Optimal. Leaf size=27 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )}{2 \sqrt{a}} \]
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Rubi [A] time = 0.0612681, antiderivative size = 27, 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 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )}{2 \sqrt{a}} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[a + b/x^4]*x),x]
[Out]
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Rubi in Sympy [A] time = 5.15627, size = 22, normalized size = 0.81 \[ \frac{\operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{4}}}}{\sqrt{a}} \right )}}{2 \sqrt{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x/(a+b/x**4)**(1/2),x)
[Out]
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Mathematica [B] time = 0.0305597, size = 55, normalized size = 2.04 \[ \frac{\sqrt{a x^4+b} \tanh ^{-1}\left (\frac{\sqrt{a} x^2}{\sqrt{a x^4+b}}\right )}{2 \sqrt{a} x^2 \sqrt{a+\frac{b}{x^4}}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[a + b/x^4]*x),x]
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Maple [B] time = 0.013, size = 49, normalized size = 1.8 \[{\frac{1}{2\,{x}^{2}}\sqrt{a{x}^{4}+b}\ln \left ({x}^{2}\sqrt{a}+\sqrt{a{x}^{4}+b} \right ){\frac{1}{\sqrt{{\frac{a{x}^{4}+b}{{x}^{4}}}}}}{\frac{1}{\sqrt{a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x/(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),x, algorithm="maxima")
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Fricas [A] time = 0.251544, size = 1, normalized size = 0.04 \[ \left [\frac{\log \left (-2 \, a x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} -{\left (2 \, a x^{4} + b\right )} \sqrt{a}\right )}{4 \, \sqrt{a}}, -\frac{\sqrt{-a} \arctan \left (\frac{\sqrt{-a}}{\sqrt{\frac{a x^{4} + b}{x^{4}}}}\right )}{2 \, a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(a + b/x^4)*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.95612, size = 20, normalized size = 0.74 \[ \frac{\operatorname{asinh}{\left (\frac{\sqrt{a} x^{2}}{\sqrt{b}} \right )}}{2 \sqrt{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x/(a+b/x**4)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + \frac{b}{x^{4}}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(a + b/x^4)*x),x, algorithm="giac")
[Out]