3.775 \(\int x^2 \cos (4 x^3) \cos (5 x^3) \, dx\)

Optimal. Leaf size=19 \[ \frac{\sin \left (x^3\right )}{6}+\frac{1}{54} \sin \left (9 x^3\right ) \]

[Out]

Sin[x^3]/6 + Sin[9*x^3]/54

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Rubi [A]  time = 0.0372446, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {4572, 3380, 2637} \[ \frac{\sin \left (x^3\right )}{6}+\frac{1}{54} \sin \left (9 x^3\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^2*Cos[4*x^3]*Cos[5*x^3],x]

[Out]

Sin[x^3]/6 + Sin[9*x^3]/54

Rule 4572

Int[Cos[v_]^(p_.)*Cos[w_]^(q_.)*(x_)^(m_.), x_Symbol] :> Int[ExpandTrigReduce[x^m, Cos[v]^p*Cos[w]^q, x], x] /
; IGtQ[m, 0] && IGtQ[p, 0] && IGtQ[q, 0] && ((PolynomialQ[v, x] && PolynomialQ[w, x]) || (BinomialQ[{v, w}, x]
&& IndependentQ[Cancel[v/w], x]))

Rule 3380

Int[((a_.) + Cos[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Cos[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simpl
ify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int x^2 \cos \left (4 x^3\right ) \cos \left (5 x^3\right ) \, dx &=\int \left (\frac{1}{2} x^2 \cos \left (x^3\right )+\frac{1}{2} x^2 \cos \left (9 x^3\right )\right ) \, dx\\ &=\frac{1}{2} \int x^2 \cos \left (x^3\right ) \, dx+\frac{1}{2} \int x^2 \cos \left (9 x^3\right ) \, dx\\ &=\frac{1}{6} \operatorname{Subst}\left (\int \cos (x) \, dx,x,x^3\right )+\frac{1}{6} \operatorname{Subst}\left (\int \cos (9 x) \, dx,x,x^3\right )\\ &=\frac{\sin \left (x^3\right )}{6}+\frac{1}{54} \sin \left (9 x^3\right )\\ \end{align*}

Mathematica [A]  time = 0.0086818, size = 19, normalized size = 1. \[ \frac{\sin \left (x^3\right )}{6}+\frac{1}{54} \sin \left (9 x^3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Cos[4*x^3]*Cos[5*x^3],x]

[Out]

Sin[x^3]/6 + Sin[9*x^3]/54

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Maple [A]  time = 0.045, size = 16, normalized size = 0.8 \begin{align*}{\frac{\sin \left ({x}^{3} \right ) }{6}}+{\frac{\sin \left ( 9\,{x}^{3} \right ) }{54}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*cos(4*x^3)*cos(5*x^3),x)

[Out]

1/6*sin(x^3)+1/54*sin(9*x^3)

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Maxima [A]  time = 0.964867, size = 20, normalized size = 1.05 \begin{align*} \frac{1}{54} \, \sin \left (9 \, x^{3}\right ) + \frac{1}{6} \, \sin \left (x^{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*cos(4*x^3)*cos(5*x^3),x, algorithm="maxima")

[Out]

1/54*sin(9*x^3) + 1/6*sin(x^3)

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Fricas [B]  time = 2.14163, size = 116, normalized size = 6.11 \begin{align*} \frac{1}{27} \,{\left (128 \, \cos \left (x^{3}\right )^{8} - 224 \, \cos \left (x^{3}\right )^{6} + 120 \, \cos \left (x^{3}\right )^{4} - 20 \, \cos \left (x^{3}\right )^{2} + 5\right )} \sin \left (x^{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*cos(4*x^3)*cos(5*x^3),x, algorithm="fricas")

[Out]

1/27*(128*cos(x^3)^8 - 224*cos(x^3)^6 + 120*cos(x^3)^4 - 20*cos(x^3)^2 + 5)*sin(x^3)

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Sympy [B]  time = 7.69553, size = 32, normalized size = 1.68 \begin{align*} - \frac{4 \sin{\left (4 x^{3} \right )} \cos{\left (5 x^{3} \right )}}{27} + \frac{5 \sin{\left (5 x^{3} \right )} \cos{\left (4 x^{3} \right )}}{27} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*cos(4*x**3)*cos(5*x**3),x)

[Out]

-4*sin(4*x**3)*cos(5*x**3)/27 + 5*sin(5*x**3)*cos(4*x**3)/27

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Giac [B]  time = 1.08441, size = 53, normalized size = 2.79 \begin{align*} \frac{128}{27} \, \sin \left (x^{3}\right )^{9} - \frac{32}{3} \, \sin \left (x^{3}\right )^{7} + 8 \, \sin \left (x^{3}\right )^{5} - \frac{20}{9} \, \sin \left (x^{3}\right )^{3} + \frac{1}{3} \, \sin \left (x^{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*cos(4*x^3)*cos(5*x^3),x, algorithm="giac")

[Out]

128/27*sin(x^3)^9 - 32/3*sin(x^3)^7 + 8*sin(x^3)^5 - 20/9*sin(x^3)^3 + 1/3*sin(x^3)