Optimal. Leaf size=8 \[ -\frac{1}{\cosh (x)+1} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0209781, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2667, 32} \[ -\frac{1}{\cosh (x)+1} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2667
Rule 32
Rubi steps
\begin{align*} \int \frac{\sinh (x)}{(1+\cosh (x))^2} \, dx &=\operatorname{Subst}\left (\int \frac{1}{(1+x)^2} \, dx,x,\cosh (x)\right )\\ &=-\frac{1}{1+\cosh (x)}\\ \end{align*}
Mathematica [A] time = 0.0105182, size = 12, normalized size = 1.5 \[ -\frac{1}{2} \text{sech}^2\left (\frac{x}{2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 9, normalized size = 1.1 \begin{align*} - \left ( 1+\cosh \left ( x \right ) \right ) ^{-1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.06784, size = 11, normalized size = 1.38 \begin{align*} -\frac{1}{\cosh \left (x\right ) + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.10478, size = 122, normalized size = 15.25 \begin{align*} -\frac{2 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}}{\cosh \left (x\right )^{2} + 2 \,{\left (\cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.431506, size = 7, normalized size = 0.88 \begin{align*} - \frac{1}{\cosh{\left (x \right )} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15624, size = 14, normalized size = 1.75 \begin{align*} -\frac{2 \, e^{x}}{{\left (e^{x} + 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]