Optimal. Leaf size=27 \[ -x+\frac {1}{2} \log \left (e^{2 x}+1\right )-e^{-x} \cot ^{-1}\left (e^x\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {2194, 5208, 2282, 36, 29, 31} \[ -x+\frac {1}{2} \log \left (e^{2 x}+1\right )-e^{-x} \cot ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 2194
Rule 2282
Rule 5208
Rubi steps
\begin {align*} \int e^{-x} \cot ^{-1}\left (e^x\right ) \, dx &=-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} \operatorname {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} \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^{2 x}\right )+\frac {1}{2} \operatorname {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*}
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Mathematica [A] time = 0.02, size = 27, normalized size = 1.00 \[ -x+\frac {1}{2} \log \left (e^{2 x}+1\right )-e^{-x} \cot ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 28, normalized size = 1.04 \[ -\frac {1}{2} \, {\left (2 \, x e^{x} - e^{x} \log \left (e^{\left (2 \, x\right )} + 1\right ) + 2 \, \operatorname {arccot}\left (e^{x}\right )\right )} e^{\left (-x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.01, size = 21, normalized size = 0.78 \[ -\arctan \left (e^{\left (-x\right )}\right ) e^{\left (-x\right )} + \frac {1}{2} \, \log \left (e^{\left (-2 \, x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 25, normalized size = 0.93 \[ -\mathrm {arccot}\left ({\mathrm e}^{x}\right ) {\mathrm e}^{-x}+\frac {\ln \left ({\mathrm e}^{2 x}+1\right )}{2}-\ln \left ({\mathrm e}^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 19, normalized size = 0.70 \[ -\operatorname {arccot}\left (e^{x}\right ) e^{\left (-x\right )} + \frac {1}{2} \, \log \left (e^{\left (-2 \, x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 22, normalized size = 0.81 \[ \frac {\ln \left ({\mathrm {e}}^{2\,x}+1\right )}{2}-x-\mathrm {acot}\left ({\mathrm {e}}^x\right )\,{\mathrm {e}}^{-x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.74, size = 19, normalized size = 0.70 \[ - x + \frac {\log {\left (e^{2 x} + 1 \right )}}{2} - e^{- x} \operatorname {acot}{\left (e^{x} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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