Optimal. Leaf size=14 \[ \frac {x}{a}-\frac {\tan ^{-1}(\sinh (x))}{a} \]
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Rubi [A] time = 0.05, antiderivative size = 14, normalized size of antiderivative = 1.00, 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 3770
Rule 3888
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*}
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Mathematica [A] time = 0.03, 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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fricas [A] time = 0.39, size = 14, normalized size = 1.00 \[ \frac {x - 2 \, \arctan \left (\cosh \relax (x) + \sinh \relax (x)\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 14, normalized size = 1.00 \[ \frac {x}{a} - \frac {2 \, \arctan \left (e^{x}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 35, normalized size = 2.50 \[ -\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a}-\frac {2 \arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 16, normalized size = 1.14 \[ \frac {x}{a} + \frac {2 \, \arctan \left (e^{\left (-x\right )}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.32, size = 25, normalized size = 1.79 \[ \frac {x}{a}-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {a^2}}{a}\right )}{\sqrt {a^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\tanh ^{2}{\relax (x )}}{\operatorname {sech}{\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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