3.685 \(\int \frac{1}{\sqrt{(2-3 x) (2+3 x)}} \, dx\)

Optimal. Leaf size=10 \[ \frac{1}{3} \sin ^{-1}\left (\frac{3 x}{2}\right ) \]

[Out]

ArcSin[(3*x)/2]/3

_______________________________________________________________________________________

Rubi [A]  time = 0.00871314, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{1}{3} \sin ^{-1}\left (\frac{3 x}{2}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[(2 - 3*x)*(2 + 3*x)],x]

[Out]

ArcSin[(3*x)/2]/3

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 0.616473, size = 7, normalized size = 0.7 \[ \frac{\operatorname{asin}{\left (\frac{3 x}{2} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

asin(3*x/2)/3

_______________________________________________________________________________________

Mathematica [A]  time = 0.00640894, size = 10, normalized size = 1. \[ \frac{1}{3} \sin ^{-1}\left (\frac{3 x}{2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[(2 - 3*x)*(2 + 3*x)],x]

[Out]

ArcSin[(3*x)/2]/3

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 7, normalized size = 0.7 \[{\frac{1}{3}\arcsin \left ({\frac{3\,x}{2}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/((2-3*x)*(2+3*x))^(1/2),x)

[Out]

1/3*arcsin(3/2*x)

_______________________________________________________________________________________

Maxima [A]  time = 0.818011, size = 8, normalized size = 0.8 \[ \frac{1}{3} \, \arcsin \left (\frac{3}{2} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*arcsin(3/2*x)

_______________________________________________________________________________________

Fricas [A]  time = 0.267666, size = 26, normalized size = 2.6 \[ -\frac{2}{3} \, \arctan \left (\frac{\sqrt{-9 \, x^{2} + 4} - 2}{3 \, x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*arctan(1/3*(sqrt(-9*x^2 + 4) - 2)/x)

_______________________________________________________________________________________

Sympy [A]  time = 4.13498, size = 7, normalized size = 0.7 \[ \frac{\operatorname{asin}{\left (\frac{3 x}{2} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

asin(3*x/2)/3

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.2718, size = 8, normalized size = 0.8 \[ \frac{1}{3} \, \arcsin \left (\frac{3}{2} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*arcsin(3/2*x)