Optimal. Leaf size=12 \[ \log \left (e^{-x}-e^x\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0398882, antiderivative size = 14, normalized size of antiderivative = 1.17, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {2282, 446, 72} \[ \log \left (1-e^{2 x}\right )-x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2282
Rule 446
Rule 72
Rubi steps
\begin{align*} \int \frac{e^{-x}+e^x}{-e^{-x}+e^x} \, dx &=\operatorname{Subst}\left (\int \frac{-1-x^2}{x \left (1-x^2\right )} \, dx,x,e^x\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{-1-x}{(1-x) x} \, dx,x,e^{2 x}\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{2}{-1+x}-\frac{1}{x}\right ) \, dx,x,e^{2 x}\right )\\ &=-x+\log \left (1-e^{2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.0082017, size = 14, normalized size = 1.17 \[ \log \left (1-e^{2 x}\right )-x \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.027, size = 17, normalized size = 1.4 \begin{align*} \ln \left ( -1+{{\rm e}^{x}} \right ) -\ln \left ({{\rm e}^{x}} \right ) +\ln \left ( 1+{{\rm e}^{x}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.963803, size = 14, normalized size = 1.17 \begin{align*} \log \left (e^{\left (-x\right )} - e^{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.961889, size = 31, normalized size = 2.58 \begin{align*} -x + \log \left (e^{\left (2 \, x\right )} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.092038, size = 8, normalized size = 0.67 \begin{align*} - x + \log{\left (e^{2 x} - 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.26354, size = 16, normalized size = 1.33 \begin{align*} -x + \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]