3.541 \(\int e^{-3 x} \cos (2 x) \, dx\)

Optimal. Leaf size=27 \[ \frac{2}{13} e^{-3 x} \sin (2 x)-\frac{3}{13} e^{-3 x} \cos (2 x) \]

[Out]

(-3*Cos[2*x])/(13*E^(3*x)) + (2*Sin[2*x])/(13*E^(3*x))

________________________________________________________________________________________

Rubi [A]  time = 0.0105735, 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, Rules used = {4433} \[ \frac{2}{13} e^{-3 x} \sin (2 x)-\frac{3}{13} e^{-3 x} \cos (2 x) \]

Antiderivative was successfully verified.

[In]

Int[Cos[2*x]/E^(3*x),x]

[Out]

(-3*Cos[2*x])/(13*E^(3*x)) + (2*Sin[2*x])/(13*E^(3*x))

Rule 4433

Int[Cos[(d_.) + (e_.)*(x_)]*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Simp[(b*c*Log[F]*F^(c*(a + b*x))*C
os[d + e*x])/(e^2 + b^2*c^2*Log[F]^2), x] + Simp[(e*F^(c*(a + b*x))*Sin[d + e*x])/(e^2 + b^2*c^2*Log[F]^2), x]
/; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 + b^2*c^2*Log[F]^2, 0]

Rubi steps

\begin{align*} \int e^{-3 x} \cos (2 x) \, dx &=-\frac{3}{13} e^{-3 x} \cos (2 x)+\frac{2}{13} e^{-3 x} \sin (2 x)\\ \end{align*}

Mathematica [A]  time = 0.0263124, size = 22, normalized size = 0.81 \[ \frac{1}{13} e^{-3 x} (2 \sin (2 x)-3 \cos (2 x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[2*x]/E^(3*x),x]

[Out]

(-3*Cos[2*x] + 2*Sin[2*x])/(13*E^(3*x))

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 22, normalized size = 0.8 \begin{align*} -{\frac{3\,{{\rm e}^{-3\,x}}\cos \left ( 2\,x \right ) }{13}}+{\frac{2\,{{\rm e}^{-3\,x}}\sin \left ( 2\,x \right ) }{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(2*x)/exp(3*x),x)

[Out]

-3/13*exp(-3*x)*cos(2*x)+2/13*exp(-3*x)*sin(2*x)

________________________________________________________________________________________

Maxima [A]  time = 0.946883, size = 26, normalized size = 0.96 \begin{align*} -\frac{1}{13} \,{\left (3 \, \cos \left (2 \, x\right ) - 2 \, \sin \left (2 \, x\right )\right )} e^{\left (-3 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(2*x)/exp(3*x),x, algorithm="maxima")

[Out]

-1/13*(3*cos(2*x) - 2*sin(2*x))*e^(-3*x)

________________________________________________________________________________________

Fricas [A]  time = 2.19539, size = 68, normalized size = 2.52 \begin{align*} -\frac{3}{13} \, \cos \left (2 \, x\right ) e^{\left (-3 \, x\right )} + \frac{2}{13} \, e^{\left (-3 \, x\right )} \sin \left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(2*x)/exp(3*x),x, algorithm="fricas")

[Out]

-3/13*cos(2*x)*e^(-3*x) + 2/13*e^(-3*x)*sin(2*x)

________________________________________________________________________________________

Sympy [A]  time = 0.477693, size = 26, normalized size = 0.96 \begin{align*} \frac{2 e^{- 3 x} \sin{\left (2 x \right )}}{13} - \frac{3 e^{- 3 x} \cos{\left (2 x \right )}}{13} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(2*x)/exp(3*x),x)

[Out]

2*exp(-3*x)*sin(2*x)/13 - 3*exp(-3*x)*cos(2*x)/13

________________________________________________________________________________________

Giac [A]  time = 1.10919, size = 26, normalized size = 0.96 \begin{align*} -\frac{1}{13} \,{\left (3 \, \cos \left (2 \, x\right ) - 2 \, \sin \left (2 \, x\right )\right )} e^{\left (-3 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(2*x)/exp(3*x),x, algorithm="giac")

[Out]

-1/13*(3*cos(2*x) - 2*sin(2*x))*e^(-3*x)