Optimal. Leaf size=14 \[ \frac{\text{sech}(x)}{a}+\frac{\log (\cosh (x))}{a} \]
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Rubi [A] time = 0.0483187, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3879, 43} \[ \frac{\text{sech}(x)}{a}+\frac{\log (\cosh (x))}{a} \]
Antiderivative was successfully verified.
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Rule 3879
Rule 43
Rubi steps
\begin{align*} \int \frac{\tanh ^3(x)}{a+a \text{sech}(x)} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{a-a x}{x^2} \, dx,x,\cosh (x)\right )}{a^2}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{a}{x^2}-\frac{a}{x}\right ) \, dx,x,\cosh (x)\right )}{a^2}\\ &=\frac{\log (\cosh (x))}{a}+\frac{\text{sech}(x)}{a}\\ \end{align*}
Mathematica [A] time = 0.0335871, size = 10, normalized size = 0.71 \[ \frac{\text{sech}(x)+\log (\cosh (x))}{a} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.027, size = 54, normalized size = 3.9 \begin{align*} -{\frac{1}{a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-{\frac{1}{a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) }+{\frac{1}{a}\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) }+2\,{\frac{1}{a \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.65543, size = 45, normalized size = 3.21 \begin{align*} \frac{x}{a} + \frac{2 \, e^{\left (-x\right )}}{a e^{\left (-2 \, x\right )} + a} + \frac{\log \left (e^{\left (-2 \, x\right )} + 1\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.52471, size = 288, normalized size = 20.57 \begin{align*} -\frac{x \cosh \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 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 2 \,{\left (x \cosh \left (x\right ) - 1\right )} \sinh \left (x\right ) + x - 2 \, \cosh \left (x\right )}{a \cosh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) \sinh \left (x\right ) + a \sinh \left (x\right )^{2} + a} \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*} \frac{\int \frac{\tanh ^{3}{\left (x \right )}}{\operatorname{sech}{\left (x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.22351, size = 47, normalized size = 3.36 \begin{align*} \frac{\log \left (e^{\left (-x\right )} + e^{x}\right )}{a} - \frac{e^{\left (-x\right )} + e^{x} - 2}{a{\left (e^{\left (-x\right )} + e^{x}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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