3.148 \(\int \sqrt{3-x^2} \, dx\)

Optimal. Leaf size=29 \[ \frac{1}{2} \sqrt{3-x^2} x+\frac{3}{2} \sin ^{-1}\left (\frac{x}{\sqrt{3}}\right ) \]

[Out]

(x*Sqrt[3 - x^2])/2 + (3*ArcSin[x/Sqrt[3]])/2

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Rubi [A]  time = 0.003629, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {195, 216} \[ \frac{1}{2} \sqrt{3-x^2} x+\frac{3}{2} \sin ^{-1}\left (\frac{x}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[3 - x^2],x]

[Out]

(x*Sqrt[3 - x^2])/2 + (3*ArcSin[x/Sqrt[3]])/2

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \sqrt{3-x^2} \, dx &=\frac{1}{2} x \sqrt{3-x^2}+\frac{3}{2} \int \frac{1}{\sqrt{3-x^2}} \, dx\\ &=\frac{1}{2} x \sqrt{3-x^2}+\frac{3}{2} \sin ^{-1}\left (\frac{x}{\sqrt{3}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0069582, size = 29, normalized size = 1. \[ \frac{1}{2} \sqrt{3-x^2} x+\frac{3}{2} \sin ^{-1}\left (\frac{x}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[3 - x^2],x]

[Out]

(x*Sqrt[3 - x^2])/2 + (3*ArcSin[x/Sqrt[3]])/2

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Maple [A]  time = 0.003, size = 23, normalized size = 0.8 \begin{align*}{\frac{3}{2}\arcsin \left ({\frac{x\sqrt{3}}{3}} \right ) }+{\frac{x}{2}\sqrt{-{x}^{2}+3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^2+3)^(1/2),x)

[Out]

3/2*arcsin(1/3*x*3^(1/2))+1/2*x*(-x^2+3)^(1/2)

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Maxima [A]  time = 1.42589, size = 30, normalized size = 1.03 \begin{align*} \frac{1}{2} \, \sqrt{-x^{2} + 3} x + \frac{3}{2} \, \arcsin \left (\frac{1}{3} \, \sqrt{3} x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+3)^(1/2),x, algorithm="maxima")

[Out]

1/2*sqrt(-x^2 + 3)*x + 3/2*arcsin(1/3*sqrt(3)*x)

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Fricas [A]  time = 0.9952, size = 72, normalized size = 2.48 \begin{align*} \frac{1}{2} \, \sqrt{-x^{2} + 3} x - \frac{3}{2} \, \arctan \left (\frac{\sqrt{-x^{2} + 3}}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+3)^(1/2),x, algorithm="fricas")

[Out]

1/2*sqrt(-x^2 + 3)*x - 3/2*arctan(sqrt(-x^2 + 3)/x)

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Sympy [A]  time = 0.192871, size = 24, normalized size = 0.83 \begin{align*} \frac{x \sqrt{3 - x^{2}}}{2} + \frac{3 \operatorname{asin}{\left (\frac{\sqrt{3} x}{3} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**2+3)**(1/2),x)

[Out]

x*sqrt(3 - x**2)/2 + 3*asin(sqrt(3)*x/3)/2

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Giac [A]  time = 1.13689, size = 30, normalized size = 1.03 \begin{align*} \frac{1}{2} \, \sqrt{-x^{2} + 3} x + \frac{3}{2} \, \arcsin \left (\frac{1}{3} \, \sqrt{3} x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+3)^(1/2),x, algorithm="giac")

[Out]

1/2*sqrt(-x^2 + 3)*x + 3/2*arcsin(1/3*sqrt(3)*x)