Integrand size = 8, antiderivative size = 12 \[ \int e^{-x} \tanh (x) \, dx=e^{-x}+2 \arctan \left (e^x\right ) \]
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Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2320, 464, 209} \[ \int e^{-x} \tanh (x) \, dx=2 \arctan \left (e^x\right )+e^{-x} \]
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Rule 209
Rule 464
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {-1+x^2}{x^2 \left (1+x^2\right )} \, dx,x,e^x\right ) \\ & = e^{-x}+2 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,e^x\right ) \\ & = e^{-x}+2 \arctan \left (e^x\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int e^{-x} \tanh (x) \, dx=e^{-x}+2 \arctan \left (e^x\right ) \]
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Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.58
method | result | size |
default | \(\frac {2}{\tanh \left (\frac {x}{2}\right )+1}+2 \arctan \left (\tanh \left (\frac {x}{2}\right )\right )\) | \(19\) |
risch | \({\mathrm e}^{-x}+i \ln \left ({\mathrm e}^{x}+i\right )-i \ln \left ({\mathrm e}^{x}-i\right )\) | \(24\) |
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Leaf count of result is larger than twice the leaf count of optimal. 23 vs. \(2 (10) = 20\).
Time = 0.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.92 \[ \int e^{-x} \tanh (x) \, dx=\frac {2 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + 1}{\cosh \left (x\right ) + \sinh \left (x\right )} \]
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\[ \int e^{-x} \tanh (x) \, dx=\int e^{- x} \tanh {\left (x \right )}\, dx \]
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none
Time = 0.29 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int e^{-x} \tanh (x) \, dx=-2 \, \arctan \left (e^{\left (-x\right )}\right ) + e^{\left (-x\right )} \]
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none
Time = 0.29 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int e^{-x} \tanh (x) \, dx=2 \, \arctan \left (e^{x}\right ) + e^{\left (-x\right )} \]
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Time = 0.05 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int e^{-x} \tanh (x) \, dx={\mathrm {e}}^{-x}-2\,\mathrm {atan}\left ({\mathrm {e}}^{-x}\right ) \]
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