Optimal. Leaf size=202 \[ -(1-x)^{5/6} \sqrt [6]{1+x}-\frac {2}{3} \text {ArcTan}\left (\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\frac {1}{3} \text {ArcTan}\left (\sqrt {3}-\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac {1}{3} \text {ArcTan}\left (\sqrt {3}+\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac {\log \left (1+\frac {\sqrt [3]{1-x}}{\sqrt [3]{1+x}}-\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )}{2 \sqrt {3}}+\frac {\log \left (1+\frac {\sqrt [3]{1-x}}{\sqrt [3]{1+x}}+\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )}{2 \sqrt {3}} \]
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Rubi [A]
time = 0.24, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 10, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.250, Rules used = {6260, 52, 65,
338, 301, 648, 632, 210, 642, 209} \begin {gather*} -\frac {2}{3} \text {ArcTan}\left (\frac {\sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )+\frac {1}{3} \text {ArcTan}\left (\sqrt {3}-\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )-\frac {1}{3} \text {ArcTan}\left (\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+\sqrt {3}\right )-(1-x)^{5/6} \sqrt [6]{x+1}-\frac {\log \left (\frac {\sqrt [3]{1-x}}{\sqrt [3]{x+1}}-\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )}{2 \sqrt {3}}+\frac {\log \left (\frac {\sqrt [3]{1-x}}{\sqrt [3]{x+1}}+\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )}{2 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 209
Rule 210
Rule 301
Rule 338
Rule 632
Rule 642
Rule 648
Rule 6260
Rubi steps
\begin {align*} \int e^{\frac {1}{3} \tanh ^{-1}(x)} \, dx &=\int \frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}} \, dx\\ &=-(1-x)^{5/6} \sqrt [6]{1+x}+\frac {1}{3} \int \frac {1}{\sqrt [6]{1-x} (1+x)^{5/6}} \, dx\\ &=-(1-x)^{5/6} \sqrt [6]{1+x}-2 \text {Subst}\left (\int \frac {x^4}{\left (2-x^6\right )^{5/6}} \, dx,x,\sqrt [6]{1-x}\right )\\ &=-(1-x)^{5/6} \sqrt [6]{1+x}-2 \text {Subst}\left (\int \frac {x^4}{1+x^6} \, dx,x,\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )\\ &=-(1-x)^{5/6} \sqrt [6]{1+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 {-\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 )-\frac {2}{3} \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 )\\ &=-(1-x)^{5/6} \sqrt [6]{1+x}-\frac {2}{3} \tan ^{-1}\left (\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac {1}{6} \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}{6} \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 {\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 \sqrt {3}}+\frac {\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 \sqrt {3}}\\ &=-(1-x)^{5/6} \sqrt [6]{1+x}-\frac {2}{3} \tan ^{-1}\left (\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac {\log \left (1+\frac {\sqrt [3]{1-x}}{\sqrt [3]{1+x}}-\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )}{2 \sqrt {3}}+\frac {\log \left (1+\frac {\sqrt [3]{1-x}}{\sqrt [3]{1+x}}+\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )}{2 \sqrt {3}}+\frac {1}{3} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,-\sqrt {3}+\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\frac {1}{3} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {3}+\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )\\ &=-(1-x)^{5/6} \sqrt [6]{1+x}-\frac {2}{3} \tan ^{-1}\left (\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\frac {1}{3} \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac {1}{3} \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac {\log \left (1+\frac {\sqrt [3]{1-x}}{\sqrt [3]{1+x}}-\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )}{2 \sqrt {3}}+\frac {\log \left (1+\frac {\sqrt [3]{1-x}}{\sqrt [3]{1+x}}+\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )}{2 \sqrt {3}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.03, size = 39, normalized size = 0.19 \begin {gather*} 2 e^{\frac {1}{3} \tanh ^{-1}(x)} \left (-\frac {1}{1+e^{2 \tanh ^{-1}(x)}}+\, _2F_1\left (\frac {1}{6},1;\frac {7}{6};-e^{2 \tanh ^{-1}(x)}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \left (\frac {1+x}{\sqrt {-x^{2}+1}}\right )^{\frac {1}{3}}\, dx\]
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 = 295, normalized size = 1.46 \begin {gather*} \frac {1}{6} \, \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}{6} \, \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 ) + {\left (x - 1\right )} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} - \frac {2}{3} \, \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 ) - \frac {2}{3} \, \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 ) + \frac {2}{3} \, \arctan \left (\left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}}\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 \sqrt [3]{\frac {x + 1}{\sqrt {1 - 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 = -0.00 \begin {gather*} \int {\left (\frac {x+1}{\sqrt {1-x^2}}\right )}^{1/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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