Optimal. Leaf size=15 \[ -\coth (x)-\log (\tanh (x))+\log (\tanh (x)+1) \]
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Rubi [A] time = 0.0530338, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {4342, 44} \[ -\coth (x)-\log (\tanh (x))+\log (\tanh (x)+1) \]
Antiderivative was successfully verified.
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Rule 4342
Rule 44
Rubi steps
\begin{align*} \int \frac{\text{sech}^2(x)}{\tanh ^2(x)+\tanh ^3(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{x^2 (1+x)} \, dx,x,\tanh (x)\right )\\ &=\operatorname{Subst}\left (\int \left (\frac{1}{x^2}-\frac{1}{x}+\frac{1}{1+x}\right ) \, dx,x,\tanh (x)\right )\\ &=-\coth (x)-\log (\tanh (x))+\log (1+\tanh (x))\\ \end{align*}
Mathematica [A] time = 0.0302431, size = 11, normalized size = 0.73 \[ x-\coth (x)-\log (\sinh (x)) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.049, size = 32, normalized size = 2.1 \begin{align*} -{\frac{1}{2}\tanh \left ({\frac{x}{2}} \right ) }+2\,\ln \left ( \tanh \left ( x/2 \right ) +1 \right ) -{\frac{1}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0052, size = 39, normalized size = 2.6 \begin{align*} \frac{2}{e^{\left (-2 \, x\right )} - 1} - \log \left (e^{\left (-x\right )} + 1\right ) - \log \left (e^{\left (-x\right )} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.06623, size = 267, normalized size = 17.8 \begin{align*} \frac{2 \, x \cosh \left (x\right )^{2} + 4 \, x \cosh \left (x\right ) \sinh \left (x\right ) + 2 \, x \sinh \left (x\right )^{2} -{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )} \log \left (\frac{2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) - 2 \, x - 2}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1} \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 \frac{\operatorname{sech}^{2}{\left (x \right )}}{\left (\tanh{\left (x \right )} + 1\right ) \tanh ^{2}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15948, size = 39, normalized size = 2.6 \begin{align*} 2 \, x + \frac{e^{\left (2 \, x\right )} - 3}{e^{\left (2 \, x\right )} - 1} - \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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