Optimal. Leaf size=10 \[ \cosh (x)-2 \log (\cosh (x)+1) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0383399, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2667, 43} \[ \cosh (x)-2 \log (\cosh (x)+1) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \frac{\sinh ^3(x)}{(1+\cosh (x))^2} \, dx &=-\operatorname{Subst}\left (\int \frac{1-x}{1+x} \, dx,x,\cosh (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (-1+\frac{2}{1+x}\right ) \, dx,x,\cosh (x)\right )\\ &=\cosh (x)-2 \log (1+\cosh (x))\\ \end{align*}
Mathematica [A] time = 0.0179428, size = 13, normalized size = 1.3 \[ \cosh (x)-4 \log \left (\cosh \left (\frac{x}{2}\right )\right )-1 \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.012, size = 11, normalized size = 1.1 \begin{align*} \cosh \left ( x \right ) -2\,\ln \left ( 1+\cosh \left ( x \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.13026, size = 31, normalized size = 3.1 \begin{align*} -2 \, x + \frac{1}{2} \, e^{\left (-x\right )} + \frac{1}{2} \, e^{x} - 4 \, \log \left (e^{\left (-x\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.2207, size = 197, normalized size = 19.7 \begin{align*} \frac{4 \, x \cosh \left (x\right ) + \cosh \left (x\right )^{2} - 8 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + 2 \,{\left (2 \, x + \cosh \left (x\right )\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1}{2 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 0.658029, size = 70, normalized size = 7. \begin{align*} - \frac{2 \log{\left (\cosh{\left (x \right )} + 1 \right )} \cosh{\left (x \right )}}{\cosh{\left (x \right )} + 1} - \frac{2 \log{\left (\cosh{\left (x \right )} + 1 \right )}}{\cosh{\left (x \right )} + 1} + \frac{\sinh ^{2}{\left (x \right )} \cosh{\left (x \right )}}{\cosh{\left (x \right )} + 1} - \frac{\cosh ^{3}{\left (x \right )}}{\cosh{\left (x \right )} + 1} + \frac{\cosh ^{2}{\left (x \right )}}{\cosh{\left (x \right )} + 1} - \frac{2}{\cosh{\left (x \right )} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.16163, size = 28, normalized size = 2.8 \begin{align*} 2 \, x + \frac{1}{2} \, e^{\left (-x\right )} + \frac{1}{2} \, e^{x} - 4 \, \log \left (e^{x} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]