Optimal. Leaf size=10 \[ -\frac{1}{2} \sin \left (e^{-2 x}\right ) \]
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Rubi [A] time = 0.0104177, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2282, 2637} \[ -\frac{1}{2} \sin \left (e^{-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2637
Rubi steps
\begin{align*} \int e^{-2 x} \cos \left (e^{-2 x}\right ) \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \cos (x) \, dx,x,e^{-2 x}\right )\right )\\ &=-\frac{1}{2} \sin \left (e^{-2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.0112636, size = 10, normalized size = 1. \[ -\frac{1}{2} \sin \left (e^{-2 x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 8, normalized size = 0.8 \begin{align*} -{\frac{\sin \left ({{\rm e}^{-2\,x}} \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.998854, size = 9, normalized size = 0.9 \begin{align*} -\frac{1}{2} \, \sin \left (e^{\left (-2 \, x\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.463653, size = 27, normalized size = 2.7 \begin{align*} -\frac{1}{2} \, \sin \left (e^{\left (-2 \, x\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.492375, size = 10, normalized size = 1. \begin{align*} - \frac{\sin{\left (e^{- 2 x} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09451, size = 9, normalized size = 0.9 \begin{align*} -\frac{1}{2} \, \sin \left (e^{\left (-2 \, x\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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