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

Optimal. Leaf size=9 \[ \sqrt {\frac {1}{x^2}-1} \]

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {25, 261} \[ \sqrt {\frac {1}{x^2}-1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 9, normalized size = 1.00 \[ \sqrt {\frac {1}{x^2}-1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[-1 + x^(-2)]

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fricas [A]  time = 0.45, size = 12, normalized size = 1.33 \[ \sqrt {-\frac {x^{2} - 1}{x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(-(x^2 - 1)/x^2)

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giac [B]  time = 0.40, size = 37, normalized size = 4.11 \[ -\frac {x \mathrm {sgn}\relax (x)}{2 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}} + \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )} \mathrm {sgn}\relax (x)}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 13, normalized size = 1.44 \[ \sqrt {-\frac {x^{2}-1}{x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.58, size = 16, normalized size = 1.78 \[ \frac {\sqrt {x + 1} \sqrt {-x + 1}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(x + 1)*sqrt(-x + 1)/x

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mupad [B]  time = 3.18, size = 14, normalized size = 1.56 \[ \frac {\sqrt {1-x^2}}{\relax |x|} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1 - x^2)^(1/2)/abs(x)

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sympy [A]  time = 2.25, size = 8, normalized size = 0.89 \[ \sqrt {-1 + \frac {1}{x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(-1 + x**(-2))

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