3.6.48 \(\int e^{3 x} \cos ^2(\frac {3 x}{2}) \sin ^2(\frac {3 x}{2}) \, dx\) [548]

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

[Out]

1/24*exp(3*x)-1/120*exp(3*x)*cos(6*x)-1/60*exp(3*x)*sin(6*x)

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Rubi [A]
time = 0.03, antiderivative size = 36, normalized size of antiderivative = 1.00, 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 = {4557, 2225, 4518} \begin {gather*} \frac {e^{3 x}}{24}-\frac {1}{60} e^{3 x} \sin (6 x)-\frac {1}{120} e^{3 x} \cos (6 x) \end {gather*}

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 2225

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 4518

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]

Rule 4557

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]

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*}

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Mathematica [A]
time = 0.03, size = 21, normalized size = 0.58 \begin {gather*} -\frac {1}{120} e^{3 x} (-5+\cos (6 x)+2 \sin (6 x)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/120*(E^(3*x)*(-5 + Cos[6*x] + 2*Sin[6*x]))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(62\) vs. \(2(27)=54\).
time = 0.06, size = 63, normalized size = 1.75

method	result	size
risch	\(\frac {{\mathrm e}^{3 x}}{24}-\frac {{\mathrm e}^{\left (3+6 i\right ) x}}{240}+\frac {i {\mathrm e}^{\left (3+6 i\right ) x}}{120}-\frac {{\mathrm e}^{\left (3-6 i\right ) x}}{240}-\frac {i {\mathrm e}^{\left (3-6 i\right ) x}}{120}\)	\(42\)
default	\(-\frac {4 \left (3 \cos \left (x \right )+6 \sin \left (x \right )\right ) {\mathrm e}^{3 x} \left (\cos ^{5}\left (x \right )\right )}{45}+\frac {2 \left (3 \cos \left (x \right )+4 \sin \left (x \right )\right ) {\mathrm e}^{3 x} \left (\cos ^{3}\left (x \right )\right )}{15}-\frac {\left (3 \cos \left (x \right )+2 \sin \left (x \right )\right ) {\mathrm e}^{3 x} \cos \left (x \right )}{20}+\frac {{\mathrm e}^{3 x}}{20}\)	\(63\)
norman	\(\frac {-\frac {2 \,{\mathrm e}^{3 x} \tan \left (\frac {3 x}{4}\right )}{15}+\frac {2 \,{\mathrm e}^{3 x} \left (\tan ^{2}\left (\frac {3 x}{4}\right )\right )}{5}+\frac {14 \,{\mathrm e}^{3 x} \left (\tan ^{3}\left (\frac {3 x}{4}\right )\right )}{15}-\frac {{\mathrm e}^{3 x} \left (\tan ^{4}\left (\frac {3 x}{4}\right )\right )}{3}-\frac {14 \,{\mathrm e}^{3 x} \left (\tan ^{5}\left (\frac {3 x}{4}\right )\right )}{15}+\frac {2 \,{\mathrm e}^{3 x} \left (\tan ^{6}\left (\frac {3 x}{4}\right )\right )}{5}+\frac {2 \,{\mathrm e}^{3 x} \left (\tan ^{7}\left (\frac {3 x}{4}\right )\right )}{15}+\frac {{\mathrm e}^{3 x} \left (\tan ^{8}\left (\frac {3 x}{4}\right )\right )}{30}+\frac {{\mathrm e}^{3 x}}{30}}{\left (1+\tan ^{2}\left (\frac {3 x}{4}\right )\right )^{4}}\)	\(113\)

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,method=_RETURNVERBOSE)

[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 = 2.01, size = 27, normalized size = 0.75 \begin {gather*} -\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 {gather*}

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 = 0.87, size = 50, normalized size = 1.39 \begin {gather*} -\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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (29) = 58\).
time = 0.68, size = 99, normalized size = 2.75 \begin {gather*} \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 {gather*}

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 = 0.72, size = 24, normalized size = 0.67 \begin {gather*} -\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 {gather*}

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)

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Mupad [B]
time = 0.39, size = 18, normalized size = 0.50 \begin {gather*} -\frac {{\mathrm {e}}^{3\,x}\,\left (\cos \left (6\,x\right )+2\,\sin \left (6\,x\right )-5\right )}{120} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(exp(3*x)*(cos(6*x) + 2*sin(6*x) - 5))/120

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