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 = 0.0200079, 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 \[ \frac{1}{2} x \sqrt{4 x^2-9}-\frac{9}{4} \tanh ^{-1}\left (\frac{2 x}{\sqrt{4 x^2-9}}\right ) \]

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

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Rubi in Sympy [A]  time = 1.50947, size = 31, normalized size = 0.86 \[ \frac{x \sqrt{4 x^{2} - 9}}{2} - \frac{9 \operatorname{atanh}{\left (\frac{2 x}{\sqrt{4 x^{2} - 9}} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sqrt(4*x**2 - 9)/2 - 9*atanh(2*x/sqrt(4*x**2 - 9))/4

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Mathematica [A]  time = 0.0106599, 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. \[{\frac{x}{2}\sqrt{4\,{x}^{2}-9}}-{\frac{9\,\sqrt{4}}{8}\ln \left ( x\sqrt{4}+\sqrt{4\,{x}^{2}-9} \right ) } \]

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 = 1.49419, size = 42, normalized size = 1.17 \[ \frac{1}{2} \, \sqrt{4 \, x^{2} - 9} x - \frac{9}{4} \, \log \left (8 \, x + 4 \, \sqrt{4 \, x^{2} - 9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(4*x^2 - 9),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 = 0.228234, size = 120, normalized size = 3.33 \[ -\frac{32 \, x^{4} - 72 \, x^{2} - 9 \,{\left (8 \, x^{2} - 4 \, \sqrt{4 \, x^{2} - 9} x - 9\right )} \log \left (-2 \, x + \sqrt{4 \, x^{2} - 9}\right ) - 2 \,{\left (8 \, x^{3} - 9 \, x\right )} \sqrt{4 \, x^{2} - 9}}{4 \,{\left (8 \, x^{2} - 4 \, \sqrt{4 \, x^{2} - 9} x - 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(32*x^4 - 72*x^2 - 9*(8*x^2 - 4*sqrt(4*x^2 - 9)*x - 9)*log(-2*x + sqrt(4*x^
2 - 9)) - 2*(8*x^3 - 9*x)*sqrt(4*x^2 - 9))/(8*x^2 - 4*sqrt(4*x^2 - 9)*x - 9)

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Sympy [A]  time = 0.491568, size = 22, normalized size = 0.61 \[ \frac{x \sqrt{4 x^{2} - 9}}{2} - \frac{9 \operatorname{acosh}{\left (\frac{2 x}{3} \right )}}{4} \]

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/XCAS [A]  time = 0.205561, size = 41, normalized size = 1.14 \[ \frac{1}{2} \, \sqrt{4 \, x^{2} - 9} x + \frac{9}{4} \,{\rm ln}\left ({\left | -2 \, x + \sqrt{4 \, x^{2} - 9} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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