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((-a+b/x^2)^(1/2)/a^(1/2))/a^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 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]

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]]

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*}

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Mathematica [B]  time = 0.04, 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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fricas [A]  time = 1.03, size = 88, normalized size = 3.26 \[ \left [-\frac {\sqrt {-a} \log \left (2 \, a x^{2} - 2 \, \sqrt {-a} x^{2} \sqrt {-\frac {a x^{2} - b}{x^{2}}} - b\right )}{2 \, a}, -\frac {\arctan \left (\frac {\sqrt {a} x^{2} \sqrt {-\frac {a x^{2} - b}{x^{2}}}}{a x^{2} - b}\right )}{\sqrt {a}}\right ] \]

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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giac [B]  time = 0.25, size = 55, normalized size = 2.04 \[ -\frac {\log \left ({\left | 2 \, {\left (\sqrt {-a} x^{2} - \sqrt {-a x^{4} + b x^{2}}\right )} \sqrt {-a} + b \right |}\right )}{2 \, \sqrt {-a}} + \frac {\log \left ({\left | b \right |}\right )}{2 \, \sqrt {-a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(abs(2*(sqrt(-a)*x^2 - sqrt(-a*x^4 + b*x^2))*sqrt(-a) + b))/sqrt(-a) + 1/2*log(abs(b))/sqrt(-a)

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maple [B]  time = 0.01, size = 50, normalized size = 1.85 \[ \frac {\sqrt {-a \,x^{2}+b}\, \arctan \left (\frac {\sqrt {a}\, x}{\sqrt {-a \,x^{2}+b}}\right )}{\sqrt {-\frac {a \,x^{2}-b}{x^{2}}}\, \sqrt {a}\, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(-a+b/x^2)^(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 [A]  time = 1.98, size = 21, normalized size = 0.78 \[ -\frac {\arctan \left (\frac {\sqrt {-a + \frac {b}{x^{2}}}}{\sqrt {a}}\right )}{\sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctan(sqrt(-a + b/x^2)/sqrt(a))/sqrt(a)

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mupad [B]  time = 1.35, size = 21, normalized size = 0.78 \[ -\frac {\mathrm {atan}\left (\frac {\sqrt {\frac {b}{x^2}-a}}{\sqrt {a}}\right )}{\sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(b/x^2 - a)^(1/2)),x)

[Out]

-atan((b/x^2 - a)^(1/2)/a^(1/2))/a^(1/2)

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sympy [A]  time = 1.31, size = 46, normalized size = 1.70 \[ \begin {cases} - \frac {i \operatorname {acosh}{\left (\frac {\sqrt {a} x}{\sqrt {b}} \right )}}{\sqrt {a}} & \text {for}\: \left |{\frac {a x^{2}}{b}}\right | > 1 \\\frac {\operatorname {asin}{\left (\frac {\sqrt {a} x}{\sqrt {b}} \right )}}{\sqrt {a}} & \text {otherwise} \end {cases} \]

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/b) > 1), (asin(sqrt(a)*x/sqrt(b))/sqrt(a), True))

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