Optimal. Leaf size=30 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b}}{x^2 \sqrt{a+\frac{b}{x^4}}}\right )}{2 \sqrt{b}} \]
[Out]
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Rubi [A] time = 0.0748344, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b}}{x^2 \sqrt{a+\frac{b}{x^4}}}\right )}{2 \sqrt{b}} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[a + b/x^4]*x^3),x]
[Out]
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Rubi in Sympy [A] time = 5.99799, size = 27, normalized size = 0.9 \[ - \frac{\operatorname{atanh}{\left (\frac{\sqrt{b}}{x^{2} \sqrt{a + \frac{b}{x^{4}}}} \right )}}{2 \sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**3/(a+b/x**4)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0767322, size = 52, normalized size = 1.73 \[ -\frac{\sqrt{a x^4+b} \tanh ^{-1}\left (\frac{\sqrt{a x^4+b}}{\sqrt{b}}\right )}{2 \sqrt{b} x^2 \sqrt{a+\frac{b}{x^4}}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[a + b/x^4]*x^3),x]
[Out]
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Maple [B] time = 0.017, size = 52, normalized size = 1.7 \[ -{\frac{1}{2\,{x}^{2}}\sqrt{a{x}^{4}+b}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{4}+b}+b}{{x}^{2}}} \right ){\frac{1}{\sqrt{{\frac{a{x}^{4}+b}{{x}^{4}}}}}}{\frac{1}{\sqrt{b}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^3/(a+b/x^4)^(1/2),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^4)*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.255224, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (-\frac{2 \, b x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}} -{\left (a x^{4} + 2 \, b\right )} \sqrt{b}}{x^{4}}\right )}{4 \, \sqrt{b}}, -\frac{\sqrt{-b} \arctan \left (\frac{b}{\sqrt{-b} x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}}}\right )}{2 \, b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(a + b/x^4)*x^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.20858, size = 22, normalized size = 0.73 \[ - \frac{\operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x^{2}} \right )}}{2 \sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**3/(a+b/x**4)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + \frac{b}{x^{4}}} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(a + b/x^4)*x^3),x, algorithm="giac")
[Out]