Optimal. Leaf size=100 \[ \frac{e^{-3 a-3 b x}}{96 b}-\frac{e^{-a-b x}}{32 b}+\frac{e^{a+b x}}{16 b}-\frac{e^{3 a+3 b x}}{48 b}-\frac{e^{5 a+5 b x}}{160 b}+\frac{e^{7 a+7 b x}}{224 b} \]
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Rubi [A] time = 0.0778496, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {2282, 12, 448} \[ \frac{e^{-3 a-3 b x}}{96 b}-\frac{e^{-a-b x}}{32 b}+\frac{e^{a+b x}}{16 b}-\frac{e^{3 a+3 b x}}{48 b}-\frac{e^{5 a+5 b x}}{160 b}+\frac{e^{7 a+7 b x}}{224 b} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 12
Rule 448
Rubi steps
\begin{align*} \int e^{2 (a+b x)} \cosh ^2(a+b x) \sinh ^3(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (-1+x^2\right )^3 \left (1+x^2\right )^2}{32 x^4} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (-1+x^2\right )^3 \left (1+x^2\right )^2}{x^4} \, dx,x,e^{a+b x}\right )}{32 b}\\ &=\frac{\operatorname{Subst}\left (\int \left (2-\frac{1}{x^4}+\frac{1}{x^2}-2 x^2-x^4+x^6\right ) \, dx,x,e^{a+b x}\right )}{32 b}\\ &=\frac{e^{-3 a-3 b x}}{96 b}-\frac{e^{-a-b x}}{32 b}+\frac{e^{a+b x}}{16 b}-\frac{e^{3 a+3 b x}}{48 b}-\frac{e^{5 a+5 b x}}{160 b}+\frac{e^{7 a+7 b x}}{224 b}\\ \end{align*}
Mathematica [A] time = 0.102981, size = 73, normalized size = 0.73 \[ \frac{e^{-3 (a+b x)} \left (-105 e^{2 (a+b x)}+210 e^{4 (a+b x)}-70 e^{6 (a+b x)}-21 e^{8 (a+b x)}+15 e^{10 (a+b x)}+35\right )}{3360 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 108, normalized size = 1.1 \begin{align*}{\frac{3\,\sinh \left ( bx+a \right ) }{32\,b}}-{\frac{\sinh \left ( 3\,bx+3\,a \right ) }{32\,b}}-{\frac{\sinh \left ( 5\,bx+5\,a \right ) }{160\,b}}+{\frac{\sinh \left ( 7\,bx+7\,a \right ) }{224\,b}}+{\frac{\cosh \left ( bx+a \right ) }{32\,b}}-{\frac{\cosh \left ( 3\,bx+3\,a \right ) }{96\,b}}-{\frac{\cosh \left ( 5\,bx+5\,a \right ) }{160\,b}}+{\frac{\cosh \left ( 7\,bx+7\,a \right ) }{224\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00432, size = 105, normalized size = 1.05 \begin{align*} -\frac{{\left (21 \, e^{\left (-2 \, b x - 2 \, a\right )} + 70 \, e^{\left (-4 \, b x - 4 \, a\right )} - 210 \, e^{\left (-6 \, b x - 6 \, a\right )} - 15\right )} e^{\left (7 \, b x + 7 \, a\right )}}{3360 \, b} - \frac{3 \, e^{\left (-b x - a\right )} - e^{\left (-3 \, b x - 3 \, a\right )}}{96 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.76445, size = 501, normalized size = 5.01 \begin{align*} \frac{25 \, \cosh \left (b x + a\right )^{5} + 125 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} - 10 \, \sinh \left (b x + a\right )^{5} - 2 \,{\left (50 \, \cosh \left (b x + a\right )^{2} - 21\right )} \sinh \left (b x + a\right )^{3} - 63 \, \cosh \left (b x + a\right )^{3} +{\left (250 \, \cosh \left (b x + a\right )^{3} - 189 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} - 2 \,{\left (25 \, \cosh \left (b x + a\right )^{4} - 63 \, \cosh \left (b x + a\right )^{2} + 70\right )} \sinh \left (b x + a\right ) + 70 \, \cosh \left (b x + a\right )}{1680 \,{\left (b \cosh \left (b x + a\right )^{2} - 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [A] time = 1.21009, size = 108, normalized size = 1.08 \begin{align*} -\frac{35 \,{\left (3 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-3 \, b x - 3 \, a\right )} -{\left (15 \, e^{\left (7 \, b x + 28 \, a\right )} - 21 \, e^{\left (5 \, b x + 26 \, a\right )} - 70 \, e^{\left (3 \, b x + 24 \, a\right )} + 210 \, e^{\left (b x + 22 \, a\right )}\right )} e^{\left (-21 \, a\right )}}{3360 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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