3.1927 \(\int \frac{1}{\sqrt{2-\frac{b}{x^2}} x^2} \, dx\)

Optimal. Leaf size=20 \[ -\frac{\csc ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{b}}\right )}{\sqrt{b}} \]

[Out]

-(ArcCsc[(Sqrt[2]*x)/Sqrt[b]]/Sqrt[b])

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Rubi [A]  time = 0.0343111, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{\csc ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{b}}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[2 - b/x^2]*x^2),x]

[Out]

-(ArcCsc[(Sqrt[2]*x)/Sqrt[b]]/Sqrt[b])

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Rubi in Sympy [A]  time = 4.13383, size = 20, normalized size = 1. \[ - \frac{\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b}}{2 x} \right )}}{\sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-asin(sqrt(2)*sqrt(b)/(2*x))/sqrt(b)

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Mathematica [C]  time = 0.054696, size = 64, normalized size = 3.2 \[ -\frac{i x \sqrt{2-\frac{b}{x^2}} \log \left (\frac{2 \left (\sqrt{2 x^2-b}-i \sqrt{b}\right )}{x}\right )}{\sqrt{b} \sqrt{2 x^2-b}} \]

Antiderivative was successfully verified.

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

[Out]

((-I)*Sqrt[2 - b/x^2]*x*Log[(2*((-I)*Sqrt[b] + Sqrt[-b + 2*x^2]))/x])/(Sqrt[b]*S
qrt[-b + 2*x^2])

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Maple [B]  time = 0.015, size = 64, normalized size = 3.2 \[ -{\frac{1}{x}\sqrt{2\,{x}^{2}-b}\ln \left ( 2\,{\frac{\sqrt{-b}\sqrt{2\,{x}^{2}-b}-b}{x}} \right ){\frac{1}{\sqrt{{\frac{2\,{x}^{2}-b}{{x}^{2}}}}}}{\frac{1}{\sqrt{-b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(2-b/x^2)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.251337, size = 1, normalized size = 0.05 \[ \left [-\frac{\sqrt{-b} \log \left (-\frac{b x \sqrt{\frac{2 \, x^{2} - b}{x^{2}}} +{\left (x^{2} - b\right )} \sqrt{-b}}{x^{2}}\right )}{2 \, b}, -\frac{\arctan \left (\frac{\sqrt{b}}{x \sqrt{\frac{2 \, x^{2} - b}{x^{2}}}}\right )}{\sqrt{b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^2*sqrt(-b/x^2 + 2)),x, algorithm="fricas")

[Out]

[-1/2*sqrt(-b)*log(-(b*x*sqrt((2*x^2 - b)/x^2) + (x^2 - b)*sqrt(-b))/x^2)/b, -ar
ctan(sqrt(b)/(x*sqrt((2*x^2 - b)/x^2)))/sqrt(b)]

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Sympy [A]  time = 4.6291, size = 49, normalized size = 2.45 \[ \begin{cases} \frac{i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{b}}{2 x} \right )}}{\sqrt{b}} & \text{for}\: \frac{\left |{\frac{b}{x^{2}}}\right |}{2} > 1 \\- \frac{\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b}}{2 x} \right )}}{\sqrt{b}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \sqrt{-\frac{b}{x^{2}} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^2*sqrt(-b/x^2 + 2)),x, algorithm="giac")

[Out]

integrate(1/(x^2*sqrt(-b/x^2 + 2)), x)