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(x*2^(1/2)/b^(1/2))/b^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [B]  time = 0.01, size = 52, normalized size = 2.60 \[ \frac {x \sqrt {2-\frac {b}{x^2}} \tan ^{-1}\left (\frac {\sqrt {2 x^2-b}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {2 x^2-b}} \]

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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fricas [B]  time = 0.73, size = 84, normalized size = 4.20 \[ \left [-\frac {\sqrt {-b} \log \left (-\frac {x^{2} - \sqrt {-b} x \sqrt {\frac {2 \, x^{2} - b}{x^{2}}} - b}{x^{2}}\right )}{2 \, b}, -\frac {\arctan \left (\frac {\sqrt {b} x \sqrt {\frac {2 \, x^{2} - b}{x^{2}}}}{2 \, x^{2} - b}\right )}{\sqrt {b}}\right ] \]

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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giac [B]  time = 0.42, size = 40, normalized size = 2.00 \[ \frac {\arctan \left (\frac {\sqrt {2 \, x^{2} - b}}{\sqrt {b}}\right ) \mathrm {sgn}\relax (x)}{\sqrt {b}} - \frac {\arctan \left (\frac {\sqrt {-b}}{\sqrt {b}}\right ) \mathrm {sgn}\relax (x)}{\sqrt {b}} \]

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)

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maple [B]  time = 0.01, size = 62, normalized size = 3.10 \[ -\frac {\sqrt {\frac {2 x^{2}-b}{x^{2}}}\, x \ln \left (\frac {-2 b +2 \sqrt {-b}\, \sqrt {2 x^{2}-b}}{x}\right )}{\sqrt {2 x^{2}-b}\, \sqrt {-b}} \]

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]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-\frac {b}{x^{2}} + 2}}{2 \, x^{2} - b}\,{d x} \]

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]

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

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mupad [B]  time = 3.47, size = 21, normalized size = 1.05 \[ -\frac {\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {-b}}{2\,x}\right )}{\sqrt {-b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2 - b/x^2)^(1/2)/(b - 2*x^2),x)

[Out]

-asinh((2^(1/2)*(-b)^(1/2))/(2*x))/(-b)^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- \frac {b}{x^{2}} + 2}}{- b + 2 x^{2}}\, dx \]

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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