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