3.1008 \(\int \frac{1}{x \sqrt{a+(2+2 b-2 (1+b)) x^2+c x^4}} \, dx\)

Optimal. Leaf size=27 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right )}{2 \sqrt{a}} \]

[Out]

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

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Rubi [A]  time = 0.048112, antiderivative size = 27, 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 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right )}{2 \sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[a + (2 + 2*b - 2*(1 + b))*x^2 + c*x^4]),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.97162, size = 24, normalized size = 0.89 \[ - \frac{\operatorname{atanh}{\left (\frac{\sqrt{a + c x^{4}}}{\sqrt{a}} \right )}}{2 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0639467, size = 27, normalized size = 1. \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right )}{2 \sqrt{a}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 29, normalized size = 1.1 \[ -{\frac{1}{2}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{4}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(c*x^4+a)^(1/2),x)

[Out]

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

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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 + a)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.28077, size = 1, normalized size = 0.04 \[ \left [\frac{\log \left (\frac{{\left (c x^{4} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{c x^{4} + a} a}{x^{4}}\right )}{4 \, \sqrt{a}}, \frac{\arctan \left (\frac{a}{\sqrt{c x^{4} + a} \sqrt{-a}}\right )}{2 \, \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.61092, size = 22, normalized size = 0.81 \[ - \frac{\operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x^{2}} \right )}}{2 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(c*x**4+a)**(1/2),x)

[Out]

-asinh(sqrt(a)/(sqrt(c)*x**2))/(2*sqrt(a))

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GIAC/XCAS [A]  time = 0.274781, size = 31, normalized size = 1.15 \[ \frac{\arctan \left (\frac{\sqrt{c x^{4} + a}}{\sqrt{-a}}\right )}{2 \, \sqrt{-a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*arctan(sqrt(c*x^4 + a)/sqrt(-a))/sqrt(-a)