Integrand size = 11, antiderivative size = 14 \[ \int \frac {\sinh ^2(x)}{(1+\cosh (x))^3} \, dx=\frac {\sinh ^3(x)}{3 (1+\cosh (x))^3} \]
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Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2750} \[ \int \frac {\sinh ^2(x)}{(1+\cosh (x))^3} \, dx=\frac {\sinh ^3(x)}{3 (\cosh (x)+1)^3} \]
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Rule 2750
Rubi steps \begin{align*} \text {integral}& = \frac {\sinh ^3(x)}{3 (1+\cosh (x))^3} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {\sinh ^2(x)}{(1+\cosh (x))^3} \, dx=\frac {1}{3} \tanh ^3\left (\frac {x}{2}\right ) \]
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Time = 0.31 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.64
method | result | size |
default | \(\frac {\tanh \left (\frac {x}{2}\right )^{3}}{3}\) | \(9\) |
risch | \(-\frac {2 \left (3 \,{\mathrm e}^{2 x}+1\right )}{3 \left ({\mathrm e}^{x}+1\right )^{3}}\) | \(17\) |
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Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (12) = 24\).
Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.36 \[ \int \frac {\sinh ^2(x)}{(1+\cosh (x))^3} \, dx=-\frac {4 \, {\left (2 \, \cosh \left (x\right ) + \sinh \left (x\right )\right )}}{3 \, {\left (\cosh \left (x\right )^{2} + 2 \, {\left (\cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) + 3\right )}} \]
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Time = 0.38 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.50 \[ \int \frac {\sinh ^2(x)}{(1+\cosh (x))^3} \, dx=\frac {\tanh ^{3}{\left (\frac {x}{2} \right )}}{3} \]
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Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (12) = 24\).
Time = 0.19 (sec) , antiderivative size = 49, normalized size of antiderivative = 3.50 \[ \int \frac {\sinh ^2(x)}{(1+\cosh (x))^3} \, dx=\frac {2 \, e^{\left (-2 \, x\right )}}{3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} + 1} + \frac {2}{3 \, {\left (3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} + 1\right )}} \]
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none
Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {\sinh ^2(x)}{(1+\cosh (x))^3} \, dx=-\frac {2 \, {\left (3 \, e^{\left (2 \, x\right )} + 1\right )}}{3 \, {\left (e^{x} + 1\right )}^{3}} \]
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Time = 0.09 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {\sinh ^2(x)}{(1+\cosh (x))^3} \, dx=-\frac {2\,\left (3\,{\mathrm {e}}^{2\,x}+1\right )}{3\,{\left ({\mathrm {e}}^x+1\right )}^3} \]
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