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

Optimal. Leaf size=8 \[ -\frac {1}{2} \cos \left (x^2\right ) \]

[Out]

-1/2*cos(x^2)

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3379, 2638} \[ -\frac {1}{2} \cos \left (x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Cos[x^2]/2

Rule 2638

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

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]))

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 8, normalized size = 1.00 \[ -\frac {1}{2} \cos \left (x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*Cos[x^2]

________________________________________________________________________________________

fricas [A]  time = 0.83, size = 6, normalized size = 0.75 \[ -\frac {1}{2} \, \cos \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*cos(x^2)

________________________________________________________________________________________

giac [A]  time = 0.13, size = 6, normalized size = 0.75 \[ -\frac {1}{2} \, \cos \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*cos(x^2)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 7, normalized size = 0.88 \[ -\frac {\cos \left (x^{2}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*cos(x^2)

________________________________________________________________________________________

maxima [A]  time = 0.31, size = 6, normalized size = 0.75 \[ -\frac {1}{2} \, \cos \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*cos(x^2)

________________________________________________________________________________________

mupad [B]  time = 0.05, size = 6, normalized size = 0.75 \[ -\frac {\cos \left (x^2\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(x^2)/2

________________________________________________________________________________________

sympy [A]  time = 0.16, size = 7, normalized size = 0.88 \[ - \frac {\cos {\left (x^{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(x**2)/2

________________________________________________________________________________________