Optimal. Leaf size=34 \[ -\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{b^{3/2}}-\frac{1}{b x} \]
[Out]
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Rubi [A] time = 0.046018, antiderivative size = 34, 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 \[ -\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{b^{3/2}}-\frac{1}{b x} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b/x^2)*x^4),x]
[Out]
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Rubi in Sympy [A] time = 7.64168, size = 29, normalized size = 0.85 \[ - \frac{\sqrt{a} \operatorname{atan}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{b^{\frac{3}{2}}} - \frac{1}{b x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x**2)/x**4,x)
[Out]
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Mathematica [A] time = 0.0229773, size = 34, normalized size = 1. \[ -\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{b^{3/2}}-\frac{1}{b x} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b/x^2)*x^4),x]
[Out]
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Maple [A] time = 0.004, size = 30, normalized size = 0.9 \[ -{\frac{a}{b}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{1}{bx}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x^2)/x^4,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/((a + b/x^2)*x^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.233231, size = 1, normalized size = 0.03 \[ \left [\frac{x \sqrt{-\frac{a}{b}} \log \left (\frac{a x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - b}{a x^{2} + b}\right ) - 2}{2 \, b x}, -\frac{x \sqrt{\frac{a}{b}} \arctan \left (\frac{a x}{b \sqrt{\frac{a}{b}}}\right ) + 1}{b x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x^2)*x^4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.32007, size = 65, normalized size = 1.91 \[ \frac{\sqrt{- \frac{a}{b^{3}}} \log{\left (x - \frac{b^{2} \sqrt{- \frac{a}{b^{3}}}}{a} \right )}}{2} - \frac{\sqrt{- \frac{a}{b^{3}}} \log{\left (x + \frac{b^{2} \sqrt{- \frac{a}{b^{3}}}}{a} \right )}}{2} - \frac{1}{b x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b/x**2)/x**4,x)
[Out]
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GIAC/XCAS [A] time = 0.223114, size = 39, normalized size = 1.15 \[ -\frac{a \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{\sqrt{a b} b} - \frac{1}{b x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x^2)*x^4),x, algorithm="giac")
[Out]