Optimal. Leaf size=17 \[ x+\frac{1}{e^x+1}-\log \left (e^x+1\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0147249, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2282, 44} \[ x+\frac{1}{e^x+1}-\log \left (e^x+1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2282
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{1+2 e^x+e^{2 x}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{x (1+x)^2} \, dx,x,e^x\right )\\ &=\operatorname{Subst}\left (\int \left (\frac{1}{-1-x}+\frac{1}{x}-\frac{1}{(1+x)^2}\right ) \, dx,x,e^x\right )\\ &=\frac{1}{1+e^x}+x-\log \left (1+e^x\right )\\ \end{align*}
Mathematica [A] time = 0.0141032, size = 17, normalized size = 1. \[ x+\frac{1}{e^x+1}-\log \left (e^x+1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.008, size = 18, normalized size = 1.1 \begin{align*} \ln \left ({{\rm e}^{x}} \right ) + \left ( 1+{{\rm e}^{x}} \right ) ^{-1}-\ln \left ( 1+{{\rm e}^{x}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.977773, size = 20, normalized size = 1.18 \begin{align*} x + \frac{1}{e^{x} + 1} - \log \left (e^{x} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.55972, size = 70, normalized size = 4.12 \begin{align*} \frac{x e^{x} -{\left (e^{x} + 1\right )} \log \left (e^{x} + 1\right ) + x + 1}{e^{x} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.079514, size = 14, normalized size = 0.82 \begin{align*} x - \log{\left (e^{x} + 1 \right )} + \frac{1}{e^{x} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.32998, size = 20, normalized size = 1.18 \begin{align*} x + \frac{1}{e^{x} + 1} - \log \left (e^{x} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]