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

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

[Out]

Sqrt[Sin[x^2]]

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Rubi [A]  time = 0.0127035, antiderivative size = 8, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0027532, size = 8, normalized size = 1. \[ \sqrt{\sin \left (x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[Sin[x^2]]

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Maple [A]  time = 0.007, size = 7, normalized size = 0.9 \begin{align*} \sqrt{\sin \left ({x}^{2} \right ) } \end{align*}

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 = 0.955825, size = 8, normalized size = 1. \begin{align*} \sqrt{\sin \left (x^{2}\right )} \end{align*}

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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Fricas [A]  time = 1.9853, size = 22, normalized size = 2.75 \begin{align*} \sqrt{\sin \left (x^{2}\right )} \end{align*}

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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Sympy [A]  time = 0.309172, size = 7, normalized size = 0.88 \begin{align*} \sqrt{\sin{\left (x^{2} \right )}} \end{align*}

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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Giac [A]  time = 1.07353, size = 8, normalized size = 1. \begin{align*} \sqrt{\sin \left (x^{2}\right )} \end{align*}

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))