3.548 \(\int e^{3 x} \cos ^2(\frac{3 x}{2}) \sin ^2(\frac{3 x}{2}) \, dx\)

Optimal. Leaf size=36 \[ \frac{e^{3 x}}{24}-\frac{1}{60} e^{3 x} \sin (6 x)-\frac{1}{120} e^{3 x} \cos (6 x) \]

[Out]

E^(3*x)/24 - (E^(3*x)*Cos[6*x])/120 - (E^(3*x)*Sin[6*x])/60

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Rubi [A]  time = 0.0433467, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {4469, 2194, 4433} \[ \frac{e^{3 x}}{24}-\frac{1}{60} e^{3 x} \sin (6 x)-\frac{1}{120} e^{3 x} \cos (6 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

E^(3*x)/24 - (E^(3*x)*Cos[6*x])/120 - (E^(3*x)*Sin[6*x])/60

Rule 4469

Int[Cos[(f_.) + (g_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sin[(d_.) + (e_.)*(x_)]^(m_.), x_Symbol] :
> Int[ExpandTrigReduce[F^(c*(a + b*x)), Sin[d + e*x]^m*Cos[f + g*x]^n, x], x] /; FreeQ[{F, a, b, c, d, e, f, g
}, x] && IGtQ[m, 0] && IGtQ[n, 0]

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 4433

Int[Cos[(d_.) + (e_.)*(x_)]*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Simp[(b*c*Log[F]*F^(c*(a + b*x))*C
os[d + e*x])/(e^2 + b^2*c^2*Log[F]^2), x] + Simp[(e*F^(c*(a + b*x))*Sin[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^{3 x} \cos ^2\left (\frac{3 x}{2}\right ) \sin ^2\left (\frac{3 x}{2}\right ) \, dx &=\int \left (\frac{e^{3 x}}{8}-\frac{1}{8} e^{3 x} \cos (6 x)\right ) \, dx\\ &=\frac{1}{8} \int e^{3 x} \, dx-\frac{1}{8} \int e^{3 x} \cos (6 x) \, dx\\ &=\frac{e^{3 x}}{24}-\frac{1}{120} e^{3 x} \cos (6 x)-\frac{1}{60} e^{3 x} \sin (6 x)\\ \end{align*}

Mathematica [A]  time = 0.0378083, size = 21, normalized size = 0.58 \[ -\frac{1}{120} e^{3 x} (2 \sin (6 x)+\cos (6 x)-5) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(E^(3*x)*(-5 + Cos[6*x] + 2*Sin[6*x]))/120

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Maple [B]  time = 0.017, size = 63, normalized size = 1.8 \begin{align*} -{\frac{ \left ( 12\,\cos \left ( x \right ) +24\,\sin \left ( x \right ) \right ){{\rm e}^{3\,x}} \left ( \cos \left ( x \right ) \right ) ^{5}}{45}}+{\frac{ \left ( 6\,\cos \left ( x \right ) +8\,\sin \left ( x \right ) \right ){{\rm e}^{3\,x}} \left ( \cos \left ( x \right ) \right ) ^{3}}{15}}-{\frac{ \left ( 3\,\cos \left ( x \right ) +2\,\sin \left ( x \right ) \right ){{\rm e}^{3\,x}}\cos \left ( x \right ) }{20}}+{\frac{ \left ({{\rm e}^{x}} \right ) ^{3}}{20}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(3*x)*cos(3/2*x)^2*sin(3/2*x)^2,x)

[Out]

-4/45*(3*cos(x)+6*sin(x))*exp(3*x)*cos(x)^5+2/15*(3*cos(x)+4*sin(x))*exp(3*x)*cos(x)^3-1/20*(3*cos(x)+2*sin(x)
)*exp(3*x)*cos(x)+1/20*exp(x)^3

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Maxima [A]  time = 0.946844, size = 36, normalized size = 1. \begin{align*} -\frac{1}{120} \, \cos \left (6 \, x\right ) e^{\left (3 \, x\right )} - \frac{1}{60} \, e^{\left (3 \, x\right )} \sin \left (6 \, x\right ) + \frac{1}{24} \, e^{\left (3 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/120*cos(6*x)*e^(3*x) - 1/60*e^(3*x)*sin(6*x) + 1/24*e^(3*x)

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Fricas [A]  time = 1.919, size = 147, normalized size = 4.08 \begin{align*} -\frac{1}{15} \,{\left (2 \, \cos \left (\frac{3}{2} \, x\right )^{3} - \cos \left (\frac{3}{2} \, x\right )\right )} e^{\left (3 \, x\right )} \sin \left (\frac{3}{2} \, x\right ) - \frac{1}{30} \,{\left (2 \, \cos \left (\frac{3}{2} \, x\right )^{4} - 2 \, \cos \left (\frac{3}{2} \, x\right )^{2} - 1\right )} e^{\left (3 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(2*cos(3/2*x)^3 - cos(3/2*x))*e^(3*x)*sin(3/2*x) - 1/30*(2*cos(3/2*x)^4 - 2*cos(3/2*x)^2 - 1)*e^(3*x)

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Sympy [B]  time = 5.62001, size = 99, normalized size = 2.75 \begin{align*} \frac{e^{3 x} \sin ^{4}{\left (\frac{3 x}{2} \right )}}{30} + \frac{e^{3 x} \sin ^{3}{\left (\frac{3 x}{2} \right )} \cos{\left (\frac{3 x}{2} \right )}}{15} + \frac{2 e^{3 x} \sin ^{2}{\left (\frac{3 x}{2} \right )} \cos ^{2}{\left (\frac{3 x}{2} \right )}}{15} - \frac{e^{3 x} \sin{\left (\frac{3 x}{2} \right )} \cos ^{3}{\left (\frac{3 x}{2} \right )}}{15} + \frac{e^{3 x} \cos ^{4}{\left (\frac{3 x}{2} \right )}}{30} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(3*x)*sin(3*x/2)**4/30 + exp(3*x)*sin(3*x/2)**3*cos(3*x/2)/15 + 2*exp(3*x)*sin(3*x/2)**2*cos(3*x/2)**2/15 -
exp(3*x)*sin(3*x/2)*cos(3*x/2)**3/15 + exp(3*x)*cos(3*x/2)**4/30

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Giac [A]  time = 1.12675, size = 32, normalized size = 0.89 \begin{align*} -\frac{1}{120} \,{\left (\cos \left (6 \, x\right ) + 2 \, \sin \left (6 \, x\right )\right )} e^{\left (3 \, x\right )} + \frac{1}{24} \, e^{\left (3 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/120*(cos(6*x) + 2*sin(6*x))*e^(3*x) + 1/24*e^(3*x)