Optimal. Leaf size=30 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{\sqrt{b}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0227959, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[2 + 2*a - 2*(1 + a) + b*x^2 + c*x^4],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 5.44413, size = 27, normalized size = 0.9 \[ - \frac{\operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{b x^{2} + c x^{4}}} \right )}}{\sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(c*x**4+b*x**2)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0425539, size = 58, normalized size = 1.93 \[ \frac{x \sqrt{b+c x^2} \left (\log (x)-\log \left (\sqrt{b} \sqrt{b+c x^2}+b\right )\right )}{\sqrt{b} \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[1/Sqrt[2 + 2*a - 2*(1 + a) + b*x^2 + c*x^4],x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0., size = 50, normalized size = 1.7 \[ -{x\sqrt{c{x}^{2}+b}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}{\frac{1}{\sqrt{b}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(c*x^4+b*x^2)^(1/2),x)
[Out]
_______________________________________________________________________________________
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),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.278009, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (-\frac{{\left (c x^{3} + 2 \, b x\right )} \sqrt{b} - 2 \, \sqrt{c x^{4} + b x^{2}} b}{x^{3}}\right )}{2 \, \sqrt{b}}, \frac{\sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{c x^{4} + b x^{2}}}\right )}{b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(c*x^4 + b*x^2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x^{2} + c x^{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(c*x**4+b*x**2)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.273501, size = 62, normalized size = 2.07 \[ -\frac{\arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right ){\rm sign}\left (x\right )}{\sqrt{-b}} + \frac{\arctan \left (\frac{\sqrt{c x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b}{\rm sign}\left (x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(c*x^4 + b*x^2),x, algorithm="giac")
[Out]