\(\int \sqrt {3-x^2} \, dx\) [148]

Optimal result

Integrand size = 11, antiderivative size = 29 \[ \int \sqrt {3-x^2} \, dx=\frac {1}{2} x \sqrt {3-x^2}+\frac {3}{2} \arcsin \left (\frac {x}{\sqrt {3}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {201, 222} \[ \int \sqrt {3-x^2} \, dx=\frac {3}{2} \arcsin \left (\frac {x}{\sqrt {3}}\right )+\frac {1}{2} \sqrt {3-x^2} x \]

[In]

Int[Sqrt[3 - x^2],x]

[Out]

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

Rule 201

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 222

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*} \text {integral}& = \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} \arcsin \left (\frac {x}{\sqrt {3}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.41 \[ \int \sqrt {3-x^2} \, dx=\frac {1}{2} x \sqrt {3-x^2}+3 \arctan \left (\frac {-\sqrt {3}+x}{\sqrt {3-x^2}}\right ) \]

[In]

Integrate[Sqrt[3 - x^2],x]

[Out]

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

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79

method	result	size
default	\(\frac {3 \arcsin \left (\frac {x \sqrt {3}}{3}\right )}{2}+\frac {x \sqrt {-x^{2}+3}}{2}\)	\(23\)
risch	\(-\frac {x \left (x^{2}-3\right )}{2 \sqrt {-x^{2}+3}}+\frac {3 \arcsin \left (\frac {x \sqrt {3}}{3}\right )}{2}\)	\(28\)
pseudoelliptic	\(\frac {x \sqrt {-x^{2}+3}}{2}-\frac {3 \arctan \left (\frac {\sqrt {-x^{2}+3}}{x}\right )}{2}\)	\(30\)
meijerg	\(\frac {3 i \left (-\frac {2 i \sqrt {\pi }\, x \sqrt {3}\, \sqrt {-\frac {x^{2}}{3}+1}}{3}-2 i \sqrt {\pi }\, \arcsin \left (\frac {x \sqrt {3}}{3}\right )\right )}{4 \sqrt {\pi }}\)	\(40\)
trager	\(\frac {x \sqrt {-x^{2}+3}}{2}+\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+3}+x \right )}{2}\)	\(41\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \sqrt {3-x^2} \, dx=\frac {1}{2} \, \sqrt {-x^{2} + 3} x - \frac {3}{2} \, \arctan \left (\frac {\sqrt {-x^{2} + 3}}{x}\right ) \]

[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)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \sqrt {3-x^2} \, dx=\frac {x \sqrt {3 - x^{2}}}{2} + \frac {3 \operatorname {asin}{\left (\frac {\sqrt {3} x}{3} \right )}}{2} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \sqrt {3-x^2} \, dx=\frac {1}{2} \, \sqrt {-x^{2} + 3} x + \frac {3}{2} \, \arcsin \left (\frac {1}{3} \, \sqrt {3} x\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \sqrt {3-x^2} \, dx=\frac {1}{2} \, \sqrt {-x^{2} + 3} x + \frac {3}{2} \, \arcsin \left (\frac {1}{3} \, \sqrt {3} x\right ) \]

[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)

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \sqrt {3-x^2} \, dx=\frac {3\,\mathrm {asin}\left (\frac {\sqrt {3}\,x}{3}\right )}{2}+\frac {x\,\sqrt {3-x^2}}{2} \]

[In]

int((3 - x^2)^(1/2),x)

[Out]

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