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 = 0.0031291, 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, Rules used = {217, 206} \[ \frac{1}{2} \tanh ^{-1}\left (\frac{2 x}{\sqrt{4 x^2-9}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-9 + 4*x^2],x]

[Out]

ArcTanh[(2*x)/Sqrt[-9 + 4*x^2]]/2

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{-9+4 x^2}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{1-4 x^2} \, dx,x,\frac{x}{\sqrt{-9+4 x^2}}\right )\\ &=\frac{1}{2} \tanh ^{-1}\left (\frac{2 x}{\sqrt{-9+4 x^2}}\right )\\ \end{align*}

Mathematica [B]  time = 0.0025181, 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 = 0.002, size = 22, normalized size = 1.2 \begin{align*}{\frac{\sqrt{4}}{4}\ln \left ( x\sqrt{4}+\sqrt{4\,{x}^{2}-9} \right ) } \end{align*}

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 = 3.60055, size = 24, normalized size = 1.26 \begin{align*} \frac{1}{2} \, \log \left (8 \, x + 4 \, \sqrt{4 \, x^{2} - 9}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(8*x + 4*sqrt(4*x^2 - 9))

________________________________________________________________________________________

Fricas [A]  time = 1.2857, size = 46, normalized size = 2.42 \begin{align*} -\frac{1}{2} \, \log \left (-2 \, x + \sqrt{4 \, x^{2} - 9}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(-2*x + sqrt(4*x^2 - 9))

________________________________________________________________________________________

Sympy [A]  time = 0.133074, size = 7, normalized size = 0.37 \begin{align*} \frac{\operatorname{acosh}{\left (\frac{2 x}{3} \right )}}{2} \end{align*}

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 [A]  time = 2.60687, size = 23, normalized size = 1.21 \begin{align*} -\frac{1}{2} \, \log \left ({\left | -2 \, x + \sqrt{4 \, x^{2} - 9} \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(abs(-2*x + sqrt(4*x^2 - 9)))