3.8.2 \(\int e^{6 x} \sin (3 x) \, dx\) [702]

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

[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 = {4517} \begin {gather*} \frac {2}{15} e^{6 x} \sin (3 x)-\frac {1}{15} e^{6 x} \cos (3 x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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^{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 \begin {gather*} -\frac {1}{15} e^{6 x} (\cos (3 x)-2 \sin (3 x)) \end {gather*}

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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Maple [A]
time = 0.03, size = 22, normalized size = 0.81

method	result	size
default	\(-\frac {{\mathrm e}^{6 x} \cos \left (3 x \right )}{15}+\frac {2 \,{\mathrm e}^{6 x} \sin \left (3 x \right )}{15}\)	\(22\)
risch	\(-\frac {{\mathrm e}^{\left (6+3 i\right ) x}}{30}-\frac {i {\mathrm e}^{\left (6+3 i\right ) x}}{15}-\frac {{\mathrm e}^{\left (6-3 i\right ) x}}{30}+\frac {i {\mathrm e}^{\left (6-3 i\right ) x}}{15}\)	\(36\)
norman	\(\frac {\frac {4 \,{\mathrm e}^{6 x} \tan \left (\frac {3 x}{2}\right )}{15}+\frac {{\mathrm e}^{6 x} \left (\tan ^{2}\left (\frac {3 x}{2}\right )\right )}{15}-\frac {{\mathrm e}^{6 x}}{15}}{1+\tan ^{2}\left (\frac {3 x}{2}\right )}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(6*x)*sin(3*x),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 17, normalized size = 0.63 \begin {gather*} -\frac {1}{15} \, {\left (\cos \left (3 \, x\right ) - 2 \, \sin \left (3 \, x\right )\right )} e^{\left (6 \, x\right )} \end {gather*}

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 = 0.36, size = 21, normalized size = 0.78 \begin {gather*} -\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 {gather*}

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 = 0.08, size = 24, normalized size = 0.89 \begin {gather*} \frac {2 e^{6 x} \sin {\left (3 x \right )}}{15} - \frac {e^{6 x} \cos {\left (3 x \right )}}{15} \end {gather*}

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 = 5.58, size = 17, normalized size = 0.63 \begin {gather*} -\frac {1}{15} \, {\left (\cos \left (3 \, x\right ) - 2 \, \sin \left (3 \, x\right )\right )} e^{\left (6 \, x\right )} \end {gather*}

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

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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