3.844 \(\int \frac{\cos (\sqrt{x}) \sin (\sqrt{x})}{\sqrt{x}} \, dx\)

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

[Out]

Sin[Sqrt[x]]^2

________________________________________________________________________________________

Rubi [A]  time = 0.0110285, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {3441} \[ \sin ^2\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0123849, size = 12, normalized size = 1.5 \[ -\frac{1}{2} \cos \left (2 \sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Cos[2*Sqrt[x]]/2

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 9, normalized size = 1.1 \begin{align*} - \left ( \cos \left ( \sqrt{x} \right ) \right ) ^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.957622, size = 11, normalized size = 1.38 \begin{align*} -\cos \left (\sqrt{x}\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(sqrt(x))^2

________________________________________________________________________________________

Fricas [A]  time = 1.89804, size = 23, normalized size = 2.88 \begin{align*} -\cos \left (\sqrt{x}\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(sqrt(x))^2

________________________________________________________________________________________

Sympy [A]  time = 0.305078, size = 8, normalized size = 1. \begin{align*} - \cos ^{2}{\left (\sqrt{x} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(sqrt(x))**2

________________________________________________________________________________________

Giac [A]  time = 1.06722, size = 11, normalized size = 1.38 \begin{align*} -\cos \left (\sqrt{x}\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(sqrt(x))^2