Optimal. Leaf size=93 \[ \frac{2 \sinh (c+d x)}{35 d (\cosh (c+d x)+1)}+\frac{2 \sinh (c+d x)}{35 d (\cosh (c+d x)+1)^2}+\frac{3 \sinh (c+d x)}{35 d (\cosh (c+d x)+1)^3}+\frac{\sinh (c+d x)}{7 d (\cosh (c+d x)+1)^4} \]
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Rubi [A] time = 0.0535939, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2650, 2648} \[ \frac{2 \sinh (c+d x)}{35 d (\cosh (c+d x)+1)}+\frac{2 \sinh (c+d x)}{35 d (\cosh (c+d x)+1)^2}+\frac{3 \sinh (c+d x)}{35 d (\cosh (c+d x)+1)^3}+\frac{\sinh (c+d x)}{7 d (\cosh (c+d x)+1)^4} \]
Antiderivative was successfully verified.
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Rule 2650
Rule 2648
Rubi steps
\begin{align*} \int \frac{1}{(1+\cosh (c+d x))^4} \, dx &=\frac{\sinh (c+d x)}{7 d (1+\cosh (c+d x))^4}+\frac{3}{7} \int \frac{1}{(1+\cosh (c+d x))^3} \, dx\\ &=\frac{\sinh (c+d x)}{7 d (1+\cosh (c+d x))^4}+\frac{3 \sinh (c+d x)}{35 d (1+\cosh (c+d x))^3}+\frac{6}{35} \int \frac{1}{(1+\cosh (c+d x))^2} \, dx\\ &=\frac{\sinh (c+d x)}{7 d (1+\cosh (c+d x))^4}+\frac{3 \sinh (c+d x)}{35 d (1+\cosh (c+d x))^3}+\frac{2 \sinh (c+d x)}{35 d (1+\cosh (c+d x))^2}+\frac{2}{35} \int \frac{1}{1+\cosh (c+d x)} \, dx\\ &=\frac{\sinh (c+d x)}{7 d (1+\cosh (c+d x))^4}+\frac{3 \sinh (c+d x)}{35 d (1+\cosh (c+d x))^3}+\frac{2 \sinh (c+d x)}{35 d (1+\cosh (c+d x))^2}+\frac{2 \sinh (c+d x)}{35 d (1+\cosh (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.0770106, size = 54, normalized size = 0.58 \[ \frac{56 \sinh (c+d x)+28 \sinh (2 (c+d x))+8 \sinh (3 (c+d x))+\sinh (4 (c+d x))}{140 d (\cosh (c+d x)+1)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 56, normalized size = 0.6 \begin{align*}{\frac{1}{d} \left ( -{\frac{1}{56} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}+{\frac{3}{40} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{1}{8} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{1}{8}\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.09567, size = 491, normalized size = 5.28 \begin{align*} \frac{4 \, e^{\left (-d x - c\right )}}{5 \, d{\left (7 \, e^{\left (-d x - c\right )} + 21 \, e^{\left (-2 \, d x - 2 \, c\right )} + 35 \, e^{\left (-3 \, d x - 3 \, c\right )} + 35 \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} + 7 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-7 \, d x - 7 \, c\right )} + 1\right )}} + \frac{12 \, e^{\left (-2 \, d x - 2 \, c\right )}}{5 \, d{\left (7 \, e^{\left (-d x - c\right )} + 21 \, e^{\left (-2 \, d x - 2 \, c\right )} + 35 \, e^{\left (-3 \, d x - 3 \, c\right )} + 35 \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} + 7 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-7 \, d x - 7 \, c\right )} + 1\right )}} + \frac{4 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d{\left (7 \, e^{\left (-d x - c\right )} + 21 \, e^{\left (-2 \, d x - 2 \, c\right )} + 35 \, e^{\left (-3 \, d x - 3 \, c\right )} + 35 \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} + 7 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-7 \, d x - 7 \, c\right )} + 1\right )}} + \frac{4}{35 \, d{\left (7 \, e^{\left (-d x - c\right )} + 21 \, e^{\left (-2 \, d x - 2 \, c\right )} + 35 \, e^{\left (-3 \, d x - 3 \, c\right )} + 35 \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} + 7 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-7 \, d x - 7 \, c\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.7497, size = 984, normalized size = 10.58 \begin{align*} -\frac{4 \,{\left (35 \, \cosh \left (d x + c\right )^{2} + 10 \,{\left (7 \, \cosh \left (d x + c\right ) + 2\right )} \sinh \left (d x + c\right ) + 35 \, \sinh \left (d x + c\right )^{2} + 22 \, \cosh \left (d x + c\right ) + 7\right )}}{35 \,{\left (d \cosh \left (d x + c\right )^{6} + d \sinh \left (d x + c\right )^{6} + 7 \, d \cosh \left (d x + c\right )^{5} +{\left (6 \, d \cosh \left (d x + c\right ) + 7 \, d\right )} \sinh \left (d x + c\right )^{5} + 21 \, d \cosh \left (d x + c\right )^{4} +{\left (15 \, d \cosh \left (d x + c\right )^{2} + 35 \, d \cosh \left (d x + c\right ) + 21 \, d\right )} \sinh \left (d x + c\right )^{4} + 35 \, d \cosh \left (d x + c\right )^{3} +{\left (20 \, d \cosh \left (d x + c\right )^{3} + 70 \, d \cosh \left (d x + c\right )^{2} + 84 \, d \cosh \left (d x + c\right ) + 35 \, d\right )} \sinh \left (d x + c\right )^{3} + 35 \, d \cosh \left (d x + c\right )^{2} +{\left (15 \, d \cosh \left (d x + c\right )^{4} + 70 \, d \cosh \left (d x + c\right )^{3} + 126 \, d \cosh \left (d x + c\right )^{2} + 105 \, d \cosh \left (d x + c\right ) + 35 \, d\right )} \sinh \left (d x + c\right )^{2} + 22 \, d \cosh \left (d x + c\right ) +{\left (6 \, d \cosh \left (d x + c\right )^{5} + 35 \, d \cosh \left (d x + c\right )^{4} + 84 \, d \cosh \left (d x + c\right )^{3} + 105 \, d \cosh \left (d x + c\right )^{2} + 70 \, d \cosh \left (d x + c\right ) + 20 \, d\right )} \sinh \left (d x + c\right ) + 7 \, d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 9.11852, size = 68, normalized size = 0.73 \begin{align*} \begin{cases} - \frac{\tanh ^{7}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{56 d} + \frac{3 \tanh ^{5}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{40 d} - \frac{\tanh ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{8 d} + \frac{\tanh{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{8 d} & \text{for}\: d \neq 0 \\\frac{x}{\left (\cosh{\left (c \right )} + 1\right )^{4}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17115, size = 63, normalized size = 0.68 \begin{align*} -\frac{4 \,{\left (35 \, e^{\left (3 \, d x + 3 \, c\right )} + 21 \, e^{\left (2 \, d x + 2 \, c\right )} + 7 \, e^{\left (d x + c\right )} + 1\right )}}{35 \, d{\left (e^{\left (d x + c\right )} + 1\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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