Optimal. Leaf size=27 \[ -x+\frac{1}{2} \log \left (e^{2 x}+1\right )-e^{-x} \cot ^{-1}\left (e^x\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0203293, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, 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.
[In]
[Out]
Rule 2194
Rule 5208
Rule 2282
Rule 36
Rule 29
Rule 31
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*}
Mathematica [A] time = 0.0208115, size = 27, normalized size = 1. \[ -x+\frac{1}{2} \log \left (e^{2 x}+1\right )-e^{-x} \cot ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.027, size = 25, normalized size = 0.9 \begin{align*} -{\frac{{\rm arccot} \left ({{\rm e}^{x}}\right )}{{{\rm e}^{x}}}}+{\frac{\ln \left ( \left ({{\rm e}^{x}} \right ) ^{2}+1 \right ) }{2}}-\ln \left ({{\rm e}^{x}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.986912, size = 26, normalized size = 0.96 \begin{align*} -\operatorname{arccot}\left (e^{x}\right ) e^{\left (-x\right )} + \frac{1}{2} \, \log \left (e^{\left (-2 \, x\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.40037, size = 84, normalized size = 3.11 \begin{align*} -\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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 12.8936, size = 19, normalized size = 0.7 \begin{align*} - x + \frac{\log{\left (e^{2 x} + 1 \right )}}{2} - e^{- x} \operatorname{acot}{\left (e^{x} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.09532, size = 28, normalized size = 1.04 \begin{align*} -\arctan \left (e^{\left (-x\right )}\right ) e^{\left (-x\right )} + \frac{1}{2} \, \log \left (e^{\left (-2 \, x\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]