Integrand size = 11, antiderivative size = 18 \[ \int \frac {\tanh (x)}{a+a \cosh (x)} \, dx=\frac {\log (\cosh (x))}{a}-\frac {\log (1+\cosh (x))}{a} \]
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Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2786, 36, 29, 31} \[ \int \frac {\tanh (x)}{a+a \cosh (x)} \, dx=\frac {\log (\cosh (x))}{a}-\frac {\log (\cosh (x)+1)}{a} \]
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Rule 29
Rule 31
Rule 36
Rule 2786
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{x (a+x)} \, dx,x,a \cosh (x)\right ) \\ & = \frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,a \cosh (x)\right )}{a}-\frac {\text {Subst}\left (\int \frac {1}{a+x} \, dx,x,a \cosh (x)\right )}{a} \\ & = \frac {\log (\cosh (x))}{a}-\frac {\log (1+\cosh (x))}{a} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.67 \[ \int \frac {\tanh (x)}{a+a \cosh (x)} \, dx=-\frac {2 \text {arctanh}(1+2 \cosh (x))}{a} \]
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Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78
method | result | size |
default | \(\frac {\ln \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )}{a}\) | \(14\) |
risch | \(-\frac {2 \ln \left ({\mathrm e}^{x}+1\right )}{a}+\frac {\ln \left (1+{\mathrm e}^{2 x}\right )}{a}\) | \(23\) |
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none
Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.56 \[ \int \frac {\tanh (x)}{a+a \cosh (x)} \, dx=\frac {\log \left (\frac {2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) - 2 \, \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right )}{a} \]
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\[ \int \frac {\tanh (x)}{a+a \cosh (x)} \, dx=\frac {\int \frac {\tanh {\left (x \right )}}{\cosh {\left (x \right )} + 1}\, dx}{a} \]
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none
Time = 0.20 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.33 \[ \int \frac {\tanh (x)}{a+a \cosh (x)} \, dx=-\frac {2 \, \log \left (e^{\left (-x\right )} + 1\right )}{a} + \frac {\log \left (e^{\left (-2 \, x\right )} + 1\right )}{a} \]
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none
Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {\tanh (x)}{a+a \cosh (x)} \, dx=\frac {\log \left (e^{\left (2 \, x\right )} + 1\right )}{a} - \frac {2 \, \log \left (e^{x} + 1\right )}{a} \]
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Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.44 \[ \int \frac {\tanh (x)}{a+a \cosh (x)} \, dx=-\frac {2\,\ln \left (36\,{\mathrm {e}}^x+36\right )-\ln \left (3\,{\mathrm {e}}^{2\,x}+3\right )}{a} \]
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