3.2082 \(\int \frac{1}{\sqrt{a+\frac{b}{x^4}} x} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0612681, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )}{2 \sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[a + b/x^4]*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.15627, size = 22, normalized size = 0.81 \[ \frac{\operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{4}}}}{\sqrt{a}} \right )}}{2 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.0305597, size = 55, normalized size = 2.04 \[ \frac{\sqrt{a x^4+b} \tanh ^{-1}\left (\frac{\sqrt{a} x^2}{\sqrt{a x^4+b}}\right )}{2 \sqrt{a} x^2 \sqrt{a+\frac{b}{x^4}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.013, size = 49, normalized size = 1.8 \[{\frac{1}{2\,{x}^{2}}\sqrt{a{x}^{4}+b}\ln \left ({x}^{2}\sqrt{a}+\sqrt{a{x}^{4}+b} \right ){\frac{1}{\sqrt{{\frac{a{x}^{4}+b}{{x}^{4}}}}}}{\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(a+b/x^4)^(1/2),x)

[Out]

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

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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(a + b/x^4)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.251544, size = 1, normalized size = 0.04 \[ \left [\frac{\log \left (-2 \, a x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} -{\left (2 \, a x^{4} + b\right )} \sqrt{a}\right )}{4 \, \sqrt{a}}, -\frac{\sqrt{-a} \arctan \left (\frac{\sqrt{-a}}{\sqrt{\frac{a x^{4} + b}{x^{4}}}}\right )}{2 \, a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.95612, size = 20, normalized size = 0.74 \[ \frac{\operatorname{asinh}{\left (\frac{\sqrt{a} x^{2}}{\sqrt{b}} \right )}}{2 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

asinh(sqrt(a)*x**2/sqrt(b))/(2*sqrt(a))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + \frac{b}{x^{4}}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(a + b/x^4)*x), x)