Optimal. Leaf size=96 \[ \sqrt [3]{\frac {1}{x}+1} \left (\frac {x-1}{x}\right )^{2/3} x-\log \left (\sqrt [3]{\frac {1}{x}+1}-\sqrt [3]{\frac {x-1}{x}}\right )-\frac {\log (x)}{3}-\frac {2 \tan ^{-1}\left (\frac {2 \sqrt [3]{\frac {x-1}{x}}}{\sqrt {3} \sqrt [3]{\frac {1}{x}+1}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.03, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6170, 94, 91} \[ \sqrt [3]{\frac {1}{x}+1} \left (\frac {x-1}{x}\right )^{2/3} x-\log \left (\sqrt [3]{\frac {1}{x}+1}-\sqrt [3]{\frac {x-1}{x}}\right )-\frac {\log (x)}{3}-\frac {2 \tan ^{-1}\left (\frac {2 \sqrt [3]{\frac {x-1}{x}}}{\sqrt {3} \sqrt [3]{\frac {1}{x}+1}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 91
Rule 94
Rule 6170
Rubi steps
\begin {align*} \int e^{\frac {2}{3} \coth ^{-1}(x)} \, dx &=-\operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x} x^2} \, dx,x,\frac {1}{x}\right )\\ &=\sqrt [3]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{2/3} x-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} x (1+x)^{2/3}} \, dx,x,\frac {1}{x}\right )\\ &=\sqrt [3]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{2/3} x-\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 )}{\sqrt {3}}-\log \left (\sqrt [3]{1+\frac {1}{x}}-\sqrt [3]{-\frac {1-x}{x}}\right )-\frac {\log (x)}{3}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 85, normalized size = 0.89 \[ \frac {1}{3} \left (\frac {6 e^{\frac {2}{3} \coth ^{-1}(x)}}{e^{2 \coth ^{-1}(x)}-1}-2 \log \left (1-e^{\frac {2}{3} \coth ^{-1}(x)}\right )+\log \left (e^{\frac {2}{3} \coth ^{-1}(x)}+e^{\frac {4}{3} \coth ^{-1}(x)}+1\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {2 e^{\frac {2}{3} \coth ^{-1}(x)}+1}{\sqrt {3}}\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.54, size = 87, normalized size = 0.91 \[ {\left (x + 1\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} - \frac {2}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{3} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) - \frac {2}{3} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 97, normalized size = 1.01 \[ -\frac {2}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right )}\right ) - \frac {2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}}{\frac {x - 1}{x + 1} - 1} + \frac {1}{3} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) - \frac {2}{3} \, \log \left ({\left | \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.54, size = 605, normalized size = 6.30 \[ \frac {-1+x}{\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}+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 -2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} 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 )}{3}-\frac {2 \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+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 +2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-3 \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}}-3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x -x^{2}-3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-2 x -1}{1+x}\right ) \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{3}+\frac {2 \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+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 +2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-3 \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}}-3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x -x^{2}-3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-2 x -1}{1+x}\right )}{3}\right ) \left (\left (-1+x \right ) \left (1+x \right )^{2}\right )^{\frac {1}{3}}}{\left (\frac {-1+x}{1+x}\right )^{\frac {1}{3}} \left (1+x \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 96, normalized size = 1.00 \[ -\frac {2}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right )}\right ) - \frac {2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}}{\frac {x - 1}{x + 1} - 1} + \frac {1}{3} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) - \frac {2}{3} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 118, normalized size = 1.23 \[ -\frac {2\,\ln \left (4\,{\left (\frac {x-1}{x+1}\right )}^{1/3}-4\right )}{3}-\ln \left (4\,{\left (\frac {x-1}{x+1}\right )}^{1/3}-9\,{\left (-\frac {1}{3}+\frac {\sqrt {3}\,1{}\mathrm {i}}{3}\right )}^2\right )\,\left (-\frac {1}{3}+\frac {\sqrt {3}\,1{}\mathrm {i}}{3}\right )+\ln \left (4\,{\left (\frac {x-1}{x+1}\right )}^{1/3}-9\,{\left (\frac {1}{3}+\frac {\sqrt {3}\,1{}\mathrm {i}}{3}\right )}^2\right )\,\left (\frac {1}{3}+\frac {\sqrt {3}\,1{}\mathrm {i}}{3}\right )-\frac {2\,{\left (\frac {x-1}{x+1}\right )}^{2/3}}{\frac {x-1}{x+1}-1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt [3]{\frac {x - 1}{x + 1}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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