Optimal. Leaf size=44 \[ -\frac{\tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{2 \sqrt{a}} \]
[Out]
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Rubi [A] time = 0.0861922, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{\tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{2 \sqrt{a}} \]
Antiderivative was successfully verified.
[In] Int[1/(x*Sqrt[a + b*x^2 + c*x^4]),x]
[Out]
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Rubi in Sympy [A] time = 10.8014, size = 39, normalized size = 0.89 \[ - \frac{\operatorname{atanh}{\left (\frac{2 a + b x^{2}}{2 \sqrt{a} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{2 \sqrt{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x/(c*x**4+b*x**2+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.11974, size = 50, normalized size = 1.14 \[ \frac{\log \left (x^2\right )-\log \left (2 \sqrt{a} \sqrt{a+x^2 \left (b+c x^2\right )}+2 a+b x^2\right )}{2 \sqrt{a}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x*Sqrt[a + b*x^2 + c*x^4]),x]
[Out]
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Maple [A] time = 0.014, size = 39, normalized size = 0.9 \[ -{\frac{1}{2}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x/(c*x^4+b*x^2+a)^(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(c*x^4 + b*x^2 + a)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.287385, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (\frac{4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (a b x^{2} + 2 \, a^{2}\right )} -{\left ({\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 8 \, a^{2}\right )} \sqrt{a}}{x^{4}}\right )}{4 \, \sqrt{a}}, -\frac{\arctan \left (\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{4} + b x^{2} + a} a}\right )}{2 \, \sqrt{-a}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*x^4 + b*x^2 + a)*x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \sqrt{a + b x^{2} + c x^{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x/(c*x**4+b*x**2+a)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{4} + b x^{2} + a} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*x^4 + b*x^2 + a)*x),x, algorithm="giac")
[Out]