3.33 \(\int e^{-t} \cos (3 t) \, dt\)

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

[Out]

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

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Rubi [A]  time = 0.0219816, 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 \[ \frac{3}{10} e^{-t} \sin (3 t)-\frac{1}{10} e^{-t} \cos (3 t) \]

Antiderivative was successfully verified.

[In]  Int[Cos[3*t]/E^t,t]

[Out]

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

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Rubi in Sympy [A]  time = 1.44892, size = 20, normalized size = 0.74 \[ \frac{3 e^{- t} \sin{\left (3 t \right )}}{10} - \frac{e^{- t} \cos{\left (3 t \right )}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0189033, size = 20, normalized size = 0.74 \[ -\frac{1}{10} e^{-t} (\cos (3 t)-3 \sin (3 t)) \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[3*t]/E^t,t]

[Out]

-(Cos[3*t] - 3*Sin[3*t])/(10*E^t)

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Maple [A]  time = 0.01, size = 22, normalized size = 0.8 \[ -{\frac{{{\rm e}^{-t}}\cos \left ( 3\,t \right ) }{10}}+{\frac{3\,{{\rm e}^{-t}}\sin \left ( 3\,t \right ) }{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(3*t)/exp(t),t)

[Out]

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

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Maxima [A]  time = 1.35266, size = 23, normalized size = 0.85 \[ -\frac{1}{10} \,{\left (\cos \left (3 \, t\right ) - 3 \, \sin \left (3 \, t\right )\right )} e^{\left (-t\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/10*(cos(3*t) - 3*sin(3*t))*e^(-t)

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Fricas [A]  time = 0.223178, size = 28, normalized size = 1.04 \[ -\frac{1}{10} \, \cos \left (3 \, t\right ) e^{\left (-t\right )} + \frac{3}{10} \, e^{\left (-t\right )} \sin \left (3 \, t\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.793812, size = 20, normalized size = 0.74 \[ \frac{3 e^{- t} \sin{\left (3 t \right )}}{10} - \frac{e^{- t} \cos{\left (3 t \right )}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(3*t)/exp(t),t)

[Out]

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

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GIAC/XCAS [A]  time = 0.199597, size = 23, normalized size = 0.85 \[ -\frac{1}{10} \,{\left (\cos \left (3 \, t\right ) - 3 \, \sin \left (3 \, t\right )\right )} e^{\left (-t\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/10*(cos(3*t) - 3*sin(3*t))*e^(-t)