Optimal. Leaf size=14 \[ \frac{\sinh ^3(x)}{3 (\cosh (x)+1)^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0322497, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2671} \[ \frac{\sinh ^3(x)}{3 (\cosh (x)+1)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2671
Rubi steps
\begin{align*} \int \frac{\sinh ^2(x)}{(1+\cosh (x))^3} \, dx &=\frac{\sinh ^3(x)}{3 (1+\cosh (x))^3}\\ \end{align*}
Mathematica [A] time = 0.0264682, size = 12, normalized size = 0.86 \[ \frac{1}{3} \tanh ^3\left (\frac{x}{2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.011, size = 9, normalized size = 0.6 \begin{align*}{\frac{1}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.06615, size = 66, normalized size = 4.71 \begin{align*} \frac{2 \, e^{\left (-2 \, x\right )}}{3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} + 1} + \frac{2}{3 \,{\left (3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.11234, size = 127, normalized size = 9.07 \begin{align*} -\frac{4 \,{\left (2 \, \cosh \left (x\right ) + \sinh \left (x\right )\right )}}{3 \,{\left (\cosh \left (x\right )^{2} + 2 \,{\left (\cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.50567, size = 7, normalized size = 0.5 \begin{align*} \frac{\tanh ^{3}{\left (\frac{x}{2} \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.16736, size = 22, normalized size = 1.57 \begin{align*} -\frac{2 \,{\left (3 \, e^{\left (2 \, x\right )} + 1\right )}}{3 \,{\left (e^{x} + 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]