Optimal. Leaf size=20 \[ -\frac {\text {csch}^{-1}\left (\frac {\sqrt {2} x}{\sqrt {b}}\right )}{\sqrt {b}} \]
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Rubi [A] time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {335, 215} \[ -\frac {\text {csch}^{-1}\left (\frac {\sqrt {2} x}{\sqrt {b}}\right )}{\sqrt {b}} \]
Antiderivative was successfully verified.
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Rule 215
Rule 335
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {2+\frac {b}{x^2}} x^2} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{\sqrt {2+b x^2}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\text {csch}^{-1}\left (\frac {\sqrt {2} x}{\sqrt {b}}\right )}{\sqrt {b}}\\ \end {align*}
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Mathematica [B] time = 0.02, size = 50, normalized size = 2.50 \[ -\frac {\sqrt {b+2 x^2} \tanh ^{-1}\left (\frac {\sqrt {b+2 x^2}}{\sqrt {b}}\right )}{\sqrt {b} x \sqrt {\frac {b}{x^2}+2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.02, size = 75, normalized size = 3.75 \[ \left [\frac {\log \left (-\frac {x^{2} - \sqrt {b} x \sqrt {\frac {2 \, x^{2} + b}{x^{2}}} + b}{x^{2}}\right )}{2 \, \sqrt {b}}, \frac {\sqrt {-b} \arctan \left (\frac {\sqrt {-b} x \sqrt {\frac {2 \, x^{2} + b}{x^{2}}}}{2 \, x^{2} + b}\right )}{b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 52, normalized size = 2.60 \[ -\frac {\sqrt {2 x^{2}+b}\, \ln \left (\frac {2 b +2 \sqrt {2 x^{2}+b}\, \sqrt {b}}{x}\right )}{\sqrt {\frac {2 x^{2}+b}{x^{2}}}\, \sqrt {b}\, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.96, size = 41, normalized size = 2.05 \[ \frac {\log \left (\frac {x \sqrt {\frac {b}{x^{2}} + 2} - \sqrt {b}}{x \sqrt {\frac {b}{x^{2}} + 2} + \sqrt {b}}\right )}{2 \, \sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.27, size = 17, normalized size = 0.85 \[ -\frac {\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {b}}{2\,x}\right )}{\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.32, size = 20, normalized size = 1.00 \[ - \frac {\operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {b}}{2 x} \right )}}{\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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