Integrand size = 10, antiderivative size = 27 \[ \int e^{-x} \cot ^{-1}\left (e^x\right ) \, dx=-x-e^{-x} \cot ^{-1}\left (e^x\right )+\frac {1}{2} \log \left (1+e^{2 x}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {2225, 5316, 2320, 36, 29, 31} \[ \int e^{-x} \cot ^{-1}\left (e^x\right ) \, dx=-x+\frac {1}{2} \log \left (e^{2 x}+1\right )-e^{-x} \cot ^{-1}\left (e^x\right ) \]
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Rule 29
Rule 31
Rule 36
Rule 2225
Rule 2320
Rule 5316
Rubi steps \begin{align*} \text {integral}& = -e^{-x} \cot ^{-1}\left (e^x\right )-\int \frac {1}{1+e^{2 x}} \, dx \\ & = -e^{-x} \cot ^{-1}\left (e^x\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{x (1+x)} \, dx,x,e^{2 x}\right ) \\ & = -e^{-x} \cot ^{-1}\left (e^x\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{x} \, dx,x,e^{2 x}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,e^{2 x}\right ) \\ & = -x-e^{-x} \cot ^{-1}\left (e^x\right )+\frac {1}{2} \log \left (1+e^{2 x}\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int e^{-x} \cot ^{-1}\left (e^x\right ) \, dx=-x-e^{-x} \cot ^{-1}\left (e^x\right )+\frac {1}{2} \log \left (1+e^{2 x}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(-\operatorname {arccot}\left ({\mathrm e}^{x}\right ) {\mathrm e}^{-x}+\frac {\ln \left (1+{\mathrm e}^{2 x}\right )}{2}-\ln \left ({\mathrm e}^{x}\right )\) | \(25\) |
default | \(-\operatorname {arccot}\left ({\mathrm e}^{x}\right ) {\mathrm e}^{-x}+\frac {\ln \left (1+{\mathrm e}^{2 x}\right )}{2}-\ln \left ({\mathrm e}^{x}\right )\) | \(25\) |
parallelrisch | \(\frac {\left (\ln \left (1+{\mathrm e}^{2 x}\right ) {\mathrm e}^{x}-2 \,{\mathrm e}^{x} x -2 \,\operatorname {arccot}\left ({\mathrm e}^{x}\right )\right ) {\mathrm e}^{-x}}{2}\) | \(28\) |
risch | \(-\frac {i {\mathrm e}^{-x} \ln \left (1+i {\mathrm e}^{x}\right )}{2}+\frac {\ln \left (1+{\mathrm e}^{2 x}\right )}{2}-x +\frac {i {\mathrm e}^{-x} \ln \left (1-i {\mathrm e}^{x}\right )}{2}-\frac {{\mathrm e}^{-x} \pi }{2}\) | \(51\) |
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Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int e^{-x} \cot ^{-1}\left (e^x\right ) \, dx=-\frac {1}{2} \, {\left (2 \, x e^{x} - e^{x} \log \left (e^{\left (2 \, x\right )} + 1\right ) + 2 \, \operatorname {arccot}\left (e^{x}\right )\right )} e^{\left (-x\right )} \]
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Time = 2.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int e^{-x} \cot ^{-1}\left (e^x\right ) \, dx=- x + \frac {\log {\left (e^{2 x} + 1 \right )}}{2} - e^{- x} \operatorname {acot}{\left (e^{x} \right )} \]
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Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int e^{-x} \cot ^{-1}\left (e^x\right ) \, dx=-\operatorname {arccot}\left (e^{x}\right ) e^{\left (-x\right )} + \frac {1}{2} \, \log \left (e^{\left (-2 \, x\right )} + 1\right ) \]
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Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int e^{-x} \cot ^{-1}\left (e^x\right ) \, dx=-\arctan \left (e^{\left (-x\right )}\right ) e^{\left (-x\right )} + \frac {1}{2} \, \log \left (e^{\left (-2 \, x\right )} + 1\right ) \]
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Time = 0.14 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int e^{-x} \cot ^{-1}\left (e^x\right ) \, dx=\frac {\ln \left ({\mathrm {e}}^{2\,x}+1\right )}{2}-x-\mathrm {acot}\left ({\mathrm {e}}^x\right )\,{\mathrm {e}}^{-x} \]
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