Integrand size = 11, antiderivative size = 39 \[ \int \frac {\tanh (x)}{a+b \tanh (x)} \, dx=-\frac {b x}{a^2-b^2}+\frac {a \log (a \cosh (x)+b \sinh (x))}{a^2-b^2} \]
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Time = 0.04 (sec) , antiderivative size = 39, 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 = {3612, 3611} \[ \int \frac {\tanh (x)}{a+b \tanh (x)} \, dx=\frac {a \log (a \cosh (x)+b \sinh (x))}{a^2-b^2}-\frac {b x}{a^2-b^2} \]
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Rule 3611
Rule 3612
Rubi steps \begin{align*} \text {integral}& = -\frac {b x}{a^2-b^2}+\frac {(i a) \int \frac {-i b-i a \tanh (x)}{a+b \tanh (x)} \, dx}{a^2-b^2} \\ & = -\frac {b x}{a^2-b^2}+\frac {a \log (a \cosh (x)+b \sinh (x))}{a^2-b^2} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.28 \[ \int \frac {\tanh (x)}{a+b \tanh (x)} \, dx=\frac {(-a+b) \log (1-\tanh (x))-(a+b) \log (1+\tanh (x))+2 a \log (a+b \tanh (x))}{2 (a-b) (a+b)} \]
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Time = 0.04 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.03
method | result | size |
parallelrisch | \(-\frac {a \ln \left (1-\tanh \left (x \right )\right )-a \ln \left (a +b \tanh \left (x \right )\right )+a x +b x}{a^{2}-b^{2}}\) | \(40\) |
risch | \(\frac {x}{a +b}-\frac {2 a x}{a^{2}-b^{2}}+\frac {a \ln \left ({\mathrm e}^{2 x}+\frac {a -b}{a +b}\right )}{a^{2}-b^{2}}\) | \(54\) |
derivativedivides | \(-\frac {\ln \left (\tanh \left (x \right )-1\right )}{2 a +2 b}+\frac {a \ln \left (a +b \tanh \left (x \right )\right )}{\left (a +b \right ) \left (a -b \right )}-\frac {\ln \left (1+\tanh \left (x \right )\right )}{2 a -2 b}\) | \(55\) |
default | \(-\frac {\ln \left (\tanh \left (x \right )-1\right )}{2 a +2 b}+\frac {a \ln \left (a +b \tanh \left (x \right )\right )}{\left (a +b \right ) \left (a -b \right )}-\frac {\ln \left (1+\tanh \left (x \right )\right )}{2 a -2 b}\) | \(55\) |
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none
Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.10 \[ \int \frac {\tanh (x)}{a+b \tanh (x)} \, dx=-\frac {{\left (a + b\right )} x - a \log \left (\frac {2 \, {\left (a \cosh \left (x\right ) + b \sinh \left (x\right )\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{a^{2} - b^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (29) = 58\).
Time = 0.21 (sec) , antiderivative size = 141, normalized size of antiderivative = 3.62 \[ \int \frac {\tanh (x)}{a+b \tanh (x)} \, dx=\begin {cases} \tilde {\infty } x & \text {for}\: a = 0 \wedge b = 0 \\\frac {x - \log {\left (\tanh {\left (x \right )} + 1 \right )}}{a} & \text {for}\: b = 0 \\\frac {x \tanh {\left (x \right )}}{2 b \tanh {\left (x \right )} - 2 b} - \frac {x}{2 b \tanh {\left (x \right )} - 2 b} + \frac {1}{2 b \tanh {\left (x \right )} - 2 b} & \text {for}\: a = - b \\\frac {x \tanh {\left (x \right )}}{2 b \tanh {\left (x \right )} + 2 b} + \frac {x}{2 b \tanh {\left (x \right )} + 2 b} + \frac {1}{2 b \tanh {\left (x \right )} + 2 b} & \text {for}\: a = b \\\frac {a x}{a^{2} - b^{2}} + \frac {a \log {\left (\frac {a}{b} + \tanh {\left (x \right )} \right )}}{a^{2} - b^{2}} - \frac {a \log {\left (\tanh {\left (x \right )} + 1 \right )}}{a^{2} - b^{2}} - \frac {b x}{a^{2} - b^{2}} & \text {otherwise} \end {cases} \]
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none
Time = 0.20 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.03 \[ \int \frac {\tanh (x)}{a+b \tanh (x)} \, dx=\frac {a \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{a^{2} - b^{2}} + \frac {x}{a + b} \]
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none
Time = 0.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.10 \[ \int \frac {\tanh (x)}{a+b \tanh (x)} \, dx=\frac {a \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a^{2} - b^{2}} - \frac {x}{a - b} \]
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Time = 1.73 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.92 \[ \int \frac {\tanh (x)}{a+b \tanh (x)} \, dx=-\frac {b\,x-a\,\left (x-\ln \left (\mathrm {tanh}\left (x\right )+1\right )+\ln \left (a+b\,\mathrm {tanh}\left (x\right )\right )\right )}{a^2-b^2} \]
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