3.78 \(\int e^{2 x} \sin (3 x) \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0093931, 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} \[ \frac{2}{13} e^{2 x} \sin (3 x)-\frac{3}{13} e^{2 x} \cos (3 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0349009, size = 22, normalized size = 0.81 \[ \frac{1}{13} e^{2 x} (2 \sin (3 x)-3 \cos (3 x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.931119, size = 26, normalized size = 0.96 \begin{align*} -\frac{1}{13} \,{\left (3 \, \cos \left (3 \, x\right ) - 2 \, \sin \left (3 \, x\right )\right )} e^{\left (2 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.32407, size = 65, normalized size = 2.41 \begin{align*} -\frac{3}{13} \, \cos \left (3 \, x\right ) e^{\left (2 \, x\right )} + \frac{2}{13} \, e^{\left (2 \, x\right )} \sin \left (3 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.331116, size = 26, normalized size = 0.96 \begin{align*} \frac{2 e^{2 x} \sin{\left (3 x \right )}}{13} - \frac{3 e^{2 x} \cos{\left (3 x \right )}}{13} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.05647, size = 26, normalized size = 0.96 \begin{align*} -\frac{1}{13} \,{\left (3 \, \cos \left (3 \, x\right ) - 2 \, \sin \left (3 \, x\right )\right )} e^{\left (2 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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