Optimal. Leaf size=56 \[ \frac{(2 A+3 B) \sinh (x)}{15 (\cosh (x)+1)}+\frac{(2 A+3 B) \sinh (x)}{15 (\cosh (x)+1)^2}+\frac{(A-B) \sinh (x)}{5 (\cosh (x)+1)^3} \]
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Rubi [A] time = 0.0482171, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2750, 2650, 2648} \[ \frac{(2 A+3 B) \sinh (x)}{15 (\cosh (x)+1)}+\frac{(2 A+3 B) \sinh (x)}{15 (\cosh (x)+1)^2}+\frac{(A-B) \sinh (x)}{5 (\cosh (x)+1)^3} \]
Antiderivative was successfully verified.
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Rule 2750
Rule 2650
Rule 2648
Rubi steps
\begin{align*} \int \frac{A+B \cosh (x)}{(1+\cosh (x))^3} \, dx &=\frac{(A-B) \sinh (x)}{5 (1+\cosh (x))^3}+\frac{1}{5} (2 A+3 B) \int \frac{1}{(1+\cosh (x))^2} \, dx\\ &=\frac{(A-B) \sinh (x)}{5 (1+\cosh (x))^3}+\frac{(2 A+3 B) \sinh (x)}{15 (1+\cosh (x))^2}+\frac{1}{15} (2 A+3 B) \int \frac{1}{1+\cosh (x)} \, dx\\ &=\frac{(A-B) \sinh (x)}{5 (1+\cosh (x))^3}+\frac{(2 A+3 B) \sinh (x)}{15 (1+\cosh (x))^2}+\frac{(2 A+3 B) \sinh (x)}{15 (1+\cosh (x))}\\ \end{align*}
Mathematica [A] time = 0.0758902, size = 42, normalized size = 0.75 \[ \frac{\sinh (x) (6 (2 A+3 B) \cosh (x)+(2 A+3 B) \cosh (2 x)+16 A+9 B)}{30 (\cosh (x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 38, normalized size = 0.7 \begin{align*}{\frac{A-B}{20} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{5}}-{\frac{A}{6} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}+{\frac{A}{4}\tanh \left ({\frac{x}{2}} \right ) }+{\frac{B}{4}\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 = 1.04365, size = 355, normalized size = 6.34 \begin{align*} \frac{4}{15} \, A{\left (\frac{5 \, e^{\left (-x\right )}}{5 \, e^{\left (-x\right )} + 10 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-3 \, x\right )} + 5 \, e^{\left (-4 \, x\right )} + e^{\left (-5 \, x\right )} + 1} + \frac{10 \, e^{\left (-2 \, x\right )}}{5 \, e^{\left (-x\right )} + 10 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-3 \, x\right )} + 5 \, e^{\left (-4 \, x\right )} + e^{\left (-5 \, x\right )} + 1} + \frac{1}{5 \, e^{\left (-x\right )} + 10 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-3 \, x\right )} + 5 \, e^{\left (-4 \, x\right )} + e^{\left (-5 \, x\right )} + 1}\right )} + \frac{2}{5} \, B{\left (\frac{5 \, e^{\left (-x\right )}}{5 \, e^{\left (-x\right )} + 10 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-3 \, x\right )} + 5 \, e^{\left (-4 \, x\right )} + e^{\left (-5 \, x\right )} + 1} + \frac{5 \, e^{\left (-2 \, x\right )}}{5 \, e^{\left (-x\right )} + 10 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-3 \, x\right )} + 5 \, e^{\left (-4 \, x\right )} + e^{\left (-5 \, x\right )} + 1} + \frac{5 \, e^{\left (-3 \, x\right )}}{5 \, e^{\left (-x\right )} + 10 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-3 \, x\right )} + 5 \, e^{\left (-4 \, x\right )} + e^{\left (-5 \, x\right )} + 1} + \frac{1}{5 \, e^{\left (-x\right )} + 10 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-3 \, x\right )} + 5 \, e^{\left (-4 \, x\right )} + e^{\left (-5 \, x\right )} + 1}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.10774, size = 427, normalized size = 7.62 \begin{align*} -\frac{2 \,{\left (15 \, B \cosh \left (x\right )^{2} + 15 \, B \sinh \left (x\right )^{2} + 2 \,{\left (11 \, A + 9 \, B\right )} \cosh \left (x\right ) + 6 \,{\left (5 \, B \cosh \left (x\right ) + 3 \, A + 2 \, B\right )} \sinh \left (x\right ) + 10 \, A + 15 \, B\right )}}{15 \,{\left (\cosh \left (x\right )^{4} +{\left (4 \, \cosh \left (x\right ) + 5\right )} \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 5 \, \cosh \left (x\right )^{3} +{\left (6 \, \cosh \left (x\right )^{2} + 15 \, \cosh \left (x\right ) + 10\right )} \sinh \left (x\right )^{2} + 10 \, \cosh \left (x\right )^{2} +{\left (4 \, \cosh \left (x\right )^{3} + 15 \, \cosh \left (x\right )^{2} + 20 \, \cosh \left (x\right ) + 9\right )} \sinh \left (x\right ) + 11 \, \cosh \left (x\right ) + 5\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.81946, size = 46, normalized size = 0.82 \begin{align*} \frac{A \tanh ^{5}{\left (\frac{x}{2} \right )}}{20} - \frac{A \tanh ^{3}{\left (\frac{x}{2} \right )}}{6} + \frac{A \tanh{\left (\frac{x}{2} \right )}}{4} - \frac{B \tanh ^{5}{\left (\frac{x}{2} \right )}}{20} + \frac{B \tanh{\left (\frac{x}{2} \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17739, size = 62, normalized size = 1.11 \begin{align*} -\frac{2 \,{\left (15 \, B e^{\left (3 \, x\right )} + 20 \, A e^{\left (2 \, x\right )} + 15 \, B e^{\left (2 \, x\right )} + 10 \, A e^{x} + 15 \, B e^{x} + 2 \, A + 3 \, B\right )}}{15 \,{\left (e^{x} + 1\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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