Optimal. Leaf size=19 \[ \frac{e^{3 x}}{6}-\frac{e^{-x}}{2} \]
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Rubi [A] time = 0.0113203, 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} \[ \frac{e^{3 x}}{6}-\frac{e^{-x}}{2} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 12
Rule 14
Rubi steps
\begin{align*} \int e^x \cosh (2 x) \, dx &=\operatorname{Subst}\left (\int \frac{1+x^4}{2 x^2} \, dx,x,e^x\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1+x^4}{x^2} \, dx,x,e^x\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{x^2}+x^2\right ) \, dx,x,e^x\right )\\ &=-\frac{e^{-x}}{2}+\frac{e^{3 x}}{6}\\ \end{align*}
Mathematica [A] time = 0.0083832, size = 16, normalized size = 0.84 \[ \frac{1}{6} e^{-x} \left (e^{4 x}-3\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 22, normalized size = 1.2 \begin{align*}{\frac{\sinh \left ( x \right ) }{2}}+{\frac{\sinh \left ( 3\,x \right ) }{6}}-{\frac{\cosh \left ( x \right ) }{2}}+{\frac{\cosh \left ( 3\,x \right ) }{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00421, size = 18, normalized size = 0.95 \begin{align*} \frac{1}{6} \, e^{\left (3 \, x\right )} - \frac{1}{2} \, e^{\left (-x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74545, size = 95, normalized size = 5. \begin{align*} -\frac{\cosh \left (x\right )^{2} - 4 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}{3 \,{\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 = 0.302073, size = 20, normalized size = 1.05 \begin{align*} \frac{2 e^{x} \sinh{\left (2 x \right )}}{3} - \frac{e^{x} \cosh{\left (2 x \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25778, size = 18, normalized size = 0.95 \begin{align*} \frac{1}{6} \, e^{\left (3 \, x\right )} - \frac{1}{2} \, e^{\left (-x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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