Optimal. Leaf size=28 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{\sqrt{b}} \]
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Rubi [A] time = 0.0135937, antiderivative size = 28, 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 = {335, 217, 206} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
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Rule 335
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+\frac{b}{x^2}} x^2} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{1}{\sqrt{a+\frac{b}{x^2}} x}\right )\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{a+\frac{b}{x^2}} x}\right )}{\sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.0141918, size = 50, normalized size = 1.79 \[ -\frac{\sqrt{a x^2+b} \tanh ^{-1}\left (\frac{\sqrt{a x^2+b}}{\sqrt{b}}\right )}{\sqrt{b} x \sqrt{a+\frac{b}{x^2}}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.005, size = 52, normalized size = 1.9 \begin{align*} -{\frac{1}{x}\sqrt{a{x}^{2}+b}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{2}+b}+b}{x}} \right ){\frac{1}{\sqrt{{\frac{a{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 [A] time = 1.51497, size = 192, normalized size = 6.86 \begin{align*} \left [\frac{\log \left (-\frac{a x^{2} - 2 \, \sqrt{b} x \sqrt{\frac{a x^{2} + b}{x^{2}}} + 2 \, b}{x^{2}}\right )}{2 \, \sqrt{b}}, \frac{\sqrt{-b} \arctan \left (\frac{\sqrt{-b} x \sqrt{\frac{a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right )}{b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.53551, size = 19, normalized size = 0.68 \begin{align*} - \frac{\operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x} \right )}}{\sqrt{b}} \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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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