Optimal. Leaf size=34 \[ -\frac {1}{3 \left (-1+\frac {1}{x^2}\right )^{3/2}}-\frac {2}{\sqrt {-1+\frac {1}{x^2}}}+\sqrt {-1+\frac {1}{x^2}} \]
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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, 272, 45}
\begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 25
Rule 45
Rule 272
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} \text {Subst}\left (\int \frac {x^2}{(-1+x)^{5/2}} \, dx,x,\frac {1}{x^2}\right )\\ &=\frac {1}{2} \text {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 = 0.04, size = 32, normalized size = 0.94 \begin {gather*} \frac {\sqrt {-1+\frac {1}{x^2}} \left (3-12 x^2+8 x^4\right )}{3 \left (-1+x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 37, normalized size = 1.09
method | result | size |
gosper | \(\frac {\left (8 x^{4}-12 x^{2}+3\right ) \sqrt {-\frac {x^{2}-1}{x^{2}}}}{3 \left (x^{2}-1\right )^{2}}\) | \(34\) |
trager | \(\frac {\left (8 x^{4}-12 x^{2}+3\right ) \sqrt {-\frac {x^{2}-1}{x^{2}}}}{3 \left (x^{2}-1\right )^{2}}\) | \(34\) |
risch | \(\frac {\left (8 x^{4}-12 x^{2}+3\right ) \sqrt {-\frac {x^{2}-1}{x^{2}}}}{3 \left (x^{2}-1\right )^{2}}\) | \(34\) |
default | \(\frac {\left (8 x^{4}-12 x^{2}+3\right ) \sqrt {-\frac {x^{2}-1}{x^{2}}}}{3 \left (1+x \right )^{2} \left (-1+x \right )^{2}}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 38, normalized size = 1.12 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 38, normalized size = 1.12 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 3.01, size = 34, normalized size = 1.00 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 68 vs.
\(2 (26) = 52\).
time = 1.83, size = 68, normalized size = 2.00 \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} - \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.09, size = 28, normalized size = 0.82 \begin {gather*} \frac {\sqrt {\frac {1}{x^2}-1}\,\left (8\,x^4-12\,x^2+3\right )}{3\,{\left (x^2-1\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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