Optimal. Leaf size=44 \[ -\frac {1}{2} \left (\frac {1}{x^2}-1\right )^{3/2} x^2+\frac {3}{2} \sqrt {\frac {1}{x^2}-1}-\frac {3}{2} \tan ^{-1}\left (\sqrt {\frac {1}{x^2}-1}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {25, 266, 47, 50, 63, 203} \[ -\frac {1}{2} \left (\frac {1}{x^2}-1\right )^{3/2} x^2+\frac {3}{2} \sqrt {\frac {1}{x^2}-1}-\frac {3}{2} \tan ^{-1}\left (\sqrt {\frac {1}{x^2}-1}\right ) \]
Antiderivative was successfully verified.
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Rule 25
Rule 47
Rule 50
Rule 63
Rule 203
Rule 266
Rubi steps
\begin {align*} \int \frac {\sqrt {-1+\frac {1}{x^2}} \left (-1+x^2\right )}{x} \, dx &=-\int \left (-1+\frac {1}{x^2}\right )^{3/2} x \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(-1+x)^{3/2}}{x^2} \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {1}{2} \left (-1+\frac {1}{x^2}\right )^{3/2} x^2+\frac {3}{4} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x}}{x} \, dx,x,\frac {1}{x^2}\right )\\ &=\frac {3}{2} \sqrt {-1+\frac {1}{x^2}}-\frac {1}{2} \left (-1+\frac {1}{x^2}\right )^{3/2} x^2-\frac {3}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,\frac {1}{x^2}\right )\\ &=\frac {3}{2} \sqrt {-1+\frac {1}{x^2}}-\frac {1}{2} \left (-1+\frac {1}{x^2}\right )^{3/2} x^2-\frac {3}{2} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+\frac {1}{x^2}}\right )\\ &=\frac {3}{2} \sqrt {-1+\frac {1}{x^2}}-\frac {1}{2} \left (-1+\frac {1}{x^2}\right )^{3/2} x^2-\frac {3}{2} \tan ^{-1}\left (\sqrt {-1+\frac {1}{x^2}}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 34, normalized size = 0.77 \[ \frac {\sqrt {\frac {1}{x^2}-1} \, _2F_1\left (-\frac {3}{2},-\frac {1}{2};\frac {1}{2};x^2\right )}{\sqrt {1-x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 43, normalized size = 0.98 \[ \frac {1}{2} \, {\left (x^{2} + 2\right )} \sqrt {-\frac {x^{2} - 1}{x^{2}}} - 3 \, \arctan \left (\frac {x \sqrt {-\frac {x^{2} - 1}{x^{2}}} - 1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 57, normalized size = 1.30 \[ \frac {1}{2} \, \sqrt {-x^{2} + 1} x \mathrm {sgn}\relax (x) + \frac {3}{2} \, \arcsin \relax (x) \mathrm {sgn}\relax (x) - \frac {x \mathrm {sgn}\relax (x)}{2 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}} + \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )} \mathrm {sgn}\relax (x)}{2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 55, normalized size = 1.25 \[ \frac {\sqrt {-\frac {x^{2}-1}{x^{2}}}\, \left (3 \sqrt {-x^{2}+1}\, x^{2}+3 x \arcsin \relax (x )+2 \left (-x^{2}+1\right )^{\frac {3}{2}}\right )}{2 \sqrt {-x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.99, size = 30, normalized size = 0.68 \[ \frac {1}{2} \, x^{2} \sqrt {\frac {1}{x^{2}} - 1} + \sqrt {\frac {1}{x^{2}} - 1} - \frac {3}{2} \, \arctan \left (\sqrt {\frac {1}{x^{2}} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.57, size = 30, normalized size = 0.68 \[ \sqrt {\frac {1}{x^2}-1}-\frac {3\,\mathrm {atan}\left (\sqrt {\frac {1}{x^2}-1}\right )}{2}+\frac {x^2\,\sqrt {\frac {1}{x^2}-1}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 43.65, size = 39, normalized size = 0.89 \[ \frac {x^{2} \sqrt {-1 + \frac {1}{x^{2}}}}{2} + \sqrt {-1 + \frac {1}{x^{2}}} - \frac {3 \operatorname {atan}{\left (\sqrt {-1 + \frac {1}{x^{2}}} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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