Optimal. Leaf size=12 \[ \frac {1}{2 (1-\cosh (x))^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2667, 32} \[ \frac {1}{2 (1-\cosh (x))^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 32
Rule 2667
Rubi steps
\begin {align*} \int \frac {\sinh (x)}{(1-\cosh (x))^3} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{(1+x)^3} \, dx,x,-\cosh (x)\right )\\ &=\frac {1}{2 (1-\cosh (x))^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 12, normalized size = 1.00 \[ \frac {1}{8} \text {csch}^4\left (\frac {x}{2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.71, size = 55, normalized size = 4.58 \[ \frac {2 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}}{\cosh \relax (x)^{3} + {\left (3 \, \cosh \relax (x) - 4\right )} \sinh \relax (x)^{2} + \sinh \relax (x)^{3} - 4 \, \cosh \relax (x)^{2} + {\left (3 \, \cosh \relax (x)^{2} - 8 \, \cosh \relax (x) + 5\right )} \sinh \relax (x) + 7 \, \cosh \relax (x) - 4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.13, size = 12, normalized size = 1.00 \[ \frac {2 \, e^{\left (2 \, x\right )}}{{\left (e^{x} - 1\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 11, normalized size = 0.92 \[ \frac {1}{2 \left (1-\cosh \relax (x )\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.30, size = 8, normalized size = 0.67 \[ \frac {1}{2 \, {\left (\cosh \relax (x) - 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.08, size = 8, normalized size = 0.67 \[ \frac {1}{2\,{\left (\mathrm {cosh}\relax (x)-1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.58, size = 14, normalized size = 1.17 \[ \frac {1}{2 \cosh ^{2}{\relax (x )} - 4 \cosh {\relax (x )} + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________