Optimal. Leaf size=346 \[ -\frac {1}{2} \sqrt {3} \log \left (\frac {\sqrt [3]{1-x}}{\sqrt [3]{x+1}}-\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )+\frac {1}{2} \sqrt {3} \log \left (\frac {\sqrt [3]{1-x}}{\sqrt [3]{x+1}}+\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )+\frac {1}{2} \log \left (\frac {\sqrt [3]{x+1}}{\sqrt [3]{1-x}}-\frac {\sqrt [6]{x+1}}{\sqrt [6]{1-x}}+1\right )-\frac {1}{2} \log \left (\frac {\sqrt [3]{x+1}}{\sqrt [3]{1-x}}+\frac {\sqrt [6]{x+1}}{\sqrt [6]{1-x}}+1\right )-2 \tan ^{-1}\left (\frac {\sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )+\tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )-\tan ^{-1}\left (\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+\sqrt {3}\right )+\sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [6]{x+1}}{\sqrt [6]{1-x}}}{\sqrt {3}}\right )-\sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [6]{x+1}}{\sqrt [6]{1-x}}+1}{\sqrt {3}}\right )-2 \tanh ^{-1}\left (\frac {\sqrt [6]{x+1}}{\sqrt [6]{1-x}}\right ) \]
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Rubi [A] time = 0.45, 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 = {6126, 105, 63, 331, 295, 634, 618, 204, 628, 203, 93, 210, 206} \[ -\frac {1}{2} \sqrt {3} \log \left (\frac {\sqrt [3]{1-x}}{\sqrt [3]{x+1}}-\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )+\frac {1}{2} \sqrt {3} \log \left (\frac {\sqrt [3]{1-x}}{\sqrt [3]{x+1}}+\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )+\frac {1}{2} \log \left (\frac {\sqrt [3]{x+1}}{\sqrt [3]{1-x}}-\frac {\sqrt [6]{x+1}}{\sqrt [6]{1-x}}+1\right )-\frac {1}{2} \log \left (\frac {\sqrt [3]{x+1}}{\sqrt [3]{1-x}}+\frac {\sqrt [6]{x+1}}{\sqrt [6]{1-x}}+1\right )-2 \tan ^{-1}\left (\frac {\sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )+\tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )-\tan ^{-1}\left (\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+\sqrt {3}\right )+\sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [6]{x+1}}{\sqrt [6]{1-x}}}{\sqrt {3}}\right )-\sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [6]{x+1}}{\sqrt [6]{1-x}}+1}{\sqrt {3}}\right )-2 \tanh ^{-1}\left (\frac {\sqrt [6]{x+1}}{\sqrt [6]{1-x}}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 93
Rule 105
Rule 203
Rule 204
Rule 206
Rule 210
Rule 295
Rule 331
Rule 618
Rule 628
Rule 634
Rule 6126
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 \operatorname {Subst}\left (\int \frac {x^4}{\left (2-x^6\right )^{5/6}} \, dx,x,\sqrt [6]{1-x}\right )\right )+6 \operatorname {Subst}\left (\int \frac {1}{-1+x^6} \, dx,x,\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )\right )-2 \operatorname {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 \operatorname {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 \operatorname {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} \operatorname {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} \operatorname {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} \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )-\frac {3}{2} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )-2 \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-2 \operatorname {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 \operatorname {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} \operatorname {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} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+3 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+\frac {2 \sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )+3 \operatorname {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} \operatorname {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} \operatorname {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 )+\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,-\sqrt {3}+\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\operatorname {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] time = 0.03, size = 74, normalized size = 0.21 \[ -\frac {3 (1-x)^{5/6} \left (\sqrt [6]{2} (x+1)^{5/6} \, _2F_1\left (\frac {5}{6},\frac {5}{6};\frac {11}{6};\frac {1-x}{2}\right )+2 \, _2F_1\left (\frac {5}{6},1;\frac {11}{6};\frac {1-x}{x+1}\right )\right )}{5 (x+1)^{5/6}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.46, size = 471, normalized size = 1.36 \[ -\sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{2} \, \sqrt {3} \log \left (4 \, \sqrt {3} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} + 4 \, \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} + 4\right ) - \frac {1}{2} \, \sqrt {3} \log \left (-4 \, \sqrt {3} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} + 4 \, \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} + 4\right ) - 2 \, \arctan \left (\sqrt {3} + \sqrt {-4 \, \sqrt {3} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} + 4 \, \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} + 4} - 2 \, \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}}\right ) - 2 \, \arctan \left (-\sqrt {3} + 2 \, \sqrt {\sqrt {3} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} + \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} + 1} - 2 \, \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}}\right ) + 2 \, \arctan \left (\left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}}\right ) - \frac {1}{2} \, \log \left (\left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} + \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} + 1\right ) + \frac {1}{2} \, \log \left (\left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} - \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} + 1\right ) - \log \left (\left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} + 1\right ) + \log \left (\left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} - 1\right ) \]
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 \[ \int \frac {\left (\frac {x + 1}{\sqrt {-x^{2} + 1}}\right )^{\frac {1}{3}}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, 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 \[ \int \frac {\left (\frac {x + 1}{\sqrt {-x^{2} + 1}}\right )^{\frac {1}{3}}}{x}\,{d x} \]
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 \[ \int \frac {{\left (\frac {x+1}{\sqrt {1-x^2}}\right )}^{1/3}}{x} \,d x \]
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 \[ \int \frac {\sqrt [3]{\frac {x + 1}{\sqrt {1 - x^{2}}}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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