\(\int e^{-3 x} \cos (x) \, dx\) [788]

Optimal result

Integrand size = 8, antiderivative size = 23 \[ \int e^{-3 x} \cos (x) \, dx=-\frac {3}{10} e^{-3 x} \cos (x)+\frac {1}{10} e^{-3 x} \sin (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4518} \[ \int e^{-3 x} \cos (x) \, dx=\frac {1}{10} e^{-3 x} \sin (x)-\frac {3}{10} e^{-3 x} \cos (x) \]

[In]

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

[Out]

(-3*Cos[x])/(10*E^(3*x)) + Sin[x]/(10*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}{10} e^{-3 x} \cos (x)+\frac {1}{10} e^{-3 x} \sin (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.70 \[ \int e^{-3 x} \cos (x) \, dx=\frac {1}{10} e^{-3 x} (-3 \cos (x)+\sin (x)) \]

[In]

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

[Out]

(-3*Cos[x] + Sin[x])/(10*E^(3*x))

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.61

method	result	size
parallelrisch	\(\frac {\left (-3 \cos \left (x \right )+\sin \left (x \right )\right ) {\mathrm e}^{-3 x}}{10}\)	\(14\)
default	\(-\frac {3 \,{\mathrm e}^{-3 x} \cos \left (x \right )}{10}+\frac {{\mathrm e}^{-3 x} \sin \left (x \right )}{10}\)	\(18\)
norman	\(\frac {\left (-\frac {3}{10}+\frac {3 \tan \left (\frac {x}{2}\right )^{2}}{10}+\frac {\tan \left (\frac {x}{2}\right )}{5}\right ) {\mathrm e}^{-3 x}}{1+\tan \left (\frac {x}{2}\right )^{2}}\)	\(34\)
risch	\(-\frac {3 \,{\mathrm e}^{\left (-3+i\right ) x}}{20}-\frac {i {\mathrm e}^{\left (-3+i\right ) x}}{20}-\frac {3 \,{\mathrm e}^{\left (-3-i\right ) x}}{20}+\frac {i {\mathrm e}^{\left (-3-i\right ) x}}{20}\)	\(36\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int e^{-3 x} \cos (x) \, dx=-\frac {3}{10} \, \cos \left (x\right ) e^{\left (-3 \, x\right )} + \frac {1}{10} \, e^{\left (-3 \, x\right )} \sin \left (x\right ) \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int e^{-3 x} \cos (x) \, dx=\frac {e^{- 3 x} \sin {\left (x \right )}}{10} - \frac {3 e^{- 3 x} \cos {\left (x \right )}}{10} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.65 \[ \int e^{-3 x} \cos (x) \, dx=-\frac {1}{10} \, {\left (3 \, \cos \left (x\right ) - \sin \left (x\right )\right )} e^{\left (-3 \, x\right )} \]

[In]

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

[Out]

-1/10*(3*cos(x) - sin(x))*e^(-3*x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.65 \[ \int e^{-3 x} \cos (x) \, dx=-\frac {1}{10} \, {\left (3 \, \cos \left (x\right ) - \sin \left (x\right )\right )} e^{\left (-3 \, x\right )} \]

[In]

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

[Out]

-1/10*(3*cos(x) - sin(x))*e^(-3*x)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.65 \[ \int e^{-3 x} \cos (x) \, dx=-\frac {{\mathrm {e}}^{-3\,x}\,\left (3\,\cos \left (x\right )-\sin \left (x\right )\right )}{10} \]

[In]

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

[Out]

-(exp(-3*x)*(3*cos(x) - sin(x)))/10