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

Optimal. Leaf size=36 \[ \frac{1}{2} x \sqrt{4 x^2-9}-\frac{9}{4} \tanh ^{-1}\left (\frac{2 x}{\sqrt{4 x^2-9}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0056792, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {195, 217, 206} \[ \frac{1}{2} x \sqrt{4 x^2-9}-\frac{9}{4} \tanh ^{-1}\left (\frac{2 x}{\sqrt{4 x^2-9}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-9 + 4*x^2],x]

[Out]

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

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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

Mathematica [A]  time = 0.0059698, size = 37, normalized size = 1.03 \[ \frac{1}{2} x \sqrt{4 x^2-9}-\frac{9}{4} \log \left (\sqrt{4 x^2-9}+2 x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-9 + 4*x^2],x]

[Out]

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

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Maple [A]  time = 0.003, size = 35, normalized size = 1. \begin{align*}{\frac{x}{2}\sqrt{4\,{x}^{2}-9}}-{\frac{9\,\sqrt{4}}{8}\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((4*x^2-9)^(1/2),x)

[Out]

1/2*x*(4*x^2-9)^(1/2)-9/8*ln(x*4^(1/2)+(4*x^2-9)^(1/2))*4^(1/2)

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Maxima [A]  time = 3.19076, size = 42, normalized size = 1.17 \begin{align*} \frac{1}{2} \, \sqrt{4 \, x^{2} - 9} x - \frac{9}{4} \, \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((4*x^2-9)^(1/2),x, algorithm="maxima")

[Out]

1/2*sqrt(4*x^2 - 9)*x - 9/4*log(8*x + 4*sqrt(4*x^2 - 9))

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Fricas [A]  time = 1.48715, size = 77, normalized size = 2.14 \begin{align*} \frac{1}{2} \, \sqrt{4 \, x^{2} - 9} x + \frac{9}{4} \, \log \left (-2 \, x + \sqrt{4 \, x^{2} - 9}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.195789, size = 22, normalized size = 0.61 \begin{align*} \frac{x \sqrt{4 x^{2} - 9}}{2} - \frac{9 \operatorname{acosh}{\left (\frac{2 x}{3} \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*sqrt(4*x**2 - 9)/2 - 9*acosh(2*x/3)/4

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Giac [A]  time = 1.58807, size = 41, normalized size = 1.14 \begin{align*} \frac{1}{2} \, \sqrt{4 \, x^{2} - 9} x + \frac{9}{4} \, \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((4*x^2-9)^(1/2),x, algorithm="giac")

[Out]

1/2*sqrt(4*x^2 - 9)*x + 9/4*log(abs(-2*x + sqrt(4*x^2 - 9)))