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

Optimal. Leaf size=10 \[ \frac{1}{12} \sin \left (4 x^3\right ) \]

[Out]

Sin[4*x^3]/12

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Rubi [A]  time = 0.0121073, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3380, 2637} \[ \frac{1}{12} \sin \left (4 x^3\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sin[4*x^3]/12

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 ) \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \cos (4 x) \, dx,x,x^3\right )\\ &=\frac{1}{12} \sin \left (4 x^3\right )\\ \end{align*}

Mathematica [A]  time = 0.003212, size = 10, normalized size = 1. \[ \frac{1}{12} \sin \left (4 x^3\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sin[4*x^3]/12

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Maple [A]  time = 0.005, size = 9, normalized size = 0.9 \begin{align*}{\frac{\sin \left ( 4\,{x}^{3} \right ) }{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*sin(4*x^3)

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Maxima [A]  time = 0.967451, size = 11, normalized size = 1.1 \begin{align*} \frac{1}{12} \, \sin \left (4 \, x^{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*sin(4*x^3)

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Fricas [A]  time = 1.99023, size = 23, normalized size = 2.3 \begin{align*} \frac{1}{12} \, \sin \left (4 \, x^{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*sin(4*x^3)

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Sympy [A]  time = 0.293403, size = 7, normalized size = 0.7 \begin{align*} \frac{\sin{\left (4 x^{3} \right )}}{12} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(4*x**3)/12

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Giac [A]  time = 1.1051, size = 11, normalized size = 1.1 \begin{align*} \frac{1}{12} \, \sin \left (4 \, x^{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*sin(4*x^3)