Integrand size = 9, antiderivative size = 10 \[ \int \frac {\text {sech}(x)}{1+\tanh (x)} \, dx=-\frac {\text {sech}(x)}{1+\tanh (x)} \]
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Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3569} \[ \int \frac {\text {sech}(x)}{1+\tanh (x)} \, dx=-\frac {\text {sech}(x)}{\tanh (x)+1} \]
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Rule 3569
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {sech}(x)}{1+\tanh (x)} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.70 \[ \int \frac {\text {sech}(x)}{1+\tanh (x)} \, dx=-\cosh (x)+\sinh (x) \]
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Time = 0.18 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.70
method | result | size |
risch | \(-{\mathrm e}^{-x}\) | \(7\) |
gosper | \(-\frac {\operatorname {sech}\left (x \right )}{1+\tanh \left (x \right )}\) | \(11\) |
default | \(-\frac {2}{\tanh \left (\frac {x}{2}\right )+1}\) | \(11\) |
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none
Time = 0.24 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.90 \[ \int \frac {\text {sech}(x)}{1+\tanh (x)} \, dx=-\frac {1}{\cosh \left (x\right ) + \sinh \left (x\right )} \]
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Time = 0.16 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {\text {sech}(x)}{1+\tanh (x)} \, dx=- \frac {\operatorname {sech}{\left (x \right )}}{\tanh {\left (x \right )} + 1} \]
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none
Time = 0.19 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.60 \[ \int \frac {\text {sech}(x)}{1+\tanh (x)} \, dx=-e^{\left (-x\right )} \]
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none
Time = 0.26 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.60 \[ \int \frac {\text {sech}(x)}{1+\tanh (x)} \, dx=-e^{\left (-x\right )} \]
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Time = 1.65 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.60 \[ \int \frac {\text {sech}(x)}{1+\tanh (x)} \, dx=-{\mathrm {e}}^{-x} \]
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