3.2083 \(\int \frac{1}{\sqrt{a+\frac{b}{x^4}} x^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0748344, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b}}{x^2 \sqrt{a+\frac{b}{x^4}}}\right )}{2 \sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[a + b/x^4]*x^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.99799, size = 27, normalized size = 0.9 \[ - \frac{\operatorname{atanh}{\left (\frac{\sqrt{b}}{x^{2} \sqrt{a + \frac{b}{x^{4}}}} \right )}}{2 \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(a+b/x**4)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0767322, size = 52, normalized size = 1.73 \[ -\frac{\sqrt{a x^4+b} \tanh ^{-1}\left (\frac{\sqrt{a x^4+b}}{\sqrt{b}}\right )}{2 \sqrt{b} x^2 \sqrt{a+\frac{b}{x^4}}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[a + b/x^4]*x^3),x]

[Out]

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

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Maple [B]  time = 0.017, size = 52, normalized size = 1.7 \[ -{\frac{1}{2\,{x}^{2}}\sqrt{a{x}^{4}+b}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{4}+b}+b}{{x}^{2}}} \right ){\frac{1}{\sqrt{{\frac{a{x}^{4}+b}{{x}^{4}}}}}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(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)/b^(1/2)*ln(2*(b^(1/2)*(a*x^4+b)^(
1/2)+b)/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(a + b/x^4)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.255224, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (-\frac{2 \, b x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}} -{\left (a x^{4} + 2 \, b\right )} \sqrt{b}}{x^{4}}\right )}{4 \, \sqrt{b}}, -\frac{\sqrt{-b} \arctan \left (\frac{b}{\sqrt{-b} x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}}}\right )}{2 \, b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x^4)*x^3),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 6.20858, size = 22, normalized size = 0.73 \[ - \frac{\operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x^{2}} \right )}}{2 \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(a+b/x**4)**(1/2),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + \frac{b}{x^{4}}} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x^4)*x^3),x, algorithm="giac")

[Out]

integrate(1/(sqrt(a + b/x^4)*x^3), x)