Optimal. Leaf size=32 \[ \frac{e^{3 a+3 b x}}{6 b}-\frac{e^{a+b x}}{2 b} \]
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Rubi [A] time = 0.0173545, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2282, 12} \[ \frac{e^{3 a+3 b x}}{6 b}-\frac{e^{a+b x}}{2 b} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 12
Rubi steps
\begin{align*} \int e^{2 (a+b x)} \sinh (a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{2} \left (-1+x^2\right ) \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \left (-1+x^2\right ) \, dx,x,e^{a+b x}\right )}{2 b}\\ &=-\frac{e^{a+b x}}{2 b}+\frac{e^{3 a+3 b x}}{6 b}\\ \end{align*}
Mathematica [A] time = 0.0127617, size = 25, normalized size = 0.78 \[ \frac{e^{a+b x} \left (e^{2 (a+b x)}-3\right )}{6 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 52, normalized size = 1.6 \begin{align*} -{\frac{\sinh \left ( bx+a \right ) }{2\,b}}+{\frac{\sinh \left ( 3\,bx+3\,a \right ) }{6\,b}}-{\frac{\cosh \left ( bx+a \right ) }{2\,b}}+{\frac{\cosh \left ( 3\,bx+3\,a \right ) }{6\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.28741, size = 35, normalized size = 1.09 \begin{align*} \frac{e^{\left (3 \, b x + 3 \, a\right )}}{6 \, b} - \frac{e^{\left (b x + a\right )}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.69578, size = 153, normalized size = 4.78 \begin{align*} \frac{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 3}{6 \,{\left (b \cosh \left (b x + a\right ) - b \sinh \left (b x + a\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.51426, size = 54, normalized size = 1.69 \begin{align*} \begin{cases} \frac{2 e^{2 a} e^{2 b x} \sinh{\left (a + b x \right )}}{3 b} - \frac{e^{2 a} e^{2 b x} \cosh{\left (a + b x \right )}}{3 b} & \text{for}\: b \neq 0 \\x e^{2 a} \sinh{\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 = 1.13691, size = 31, normalized size = 0.97 \begin{align*} \frac{e^{\left (3 \, b x + 3 \, a\right )} - 3 \, e^{\left (b x + a\right )}}{6 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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