Integrand size = 13, antiderivative size = 14 \[ \int \frac {\sinh ^2(x)}{(1-\cosh (x))^2} \, dx=x+\frac {2 \sinh (x)}{1-\cosh (x)} \]
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Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2759, 8} \[ \int \frac {\sinh ^2(x)}{(1-\cosh (x))^2} \, dx=x+\frac {2 \sinh (x)}{1-\cosh (x)} \]
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Rule 8
Rule 2759
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sinh (x)}{1-\cosh (x)}+\int 1 \, dx \\ & = x+\frac {2 \sinh (x)}{1-\cosh (x)} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.71 \[ \int \frac {\sinh ^2(x)}{(1-\cosh (x))^2} \, dx=-2 \coth \left (\frac {x}{2}\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\tanh ^2\left (\frac {x}{2}\right )\right ) \]
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Time = 0.16 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.79
method | result | size |
risch | \(x -\frac {4}{{\mathrm e}^{x}-1}\) | \(11\) |
default | \(-\frac {2}{\tanh \left (\frac {x}{2}\right )}-\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )\) | \(26\) |
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none
Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.57 \[ \int \frac {\sinh ^2(x)}{(1-\cosh (x))^2} \, dx=\frac {x \cosh \left (x\right ) + x \sinh \left (x\right ) - x - 4}{\cosh \left (x\right ) + \sinh \left (x\right ) - 1} \]
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Time = 0.35 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.50 \[ \int \frac {\sinh ^2(x)}{(1-\cosh (x))^2} \, dx=x - \frac {2}{\tanh {\left (\frac {x}{2} \right )}} \]
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none
Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {\sinh ^2(x)}{(1-\cosh (x))^2} \, dx=x + \frac {4}{e^{\left (-x\right )} - 1} \]
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none
Time = 0.25 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \frac {\sinh ^2(x)}{(1-\cosh (x))^2} \, dx=x - \frac {4}{e^{x} - 1} \]
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Time = 0.05 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \frac {\sinh ^2(x)}{(1-\cosh (x))^2} \, dx=x-\frac {4}{{\mathrm {e}}^x-1} \]
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