Integrand size = 13, antiderivative size = 15 \[ \int \frac {\tanh ^2(x)}{a+a \cosh (x)} \, dx=\frac {\arctan (\sinh (x))}{a}-\frac {\tanh (x)}{a} \]
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Time = 0.04 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2785, 3852, 8, 3855} \[ \int \frac {\tanh ^2(x)}{a+a \cosh (x)} \, dx=\frac {\arctan (\sinh (x))}{a}-\frac {\tanh (x)}{a} \]
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Rule 8
Rule 2785
Rule 3852
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\int \text {sech}(x) \, dx}{a}-\frac {\int \text {sech}^2(x) \, dx}{a} \\ & = \frac {\arctan (\sinh (x))}{a}-\frac {i \text {Subst}(\int 1 \, dx,x,-i \tanh (x))}{a} \\ & = \frac {\arctan (\sinh (x))}{a}-\frac {\tanh (x)}{a} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.20 \[ \int \frac {\tanh ^2(x)}{a+a \cosh (x)} \, dx=\frac {2 \arctan \left (\tanh \left (\frac {x}{2}\right )\right )-\tanh (x)}{a} \]
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Time = 0.13 (sec) , antiderivative size = 30, normalized size of antiderivative = 2.00
method | result | size |
default | \(\frac {-\frac {2 \tanh \left (\frac {x}{2}\right )}{1+\tanh \left (\frac {x}{2}\right )^{2}}+2 \arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{a}\) | \(30\) |
risch | \(\frac {2}{a \left (1+{\mathrm e}^{2 x}\right )}+\frac {i \ln \left ({\mathrm e}^{x}+i\right )}{a}-\frac {i \ln \left ({\mathrm e}^{x}-i\right )}{a}\) | \(39\) |
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Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (15) = 30\).
Time = 0.25 (sec) , antiderivative size = 50, normalized size of antiderivative = 3.33 \[ \int \frac {\tanh ^2(x)}{a+a \cosh (x)} \, dx=\frac {2 \, {\left ({\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + 1\right )}}{a \cosh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) \sinh \left (x\right ) + a \sinh \left (x\right )^{2} + a} \]
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\[ \int \frac {\tanh ^2(x)}{a+a \cosh (x)} \, dx=\frac {\int \frac {\tanh ^{2}{\left (x \right )}}{\cosh {\left (x \right )} + 1}\, dx}{a} \]
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none
Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.53 \[ \int \frac {\tanh ^2(x)}{a+a \cosh (x)} \, dx=-\frac {2 \, \arctan \left (e^{\left (-x\right )}\right )}{a} - \frac {2}{a e^{\left (-2 \, x\right )} + a} \]
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none
Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.47 \[ \int \frac {\tanh ^2(x)}{a+a \cosh (x)} \, dx=\frac {2 \, \arctan \left (e^{x}\right )}{a} + \frac {2}{a {\left (e^{\left (2 \, x\right )} + 1\right )}} \]
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Time = 1.65 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.20 \[ \int \frac {\tanh ^2(x)}{a+a \cosh (x)} \, dx=\frac {2}{a\,\left ({\mathrm {e}}^{2\,x}+1\right )}+\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {a^2}}{a}\right )}{\sqrt {a^2}} \]
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