3.2.28 \(\int \sqrt {1-4 x^2} \, dx\) [128]

Optimal. Leaf size=25 \[ \frac {1}{2} x \sqrt {1-4 x^2}+\frac {1}{4} \sin ^{-1}(2 x) \]

[Out]

1/4*arcsin(2*x)+1/2*x*(-4*x^2+1)^(1/2)

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Rubi [A]
time = 0.00, antiderivative size = 25, 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 {1}{4} \text {ArcSin}(2 x)+\frac {1}{2} \sqrt {1-4 x^2} x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - 4*x^2],x]

[Out]

(x*Sqrt[1 - 4*x^2])/2 + ArcSin[2*x]/4

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 {1-4 x^2} \, dx &=\frac {1}{2} x \sqrt {1-4 x^2}+\frac {1}{2} \int \frac {1}{\sqrt {1-4 x^2}} \, dx\\ &=\frac {1}{2} x \sqrt {1-4 x^2}+\frac {1}{4} \sin ^{-1}(2 x)\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 41, normalized size = 1.64 \begin {gather*} \frac {1}{2} x \sqrt {1-4 x^2}-\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {1-4 x^2}}{1+2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - 4*x^2],x]

[Out]

(x*Sqrt[1 - 4*x^2])/2 - ArcTan[Sqrt[1 - 4*x^2]/(1 + 2*x)]/2

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Maple [A]
time = 0.09, size = 20, normalized size = 0.80

method	result	size
default	\(\frac {\arcsin \left (2 x \right )}{4}+\frac {x \sqrt {-4 x^{2}+1}}{2}\)	\(20\)
risch	\(-\frac {\left (4 x^{2}-1\right ) x}{2 \sqrt {-4 x^{2}+1}}+\frac {\arcsin \left (2 x \right )}{4}\)	\(27\)
meijerg	\(\frac {i \left (-4 i \sqrt {\pi }\, x \sqrt {-4 x^{2}+1}-2 i \sqrt {\pi }\, \arcsin \left (2 x \right )\right )}{8 \sqrt {\pi }}\)	\(34\)
trager	\(\frac {x \sqrt {-4 x^{2}+1}}{2}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-4 x^{2}+1}+2 x \right )}{4}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*x^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/4*arcsin(2*x)+1/2*x*(-4*x^2+1)^(1/2)

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Maxima [A]
time = 2.21, size = 19, normalized size = 0.76 \begin {gather*} \frac {1}{2} \, \sqrt {-4 \, x^{2} + 1} x + \frac {1}{4} \, \arcsin \left (2 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

1/2*sqrt(-4*x^2 + 1)*x + 1/4*arcsin(2*x)

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Fricas [A]
time = 1.11, size = 32, normalized size = 1.28 \begin {gather*} \frac {1}{2} \, \sqrt {-4 \, x^{2} + 1} x - \frac {1}{2} \, \arctan \left (\frac {\sqrt {-4 \, x^{2} + 1} - 1}{2 \, x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

1/2*sqrt(-4*x^2 + 1)*x - 1/2*arctan(1/2*(sqrt(-4*x^2 + 1) - 1)/x)

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Sympy [A]
time = 0.07, size = 19, normalized size = 0.76 \begin {gather*} \frac {x \sqrt {1 - 4 x^{2}}}{2} + \frac {\operatorname {asin}{\left (2 x \right )}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x**2+1)**(1/2),x)

[Out]

x*sqrt(1 - 4*x**2)/2 + asin(2*x)/4

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Giac [A]
time = 0.65, size = 19, normalized size = 0.76 \begin {gather*} \frac {1}{2} \, \sqrt {-4 \, x^{2} + 1} x + \frac {1}{4} \, \arcsin \left (2 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^2+1)^(1/2),x, algorithm="giac")

[Out]

1/2*sqrt(-4*x^2 + 1)*x + 1/4*arcsin(2*x)

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Mupad [B]
time = 0.03, size = 18, normalized size = 0.72 \begin {gather*} \frac {\mathrm {asin}\left (2\,x\right )}{4}+x\,\sqrt {\frac {1}{4}-x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - 4*x^2)^(1/2),x)

[Out]

asin(2*x)/4 + x*(1/4 - x^2)^(1/2)

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