3.820 \(\int x \sin (2 x^2) \, dx\)

Optimal. Leaf size=10 \[ -\frac{1}{4} \cos \left (2 x^2\right ) \]

[Out]

-Cos[2*x^2]/4

________________________________________________________________________________________

Rubi [A]  time = 0.0077968, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3379, 2638} \[ -\frac{1}{4} \cos \left (2 x^2\right ) \]

Antiderivative was successfully verified.

[In]

Int[x*Sin[2*x^2],x]

[Out]

-Cos[2*x^2]/4

Rule 3379

Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Sin[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 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 \sin \left (2 x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \sin (2 x) \, dx,x,x^2\right )\\ &=-\frac{1}{4} \cos \left (2 x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0090049, size = 10, normalized size = 1. \[ -\frac{1}{4} \cos \left (2 x^2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*Sin[2*x^2],x]

[Out]

-Cos[2*x^2]/4

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 9, normalized size = 0.9 \begin{align*} -{\frac{\cos \left ( 2\,{x}^{2} \right ) }{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*sin(2*x^2),x)

[Out]

-1/4*cos(2*x^2)

________________________________________________________________________________________

Maxima [A]  time = 0.952914, size = 11, normalized size = 1.1 \begin{align*} -\frac{1}{4} \, \cos \left (2 \, x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*cos(2*x^2)

________________________________________________________________________________________

Fricas [A]  time = 2.14754, size = 23, normalized size = 2.3 \begin{align*} -\frac{1}{4} \, \cos \left (2 \, x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*cos(2*x^2)

________________________________________________________________________________________

Sympy [A]  time = 0.166031, size = 8, normalized size = 0.8 \begin{align*} - \frac{\cos{\left (2 x^{2} \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sin(2*x**2),x)

[Out]

-cos(2*x**2)/4

________________________________________________________________________________________

Giac [A]  time = 1.07296, size = 11, normalized size = 1.1 \begin{align*} -\frac{1}{4} \, \cos \left (2 \, x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*cos(2*x^2)