Optimal. Leaf size=38 \[ \frac{\sinh ^3(a+b x)}{3 b}-\frac{\sinh (a+b x)}{b}+\frac{\tan ^{-1}(\sinh (a+b x))}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0264162, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2592, 302, 203} \[ \frac{\sinh ^3(a+b x)}{3 b}-\frac{\sinh (a+b x)}{b}+\frac{\tan ^{-1}(\sinh (a+b x))}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2592
Rule 302
Rule 203
Rubi steps
\begin{align*} \int \sinh ^3(a+b x) \tanh (a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^4}{1+x^2} \, dx,x,\sinh (a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \left (-1+x^2+\frac{1}{1+x^2}\right ) \, dx,x,\sinh (a+b x)\right )}{b}\\ &=-\frac{\sinh (a+b x)}{b}+\frac{\sinh ^3(a+b x)}{3 b}+\frac{\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sinh (a+b x)\right )}{b}\\ &=\frac{\tan ^{-1}(\sinh (a+b x))}{b}-\frac{\sinh (a+b x)}{b}+\frac{\sinh ^3(a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0144086, size = 38, normalized size = 1. \[ \frac{\sinh ^3(a+b x)}{3 b}-\frac{\sinh (a+b x)}{b}+\frac{\tan ^{-1}(\sinh (a+b x))}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.019, size = 38, normalized size = 1. \begin{align*}{\frac{ \left ( \sinh \left ( bx+a \right ) \right ) ^{3}}{3\,b}}-{\frac{\sinh \left ( bx+a \right ) }{b}}+2\,{\frac{\arctan \left ({{\rm e}^{bx+a}} \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.56535, size = 96, normalized size = 2.53 \begin{align*} -\frac{{\left (15 \, e^{\left (-2 \, b x - 2 \, a\right )} - 1\right )} e^{\left (3 \, b x + 3 \, a\right )}}{24 \, b} + \frac{15 \, e^{\left (-b x - a\right )} - e^{\left (-3 \, b x - 3 \, a\right )}}{24 \, b} - \frac{2 \, \arctan \left (e^{\left (-b x - a\right )}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.87216, size = 836, normalized size = 22. \begin{align*} \frac{\cosh \left (b x + a\right )^{6} + 6 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + \sinh \left (b x + a\right )^{6} + 15 \,{\left (\cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{4} - 15 \, \cosh \left (b x + a\right )^{4} + 20 \,{\left (\cosh \left (b x + a\right )^{3} - 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 15 \,{\left (\cosh \left (b x + a\right )^{4} - 6 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{2} + 48 \,{\left (\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right ) + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3}\right )} \arctan \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 15 \, \cosh \left (b x + a\right )^{2} + 6 \,{\left (\cosh \left (b x + a\right )^{5} - 10 \, \cosh \left (b x + a\right )^{3} + 5 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - 1}{24 \,{\left (b \cosh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right ) + 3 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b \sinh \left (b x + a\right )^{3}\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 \sinh ^{3}{\left (a + b x \right )} \tanh{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.18248, size = 85, normalized size = 2.24 \begin{align*} \frac{{\left (15 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-3 \, b x - 3 \, a\right )} +{\left (e^{\left (3 \, b x + 18 \, a\right )} - 15 \, e^{\left (b x + 16 \, a\right )}\right )} e^{\left (-15 \, a\right )} + 48 \, \arctan \left (e^{\left (b x + a\right )}\right )}{24 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]