Optimal. Leaf size=17 \[ \frac{\cosh (x)}{a}-\frac{\log (\cosh (x)+1)}{a} \]
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Rubi [A] time = 0.0730798, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {3872, 2833, 12, 43} \[ \frac{\cosh (x)}{a}-\frac{\log (\cosh (x)+1)}{a} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2833
Rule 12
Rule 43
Rubi steps
\begin{align*} \int \frac{\sinh (x)}{a+a \text{sech}(x)} \, dx &=-\int \frac{\cosh (x) \sinh (x)}{-a-a \cosh (x)} \, dx\\ &=-\frac{\operatorname{Subst}\left (\int \frac{x}{a (-a+x)} \, dx,x,-a \cosh (x)\right )}{a}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{x}{-a+x} \, dx,x,-a \cosh (x)\right )}{a^2}\\ &=-\frac{\operatorname{Subst}\left (\int \left (1-\frac{a}{a-x}\right ) \, dx,x,-a \cosh (x)\right )}{a^2}\\ &=\frac{\cosh (x)}{a}-\frac{\log (1+\cosh (x))}{a}\\ \end{align*}
Mathematica [A] time = 0.0180007, size = 16, normalized size = 0.94 \[ \frac{\cosh (x)-2 \log \left (\cosh \left (\frac{x}{2}\right )\right )}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 27, normalized size = 1.6 \begin{align*} -{\frac{\ln \left ( 1+{\rm sech} \left (x\right ) \right ) }{a}}+{\frac{1}{a{\rm sech} \left (x\right )}}+{\frac{\ln \left ({\rm sech} \left (x\right ) \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.12734, size = 47, normalized size = 2.76 \begin{align*} -\frac{x}{a} + \frac{e^{\left (-x\right )}}{2 \, a} + \frac{e^{x}}{2 \, a} - \frac{2 \, \log \left (e^{\left (-x\right )} + 1\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.40772, size = 200, normalized size = 11.76 \begin{align*} \frac{2 \, x \cosh \left (x\right ) + \cosh \left (x\right )^{2} - 4 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + 2 \,{\left (x + \cosh \left (x\right )\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1}{2 \,{\left (a \cosh \left (x\right ) + a \sinh \left (x\right )\right )}} \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{\sinh{\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 = 1.15025, size = 43, normalized size = 2.53 \begin{align*} \frac{x}{a} + \frac{e^{\left (-x\right )}}{2 \, a} + \frac{e^{x}}{2 \, a} - \frac{2 \, \log \left (e^{x} + 1\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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