Optimal. Leaf size=20 \[ \frac{\tan ^{-1}(\sinh (x))}{a}-\frac{\sinh (x)}{a \cosh (x)+a} \]
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Rubi [A] time = 0.0415903, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {2747, 3770, 2648} \[ \frac{\tan ^{-1}(\sinh (x))}{a}-\frac{\sinh (x)}{a \cosh (x)+a} \]
Antiderivative was successfully verified.
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Rule 2747
Rule 3770
Rule 2648
Rubi steps
\begin{align*} \int \frac{\text{sech}(x)}{a+a \cosh (x)} \, dx &=\frac{\int \text{sech}(x) \, dx}{a}-\int \frac{1}{a+a \cosh (x)} \, dx\\ &=\frac{\tan ^{-1}(\sinh (x))}{a}-\frac{\sinh (x)}{a+a \cosh (x)}\\ \end{align*}
Mathematica [A] time = 0.0235633, size = 22, normalized size = 1.1 \[ \frac{2 \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )-\tanh \left (\frac{x}{2}\right )}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 21, normalized size = 1.1 \begin{align*} -{\frac{1}{a}\tanh \left ({\frac{x}{2}} \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 = 1.81931, size = 31, normalized size = 1.55 \begin{align*} -\frac{2 \, \arctan \left (e^{\left (-x\right )}\right )}{a} - \frac{2}{a e^{\left (-x\right )} + a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78969, size = 117, normalized size = 5.85 \begin{align*} \frac{2 \,{\left ({\left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + 1\right )}}{a \cosh \left (x\right ) + a \sinh \left (x\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{\operatorname{sech}{\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 = 1.17131, size = 27, normalized size = 1.35 \begin{align*} \frac{2 \, \arctan \left (e^{x}\right )}{a} + \frac{2}{a{\left (e^{x} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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