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

Optimal. Leaf size=23 \[ \frac{1}{10} e^{-3 x} \sin (x)-\frac{3}{10} e^{-3 x} \cos (x) \]

[Out]

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

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Rubi [A]  time = 0.0089591, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {4433} \[ \frac{1}{10} e^{-3 x} \sin (x)-\frac{3}{10} e^{-3 x} \cos (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0131711, size = 16, normalized size = 0.7 \[ \frac{1}{10} e^{-3 x} (\sin (x)-3 \cos (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*Cos[x] + Sin[x])/(10*E^(3*x))

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Maple [A]  time = 0.004, size = 18, normalized size = 0.8 \begin{align*} -{\frac{3\,{{\rm e}^{-3\,x}}\cos \left ( x \right ) }{10}}+{\frac{{{\rm e}^{-3\,x}}\sin \left ( x \right ) }{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.960083, size = 20, normalized size = 0.87 \begin{align*} -\frac{1}{10} \,{\left (3 \, \cos \left (x\right ) - \sin \left (x\right )\right )} e^{\left (-3 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*(3*cos(x) - sin(x))*e^(-3*x)

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Fricas [A]  time = 2.17241, size = 62, normalized size = 2.7 \begin{align*} -\frac{3}{10} \, \cos \left (x\right ) e^{\left (-3 \, x\right )} + \frac{1}{10} \, e^{\left (-3 \, x\right )} \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 0.4711, size = 20, normalized size = 0.87 \begin{align*} \frac{e^{- 3 x} \sin{\left (x \right )}}{10} - \frac{3 e^{- 3 x} \cos{\left (x \right )}}{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.0824, size = 20, normalized size = 0.87 \begin{align*} -\frac{1}{10} \,{\left (3 \, \cos \left (x\right ) - \sin \left (x\right )\right )} e^{\left (-3 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*(3*cos(x) - sin(x))*e^(-3*x)