Optimal. Leaf size=31 \[ \frac{3 x}{2}+\frac{\tanh ^2(x)}{2 (\tanh (x)+1)}-\frac{3 \tanh (x)}{2}-\log (\cosh (x)) \]
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Rubi [A] time = 0.0512713, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {3550, 3525, 3475} \[ \frac{3 x}{2}+\frac{\tanh ^2(x)}{2 (\tanh (x)+1)}-\frac{3 \tanh (x)}{2}-\log (\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 3550
Rule 3525
Rule 3475
Rubi steps
\begin{align*} \int \frac{\tanh ^3(x)}{1+\tanh (x)} \, dx &=\frac{\tanh ^2(x)}{2 (1+\tanh (x))}-\frac{1}{2} \int (2-3 \tanh (x)) \tanh (x) \, dx\\ &=\frac{3 x}{2}-\frac{3 \tanh (x)}{2}+\frac{\tanh ^2(x)}{2 (1+\tanh (x))}-\int \tanh (x) \, dx\\ &=\frac{3 x}{2}-\log (\cosh (x))-\frac{3 \tanh (x)}{2}+\frac{\tanh ^2(x)}{2 (1+\tanh (x))}\\ \end{align*}
Mathematica [A] time = 0.0453772, size = 27, normalized size = 0.87 \[ \frac{1}{4} (6 x-\sinh (2 x)+\cosh (2 x)-4 \tanh (x)-4 \log (\cosh (x))) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 28, normalized size = 0.9 \begin{align*} -\tanh \left ( x \right ) +{\frac{1}{2+2\,\tanh \left ( x \right ) }}+{\frac{5\,\ln \left ( 1+\tanh \left ( x \right ) \right ) }{4}}-{\frac{\ln \left ( \tanh \left ( x \right ) -1 \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.92124, size = 39, normalized size = 1.26 \begin{align*} \frac{1}{2} \, x - \frac{2}{e^{\left (-2 \, x\right )} + 1} + \frac{1}{4} \, e^{\left (-2 \, x\right )} - \log \left (e^{\left (-2 \, x\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.28245, size = 632, normalized size = 20.39 \begin{align*} \frac{10 \, x \cosh \left (x\right )^{4} + 40 \, x \cosh \left (x\right ) \sinh \left (x\right )^{3} + 10 \, x \sinh \left (x\right )^{4} +{\left (10 \, x + 9\right )} \cosh \left (x\right )^{2} +{\left (60 \, x \cosh \left (x\right )^{2} + 10 \, x + 9\right )} \sinh \left (x\right )^{2} - 4 \,{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} +{\left (6 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + \cosh \left (x\right )^{2} + 2 \,{\left (2 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac{2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 2 \,{\left (20 \, x \cosh \left (x\right )^{3} +{\left (10 \, x + 9\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1}{4 \,{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} +{\left (6 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + \cosh \left (x\right )^{2} + 2 \,{\left (2 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.470005, size = 75, normalized size = 2.42 \begin{align*} \frac{x \tanh{\left (x \right )}}{2 \tanh{\left (x \right )} + 2} + \frac{x}{2 \tanh{\left (x \right )} + 2} + \frac{2 \log{\left (\tanh{\left (x \right )} + 1 \right )} \tanh{\left (x \right )}}{2 \tanh{\left (x \right )} + 2} + \frac{2 \log{\left (\tanh{\left (x \right )} + 1 \right )}}{2 \tanh{\left (x \right )} + 2} - \frac{2 \tanh ^{2}{\left (x \right )}}{2 \tanh{\left (x \right )} + 2} + \frac{3}{2 \tanh{\left (x \right )} + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19976, size = 47, normalized size = 1.52 \begin{align*} \frac{5}{2} \, x + \frac{{\left (9 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-2 \, x\right )}}{4 \,{\left (e^{\left (2 \, x\right )} + 1\right )}} - \log \left (e^{\left (2 \, x\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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