Optimal. Leaf size=19 \[ \frac{e^{-2 x}}{4}+\frac{e^{4 x}}{8} \]
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Rubi [A] time = 0.0119808, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {2282, 12, 14} \[ \frac{e^{-2 x}}{4}+\frac{e^{4 x}}{8} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 12
Rule 14
Rubi steps
\begin{align*} \int e^x \sinh (3 x) \, dx &=\operatorname{Subst}\left (\int \frac{-1+x^6}{2 x^3} \, dx,x,e^x\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{-1+x^6}{x^3} \, dx,x,e^x\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{1}{x^3}+x^3\right ) \, dx,x,e^x\right )\\ &=\frac{e^{-2 x}}{4}+\frac{e^{4 x}}{8}\\ \end{align*}
Mathematica [A] time = 0.0091856, size = 16, normalized size = 0.84 \[ \frac{1}{8} e^{-2 x} \left (e^{6 x}+2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 26, normalized size = 1.4 \begin{align*} -{\frac{\sinh \left ( 2\,x \right ) }{4}}+{\frac{\sinh \left ( 4\,x \right ) }{8}}+{\frac{\cosh \left ( 2\,x \right ) }{4}}+{\frac{\cosh \left ( 4\,x \right ) }{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10799, size = 18, normalized size = 0.95 \begin{align*} \frac{1}{8} \, e^{\left (4 \, x\right )} + \frac{1}{4} \, e^{\left (-2 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.96551, size = 128, normalized size = 6.74 \begin{align*} \frac{3 \, \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )^{2} \sinh \left (x\right ) + 9 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} - \sinh \left (x\right )^{3}}{8 \,{\left (\cosh \left (x\right ) - \sinh \left (x\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.329055, size = 20, normalized size = 1.05 \begin{align*} - \frac{e^{x} \sinh{\left (3 x \right )}}{8} + \frac{3 e^{x} \cosh{\left (3 x \right )}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11711, size = 18, normalized size = 0.95 \begin{align*} \frac{1}{8} \, e^{\left (4 \, x\right )} + \frac{1}{4} \, e^{\left (-2 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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