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]

1/4*sin(x^2)^2

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Rubi [A]  time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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*}

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Mathematica [A]  time = 0.00, size = 10, normalized size = 1.00 \[ -\frac {1}{4} \cos ^2\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*Cos[x^2]^2

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fricas [A]  time = 0.42, size = 8, normalized size = 0.80 \[ -\frac {1}{4} \, \cos \left (x^{2}\right )^{2} \]

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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giac [A]  time = 0.01, size = 8, normalized size = 0.80 \[ -\frac {1}{4} \, \cos \left (x^{2}\right )^{2} \]

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

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maple [A]  time = 0.00, size = 9, normalized size = 0.90 \[ -\frac {\left (\cos ^{2}\left (x^{2}\right )\right )}{4} \]

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 = 0.53, size = 8, normalized size = 0.80 \[ -\frac {1}{4} \, \cos \left (x^{2}\right )^{2} \]

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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mupad [B]  time = 0.05, size = 8, normalized size = 0.80 \[ \frac {{\sin \left (x^2\right )}^2}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(x^2)^2/4

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sympy [A]  time = 0.33, size = 8, normalized size = 0.80 \[ - \frac {\cos ^{2}{\left (x^{2} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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