Optimal. Leaf size=15 \[ \frac {\tan ^{-1}(\sinh (x))}{a}-\frac {\tanh (x)}{a} \]
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Rubi [A] time = 0.05, antiderivative size = 15, normalized size of antiderivative = 1.00, 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 8
Rule 2706
Rule 3767
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*}
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Mathematica [A] time = 0.05, size = 18, normalized size = 1.20 \[ \frac {2 \tan ^{-1}\left (\tanh \left (\frac {x}{2}\right )\right )-\tanh (x)}{a} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.54, size = 50, normalized size = 3.33 \[ \frac {2 \, {\left ({\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1\right )} \arctan \left (\cosh \relax (x) + \sinh \relax (x)\right ) + 1\right )}}{a \cosh \relax (x)^{2} + 2 \, a \cosh \relax (x) \sinh \relax (x) + a \sinh \relax (x)^{2} + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 22, normalized size = 1.47 \[ \frac {2 \, \arctan \left (e^{x}\right )}{a} + \frac {2}{a {\left (e^{\left (2 \, x\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 31, normalized size = 2.07 \[ -\frac {2 \tanh \left (\frac {x}{2}\right )}{a \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )}+\frac {2 \arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 23, normalized size = 1.53 \[ -\frac {2 \, \arctan \left (e^{\left (-x\right )}\right )}{a} - \frac {2}{a e^{\left (-2 \, x\right )} + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.92, size = 33, normalized size = 2.20 \[ \frac {2}{a\,\left ({\mathrm {e}}^{2\,x}+1\right )}+\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {a^2}}{a}\right )}{\sqrt {a^2}} \]
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 \[ \frac {\int \frac {\tanh ^{2}{\relax (x )}}{\cosh {\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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