Optimal. Leaf size=24 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{\sqrt{a}} \]
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Rubi [A] time = 0.0170164, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {266, 63, 208} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 63
Rule 208
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{\tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{\sqrt{a}}\\ \end{align*}
Mathematica [B] time = 0.0130868, size = 50, normalized size = 2.08 \[ \frac{\sqrt{a x^2+b} \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b}}\right )}{\sqrt{a} x \sqrt{a+\frac{b}{x^2}}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 46, normalized size = 1.9 \begin{align*}{\frac{1}{x}\sqrt{a{x}^{2}+b}\ln \left ( x\sqrt{a}+\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 [B] time = 1.53614, size = 190, normalized size = 7.92 \begin{align*} \left [\frac{\log \left (-2 \, a x^{2} - 2 \, \sqrt{a} x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} - b\right )}{2 \, \sqrt{a}}, -\frac{\sqrt{-a} \arctan \left (\frac{\sqrt{-a} x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right )}{a}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.61994, size = 17, normalized size = 0.71 \begin{align*} \frac{\operatorname{asinh}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{\sqrt{a}} \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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