3.62 \(\int \cos \left (\sqrt{x}\right ) \, dx\)

Optimal. Leaf size=22 \[ 2 \sqrt{x} \sin \left (\sqrt{x}\right )+2 \cos \left (\sqrt{x}\right ) \]

[Out]

2*Cos[Sqrt[x]] + 2*Sqrt[x]*Sin[Sqrt[x]]

_______________________________________________________________________________________

Rubi [A]  time = 0.0174071, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ 2 \sqrt{x} \sin \left (\sqrt{x}\right )+2 \cos \left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Cos[Sqrt[x]],x]

[Out]

2*Cos[Sqrt[x]] + 2*Sqrt[x]*Sin[Sqrt[x]]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.61152, size = 48, normalized size = 2.18 \[ - i \sqrt{x} e^{i \sqrt{x}} + i \sqrt{x} e^{- i \sqrt{x}} + e^{i \sqrt{x}} + e^{- i \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(x**(1/2)),x)

[Out]

-I*sqrt(x)*exp(I*sqrt(x)) + I*sqrt(x)*exp(-I*sqrt(x)) + exp(I*sqrt(x)) + exp(-I*
sqrt(x))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0102961, size = 22, normalized size = 1. \[ 2 \sqrt{x} \sin \left (\sqrt{x}\right )+2 \cos \left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[Sqrt[x]],x]

[Out]

2*Cos[Sqrt[x]] + 2*Sqrt[x]*Sin[Sqrt[x]]

_______________________________________________________________________________________

Maple [A]  time = 0., size = 17, normalized size = 0.8 \[ 2\,\cos \left ( \sqrt{x} \right ) +2\,\sin \left ( \sqrt{x} \right ) \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(x^(1/2)),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.34485, size = 22, normalized size = 1. \[ 2 \, \sqrt{x} \sin \left (\sqrt{x}\right ) + 2 \, \cos \left (\sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.23025, size = 22, normalized size = 1. \[ 2 \, \sqrt{x} \sin \left (\sqrt{x}\right ) + 2 \, \cos \left (\sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.401382, size = 20, normalized size = 0.91 \[ 2 \sqrt{x} \sin{\left (\sqrt{x} \right )} + 2 \cos{\left (\sqrt{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x**(1/2)),x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.228569, size = 22, normalized size = 1. \[ 2 \, \sqrt{x} \sin \left (\sqrt{x}\right ) + 2 \, \cos \left (\sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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