3.343 \(\int e^{-3 x} \cos (4 x) \, dx\)

Optimal. Leaf size=27 \[ \frac{4}{25} e^{-3 x} \sin (4 x)-\frac{3}{25} e^{-3 x} \cos (4 x) \]

[Out]

(-3*Cos[4*x])/(25*E^(3*x)) + (4*Sin[4*x])/(25*E^(3*x))

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Rubi [A]  time = 0.0199228, 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 \[ \frac{4}{25} e^{-3 x} \sin (4 x)-\frac{3}{25} e^{-3 x} \cos (4 x) \]

Antiderivative was successfully verified.

[In]  Int[Cos[4*x]/E^(3*x),x]

[Out]

(-3*Cos[4*x])/(25*E^(3*x)) + (4*Sin[4*x])/(25*E^(3*x))

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Rubi in Sympy [A]  time = 1.55764, size = 26, normalized size = 0.96 \[ \frac{4 e^{- 3 x} \sin{\left (4 x \right )}}{25} - \frac{3 e^{- 3 x} \cos{\left (4 x \right )}}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(4*x)/exp(3*x),x)

[Out]

4*exp(-3*x)*sin(4*x)/25 - 3*exp(-3*x)*cos(4*x)/25

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Mathematica [A]  time = 0.021291, size = 22, normalized size = 0.81 \[ \frac{1}{25} e^{-3 x} (4 \sin (4 x)-3 \cos (4 x)) \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[4*x]/E^(3*x),x]

[Out]

(-3*Cos[4*x] + 4*Sin[4*x])/(25*E^(3*x))

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Maple [A]  time = 0.012, size = 22, normalized size = 0.8 \[ -{\frac{3\,{{\rm e}^{-3\,x}}\cos \left ( 4\,x \right ) }{25}}+{\frac{4\,{{\rm e}^{-3\,x}}\sin \left ( 4\,x \right ) }{25}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(4*x)/exp(3*x),x)

[Out]

-3/25*exp(-3*x)*cos(4*x)+4/25*exp(-3*x)*sin(4*x)

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Maxima [A]  time = 1.33831, size = 26, normalized size = 0.96 \[ -\frac{1}{25} \,{\left (3 \, \cos \left (4 \, x\right ) - 4 \, \sin \left (4 \, x\right )\right )} e^{\left (-3 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(4*x)*e^(-3*x),x, algorithm="maxima")

[Out]

-1/25*(3*cos(4*x) - 4*sin(4*x))*e^(-3*x)

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Fricas [A]  time = 0.210979, size = 28, normalized size = 1.04 \[ -\frac{3}{25} \, \cos \left (4 \, x\right ) e^{\left (-3 \, x\right )} + \frac{4}{25} \, e^{\left (-3 \, x\right )} \sin \left (4 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(4*x)*e^(-3*x),x, algorithm="fricas")

[Out]

-3/25*cos(4*x)*e^(-3*x) + 4/25*e^(-3*x)*sin(4*x)

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Sympy [A]  time = 0.785049, size = 26, normalized size = 0.96 \[ \frac{4 e^{- 3 x} \sin{\left (4 x \right )}}{25} - \frac{3 e^{- 3 x} \cos{\left (4 x \right )}}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(4*x)/exp(3*x),x)

[Out]

4*exp(-3*x)*sin(4*x)/25 - 3*exp(-3*x)*cos(4*x)/25

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GIAC/XCAS [A]  time = 0.209619, size = 26, normalized size = 0.96 \[ -\frac{1}{25} \,{\left (3 \, \cos \left (4 \, x\right ) - 4 \, \sin \left (4 \, x\right )\right )} e^{\left (-3 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(4*x)*e^(-3*x),x, algorithm="giac")

[Out]

-1/25*(3*cos(4*x) - 4*sin(4*x))*e^(-3*x)