Optimal. Leaf size=20 \[ -\frac{\csc ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{b}}\right )}{\sqrt{b}} \]
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Rubi [A] time = 0.00904, antiderivative size = 20, normalized size of antiderivative = 1., 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 335
Rule 216
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*}
Mathematica [B] time = 0.0133659, size = 52, normalized size = 2.6 \[ \frac{x \sqrt{2-\frac{b}{x^2}} \tan ^{-1}\left (\frac{\sqrt{2 x^2-b}}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{2 x^2-b}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.008, size = 62, normalized size = 3.1 \begin{align*} -{x\sqrt{{\frac{2\,{x}^{2}-b}{{x}^{2}}}}\ln \left ( 2\,{\frac{\sqrt{-b}\sqrt{2\,{x}^{2}-b}-b}{x}} \right ){\frac{1}{\sqrt{-b}}}{\frac{1}{\sqrt{2\,{x}^{2}-b}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.55005, size = 186, normalized size = 9.3 \begin{align*} \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 ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- \frac{b}{x^{2}} + 2}}{- b + 2 x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.10405, size = 54, normalized size = 2.7 \begin{align*} \frac{\arctan \left (\frac{\sqrt{2 \, x^{2} - b}}{\sqrt{b}}\right ) \mathrm{sgn}\left (x\right )}{\sqrt{b}} - \frac{\arctan \left (\frac{\sqrt{-b}}{\sqrt{b}}\right ) \mathrm{sgn}\left (x\right )}{\sqrt{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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