3.1917 \(\int \frac{1}{\sqrt{a+\frac{b}{x^2}} x} \, dx\)

Optimal. Leaf size=24 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{\sqrt{a}} \]

[Out]

ArcTanh[Sqrt[a + b/x^2]/Sqrt[a]]/Sqrt[a]

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Rubi [A]  time = 0.0170164, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {266, 63, 208} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b/x^2]*x),x]

[Out]

ArcTanh[Sqrt[a + b/x^2]/Sqrt[a]]/Sqrt[a]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{a+\frac{b}{x^2}} x} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x^2}}\right )}{b}\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{\sqrt{a}}\\ \end{align*}

Mathematica [B]  time = 0.0130868, size = 50, normalized size = 2.08 \[ \frac{\sqrt{a x^2+b} \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b}}\right )}{\sqrt{a} x \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b/x^2]*x),x]

[Out]

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

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Maple [B]  time = 0.003, size = 46, normalized size = 1.9 \begin{align*}{\frac{1}{x}\sqrt{a{x}^{2}+b}\ln \left ( x\sqrt{a}+\sqrt{a{x}^{2}+b} \right ){\frac{1}{\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}}}}{\frac{1}{\sqrt{a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+1/x^2*b)^(1/2)/x,x)

[Out]

1/((a*x^2+b)/x^2)^(1/2)/x*(a*x^2+b)^(1/2)*ln(x*a^(1/2)+(a*x^2+b)^(1/2))/a^(1/2)

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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(1/(a+b/x^2)^(1/2)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.53614, size = 190, normalized size = 7.92 \begin{align*} \left [\frac{\log \left (-2 \, a x^{2} - 2 \, \sqrt{a} x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} - b\right )}{2 \, \sqrt{a}}, -\frac{\sqrt{-a} \arctan \left (\frac{\sqrt{-a} x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right )}{a}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^(1/2)/x,x, algorithm="fricas")

[Out]

[1/2*log(-2*a*x^2 - 2*sqrt(a)*x^2*sqrt((a*x^2 + b)/x^2) - b)/sqrt(a), -sqrt(-a)*arctan(sqrt(-a)*x^2*sqrt((a*x^
2 + b)/x^2)/(a*x^2 + b))/a]

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Sympy [A]  time = 1.61994, size = 17, normalized size = 0.71 \begin{align*} \frac{\operatorname{asinh}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{\sqrt{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

asinh(sqrt(a)*x/sqrt(b))/sqrt(a)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a + \frac{b}{x^{2}}} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(a + b/x^2)*x), x)