\(\int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x^3} \, dx\) [125]

Optimal result

Integrand size = 12, antiderivative size = 130 \[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x^3} \, dx=\frac {1}{3} \sqrt [3]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{2/3}+\frac {1}{2} \left (1+\frac {1}{x}\right )^{4/3} \left (\frac {-1+x}{x}\right )^{2/3}-\frac {2 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{\frac {-1+x}{x}}}{\sqrt {3} \sqrt [3]{1+\frac {1}{x}}}\right )}{3 \sqrt {3}}-\frac {1}{3} \log \left (1+\frac {\sqrt [3]{\frac {-1+x}{x}}}{\sqrt [3]{1+\frac {1}{x}}}\right )-\frac {1}{9} \log \left (1+\frac {1}{x}\right ) \]

[Out]

1/3*(1+1/x)^(1/3)*((-1+x)/x)^(2/3)+1/2*(1+1/x)^(4/3)*((-1+x)/x)^(2/3)-1/3*ln(1+((-1+x)/x)^(1/3)/(1+1/x)^(1/3))
-1/9*ln(1+1/x)+2/9*arctan(-1/3*3^(1/2)+2/3*((-1+x)/x)^(1/3)/(1+1/x)^(1/3)*3^(1/2))*3^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6306, 81, 52, 62} \[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x^3} \, dx=-\frac {2 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{\frac {x-1}{x}}}{\sqrt {3} \sqrt [3]{\frac {1}{x}+1}}\right )}{3 \sqrt {3}}+\frac {1}{2} \left (\frac {x-1}{x}\right )^{2/3} \left (\frac {1}{x}+1\right )^{4/3}+\frac {1}{3} \left (\frac {x-1}{x}\right )^{2/3} \sqrt [3]{\frac {1}{x}+1}-\frac {1}{3} \log \left (\frac {\sqrt [3]{\frac {x-1}{x}}}{\sqrt [3]{\frac {1}{x}+1}}+1\right )-\frac {1}{9} \log \left (\frac {1}{x}+1\right ) \]

[In]

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

[Out]

((1 + x^(-1))^(1/3)*((-1 + x)/x)^(2/3))/3 + ((1 + x^(-1))^(4/3)*((-1 + x)/x)^(2/3))/2 - (2*ArcTan[1/Sqrt[3] -
(2*((-1 + x)/x)^(1/3))/(Sqrt[3]*(1 + x^(-1))^(1/3))])/(3*Sqrt[3]) - Log[1 + ((-1 + x)/x)^(1/3)/(1 + x^(-1))^(1
/3)]/3 - Log[1 + x^(-1)]/9

Rule 52

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] + Dist[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] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 62

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])] /; FreeQ[{a, b, c, d}, x] && NeQ[b
*c - a*d, 0] && NegQ[d/b]

Rule 81

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Dist[(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]

Rule 6306

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]

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {x \sqrt [3]{1+x}}{\sqrt [3]{1-x}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{2} \left (1+\frac {1}{x}\right )^{4/3} \left (\frac {-1+x}{x}\right )^{2/3}-\frac {1}{3} \text {Subst}\left (\int \frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{3} \sqrt [3]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{2/3}+\frac {1}{2} \left (1+\frac {1}{x}\right )^{4/3} \left (\frac {-1+x}{x}\right )^{2/3}-\frac {2}{9} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} (1+x)^{2/3}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{3} \sqrt [3]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{2/3}+\frac {1}{2} \left (1+\frac {1}{x}\right )^{4/3} \left (\frac {-1+x}{x}\right )^{2/3}-\frac {2 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-\frac {1-x}{x}}}{\sqrt {3} \sqrt [3]{1+\frac {1}{x}}}\right )}{3 \sqrt {3}}-\frac {1}{3} \log \left (1+\frac {\sqrt [3]{-\frac {1-x}{x}}}{\sqrt [3]{1+\frac {1}{x}}}\right )-\frac {1}{9} \log \left (1+\frac {1}{x}\right ) \\ \end{align*}

Mathematica [C] (verified)

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

Time = 0.47 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.03 \[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x^3} \, dx=-\frac {2}{27} \left (\frac {27 e^{\frac {2}{3} \coth ^{-1}(x)}}{\left (1+e^{2 \coth ^{-1}(x)}\right )^2}-\frac {36 e^{\frac {2}{3} \coth ^{-1}(x)}}{1+e^{2 \coth ^{-1}(x)}}-2 \coth ^{-1}(x)+3 \log \left (1+e^{\frac {2}{3} \coth ^{-1}(x)}\right )-\text {RootSum}\left [1-\text {$\#$1}^2+\text {$\#$1}^4\&,\frac {\coth ^{-1}(x)-3 \log \left (e^{\frac {1}{3} \coth ^{-1}(x)}-\text {$\#$1}\right )+\coth ^{-1}(x) \text {$\#$1}^2-3 \log \left (e^{\frac {1}{3} \coth ^{-1}(x)}-\text {$\#$1}\right ) \text {$\#$1}^2}{-2+\text {$\#$1}^2}\&\right ]\right ) \]

[In]

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

[Out]

(-2*((27*E^((2*ArcCoth[x])/3))/(1 + E^(2*ArcCoth[x]))^2 - (36*E^((2*ArcCoth[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

Maple [C] (verified)

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

Time = 0.78 (sec) , antiderivative size = 506, normalized size of antiderivative = 3.89

method	result	size
risch	\(\frac {\left (x -1\right ) \left (5 x +3\right )}{6 x^{2} \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}-8 \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 ) \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 -30 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) x^{2}-16 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2}-45 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-54 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) x -216 \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}-81 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x -27 x^{2}-24 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )-81 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-36 x -9}{x \left (1+x \right )}\right )}{9}+\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}-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 ) \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 +27 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) x^{2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2}-72 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) x -135 \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}+81 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x +36 x^{2}-33 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )+81 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}+216 x +180}{x \left (1+x \right )}\right )}{27}\right ) \left (\left (1+x \right )^{2} \left (x -1\right )\right )^{\frac {1}{3}}}{\left (\frac {x -1}{1+x}\right )^{\frac {1}{3}} \left (1+x \right )}\)	\(506\)
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}\)	\(675\)

[In]

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

[Out]

1/6*(x-1)*(5*x+3)/x^2/((x-1)/(1+x))^(1/3)+(-2/9*ln((8*RootOf(_Z^2-3*_Z+9)^2*x^2-8*RootOf(_Z^2-3*_Z+9)^2*x+27*R
ootOf(_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-30*RootOf(_Z^2-3*_Z+9)*x^2
-16*RootOf(_Z^2-3*_Z+9)^2-45*RootOf(_Z^2-3*_Z+9)*(x^3+x^2-x-1)^(1/3)-54*RootOf(_Z^2-3*_Z+9)*x-216*(x^3+x^2-x-1
)^(2/3)-81*(x^3+x^2-x-1)^(1/3)*x-27*x^2-24*RootOf(_Z^2-3*_Z+9)-81*(x^3+x^2-x-1)^(1/3)-36*x-9)/x/(1+x))+2/27*Ro
otOf(_Z^2-3*_Z+9)*ln((2*RootOf(_Z^2-3*_Z+9)^2*x^2-2*RootOf(_Z^2-3*_Z+9)^2*x-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+27*RootOf(_Z^2-3*_Z+9)*x^2-4*RootOf(_Z^2-3*_Z+9)^2-72*Ro
otOf(_Z^2-3*_Z+9)*(x^3+x^2-x-1)^(1/3)-6*RootOf(_Z^2-3*_Z+9)*x-135*(x^3+x^2-x-1)^(2/3)+81*(x^3+x^2-x-1)^(1/3)*x
+36*x^2-33*RootOf(_Z^2-3*_Z+9)+81*(x^3+x^2-x-1)^(1/3)+216*x+180)/x/(1+x)))/((x-1)/(1+x))^(1/3)*((1+x)^2*(x-1))
^(1/3)/(1+x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.85 \[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x^3} \, dx=\frac {4 \, \sqrt {3} x^{2} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + 2 \, x^{2} \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) - 4 \, x^{2} \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + 3 \, {\left (5 \, x^{2} + 8 \, x + 3\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}}{18 \, x^{2}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.95 \[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x^3} \, dx=\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - 1\right )}\right ) + \frac {2 \, {\left (\left (\frac {x - 1}{x + 1}\right )^{\frac {5}{3}} + 4 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}\right )}}{3 \, {\left (\frac {2 \, {\left (x - 1\right )}}{x + 1} + \frac {{\left (x - 1\right )}^{2}}{{\left (x + 1\right )}^{2}} + 1\right )}} + \frac {1}{9} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) - \frac {2}{9} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) \]

[In]

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

[Out]

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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.94 \[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x^3} \, dx=\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - 1\right )}\right ) + \frac {2 \, {\left (\frac {{\left (x - 1\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}}{x + 1} + 4 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}\right )}}{3 \, {\left (\frac {x - 1}{x + 1} + 1\right )}^{2}} + \frac {1}{9} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) - \frac {2}{9} \, \log \left ({\left | \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1 \right |}\right ) \]

[In]

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

[Out]

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))

Mupad [B] (verification not implemented)

Time = 4.27 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.12 \[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x^3} \, dx=\frac {\frac {8\,{\left (\frac {x-1}{x+1}\right )}^{2/3}}{3}+\frac {2\,{\left (\frac {x-1}{x+1}\right )}^{5/3}}{3}}{\frac {2\,\left (x-1\right )}{x+1}+\frac {{\left (x-1\right )}^2}{{\left (x+1\right )}^2}+1}-\frac {2\,\ln \left (\frac {4\,{\left (\frac {x-1}{x+1}\right )}^{1/3}}{9}+\frac {4}{9}\right )}{9}-\ln \left (9\,{\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )}^2+\frac {4\,{\left (\frac {x-1}{x+1}\right )}^{1/3}}{9}\right )\,\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )+\ln \left (9\,{\left (\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )}^2+\frac {4\,{\left (\frac {x-1}{x+1}\right )}^{1/3}}{9}\right )\,\left (\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right ) \]

[In]

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

[Out]

((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)