Integrand size = 12, antiderivative size = 130 \[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x^3} \, dx=\frac {1}{3} \left (1-\frac {1}{x}\right )^{2/3} \sqrt [3]{1+\frac {1}{x}}+\frac {1}{2} \left (1-\frac {1}{x}\right )^{2/3} \left (1+\frac {1}{x}\right )^{4/3}-\frac {2 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-\frac {1}{x}}}{\sqrt {3} \sqrt [3]{1+\frac {1}{x}}}\right )}{3 \sqrt {3}}-\frac {1}{3} \log \left (1+\frac {\sqrt [3]{1-\frac {1}{x}}}{\sqrt [3]{1+\frac {1}{x}}}\right )-\frac {1}{9} \log \left (1+\frac {1}{x}\right ) \] Output:
1/3*(1-1/x)^(2/3)*(1+1/x)^(1/3)+1/2*(1-1/x)^(2/3)*(1+1/x)^(4/3)+2/9*3^(1/2 )*arctan(-1/3*3^(1/2)+2/3*(1-1/x)^(1/3)*3^(1/2)/(1+1/x)^(1/3))-1/3*ln(1+(1 -1/x)^(1/3)/(1+1/x)^(1/3))-1/9*ln(1+1/x)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.50 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.03 \[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x^3} \, dx=-\frac {2}{27} \left (\frac {27 e^{\frac {2}{3} \coth ^{-1}(x)}}{\left (1+e^{2 \coth ^{-1}(x)}\right )^2}-\frac {36 e^{\frac {2}{3} \coth ^{-1}(x)}}{1+e^{2 \coth ^{-1}(x)}}-2 \coth ^{-1}(x)+3 \log \left (1+e^{\frac {2}{3} \coth ^{-1}(x)}\right )-\text {RootSum}\left [1-\text {$\#$1}^2+\text {$\#$1}^4\&,\frac {\coth ^{-1}(x)-3 \log \left (e^{\frac {1}{3} \coth ^{-1}(x)}-\text {$\#$1}\right )+\coth ^{-1}(x) \text {$\#$1}^2-3 \log \left (e^{\frac {1}{3} \coth ^{-1}(x)}-\text {$\#$1}\right ) \text {$\#$1}^2}{-2+\text {$\#$1}^2}\&\right ]\right ) \] Input:
Integrate[E^((2*ArcCoth[x])/3)/x^3,x]
Output:
(-2*((27*E^((2*ArcCoth[x])/3))/(1 + E^(2*ArcCoth[x]))^2 - (36*E^((2*ArcCot h[x])/3))/(1 + E^(2*ArcCoth[x])) - 2*ArcCoth[x] + 3*Log[1 + E^((2*ArcCoth[ x])/3)] - RootSum[1 - #1^2 + #1^4 & , (ArcCoth[x] - 3*Log[E^(ArcCoth[x]/3) - #1] + ArcCoth[x]*#1^2 - 3*Log[E^(ArcCoth[x]/3) - #1]*#1^2)/(-2 + #1^2) & ]))/27
Time = 0.41 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6721, 90, 60, 72}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x^3} \, dx\) |
| \(\Big \downarrow \) 6721 |
| \(\displaystyle -\int \frac {\sqrt [3]{1+\frac {1}{x}}}{\sqrt [3]{1-\frac {1}{x}} x}d\frac {1}{x}\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {1}{2} \left (1-\frac {1}{x}\right )^{2/3} \left (\frac {1}{x}+1\right )^{4/3}-\frac {1}{3} \int \frac {\sqrt [3]{1+\frac {1}{x}}}{\sqrt [3]{1-\frac {1}{x}}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{3} \left (\left (1-\frac {1}{x}\right )^{2/3} \sqrt [3]{\frac {1}{x}+1}-\frac {2}{3} \int \frac {1}{\sqrt [3]{1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{2/3}}d\frac {1}{x}\right )+\frac {1}{2} \left (1-\frac {1}{x}\right )^{2/3} \left (\frac {1}{x}+1\right )^{4/3}\) |
| \(\Big \downarrow \) 72 |
| \(\displaystyle \frac {1}{3} \left (\left (1-\frac {1}{x}\right )^{2/3} \sqrt [3]{\frac {1}{x}+1}-\frac {2}{3} \left (\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-\frac {1}{x}}}{\sqrt {3} \sqrt [3]{\frac {1}{x}+1}}\right )+\frac {3}{2} \log \left (\frac {\sqrt [3]{1-\frac {1}{x}}}{\sqrt [3]{\frac {1}{x}+1}}+1\right )+\frac {1}{2} \log \left (\frac {1}{x}+1\right )\right )\right )+\frac {1}{2} \left (1-\frac {1}{x}\right )^{2/3} \left (\frac {1}{x}+1\right )^{4/3}\) |
Input:
Int[E^((2*ArcCoth[x])/3)/x^3,x]
Output:
((1 - x^(-1))^(2/3)*(1 + x^(-1))^(4/3))/2 + ((1 - x^(-1))^(2/3)*(1 + x^(-1 ))^(1/3) - (2*(Sqrt[3]*ArcTan[1/Sqrt[3] - (2*(1 - x^(-1))^(1/3))/(Sqrt[3]* (1 + x^(-1))^(1/3))] + (3*Log[1 + (1 - x^(-1))^(1/3)/(1 + x^(-1))^(1/3)])/ 2 + Log[1 + x^(-1)]/2))/3)/3
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[-d/b, 3]}, Simp[Sqrt[3]*(q/d)*ArcTan[1/Sqrt[3] - 2*q*((a + b* x)^(1/3)/(Sqrt[3]*(c + d*x)^(1/3)))], x] + (Simp[3*(q/(2*d))*Log[q*((a + b* x)^(1/3)/(c + d*x)^(1/3)) + 1], x] + Simp[(q/(2*d))*Log[c + d*x], x])] /; F reeQ[{a, b, c, d}, x] && NegQ[d/b]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(x_)^(m_.), x_Symbol] :> -Subst[Int[(1 + x /a)^(n/2)/(x^(m + 2)*(1 - x/a)^(n/2)), x], x, 1/x] /; FreeQ[{a, n}, x] && !IntegerQ[n] && IntegerQ[m]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.64 (sec) , antiderivative size = 675, normalized size of antiderivative = 5.19
| method | result | size |
| trager | \(\frac {\left (1+x \right ) \left (5 x +3\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}}{6 x^{2}}+\frac {2 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (\frac {-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}} x -9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}} x +18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -15 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-2 x +2}{x}\right )}{3}+\frac {2 \ln \left (\frac {9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}} x +9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}} x -3 \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}} x +18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -3 \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}+3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}} x +3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}}-x -1}{x}\right )}{9}-\frac {2 \ln \left (\frac {9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}} x +9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}} x -3 \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}} x +18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -3 \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}+3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}} x +3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}}-x -1}{x}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )}{3}\) | \(675\) |
| risch | \(\text {Expression too large to display}\) | \(739\) |
Input:
int(1/((x-1)/(1+x))^(1/3)/x^3,x,method=_RETURNVERBOSE)
Output:
1/6*(1+x)*(5*x+3)/x^2*(-(1-x)/(1+x))^(2/3)+2/3*RootOf(9*_Z^2-3*_Z+1)*ln((- 9*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(2/3)*x-9*RootOf(9*_Z^2-3*_Z+1)*(-( 1-x)/(1+x))^(2/3)+9*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(1/3)*x+18*RootOf (9*_Z^2-3*_Z+1)^2+9*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(1/3)+3*RootOf(9* _Z^2-3*_Z+1)*x-15*RootOf(9*_Z^2-3*_Z+1)-2*x+2)/x)+2/9*ln((9*RootOf(9*_Z^2- 3*_Z+1)*(-(1-x)/(1+x))^(2/3)*x+9*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(2/3 )-9*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(1/3)*x-3*(-(1-x)/(1+x))^(2/3)*x+ 18*RootOf(9*_Z^2-3*_Z+1)^2-9*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(1/3)-3* RootOf(9*_Z^2-3*_Z+1)*x-3*(-(1-x)/(1+x))^(2/3)+3*(-(1-x)/(1+x))^(1/3)*x+3* RootOf(9*_Z^2-3*_Z+1)+3*(-(1-x)/(1+x))^(1/3)-x-1)/x)-2/3*ln((9*RootOf(9*_Z ^2-3*_Z+1)*(-(1-x)/(1+x))^(2/3)*x+9*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^( 2/3)-9*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(1/3)*x-3*(-(1-x)/(1+x))^(2/3) *x+18*RootOf(9*_Z^2-3*_Z+1)^2-9*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(1/3) -3*RootOf(9*_Z^2-3*_Z+1)*x-3*(-(1-x)/(1+x))^(2/3)+3*(-(1-x)/(1+x))^(1/3)*x +3*RootOf(9*_Z^2-3*_Z+1)+3*(-(1-x)/(1+x))^(1/3)-x-1)/x)*RootOf(9*_Z^2-3*_Z +1)
Time = 0.09 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.85 \[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x^3} \, dx=\frac {4 \, \sqrt {3} x^{2} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + 2 \, x^{2} \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) - 4 \, x^{2} \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + 3 \, {\left (5 \, x^{2} + 8 \, x + 3\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}}{18 \, x^{2}} \] Input:
integrate(1/((x-1)/(1+x))^(1/3)/x^3,x, algorithm="fricas")
Output:
1/18*(4*sqrt(3)*x^2*arctan(2/3*sqrt(3)*((x - 1)/(x + 1))^(1/3) - 1/3*sqrt( 3)) + 2*x^2*log(((x - 1)/(x + 1))^(2/3) - ((x - 1)/(x + 1))^(1/3) + 1) - 4 *x^2*log(((x - 1)/(x + 1))^(1/3) + 1) + 3*(5*x^2 + 8*x + 3)*((x - 1)/(x + 1))^(2/3))/x^2
\[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x^3} \, dx=\int \frac {1}{x^{3} \sqrt [3]{\frac {x - 1}{x + 1}}}\, dx \] Input:
integrate(1/((x-1)/(1+x))**(1/3)/x**3,x)
Output:
Integral(1/(x**3*((x - 1)/(x + 1))**(1/3)), x)
Time = 0.11 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.95 \[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x^3} \, dx=\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - 1\right )}\right ) + \frac {2 \, {\left (\left (\frac {x - 1}{x + 1}\right )^{\frac {5}{3}} + 4 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}\right )}}{3 \, {\left (\frac {2 \, {\left (x - 1\right )}}{x + 1} + \frac {{\left (x - 1\right )}^{2}}{{\left (x + 1\right )}^{2}} + 1\right )}} + \frac {1}{9} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) - \frac {2}{9} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) \] Input:
integrate(1/((x-1)/(1+x))^(1/3)/x^3,x, algorithm="maxima")
Output:
2/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/3) - 1)) + 2/3*(((x - 1)/(x + 1))^(5/3) + 4*((x - 1)/(x + 1))^(2/3))/(2*(x - 1)/(x + 1) + (x - 1)^2/(x + 1)^2 + 1) + 1/9*log(((x - 1)/(x + 1))^(2/3) - ((x - 1)/(x + 1) )^(1/3) + 1) - 2/9*log(((x - 1)/(x + 1))^(1/3) + 1)
Time = 0.13 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.94 \[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x^3} \, dx=\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - 1\right )}\right ) + \frac {2 \, {\left (\frac {{\left (x - 1\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}}{x + 1} + 4 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}\right )}}{3 \, {\left (\frac {x - 1}{x + 1} + 1\right )}^{2}} + \frac {1}{9} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) - \frac {2}{9} \, \log \left ({\left | \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1 \right |}\right ) \] Input:
integrate(1/((x-1)/(1+x))^(1/3)/x^3,x, algorithm="giac")
Output:
2/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/3) - 1)) + 2/3*((x - 1)*((x - 1)/(x + 1))^(2/3)/(x + 1) + 4*((x - 1)/(x + 1))^(2/3))/((x - 1) /(x + 1) + 1)^2 + 1/9*log(((x - 1)/(x + 1))^(2/3) - ((x - 1)/(x + 1))^(1/3 ) + 1) - 2/9*log(abs(((x - 1)/(x + 1))^(1/3) + 1))
Time = 0.03 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.12 \[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x^3} \, dx=\frac {\frac {8\,{\left (\frac {x-1}{x+1}\right )}^{2/3}}{3}+\frac {2\,{\left (\frac {x-1}{x+1}\right )}^{5/3}}{3}}{\frac {2\,\left (x-1\right )}{x+1}+\frac {{\left (x-1\right )}^2}{{\left (x+1\right )}^2}+1}-\frac {2\,\ln \left (\frac {4\,{\left (\frac {x-1}{x+1}\right )}^{1/3}}{9}+\frac {4}{9}\right )}{9}-\ln \left (9\,{\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )}^2+\frac {4\,{\left (\frac {x-1}{x+1}\right )}^{1/3}}{9}\right )\,\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )+\ln \left (9\,{\left (\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )}^2+\frac {4\,{\left (\frac {x-1}{x+1}\right )}^{1/3}}{9}\right )\,\left (\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right ) \] Input:
int(1/(x^3*((x - 1)/(x + 1))^(1/3)),x)
Output:
((8*((x - 1)/(x + 1))^(2/3))/3 + (2*((x - 1)/(x + 1))^(5/3))/3)/((2*(x - 1 ))/(x + 1) + (x - 1)^2/(x + 1)^2 + 1) - (2*log((4*((x - 1)/(x + 1))^(1/3)) /9 + 4/9))/9 - log(9*((3^(1/2)*1i)/9 - 1/9)^2 + (4*((x - 1)/(x + 1))^(1/3) )/9)*((3^(1/2)*1i)/9 - 1/9) + log(9*((3^(1/2)*1i)/9 + 1/9)^2 + (4*((x - 1) /(x + 1))^(1/3))/9)*((3^(1/2)*1i)/9 + 1/9)
\[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x^3} \, dx=\int \frac {\left (x +1\right )^{\frac {1}{3}}}{\left (x -1\right )^{\frac {1}{3}} x^{3}}d x \] Input:
int(1/((x-1)/(1+x))^(1/3)/x^3,x)
Output:
int((x + 1)**(1/3)/((x - 1)**(1/3)*x**3),x)