3.192 \(\int \sin \left (\sqrt{x}\right ) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0173069, 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 \[ 2 \sin \left (\sqrt{x}\right )-2 \sqrt{x} \cos \left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sin[Sqrt[x]],x]

[Out]

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

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Rubi in Sympy [A]  time = 2.86635, size = 46, normalized size = 2.09 \[ - \sqrt{x} e^{i \sqrt{x}} - \sqrt{x} e^{- i \sqrt{x}} - i e^{i \sqrt{x}} + i e^{- i \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(sin(x**(1/2)),x)

[Out]

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

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Mathematica [A]  time = 0.00893873, size = 22, normalized size = 1. \[ 2 \sin \left (\sqrt{x}\right )-2 \sqrt{x} \cos \left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sin[Sqrt[x]],x]

[Out]

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

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Maple [A]  time = 0.002, size = 17, normalized size = 0.8 \[ 2\,\sin \left ( \sqrt{x} \right ) -2\,\cos \left ( \sqrt{x} \right ) \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(sin(x^(1/2)),x)

[Out]

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

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Maxima [A]  time = 1.43035, size = 22, normalized size = 1. \[ -2 \, \sqrt{x} \cos \left (\sqrt{x}\right ) + 2 \, \sin \left (\sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.233666, size = 22, normalized size = 1. \[ -2 \, \sqrt{x} \cos \left (\sqrt{x}\right ) + 2 \, \sin \left (\sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.389953, size = 20, normalized size = 0.91 \[ - 2 \sqrt{x} \cos{\left (\sqrt{x} \right )} + 2 \sin{\left (\sqrt{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(x**(1/2)),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.198767, size = 22, normalized size = 1. \[ -2 \, \sqrt{x} \cos \left (\sqrt{x}\right ) + 2 \, \sin \left (\sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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