3.778 \(\int 2 x \cos (x^2) \, dx\)

Optimal. Leaf size=4 \[ \sin \left (x^2\right ) \]

[Out]

Sin[x^2]

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Rubi [A]  time = 0.0065855, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {12, 3380, 2637} \[ \sin \left (x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sin[x^2]

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

Mathematica [A]  time = 0.0014906, size = 4, normalized size = 1. \[ \sin \left (x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sin[x^2]

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Maple [A]  time = 0.001, size = 5, normalized size = 1.3 \begin{align*} \sin \left ({x}^{2} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(x^2)

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Maxima [A]  time = 0.960729, size = 5, normalized size = 1.25 \begin{align*} \sin \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(x^2)

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Fricas [A]  time = 2.01673, size = 14, normalized size = 3.5 \begin{align*} \sin \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(x^2)

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Sympy [A]  time = 0.163815, size = 3, normalized size = 0.75 \begin{align*} \sin{\left (x^{2} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(x**2)

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Giac [A]  time = 1.09887, size = 5, normalized size = 1.25 \begin{align*} \sin \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(x^2)