Optimal. Leaf size=20 \[ -\frac{\csc ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{b}}\right )}{\sqrt{b}} \]
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Rubi [A] time = 0.0079047, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {335, 216} \[ -\frac{\csc ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{b}}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
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Rule 335
Rule 216
Rubi steps
\begin{align*} \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 = 0.0174186, size = 54, normalized size = 2.7 \[ \frac{\sqrt{2 x^2-b} \tan ^{-1}\left (\frac{\sqrt{2 x^2-b}}{\sqrt{b}}\right )}{\sqrt{b} x \sqrt{2-\frac{b}{x^2}}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.009, size = 64, normalized size = 3.2 \begin{align*} -{\frac{1}{x}\sqrt{2\,{x}^{2}-b}\ln \left ( 2\,{\frac{\sqrt{-b}\sqrt{2\,{x}^{2}-b}-b}{x}} \right ){\frac{1}{\sqrt{{\frac{2\,{x}^{2}-b}{{x}^{2}}}}}}{\frac{1}{\sqrt{-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 = 1.52683, 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 [A] time = 1.25341, size = 51, normalized size = 2.55 \begin{align*} \begin{cases} \frac{i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{b}}{2 x} \right )}}{\sqrt{b}} & \text{for}\: \frac{\left |{b}\right |}{2 \left |{x^{2}}\right |} > 1 \\- \frac{\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b}}{2 x} \right )}}{\sqrt{b}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \sqrt{-\frac{b}{x^{2}} + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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