3.790 \(\int x^3 \cos (x^2) \, dx\)

Optimal. Leaf size=20 \[ \frac{1}{2} x^2 \sin \left (x^2\right )+\frac{\cos \left (x^2\right )}{2} \]

[Out]

Cos[x^2]/2 + (x^2*Sin[x^2])/2

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Rubi [A]  time = 0.0168783, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {3380, 3296, 2638} \[ \frac{1}{2} x^2 \sin \left (x^2\right )+\frac{\cos \left (x^2\right )}{2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Cos[x^2]/2 + (x^2*Sin[x^2])/2

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 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2638

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

Rubi steps

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

Mathematica [A]  time = 0.0075433, size = 20, normalized size = 1. \[ \frac{1}{2} x^2 \sin \left (x^2\right )+\frac{\cos \left (x^2\right )}{2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Cos[x^2]/2 + (x^2*Sin[x^2])/2

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Maple [A]  time = 0.005, size = 17, normalized size = 0.9 \begin{align*}{\frac{\cos \left ({x}^{2} \right ) }{2}}+{\frac{{x}^{2}\sin \left ({x}^{2} \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*cos(x^2)+1/2*x^2*sin(x^2)

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Maxima [A]  time = 0.974607, size = 22, normalized size = 1.1 \begin{align*} \frac{1}{2} \, x^{2} \sin \left (x^{2}\right ) + \frac{1}{2} \, \cos \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2*sin(x^2) + 1/2*cos(x^2)

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Fricas [A]  time = 1.96105, size = 45, normalized size = 2.25 \begin{align*} \frac{1}{2} \, x^{2} \sin \left (x^{2}\right ) + \frac{1}{2} \, \cos \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2*sin(x^2) + 1/2*cos(x^2)

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Sympy [A]  time = 0.549555, size = 15, normalized size = 0.75 \begin{align*} \frac{x^{2} \sin{\left (x^{2} \right )}}{2} + \frac{\cos{\left (x^{2} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2*sin(x**2)/2 + cos(x**2)/2

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Giac [A]  time = 1.09492, size = 22, normalized size = 1.1 \begin{align*} \frac{1}{2} \, x^{2} \sin \left (x^{2}\right ) + \frac{1}{2} \, \cos \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2*sin(x^2) + 1/2*cos(x^2)