3.1.32 \(\int e^{2 t} \sin (3 t) \, dt\) [32]

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

[Out]

-3/13*exp(2*t)*cos(3*t)+2/13*exp(2*t)*sin(3*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 = {4517} \begin {gather*} \frac {2}{13} e^{2 t} \sin (3 t)-\frac {3}{13} e^{2 t} \cos (3 t) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(2*t)*Sin[3*t],t]

[Out]

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

Rule 4517

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sin[(d_.) + (e_.)*(x_)], x_Symbol] :> Simp[b*c*Log[F]*F^(c*(a + b*x))*(S
in[d + e*x]/(e^2 + b^2*c^2*Log[F]^2)), x] - Simp[e*F^(c*(a + b*x))*(Cos[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^{2 t} \sin (3 t) \, dt &=-\frac {3}{13} e^{2 t} \cos (3 t)+\frac {2}{13} e^{2 t} \sin (3 t)\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 22, normalized size = 0.81 \begin {gather*} \frac {1}{13} e^{2 t} (-3 \cos (3 t)+2 \sin (3 t)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(2*t)*Sin[3*t],t]

[Out]

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

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Mathics [A]
time = 1.89, size = 20, normalized size = 0.74 \begin {gather*} \frac {\left (-3 \text {Cos}\left [3 t\right ]+2 \text {Sin}\left [3 t\right ]\right ) E^{2 t}}{13} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[E^(2*t)*Sin[3*t],t]')

[Out]

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

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Maple [A]
time = 0.02, size = 22, normalized size = 0.81

method	result	size
default	\(-\frac {3 \,{\mathrm e}^{2 t} \cos \left (3 t \right )}{13}+\frac {2 \,{\mathrm e}^{2 t} \sin \left (3 t \right )}{13}\)	\(22\)
risch	\(-\frac {3 \,{\mathrm e}^{\left (2+3 i\right ) t}}{26}-\frac {i {\mathrm e}^{\left (2+3 i\right ) t}}{13}-\frac {3 \,{\mathrm e}^{\left (2-3 i\right ) t}}{26}+\frac {i {\mathrm e}^{\left (2-3 i\right ) t}}{13}\)	\(36\)
norman	\(\frac {\frac {4 \,{\mathrm e}^{2 t} \tan \left (\frac {3 t}{2}\right )}{13}+\frac {3 \,{\mathrm e}^{2 t} \left (\tan ^{2}\left (\frac {3 t}{2}\right )\right )}{13}-\frac {3 \,{\mathrm e}^{2 t}}{13}}{1+\tan ^{2}\left (\frac {3 t}{2}\right )}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*t)*sin(3*t),t,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.26, size = 19, normalized size = 0.70 \begin {gather*} -\frac {1}{13} \, {\left (3 \, \cos \left (3 \, t\right ) - 2 \, \sin \left (3 \, t\right )\right )} e^{\left (2 \, t\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*t)*sin(3*t),t, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.37, size = 21, normalized size = 0.78 \begin {gather*} -\frac {3}{13} \, \cos \left (3 \, t\right ) e^{\left (2 \, t\right )} + \frac {2}{13} \, e^{\left (2 \, t\right )} \sin \left (3 \, t\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*t)*sin(3*t),t, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.11, size = 26, normalized size = 0.96 \begin {gather*} \frac {2 e^{2 t} \sin {\left (3 t \right )}}{13} - \frac {3 e^{2 t} \cos {\left (3 t \right )}}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*t)*sin(3*t),t)

[Out]

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

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Giac [A]
time = 0.00, size = 22, normalized size = 0.81 \begin {gather*} \mathrm {e}^{2 t} \left (-\frac {3}{13} \cos \left (3 t\right )+\frac {2}{13} \sin \left (3 t\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*t)*sin(3*t),t)

[Out]

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

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Mupad [B]
time = 0.03, size = 19, normalized size = 0.70 \begin {gather*} -\frac {{\mathrm {e}}^{2\,t}\,\left (3\,\cos \left (3\,t\right )-2\,\sin \left (3\,t\right )\right )}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(3*t)*exp(2*t),t)

[Out]

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

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