3.559 \(\int \frac{1}{\sqrt{-9+4 x^2}} \, dx\)

Optimal. Leaf size=19 \[ \frac{1}{2} \tanh ^{-1}\left (\frac{2 x}{\sqrt{4 x^2-9}}\right ) \]

[Out]

ArcTanh[(2*x)/Sqrt[-9 + 4*x^2]]/2

_______________________________________________________________________________________

Rubi [A]  time = 0.0123318, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{1}{2} \tanh ^{-1}\left (\frac{2 x}{\sqrt{4 x^2-9}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[-9 + 4*x^2],x]

[Out]

ArcTanh[(2*x)/Sqrt[-9 + 4*x^2]]/2

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 1.21891, size = 15, normalized size = 0.79 \[ \frac{\operatorname{atanh}{\left (\frac{2 x}{\sqrt{4 x^{2} - 9}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(4*x**2-9)**(1/2),x)

[Out]

atanh(2*x/sqrt(4*x**2 - 9))/2

_______________________________________________________________________________________

Mathematica [B]  time = 0.00505957, size = 43, normalized size = 2.26 \[ \frac{1}{4} \log \left (\frac{2 x}{\sqrt{4 x^2-9}}+1\right )-\frac{1}{4} \log \left (1-\frac{2 x}{\sqrt{4 x^2-9}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[-9 + 4*x^2],x]

[Out]

-Log[1 - (2*x)/Sqrt[-9 + 4*x^2]]/4 + Log[1 + (2*x)/Sqrt[-9 + 4*x^2]]/4

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 22, normalized size = 1.2 \[{\frac{\sqrt{4}}{4}\ln \left ( x\sqrt{4}+\sqrt{4\,{x}^{2}-9} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(4*x^2-9)^(1/2),x)

[Out]

1/4*ln(x*4^(1/2)+(4*x^2-9)^(1/2))*4^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.49784, size = 24, normalized size = 1.26 \[ \frac{1}{2} \, \log \left (8 \, x + 4 \, \sqrt{4 \, x^{2} - 9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(4*x^2 - 9),x, algorithm="maxima")

[Out]

1/2*log(8*x + 4*sqrt(4*x^2 - 9))

_______________________________________________________________________________________

Fricas [A]  time = 0.211873, size = 22, normalized size = 1.16 \[ -\frac{1}{2} \, \log \left (-2 \, x + \sqrt{4 \, x^{2} - 9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(4*x^2 - 9),x, algorithm="fricas")

[Out]

-1/2*log(-2*x + sqrt(4*x^2 - 9))

_______________________________________________________________________________________

Sympy [A]  time = 0.31438, size = 7, normalized size = 0.37 \[ \frac{\operatorname{acosh}{\left (\frac{2 x}{3} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(4*x**2-9)**(1/2),x)

[Out]

acosh(2*x/3)/2

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.215207, size = 23, normalized size = 1.21 \[ -\frac{1}{2} \,{\rm ln}\left ({\left | -2 \, x + \sqrt{4 \, x^{2} - 9} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(4*x^2 - 9),x, algorithm="giac")

[Out]

-1/2*ln(abs(-2*x + sqrt(4*x^2 - 9)))