Optimal. Leaf size=27 \[ -\frac{\text{csch}^3(a+b x)}{3 b}-\frac{\text{csch}(a+b x)}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0206844, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {2606} \[ -\frac{\text{csch}^3(a+b x)}{3 b}-\frac{\text{csch}(a+b x)}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2606
Rubi steps
\begin{align*} \int \coth ^3(a+b x) \text{csch}(a+b x) \, dx &=\frac{i \operatorname{Subst}\left (\int \left (-1+x^2\right ) \, dx,x,-i \text{csch}(a+b x)\right )}{b}\\ &=-\frac{\text{csch}(a+b x)}{b}-\frac{\text{csch}^3(a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0137484, size = 27, normalized size = 1. \[ -\frac{\text{csch}^3(a+b x)}{3 b}-\frac{\text{csch}(a+b x)}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.014, size = 50, normalized size = 1.9 \begin{align*}{\frac{1}{b} \left ( -{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{3\, \left ( \sinh \left ( bx+a \right ) \right ) ^{3}}}-{\frac{2\, \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{3\,\sinh \left ( bx+a \right ) }}+{\frac{2\,\sinh \left ( bx+a \right ) }{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.05293, size = 200, normalized size = 7.41 \begin{align*} \frac{2 \, e^{\left (-b x - a\right )}}{b{\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} - 3 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )} - 1\right )}} - \frac{4 \, e^{\left (-3 \, b x - 3 \, a\right )}}{3 \, b{\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} - 3 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )} - 1\right )}} + \frac{2 \, e^{\left (-5 \, b x - 5 \, a\right )}}{b{\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} - 3 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.7614, size = 464, normalized size = 17.19 \begin{align*} -\frac{2 \,{\left (3 \, \cosh \left (b x + a\right )^{3} + 9 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + 3 \, \sinh \left (b x + a\right )^{3} +{\left (9 \, \cosh \left (b x + a\right )^{2} - 5\right )} \sinh \left (b x + a\right ) + \cosh \left (b x + a\right )\right )}}{3 \,{\left (b \cosh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b \sinh \left (b x + a\right )^{4} - 4 \, b \cosh \left (b x + a\right )^{2} + 2 \,{\left (3 \, b \cosh \left (b x + a\right )^{2} - 2 \, b\right )} \sinh \left (b x + a\right )^{2} + 4 \,{\left (b \cosh \left (b x + a\right )^{3} - b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 3 \, b\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 \coth ^{3}{\left (a + b x \right )} \operatorname{csch}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.21634, size = 66, normalized size = 2.44 \begin{align*} -\frac{2 \,{\left (3 \, e^{\left (5 \, b x + 5 \, a\right )} - 2 \, e^{\left (3 \, b x + 3 \, a\right )} + 3 \, e^{\left (b x + a\right )}\right )}}{3 \, b{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]