3.1923 \(\int \frac{1}{\sqrt{a+\frac{b}{x^2}} x^2} \, dx\)

Optimal. Leaf size=28 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{\sqrt{b}} \]

[Out]

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

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Rubi [A]  time = 0.0472439, antiderivative size = 28, 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 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[a + b/x^2]*x^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.34709, size = 24, normalized size = 0.86 \[ - \frac{\operatorname{atanh}{\left (\frac{\sqrt{b}}{x \sqrt{a + \frac{b}{x^{2}}}} \right )}}{\sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0449711, size = 56, normalized size = 2. \[ \frac{\sqrt{a x^2+b} \left (\log (x)-\log \left (\sqrt{b} \sqrt{a x^2+b}+b\right )\right )}{\sqrt{b} x \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.011, size = 52, normalized size = 1.9 \[ -{\frac{1}{x}\sqrt{a{x}^{2}+b}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{2}+b}+b}{x}} \right ){\frac{1}{\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}}}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^(1/2)/x^2,x)

[Out]

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

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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^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.241416, size = 1, normalized size = 0.04 \[ \left [\frac{\log \left (\frac{2 \, b x \sqrt{\frac{a x^{2} + b}{x^{2}}} -{\left (a x^{2} + 2 \, b\right )} \sqrt{b}}{x^{2}}\right )}{2 \, \sqrt{b}}, \frac{\sqrt{-b} \arctan \left (\frac{\sqrt{-b}}{x \sqrt{\frac{a x^{2} + b}{x^{2}}}}\right )}{b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.50102, size = 19, normalized size = 0.68 \[ - \frac{\operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x} \right )}}{\sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-asinh(sqrt(b)/(sqrt(a)*x))/sqrt(b)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(a + b/x^2)*x^2), x)