3.660 \(\int e^{-x} \cos (3 x) \, dx\)

Optimal. Leaf size=27 \[ \frac{3}{10} e^{-x} \sin (3 x)-\frac{1}{10} e^{-x} \cos (3 x) \]

[Out]

-Cos[3*x]/(10*E^x) + (3*Sin[3*x])/(10*E^x)

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Rubi [A]  time = 0.0196015, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{3}{10} e^{-x} \sin (3 x)-\frac{1}{10} e^{-x} \cos (3 x) \]

Antiderivative was successfully verified.

[In]  Int[Cos[3*x]/E^x,x]

[Out]

-Cos[3*x]/(10*E^x) + (3*Sin[3*x])/(10*E^x)

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Rubi in Sympy [A]  time = 2.80563, size = 20, normalized size = 0.74 \[ \frac{3 e^{- x} \sin{\left (3 x \right )}}{10} - \frac{e^{- x} \cos{\left (3 x \right )}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(3*x)/exp(x),x)

[Out]

3*exp(-x)*sin(3*x)/10 - exp(-x)*cos(3*x)/10

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Mathematica [A]  time = 0.0192131, size = 20, normalized size = 0.74 \[ -\frac{1}{10} e^{-x} (\cos (3 x)-3 \sin (3 x)) \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[3*x]/E^x,x]

[Out]

-(Cos[3*x] - 3*Sin[3*x])/(10*E^x)

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Maple [A]  time = 0.013, size = 22, normalized size = 0.8 \[ -{\frac{{{\rm e}^{-x}}\cos \left ( 3\,x \right ) }{10}}+{\frac{3\,{{\rm e}^{-x}}\sin \left ( 3\,x \right ) }{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(3*x)/exp(x),x)

[Out]

-1/10*exp(-x)*cos(3*x)+3/10*exp(-x)*sin(3*x)

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Maxima [A]  time = 0.76099, size = 23, normalized size = 0.85 \[ -\frac{1}{10} \,{\left (\cos \left (3 \, x\right ) - 3 \, \sin \left (3 \, x\right )\right )} e^{\left (-x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(3*x)*e^(-x),x, algorithm="maxima")

[Out]

-1/10*(cos(3*x) - 3*sin(3*x))*e^(-x)

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Fricas [A]  time = 0.246223, size = 28, normalized size = 1.04 \[ -\frac{1}{10} \, \cos \left (3 \, x\right ) e^{\left (-x\right )} + \frac{3}{10} \, e^{\left (-x\right )} \sin \left (3 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(3*x)*e^(-x),x, algorithm="fricas")

[Out]

-1/10*cos(3*x)*e^(-x) + 3/10*e^(-x)*sin(3*x)

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Sympy [A]  time = 0.747825, size = 20, normalized size = 0.74 \[ \frac{3 e^{- x} \sin{\left (3 x \right )}}{10} - \frac{e^{- x} \cos{\left (3 x \right )}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(3*x)/exp(x),x)

[Out]

3*exp(-x)*sin(3*x)/10 - exp(-x)*cos(3*x)/10

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GIAC/XCAS [A]  time = 0.224404, size = 23, normalized size = 0.85 \[ -\frac{1}{10} \,{\left (\cos \left (3 \, x\right ) - 3 \, \sin \left (3 \, x\right )\right )} e^{\left (-x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(3*x)*e^(-x),x, algorithm="giac")

[Out]

-1/10*(cos(3*x) - 3*sin(3*x))*e^(-x)