Optimal. Leaf size=26 \[ -\frac{1}{28} e^{-7 x}+\frac{e^x}{2}+\frac{e^{9 x}}{36} \]
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Rubi [A] time = 0.0190408, 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} \[ -\frac{1}{28} e^{-7 x}+\frac{e^x}{2}+\frac{e^{9 x}}{36} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 12
Rule 270
Rubi steps
\begin{align*} \int e^x \cosh ^2(4 x) \, dx &=\operatorname{Subst}\left (\int \frac{\left (1+x^8\right )^2}{4 x^8} \, dx,x,e^x\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{\left (1+x^8\right )^2}{x^8} \, dx,x,e^x\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (2+\frac{1}{x^8}+x^8\right ) \, dx,x,e^x\right )\\ &=-\frac{1}{28} e^{-7 x}+\frac{e^x}{2}+\frac{e^{9 x}}{36}\\ \end{align*}
Mathematica [A] time = 0.014904, size = 26, normalized size = 1. \[ -\frac{1}{28} e^{-7 x}+\frac{e^x}{2}+\frac{e^{9 x}}{36} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 34, normalized size = 1.3 \begin{align*}{\frac{\sinh \left ( x \right ) }{2}}+{\frac{\sinh \left ( 7\,x \right ) }{28}}+{\frac{\sinh \left ( 9\,x \right ) }{36}}+{\frac{\cosh \left ( x \right ) }{2}}-{\frac{\cosh \left ( 7\,x \right ) }{28}}+{\frac{\cosh \left ( 9\,x \right ) }{36}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0343, size = 23, normalized size = 0.88 \begin{align*} \frac{1}{36} \, e^{\left (9 \, x\right )} - \frac{1}{28} \, e^{\left (-7 \, 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 = 1.9059, size = 311, normalized size = 11.96 \begin{align*} -\frac{\cosh \left (x\right )^{8} - 64 \, \cosh \left (x\right )^{7} \sinh \left (x\right ) + 28 \, \cosh \left (x\right )^{6} \sinh \left (x\right )^{2} - 448 \, \cosh \left (x\right )^{5} \sinh \left (x\right )^{3} + 70 \, \cosh \left (x\right )^{4} \sinh \left (x\right )^{4} - 448 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{5} + 28 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{6} - 64 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} - 63}{126 \,{\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 = 0.816193, size = 42, normalized size = 1.62 \begin{align*} - \frac{32 e^{x} \sinh ^{2}{\left (4 x \right )}}{63} + \frac{8 e^{x} \sinh{\left (4 x \right )} \cosh{\left (4 x \right )}}{63} + \frac{31 e^{x} \cosh ^{2}{\left (4 x \right )}}{63} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.3319, size = 23, normalized size = 0.88 \begin{align*} \frac{1}{36} \, e^{\left (9 \, x\right )} - \frac{1}{28} \, e^{\left (-7 \, x\right )} + \frac{1}{2} \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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