Integrand size = 13, antiderivative size = 27 \[ \int \frac {\sinh ^2(x)}{a+a \text {sech}(x)} \, dx=\frac {x}{2 a}-\frac {\sinh (x)}{a}+\frac {\cosh (x) \sinh (x)}{2 a} \]
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Time = 0.07 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3957, 2918, 2717, 2715, 8} \[ \int \frac {\sinh ^2(x)}{a+a \text {sech}(x)} \, dx=\frac {x}{2 a}-\frac {\sinh (x)}{a}+\frac {\sinh (x) \cosh (x)}{2 a} \]
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Rule 8
Rule 2715
Rule 2717
Rule 2918
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int \frac {\cosh (x) \sinh ^2(x)}{-a-a \cosh (x)} \, dx \\ & = -\frac {\int \cosh (x) \, dx}{a}+\frac {\int \cosh ^2(x) \, dx}{a} \\ & = -\frac {\sinh (x)}{a}+\frac {\cosh (x) \sinh (x)}{2 a}+\frac {\int 1 \, dx}{2 a} \\ & = \frac {x}{2 a}-\frac {\sinh (x)}{a}+\frac {\cosh (x) \sinh (x)}{2 a} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.59 \[ \int \frac {\sinh ^2(x)}{a+a \text {sech}(x)} \, dx=\frac {x+(-2+\cosh (x)) \sinh (x)}{2 a} \]
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Time = 0.82 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56
method | result | size |
risch | \(\frac {x}{2 a}+\frac {{\mathrm e}^{2 x}}{8 a}-\frac {{\mathrm e}^{x}}{2 a}+\frac {{\mathrm e}^{-x}}{2 a}-\frac {{\mathrm e}^{-2 x}}{8 a}\) | \(42\) |
default | \(\frac {-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {3}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2}+\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {3}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}}{a}\) | \(65\) |
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none
Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.52 \[ \int \frac {\sinh ^2(x)}{a+a \text {sech}(x)} \, dx=\frac {{\left (\cosh \left (x\right ) - 2\right )} \sinh \left (x\right ) + x}{2 \, a} \]
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\[ \int \frac {\sinh ^2(x)}{a+a \text {sech}(x)} \, dx=\frac {\int \frac {\sinh ^{2}{\left (x \right )}}{\operatorname {sech}{\left (x \right )} + 1}\, dx}{a} \]
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none
Time = 0.20 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \[ \int \frac {\sinh ^2(x)}{a+a \text {sech}(x)} \, dx=-\frac {{\left (4 \, e^{\left (-x\right )} - 1\right )} e^{\left (2 \, x\right )}}{8 \, a} + \frac {x}{2 \, a} + \frac {4 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )}}{8 \, a} \]
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none
Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \frac {\sinh ^2(x)}{a+a \text {sech}(x)} \, dx=\frac {{\left (4 \, e^{x} - 1\right )} e^{\left (-2 \, x\right )} + 4 \, x + e^{\left (2 \, x\right )} - 4 \, e^{x}}{8 \, a} \]
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Time = 1.97 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52 \[ \int \frac {\sinh ^2(x)}{a+a \text {sech}(x)} \, dx=\frac {{\mathrm {e}}^{-x}}{2\,a}-\frac {{\mathrm {e}}^{-2\,x}}{8\,a}+\frac {{\mathrm {e}}^{2\,x}}{8\,a}+\frac {x}{2\,a}-\frac {{\mathrm {e}}^x}{2\,a} \]
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