3.184 \(\int \frac{1}{b x^2+c x^4} \, dx\)

Optimal. Leaf size=34 \[ -\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{3/2}}-\frac{1}{b x} \]

[Out]

-(1/(b*x)) - (Sqrt[c]*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/b^(3/2)

_______________________________________________________________________________________

Rubi [A]  time = 0.0372819, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{3/2}}-\frac{1}{b x} \]

Antiderivative was successfully verified.

[In]  Int[(b*x^2 + c*x^4)^(-1),x]

[Out]

-(1/(b*x)) - (Sqrt[c]*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/b^(3/2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 11.274, size = 29, normalized size = 0.85 \[ - \frac{1}{b x} - \frac{\sqrt{c} \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{b}} \right )}}{b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x**4+b*x**2),x)

[Out]

-1/(b*x) - sqrt(c)*atan(sqrt(c)*x/sqrt(b))/b**(3/2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0217809, size = 34, normalized size = 1. \[ -\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{3/2}}-\frac{1}{b x} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x^2 + c*x^4)^(-1),x]

[Out]

-(1/(b*x)) - (Sqrt[c]*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/b^(3/2)

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 30, normalized size = 0.9 \[ -{\frac{c}{b}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}-{\frac{1}{bx}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x^4+b*x^2),x)

[Out]

-c/b/(b*c)^(1/2)*arctan(c*x/(b*c)^(1/2))-1/b/x

_______________________________________________________________________________________

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/(c*x^4 + b*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.25905, size = 1, normalized size = 0.03 \[ \left [\frac{x \sqrt{-\frac{c}{b}} \log \left (\frac{c x^{2} - 2 \, b x \sqrt{-\frac{c}{b}} - b}{c x^{2} + b}\right ) - 2}{2 \, b x}, -\frac{x \sqrt{\frac{c}{b}} \arctan \left (\frac{c x}{b \sqrt{\frac{c}{b}}}\right ) + 1}{b x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c*x^4 + b*x^2),x, algorithm="fricas")

[Out]

[1/2*(x*sqrt(-c/b)*log((c*x^2 - 2*b*x*sqrt(-c/b) - b)/(c*x^2 + b)) - 2)/(b*x), -
(x*sqrt(c/b)*arctan(c*x/(b*sqrt(c/b))) + 1)/(b*x)]

_______________________________________________________________________________________

Sympy [A]  time = 1.27202, size = 65, normalized size = 1.91 \[ \frac{\sqrt{- \frac{c}{b^{3}}} \log{\left (- \frac{b^{2} \sqrt{- \frac{c}{b^{3}}}}{c} + x \right )}}{2} - \frac{\sqrt{- \frac{c}{b^{3}}} \log{\left (\frac{b^{2} \sqrt{- \frac{c}{b^{3}}}}{c} + x \right )}}{2} - \frac{1}{b x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c*x**4+b*x**2),x)

[Out]

sqrt(-c/b**3)*log(-b**2*sqrt(-c/b**3)/c + x)/2 - sqrt(-c/b**3)*log(b**2*sqrt(-c/
b**3)/c + x)/2 - 1/(b*x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.269403, size = 39, normalized size = 1.15 \[ -\frac{c \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{\sqrt{b c} b} - \frac{1}{b x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c*x^4 + b*x^2),x, algorithm="giac")

[Out]

-c*arctan(c*x/sqrt(b*c))/(sqrt(b*c)*b) - 1/(b*x)