Optimal. Leaf size=223 \[ \sqrt [6]{\frac {1}{x}+1} \left (\frac {x-1}{x}\right )^{5/6} x-\frac {1}{6} \log \left (\frac {\sqrt [3]{\frac {1}{x}+1}}{\sqrt [3]{\frac {x-1}{x}}}-\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}+1\right )+\frac {1}{6} \log \left (\frac {\sqrt [3]{\frac {1}{x}+1}}{\sqrt [3]{\frac {x-1}{x}}}+\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}+1\right )-\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\tan ^{-1}\left (\frac {\frac {2 \sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {2}{3} \tanh ^{-1}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}\right ) \]
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Rubi [A] time = 0.18, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.125, Rules used = {6170, 94, 93, 210, 634, 618, 204, 628, 206} \[ \sqrt [6]{\frac {1}{x}+1} \left (\frac {x-1}{x}\right )^{5/6} x-\frac {1}{6} \log \left (\frac {\sqrt [3]{\frac {1}{x}+1}}{\sqrt [3]{\frac {x-1}{x}}}-\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}+1\right )+\frac {1}{6} \log \left (\frac {\sqrt [3]{\frac {1}{x}+1}}{\sqrt [3]{\frac {x-1}{x}}}+\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}+1\right )-\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\tan ^{-1}\left (\frac {\frac {2 \sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {2}{3} \tanh ^{-1}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}\right ) \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 204
Rule 206
Rule 210
Rule 618
Rule 628
Rule 634
Rule 6170
Rubi steps
\begin {align*} \int e^{\frac {1}{3} \coth ^{-1}(x)} \, dx &=-\operatorname {Subst}\left (\int \frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x} x^2} \, dx,x,\frac {1}{x}\right )\\ &=\sqrt [6]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{5/6} x-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt [6]{1-x} x (1+x)^{5/6}} \, dx,x,\frac {1}{x}\right )\\ &=\sqrt [6]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{5/6} x-2 \operatorname {Subst}\left (\int \frac {1}{-1+x^6} \, dx,x,\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )\\ &=\sqrt [6]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{5/6} x+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1-\frac {x}{2}}{1-x+x^2} \, dx,x,\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1+\frac {x}{2}}{1+x+x^2} \, dx,x,\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )\\ &=\sqrt [6]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{5/6} x+\frac {2}{3} \tanh ^{-1}\left (\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{-\frac {1-x}{x}}}\right )-\frac {1}{6} \operatorname {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )\\ &=\sqrt [6]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{5/6} x+\frac {2}{3} \tanh ^{-1}\left (\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{-\frac {1-x}{x}}}\right )-\frac {1}{6} \log \left (1+\frac {\sqrt [3]{1+\frac {1}{x}}}{\sqrt [3]{-\frac {1-x}{x}}}-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{-\frac {1-x}{x}}}\right )+\frac {1}{6} \log \left (1+\frac {\sqrt [3]{1+\frac {1}{x}}}{\sqrt [3]{-\frac {1-x}{x}}}+\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{-\frac {1-x}{x}}}\right )-\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )-\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )\\ &=\sqrt [6]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{5/6} x+\frac {\tan ^{-1}\left (\frac {-1+\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\tan ^{-1}\left (\frac {1+\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {2}{3} \tanh ^{-1}\left (\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{-\frac {1-x}{x}}}\right )-\frac {1}{6} \log \left (1+\frac {\sqrt [3]{1+\frac {1}{x}}}{\sqrt [3]{-\frac {1-x}{x}}}-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{-\frac {1-x}{x}}}\right )+\frac {1}{6} \log \left (1+\frac {\sqrt [3]{1+\frac {1}{x}}}{\sqrt [3]{-\frac {1-x}{x}}}+\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{-\frac {1-x}{x}}}\right )\\ \end {align*}
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Mathematica [C] time = 0.05, size = 35, normalized size = 0.16 \[ 2 e^{\frac {1}{3} \coth ^{-1}(x)} \left (\, _2F_1\left (\frac {1}{6},1;\frac {7}{6};e^{2 \coth ^{-1}(x)}\right )+\frac {1}{e^{2 \coth ^{-1}(x)}-1}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.61, size = 160, normalized size = 0.72 \[ {\left (x + 1\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}} - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \frac {1}{3} \, \sqrt {3}\right ) - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{6} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {1}{6} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) + \frac {1}{3} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {1}{3} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 168, normalized size = 0.75 \[ -\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right )}\right ) - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right )}\right ) - \frac {2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{\frac {x - 1}{x + 1} - 1} + \frac {1}{6} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {1}{6} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) + \frac {1}{3} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {1}{3} \, \log \left ({\left | \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 4.10, size = 1700, normalized size = 7.62 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 167, normalized size = 0.75 \[ -\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right )}\right ) - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right )}\right ) - \frac {2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{\frac {x - 1}{x + 1} - 1} + \frac {1}{6} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {1}{6} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) + \frac {1}{3} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {1}{3} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 115, normalized size = 0.52 \[ -\frac {\mathrm {atan}\left ({\left (\frac {x-1}{x+1}\right )}^{1/6}\,1{}\mathrm {i}\right )\,2{}\mathrm {i}}{3}-\frac {2\,{\left (\frac {x-1}{x+1}\right )}^{5/6}}{\frac {x-1}{x+1}-1}-\mathrm {atan}\left (\frac {{\left (\frac {x-1}{x+1}\right )}^{1/6}\,64{}\mathrm {i}}{-32+\sqrt {3}\,32{}\mathrm {i}}\right )\,\left (\frac {\sqrt {3}}{3}-\frac {1}{3}{}\mathrm {i}\right )-\mathrm {atan}\left (\frac {{\left (\frac {x-1}{x+1}\right )}^{1/6}\,64{}\mathrm {i}}{32+\sqrt {3}\,32{}\mathrm {i}}\right )\,\left (\frac {\sqrt {3}}{3}+\frac {1}{3}{}\mathrm {i}\right ) \]
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 \[ \int \frac {1}{\sqrt [6]{\frac {x - 1}{x + 1}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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