Optimal. Leaf size=57 \[ \frac{e^{-4 a-4 b x}}{256 b}-\frac{3 e^{4 a+4 b x}}{256 b}+\frac{e^{8 a+8 b x}}{512 b}+\frac{3 x}{64} \]
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Rubi [A] time = 0.0624658, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2282, 12, 266, 43} \[ \frac{e^{-4 a-4 b x}}{256 b}-\frac{3 e^{4 a+4 b x}}{256 b}+\frac{e^{8 a+8 b x}}{512 b}+\frac{3 x}{64} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 12
Rule 266
Rule 43
Rubi steps
\begin{align*} \int e^{2 (a+b x)} \cosh ^3(a+b x) \sinh ^3(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (-1+x^4\right )^3}{64 x^5} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (-1+x^4\right )^3}{x^5} \, dx,x,e^{a+b x}\right )}{64 b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(-1+x)^3}{x^2} \, dx,x,e^{4 a+4 b x}\right )}{256 b}\\ &=\frac{\operatorname{Subst}\left (\int \left (-3-\frac{1}{x^2}+\frac{3}{x}+x\right ) \, dx,x,e^{4 a+4 b x}\right )}{256 b}\\ &=\frac{e^{-4 a-4 b x}}{256 b}-\frac{3 e^{4 a+4 b x}}{256 b}+\frac{e^{8 a+8 b x}}{512 b}+\frac{3 x}{64}\\ \end{align*}
Mathematica [A] time = 0.0435625, size = 45, normalized size = 0.79 \[ \frac{e^{-4 (a+b x)}-3 e^{4 (a+b x)}+\frac{1}{2} e^{8 (a+b x)}+12 b x}{256 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 61, normalized size = 1.1 \begin{align*}{\frac{3\,x}{64}}-{\frac{\sinh \left ( 4\,bx+4\,a \right ) }{64\,b}}+{\frac{\sinh \left ( 8\,bx+8\,a \right ) }{512\,b}}-{\frac{\cosh \left ( 4\,bx+4\,a \right ) }{128\,b}}+{\frac{\cosh \left ( 8\,bx+8\,a \right ) }{512\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14565, size = 70, normalized size = 1.23 \begin{align*} -\frac{{\left (6 \, e^{\left (-4 \, b x - 4 \, a\right )} - 1\right )} e^{\left (8 \, b x + 8 \, a\right )}}{512 \, b} + \frac{3 \,{\left (b x + a\right )}}{64 \, b} + \frac{e^{\left (-4 \, b x - 4 \, a\right )}}{256 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.73983, size = 513, normalized size = 9. \begin{align*} \frac{3 \, \cosh \left (b x + a\right )^{6} - 20 \, \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{3} + 45 \, \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{4} - 6 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + 3 \, \sinh \left (b x + a\right )^{6} + 6 \,{\left (4 \, b x - 1\right )} \cosh \left (b x + a\right )^{2} + 3 \,{\left (15 \, \cosh \left (b x + a\right )^{4} + 8 \, b x - 2\right )} \sinh \left (b x + a\right )^{2} - 6 \,{\left (\cosh \left (b x + a\right )^{5} + 2 \,{\left (4 \, b x + 1\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{512 \,{\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 = 1.20434, size = 81, normalized size = 1.42 \begin{align*} \frac{24 \, b x - 2 \,{\left (3 \, e^{\left (4 \, b x + 4 \, a\right )} - 1\right )} e^{\left (-4 \, b x - 4 \, a\right )} +{\left (e^{\left (8 \, b x + 16 \, a\right )} - 6 \, e^{\left (4 \, b x + 12 \, a\right )}\right )} e^{\left (-8 \, a\right )}}{512 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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