Optimal. Leaf size=10 \[ -\frac {1}{2} \sin \left (e^{-2 x}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.01, size = 10, normalized size = 1.00 \[ -\frac {1}{2} \sin \left (e^{-2 x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 7, normalized size = 0.70 \[ -\frac {1}{2} \, \sin \left (e^{\left (-2 \, x\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 7, normalized size = 0.70 \[ -\frac {1}{2} \, \sin \left (e^{\left (-2 \, x\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 8, normalized size = 0.80 \[ -\frac {\sin \left ({\mathrm e}^{-2 x}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 7, normalized size = 0.70 \[ -\frac {1}{2} \, \sin \left (e^{\left (-2 \, x\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.20, size = 7, normalized size = 0.70 \[ -\frac {\sin \left ({\mathrm {e}}^{-2\,x}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.37, size = 10, normalized size = 1.00 \[ - \frac {\sin {\left (e^{- 2 x} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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