3.1929 \(\int \frac {1}{\sqrt {2-\frac {b}{x^2}} x^2} \, dx\)

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

[Out]

-arccsc(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 = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {335, 216} \[ -\frac {\csc ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {b}}\right )}{\sqrt {b}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[2 - b/x^2]*x^2),x]

[Out]

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

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 {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 = 0.02, size = 54, normalized size = 2.70 \[ \frac {\sqrt {2 x^2-b} \tan ^{-1}\left (\frac {\sqrt {2 x^2-b}}{\sqrt {b}}\right )}{\sqrt {b} x \sqrt {2-\frac {b}{x^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[2 - b/x^2]*x^2),x]

[Out]

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

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fricas [B]  time = 0.78, 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(1/x^2/(2-b/x^2)^(1/2),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 [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(t_nostep)]Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is
real):Check [sign(t_nostep),sign(t_nostep+sqrt(-b)/2*sign(t_nostep))]sym2poly/r2sym(const gen & e,const index
_m & i,const vecteur & l) Error: Bad Argument Value

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maple [B]  time = 0.01, size = 64, normalized size = 3.20 \[ -\frac {\sqrt {2 x^{2}-b}\, \ln \left (\frac {-2 b +2 \sqrt {-b}\, \sqrt {2 x^{2}-b}}{x}\right )}{\sqrt {\frac {2 x^{2}-b}{x^{2}}}\, \sqrt {-b}\, x} \]

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 [A]  time = 1.97, size = 20, normalized size = 1.00 \[ \frac {\arctan \left (\frac {x \sqrt {-\frac {b}{x^{2}} + 2}}{\sqrt {b}}\right )}{\sqrt {b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.25, 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(1/(x^2*(2 - b/x^2)^(1/2)),x)

[Out]

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

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sympy [A]  time = 1.33, 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(sqrt(2)*sqrt(b)/(2*x))/sqrt(b),
True))

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