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 = 0.04, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 2648
Rule 2650
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*}
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Mathematica [A] time = 0.06, 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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fricas [B] time = 0.50, size = 174, normalized size = 2.49 \[ -\frac {4 \, {\left (11 \, \cosh \left (d x + c\right ) + 9 \, \sinh \left (d x + c\right ) + 5\right )}}{15 \, {\left (d \cosh \left (d x + c\right )^{4} + d \sinh \left (d x + c\right )^{4} + 5 \, d \cosh \left (d x + c\right )^{3} + {\left (4 \, d \cosh \left (d x + c\right ) + 5 \, d\right )} \sinh \left (d x + c\right )^{3} + 10 \, d \cosh \left (d x + c\right )^{2} + {\left (6 \, d \cosh \left (d x + c\right )^{2} + 15 \, d \cosh \left (d x + c\right ) + 10 \, d\right )} \sinh \left (d x + c\right )^{2} + 11 \, d \cosh \left (d x + c\right ) + {\left (4 \, d \cosh \left (d x + c\right )^{3} + 15 \, d \cosh \left (d x + c\right )^{2} + 20 \, d \cosh \left (d x + c\right ) + 9 \, d\right )} \sinh \left (d x + c\right ) + 5 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 36, normalized size = 0.51 \[ -\frac {4 \, {\left (10 \, e^{\left (2 \, d x + 2 \, c\right )} + 5 \, e^{\left (d x + c\right )} + 1\right )}}{15 \, d {\left (e^{\left (d x + c\right )} + 1\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 43, normalized size = 0.61 \[ \frac {\frac {\left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20}-\frac {\left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}+\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 205, normalized size = 2.93 \[ \frac {4 \, e^{\left (-d x - c\right )}}{3 \, d {\left (5 \, e^{\left (-d x - c\right )} + 10 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-3 \, d x - 3 \, c\right )} + 5 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-5 \, d x - 5 \, c\right )} + 1\right )}} + \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{3 \, d {\left (5 \, e^{\left (-d x - c\right )} + 10 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-3 \, d x - 3 \, c\right )} + 5 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-5 \, d x - 5 \, c\right )} + 1\right )}} + \frac {4}{15 \, d {\left (5 \, e^{\left (-d x - c\right )} + 10 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-3 \, d x - 3 \, c\right )} + 5 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-5 \, d x - 5 \, c\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.93, size = 36, normalized size = 0.51 \[ -\frac {4\,\left (5\,{\mathrm {e}}^{c+d\,x}+10\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}{15\,d\,{\left ({\mathrm {e}}^{c+d\,x}+1\right )}^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.17, size = 51, normalized size = 0.73 \[ \begin {cases} \frac {\tanh ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{20 d} - \frac {\tanh ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 d} + \frac {\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{4 d} & \text {for}\: d \neq 0 \\\frac {x}{\left (\cosh {\relax (c )} + 1\right )^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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