Optimal. Leaf size=70 \[ \frac{2 \sinh (c+d x)}{15 d (\cosh (c+d x)+1)}+\frac{2 \sinh (c+d x)}{15 d (\cosh (c+d x)+1)^2}+\frac{\sinh (c+d x)}{5 d (\cosh (c+d x)+1)^3} \]
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Rubi [A] time = 0.036479, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2650, 2648} \[ \frac{2 \sinh (c+d x)}{15 d (\cosh (c+d x)+1)}+\frac{2 \sinh (c+d x)}{15 d (\cosh (c+d x)+1)^2}+\frac{\sinh (c+d x)}{5 d (\cosh (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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Rule 2650
Rule 2648
Rubi steps
\begin{align*} \int \frac{1}{(1+\cosh (c+d x))^3} \, dx &=\frac{\sinh (c+d x)}{5 d (1+\cosh (c+d x))^3}+\frac{2}{5} \int \frac{1}{(1+\cosh (c+d x))^2} \, dx\\ &=\frac{\sinh (c+d x)}{5 d (1+\cosh (c+d x))^3}+\frac{2 \sinh (c+d x)}{15 d (1+\cosh (c+d x))^2}+\frac{2}{15} \int \frac{1}{1+\cosh (c+d x)} \, dx\\ &=\frac{\sinh (c+d x)}{5 d (1+\cosh (c+d x))^3}+\frac{2 \sinh (c+d x)}{15 d (1+\cosh (c+d x))^2}+\frac{2 \sinh (c+d x)}{15 d (1+\cosh (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.055126, size = 44, normalized size = 0.63 \[ \frac{15 \sinh (c+d x)+6 \sinh (2 (c+d x))+\sinh (3 (c+d x))}{30 d (\cosh (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 43, normalized size = 0.6 \begin{align*}{\frac{1}{d} \left ({\frac{1}{20} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{1}{6} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{1}{4}\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 = 1.05483, size = 277, normalized size = 3.96 \begin{align*} \frac{4 \, e^{\left (-d x - c\right )}}{3 \, d{\left (5 \, e^{\left (-d x - c\right )} + 10 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-3 \, d x - 3 \, c\right )} + 5 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-5 \, d x - 5 \, c\right )} + 1\right )}} + \frac{8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{3 \, d{\left (5 \, e^{\left (-d x - c\right )} + 10 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-3 \, d x - 3 \, c\right )} + 5 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-5 \, d x - 5 \, c\right )} + 1\right )}} + \frac{4}{15 \, d{\left (5 \, e^{\left (-d x - c\right )} + 10 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-3 \, d x - 3 \, c\right )} + 5 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-5 \, d x - 5 \, c\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.79728, size = 486, normalized size = 6.94 \begin{align*} -\frac{4 \,{\left (11 \, \cosh \left (d x + c\right ) + 9 \, \sinh \left (d x + c\right ) + 5\right )}}{15 \,{\left (d \cosh \left (d x + c\right )^{4} + d \sinh \left (d x + c\right )^{4} + 5 \, d \cosh \left (d x + c\right )^{3} +{\left (4 \, d \cosh \left (d x + c\right ) + 5 \, d\right )} \sinh \left (d x + c\right )^{3} + 10 \, d \cosh \left (d x + c\right )^{2} +{\left (6 \, d \cosh \left (d x + c\right )^{2} + 15 \, d \cosh \left (d x + c\right ) + 10 \, d\right )} \sinh \left (d x + c\right )^{2} + 11 \, d \cosh \left (d x + c\right ) +{\left (4 \, d \cosh \left (d x + c\right )^{3} + 15 \, d \cosh \left (d x + c\right )^{2} + 20 \, d \cosh \left (d x + c\right ) + 9 \, d\right )} \sinh \left (d x + c\right ) + 5 \, d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.49056, size = 51, normalized size = 0.73 \begin{align*} \begin{cases} \frac{\tanh ^{5}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{20 d} - \frac{\tanh ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{6 d} + \frac{\tanh{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{4 d} & \text{for}\: d \neq 0 \\\frac{x}{\left (\cosh{\left (c \right )} + 1\right )^{3}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14721, size = 49, normalized size = 0.7 \begin{align*} -\frac{4 \,{\left (10 \, e^{\left (2 \, d x + 2 \, c\right )} + 5 \, e^{\left (d x + c\right )} + 1\right )}}{15 \, d{\left (e^{\left (d x + c\right )} + 1\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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