Optimal. Leaf size=55 \[ \frac{b (2 a-b) \sinh (c+d x)}{d}+\frac{(a-b)^2 \tan ^{-1}(\sinh (c+d x))}{d}+\frac{b^2 \sinh ^3(c+d x)}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0577477, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3190, 390, 203} \[ \frac{b (2 a-b) \sinh (c+d x)}{d}+\frac{(a-b)^2 \tan ^{-1}(\sinh (c+d x))}{d}+\frac{b^2 \sinh ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3190
Rule 390
Rule 203
Rubi steps
\begin{align*} \int \text{sech}(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^2}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left ((2 a-b) b+b^2 x^2+\frac{(a-b)^2}{1+x^2}\right ) \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{(2 a-b) b \sinh (c+d x)}{d}+\frac{b^2 \sinh ^3(c+d x)}{3 d}+\frac{(a-b)^2 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{(a-b)^2 \tan ^{-1}(\sinh (c+d x))}{d}+\frac{(2 a-b) b \sinh (c+d x)}{d}+\frac{b^2 \sinh ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.227902, size = 70, normalized size = 1.27 \[ \frac{\sinh (c+d x) \left (b \left (6 a+b \left (\sinh ^2(c+d x)-3\right )\right )+\frac{3 (a-b)^2 \tanh ^{-1}\left (\sqrt{-\sinh ^2(c+d x)}\right )}{\sqrt{-\sinh ^2(c+d x)}}\right )}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.038, size = 89, normalized size = 1.6 \begin{align*} 2\,{\frac{{a}^{2}\arctan \left ({{\rm e}^{dx+c}} \right ) }{d}}+2\,{\frac{ab\sinh \left ( dx+c \right ) }{d}}-4\,{\frac{ab\arctan \left ({{\rm e}^{dx+c}} \right ) }{d}}+{\frac{{b}^{2} \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-{\frac{{b}^{2}\sinh \left ( dx+c \right ) }{d}}+2\,{\frac{{b}^{2}\arctan \left ({{\rm e}^{dx+c}} \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.72436, size = 180, normalized size = 3.27 \begin{align*} -\frac{1}{24} \, b^{2}{\left (\frac{{\left (15 \, e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )} e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac{15 \, e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac{48 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d}\right )} + a b{\left (\frac{4 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} + \frac{e^{\left (d x + c\right )}}{d} - \frac{e^{\left (-d x - c\right )}}{d}\right )} + \frac{a^{2} \arctan \left (\sinh \left (d x + c\right )\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.60384, size = 1143, normalized size = 20.78 \begin{align*} \frac{b^{2} \cosh \left (d x + c\right )^{6} + 6 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + b^{2} \sinh \left (d x + c\right )^{6} + 3 \,{\left (8 \, a b - 5 \, b^{2}\right )} \cosh \left (d x + c\right )^{4} + 3 \,{\left (5 \, b^{2} \cosh \left (d x + c\right )^{2} + 8 \, a b - 5 \, b^{2}\right )} \sinh \left (d x + c\right )^{4} + 4 \,{\left (5 \, b^{2} \cosh \left (d x + c\right )^{3} + 3 \,{\left (8 \, a b - 5 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - 3 \,{\left (8 \, a b - 5 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 3 \,{\left (5 \, b^{2} \cosh \left (d x + c\right )^{4} + 6 \,{\left (8 \, a b - 5 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} - 8 \, a b + 5 \, b^{2}\right )} \sinh \left (d x + c\right )^{2} - b^{2} + 48 \,{\left ({\left (a^{2} - 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{3} + 3 \,{\left (a^{2} - 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \,{\left (a^{2} - 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} +{\left (a^{2} - 2 \, a b + b^{2}\right )} \sinh \left (d x + c\right )^{3}\right )} \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) + 6 \,{\left (b^{2} \cosh \left (d x + c\right )^{5} + 2 \,{\left (8 \, a b - 5 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} -{\left (8 \, a b - 5 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{24 \,{\left (d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + d \sinh \left (d x + c\right )^{3}\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 = 1.18227, size = 159, normalized size = 2.89 \begin{align*} \frac{2 \,{\left (a^{2} - 2 \, a b + b^{2}\right )} \arctan \left (e^{\left (d x + c\right )}\right )}{d} - \frac{{\left (24 \, a b e^{\left (2 \, d x + 2 \, c\right )} - 15 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + b^{2}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{24 \, d} + \frac{b^{2} d^{2} e^{\left (3 \, d x + 3 \, c\right )} + 24 \, a b d^{2} e^{\left (d x + c\right )} - 15 \, b^{2} d^{2} e^{\left (d x + c\right )}}{24 \, d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]