*(x - 1)^(1/3))/x^4 - 55/243
*ln((x - 1)^(2/3) - (x - 1)^(1/3) + 1) + 110/243*ln((x - 1)^(1/3) + 1)