\(\int \frac {x \cos (x^2)}{\sqrt {\sin (x^2)}} \, dx\) [852]

Optimal result

Integrand size = 14, antiderivative size = 8 \[ \int \frac {x \cos \left (x^2\right )}{\sqrt {\sin \left (x^2\right )}} \, dx=\sqrt {\sin \left (x^2\right )} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {3522} \[ \int \frac {x \cos \left (x^2\right )}{\sqrt {\sin \left (x^2\right )}} \, dx=\sqrt {\sin \left (x^2\right )} \]

[In]

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

[Out]

Sqrt[Sin[x^2]]

Rule 3522

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*} \text {integral}& = \sqrt {\sin \left (x^2\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {x \cos \left (x^2\right )}{\sqrt {\sin \left (x^2\right )}} \, dx=\sqrt {\sin \left (x^2\right )} \]

[In]

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

[Out]

Sqrt[Sin[x^2]]

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.88

method	result	size
derivativedivides	\(\sqrt {\sin \left (x^{2}\right )}\)	\(7\)
default	\(\sqrt {\sin \left (x^{2}\right )}\)	\(7\)
risch	\(\sqrt {\sin \left (x^{2}\right )}\)	\(7\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \frac {x \cos \left (x^2\right )}{\sqrt {\sin \left (x^2\right )}} \, dx=\sqrt {\sin \left (x^{2}\right )} \]

[In]

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

[Out]

sqrt(sin(x^2))

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.88 \[ \int \frac {x \cos \left (x^2\right )}{\sqrt {\sin \left (x^2\right )}} \, dx=\sqrt {\sin {\left (x^{2} \right )}} \]

[In]

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

[Out]

sqrt(sin(x**2))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \frac {x \cos \left (x^2\right )}{\sqrt {\sin \left (x^2\right )}} \, dx=\sqrt {\sin \left (x^{2}\right )} \]

[In]

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

[Out]

sqrt(sin(x^2))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \frac {x \cos \left (x^2\right )}{\sqrt {\sin \left (x^2\right )}} \, dx=\sqrt {\sin \left (x^{2}\right )} \]

[In]

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

[Out]

sqrt(sin(x^2))

Mupad [B] (verification not implemented)

Time = 27.19 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \frac {x \cos \left (x^2\right )}{\sqrt {\sin \left (x^2\right )}} \, dx=\sqrt {\sin \left (x^2\right )} \]

[In]

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

[Out]

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