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

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)

________________________________________________________________________________________

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 = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {25, 335, 215} \[ -\frac {\text {csch}^{-1}\left (\frac {\sqrt {2} x}{\sqrt {b}}\right )}{\sqrt {b}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[2 + b/x^2]/(b + 2*x^2),x]

[Out]

-(ArcCsch[(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 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 {\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 {\text {csch}^{-1}\left (\frac {\sqrt {2} x}{\sqrt {b}}\right )}{\sqrt {b}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B]  time = 0.01, size = 48, normalized size = 2.40 \[ -\frac {x \sqrt {\frac {b}{x^2}+2} \tanh ^{-1}\left (\frac {\sqrt {b+2 x^2}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {b+2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.85, 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((2+b/x^2)^(1/2)/(2*x^2+b),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]

________________________________________________________________________________________

giac [B]  time = 0.35, size = 44, normalized size = 2.20 \[ \frac {\arctan \left (\frac {\sqrt {2 \, x^{2} + b}}{\sqrt {-b}}\right ) \mathrm {sgn}\relax (x)}{\sqrt {-b}} - \frac {\arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right ) \mathrm {sgn}\relax (x)}{\sqrt {-b}} \]

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)

________________________________________________________________________________________

maple [B]  time = 0.01, size = 50, normalized size = 2.50 \[ -\frac {\sqrt {\frac {2 x^{2}+b}{x^{2}}}\, x \ln \left (\frac {2 b +2 \sqrt {2 x^{2}+b}\, \sqrt {b}}{x}\right )}{\sqrt {2 x^{2}+b}\, \sqrt {b}} \]

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)

________________________________________________________________________________________

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)

________________________________________________________________________________________

mupad [B]  time = 3.46, 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((b/x^2 + 2)^(1/2)/(b + 2*x^2),x)

[Out]

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

________________________________________________________________________________________

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)

________________________________________________________________________________________