Optimal. Leaf size=38 \[ \frac{\sinh ^3(a+b x)}{3 b}+\frac{2 \sinh (a+b x)}{b}-\frac{\text{csch}(a+b x)}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0353186, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2590, 270} \[ \frac{\sinh ^3(a+b x)}{3 b}+\frac{2 \sinh (a+b x)}{b}-\frac{\text{csch}(a+b x)}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2590
Rule 270
Rubi steps
\begin{align*} \int \cosh ^3(a+b x) \coth ^2(a+b x) \, dx &=-\frac{i \operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^2}{x^2} \, dx,x,-i \sinh (a+b x)\right )}{b}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (-2+\frac{1}{x^2}+x^2\right ) \, dx,x,-i \sinh (a+b x)\right )}{b}\\ &=-\frac{\text{csch}(a+b x)}{b}+\frac{2 \sinh (a+b x)}{b}+\frac{\sinh ^3(a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0182505, size = 38, normalized size = 1. \[ \frac{\sinh ^3(a+b x)}{3 b}+\frac{2 \sinh (a+b x)}{b}-\frac{\text{csch}(a+b x)}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.017, size = 50, normalized size = 1.3 \begin{align*}{\frac{1}{b} \left ({\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{4}}{3\,\sinh \left ( bx+a \right ) }}-{\frac{4\, \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{3\,\sinh \left ( bx+a \right ) }}+{\frac{8\,\sinh \left ( bx+a \right ) }{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.0822, size = 107, normalized size = 2.82 \begin{align*} -\frac{21 \, e^{\left (-b x - a\right )} + e^{\left (-3 \, b x - 3 \, a\right )}}{24 \, b} + \frac{20 \, e^{\left (-2 \, b x - 2 \, a\right )} - 69 \, e^{\left (-4 \, b x - 4 \, a\right )} + 1}{24 \, b{\left (e^{\left (-3 \, b x - 3 \, a\right )} - e^{\left (-5 \, b x - 5 \, a\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.85744, size = 177, normalized size = 4.66 \begin{align*} \frac{\cosh \left (b x + a\right )^{4} + \sinh \left (b x + a\right )^{4} + 2 \,{\left (3 \, \cosh \left (b x + a\right )^{2} + 10\right )} \sinh \left (b x + a\right )^{2} + 20 \, \cosh \left (b x + a\right )^{2} - 45}{24 \, b \sinh \left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.24665, size = 103, normalized size = 2.71 \begin{align*} -\frac{{\left (21 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )} e^{\left (-3 \, b x - 3 \, a\right )} -{\left (e^{\left (3 \, b x + 24 \, a\right )} + 21 \, e^{\left (b x + 22 \, a\right )}\right )} e^{\left (-21 \, a\right )} + \frac{48 \, e^{\left (b x + a\right )}}{e^{\left (2 \, b x + 2 \, a\right )} - 1}}{24 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]