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(x^(1/2))^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 = 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*}

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Mathematica [A]  time = 0.01, size = 12, normalized size = 1.50 \[ -\frac {1}{2} \cos \left (2 \sqrt {x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*Cos[2*Sqrt[x]]

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fricas [A]  time = 0.73, size = 8, normalized size = 1.00 \[ -\cos \left (\sqrt {x}\right )^{2} \]

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

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giac [A]  time = 0.12, size = 6, normalized size = 0.75 \[ \sin \left (\sqrt {x}\right )^{2} \]

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]

sin(sqrt(x))^2

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maple [A]  time = 0.04, size = 9, normalized size = 1.12 \[ -\left (\cos ^{2}\left (\sqrt {x}\right )\right ) \]

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

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maxima [A]  time = 0.31, size = 8, normalized size = 1.00 \[ -\cos \left (\sqrt {x}\right )^{2} \]

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

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mupad [B]  time = 3.05, size = 8, normalized size = 1.00 \[ -{\cos \left (\sqrt {x}\right )}^2 \]

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

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sympy [A]  time = 0.27, size = 8, normalized size = 1.00 \[ - \cos ^{2}{\left (\sqrt {x} \right )} \]

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

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