Optimal. Leaf size=75 \[ \frac{2 (3 A+4 B) \sinh (x)}{105 (\cosh (x)+1)}+\frac{2 (3 A+4 B) \sinh (x)}{105 (\cosh (x)+1)^2}+\frac{(3 A+4 B) \sinh (x)}{35 (\cosh (x)+1)^3}+\frac{(A-B) \sinh (x)}{7 (\cosh (x)+1)^4} \]
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Rubi [A] time = 0.0594869, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2750, 2650, 2648} \[ \frac{2 (3 A+4 B) \sinh (x)}{105 (\cosh (x)+1)}+\frac{2 (3 A+4 B) \sinh (x)}{105 (\cosh (x)+1)^2}+\frac{(3 A+4 B) \sinh (x)}{35 (\cosh (x)+1)^3}+\frac{(A-B) \sinh (x)}{7 (\cosh (x)+1)^4} \]
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))^4} \, dx &=\frac{(A-B) \sinh (x)}{7 (1+\cosh (x))^4}+\frac{1}{7} (3 A+4 B) \int \frac{1}{(1+\cosh (x))^3} \, dx\\ &=\frac{(A-B) \sinh (x)}{7 (1+\cosh (x))^4}+\frac{(3 A+4 B) \sinh (x)}{35 (1+\cosh (x))^3}+\frac{1}{35} (2 (3 A+4 B)) \int \frac{1}{(1+\cosh (x))^2} \, dx\\ &=\frac{(A-B) \sinh (x)}{7 (1+\cosh (x))^4}+\frac{(3 A+4 B) \sinh (x)}{35 (1+\cosh (x))^3}+\frac{2 (3 A+4 B) \sinh (x)}{105 (1+\cosh (x))^2}+\frac{1}{105} (2 (3 A+4 B)) \int \frac{1}{1+\cosh (x)} \, dx\\ &=\frac{(A-B) \sinh (x)}{7 (1+\cosh (x))^4}+\frac{(3 A+4 B) \sinh (x)}{35 (1+\cosh (x))^3}+\frac{2 (3 A+4 B) \sinh (x)}{105 (1+\cosh (x))^2}+\frac{2 (3 A+4 B) \sinh (x)}{105 (1+\cosh (x))}\\ \end{align*}
Mathematica [A] time = 0.0918554, size = 57, normalized size = 0.76 \[ \frac{\sinh (x) (29 (3 A+4 B) \cosh (x)+8 (3 A+4 B) \cosh (2 x)+3 A \cosh (3 x)+96 A+4 B \cosh (3 x)+58 B)}{210 (\cosh (x)+1)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 55, normalized size = 0.7 \begin{align*} -{\frac{A-B}{56} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{7}}-{\frac{-3\,A+B}{40} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{5}}-{\frac{3\,A+B}{24} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}+{\frac{A}{8}\tanh \left ({\frac{x}{2}} \right ) }+{\frac{B}{8}\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 = 1.0755, size = 606, normalized size = 8.08 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.09743, size = 586, normalized size = 7.81 \begin{align*} -\frac{4 \,{\left ({\left (3 \, A + 74 \, B\right )} \cosh \left (x\right )^{2} +{\left (3 \, A + 74 \, B\right )} \sinh \left (x\right )^{2} + 14 \,{\left (9 \, A + 7 \, B\right )} \cosh \left (x\right ) - 6 \,{\left ({\left (A - 22 \, B\right )} \cosh \left (x\right ) - 14 \, A - 7 \, B\right )} \sinh \left (x\right ) + 63 \, A + 84 \, B\right )}}{105 \,{\left (\cosh \left (x\right )^{5} +{\left (5 \, \cosh \left (x\right ) + 7\right )} \sinh \left (x\right )^{4} + \sinh \left (x\right )^{5} + 7 \, \cosh \left (x\right )^{4} +{\left (10 \, \cosh \left (x\right )^{2} + 28 \, \cosh \left (x\right ) + 21\right )} \sinh \left (x\right )^{3} + 21 \, \cosh \left (x\right )^{3} +{\left (10 \, \cosh \left (x\right )^{3} + 42 \, \cosh \left (x\right )^{2} + 63 \, \cosh \left (x\right ) + 36\right )} \sinh \left (x\right )^{2} + 36 \, \cosh \left (x\right )^{2} +{\left (5 \, \cosh \left (x\right )^{4} + 28 \, \cosh \left (x\right )^{3} + 63 \, \cosh \left (x\right )^{2} + 68 \, \cosh \left (x\right ) + 28\right )} \sinh \left (x\right ) + 42 \, \cosh \left (x\right ) + 21\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.36389, size = 78, normalized size = 1.04 \begin{align*} - \frac{A \tanh ^{7}{\left (\frac{x}{2} \right )}}{56} + \frac{3 A \tanh ^{5}{\left (\frac{x}{2} \right )}}{40} - \frac{A \tanh ^{3}{\left (\frac{x}{2} \right )}}{8} + \frac{A \tanh{\left (\frac{x}{2} \right )}}{8} + \frac{B \tanh ^{7}{\left (\frac{x}{2} \right )}}{56} - \frac{B \tanh ^{5}{\left (\frac{x}{2} \right )}}{40} - \frac{B \tanh ^{3}{\left (\frac{x}{2} \right )}}{24} + \frac{B \tanh{\left (\frac{x}{2} \right )}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19734, size = 81, normalized size = 1.08 \begin{align*} -\frac{4 \,{\left (70 \, B e^{\left (4 \, x\right )} + 105 \, A e^{\left (3 \, x\right )} + 70 \, B e^{\left (3 \, x\right )} + 63 \, A e^{\left (2 \, x\right )} + 84 \, B e^{\left (2 \, x\right )} + 21 \, A e^{x} + 28 \, B e^{x} + 3 \, A + 4 \, B\right )}}{105 \,{\left (e^{x} + 1\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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