3.1.33 \(\int e^{-t} \cos (3 t) \, dt\) [33]

Optimal. Leaf size=27 \[ -\frac {1}{10} e^{-t} \cos (3 t)+\frac {3}{10} e^{-t} \sin (3 t) \]

[Out]

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

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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 {3}{10} e^{-t} \sin (3 t)-\frac {1}{10} e^{-t} \cos (3 t) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[3*t]/E^t,t]

[Out]

-1/10*Cos[3*t]/E^t + (3*Sin[3*t])/(10*E^t)

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^{-t} \cos (3 t) \, dt &=-\frac {1}{10} e^{-t} \cos (3 t)+\frac {3}{10} e^{-t} \sin (3 t)\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 20, normalized size = 0.74 \begin {gather*} -\frac {1}{10} e^{-t} (\cos (3 t)-3 \sin (3 t)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[3*t]/E^t,t]

[Out]

-1/10*(Cos[3*t] - 3*Sin[3*t])/E^t

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Mathics [A]
time = 1.89, size = 20, normalized size = 0.74 \begin {gather*} \frac {\left (-\text {Cos}\left [3 t\right ]+3 \text {Sin}\left [3 t\right ]\right ) E^{-t}}{10} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[Cos[3*t]/E^t,t]')

[Out]

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

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

method	result	size
default	\(-\frac {{\mathrm e}^{-t} \cos \left (3 t \right )}{10}+\frac {3 \,{\mathrm e}^{-t} \sin \left (3 t \right )}{10}\)	\(22\)
norman	\(\frac {\left (-\frac {1}{10}+\frac {\left (\tan ^{2}\left (\frac {3 t}{2}\right )\right )}{10}+\frac {3 \tan \left (\frac {3 t}{2}\right )}{5}\right ) {\mathrm e}^{-t}}{1+\tan ^{2}\left (\frac {3 t}{2}\right )}\)	\(32\)
risch	\(-\frac {{\mathrm e}^{\left (-1+3 i\right ) t}}{20}-\frac {3 i {\mathrm e}^{\left (-1+3 i\right ) t}}{20}-\frac {{\mathrm e}^{\left (-1-3 i\right ) t}}{20}+\frac {3 i {\mathrm e}^{\left (-1-3 i\right ) t}}{20}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 17, normalized size = 0.63 \begin {gather*} -\frac {1}{10} \, {\left (\cos \left (3 \, t\right ) - 3 \, \sin \left (3 \, t\right )\right )} e^{\left (-t\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*(cos(3*t) - 3*sin(3*t))*e^(-t)

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Fricas [A]
time = 0.33, size = 21, normalized size = 0.78 \begin {gather*} -\frac {1}{10} \, \cos \left (3 \, t\right ) e^{\left (-t\right )} + \frac {3}{10} \, e^{\left (-t\right )} \sin \left (3 \, t\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*cos(3*t)*e^(-t) + 3/10*e^(-t)*sin(3*t)

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Sympy [A]
time = 0.18, size = 20, normalized size = 0.74 \begin {gather*} \frac {3 e^{- t} \sin {\left (3 t \right )}}{10} - \frac {e^{- t} \cos {\left (3 t \right )}}{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.00, size = 22, normalized size = 0.81 \begin {gather*} \mathrm {e}^{-t} \left (-\frac {\cos \left (3 t\right )}{10}+\frac {3}{10} \sin \left (3 t\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*(cos(3*t) - 3*sin(3*t))/e^t

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Mupad [B]
time = 0.03, size = 17, normalized size = 0.63 \begin {gather*} -\frac {{\mathrm {e}}^{-t}\,\left (\cos \left (3\,t\right )-3\,\sin \left (3\,t\right )\right )}{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(exp(-t)*(cos(3*t) - 3*sin(3*t)))/10

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