3.832 \(\int \frac{1}{x \sqrt{a-b x^4}} \, dx\)

Optimal. Leaf size=28 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a-b x^4}}{\sqrt{a}}\right )}{2 \sqrt{a}} \]

[Out]

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

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Rubi [A]  time = 0.0505528, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a-b x^4}}{\sqrt{a}}\right )}{2 \sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[a - b*x^4]),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.64752, size = 24, normalized size = 0.86 \[ - \frac{\operatorname{atanh}{\left (\frac{\sqrt{a - b x^{4}}}{\sqrt{a}} \right )}}{2 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-atanh(sqrt(a - b*x**4)/sqrt(a))/(2*sqrt(a))

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Mathematica [A]  time = 0.0758824, size = 28, normalized size = 1. \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a-b x^4}}{\sqrt{a}}\right )}{2 \sqrt{a}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.016, size = 30, normalized size = 1.1 \[ -{\frac{1}{2}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{-b{x}^{4}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(-b*x^4+a)^(1/2),x)

[Out]

-1/2/a^(1/2)*ln((2*a+2*a^(1/2)*(-b*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(-b*x^4 + a)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.26438, size = 1, normalized size = 0.04 \[ \left [\frac{\log \left (\frac{{\left (b x^{4} - 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{-b x^{4} + a} a}{x^{4}}\right )}{4 \, \sqrt{a}}, \frac{\arctan \left (\frac{a}{\sqrt{-b x^{4} + a} \sqrt{-a}}\right )}{2 \, \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.89452, size = 53, normalized size = 1.89 \[ \begin{cases} - \frac{\operatorname{acosh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2 \sqrt{a}} & \text{for}\: \left |{\frac{a}{b x^{4}}}\right | > 1 \\\frac{i \operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2 \sqrt{a}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-acosh(sqrt(a)/(sqrt(b)*x**2))/(2*sqrt(a)), Abs(a/(b*x**4)) > 1), (I*
asin(sqrt(a)/(sqrt(b)*x**2))/(2*sqrt(a)), True))

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GIAC/XCAS [A]  time = 0.223323, size = 32, normalized size = 1.14 \[ \frac{\arctan \left (\frac{\sqrt{-b x^{4} + a}}{\sqrt{-a}}\right )}{2 \, \sqrt{-a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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