3.852 \(\int \frac {x \cos (x^2)}{\sqrt {\sin (x^2)}} \, dx\)

Optimal. Leaf size=8 \[ \sqrt {\sin \left (x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {3441} \[ \sqrt {\sin \left (x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(x*Cos[x^2])/Sqrt[Sin[x^2]],x]

[Out]

Sqrt[Sin[x^2]]

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

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Mathematica [A]  time = 0.00, size = 8, normalized size = 1.00 \[ \sqrt {\sin \left (x^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*Cos[x^2])/Sqrt[Sin[x^2]],x]

[Out]

Sqrt[Sin[x^2]]

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fricas [A]  time = 2.08, size = 6, normalized size = 0.75 \[ \sqrt {\sin \left (x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(sin(x^2))

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giac [A]  time = 0.14, size = 6, normalized size = 0.75 \[ \sqrt {\sin \left (x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(sin(x^2))

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maple [A]  time = 0.02, size = 7, normalized size = 0.88 \[ \sqrt {\sin }\left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.31, size = 6, normalized size = 0.75 \[ \sqrt {\sin \left (x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(sin(x^2))

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mupad [B]  time = 3.19, size = 6, normalized size = 0.75 \[ \sqrt {\sin \left (x^2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.29, size = 7, normalized size = 0.88 \[ \sqrt {\sin {\left (x^{2} \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(sin(x**2))

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