3.541 \(\int e^{-3 x} \cos (2 x) \, dx\)

Optimal. Leaf size=27 \[ \frac {2}{13} e^{-3 x} \sin (2 x)-\frac {3}{13} e^{-3 x} \cos (2 x) \]

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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} \[ \frac {2}{13} e^{-3 x} \sin (2 x)-\frac {3}{13} e^{-3 x} \cos (2 x) \]

Antiderivative was successfully verified.

[In]

Int[Cos[2*x]/E^(3*x),x]

[Out]

(-3*Cos[2*x])/(13*E^(3*x)) + (2*Sin[2*x])/(13*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 (2 x) \, dx &=-\frac {3}{13} e^{-3 x} \cos (2 x)+\frac {2}{13} e^{-3 x} \sin (2 x)\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 22, normalized size = 0.81 \[ \frac {1}{13} e^{-3 x} (2 \sin (2 x)-3 \cos (2 x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[2*x]/E^(3*x),x]

[Out]

(-3*Cos[2*x] + 2*Sin[2*x])/(13*E^(3*x))

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{-3 x} \cos (2 x) \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[Cos[2*x]/E^(3*x),x]

[Out]

Could not integrate

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fricas [A]  time = 1.19, size = 21, normalized size = 0.78 \[ -\frac {3}{13} \, \cos \left (2 \, x\right ) e^{\left (-3 \, x\right )} + \frac {2}{13} \, e^{\left (-3 \, x\right )} \sin \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(2*x)/exp(3*x),x, algorithm="fricas")

[Out]

-3/13*cos(2*x)*e^(-3*x) + 2/13*e^(-3*x)*sin(2*x)

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giac [A]  time = 0.64, size = 19, normalized size = 0.70 \[ -\frac {1}{13} \, {\left (3 \, \cos \left (2 \, x\right ) - 2 \, \sin \left (2 \, x\right )\right )} e^{\left (-3 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(2*x)/exp(3*x),x, algorithm="giac")

[Out]

-1/13*(3*cos(2*x) - 2*sin(2*x))*e^(-3*x)

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maple [A]  time = 0.05, size = 22, normalized size = 0.81

method	result	size
default	\(-\frac {3 \,{\mathrm e}^{-3 x} \cos \left (2 x \right )}{13}+\frac {2 \,{\mathrm e}^{-3 x} \sin \left (2 x \right )}{13}\)	\(22\)
norman	\(\frac {\left (-\frac {3}{13}+\frac {3 \left (\tan ^{2}\relax (x )\right )}{13}+\frac {4 \tan \relax (x )}{13}\right ) {\mathrm e}^{-3 x}}{1+\tan ^{2}\relax (x )}\)	\(28\)
risch	\(-\frac {3 \,{\mathrm e}^{\left (-3+2 i\right ) x}}{26}-\frac {i {\mathrm e}^{\left (-3+2 i\right ) x}}{13}-\frac {3 \,{\mathrm e}^{\left (-3-2 i\right ) x}}{26}+\frac {i {\mathrm e}^{\left (-3-2 i\right ) x}}{13}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(2*x)/exp(3*x),x,method=_RETURNVERBOSE)

[Out]

-3/13*exp(-3*x)*cos(2*x)+2/13*exp(-3*x)*sin(2*x)

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maxima [A]  time = 0.55, size = 19, normalized size = 0.70 \[ -\frac {1}{13} \, {\left (3 \, \cos \left (2 \, x\right ) - 2 \, \sin \left (2 \, x\right )\right )} e^{\left (-3 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(2*x)/exp(3*x),x, algorithm="maxima")

[Out]

-1/13*(3*cos(2*x) - 2*sin(2*x))*e^(-3*x)

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mupad [B]  time = 0.03, size = 19, normalized size = 0.70 \[ -\frac {{\mathrm {e}}^{-3\,x}\,\left (3\,\cos \left (2\,x\right )-2\,\sin \left (2\,x\right )\right )}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(2*x)*exp(-3*x),x)

[Out]

-(exp(-3*x)*(3*cos(2*x) - 2*sin(2*x)))/13

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sympy [A]  time = 0.47, size = 26, normalized size = 0.96 \[ \frac {2 e^{- 3 x} \sin {\left (2 x \right )}}{13} - \frac {3 e^{- 3 x} \cos {\left (2 x \right )}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(2*x)/exp(3*x),x)

[Out]

2*exp(-3*x)*sin(2*x)/13 - 3*exp(-3*x)*cos(2*x)/13

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