3.21 \(\int \frac{1}{\sqrt{3 x-4 x^2}} \, dx\)

Optimal. Leaf size=12 \[ -\frac{1}{2} \sin ^{-1}\left (1-\frac{8 x}{3}\right ) \]

[Out]

-ArcSin[1 - (8*x)/3]/2

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Rubi [A]  time = 0.0056575, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {619, 216} \[ -\frac{1}{2} \sin ^{-1}\left (1-\frac{8 x}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[3*x - 4*x^2],x]

[Out]

-ArcSin[1 - (8*x)/3]/2

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

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 \frac{1}{\sqrt{3 x-4 x^2}} \, dx &=-\left (\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{9}}} \, dx,x,3-8 x\right )\right )\\ &=-\frac{1}{2} \sin ^{-1}\left (1-\frac{8 x}{3}\right )\\ \end{align*}

Mathematica [B]  time = 0.0138426, size = 40, normalized size = 3.33 \[ -\frac{\sqrt{-x (4 x-3)} \sin ^{-1}\left (\sqrt{1-\frac{4 x}{3}}\right )}{\sqrt{3-4 x} \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[3*x - 4*x^2],x]

[Out]

-((Sqrt[-(x*(-3 + 4*x))]*ArcSin[Sqrt[1 - (4*x)/3]])/(Sqrt[3 - 4*x]*Sqrt[x]))

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Maple [A]  time = 0.048, size = 9, normalized size = 0.8 \begin{align*}{\frac{1}{2}\arcsin \left ( -1+{\frac{8\,x}{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-4*x^2+3*x)^(1/2),x)

[Out]

1/2*arcsin(-1+8/3*x)

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Maxima [A]  time = 1.78698, size = 11, normalized size = 0.92 \begin{align*} -\frac{1}{2} \, \arcsin \left (-\frac{8}{3} \, x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-4*x^2+3*x)^(1/2),x, algorithm="maxima")

[Out]

-1/2*arcsin(-8/3*x + 1)

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Fricas [B]  time = 2.33111, size = 47, normalized size = 3.92 \begin{align*} -\arctan \left (\frac{\sqrt{-4 \, x^{2} + 3 \, x}}{2 \, x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-4*x^2+3*x)^(1/2),x, algorithm="fricas")

[Out]

-arctan(1/2*sqrt(-4*x^2 + 3*x)/x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- 4 x^{2} + 3 x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-4*x**2+3*x)**(1/2),x)

[Out]

Integral(1/sqrt(-4*x**2 + 3*x), x)

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Giac [A]  time = 1.2208, size = 11, normalized size = 0.92 \begin{align*} \frac{1}{2} \, \arcsin \left (\frac{8}{3} \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-4*x^2+3*x)^(1/2),x, algorithm="giac")

[Out]

1/2*arcsin(8/3*x - 1)