Optimal. Leaf size=18 \[ \frac{x}{a}-\frac{\sinh (x)}{a \cosh (x)+a} \]
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Rubi [A] time = 0.0315345, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2735, 2648} \[ \frac{x}{a}-\frac{\sinh (x)}{a \cosh (x)+a} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2648
Rubi steps
\begin{align*} \int \frac{\cosh (x)}{a+a \cosh (x)} \, dx &=\frac{x}{a}-\int \frac{1}{a+a \cosh (x)} \, dx\\ &=\frac{x}{a}-\frac{\sinh (x)}{a+a \cosh (x)}\\ \end{align*}
Mathematica [A] time = 0.0239296, size = 14, normalized size = 0.78 \[ \frac{x-\tanh \left (\frac{x}{2}\right )}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 34, normalized size = 1.9 \begin{align*} -{\frac{1}{a}\tanh \left ({\frac{x}{2}} \right ) }-{\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 ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.26951, size = 24, normalized size = 1.33 \begin{align*} \frac{x}{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 = 1.92144, size = 82, normalized size = 4.56 \begin{align*} \frac{x \cosh \left (x\right ) + x \sinh \left (x\right ) + x + 2}{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 [A] time = 0.414297, size = 8, normalized size = 0.44 \begin{align*} \frac{x}{a} - \frac{\tanh{\left (\frac{x}{2} \right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16483, size = 23, normalized size = 1.28 \begin{align*} \frac{x}{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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