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

Optimal. Leaf size=21 \[ \frac{1}{\sqrt{\frac{1}{x^2}-1}}-\sqrt{\frac{1}{x^2}-1} \]

[Out]

1/Sqrt[-1 + x^(-2)] - Sqrt[-1 + x^(-2)]

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Rubi [A]  time = 0.0107285, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {25, 266, 43} \[ \frac{1}{\sqrt{\frac{1}{x^2}-1}}-\sqrt{\frac{1}{x^2}-1} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-1 + x^(-2)]/(x*(-1 + x^2)^2),x]

[Out]

1/Sqrt[-1 + x^(-2)] - Sqrt[-1 + x^(-2)]

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 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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\sqrt{-1+\frac{1}{x^2}}}{x \left (-1+x^2\right )^2} \, dx &=\int \frac{1}{\left (-1+\frac{1}{x^2}\right )^{3/2} x^5} \, dx\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{(-1+x)^{3/2}} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{(-1+x)^{3/2}}+\frac{1}{\sqrt{-1+x}}\right ) \, dx,x,\frac{1}{x^2}\right )\right )\\ &=\frac{1}{\sqrt{-1+\frac{1}{x^2}}}-\sqrt{-1+\frac{1}{x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0057347, size = 24, normalized size = 1.14 \[ \frac{\sqrt{\frac{1}{x^2}-1} \left (1-2 x^2\right )}{x^2-1} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-1 + x^(-2)]/(x*(-1 + x^2)^2),x]

[Out]

(Sqrt[-1 + x^(-2)]*(1 - 2*x^2))/(-1 + x^2)

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Maple [A]  time = 0.003, size = 29, normalized size = 1.4 \begin{align*} -{\frac{2\,{x}^{2}-1}{{x}^{2}-1}\sqrt{-{\frac{{x}^{2}-1}{{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1+1/x^2)^(1/2)/x/(x^2-1)^2,x)

[Out]

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

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Maxima [A]  time = 1.04155, size = 41, normalized size = 1.95 \begin{align*} -\frac{{\left (2 \, x^{2} - 1\right )} \sqrt{x + 1} \sqrt{-x + 1}}{x^{3} - x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*x^2 - 1)*sqrt(x + 1)*sqrt(-x + 1)/(x^3 - x)

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Fricas [A]  time = 1.45863, size = 61, normalized size = 2.9 \begin{align*} -\frac{{\left (2 \, x^{2} - 1\right )} \sqrt{-\frac{x^{2} - 1}{x^{2}}}}{x^{2} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*x^2 - 1)*sqrt(-(x^2 - 1)/x^2)/(x^2 - 1)

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Sympy [A]  time = 3.16238, size = 20, normalized size = 0.95 \begin{align*} - \sqrt{-1 + \frac{1}{x^{2}}} + \frac{1}{\sqrt{-1 + \frac{1}{x^{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-1 + x**(-2)) + 1/sqrt(-1 + x**(-2))

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Giac [B]  time = 1.17584, size = 78, normalized size = 3.71 \begin{align*} -\frac{\sqrt{-x^{2} + 1} x \mathrm{sgn}\left (x\right )}{x^{2} - 1} + \frac{x \mathrm{sgn}\left (x\right )}{2 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}} - \frac{{\left (\sqrt{-x^{2} + 1} - 1\right )} \mathrm{sgn}\left (x\right )}{2 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 1)*x*sgn(x)/(x^2 - 1) + 1/2*x*sgn(x)/(sqrt(-x^2 + 1) - 1) - 1/2*(sqrt(-x^2 + 1) - 1)*sgn(x)/x