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

Optimal. Leaf size=34 \[ 3 \tanh ^{-1}\left (\frac{3 x}{\sqrt{9 x^2-1}}\right )-\frac{\sqrt{9 x^2-1}}{x} \]

[Out]

-(Sqrt[-1 + 9*x^2]/x) + 3*ArcTanh[(3*x)/Sqrt[-1 + 9*x^2]]

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Rubi [A]  time = 0.0222532, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ 3 \tanh ^{-1}\left (\frac{3 x}{\sqrt{9 x^2-1}}\right )-\frac{\sqrt{9 x^2-1}}{x} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[-1 + 9*x^2]/x^2,x]

[Out]

-(Sqrt[-1 + 9*x^2]/x) + 3*ArcTanh[(3*x)/Sqrt[-1 + 9*x^2]]

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Rubi in Sympy [A]  time = 1.68437, size = 27, normalized size = 0.79 \[ 3 \operatorname{atanh}{\left (\frac{3 x}{\sqrt{9 x^{2} - 1}} \right )} - \frac{\sqrt{9 x^{2} - 1}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*atanh(3*x/sqrt(9*x**2 - 1)) - sqrt(9*x**2 - 1)/x

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Mathematica [A]  time = 0.0155297, size = 35, normalized size = 1.03 \[ 3 \log \left (\sqrt{9 x^2-1}+3 x\right )-\frac{\sqrt{9 x^2-1}}{x} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[-1 + 9*x^2]/x^2,x]

[Out]

-(Sqrt[-1 + 9*x^2]/x) + 3*Log[3*x + Sqrt[-1 + 9*x^2]]

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Maple [A]  time = 0.01, size = 47, normalized size = 1.4 \[{\frac{1}{x} \left ( 9\,{x}^{2}-1 \right ) ^{{\frac{3}{2}}}}-9\,x\sqrt{9\,{x}^{2}-1}+\ln \left ( x\sqrt{9}+\sqrt{9\,{x}^{2}-1} \right ) \sqrt{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((9*x^2-1)^(1/2)/x^2,x)

[Out]

1/x*(9*x^2-1)^(3/2)-9*x*(9*x^2-1)^(1/2)+ln(x*9^(1/2)+(9*x^2-1)^(1/2))*9^(1/2)

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Maxima [A]  time = 1.48653, size = 45, normalized size = 1.32 \[ -\frac{\sqrt{9 \, x^{2} - 1}}{x} + 3 \, \log \left (18 \, x + 6 \, \sqrt{9 \, x^{2} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(9*x^2 - 1)/x + 3*log(18*x + 6*sqrt(9*x^2 - 1))

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Fricas [A]  time = 0.211885, size = 78, normalized size = 2.29 \[ -\frac{3 \,{\left (3 \, x^{2} - \sqrt{9 \, x^{2} - 1} x\right )} \log \left (-3 \, x + \sqrt{9 \, x^{2} - 1}\right ) + 1}{3 \, x^{2} - \sqrt{9 \, x^{2} - 1} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.343018, size = 17, normalized size = 0.5 \[ 3 \operatorname{acosh}{\left (3 x \right )} - \frac{\sqrt{9 x^{2} - 1}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*acosh(3*x) - sqrt(9*x**2 - 1)/x

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GIAC/XCAS [A]  time = 0.21771, size = 59, normalized size = 1.74 \[ -\frac{6}{{\left (3 \, x - \sqrt{9 \, x^{2} - 1}\right )}^{2} + 1} - \frac{3}{2} \,{\rm ln}\left ({\left (3 \, x - \sqrt{9 \, x^{2} - 1}\right )}^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-6/((3*x - sqrt(9*x^2 - 1))^2 + 1) - 3/2*ln((3*x - sqrt(9*x^2 - 1))^2)