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

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Rubi [A]  time = 0.0081978, antiderivative size = 20, normalized size of antiderivative = 1., 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 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 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]

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 = 0.0131397, size = 48, normalized size = 2.4 \[ -\frac{x \sqrt{\frac{b}{x^2}+2} \tanh ^{-1}\left (\frac{\sqrt{b+2 x^2}}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{b+2 x^2}} \]

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]))

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Maple [B]  time = 0.01, size = 50, normalized size = 2.5 \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 = 1.78051, size = 184, normalized size = 9.2 \begin{align*} \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 ] \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*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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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 = 1.09049, size = 59, normalized size = 2.95 \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)