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 = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 63
Rule 205
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 {\tan ^{-1}\left (\frac {\sqrt {-a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{\sqrt {a}}\\ \end {align*}
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Mathematica [B] time = 0.04, 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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fricas [A] time = 1.03, size = 88, normalized size = 3.26 \[ \left [-\frac {\sqrt {-a} \log \left (2 \, a x^{2} - 2 \, \sqrt {-a} x^{2} \sqrt {-\frac {a x^{2} - b}{x^{2}}} - b\right )}{2 \, a}, -\frac {\arctan \left (\frac {\sqrt {a} x^{2} \sqrt {-\frac {a x^{2} - b}{x^{2}}}}{a x^{2} - b}\right )}{\sqrt {a}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 55, normalized size = 2.04 \[ -\frac {\log \left ({\left | 2 \, {\left (\sqrt {-a} x^{2} - \sqrt {-a x^{4} + b x^{2}}\right )} \sqrt {-a} + b \right |}\right )}{2 \, \sqrt {-a}} + \frac {\log \left ({\left | b \right |}\right )}{2 \, \sqrt {-a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 50, normalized size = 1.85 \[ \frac {\sqrt {-a \,x^{2}+b}\, \arctan \left (\frac {\sqrt {a}\, x}{\sqrt {-a \,x^{2}+b}}\right )}{\sqrt {-\frac {a \,x^{2}-b}{x^{2}}}\, \sqrt {a}\, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.98, size = 21, normalized size = 0.78 \[ -\frac {\arctan \left (\frac {\sqrt {-a + \frac {b}{x^{2}}}}{\sqrt {a}}\right )}{\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.35, size = 21, normalized size = 0.78 \[ -\frac {\mathrm {atan}\left (\frac {\sqrt {\frac {b}{x^2}-a}}{\sqrt {a}}\right )}{\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.31, size = 46, normalized size = 1.70 \[ \begin {cases} - \frac {i \operatorname {acosh}{\left (\frac {\sqrt {a} x}{\sqrt {b}} \right )}}{\sqrt {a}} & \text {for}\: \left |{\frac {a x^{2}}{b}}\right | > 1 \\\frac {\operatorname {asin}{\left (\frac {\sqrt {a} x}{\sqrt {b}} \right )}}{\sqrt {a}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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