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]

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

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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 = {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*}

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Mathematica [A]  time = 0.02, 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]

-1/15*(E^(6*x)*(Cos[3*x] - 2*Sin[3*x]))

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fricas [A]  time = 0.40, size = 21, normalized size = 0.78 \[ -\frac {1}{15} \, \cos \left (3 \, x\right ) e^{\left (6 \, x\right )} + \frac {2}{15} \, e^{\left (6 \, x\right )} \sin \left (3 \, x\right ) \]

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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giac [A]  time = 0.21, size = 17, normalized size = 0.63 \[ -\frac {1}{15} \, {\left (\cos \left (3 \, x\right ) - 2 \, \sin \left (3 \, x\right )\right )} e^{\left (6 \, x\right )} \]

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)

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maple [A]  time = 0.08, size = 22, normalized size = 0.81 \[ -\frac {\cos \left (3 x \right ) {\mathrm e}^{6 x}}{15}+\frac {2 \,{\mathrm e}^{6 x} \sin \left (3 x \right )}{15} \]

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 = 0.93, size = 17, normalized size = 0.63 \[ -\frac {1}{15} \, {\left (\cos \left (3 \, x\right ) - 2 \, \sin \left (3 \, x\right )\right )} e^{\left (6 \, x\right )} \]

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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mupad [B]  time = 0.03, size = 19, normalized size = 0.70 \[ -\frac {{\mathrm {e}}^{6\,x}\,\left (3\,\cos \left (3\,x\right )-6\,\sin \left (3\,x\right )\right )}{45} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(exp(6*x)*(3*cos(3*x) - 6*sin(3*x)))/45

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sympy [A]  time = 0.30, size = 24, normalized size = 0.89 \[ \frac {2 e^{6 x} \sin {\left (3 x \right )}}{15} - \frac {e^{6 x} \cos {\left (3 x \right )}}{15} \]

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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