Optimal. Leaf size=19 \[ \frac{e^{4 x}}{8}-\frac{1}{4} e^{-2 x} \]
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Rubi [A] time = 0.0116903, 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} \[ \frac{e^{4 x}}{8}-\frac{1}{4} e^{-2 x} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 12
Rule 14
Rubi steps
\begin{align*} \int e^x \cosh (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{1}{4} e^{-2 x}+\frac{e^{4 x}}{8}\\ \end{align*}
Mathematica [A] time = 0.0092524, 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 = 0.018, 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 = 1.05605, 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 = 1.9006, size = 130, normalized size = 6.84 \begin{align*} -\frac{\cosh \left (x\right )^{3} - 9 \, \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} - 3 \, \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 = 0.301793, size = 20, normalized size = 1.05 \begin{align*} \frac{3 e^{x} \sinh{\left (3 x \right )}}{8} - \frac{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 = 1.22303, 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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