3.76 \(\int \frac{\cos ^2\left (\sqrt{x}\right )}{\sqrt{x}} \, dx\)

Optimal. Leaf size=19 \[ \sqrt{x}+\sin \left (\sqrt{x}\right ) \cos \left (\sqrt{x}\right ) \]

[Out]

Sqrt[x] + Cos[Sqrt[x]]*Sin[Sqrt[x]]

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Rubi [A]  time = 0.03187, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \sqrt{x}+\sin \left (\sqrt{x}\right ) \cos \left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Cos[Sqrt[x]]^2/Sqrt[x],x]

[Out]

Sqrt[x] + Cos[Sqrt[x]]*Sin[Sqrt[x]]

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Rubi in Sympy [A]  time = 1.6876, size = 17, normalized size = 0.89 \[ \sqrt{x} + \sin{\left (\sqrt{x} \right )} \cos{\left (\sqrt{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(x**(1/2))**2/x**(1/2),x)

[Out]

sqrt(x) + sin(sqrt(x))*cos(sqrt(x))

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Mathematica [A]  time = 0.0100187, size = 18, normalized size = 0.95 \[ \sqrt{x}+\frac{1}{2} \sin \left (2 \sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[Sqrt[x]]^2/Sqrt[x],x]

[Out]

Sqrt[x] + Sin[2*Sqrt[x]]/2

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Maple [A]  time = 0.017, size = 14, normalized size = 0.7 \[ \cos \left ( \sqrt{x} \right ) \sin \left ( \sqrt{x} \right ) +\sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(x^(1/2))^2/x^(1/2),x)

[Out]

cos(x^(1/2))*sin(x^(1/2))+x^(1/2)

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Maxima [A]  time = 1.33552, size = 16, normalized size = 0.84 \[ \sqrt{x} + \frac{1}{2} \, \sin \left (2 \, \sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(sqrt(x))^2/sqrt(x),x, algorithm="maxima")

[Out]

sqrt(x) + 1/2*sin(2*sqrt(x))

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Fricas [A]  time = 0.226285, size = 18, normalized size = 0.95 \[ \cos \left (\sqrt{x}\right ) \sin \left (\sqrt{x}\right ) + \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(sqrt(x))^2/sqrt(x),x, algorithm="fricas")

[Out]

cos(sqrt(x))*sin(sqrt(x)) + sqrt(x)

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Sympy [A]  time = 0.455867, size = 39, normalized size = 2.05 \[ \sqrt{x} \sin ^{2}{\left (\sqrt{x} \right )} + \sqrt{x} \cos ^{2}{\left (\sqrt{x} \right )} + \sin{\left (\sqrt{x} \right )} \cos{\left (\sqrt{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x**(1/2))**2/x**(1/2),x)

[Out]

sqrt(x)*sin(sqrt(x))**2 + sqrt(x)*cos(sqrt(x))**2 + sin(sqrt(x))*cos(sqrt(x))

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GIAC/XCAS [A]  time = 0.198724, size = 16, normalized size = 0.84 \[ \sqrt{x} + \frac{1}{2} \, \sin \left (2 \, \sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(sqrt(x))^2/sqrt(x),x, algorithm="giac")

[Out]

sqrt(x) + 1/2*sin(2*sqrt(x))