3.9.72 \(\int \sqrt {1-x^2} \, dx\) [872]

Optimal. Leaf size=23 \[ \frac {1}{2} x \sqrt {1-x^2}+\frac {1}{2} \sin ^{-1}(x) \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.00, 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 = {201, 222} \begin {gather*} \frac {\text {ArcSin}(x)}{2}+\frac {1}{2} \sqrt {1-x^2} x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x^2],x]

[Out]

(x*Sqrt[1 - x^2])/2 + ArcSin[x]/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*} \int \sqrt {1-x^2} \, dx &=\frac {1}{2} x \sqrt {1-x^2}+\frac {1}{2} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=\frac {1}{2} x \sqrt {1-x^2}+\frac {1}{2} \sin ^{-1}(x)\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 37, normalized size = 1.61 \begin {gather*} \frac {1}{2} x \sqrt {1-x^2}-\tan ^{-1}\left (\frac {\sqrt {1-x^2}}{1+x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - x^2],x]

[Out]

(x*Sqrt[1 - x^2])/2 - ArcTan[Sqrt[1 - x^2]/(1 + x)]

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Maple [A]
time = 0.50, size = 18, normalized size = 0.78

method	result	size
default	\(\frac {\arcsin \left (x \right )}{2}+\frac {x \sqrt {-x^{2}+1}}{2}\)	\(18\)
risch	\(-\frac {x \left (x^{2}-1\right )}{2 \sqrt {-x^{2}+1}}+\frac {\arcsin \left (x \right )}{2}\)	\(23\)
meijerg	\(\frac {i \left (-2 i \sqrt {\pi }\, x \sqrt {-x^{2}+1}-2 i \sqrt {\pi }\, \arcsin \left (x \right )\right )}{4 \sqrt {\pi }}\)	\(32\)
trager	\(\frac {x \sqrt {-x^{2}+1}}{2}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+1}+x \right )}{2}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.49, size = 17, normalized size = 0.74 \begin {gather*} \frac {1}{2} \, \sqrt {-x^{2} + 1} x + \frac {1}{2} \, \arcsin \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(-x^2 + 1)*x + 1/2*arcsin(x)

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Fricas [A]
time = 0.38, size = 31, normalized size = 1.35 \begin {gather*} \frac {1}{2} \, \sqrt {-x^{2} + 1} x - \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.06, size = 15, normalized size = 0.65 \begin {gather*} \frac {x \sqrt {1 - x^{2}}}{2} + \frac {\operatorname {asin}{\left (x \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*sqrt(1 - x**2)/2 + asin(x)/2

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Giac [A]
time = 3.33, size = 17, normalized size = 0.74 \begin {gather*} \frac {1}{2} \, \sqrt {-x^{2} + 1} x + \frac {1}{2} \, \arcsin \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(-x^2 + 1)*x + 1/2*arcsin(x)

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Mupad [B]
time = 0.03, size = 17, normalized size = 0.74 \begin {gather*} \frac {\mathrm {asin}\left (x\right )}{2}+\frac {x\,\sqrt {1-x^2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x^2)^(1/2),x)

[Out]

asin(x)/2 + (x*(1 - x^2)^(1/2))/2

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