\(\int e^{-3 x} \cos (2 x) \, dx\) [541]

Optimal result

Integrand size = 10, antiderivative size = 27 \[ \int e^{-3 x} \cos (2 x) \, dx=-\frac {3}{13} e^{-3 x} \cos (2 x)+\frac {2}{13} e^{-3 x} \sin (2 x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4518} \[ \int e^{-3 x} \cos (2 x) \, dx=\frac {2}{13} e^{-3 x} \sin (2 x)-\frac {3}{13} e^{-3 x} \cos (2 x) \]

[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 4518

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*} \text {integral}& = -\frac {3}{13} e^{-3 x} \cos (2 x)+\frac {2}{13} e^{-3 x} \sin (2 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int e^{-3 x} \cos (2 x) \, dx=\frac {1}{13} e^{-3 x} (-3 \cos (2 x)+2 \sin (2 x)) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74

method	result	size
parallelrisch	\(\frac {{\mathrm e}^{-3 x} \left (-3 \cos \left (2 x \right )+2 \sin \left (2 x \right )\right )}{13}\)	\(20\)
default	\(-\frac {3 \,{\mathrm e}^{-3 x} \cos \left (2 x \right )}{13}+\frac {2 \,{\mathrm e}^{-3 x} \sin \left (2 x \right )}{13}\)	\(22\)
norman	\(\frac {\left (-\frac {3}{13}+\frac {3 \left (\tan ^{2}\left (x \right )\right )}{13}+\frac {4 \tan \left (x \right )}{13}\right ) {\mathrm e}^{-3 x}}{1+\tan ^{2}\left (x \right )}\)	\(28\)
risch	\(-\frac {3 \,{\mathrm e}^{\left (-3+2 i\right ) x}}{26}-\frac {i {\mathrm e}^{\left (-3+2 i\right ) x}}{13}-\frac {3 \,{\mathrm e}^{\left (-3-2 i\right ) x}}{26}+\frac {i {\mathrm e}^{\left (-3-2 i\right ) x}}{13}\)	\(36\)

[In]

int(cos(2*x)/exp(3*x),x,method=_RETURNVERBOSE)

[Out]

1/13*exp(-3*x)*(-3*cos(2*x)+2*sin(2*x))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int e^{-3 x} \cos (2 x) \, dx=-\frac {3}{13} \, \cos \left (2 \, x\right ) e^{\left (-3 \, x\right )} + \frac {2}{13} \, e^{\left (-3 \, x\right )} \sin \left (2 \, x\right ) \]

[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] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int e^{-3 x} \cos (2 x) \, dx=\frac {2 e^{- 3 x} \sin {\left (2 x \right )}}{13} - \frac {3 e^{- 3 x} \cos {\left (2 x \right )}}{13} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int e^{-3 x} \cos (2 x) \, dx=-\frac {1}{13} \, {\left (3 \, \cos \left (2 \, x\right ) - 2 \, \sin \left (2 \, x\right )\right )} e^{\left (-3 \, x\right )} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int e^{-3 x} \cos (2 x) \, dx=-\frac {1}{13} \, {\left (3 \, \cos \left (2 \, x\right ) - 2 \, \sin \left (2 \, x\right )\right )} e^{\left (-3 \, x\right )} \]

[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)

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int e^{-3 x} \cos (2 x) \, dx=-\frac {{\mathrm {e}}^{-3\,x}\,\left (3\,\cos \left (2\,x\right )-2\,\sin \left (2\,x\right )\right )}{13} \]

[In]

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

[Out]

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