Integrand size = 11, antiderivative size = 12 \[ \int \frac {\sinh ^2(x)}{(1+\cosh (x))^2} \, dx=x-\frac {2 \sinh (x)}{1+\cosh (x)} \]
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Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2759, 8} \[ \int \frac {\sinh ^2(x)}{(1+\cosh (x))^2} \, dx=x-\frac {2 \sinh (x)}{\cosh (x)+1} \]
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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*}
Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.50 \[ \int \frac {\sinh ^2(x)}{(1+\cosh (x))^2} \, dx=2 \text {arctanh}\left (\tanh \left (\frac {x}{2}\right )\right )-2 \tanh \left (\frac {x}{2}\right ) \]
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Time = 0.14 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.92
method | result | size |
risch | \(x +\frac {4}{{\mathrm e}^{x}+1}\) | \(11\) |
default | \(-2 \tanh \left (\frac {x}{2}\right )-\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )\) | \(24\) |
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none
Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.67 \[ \int \frac {\sinh ^2(x)}{(1+\cosh (x))^2} \, dx=\frac {x \cosh \left (x\right ) + x \sinh \left (x\right ) + x + 4}{\cosh \left (x\right ) + \sinh \left (x\right ) + 1} \]
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Time = 0.21 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.58 \[ \int \frac {\sinh ^2(x)}{(1+\cosh (x))^2} \, dx=x - 2 \tanh {\left (\frac {x}{2} \right )} \]
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none
Time = 0.17 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {\sinh ^2(x)}{(1+\cosh (x))^2} \, dx=x - \frac {4}{e^{\left (-x\right )} + 1} \]
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none
Time = 0.26 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {\sinh ^2(x)}{(1+\cosh (x))^2} \, dx=x + \frac {4}{e^{x} + 1} \]
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Time = 1.65 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {\sinh ^2(x)}{(1+\cosh (x))^2} \, dx=x+\frac {4}{{\mathrm {e}}^x+1} \]
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