Optimal. Leaf size=25 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{\sqrt{a}} \]
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Rubi [A] time = 0.0475632, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
[In] Int[1/(x*Sqrt[a + b*x^2 + (2 + 2*c - 2*(1 + c))*x^4]),x]
[Out]
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Rubi in Sympy [A] time = 5.42323, size = 22, normalized size = 0.88 \[ - \frac{\operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{\sqrt{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x/(b*x**2+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0273563, size = 31, normalized size = 1.24 \[ \frac{\log (x)-\log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x*Sqrt[a + b*x^2 + (2 + 2*c - 2*(1 + c))*x^4]),x]
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Maple [A] time = 0.006, size = 29, normalized size = 1.2 \[ -{1\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x/(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(b*x^2 + a)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.279876, size = 1, normalized size = 0.04 \[ \left [\frac{\log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right )}{2 \, \sqrt{a}}, -\frac{\arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right )}{\sqrt{-a}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x^2 + a)*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.47171, size = 19, normalized size = 0.76 \[ - \frac{\operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{\sqrt{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x/(b*x**2+a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.265499, size = 30, normalized size = 1.2 \[ \frac{\arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x^2 + a)*x),x, algorithm="giac")
[Out]