Optimal. Leaf size=18 \[ \frac{(A-B) \sinh (x)}{\cosh (x)+1}+B x \]
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Rubi [A] time = 0.0359926, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2735, 2648} \[ \frac{(A-B) \sinh (x)}{\cosh (x)+1}+B x \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2648
Rubi steps
\begin{align*} \int \frac{A+B \cosh (x)}{1+\cosh (x)} \, dx &=B x-(-A+B) \int \frac{1}{1+\cosh (x)} \, dx\\ &=B x+\frac{(A-B) \sinh (x)}{1+\cosh (x)}\\ \end{align*}
Mathematica [A] time = 0.0578666, size = 23, normalized size = 1.28 \[ \frac{\sinh (x) \left (A+B x \coth \left (\frac{x}{2}\right )-B\right )}{\cosh (x)+1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 34, normalized size = 1.9 \begin{align*} A\tanh \left ({\frac{x}{2}} \right ) -B\tanh \left ({\frac{x}{2}} \right ) -B\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) +B\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 = 1.04029, size = 35, normalized size = 1.94 \begin{align*} B{\left (x - \frac{2}{e^{\left (-x\right )} + 1}\right )} + \frac{2 \, A}{e^{\left (-x\right )} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.14689, size = 96, normalized size = 5.33 \begin{align*} \frac{B x \cosh \left (x\right ) + B x \sinh \left (x\right ) + B x - 2 \, A + 2 \, B}{\cosh \left (x\right ) + \sinh \left (x\right ) + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.411026, size = 15, normalized size = 0.83 \begin{align*} A \tanh{\left (\frac{x}{2} \right )} + B x - B \tanh{\left (\frac{x}{2} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16183, size = 23, normalized size = 1.28 \begin{align*} B x - \frac{2 \,{\left (A - B\right )}}{e^{x} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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