Optimal. Leaf size=23 \[ \frac {1}{3} \log \left (1-2 e^x\right )-\frac {1}{3} \log \left (e^x+1\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2282, 616, 31} \[ \frac {1}{3} \log \left (1-2 e^x\right )-\frac {1}{3} \log \left (e^x+1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 616
Rule 2282
Rubi steps
\begin {align*} \int \frac {1}{1-e^{-x}+2 e^x} \, dx &=\operatorname {Subst}\left (\int \frac {1}{-1+x+2 x^2} \, dx,x,e^x\right )\\ &=\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{-1+2 x} \, dx,x,e^x\right )-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{2+2 x} \, dx,x,e^x\right )\\ &=\frac {1}{3} \log \left (1-2 e^x\right )-\frac {1}{3} \log \left (1+e^x\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 16, normalized size = 0.70 \[ -\frac {2}{3} \tanh ^{-1}\left (\frac {1}{3} \left (4 e^x+1\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.40, size = 17, normalized size = 0.74 \[ \frac {1}{3} \, \log \left (2 \, e^{x} - 1\right ) - \frac {1}{3} \, \log \left (e^{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.97, size = 18, normalized size = 0.78 \[ -\frac {1}{3} \, \log \left (e^{x} + 1\right ) + \frac {1}{3} \, \log \left ({\left | 2 \, e^{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 18, normalized size = 0.78 \[ -\frac {\ln \left ({\mathrm e}^{x}+1\right )}{3}+\frac {\ln \left (2 \,{\mathrm e}^{x}-1\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.44, size = 19, normalized size = 0.83 \[ -\frac {1}{3} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac {1}{3} \, \log \left (e^{\left (-x\right )} - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.25, size = 17, normalized size = 0.74 \[ \frac {\ln \left (2\,{\mathrm {e}}^x-1\right )}{3}-\frac {\ln \left ({\mathrm {e}}^x+1\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.12, size = 17, normalized size = 0.74 \[ \frac {\log {\left (e^{x} - \frac {1}{2} \right )}}{3} - \frac {\log {\left (e^{x} + 1 \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________