Optimal. Leaf size=112 \[ -\frac {1}{3} (1-x)^{2/3} \sqrt [3]{1+x}-\frac {1}{2} (1-x)^{2/3} (1+x)^{4/3}+\frac {2 \text {ArcTan}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{1+x}}\right )}{3 \sqrt {3}}+\frac {1}{9} \log (1+x)+\frac {1}{3} \log \left (1+\frac {\sqrt [3]{1-x}}{\sqrt [3]{1+x}}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6261, 81, 52,
62} \begin {gather*} \frac {2 \text {ArcTan}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{x+1}}\right )}{3 \sqrt {3}}-\frac {1}{2} (1-x)^{2/3} (x+1)^{4/3}-\frac {1}{3} (1-x)^{2/3} \sqrt [3]{x+1}+\frac {1}{9} \log (x+1)+\frac {1}{3} \log \left (\frac {\sqrt [3]{1-x}}{\sqrt [3]{x+1}}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 62
Rule 81
Rule 6261
Rubi steps
\begin {align*} \int e^{\frac {2}{3} \tanh ^{-1}(x)} x \, dx &=\int \frac {x \sqrt [3]{1+x}}{\sqrt [3]{1-x}} \, dx\\ &=-\frac {1}{2} (1-x)^{2/3} (1+x)^{4/3}+\frac {1}{3} \int \frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x}} \, dx\\ &=-\frac {1}{3} (1-x)^{2/3} \sqrt [3]{1+x}-\frac {1}{2} (1-x)^{2/3} (1+x)^{4/3}+\frac {2}{9} \int \frac {1}{\sqrt [3]{1-x} (1+x)^{2/3}} \, dx\\ &=-\frac {1}{3} (1-x)^{2/3} \sqrt [3]{1+x}-\frac {1}{2} (1-x)^{2/3} (1+x)^{4/3}+\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{1+x}}\right )}{3 \sqrt {3}}+\frac {1}{9} \log (1+x)+\frac {1}{3} \log \left (1+\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 = 46, normalized size = 0.41 \begin {gather*} -\frac {1}{2} (1-x)^{2/3} \left ((1+x)^{4/3}+\sqrt [3]{2} \, _2F_1\left (-\frac {1}{3},\frac {2}{3};\frac {5}{3};\frac {1-x}{2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \left (\frac {1+x}{\sqrt {-x^{2}+1}}\right )^{\frac {2}{3}} x\, dx\]
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 = 0.36, size = 154, normalized size = 1.38 \begin {gather*} \frac {1}{6} \, {\left (3 \, x^{2} + 2 \, x - 5\right )} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} + \frac {2}{9} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + \frac {2}{9} \, \log \left (\left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} + 1\right ) - \frac {1}{9} \, \log \left (-\frac {{\left (x - 1\right )} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} - x + \sqrt {-x^{2} + 1} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} + 1}{x - 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 x \left (\frac {x + 1}{\sqrt {1 - x^{2}}}\right )^{\frac {2}{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 = -0.01 \begin {gather*} \int x\,{\left (\frac {x+1}{\sqrt {1-x^2}}\right )}^{2/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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