Optimal. Leaf size=346 \[ -2 \text {ArcTan}\left (\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\text {ArcTan}\left (\sqrt {3}-\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\text {ArcTan}\left (\sqrt {3}+\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\sqrt {3} \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [6]{1+x}}{\sqrt [6]{1-x}}}{\sqrt {3}}\right )-\sqrt {3} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [6]{1+x}}{\sqrt [6]{1-x}}}{\sqrt {3}}\right )-2 \tanh ^{-1}\left (\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )-\frac {1}{2} \sqrt {3} \log \left (1+\frac {\sqrt [3]{1-x}}{\sqrt [3]{1+x}}-\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\frac {1}{2} \sqrt {3} \log \left (1+\frac {\sqrt [3]{1-x}}{\sqrt [3]{1+x}}+\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\frac {1}{2} \log \left (1-\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}+\frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right )-\frac {1}{2} \log \left (1+\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}+\frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right ) \]
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Rubi [A]
time = 0.33, antiderivative size = 346, normalized size of antiderivative = 1.00, number of steps
used = 25, number of rules used = 13, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.083, Rules used = {6261, 132,
65, 338, 301, 648, 632, 210, 642, 209, 95, 216, 212} \begin {gather*} -2 \text {ArcTan}\left (\frac {\sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )+\text {ArcTan}\left (\sqrt {3}-\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )-\text {ArcTan}\left (\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+\sqrt {3}\right )+\sqrt {3} \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [6]{x+1}}{\sqrt [6]{1-x}}}{\sqrt {3}}\right )-\sqrt {3} \text {ArcTan}\left (\frac {\frac {2 \sqrt [6]{x+1}}{\sqrt [6]{1-x}}+1}{\sqrt {3}}\right )-\frac {1}{2} \sqrt {3} \log \left (\frac {\sqrt [3]{1-x}}{\sqrt [3]{x+1}}-\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )+\frac {1}{2} \sqrt {3} \log \left (\frac {\sqrt [3]{1-x}}{\sqrt [3]{x+1}}+\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )+\frac {1}{2} \log \left (\frac {\sqrt [3]{x+1}}{\sqrt [3]{1-x}}-\frac {\sqrt [6]{x+1}}{\sqrt [6]{1-x}}+1\right )-\frac {1}{2} \log \left (\frac {\sqrt [3]{x+1}}{\sqrt [3]{1-x}}+\frac {\sqrt [6]{x+1}}{\sqrt [6]{1-x}}+1\right )-2 \tanh ^{-1}\left (\frac {\sqrt [6]{x+1}}{\sqrt [6]{1-x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 95
Rule 132
Rule 209
Rule 210
Rule 212
Rule 216
Rule 301
Rule 338
Rule 632
Rule 642
Rule 648
Rule 6261
Rubi steps
\begin {align*} \int \frac {e^{\frac {1}{3} \tanh ^{-1}(x)}}{x} \, dx &=\int \frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x} x} \, dx\\ &=\int \frac {1}{\sqrt [6]{1-x} (1+x)^{5/6}} \, dx+\int \frac {1}{\sqrt [6]{1-x} x (1+x)^{5/6}} \, dx\\ &=-\left (6 \text {Subst}\left (\int \frac {x^4}{\left (2-x^6\right )^{5/6}} \, dx,x,\sqrt [6]{1-x}\right )\right )+6 \text {Subst}\left (\int \frac {1}{-1+x^6} \, dx,x,\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )\\ &=-\left (2 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )\right )-2 \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 )-2 \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 )-6 \text {Subst}\left (\int \frac {x^4}{1+x^6} \, dx,x,\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )\\ &=-2 \tanh ^{-1}\left (\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )+\frac {1}{2} \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+2 x}{1+x+x^2} \, dx,x,\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )-\frac {3}{2} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )-\frac {3}{2} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )-2 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-2 \text {Subst}\left (\int \frac {-\frac {1}{2}+\frac {\sqrt {3} x}{2}}{1-\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-2 \text {Subst}\left (\int \frac {-\frac {1}{2}-\frac {\sqrt {3} x}{2}}{1+\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )\\ &=-2 \tan ^{-1}\left (\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-2 \tanh ^{-1}\left (\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )+\frac {1}{2} \log \left (1-\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}+\frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right )-\frac {1}{2} \log \left (1+\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}+\frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-\sqrt {3} 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+\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+3 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+\frac {2 \sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )+3 \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}{2} \sqrt {3} \text {Subst}\left (\int \frac {-\sqrt {3}+2 x}{1-\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\frac {1}{2} \sqrt {3} \text {Subst}\left (\int \frac {\sqrt {3}+2 x}{1+\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )\\ &=-2 \tan ^{-1}\left (\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [6]{1+x}}{\sqrt [6]{1-x}}}{\sqrt {3}}\right )-\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [6]{1+x}}{\sqrt [6]{1-x}}}{\sqrt {3}}\right )-2 \tanh ^{-1}\left (\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )-\frac {1}{2} \sqrt {3} \log \left (1+\frac {\sqrt [3]{1-x}}{\sqrt [3]{1+x}}-\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\frac {1}{2} \sqrt {3} \log \left (1+\frac {\sqrt [3]{1-x}}{\sqrt [3]{1+x}}+\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\frac {1}{2} \log \left (1-\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}+\frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right )-\frac {1}{2} \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}{-1-x^2} \, dx,x,-\sqrt {3}+\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {3}+\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )\\ &=-2 \tan ^{-1}\left (\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [6]{1+x}}{\sqrt [6]{1-x}}}{\sqrt {3}}\right )-\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [6]{1+x}}{\sqrt [6]{1-x}}}{\sqrt {3}}\right )-2 \tanh ^{-1}\left (\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )-\frac {1}{2} \sqrt {3} \log \left (1+\frac {\sqrt [3]{1-x}}{\sqrt [3]{1+x}}-\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\frac {1}{2} \sqrt {3} \log \left (1+\frac {\sqrt [3]{1-x}}{\sqrt [3]{1+x}}+\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\frac {1}{2} \log \left (1-\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}+\frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right )-\frac {1}{2} \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 = 0.04, size = 74, normalized size = 0.21 \begin {gather*} -\frac {3 (1-x)^{5/6} \left (\sqrt [6]{2} (1+x)^{5/6} \, _2F_1\left (\frac {5}{6},\frac {5}{6};\frac {11}{6};\frac {1-x}{2}\right )+2 \, _2F_1\left (\frac {5}{6},1;\frac {11}{6};\frac {1-x}{1+x}\right )\right )}{5 (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 \[\int \frac {\left (\frac {1+x}{\sqrt {-x^{2}+1}}\right )^{\frac {1}{3}}}{x}\, dx\]
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 = 0.37, size = 471, normalized size = 1.36 \begin {gather*} -\sqrt {3} \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 ) - \sqrt {3} \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 ) + \frac {1}{2} \, \sqrt {3} \log \left (4 \, \sqrt {3} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} + 4 \, \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} + 4\right ) - \frac {1}{2} \, \sqrt {3} \log \left (-4 \, \sqrt {3} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} + 4 \, \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} + 4\right ) - 2 \, \arctan \left (\sqrt {3} + \sqrt {-4 \, \sqrt {3} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} + 4 \, \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} + 4} - 2 \, \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}}\right ) - 2 \, \arctan \left (-\sqrt {3} + 2 \, \sqrt {\sqrt {3} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} + \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} + 1} - 2 \, \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}}\right ) + 2 \, \arctan \left (\left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}}\right ) - \frac {1}{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 ) + \frac {1}{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 ) - \log \left (\left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} + 1\right ) + \log \left (\left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} - 1\right ) \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}\, 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} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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