Optimal. Leaf size=224 \[ -\frac {(1-x)^{5/6} \sqrt [6]{1+x}}{6 x}-\frac {(1-x)^{5/6} (1+x)^{7/6}}{2 x^2}+\frac {\text {ArcTan}\left (\frac {1-\frac {2 \sqrt [6]{1+x}}{\sqrt [6]{1-x}}}{\sqrt {3}}\right )}{6 \sqrt {3}}-\frac {\text {ArcTan}\left (\frac {1+\frac {2 \sqrt [6]{1+x}}{\sqrt [6]{1-x}}}{\sqrt {3}}\right )}{6 \sqrt {3}}-\frac {1}{9} \tanh ^{-1}\left (\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )+\frac {1}{36} \log \left (1-\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}+\frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right )-\frac {1}{36} \log \left (1+\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}+\frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right ) \]
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Rubi [A]
time = 0.12, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 10, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6261, 98, 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 )}{6 \sqrt {3}}-\frac {\text {ArcTan}\left (\frac {\frac {2 \sqrt [6]{x+1}}{\sqrt [6]{1-x}}+1}{\sqrt {3}}\right )}{6 \sqrt {3}}-\frac {(1-x)^{5/6} (x+1)^{7/6}}{2 x^2}-\frac {(1-x)^{5/6} \sqrt [6]{x+1}}{6 x}+\frac {1}{36} \log \left (\frac {\sqrt [3]{x+1}}{\sqrt [3]{1-x}}-\frac {\sqrt [6]{x+1}}{\sqrt [6]{1-x}}+1\right )-\frac {1}{36} \log \left (\frac {\sqrt [3]{x+1}}{\sqrt [3]{1-x}}+\frac {\sqrt [6]{x+1}}{\sqrt [6]{1-x}}+1\right )-\frac {1}{9} \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 98
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^3} \, dx &=\int \frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x} x^3} \, dx\\ &=-\frac {(1-x)^{5/6} (1+x)^{7/6}}{2 x^2}+\frac {1}{6} \int \frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x} x^2} \, dx\\ &=-\frac {(1-x)^{5/6} \sqrt [6]{1+x}}{6 x}-\frac {(1-x)^{5/6} (1+x)^{7/6}}{2 x^2}+\frac {1}{18} \int \frac {1}{\sqrt [6]{1-x} x (1+x)^{5/6}} \, dx\\ &=-\frac {(1-x)^{5/6} \sqrt [6]{1+x}}{6 x}-\frac {(1-x)^{5/6} (1+x)^{7/6}}{2 x^2}+\frac {1}{3} \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}}{6 x}-\frac {(1-x)^{5/6} (1+x)^{7/6}}{2 x^2}-\frac {1}{9} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )-\frac {1}{9} \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}{9} \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}}{6 x}-\frac {(1-x)^{5/6} (1+x)^{7/6}}{2 x^2}-\frac {1}{9} \tanh ^{-1}\left (\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )+\frac {1}{36} \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}{36} \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}{12} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )-\frac {1}{12} \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}}{6 x}-\frac {(1-x)^{5/6} (1+x)^{7/6}}{2 x^2}-\frac {1}{9} \tanh ^{-1}\left (\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )+\frac {1}{36} \log \left (1-\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}+\frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right )-\frac {1}{36} \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} \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}{6} \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}}{6 x}-\frac {(1-x)^{5/6} (1+x)^{7/6}}{2 x^2}+\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [6]{1+x}}{\sqrt [6]{1-x}}}{\sqrt {3}}\right )}{6 \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 \sqrt [6]{1+x}}{\sqrt [6]{1-x}}}{\sqrt {3}}\right )}{6 \sqrt {3}}-\frac {1}{9} \tanh ^{-1}\left (\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )+\frac {1}{36} \log \left (1-\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}+\frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right )-\frac {1}{36} \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 = 60, normalized size = 0.27 \begin {gather*} -\frac {(1-x)^{5/6} \left (5 \left (3+7 x+4 x^2\right )+2 x^2 \, _2F_1\left (\frac {5}{6},1;\frac {11}{6};\frac {1-x}{1+x}\right )\right )}{30 x^2 (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 \[\int \frac {\left (\frac {1+x}{\sqrt {-x^{2}+1}}\right )^{\frac {1}{3}}}{x^{3}}\, dx\]
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 = 0.38, size = 253, normalized size = 1.13 \begin {gather*} -\frac {2 \, \sqrt {3} x^{2} \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^{2} \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^{2} \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^{2} \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^{2} \log \left (\left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} + 1\right ) - 2 \, x^{2} \log \left (\left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} - 1\right ) - 6 \, {\left (4 \, x^{2} - x - 3\right )} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}}}{36 \, x^{2}} \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^{3}}\, 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 = -0.00 \begin {gather*} \int \frac {{\left (\frac {x+1}{\sqrt {1-x^2}}\right )}^{1/3}}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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