3.785 \(\int 12 x^2 \cos (x^3) \, dx\)

Optimal. Leaf size=6 \[ 4 \sin \left (x^3\right ) \]

[Out]

4*Sin[x^3]

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Rubi [A]  time = 0.0091064, antiderivative size = 6, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {12, 3380, 2637} \[ 4 \sin \left (x^3\right ) \]

Antiderivative was successfully verified.

[In]

Int[12*x^2*Cos[x^3],x]

[Out]

4*Sin[x^3]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 12 x^2 \cos \left (x^3\right ) \, dx &=12 \int x^2 \cos \left (x^3\right ) \, dx\\ &=4 \operatorname{Subst}\left (\int \cos (x) \, dx,x,x^3\right )\\ &=4 \sin \left (x^3\right )\\ \end{align*}

Mathematica [A]  time = 0.0016396, size = 6, normalized size = 1. \[ 4 \sin \left (x^3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[12*x^2*Cos[x^3],x]

[Out]

4*Sin[x^3]

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Maple [A]  time = 0.001, size = 7, normalized size = 1.2 \begin{align*} 4\,\sin \left ({x}^{3} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(12*x^2*cos(x^3),x)

[Out]

4*sin(x^3)

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Maxima [A]  time = 0.962417, size = 8, normalized size = 1.33 \begin{align*} 4 \, \sin \left (x^{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(12*x^2*cos(x^3),x, algorithm="maxima")

[Out]

4*sin(x^3)

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Fricas [A]  time = 2.03581, size = 16, normalized size = 2.67 \begin{align*} 4 \, \sin \left (x^{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(12*x^2*cos(x^3),x, algorithm="fricas")

[Out]

4*sin(x^3)

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Sympy [A]  time = 0.296115, size = 5, normalized size = 0.83 \begin{align*} 4 \sin{\left (x^{3} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(12*x**2*cos(x**3),x)

[Out]

4*sin(x**3)

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Giac [A]  time = 1.06525, size = 8, normalized size = 1.33 \begin{align*} 4 \, \sin \left (x^{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(12*x^2*cos(x^3),x, algorithm="giac")

[Out]

4*sin(x^3)