Optimal. Leaf size=17 \[ \frac{\tanh ^2(x)}{2}-\frac{\tanh ^3(x)}{3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0454175, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {3516, 848, 43} \[ \frac{\tanh ^2(x)}{2}-\frac{\tanh ^3(x)}{3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3516
Rule 848
Rule 43
Rubi steps
\begin{align*} \int \frac{\text{sech}^4(x)}{1+\coth (x)} \, dx &=-\operatorname{Subst}\left (\int \frac{-1+x^2}{x^4 (1+x)} \, dx,x,\coth (x)\right )\\ &=-\operatorname{Subst}\left (\int \frac{-1+x}{x^4} \, dx,x,\coth (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (-\frac{1}{x^4}+\frac{1}{x^3}\right ) \, dx,x,\coth (x)\right )\\ &=\frac{\tanh ^2(x)}{2}-\frac{\tanh ^3(x)}{3}\\ \end{align*}
Mathematica [A] time = 0.0498663, size = 17, normalized size = 1. \[ \frac{1}{6} \left (-2 \tanh ^3(x)-3 \text{sech}^2(x)\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.029, size = 38, normalized size = 2.2 \begin{align*} -4\,{\frac{-1/2\, \left ( \tanh \left ( x/2 \right ) \right ) ^{4}+2/3\, \left ( \tanh \left ( x/2 \right ) \right ) ^{3}-1/2\, \left ( \tanh \left ( x/2 \right ) \right ) ^{2}}{ \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.03484, size = 101, normalized size = 5.94 \begin{align*} -\frac{2 \, e^{\left (-2 \, x\right )}}{3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1} - \frac{4 \, e^{\left (-4 \, x\right )}}{3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1} - \frac{2}{3 \,{\left (3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.52306, size = 286, normalized size = 16.82 \begin{align*} -\frac{4 \,{\left (\cosh \left (x\right ) + 2 \, \sinh \left (x\right )\right )}}{3 \,{\left (\cosh \left (x\right )^{5} + 5 \, \cosh \left (x\right ) \sinh \left (x\right )^{4} + \sinh \left (x\right )^{5} +{\left (10 \, \cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right )^{3} + 3 \, \cosh \left (x\right )^{3} +{\left (10 \, \cosh \left (x\right )^{3} + 9 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} +{\left (5 \, \cosh \left (x\right )^{4} + 9 \, \cosh \left (x\right )^{2} + 2\right )} \sinh \left (x\right ) + 4 \, \cosh \left (x\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}^{4}{\left (x \right )}}{\coth{\left (x \right )} + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.14445, size = 24, normalized size = 1.41 \begin{align*} -\frac{2 \,{\left (3 \, e^{\left (2 \, x\right )} - 1\right )}}{3 \,{\left (e^{\left (2 \, x\right )} + 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]