3.1927 \(\int \frac{1}{\sqrt{-a+\frac{b}{x^2}} x} \, dx\)

Optimal. Leaf size=27 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{\frac{b}{x^2}-a}}{\sqrt{a}}\right )}{\sqrt{a}} \]

[Out]

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

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Rubi [A]  time = 0.0180662, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {266, 63, 205} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{\frac{b}{x^2}-a}}{\sqrt{a}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[-a + b/x^2]*x),x]

[Out]

-(ArcTan[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 205

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

Mathematica [B]  time = 0.0318742, size = 56, normalized size = 2.07 \[ \frac{\sqrt{a x^2-b} \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2-b}}\right )}{\sqrt{a} x \sqrt{\frac{b}{x^2}-a}} \]

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 = 0.009, size = 50, normalized size = 1.9 \begin{align*}{\frac{1}{x}\sqrt{-a{x}^{2}+b}\arctan \left ({x\sqrt{a}{\frac{1}{\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/x/(-a+1/x^2*b)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.53107, size = 193, normalized size = 7.15 \begin{align*} \left [-\frac{\sqrt{-a} \log \left (2 \, a x^{2} - 2 \, \sqrt{-a} x^{2} \sqrt{-\frac{a x^{2} - b}{x^{2}}} - b\right )}{2 \, a}, -\frac{\arctan \left (\frac{\sqrt{a} x^{2} \sqrt{-\frac{a x^{2} - b}{x^{2}}}}{a x^{2} - b}\right )}{\sqrt{a}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.24153, size = 48, normalized size = 1.78 \begin{align*} \begin{cases} - \frac{i \operatorname{acosh}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{\sqrt{a}} & \text{for}\: \frac{\left |{a x^{2}}\right |}{\left |{b}\right |} > 1 \\\frac{\operatorname{asin}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{\sqrt{a}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-I*acosh(sqrt(a)*x/sqrt(b))/sqrt(a), Abs(a*x**2)/Abs(b) > 1), (asin(sqrt(a)*x/sqrt(b))/sqrt(a), Tru
e))

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

[Out]

integrate(1/(sqrt(-a + b/x^2)*x), x)