3.998 \(\int \frac{1}{\sqrt{2+2 a-2 (1+a)+b x^2+c x^4}} \, dx\)

Optimal. Leaf size=30 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{\sqrt{b}} \]

[Out]

-(ArcTanh[(Sqrt[b]*x)/Sqrt[b*x^2 + c*x^4]]/Sqrt[b])

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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]

-(ArcTanh[(Sqrt[b]*x)/Sqrt[b*x^2 + c*x^4]]/Sqrt[b])

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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]

-atanh(sqrt(b)*x/sqrt(b*x**2 + c*x**4))/sqrt(b)

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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]

(x*Sqrt[b + c*x^2]*(Log[x] - Log[b + Sqrt[b]*Sqrt[b + c*x^2]]))/(Sqrt[b]*Sqrt[x^
2*(b + c*x^2)])

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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]

-1/(c*x^4+b*x^2)^(1/2)*x*(c*x^2+b)^(1/2)/b^(1/2)*ln(2*(b^(1/2)*(c*x^2+b)^(1/2)+b
)/x)

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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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]

[1/2*log(-((c*x^3 + 2*b*x)*sqrt(b) - 2*sqrt(c*x^4 + b*x^2)*b)/x^3)/sqrt(b), sqrt
(-b)*arctan(sqrt(-b)*x/sqrt(c*x^4 + b*x^2))/b]

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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]

Integral(1/sqrt(b*x**2 + c*x**4), x)

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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]

-arctan(sqrt(b)/sqrt(-b))*sign(x)/sqrt(-b) + arctan(sqrt(c*x^2 + b)/sqrt(-b))/(s
qrt(-b)*sign(x))