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]

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

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Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 0.70 \[ \frac {1}{10} e^{-3 x} (\sin (x)-3 \cos (x)) \]

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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fricas [A]  time = 0.69, size = 17, normalized size = 0.74 \[ -\frac {3}{10} \, \cos \relax (x) e^{\left (-3 \, x\right )} + \frac {1}{10} \, e^{\left (-3 \, x\right )} \sin \relax (x) \]

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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giac [A]  time = 0.14, size = 15, normalized size = 0.65 \[ -\frac {1}{10} \, {\left (3 \, \cos \relax (x) - \sin \relax (x)\right )} e^{\left (-3 \, x\right )} \]

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)

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maple [A]  time = 0.02, size = 18, normalized size = 0.78 \[ -\frac {3 \,{\mathrm e}^{-3 x} \cos \relax (x )}{10}+\frac {{\mathrm e}^{-3 x} \sin \relax (x )}{10} \]

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 = 0.33, size = 15, normalized size = 0.65 \[ -\frac {1}{10} \, {\left (3 \, \cos \relax (x) - \sin \relax (x)\right )} e^{\left (-3 \, x\right )} \]

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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mupad [B]  time = 0.02, size = 15, normalized size = 0.65 \[ -\frac {{\mathrm {e}}^{-3\,x}\,\left (3\,\cos \relax (x)-\sin \relax (x)\right )}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(exp(-3*x)*(3*cos(x) - sin(x)))/10

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sympy [A]  time = 0.40, size = 20, normalized size = 0.87 \[ \frac {e^{- 3 x} \sin {\relax (x )}}{10} - \frac {3 e^{- 3 x} \cos {\relax (x )}}{10} \]

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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