Integrand size = 11, antiderivative size = 9 \[ \int \frac {\tanh (x)}{a+a \text {sech}(x)} \, dx=\frac {\log (1+\cosh (x))}{a} \]
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Time = 0.02 (sec) , antiderivative size = 9, 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 = {3964, 31} \[ \int \frac {\tanh (x)}{a+a \text {sech}(x)} \, dx=\frac {\log (\cosh (x)+1)}{a} \]
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Rule 31
Rule 3964
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{a+a x} \, dx,x,\cosh (x)\right ) \\ & = \frac {\log (1+\cosh (x))}{a} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.33 \[ \int \frac {\tanh (x)}{a+a \text {sech}(x)} \, dx=\frac {2 \log \left (\cosh \left (\frac {x}{2}\right )\right )}{a} \]
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Time = 0.14 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.89
method | result | size |
derivativedivides | \(-\frac {-\ln \left (1+\operatorname {sech}\left (x \right )\right )+\ln \left (\operatorname {sech}\left (x \right )\right )}{a}\) | \(17\) |
default | \(-\frac {-\ln \left (1+\operatorname {sech}\left (x \right )\right )+\ln \left (\operatorname {sech}\left (x \right )\right )}{a}\) | \(17\) |
risch | \(-\frac {x}{a}+\frac {2 \ln \left ({\mathrm e}^{x}+1\right )}{a}\) | \(18\) |
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none
Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.78 \[ \int \frac {\tanh (x)}{a+a \text {sech}(x)} \, dx=-\frac {x - 2 \, \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right )}{a} \]
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Leaf count of result is larger than twice the leaf count of optimal. 19 vs. \(2 (7) = 14\).
Time = 0.08 (sec) , antiderivative size = 19, normalized size of antiderivative = 2.11 \[ \int \frac {\tanh (x)}{a+a \text {sech}(x)} \, dx=\frac {x}{a} - \frac {\log {\left (\tanh {\left (x \right )} + 1 \right )}}{a} + \frac {\log {\left (\operatorname {sech}{\left (x \right )} + 1 \right )}}{a} \]
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none
Time = 0.20 (sec) , antiderivative size = 18, normalized size of antiderivative = 2.00 \[ \int \frac {\tanh (x)}{a+a \text {sech}(x)} \, dx=\frac {x}{a} + \frac {2 \, \log \left (e^{\left (-x\right )} + 1\right )}{a} \]
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none
Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.89 \[ \int \frac {\tanh (x)}{a+a \text {sech}(x)} \, dx=-\frac {x}{a} + \frac {2 \, \log \left (e^{x} + 1\right )}{a} \]
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Time = 0.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.56 \[ \int \frac {\tanh (x)}{a+a \text {sech}(x)} \, dx=-\frac {x-2\,\ln \left ({\mathrm {e}}^x+1\right )}{a} \]
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