Integrand size = 13, antiderivative size = 14 \[ \int \frac {\tanh ^2(x)}{a+a \text {sech}(x)} \, dx=\frac {x}{a}-\frac {\arctan (\sinh (x))}{a} \]
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Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3973, 3855} \[ \int \frac {\tanh ^2(x)}{a+a \text {sech}(x)} \, dx=\frac {x}{a}-\frac {\arctan (\sinh (x))}{a} \]
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Rule 3855
Rule 3973
Rubi steps \begin{align*} \text {integral}& = -\frac {\int (-a+a \text {sech}(x)) \, dx}{a^2} \\ & = \frac {x}{a}-\frac {\int \text {sech}(x) \, dx}{a} \\ & = \frac {x}{a}-\frac {\arctan (\sinh (x))}{a} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07 \[ \int \frac {\tanh ^2(x)}{a+a \text {sech}(x)} \, dx=\frac {x-2 \arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{a} \]
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Result contains complex when optimal does not.
Time = 0.17 (sec) , antiderivative size = 31, normalized size of antiderivative = 2.21
method | result | size |
risch | \(\frac {x}{a}+\frac {i \ln \left ({\mathrm e}^{x}-i\right )}{a}-\frac {i \ln \left ({\mathrm e}^{x}+i\right )}{a}\) | \(31\) |
default | \(\frac {-\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )-2 \arctan \left (\tanh \left (\frac {x}{2}\right )\right )+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a}\) | \(32\) |
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none
Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {\tanh ^2(x)}{a+a \text {sech}(x)} \, dx=\frac {x - 2 \, \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}{a} \]
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\[ \int \frac {\tanh ^2(x)}{a+a \text {sech}(x)} \, dx=\frac {\int \frac {\tanh ^{2}{\left (x \right )}}{\operatorname {sech}{\left (x \right )} + 1}\, dx}{a} \]
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none
Time = 0.28 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {\tanh ^2(x)}{a+a \text {sech}(x)} \, dx=\frac {x}{a} + \frac {2 \, \arctan \left (e^{\left (-x\right )}\right )}{a} \]
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none
Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {\tanh ^2(x)}{a+a \text {sech}(x)} \, dx=\frac {x}{a} - \frac {2 \, \arctan \left (e^{x}\right )}{a} \]
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Time = 2.11 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.79 \[ \int \frac {\tanh ^2(x)}{a+a \text {sech}(x)} \, dx=\frac {x}{a}-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {a^2}}{a}\right )}{\sqrt {a^2}} \]
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