3.702 \(\int e^{6 x} \sin (3 x) \, dx\)

Optimal. Leaf size=27 \[ \frac{2}{15} e^{6 x} \sin (3 x)-\frac{1}{15} e^{6 x} \cos (3 x) \]

[Out]

-(E^(6*x)*Cos[3*x])/15 + (2*E^(6*x)*Sin[3*x])/15

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Rubi [A]  time = 0.011199, 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} \[ \frac{2}{15} e^{6 x} \sin (3 x)-\frac{1}{15} e^{6 x} \cos (3 x) \]

Antiderivative was successfully verified.

[In]

Int[E^(6*x)*Sin[3*x],x]

[Out]

-(E^(6*x)*Cos[3*x])/15 + (2*E^(6*x)*Sin[3*x])/15

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^{6 x} \sin (3 x) \, dx &=-\frac{1}{15} e^{6 x} \cos (3 x)+\frac{2}{15} e^{6 x} \sin (3 x)\\ \end{align*}

Mathematica [A]  time = 0.0228407, size = 20, normalized size = 0.74 \[ -\frac{1}{15} e^{6 x} (\cos (3 x)-2 \sin (3 x)) \]

Antiderivative was successfully verified.

[In]

Integrate[E^(6*x)*Sin[3*x],x]

[Out]

-(E^(6*x)*(Cos[3*x] - 2*Sin[3*x]))/15

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Maple [A]  time = 0.022, size = 22, normalized size = 0.8 \begin{align*} -{\frac{{{\rm e}^{6\,x}}\cos \left ( 3\,x \right ) }{15}}+{\frac{2\,{{\rm e}^{6\,x}}\sin \left ( 3\,x \right ) }{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(6*x)*sin(3*x),x)

[Out]

-1/15*exp(6*x)*cos(3*x)+2/15*exp(6*x)*sin(3*x)

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Maxima [A]  time = 0.956844, size = 23, normalized size = 0.85 \begin{align*} -\frac{1}{15} \,{\left (\cos \left (3 \, x\right ) - 2 \, \sin \left (3 \, x\right )\right )} e^{\left (6 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(6*x)*sin(3*x),x, algorithm="maxima")

[Out]

-1/15*(cos(3*x) - 2*sin(3*x))*e^(6*x)

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Fricas [A]  time = 0.917258, size = 65, normalized size = 2.41 \begin{align*} -\frac{1}{15} \, \cos \left (3 \, x\right ) e^{\left (6 \, x\right )} + \frac{2}{15} \, e^{\left (6 \, x\right )} \sin \left (3 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(6*x)*sin(3*x),x, algorithm="fricas")

[Out]

-1/15*cos(3*x)*e^(6*x) + 2/15*e^(6*x)*sin(3*x)

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Sympy [A]  time = 0.318589, size = 24, normalized size = 0.89 \begin{align*} \frac{2 e^{6 x} \sin{\left (3 x \right )}}{15} - \frac{e^{6 x} \cos{\left (3 x \right )}}{15} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(6*x)*sin(3*x),x)

[Out]

2*exp(6*x)*sin(3*x)/15 - exp(6*x)*cos(3*x)/15

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Giac [A]  time = 1.2606, size = 23, normalized size = 0.85 \begin{align*} -\frac{1}{15} \,{\left (\cos \left (3 \, x\right ) - 2 \, \sin \left (3 \, x\right )\right )} e^{\left (6 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(6*x)*sin(3*x),x, algorithm="giac")

[Out]

-1/15*(cos(3*x) - 2*sin(3*x))*e^(6*x)