3.1864 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^2 x^2} \, dx\)

Optimal. Leaf size=45 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 a^{3/2} \sqrt{b}}-\frac{x}{2 a \left (a x^2+b\right )} \]

[Out]

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

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Rubi [A]  time = 0.0505717, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 a^{3/2} \sqrt{b}}-\frac{x}{2 a \left (a x^2+b\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^2)^2*x^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.60514, size = 36, normalized size = 0.8 \[ - \frac{x}{2 a \left (a x^{2} + b\right )} + \frac{\operatorname{atan}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{2 a^{\frac{3}{2}} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0377234, size = 45, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 a^{3/2} \sqrt{b}}-\frac{x}{2 a \left (a x^2+b\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 36, normalized size = 0.8 \[ -{\frac{x}{2\,a \left ( a{x}^{2}+b \right ) }}+{\frac{1}{2\,a}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^2/x^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232293, size = 1, normalized size = 0.02 \[ \left [\frac{{\left (a x^{2} + b\right )} \log \left (\frac{2 \, a b x +{\left (a x^{2} - b\right )} \sqrt{-a b}}{a x^{2} + b}\right ) - 2 \, \sqrt{-a b} x}{4 \,{\left (a^{2} x^{2} + a b\right )} \sqrt{-a b}}, \frac{{\left (a x^{2} + b\right )} \arctan \left (\frac{\sqrt{a b} x}{b}\right ) - \sqrt{a b} x}{2 \,{\left (a^{2} x^{2} + a b\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*((a*x^2 + b)*log((2*a*b*x + (a*x^2 - b)*sqrt(-a*b))/(a*x^2 + b)) - 2*sqrt(-
a*b)*x)/((a^2*x^2 + a*b)*sqrt(-a*b)), 1/2*((a*x^2 + b)*arctan(sqrt(a*b)*x/b) - s
qrt(a*b)*x)/((a^2*x^2 + a*b)*sqrt(a*b))]

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Sympy [A]  time = 1.40102, size = 78, normalized size = 1.73 \[ - \frac{x}{2 a^{2} x^{2} + 2 a b} - \frac{\sqrt{- \frac{1}{a^{3} b}} \log{\left (- a b \sqrt{- \frac{1}{a^{3} b}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a^{3} b}} \log{\left (a b \sqrt{- \frac{1}{a^{3} b}} + x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x/(2*a**2*x**2 + 2*a*b) - sqrt(-1/(a**3*b))*log(-a*b*sqrt(-1/(a**3*b)) + x)/4 +
sqrt(-1/(a**3*b))*log(a*b*sqrt(-1/(a**3*b)) + x)/4

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GIAC/XCAS [A]  time = 0.223468, size = 47, normalized size = 1.04 \[ \frac{\arctan \left (\frac{a x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a} - \frac{x}{2 \,{\left (a x^{2} + b\right )} a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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