3.2.21 \(\int e^{\frac {2}{3} \coth ^{-1}(x)} x \, dx\) [121]

Optimal. Leaf size=130 \[ \frac {1}{3} \sqrt [3]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{2/3} x+\frac {1}{2} \left (1+\frac {1}{x}\right )^{4/3} \left (\frac {-1+x}{x}\right )^{2/3} x^2-\frac {2 \text {ArcTan}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{\frac {-1+x}{x}}}{\sqrt {3} \sqrt [3]{1+\frac {1}{x}}}\right )}{3 \sqrt {3}}-\frac {1}{3} \log \left (\sqrt [3]{1+\frac {1}{x}}-\sqrt [3]{\frac {-1+x}{x}}\right )-\frac {\log (x)}{9} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6306, 98, 96, 93} \begin {gather*} -\frac {2 \text {ArcTan}\left (\frac {2 \sqrt [3]{\frac {x-1}{x}}}{\sqrt {3} \sqrt [3]{\frac {1}{x}+1}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{2} \left (\frac {1}{x}+1\right )^{4/3} \left (\frac {x-1}{x}\right )^{2/3} x^2+\frac {1}{3} \sqrt [3]{\frac {1}{x}+1} \left (\frac {x-1}{x}\right )^{2/3} x-\frac {1}{3} \log \left (\sqrt [3]{\frac {1}{x}+1}-\sqrt [3]{\frac {x-1}{x}}\right )-\frac {\log (x)}{9} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 93

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)*((e_.) + (f_.)*(x_))), x_Symbol] :> With[{q = Rt[
(d*e - c*f)/(b*e - a*f), 3]}, Simp[(-Sqrt[3])*q*(ArcTan[1/Sqrt[3] + 2*q*((a + b*x)^(1/3)/(Sqrt[3]*(c + d*x)^(1
/3)))]/(d*e - c*f)), x] + (Simp[q*(Log[e + f*x]/(2*(d*e - c*f))), x] - Simp[3*q*(Log[q*(a + b*x)^(1/3) - (c +
d*x)^(1/3)]/(2*(d*e - c*f))), x])] /; FreeQ[{a, b, c, d, e, f}, x]

Rule 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*
x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Dist[n*((d*e - c*f)/((m + 1)*(b*e - a*f
))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]

Rule 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

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

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Mathematica [A]
time = 0.23, size = 165, normalized size = 1.27 \begin {gather*} \frac {1}{9} \left (\frac {18 e^{\frac {2}{3} \coth ^{-1}(x)}}{\left (-1+e^{2 \coth ^{-1}(x)}\right )^2}+\frac {24 e^{\frac {2}{3} \coth ^{-1}(x)}}{-1+e^{2 \coth ^{-1}(x)}}+2 \sqrt {3} \text {ArcTan}\left (\frac {-1+2 e^{\frac {1}{3} \coth ^{-1}(x)}}{\sqrt {3}}\right )-2 \sqrt {3} \text {ArcTan}\left (\frac {1+2 e^{\frac {1}{3} \coth ^{-1}(x)}}{\sqrt {3}}\right )-2 \log \left (1-e^{\frac {1}{3} \coth ^{-1}(x)}\right )-2 \log \left (1+e^{\frac {1}{3} \coth ^{-1}(x)}\right )+\log \left (1-e^{\frac {1}{3} \coth ^{-1}(x)}+e^{\frac {2}{3} \coth ^{-1}(x)}\right )+\log \left (1+e^{\frac {1}{3} \coth ^{-1}(x)}+e^{\frac {2}{3} \coth ^{-1}(x)}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((18*E^((2*ArcCoth[x])/3))/(-1 + E^(2*ArcCoth[x]))^2 + (24*E^((2*ArcCoth[x])/3))/(-1 + E^(2*ArcCoth[x])) + 2*S
qrt[3]*ArcTan[(-1 + 2*E^(ArcCoth[x]/3))/Sqrt[3]] - 2*Sqrt[3]*ArcTan[(1 + 2*E^(ArcCoth[x]/3))/Sqrt[3]] - 2*Log[
1 - E^(ArcCoth[x]/3)] - 2*Log[1 + E^(ArcCoth[x]/3)] + Log[1 - E^(ArcCoth[x]/3) + E^((2*ArcCoth[x])/3)] + Log[1
+ E^(ArcCoth[x]/3) + E^((2*ArcCoth[x])/3)])/9

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.95, size = 403, normalized size = 3.10

method	result	size
trager	\(\frac {\left (1+x \right ) \left (5+3 x \right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}}{6}-\frac {2 \ln \left (-9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}} x -9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}+9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}} x -36 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x +9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}}-3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}} x +12 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}}-6 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-x +1\right )}{9}+\frac {2 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}} x +9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}-3 \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}} x +18 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x -3 \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}+3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}} x -15 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x +3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}}-3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+2 x +2\right )}{3}\)	\(403\)
risch	\(\frac {\left (5+3 x \right ) \left (-1+x \right )}{6 \left (\frac {-1+x}{1+x}\right )^{\frac {1}{3}}}+\frac {\left (\frac {2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x -3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x -5 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-4 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +2 x^{2}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-2}{1+x}\right )}{9}-\frac {2 \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x +\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-3 \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}-3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x -x^{2}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-2 x -1}{1+x}\right ) \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{9}+\frac {2 \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x +\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-3 \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}-3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x -x^{2}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-2 x -1}{1+x}\right )}{9}\right ) \left (\left (1+x \right )^{2} \left (-1+x \right )\right )^{\frac {1}{3}}}{\left (\frac {-1+x}{1+x}\right )^{\frac {1}{3}} \left (1+x \right )}\)	\(608\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(1+x)*(5+3*x)*(-(1-x)/(1+x))^(2/3)-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-36*RootOf(9*_Z^2-3*_Z+1)^2*x+9*Roo
tOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(1/3)-3*(-(1-x)/(1+x))^(1/3)*x+12*RootOf(9*_Z^2-3*_Z+1)*x-3*(-(1-x)/(1+x))^(
1/3)-6*RootOf(9*_Z^2-3*_Z+1)-x+1)+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)-3*(-(1-x)/(1+x))^(2/3)*x+18*RootOf(9*_Z^2-3*_Z+1)^2*x-3*(-(1-x)/(
1+x))^(2/3)+3*(-(1-x)/(1+x))^(1/3)*x-15*RootOf(9*_Z^2-3*_Z+1)*x+3*(-(1-x)/(1+x))^(1/3)-3*RootOf(9*_Z^2-3*_Z+1)
+2*x+2)

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Maxima [A]
time = 0.47, size = 123, normalized size = 0.95 \begin {gather*} -\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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [A]
time = 0.34, size = 95, normalized size = 0.73 \begin {gather*} \frac {1}{6} \, {\left (3 \, x^{2} + 8 \, x + 5\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} - \frac {2}{9} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(3*x^2 + 8*x + 5)*((x - 1)/(x + 1))^(2/3) - 2/9*sqrt(3)*arctan(2/3*sqrt(3)*((x - 1)/(x + 1))^(1/3) + 1/3*s
qrt(3)) + 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
)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt [3]{\frac {x - 1}{x + 1}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.42, size = 120, normalized size = 0.92 \begin {gather*} -\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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Mupad [B]
time = 0.05, size = 145, normalized size = 1.12 \begin {gather*} \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 {{\left (x-1\right )}^2}{{\left (x+1\right )}^2}-\frac {2\,\left (x-1\right )}{x+1}+1}-\frac {2\,\ln \left (\frac {4\,{\left (\frac {x-1}{x+1}\right )}^{1/3}}{9}-\frac {4}{9}\right )}{9}-\ln \left (\frac {4\,{\left (\frac {x-1}{x+1}\right )}^{1/3}}{9}-9\,{\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )}^2\right )\,\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )+\ln \left (\frac {4\,{\left (\frac {x-1}{x+1}\right )}^{1/3}}{9}-9\,{\left (\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )}^2\right )\,\left (\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/((x - 1)/(x + 1))^(1/3),x)

[Out]

((8*((x - 1)/(x + 1))^(2/3))/3 - (2*((x - 1)/(x + 1))^(5/3))/3)/((x - 1)^2/(x + 1)^2 - (2*(x - 1))/(x + 1) + 1
) - (2*log((4*((x - 1)/(x + 1))^(1/3))/9 - 4/9))/9 - log((4*((x - 1)/(x + 1))^(1/3))/9 - 9*((3^(1/2)*1i)/9 - 1
/9)^2)*((3^(1/2)*1i)/9 - 1/9) + log((4*((x - 1)/(x + 1))^(1/3))/9 - 9*((3^(1/2)*1i)/9 + 1/9)^2)*((3^(1/2)*1i)/
9 + 1/9)

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