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

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Rubi [A]  time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, 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 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])

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 {\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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.46, size = 38, normalized size = 1.12 \[ \frac {{\left (8 \, x^{4} - 12 \, x^{2} + 3\right )} \sqrt {-\frac {x^{2} - 1}{x^{2}}}}{3 \, {\left (x^{4} - 2 \, x^{2} + 1\right )}} \]

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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giac [B]  time = 0.44, size = 68, normalized size = 2.00 \[ -\frac {x \mathrm {sgn}\relax (x)}{2 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}} + \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )} \mathrm {sgn}\relax (x)}{2 \, x} - \frac {{\left (5 \, x^{2} \mathrm {sgn}\relax (x) - 6 \, \mathrm {sgn}\relax (x)\right )} x}{3 \, {\left (x^{2} - 1\right )} \sqrt {-x^{2} + 1}} \]

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))

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maple [A]  time = 0.00, size = 34, normalized size = 1.00 \[ \frac {\left (8 x^{4}-12 x^{2}+3\right ) \sqrt {-\frac {x^{2}-1}{x^{2}}}}{3 \left (x^{2}-1\right )^{2}} \]

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 = 0.64, size = 38, normalized size = 1.12 \[ \frac {{\left (8 \, x^{4} - 12 \, x^{2} + 3\right )} \sqrt {x + 1} \sqrt {-x + 1}}{3 \, {\left (x^{5} - 2 \, x^{3} + x\right )}} \]

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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mupad [B]  time = 3.09, size = 28, normalized size = 0.82 \[ \frac {\sqrt {\frac {1}{x^2}-1}\,\left (8\,x^4-12\,x^2+3\right )}{3\,{\left (x^2-1\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/x^2 - 1)^(1/2)/(x*(x^2 - 1)^3),x)

[Out]

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

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sympy [A]  time = 5.12, size = 34, normalized size = 1.00 \[ \sqrt {-1 + \frac {1}{x^{2}}} - \frac {2}{\sqrt {-1 + \frac {1}{x^{2}}}} - \frac {1}{3 \left (-1 + \frac {1}{x^{2}}\right )^{\frac {3}{2}}} \]

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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