Optimal. Leaf size=27 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{\frac{b}{x^2}-a}}{\sqrt{a}}\right )}{\sqrt{a}} \]
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Rubi [A] time = 0.0180662, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {266, 63, 205} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{\frac{b}{x^2}-a}}{\sqrt{a}}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-a+\frac{b}{x^2}} x} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{-a+b x}} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{-a+\frac{b}{x^2}}\right )}{b}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{-a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{\sqrt{a}}\\ \end{align*}
Mathematica [B] time = 0.0318742, size = 56, normalized size = 2.07 \[ \frac{\sqrt{a x^2-b} \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2-b}}\right )}{\sqrt{a} x \sqrt{\frac{b}{x^2}-a}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.009, size = 50, normalized size = 1.9 \begin{align*}{\frac{1}{x}\sqrt{-a{x}^{2}+b}\arctan \left ({x\sqrt{a}{\frac{1}{\sqrt{-a{x}^{2}+b}}}} \right ){\frac{1}{\sqrt{-{\frac{a{x}^{2}-b}{{x}^{2}}}}}}{\frac{1}{\sqrt{a}}}} \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 [A] time = 1.53107, size = 193, normalized size = 7.15 \begin{align*} \left [-\frac{\sqrt{-a} \log \left (2 \, a x^{2} - 2 \, \sqrt{-a} x^{2} \sqrt{-\frac{a x^{2} - b}{x^{2}}} - b\right )}{2 \, a}, -\frac{\arctan \left (\frac{\sqrt{a} x^{2} \sqrt{-\frac{a x^{2} - b}{x^{2}}}}{a x^{2} - b}\right )}{\sqrt{a}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.24153, size = 48, normalized size = 1.78 \begin{align*} \begin{cases} - \frac{i \operatorname{acosh}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{\sqrt{a}} & \text{for}\: \frac{\left |{a x^{2}}\right |}{\left |{b}\right |} > 1 \\\frac{\operatorname{asin}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{\sqrt{a}} & \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}{\sqrt{-a + \frac{b}{x^{2}}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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