3.343 \(\int e^{-3 x} \cos (4 x) \, dx\)

Optimal. Leaf size=27 \[ \frac {4}{25} e^{-3 x} \sin (4 x)-\frac {3}{25} e^{-3 x} \cos (4 x) \]

[Out]

-3/25*cos(4*x)/exp(3*x)+4/25*sin(4*x)/exp(3*x)

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Rubi [A]  time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4433} \[ \frac {4}{25} e^{-3 x} \sin (4 x)-\frac {3}{25} e^{-3 x} \cos (4 x) \]

Antiderivative was successfully verified.

[In]

Int[Cos[4*x]/E^(3*x),x]

[Out]

(-3*Cos[4*x])/(25*E^(3*x)) + (4*Sin[4*x])/(25*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 (4 x) \, dx &=-\frac {3}{25} e^{-3 x} \cos (4 x)+\frac {4}{25} e^{-3 x} \sin (4 x)\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 22, normalized size = 0.81 \[ \frac {1}{25} e^{-3 x} (4 \sin (4 x)-3 \cos (4 x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[4*x]/E^(3*x),x]

[Out]

(-3*Cos[4*x] + 4*Sin[4*x])/(25*E^(3*x))

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fricas [A]  time = 0.42, size = 21, normalized size = 0.78 \[ -\frac {3}{25} \, \cos \left (4 \, x\right ) e^{\left (-3 \, x\right )} + \frac {4}{25} \, e^{\left (-3 \, x\right )} \sin \left (4 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(4*x)/exp(3*x),x, algorithm="fricas")

[Out]

-3/25*cos(4*x)*e^(-3*x) + 4/25*e^(-3*x)*sin(4*x)

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giac [A]  time = 0.92, size = 19, normalized size = 0.70 \[ -\frac {1}{25} \, {\left (3 \, \cos \left (4 \, x\right ) - 4 \, \sin \left (4 \, x\right )\right )} e^{\left (-3 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(4*x)/exp(3*x),x, algorithm="giac")

[Out]

-1/25*(3*cos(4*x) - 4*sin(4*x))*e^(-3*x)

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maple [A]  time = 0.02, size = 22, normalized size = 0.81 \[ -\frac {3 \cos \left (4 x \right ) {\mathrm e}^{-3 x}}{25}+\frac {4 \,{\mathrm e}^{-3 x} \sin \left (4 x \right )}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(4*x)/exp(3*x),x)

[Out]

-3/25*exp(-3*x)*cos(4*x)+4/25*exp(-3*x)*sin(4*x)

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maxima [A]  time = 0.49, size = 19, normalized size = 0.70 \[ -\frac {1}{25} \, {\left (3 \, \cos \left (4 \, x\right ) - 4 \, \sin \left (4 \, x\right )\right )} e^{\left (-3 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(4*x)/exp(3*x),x, algorithm="maxima")

[Out]

-1/25*(3*cos(4*x) - 4*sin(4*x))*e^(-3*x)

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mupad [B]  time = 0.03, size = 19, normalized size = 0.70 \[ -\frac {{\mathrm {e}}^{-3\,x}\,\left (3\,\cos \left (4\,x\right )-4\,\sin \left (4\,x\right )\right )}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(4*x)*exp(-3*x),x)

[Out]

-(exp(-3*x)*(3*cos(4*x) - 4*sin(4*x)))/25

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sympy [A]  time = 0.49, size = 26, normalized size = 0.96 \[ \frac {4 e^{- 3 x} \sin {\left (4 x \right )}}{25} - \frac {3 e^{- 3 x} \cos {\left (4 x \right )}}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(4*x)/exp(3*x),x)

[Out]

4*exp(-3*x)*sin(4*x)/25 - 3*exp(-3*x)*cos(4*x)/25

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