Optimal. Leaf size=19 \[ -\frac{x}{2}-\frac{1}{2 (\tanh (x)+1)}+\log (\cosh (x)) \]
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Rubi [A] time = 0.0375599, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3540, 3475} \[ -\frac{x}{2}-\frac{1}{2 (\tanh (x)+1)}+\log (\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 3540
Rule 3475
Rubi steps
\begin{align*} \int \frac{\tanh ^2(x)}{1+\tanh (x)} \, dx &=-\frac{1}{2 (1+\tanh (x))}-\frac{1}{2} \int (1-2 \tanh (x)) \, dx\\ &=-\frac{x}{2}-\frac{1}{2 (1+\tanh (x))}+\int \tanh (x) \, dx\\ &=-\frac{x}{2}+\log (\cosh (x))-\frac{1}{2 (1+\tanh (x))}\\ \end{align*}
Mathematica [A] time = 0.0276375, size = 23, normalized size = 1.21 \[ \frac{1}{4} (-2 x+\sinh (2 x)-\cosh (2 x)+4 \log (\cosh (x))) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 24, normalized size = 1.3 \begin{align*} -{\frac{1}{2+2\,\tanh \left ( x \right ) }}-{\frac{3\,\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 = 1.84957, size = 23, normalized size = 1.21 \begin{align*} \frac{1}{2} \, x - \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 = 2.32245, size = 259, normalized size = 13.63 \begin{align*} -\frac{6 \, x \cosh \left (x\right )^{2} + 12 \, x \cosh \left (x\right ) \sinh \left (x\right ) + 6 \, x \sinh \left (x\right )^{2} - 4 \,{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )} \log \left (\frac{2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 1}{4 \,{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.428343, size = 61, normalized size = 3.21 \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{1}{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 = 1.20933, size = 23, normalized size = 1.21 \begin{align*} -\frac{3}{2} \, x - \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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