### 3.32 $$\int e^{2 t} \sin (3 t) \, dt$$

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

[Out]

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

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Rubi [A]  time = 0.011464, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.1, Rules used = {4432} $\frac{2}{13} e^{2 t} \sin (3 t)-\frac{3}{13} e^{2 t} \cos (3 t)$

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 4432

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*}

Mathematica [A]  time = 0.0324587, size = 22, normalized size = 0.81 $\frac{1}{13} e^{2 t} (2 \sin (3 t)-3 \cos (3 t))$

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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Maple [A]  time = 0.004, size = 22, normalized size = 0.8 \begin{align*} -{\frac{3\,{{\rm e}^{2\,t}}\cos \left ( 3\,t \right ) }{13}}+{\frac{2\,{{\rm e}^{2\,t}}\sin \left ( 3\,t \right ) }{13}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.925916, size = 26, normalized size = 0.96 \begin{align*} -\frac{1}{13} \,{\left (3 \, \cos \left (3 \, t\right ) - 2 \, \sin \left (3 \, t\right )\right )} e^{\left (2 \, t\right )} \end{align*}

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 = 1.92505, size = 65, normalized size = 2.41 \begin{align*} -\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{align*}

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 = 0.293081, size = 26, normalized size = 0.96 \begin{align*} \frac{2 e^{2 t} \sin{\left (3 t \right )}}{13} - \frac{3 e^{2 t} \cos{\left (3 t \right )}}{13} \end{align*}

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 = 1.0594, size = 26, normalized size = 0.96 \begin{align*} -\frac{1}{13} \,{\left (3 \, \cos \left (3 \, t\right ) - 2 \, \sin \left (3 \, t\right )\right )} e^{\left (2 \, t\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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