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

_______________________________________________________________________________________

Rubi [A]  time = 0.0142831, 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 \[ -\frac{1}{2} \sin ^{-1}\left (1-\frac{8 x}{3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[3*x - 4*x^2],x]

[Out]

-ArcSin[1 - (8*x)/3]/2

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 1.35323, size = 8, normalized size = 0.67 \[ \frac{\operatorname{asin}{\left (\frac{8 x}{3} - 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-4*x**2+3*x)**(1/2),x)

[Out]

asin(8*x/3 - 1)/2

_______________________________________________________________________________________

Mathematica [B]  time = 0.0159947, size = 45, normalized size = 3.75 \[ \frac{\sqrt{x} \sqrt{4 x-3} \log \left (2 \sqrt{x}+\sqrt{4 x-3}\right )}{\sqrt{-x (4 x-3)}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[3*x - 4*x^2],x]

[Out]

(Sqrt[x]*Sqrt[-3 + 4*x]*Log[2*Sqrt[x] + Sqrt[-3 + 4*x]])/Sqrt[-(x*(-3 + 4*x))]

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 9, normalized size = 0.8 \[{\frac{1}{2}\arcsin \left ( -1+{\frac{8\,x}{3}} \right ) } \]

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)

_______________________________________________________________________________________

Maxima [A]  time = 0.778173, size = 11, normalized size = 0.92 \[ -\frac{1}{2} \, \arcsin \left (-\frac{8}{3} \, x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*arcsin(-8/3*x + 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.215647, size = 26, normalized size = 2.17 \[ -\arctan \left (\frac{\sqrt{-4 \, x^{2} + 3 \, x}}{2 \, x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-arctan(1/2*sqrt(-4*x^2 + 3*x)/x)

_______________________________________________________________________________________

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

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)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.214183, size = 11, normalized size = 0.92 \[ \frac{1}{2} \, \arcsin \left (\frac{8}{3} \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*arcsin(8/3*x - 1)