Optimal. Leaf size=20 \[ -\frac {\csc ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {b}}\right )}{\sqrt {b}} \]
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Rubi [A] time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {25, 335, 216} \[ -\frac {\csc ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {b}}\right )}{\sqrt {b}} \]
Antiderivative was successfully verified.
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Rule 25
Rule 216
Rule 335
Rubi steps
\begin {align*} \int \frac {\sqrt {2-\frac {b}{x^2}}}{-b+2 x^2} \, dx &=\int \frac {1}{\sqrt {2-\frac {b}{x^2}} x^2} \, dx\\ &=-\operatorname {Subst}\left (\int \frac {1}{\sqrt {2-b x^2}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\csc ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {b}}\right )}{\sqrt {b}}\\ \end {align*}
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Mathematica [B] time = 0.01, size = 52, normalized size = 2.60 \[ \frac {x \sqrt {2-\frac {b}{x^2}} \tan ^{-1}\left (\frac {\sqrt {2 x^2-b}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {2 x^2-b}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.73, size = 84, normalized size = 4.20 \[ \left [-\frac {\sqrt {-b} \log \left (-\frac {x^{2} - \sqrt {-b} x \sqrt {\frac {2 \, x^{2} - b}{x^{2}}} - b}{x^{2}}\right )}{2 \, b}, -\frac {\arctan \left (\frac {\sqrt {b} x \sqrt {\frac {2 \, x^{2} - b}{x^{2}}}}{2 \, x^{2} - b}\right )}{\sqrt {b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.42, size = 40, normalized size = 2.00 \[ \frac {\arctan \left (\frac {\sqrt {2 \, x^{2} - b}}{\sqrt {b}}\right ) \mathrm {sgn}\relax (x)}{\sqrt {b}} - \frac {\arctan \left (\frac {\sqrt {-b}}{\sqrt {b}}\right ) \mathrm {sgn}\relax (x)}{\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 62, normalized size = 3.10 \[ -\frac {\sqrt {\frac {2 x^{2}-b}{x^{2}}}\, x \ln \left (\frac {-2 b +2 \sqrt {-b}\, \sqrt {2 x^{2}-b}}{x}\right )}{\sqrt {2 x^{2}-b}\, \sqrt {-b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-\frac {b}{x^{2}} + 2}}{2 \, x^{2} - b}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.47, size = 21, normalized size = 1.05 \[ -\frac {\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {-b}}{2\,x}\right )}{\sqrt {-b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- \frac {b}{x^{2}} + 2}}{- b + 2 x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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