Optimal. Leaf size=14 \[ \frac{x}{a}-\frac{\tan ^{-1}(\sinh (x))}{a} \]
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Rubi [A] time = 0.0457572, 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 = {3888, 3770} \[ \frac{x}{a}-\frac{\tan ^{-1}(\sinh (x))}{a} \]
Antiderivative was successfully verified.
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Rule 3888
Rule 3770
Rubi steps
\begin{align*} \int \frac{\tanh ^2(x)}{a+a \text{sech}(x)} \, dx &=-\frac{\int (-a+a \text{sech}(x)) \, dx}{a^2}\\ &=\frac{x}{a}-\frac{\int \text{sech}(x) \, dx}{a}\\ &=\frac{x}{a}-\frac{\tan ^{-1}(\sinh (x))}{a}\\ \end{align*}
Mathematica [A] time = 0.0295558, size = 15, normalized size = 1.07 \[ \frac{x-2 \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )}{a} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.019, size = 35, normalized size = 2.5 \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 ) }-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 = 1.65944, size = 22, normalized size = 1.57 \begin{align*} \frac{x}{a} + \frac{2 \, \arctan \left (e^{\left (-x\right )}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.42832, size = 50, normalized size = 3.57 \begin{align*} \frac{x - 2 \, \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}{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 )}}{\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 [A] time = 1.15041, size = 19, normalized size = 1.36 \begin{align*} \frac{x}{a} - \frac{2 \, \arctan \left (e^{x}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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