3.54 \(\int x \cos (x^2) \sin (x^2) \, dx\)

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

[Out]

Sin[x^2]^2/4

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Rubi [A]  time = 0.0071152, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {3441} \[ \frac{1}{4} \sin ^2\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sin[x^2]^2/4

Rule 3441

Int[Cos[(a_.) + (b_.)*(x_)^(n_.)]*(x_)^(m_.)*Sin[(a_.) + (b_.)*(x_)^(n_.)]^(p_.), x_Symbol] :> Simp[Sin[a + b*
x^n]^(p + 1)/(b*n*(p + 1)), x] /; FreeQ[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int x \cos \left (x^2\right ) \sin \left (x^2\right ) \, dx &=\frac{1}{4} \sin ^2\left (x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0024186, size = 10, normalized size = 1. \[ -\frac{1}{4} \cos ^2\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Cos[x^2]^2/4

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.959578, size = 11, normalized size = 1.1 \begin{align*} -\frac{1}{4} \, \cos \left (x^{2}\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.517447, size = 23, normalized size = 2.3 \begin{align*} -\frac{1}{4} \, \cos \left (x^{2}\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.301366, size = 7, normalized size = 0.7 \begin{align*} \frac{\sin ^{2}{\left (x^{2} \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(x**2)**2/4

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Giac [A]  time = 1.10867, size = 11, normalized size = 1.1 \begin{align*} -\frac{1}{4} \, \cos \left (x^{2}\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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