Optimal. Leaf size=26 \[ -\frac{1}{12} e^{-3 x}+\frac{e^x}{2}+\frac{e^{5 x}}{20} \]
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Rubi [A] time = 0.0202289, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2282, 12, 270} \[ -\frac{1}{12} e^{-3 x}+\frac{e^x}{2}+\frac{e^{5 x}}{20} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 12
Rule 270
Rubi steps
\begin{align*} \int e^x \cosh ^2(2 x) \, dx &=\operatorname{Subst}\left (\int \frac{\left (1+x^4\right )^2}{4 x^4} \, dx,x,e^x\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{\left (1+x^4\right )^2}{x^4} \, dx,x,e^x\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (2+\frac{1}{x^4}+x^4\right ) \, dx,x,e^x\right )\\ &=-\frac{1}{12} e^{-3 x}+\frac{e^x}{2}+\frac{e^{5 x}}{20}\\ \end{align*}
Mathematica [A] time = 0.0133952, size = 26, normalized size = 1. \[ -\frac{1}{12} e^{-3 x}+\frac{e^x}{2}+\frac{e^{5 x}}{20} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 34, normalized size = 1.3 \begin{align*}{\frac{\sinh \left ( x \right ) }{2}}+{\frac{\sinh \left ( 3\,x \right ) }{12}}+{\frac{\sinh \left ( 5\,x \right ) }{20}}+{\frac{\cosh \left ( x \right ) }{2}}-{\frac{\cosh \left ( 3\,x \right ) }{12}}+{\frac{\cosh \left ( 5\,x \right ) }{20}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06247, size = 23, normalized size = 0.88 \begin{align*} \frac{1}{20} \, e^{\left (5 \, x\right )} - \frac{1}{12} \, e^{\left (-3 \, x\right )} + \frac{1}{2} \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.82604, size = 170, normalized size = 6.54 \begin{align*} -\frac{\cosh \left (x\right )^{4} - 16 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} - 16 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 15}{30 \,{\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 [B] time = 0.814416, size = 42, normalized size = 1.62 \begin{align*} - \frac{8 e^{x} \sinh ^{2}{\left (2 x \right )}}{15} + \frac{4 e^{x} \sinh{\left (2 x \right )} \cosh{\left (2 x \right )}}{15} + \frac{7 e^{x} \cosh ^{2}{\left (2 x \right )}}{15} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23804, size = 23, normalized size = 0.88 \begin{align*} \frac{1}{20} \, e^{\left (5 \, x\right )} - \frac{1}{12} \, e^{\left (-3 \, x\right )} + \frac{1}{2} \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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