Optimal. Leaf size=35 \[ a \coth (x) \sqrt{a \tanh ^2(x)} \log (\cosh (x))-\frac{1}{2} a \tanh (x) \sqrt{a \tanh ^2(x)} \]
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Rubi [A] time = 0.020685, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3658, 3473, 3475} \[ a \coth (x) \sqrt{a \tanh ^2(x)} \log (\cosh (x))-\frac{1}{2} a \tanh (x) \sqrt{a \tanh ^2(x)} \]
Antiderivative was successfully verified.
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Rule 3658
Rule 3473
Rule 3475
Rubi steps
\begin{align*} \int \left (a \tanh ^2(x)\right )^{3/2} \, dx &=\left (a \coth (x) \sqrt{a \tanh ^2(x)}\right ) \int \tanh ^3(x) \, dx\\ &=-\frac{1}{2} a \tanh (x) \sqrt{a \tanh ^2(x)}+\left (a \coth (x) \sqrt{a \tanh ^2(x)}\right ) \int \tanh (x) \, dx\\ &=a \coth (x) \log (\cosh (x)) \sqrt{a \tanh ^2(x)}-\frac{1}{2} a \tanh (x) \sqrt{a \tanh ^2(x)}\\ \end{align*}
Mathematica [A] time = 0.0234227, size = 28, normalized size = 0.8 \[ \frac{1}{2} a \sqrt{a \tanh ^2(x)} (\text{csch}(x) \text{sech}(x)+2 \coth (x) \log (\cosh (x))) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 30, normalized size = 0.9 \begin{align*} -{\frac{ \left ( \tanh \left ( x \right ) \right ) ^{2}+\ln \left ( \tanh \left ( x \right ) -1 \right ) +\ln \left ( 1+\tanh \left ( x \right ) \right ) }{2\, \left ( \tanh \left ( x \right ) \right ) ^{3}} \left ( a \left ( \tanh \left ( x \right ) \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.57375, size = 57, normalized size = 1.63 \begin{align*} -a^{\frac{3}{2}} x - a^{\frac{3}{2}} \log \left (e^{\left (-2 \, x\right )} + 1\right ) - \frac{2 \, a^{\frac{3}{2}} e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} + e^{\left (-4 \, x\right )} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.42579, size = 1368, normalized size = 39.09 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \tanh ^{2}{\left (x \right )}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20896, size = 70, normalized size = 2. \begin{align*} -{\left (x \mathrm{sgn}\left (e^{\left (4 \, x\right )} - 1\right ) - \log \left (e^{\left (2 \, x\right )} + 1\right ) \mathrm{sgn}\left (e^{\left (4 \, x\right )} - 1\right ) - \frac{2 \, e^{\left (2 \, x\right )} \mathrm{sgn}\left (e^{\left (4 \, x\right )} - 1\right )}{{\left (e^{\left (2 \, x\right )} + 1\right )}^{2}}\right )} a^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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