3.4.43 \(\int e^{-3 x} \cos (4 x) \, dx\) [343]

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

[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 = {4518} \begin {gather*} \frac {4}{25} e^{-3 x} \sin (4 x)-\frac {3}{25} e^{-3 x} \cos (4 x) \end {gather*}

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 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*} \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 = 0.02, size = 22, normalized size = 0.81 \begin {gather*} \frac {1}{25} e^{-3 x} (-3 \cos (4 x)+4 \sin (4 x)) \end {gather*}

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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Mathics [A]
time = 2.00, size = 20, normalized size = 0.74 \begin {gather*} \frac {\left (-3 \text {Cos}\left [4 x\right ]+4 \text {Sin}\left [4 x\right ]\right ) E^{-3 x}}{25} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.03, size = 22, normalized size = 0.81

method	result	size
default	\(-\frac {3 \,{\mathrm e}^{-3 x} \cos \left (4 x \right )}{25}+\frac {4 \,{\mathrm e}^{-3 x} \sin \left (4 x \right )}{25}\)	\(22\)
norman	\(\frac {\left (-\frac {3}{25}+\frac {3 \left (\tan ^{2}\left (2 x \right )\right )}{25}+\frac {8 \tan \left (2 x \right )}{25}\right ) {\mathrm e}^{-3 x}}{1+\tan ^{2}\left (2 x \right )}\)	\(34\)
risch	\(-\frac {3 \,{\mathrm e}^{\left (-3+4 i\right ) x}}{50}-\frac {2 i {\mathrm e}^{\left (-3+4 i\right ) x}}{25}-\frac {3 \,{\mathrm e}^{\left (-3-4 i\right ) x}}{50}+\frac {2 i {\mathrm e}^{\left (-3-4 i\right ) x}}{25}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(4*x)/exp(3*x),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.31, size = 19, normalized size = 0.70 \begin {gather*} -\frac {1}{25} \, {\left (3 \, \cos \left (4 \, x\right ) - 4 \, \sin \left (4 \, x\right )\right )} e^{\left (-3 \, x\right )} \end {gather*}

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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Fricas [A]
time = 0.33, size = 21, normalized size = 0.78 \begin {gather*} -\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 ) \end {gather*}

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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Sympy [A]
time = 0.19, size = 26, normalized size = 0.96 \begin {gather*} \frac {4 e^{- 3 x} \sin {\left (4 x \right )}}{25} - \frac {3 e^{- 3 x} \cos {\left (4 x \right )}}{25} \end {gather*}

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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Giac [A]
time = 0.00, size = 24, normalized size = 0.89 \begin {gather*} \mathrm {e}^{-3 x} \left (-\frac {3}{25} \cos \left (4 x\right )+\frac {4}{25} \sin \left (4 x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 \begin {gather*} -\frac {{\mathrm {e}}^{-3\,x}\,\left (3\,\cos \left (4\,x\right )-4\,\sin \left (4\,x\right )\right )}{25} \end {gather*}

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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