3.1847 \(\int \frac{1}{a+\frac{b}{x^2}} \, dx\)

Optimal. Leaf size=31 \[ \frac{x}{a}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{a^{3/2}} \]

[Out]

x/a - (Sqrt[b]*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/a^(3/2)

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Rubi [A]  time = 0.0393342, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{x}{a}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{a^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^(-1),x]

[Out]

x/a - (Sqrt[b]*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/a^(3/2)

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Rubi in Sympy [A]  time = 6.44295, size = 26, normalized size = 0.84 \[ \frac{x}{a} - \frac{\sqrt{b} \operatorname{atan}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**2),x)

[Out]

x/a - sqrt(b)*atan(sqrt(a)*x/sqrt(b))/a**(3/2)

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Mathematica [A]  time = 0.0157899, size = 31, normalized size = 1. \[ \frac{x}{a}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{a^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)^(-1),x]

[Out]

x/a - (Sqrt[b]*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/a^(3/2)

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Maple [A]  time = 0.002, size = 27, normalized size = 0.9 \[{\frac{x}{a}}-{\frac{b}{a}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2),x)

[Out]

x/a-b/a/(a*b)^(1/2)*arctan(a*x/(a*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/(a + b/x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237185, size = 1, normalized size = 0.03 \[ \left [\frac{\sqrt{-\frac{b}{a}} \log \left (\frac{a x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - b}{a x^{2} + b}\right ) + 2 \, x}{2 \, a}, -\frac{\sqrt{\frac{b}{a}} \arctan \left (\frac{x}{\sqrt{\frac{b}{a}}}\right ) - x}{a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.18514, size = 56, normalized size = 1.81 \[ \frac{\sqrt{- \frac{b}{a^{3}}} \log{\left (- a \sqrt{- \frac{b}{a^{3}}} + x \right )}}{2} - \frac{\sqrt{- \frac{b}{a^{3}}} \log{\left (a \sqrt{- \frac{b}{a^{3}}} + x \right )}}{2} + \frac{x}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**2),x)

[Out]

sqrt(-b/a**3)*log(-a*sqrt(-b/a**3) + x)/2 - sqrt(-b/a**3)*log(a*sqrt(-b/a**3) +
x)/2 + x/a

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GIAC/XCAS [A]  time = 0.225645, size = 35, normalized size = 1.13 \[ -\frac{b \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{\sqrt{a b} a} + \frac{x}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*arctan(a*x/sqrt(a*b))/(sqrt(a*b)*a) + x/a