3.769 \(\int x \sin (\frac{x^2}{2}) \, dx\)

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

[Out]

-Cos[x^2/2]

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Rubi [A]  time = 0.0077483, 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 = {3379, 2638} \[ -\cos \left (\frac{x^2}{2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Cos[x^2/2]

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

Mathematica [A]  time = 0.0104019, size = 10, normalized size = 1. \[ -\cos \left (\frac{x^2}{2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Cos[x^2/2]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*sin(1/2*x^2),x)

[Out]

-cos(1/2*x^2)

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Maxima [A]  time = 0.956508, size = 11, normalized size = 1.1 \begin{align*} -\cos \left (\frac{1}{2} \, x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sin(1/2*x^2),x, algorithm="maxima")

[Out]

-cos(1/2*x^2)

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Fricas [A]  time = 1.8549, size = 20, normalized size = 2. \begin{align*} -\cos \left (\frac{1}{2} \, x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sin(1/2*x^2),x, algorithm="fricas")

[Out]

-cos(1/2*x^2)

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Sympy [A]  time = 0.163217, size = 7, normalized size = 0.7 \begin{align*} - \cos{\left (\frac{x^{2}}{2} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sin(1/2*x**2),x)

[Out]

-cos(x**2/2)

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Giac [A]  time = 1.08726, size = 11, normalized size = 1.1 \begin{align*} -\cos \left (\frac{1}{2} \, x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sin(1/2*x^2),x, algorithm="giac")

[Out]

-cos(1/2*x^2)