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 = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 63
Rule 208
Rule 266
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*}
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Mathematica [B] time = 0.02, 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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fricas [B] time = 0.85, size = 80, normalized size = 3.33 \[ \left [\frac {\log \left (-2 \, a x^{2} - 2 \, \sqrt {a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}} - b\right )}{2 \, \sqrt {a}}, -\frac {\sqrt {-a} \arctan \left (\frac {\sqrt {-a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right )}{a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 48, normalized size = 2.00 \[ -\frac {\log \left ({\left | -2 \, {\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b x^{2}}\right )} \sqrt {a} - b \right |}\right )}{2 \, \sqrt {a}} + \frac {\log \left ({\left | b \right |}\right )}{2 \, \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 46, normalized size = 1.92 \[ \frac {\sqrt {a \,x^{2}+b}\, \ln \left (\sqrt {a}\, x +\sqrt {a \,x^{2}+b}\right )}{\sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, \sqrt {a}\, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.90, size = 37, normalized size = 1.54 \[ -\frac {\log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{2}}} + \sqrt {a}}\right )}{2 \, \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.31, size = 18, normalized size = 0.75 \[ \frac {\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.29, size = 17, normalized size = 0.71 \[ \frac {\operatorname {asinh}{\left (\frac {\sqrt {a} x}{\sqrt {b}} \right )}}{\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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