3.7.78 \(\int \frac {\sqrt {-1+\frac {1}{x^2}}}{x (-1+x^2)} \, dx\) [678]

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

[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, 267} \begin {gather*} \sqrt {\frac {1}{x^2}-1} \end {gather*}

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 267

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 = 0.02, size = 9, normalized size = 1.00 \begin {gather*} \sqrt {-1+\frac {1}{x^2}} \end {gather*}

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 = 0.41, size = 13, normalized size = 1.44

method	result	size
gosper	\(\sqrt {-\frac {x^{2}-1}{x^{2}}}\)	\(13\)
default	\(\sqrt {-\frac {x^{2}-1}{x^{2}}}\)	\(13\)
trager	\(\sqrt {-\frac {x^{2}-1}{x^{2}}}\)	\(13\)
risch	\(\sqrt {-\frac {x^{2}-1}{x^{2}}}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1+1/x^2)^(1/2)/x/(x^2-1),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 16 vs. \(2 (7) = 14\).
time = 0.27, size = 16, normalized size = 1.78 \begin {gather*} \frac {\sqrt {x + 1} \sqrt {-x + 1}}{x} \end {gather*}

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 = 0.36, size = 12, normalized size = 1.33 \begin {gather*} \sqrt {-\frac {x^{2} - 1}{x^{2}}} \end {gather*}

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 = 1.27, size = 8, normalized size = 0.89 \begin {gather*} \sqrt {-1 + \frac {1}{x^{2}}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (7) = 14\).
time = 1.89, size = 37, normalized size = 4.11 \begin {gather*} -\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 {gather*}

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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Mupad [B]
time = 3.18, size = 14, normalized size = 1.56 \begin {gather*} \frac {\sqrt {1-x^2}}{\left |x\right |} \end {gather*}

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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