3.1928 \(\int \frac {1}{\sqrt {2+\frac {b}{x^2}} x^2} \, dx\)

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

[Out]

-arccsch(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 = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {335, 215} \[ -\frac {\text {csch}^{-1}\left (\frac {\sqrt {2} x}{\sqrt {b}}\right )}{\sqrt {b}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[2 + b/x^2]*x^2),x]

[Out]

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

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[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 {\text {csch}^{-1}\left (\frac {\sqrt {2} x}{\sqrt {b}}\right )}{\sqrt {b}}\\ \end {align*}

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Mathematica [B]  time = 0.02, size = 50, normalized size = 2.50 \[ -\frac {\sqrt {b+2 x^2} \tanh ^{-1}\left (\frac {\sqrt {b+2 x^2}}{\sqrt {b}}\right )}{\sqrt {b} x \sqrt {\frac {b}{x^2}+2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[2 + b/x^2]*x^2),x]

[Out]

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

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fricas [B]  time = 1.02, size = 75, normalized size = 3.75 \[ \left [\frac {\log \left (-\frac {x^{2} - \sqrt {b} x \sqrt {\frac {2 \, x^{2} + b}{x^{2}}} + b}{x^{2}}\right )}{2 \, \sqrt {b}}, \frac {\sqrt {-b} \arctan \left (\frac {\sqrt {-b} x \sqrt {\frac {2 \, x^{2} + b}{x^{2}}}}{2 \, x^{2} + b}\right )}{b}\right ] \]

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*log(-(x^2 - sqrt(b)*x*sqrt((2*x^2 + b)/x^2) + b)/x^2)/sqrt(b), sqrt(-b)*arctan(sqrt(-b)*x*sqrt((2*x^2 + b
)/x^2)/(2*x^2 + b))/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 = 52, normalized size = 2.60 \[ -\frac {\sqrt {2 x^{2}+b}\, \ln \left (\frac {2 b +2 \sqrt {2 x^{2}+b}\, \sqrt {b}}{x}\right )}{\sqrt {\frac {2 x^{2}+b}{x^{2}}}\, \sqrt {b}\, x} \]

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 [B]  time = 1.96, size = 41, normalized size = 2.05 \[ \frac {\log \left (\frac {x \sqrt {\frac {b}{x^{2}} + 2} - \sqrt {b}}{x \sqrt {\frac {b}{x^{2}} + 2} + \sqrt {b}}\right )}{2 \, \sqrt {b}} \]

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]

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

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mupad [B]  time = 1.27, size = 17, normalized size = 0.85 \[ -\frac {\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {b}}{2\,x}\right )}{\sqrt {b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(b/x^2 + 2)^(1/2)),x)

[Out]

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

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sympy [A]  time = 1.32, size = 20, normalized size = 1.00 \[ - \frac {\operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {b}}{2 x} \right )}}{\sqrt {b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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