3.1915 \(\int \frac{1}{\sqrt{a+\frac{b}{x^2}} x} \, dx\)

Optimal. Leaf size=24 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{\sqrt{a}} \]

[Out]

ArcTanh[Sqrt[a + b/x^2]/Sqrt[a]]/Sqrt[a]

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Rubi [A]  time = 0.057053, antiderivative size = 24, 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 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[a + b/x^2]*x),x]

[Out]

ArcTanh[Sqrt[a + b/x^2]/Sqrt[a]]/Sqrt[a]

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Rubi in Sympy [A]  time = 5.13101, size = 20, normalized size = 0.83 \[ \frac{\operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{2}}}}{\sqrt{a}} \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.0293485, size = 50, normalized size = 2.08 \[ \frac{\sqrt{a x^2+b} \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b}}\right )}{\sqrt{a} x \sqrt{a+\frac{b}{x^2}}} \]

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 = 0.008, size = 46, normalized size = 1.9 \[{\frac{1}{x}\sqrt{a{x}^{2}+b}\ln \left ( \sqrt{a}x+\sqrt{a{x}^{2}+b} \right ){\frac{1}{\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}}}}{\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^(1/2)/x,x)

[Out]

1/((a*x^2+b)/x^2)^(1/2)/x*(a*x^2+b)^(1/2)*ln(a^(1/2)*x+(a*x^2+b)^(1/2))/a^(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 = 0.241666, size = 1, normalized size = 0.04 \[ \left [\frac{\log \left (-2 \, a x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} -{\left (2 \, a x^{2} + b\right )} \sqrt{a}\right )}{2 \, \sqrt{a}}, -\frac{\sqrt{-a} \arctan \left (\frac{\sqrt{-a}}{\sqrt{\frac{a x^{2} + b}{x^{2}}}}\right )}{a}\right ] \]

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

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Sympy [A]  time = 4.25846, size = 17, normalized size = 0.71 \[ \frac{\operatorname{asinh}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)