Optimal. Leaf size=194 \[ -\frac {(1-x)^{5/6} \sqrt [6]{1+x}}{x}+\frac {\text {ArcTan}\left (\frac {1-\frac {2 \sqrt [6]{1+x}}{\sqrt [6]{1-x}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\text {ArcTan}\left (\frac {1+\frac {2 \sqrt [6]{1+x}}{\sqrt [6]{1-x}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2}{3} \tanh ^{-1}\left (\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )+\frac {1}{6} \log \left (1-\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}+\frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right )-\frac {1}{6} \log \left (1+\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}+\frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right ) \]
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Rubi [A]
time = 0.12, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6261, 96, 95,
216, 648, 632, 210, 642, 212} \begin {gather*} \frac {\text {ArcTan}\left (\frac {1-\frac {2 \sqrt [6]{x+1}}{\sqrt [6]{1-x}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\text {ArcTan}\left (\frac {\frac {2 \sqrt [6]{x+1}}{\sqrt [6]{1-x}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {(1-x)^{5/6} \sqrt [6]{x+1}}{x}+\frac {1}{6} \log \left (\frac {\sqrt [3]{x+1}}{\sqrt [3]{1-x}}-\frac {\sqrt [6]{x+1}}{\sqrt [6]{1-x}}+1\right )-\frac {1}{6} \log \left (\frac {\sqrt [3]{x+1}}{\sqrt [3]{1-x}}+\frac {\sqrt [6]{x+1}}{\sqrt [6]{1-x}}+1\right )-\frac {2}{3} \tanh ^{-1}\left (\frac {\sqrt [6]{x+1}}{\sqrt [6]{1-x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 95
Rule 96
Rule 210
Rule 212
Rule 216
Rule 632
Rule 642
Rule 648
Rule 6261
Rubi steps
\begin {align*} \int \frac {e^{\frac {1}{3} \tanh ^{-1}(x)}}{x^2} \, dx &=\int \frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x} x^2} \, dx\\ &=-\frac {(1-x)^{5/6} \sqrt [6]{1+x}}{x}+\frac {1}{3} \int \frac {1}{\sqrt [6]{1-x} x (1+x)^{5/6}} \, dx\\ &=-\frac {(1-x)^{5/6} \sqrt [6]{1+x}}{x}+2 \text {Subst}\left (\int \frac {1}{-1+x^6} \, dx,x,\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )\\ &=-\frac {(1-x)^{5/6} \sqrt [6]{1+x}}{x}-\frac {2}{3} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )-\frac {2}{3} \text {Subst}\left (\int \frac {1-\frac {x}{2}}{1-x+x^2} \, dx,x,\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )-\frac {2}{3} \text {Subst}\left (\int \frac {1+\frac {x}{2}}{1+x+x^2} \, dx,x,\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )\\ &=-\frac {(1-x)^{5/6} \sqrt [6]{1+x}}{x}-\frac {2}{3} \tanh ^{-1}\left (\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )+\frac {1}{6} \text {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )-\frac {1}{6} \text {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )\\ &=-\frac {(1-x)^{5/6} \sqrt [6]{1+x}}{x}-\frac {2}{3} \tanh ^{-1}\left (\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )+\frac {1}{6} \log \left (1-\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}+\frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right )-\frac {1}{6} \log \left (1+\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}+\frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right )+\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+\frac {2 \sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )+\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )\\ &=-\frac {(1-x)^{5/6} \sqrt [6]{1+x}}{x}-\frac {\tan ^{-1}\left (\frac {-1+\frac {2 \sqrt [6]{1+x}}{\sqrt [6]{1-x}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 \sqrt [6]{1+x}}{\sqrt [6]{1-x}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2}{3} \tanh ^{-1}\left (\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )+\frac {1}{6} \log \left (1-\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}+\frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right )-\frac {1}{6} \log \left (1+\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}+\frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right )\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.01, size = 50, normalized size = 0.26 \begin {gather*} -\frac {(1-x)^{5/6} \left (5+5 x+2 x \, _2F_1\left (\frac {5}{6},1;\frac {11}{6};\frac {1-x}{1+x}\right )\right )}{5 x (1+x)^{5/6}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (\frac {1+x}{\sqrt {-x^{2}+1}}\right )^{\frac {1}{3}}}{x^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 234, normalized size = 1.21 \begin {gather*} -\frac {2 \, \sqrt {3} x \arctan \left (\frac {2}{3} \, \sqrt {3} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + 2 \, \sqrt {3} x \arctan \left (\frac {2}{3} \, \sqrt {3} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + x \log \left (\left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} + \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} + 1\right ) - x \log \left (\left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} - \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} + 1\right ) + 2 \, x \log \left (\left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} + 1\right ) - 2 \, x \log \left (\left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} - 1\right ) - 6 \, {\left (x - 1\right )} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}}}{6 \, x} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt [3]{\frac {x + 1}{\sqrt {1 - x^{2}}}}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (\frac {x+1}{\sqrt {1-x^2}}\right )}^{1/3}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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