\(\int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x^2} \, dx\) [132]

Optimal result

Integrand size = 12, antiderivative size = 99 \[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x^2} \, dx=\left (1-\frac {1}{x}\right )^{2/3} \sqrt [3]{1+\frac {1}{x}}-\frac {2 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-\frac {1}{x}}}{\sqrt {3} \sqrt [3]{1+\frac {1}{x}}}\right )}{\sqrt {3}}-\log \left (1+\frac {\sqrt [3]{1-\frac {1}{x}}}{\sqrt [3]{1+\frac {1}{x}}}\right )-\frac {1}{3} \log \left (1+\frac {1}{x}\right ) \] Output:

(1-1/x)^(2/3)*(1+1/x)^(1/3)+2/3*3^(1/2)*arctan(-1/3*3^(1/2)+2/3*(1-1/x)^(1 
/3)*3^(1/2)/(1+1/x)^(1/3))-ln(1+(1-1/x)^(1/3)/(1+1/x)^(1/3))-1/3*ln(1+1/x)
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.88 \[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x^2} \, dx=\frac {2 e^{\frac {2}{3} \coth ^{-1}(x)}}{1+e^{2 \coth ^{-1}(x)}}-\frac {2 \arctan \left (\frac {-1+2 e^{\frac {2}{3} \coth ^{-1}(x)}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2}{3} \log \left (1+e^{\frac {2}{3} \coth ^{-1}(x)}\right )+\frac {1}{3} \log \left (1-e^{\frac {2}{3} \coth ^{-1}(x)}+e^{\frac {4}{3} \coth ^{-1}(x)}\right ) \] Input:

Integrate[E^((2*ArcCoth[x])/3)/x^2,x]
 

Output:

(2*E^((2*ArcCoth[x])/3))/(1 + E^(2*ArcCoth[x])) - (2*ArcTan[(-1 + 2*E^((2* 
ArcCoth[x])/3))/Sqrt[3]])/Sqrt[3] - (2*Log[1 + E^((2*ArcCoth[x])/3)])/3 + 
Log[1 - E^((2*ArcCoth[x])/3) + E^((4*ArcCoth[x])/3)]/3
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6721, 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.

\(\displaystyle \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x^2} \, dx\)

\(\Big \downarrow \) 6721

\(\displaystyle -\int \frac {\sqrt [3]{1+\frac {1}{x}}}{\sqrt [3]{1-\frac {1}{x}}}d\frac {1}{x}\)

\(\Big \downarrow \) 60

\(\displaystyle \left (1-\frac {1}{x}\right )^{2/3} \sqrt [3]{\frac {1}{x}+1}-\frac {2}{3} \int \frac {1}{\sqrt [3]{1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{2/3}}d\frac {1}{x}\)

\(\Big \downarrow \) 72

\(\displaystyle \left (1-\frac {1}{x}\right )^{2/3} \sqrt [3]{\frac {1}{x}+1}-\frac {2}{3} \left (\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-\frac {1}{x}}}{\sqrt {3} \sqrt [3]{\frac {1}{x}+1}}\right )+\frac {3}{2} \log \left (\frac {\sqrt [3]{1-\frac {1}{x}}}{\sqrt [3]{\frac {1}{x}+1}}+1\right )+\frac {1}{2} \log \left (\frac {1}{x}+1\right )\right )\)

Input:

Int[E^((2*ArcCoth[x])/3)/x^2,x]
 

Output:

(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
 

Defintions of rubi rules used

rule 60

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]
 

rule 72

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]
 

rule 6721

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]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.70 (sec) , antiderivative size = 501, normalized size of antiderivative = 5.06

method	result	size
risch	\(\frac {x -1}{x \left (\frac {x -1}{1+x}\right )^{\frac {1}{3}}}+\frac {\left (-\frac {2 \ln \left (\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2} x^{2}+27 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}-45 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x -8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2} x -30 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) x^{2}-216 \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}-45 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-81 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x -16 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2}-54 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) x -27 x^{2}-81 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-24 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )-36 x -9}{x \left (1+x \right )}\right )}{3}+\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \ln \left (-\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2} x^{2}+27 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}+72 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x +2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2} x -27 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) x^{2}+135 \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}+72 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-81 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x +4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2}+6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) x -36 x^{2}-81 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}+33 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )-216 x -180}{x \left (1+x \right )}\right )}{9}\right ) \left (\left (x -1\right ) \left (1+x \right )^{2}\right )^{\frac {1}{3}}}{\left (\frac {x -1}{1+x}\right )^{\frac {1}{3}} \left (1+x \right )}\)	\(501\)
trager	\(\frac {\left (1+x \right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}}{x}+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 )+\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 )}{3}-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 )\)	\(669\)

Input:

int(1/((x-1)/(1+x))^(1/3)/x^2,x,method=_RETURNVERBOSE)
 

Output:

(x-1)/x/((x-1)/(1+x))^(1/3)+(-2/3*ln((8*RootOf(_Z^2-3*_Z+9)^2*x^2+27*RootO 
f(_Z^2-3*_Z+9)*(x^3+x^2-x-1)^(2/3)-45*RootOf(_Z^2-3*_Z+9)*(x^3+x^2-x-1)^(1 
/3)*x-8*RootOf(_Z^2-3*_Z+9)^2*x-30*RootOf(_Z^2-3*_Z+9)*x^2-216*(x^3+x^2-x- 
1)^(2/3)-45*RootOf(_Z^2-3*_Z+9)*(x^3+x^2-x-1)^(1/3)-81*(x^3+x^2-x-1)^(1/3) 
*x-16*RootOf(_Z^2-3*_Z+9)^2-54*RootOf(_Z^2-3*_Z+9)*x-27*x^2-81*(x^3+x^2-x- 
1)^(1/3)-24*RootOf(_Z^2-3*_Z+9)-36*x-9)/x/(1+x))+2/9*RootOf(_Z^2-3*_Z+9)*l 
n(-(-2*RootOf(_Z^2-3*_Z+9)^2*x^2+27*RootOf(_Z^2-3*_Z+9)*(x^3+x^2-x-1)^(2/3 
)+72*RootOf(_Z^2-3*_Z+9)*(x^3+x^2-x-1)^(1/3)*x+2*RootOf(_Z^2-3*_Z+9)^2*x-2 
7*RootOf(_Z^2-3*_Z+9)*x^2+135*(x^3+x^2-x-1)^(2/3)+72*RootOf(_Z^2-3*_Z+9)*( 
x^3+x^2-x-1)^(1/3)-81*(x^3+x^2-x-1)^(1/3)*x+4*RootOf(_Z^2-3*_Z+9)^2+6*Root 
Of(_Z^2-3*_Z+9)*x-36*x^2-81*(x^3+x^2-x-1)^(1/3)+33*RootOf(_Z^2-3*_Z+9)-216 
*x-180)/x/(1+x)))/((x-1)/(1+x))^(1/3)*((x-1)*(1+x)^2)^(1/3)/(1+x)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.98 \[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x^2} \, dx=\frac {2 \, \sqrt {3} x \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + x \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) - 2 \, x \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + 3 \, {\left (x + 1\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}}{3 \, x} \] Input:

integrate(1/((x-1)/(1+x))^(1/3)/x^2,x, algorithm="fricas")
 

Output:

1/3*(2*sqrt(3)*x*arctan(2/3*sqrt(3)*((x - 1)/(x + 1))^(1/3) - 1/3*sqrt(3)) 
 + x*log(((x - 1)/(x + 1))^(2/3) - ((x - 1)/(x + 1))^(1/3) + 1) - 2*x*log( 
((x - 1)/(x + 1))^(1/3) + 1) + 3*(x + 1)*((x - 1)/(x + 1))^(2/3))/x
 

Sympy [F]

\[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x^2} \, dx=\int \frac {1}{x^{2} \sqrt [3]{\frac {x - 1}{x + 1}}}\, dx \] Input:

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

Output:

Integral(1/(x**2*((x - 1)/(x + 1))**(1/3)), x)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.99 \[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x^2} \, dx=\frac {2}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - 1\right )}\right ) + \frac {2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}}{\frac {x - 1}{x + 1} + 1} + \frac {1}{3} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) - \frac {2}{3} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) \] Input:

integrate(1/((x-1)/(1+x))^(1/3)/x^2,x, algorithm="maxima")
 

Output:

2/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/3) - 1)) + 2*((x - 
1)/(x + 1))^(2/3)/((x - 1)/(x + 1) + 1) + 1/3*log(((x - 1)/(x + 1))^(2/3) 
- ((x - 1)/(x + 1))^(1/3) + 1) - 2/3*log(((x - 1)/(x + 1))^(1/3) + 1)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x^2} \, dx=\frac {2}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - 1\right )}\right ) + \frac {2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}}{\frac {x - 1}{x + 1} + 1} + \frac {1}{3} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) - \frac {2}{3} \, \log \left ({\left | \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1 \right |}\right ) \] Input:

integrate(1/((x-1)/(1+x))^(1/3)/x^2,x, algorithm="giac")
 

Output:

2/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/3) - 1)) + 2*((x - 
1)/(x + 1))^(2/3)/((x - 1)/(x + 1) + 1) + 1/3*log(((x - 1)/(x + 1))^(2/3) 
- ((x - 1)/(x + 1))^(1/3) + 1) - 2/3*log(abs(((x - 1)/(x + 1))^(1/3) + 1))
 

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.19 \[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x^2} \, dx=\frac {2\,{\left (\frac {x-1}{x+1}\right )}^{2/3}}{\frac {x-1}{x+1}+1}-\ln \left (9\,{\left (-\frac {1}{3}+\frac {\sqrt {3}\,1{}\mathrm {i}}{3}\right )}^2+4\,{\left (\frac {x-1}{x+1}\right )}^{1/3}\right )\,\left (-\frac {1}{3}+\frac {\sqrt {3}\,1{}\mathrm {i}}{3}\right )+\ln \left (9\,{\left (\frac {1}{3}+\frac {\sqrt {3}\,1{}\mathrm {i}}{3}\right )}^2+4\,{\left (\frac {x-1}{x+1}\right )}^{1/3}\right )\,\left (\frac {1}{3}+\frac {\sqrt {3}\,1{}\mathrm {i}}{3}\right )-\frac {2\,\ln \left (4\,{\left (\frac {x-1}{x+1}\right )}^{1/3}+4\right )}{3} \] Input:

int(1/(x^2*((x - 1)/(x + 1))^(1/3)),x)
 

Output:

log(9*((3^(1/2)*1i)/3 + 1/3)^2 + 4*((x - 1)/(x + 1))^(1/3))*((3^(1/2)*1i)/ 
3 + 1/3) - log(9*((3^(1/2)*1i)/3 - 1/3)^2 + 4*((x - 1)/(x + 1))^(1/3))*((3 
^(1/2)*1i)/3 - 1/3) - (2*log(4*((x - 1)/(x + 1))^(1/3) + 4))/3 + (2*((x - 
1)/(x + 1))^(2/3))/((x - 1)/(x + 1) + 1)
 

Reduce [F]

\[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x^2} \, dx=\int \frac {\left (x +1\right )^{\frac {1}{3}}}{\left (x -1\right )^{\frac {1}{3}} x^{2}}d x \] Input:

int(1/((x-1)/(1+x))^(1/3)/x^2,x)
 

Output:

int((x + 1)**(1/3)/((x - 1)**(1/3)*x**2),x)