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]

Sqrt[-1 + x^(-2)]

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Rubi [A]  time = 0.0040838, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, 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*}

Mathematica [A]  time = 0.0025487, size = 9, normalized size = 1. \[ \sqrt{\frac{1}{x^2}-1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[-1 + x^(-2)]

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Maple [A]  time = 0.004, size = 13, normalized size = 1.4 \begin{align*} \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),x)

[Out]

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

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Maxima [B]  time = 1.50257, size = 22, normalized size = 2.44 \begin{align*} \frac{\sqrt{x + 1} \sqrt{-x + 1}}{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),x, algorithm="maxima")

[Out]

sqrt(x + 1)*sqrt(-x + 1)/x

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Fricas [A]  time = 1.4282, size = 30, normalized size = 3.33 \begin{align*} \sqrt{-\frac{x^{2} - 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),x, algorithm="fricas")

[Out]

sqrt(-(x^2 - 1)/x^2)

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Sympy [A]  time = 1.88333, size = 8, normalized size = 0.89 \begin{align*} \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),x)

[Out]

sqrt(-1 + x**(-2))

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Giac [B]  time = 1.16072, size = 50, normalized size = 5.56 \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} \end{align*}

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