Optimal. Leaf size=52 \[ -\frac{e^{-2 a-2 b x}}{32 b}-\frac{e^{2 a+2 b x}}{16 b}+\frac{e^{6 a+6 b x}}{96 b} \]
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Rubi [A] time = 0.0592756, antiderivative size = 52, 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, 270} \[ -\frac{e^{-2 a-2 b x}}{32 b}-\frac{e^{2 a+2 b x}}{16 b}+\frac{e^{6 a+6 b x}}{96 b} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 12
Rule 270
Rubi steps
\begin{align*} \int e^{2 (a+b x)} \cosh ^2(a+b x) \sinh ^2(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^4\right )^2}{16 x^3} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^4\right )^2}{x^3} \, dx,x,e^{a+b x}\right )}{16 b}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{x^3}-2 x+x^5\right ) \, dx,x,e^{a+b x}\right )}{16 b}\\ &=-\frac{e^{-2 a-2 b x}}{32 b}-\frac{e^{2 a+2 b x}}{16 b}+\frac{e^{6 a+6 b x}}{96 b}\\ \end{align*}
Mathematica [A] time = 0.0378905, size = 38, normalized size = 0.73 \[ \frac{e^{-2 (a+b x)} \left (-6 e^{4 (a+b x)}+e^{8 (a+b x)}-3\right )}{96 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 58, normalized size = 1.1 \begin{align*} -{\frac{\sinh \left ( 2\,bx+2\,a \right ) }{32\,b}}+{\frac{\sinh \left ( 6\,bx+6\,a \right ) }{96\,b}}-{\frac{3\,\cosh \left ( 2\,bx+2\,a \right ) }{32\,b}}+{\frac{\cosh \left ( 6\,bx+6\,a \right ) }{96\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01396, size = 57, normalized size = 1.1 \begin{align*} -\frac{{\left (6 \, e^{\left (-4 \, b x - 4 \, a\right )} - 1\right )} e^{\left (6 \, b x + 6 \, a\right )}}{96 \, b} - \frac{e^{\left (-2 \, b x - 2 \, a\right )}}{32 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.75634, size = 304, normalized size = 5.85 \begin{align*} -\frac{\cosh \left (b x + a\right )^{4} - 8 \, \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + 6 \, \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} - 8 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 3}{48 \,{\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 [A] time = 111.514, size = 70, normalized size = 1.35 \begin{align*} \begin{cases} \frac{e^{2 a} e^{2 b x} \sinh{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{3 b} - \frac{e^{2 a} e^{2 b x} \cosh ^{4}{\left (a + b x \right )}}{6 b} & \text{for}\: b \neq 0 \\x e^{2 a} \sinh ^{2}{\left (a \right )} \cosh ^{2}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1451, size = 58, normalized size = 1.12 \begin{align*} \frac{{\left (e^{\left (6 \, b x + 12 \, a\right )} - 6 \, e^{\left (2 \, b x + 8 \, a\right )}\right )} e^{\left (-6 \, a\right )} - 3 \, e^{\left (-2 \, b x - 2 \, a\right )}}{96 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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