Optimal. Leaf size=15 \[ \frac{\tan ^{-1}(\sinh (x))}{a}-\frac{\tanh (x)}{a} \]
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Rubi [A] time = 0.0488648, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {2706, 3767, 8, 3770} \[ \frac{\tan ^{-1}(\sinh (x))}{a}-\frac{\tanh (x)}{a} \]
Antiderivative was successfully verified.
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Rule 2706
Rule 3767
Rule 8
Rule 3770
Rubi steps
\begin{align*} \int \frac{\tanh ^2(x)}{a+a \cosh (x)} \, dx &=\frac{\int \text{sech}(x) \, dx}{a}-\frac{\int \text{sech}^2(x) \, dx}{a}\\ &=\frac{\tan ^{-1}(\sinh (x))}{a}-\frac{i \operatorname{Subst}(\int 1 \, dx,x,-i \tanh (x))}{a}\\ &=\frac{\tan ^{-1}(\sinh (x))}{a}-\frac{\tanh (x)}{a}\\ \end{align*}
Mathematica [A] time = 0.045499, size = 18, normalized size = 1.2 \[ \frac{2 \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )-\tanh (x)}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 31, normalized size = 2.1 \begin{align*} -2\,{\frac{\tanh \left ( x/2 \right ) }{a \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1 \right ) }}+2\,{\frac{\arctan \left ( \tanh \left ( x/2 \right ) \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.54332, size = 31, normalized size = 2.07 \begin{align*} -\frac{2 \, \arctan \left (e^{\left (-x\right )}\right )}{a} - \frac{2}{a e^{\left (-2 \, x\right )} + a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.92385, size = 185, normalized size = 12.33 \begin{align*} \frac{2 \,{\left ({\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + 1\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 ^{2}{\left (x \right )}}{\cosh{\left (x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16883, size = 30, normalized size = 2. \begin{align*} \frac{2 \, \arctan \left (e^{x}\right )}{a} + \frac{2}{a{\left (e^{\left (2 \, x\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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