3.1925 \(\int \frac{1}{\sqrt{-a+\frac{b}{x^2}} x} \, dx\)

Optimal. Leaf size=27 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{\frac{b}{x^2}-a}}{\sqrt{a}}\right )}{\sqrt{a}} \]

[Out]

-(ArcTan[Sqrt[-a + b/x^2]/Sqrt[a]]/Sqrt[a])

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Rubi [A]  time = 0.0567394, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{\frac{b}{x^2}-a}}{\sqrt{a}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[-a + b/x^2]*x),x]

[Out]

-(ArcTan[Sqrt[-a + b/x^2]/Sqrt[a]]/Sqrt[a])

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Rubi in Sympy [A]  time = 5.5592, size = 22, normalized size = 0.81 \[ - \frac{\operatorname{atan}{\left (\frac{\sqrt{- a + \frac{b}{x^{2}}}}{\sqrt{a}} \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(-a+b/x**2)**(1/2),x)

[Out]

-atan(sqrt(-a + b/x**2)/sqrt(a))/sqrt(a)

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Mathematica [B]  time = 0.0605379, size = 56, normalized size = 2.07 \[ \frac{\sqrt{a x^2-b} \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2-b}}\right )}{\sqrt{a} x \sqrt{\frac{b}{x^2}-a}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[-a + b/x^2]*x),x]

[Out]

(Sqrt[-b + a*x^2]*ArcTanh[(Sqrt[a]*x)/Sqrt[-b + a*x^2]])/(Sqrt[a]*Sqrt[-a + b/x^
2]*x)

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Maple [B]  time = 0.013, size = 50, normalized size = 1.9 \[{\frac{1}{x}\sqrt{-a{x}^{2}+b}\arctan \left ({x\sqrt{a}{\frac{1}{\sqrt{-a{x}^{2}+b}}}} \right ){\frac{1}{\sqrt{-{\frac{a{x}^{2}-b}{{x}^{2}}}}}}{\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(-a+b/x^2)^(1/2),x)

[Out]

1/(-(a*x^2-b)/x^2)^(1/2)/x*(-a*x^2+b)^(1/2)/a^(1/2)*arctan(a^(1/2)*x/(-a*x^2+b)^
(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.247084, size = 1, normalized size = 0.04 \[ \left [-\frac{\sqrt{-a} \log \left (2 \, a x^{2} \sqrt{-\frac{a x^{2} - b}{x^{2}}} +{\left (2 \, a x^{2} - b\right )} \sqrt{-a}\right )}{2 \, a}, \frac{\arctan \left (\frac{\sqrt{a}}{\sqrt{-\frac{a x^{2} - b}{x^{2}}}}\right )}{\sqrt{a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/2*sqrt(-a)*log(2*a*x^2*sqrt(-(a*x^2 - b)/x^2) + (2*a*x^2 - b)*sqrt(-a))/a, a
rctan(sqrt(a)/sqrt(-(a*x^2 - b)/x^2))/sqrt(a)]

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Sympy [A]  time = 4.48985, size = 46, normalized size = 1.7 \[ \begin{cases} - \frac{i \operatorname{acosh}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{\sqrt{a}} & \text{for}\: \left |{\frac{a x^{2}}{b}}\right | > 1 \\\frac{\operatorname{asin}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{\sqrt{a}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(-a+b/x**2)**(1/2),x)

[Out]

Piecewise((-I*acosh(sqrt(a)*x/sqrt(b))/sqrt(a), Abs(a*x**2/b) > 1), (asin(sqrt(a
)*x/sqrt(b))/sqrt(a), True))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-a + \frac{b}{x^{2}}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(-a + b/x^2)*x), x)