Optimal. Leaf size=35 \[ \frac{(A+2 B) \sinh (x)}{3 (\cosh (x)+1)}+\frac{(A-B) \sinh (x)}{3 (\cosh (x)+1)^2} \]
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Rubi [A] time = 0.0375198, antiderivative size = 35, 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 = {2750, 2648} \[ \frac{(A+2 B) \sinh (x)}{3 (\cosh (x)+1)}+\frac{(A-B) \sinh (x)}{3 (\cosh (x)+1)^2} \]
Antiderivative was successfully verified.
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Rule 2750
Rule 2648
Rubi steps
\begin{align*} \int \frac{A+B \cosh (x)}{(1+\cosh (x))^2} \, dx &=\frac{(A-B) \sinh (x)}{3 (1+\cosh (x))^2}+\frac{1}{3} (A+2 B) \int \frac{1}{1+\cosh (x)} \, dx\\ &=\frac{(A-B) \sinh (x)}{3 (1+\cosh (x))^2}+\frac{(A+2 B) \sinh (x)}{3 (1+\cosh (x))}\\ \end{align*}
Mathematica [A] time = 0.0504652, size = 25, normalized size = 0.71 \[ \frac{\sinh (x) ((A+2 B) \cosh (x)+2 A+B)}{3 (\cosh (x)+1)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 34, normalized size = 1. \begin{align*} -{\frac{A}{6} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}+{\frac{B}{6} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}+{\frac{A}{2}\tanh \left ({\frac{x}{2}} \right ) }+{\frac{B}{2}\tanh \left ({\frac{x}{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.03681, size = 174, normalized size = 4.97 \begin{align*} \frac{2}{3} \, B{\left (\frac{3 \, e^{\left (-x\right )}}{3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} + 1} + \frac{3 \, e^{\left (-2 \, x\right )}}{3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} + 1} + \frac{2}{3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} + 1}\right )} + \frac{2}{3} \, A{\left (\frac{3 \, e^{\left (-x\right )}}{3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} + 1} + \frac{1}{3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} + 1}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.07598, size = 165, normalized size = 4.71 \begin{align*} -\frac{2 \,{\left ({\left (A + 5 \, B\right )} \cosh \left (x\right ) -{\left (A - B\right )} \sinh \left (x\right ) + 3 \, A + 3 \, B\right )}}{3 \,{\left (\cosh \left (x\right )^{2} + 2 \,{\left (\cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.856958, size = 36, normalized size = 1.03 \begin{align*} - \frac{A \tanh ^{3}{\left (\frac{x}{2} \right )}}{6} + \frac{A \tanh{\left (\frac{x}{2} \right )}}{2} + \frac{B \tanh ^{3}{\left (\frac{x}{2} \right )}}{6} + \frac{B \tanh{\left (\frac{x}{2} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21306, size = 41, normalized size = 1.17 \begin{align*} -\frac{2 \,{\left (3 \, B e^{\left (2 \, x\right )} + 3 \, A e^{x} + 3 \, B e^{x} + A + 2 \, B\right )}}{3 \,{\left (e^{x} + 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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