3.919 \(\int \frac{\sqrt{2-\frac{b}{x^2}}}{-b+2 x^2} \, dx\)

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 = 0.00904, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {25, 335, 216} \[ -\frac{\csc ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{b}}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[2 - b/x^2]/(-b + 2*x^2),x]

[Out]

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

Rule 25

Int[(u_.)*((a_) + (b_.)*(x_)^(n_.))^(m_.)*((c_) + (d_.)*(x_)^(q_.))^(p_.), x_Symbol] :> Dist[(d/a)^p, Int[(u*(
a + b*x^n)^(m + p))/x^(n*p), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[q, -n] && IntegerQ[p] && EqQ[a*c -
b*d, 0] &&  !(IntegerQ[m] && NegQ[n])

Rule 335

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /;
FreeQ[{a, b, p}, x] && ILtQ[n, 0] && IntegerQ[m]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{\sqrt{2-\frac{b}{x^2}}}{-b+2 x^2} \, dx &=\int \frac{1}{\sqrt{2-\frac{b}{x^2}} x^2} \, dx\\ &=-\operatorname{Subst}\left (\int \frac{1}{\sqrt{2-b x^2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\csc ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{b}}\right )}{\sqrt{b}}\\ \end{align*}

Mathematica [B]  time = 0.0133659, size = 52, normalized size = 2.6 \[ \frac{x \sqrt{2-\frac{b}{x^2}} \tan ^{-1}\left (\frac{\sqrt{2 x^2-b}}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{2 x^2-b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2 - b/x^2]*x*ArcTan[Sqrt[-b + 2*x^2]/Sqrt[b]])/(Sqrt[b]*Sqrt[-b + 2*x^2])

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Maple [B]  time = 0.008, size = 62, normalized size = 3.1 \begin{align*} -{x\sqrt{{\frac{2\,{x}^{2}-b}{{x}^{2}}}}\ln \left ( 2\,{\frac{\sqrt{-b}\sqrt{2\,{x}^{2}-b}-b}{x}} \right ){\frac{1}{\sqrt{-b}}}{\frac{1}{\sqrt{2\,{x}^{2}-b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2-b/x^2)^(1/2)/(2*x^2-b),x)

[Out]

-((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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2-b/x^2)^(1/2)/(2*x^2-b),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.55005, size = 186, normalized size = 9.3 \begin{align*} \left [-\frac{\sqrt{-b} \log \left (-\frac{x^{2} - \sqrt{-b} x \sqrt{\frac{2 \, x^{2} - b}{x^{2}}} - b}{x^{2}}\right )}{2 \, b}, -\frac{\arctan \left (\frac{\sqrt{b} x \sqrt{\frac{2 \, x^{2} - b}{x^{2}}}}{2 \, x^{2} - b}\right )}{\sqrt{b}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2-b/x^2)^(1/2)/(2*x^2-b),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- \frac{b}{x^{2}} + 2}}{- b + 2 x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2-b/x**2)**(1/2)/(2*x**2-b),x)

[Out]

Integral(sqrt(-b/x**2 + 2)/(-b + 2*x**2), x)

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Giac [B]  time = 1.10405, size = 54, normalized size = 2.7 \begin{align*} \frac{\arctan \left (\frac{\sqrt{2 \, x^{2} - b}}{\sqrt{b}}\right ) \mathrm{sgn}\left (x\right )}{\sqrt{b}} - \frac{\arctan \left (\frac{\sqrt{-b}}{\sqrt{b}}\right ) \mathrm{sgn}\left (x\right )}{\sqrt{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2-b/x^2)^(1/2)/(2*x^2-b),x, algorithm="giac")

[Out]

arctan(sqrt(2*x^2 - b)/sqrt(b))*sgn(x)/sqrt(b) - arctan(sqrt(-b)/sqrt(b))*sgn(x)/sqrt(b)