3.680 \(\int \frac{\sqrt{-1+\frac{1}{x^2}}}{x (-1+x^2)^3} \, dx\)

Optimal. Leaf size=34 \[ \sqrt{\frac{1}{x^2}-1}-\frac{2}{\sqrt{\frac{1}{x^2}-1}}-\frac{1}{3 \left (\frac{1}{x^2}-1\right )^{3/2}} \]

[Out]

-1/(3*(-1 + x^(-2))^(3/2)) - 2/Sqrt[-1 + x^(-2)] + Sqrt[-1 + x^(-2)]

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Rubi [A]  time = 0.0142253, antiderivative size = 34, 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} \[ \sqrt{\frac{1}{x^2}-1}-\frac{2}{\sqrt{\frac{1}{x^2}-1}}-\frac{1}{3 \left (\frac{1}{x^2}-1\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-1 + x^(-2)]/(x*(-1 + x^2)^3),x]

[Out]

-1/(3*(-1 + x^(-2))^(3/2)) - 2/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 )^3} \, dx &=-\int \frac{1}{\left (-1+\frac{1}{x^2}\right )^{5/2} x^7} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{(-1+x)^{5/2}} \, dx,x,\frac{1}{x^2}\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{(-1+x)^{5/2}}+\frac{2}{(-1+x)^{3/2}}+\frac{1}{\sqrt{-1+x}}\right ) \, dx,x,\frac{1}{x^2}\right )\\ &=-\frac{1}{3 \left (-1+\frac{1}{x^2}\right )^{3/2}}-\frac{2}{\sqrt{-1+\frac{1}{x^2}}}+\sqrt{-1+\frac{1}{x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0072378, size = 32, normalized size = 0.94 \[ \frac{\sqrt{\frac{1}{x^2}-1} \left (8 x^4-12 x^2+3\right )}{3 \left (x^2-1\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-1 + x^(-2)]/(x*(-1 + x^2)^3),x]

[Out]

(Sqrt[-1 + x^(-2)]*(3 - 12*x^2 + 8*x^4))/(3*(-1 + x^2)^2)

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Maple [A]  time = 0.003, size = 34, normalized size = 1. \begin{align*}{\frac{8\,{x}^{4}-12\,{x}^{2}+3}{3\, \left ({x}^{2}-1 \right ) ^{2}}\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)^3,x)

[Out]

1/3*(8*x^4-12*x^2+3)*(-(x^2-1)/x^2)^(1/2)/(x^2-1)^2

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Maxima [A]  time = 1.02952, size = 51, normalized size = 1.5 \begin{align*} \frac{{\left (8 \, x^{4} - 12 \, x^{2} + 3\right )} \sqrt{x + 1} \sqrt{-x + 1}}{3 \,{\left (x^{5} - 2 \, x^{3} + x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+1/x^2)^(1/2)/x/(x^2-1)^3,x, algorithm="maxima")

[Out]

1/3*(8*x^4 - 12*x^2 + 3)*sqrt(x + 1)*sqrt(-x + 1)/(x^5 - 2*x^3 + x)

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Fricas [A]  time = 1.41155, size = 88, normalized size = 2.59 \begin{align*} \frac{{\left (8 \, x^{4} - 12 \, x^{2} + 3\right )} \sqrt{-\frac{x^{2} - 1}{x^{2}}}}{3 \,{\left (x^{4} - 2 \, x^{2} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+1/x^2)^(1/2)/x/(x^2-1)^3,x, algorithm="fricas")

[Out]

1/3*(8*x^4 - 12*x^2 + 3)*sqrt(-(x^2 - 1)/x^2)/(x^4 - 2*x^2 + 1)

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Sympy [A]  time = 4.48906, size = 34, normalized size = 1. \begin{align*} \sqrt{-1 + \frac{1}{x^{2}}} - \frac{2}{\sqrt{-1 + \frac{1}{x^{2}}}} - \frac{1}{3 \left (-1 + \frac{1}{x^{2}}\right )^{\frac{3}{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)**3,x)

[Out]

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

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Giac [B]  time = 1.15893, size = 92, normalized size = 2.71 \begin{align*} -\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} - \frac{{\left (5 \, x^{2} \mathrm{sgn}\left (x\right ) - 6 \, \mathrm{sgn}\left (x\right )\right )} x}{3 \,{\left (x^{2} - 1\right )} \sqrt{-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)^3,x, algorithm="giac")

[Out]

-1/2*x*sgn(x)/(sqrt(-x^2 + 1) - 1) + 1/2*(sqrt(-x^2 + 1) - 1)*sgn(x)/x - 1/3*(5*x^2*sgn(x) - 6*sgn(x))*x/((x^2
- 1)*sqrt(-x^2 + 1))