Optimal. Leaf size=23 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {x^2+3}}{\sqrt {3}}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {266, 63, 207} \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {x^2+3}}{\sqrt {3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 207
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {3+x^2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {3+x}} \, dx,x,x^2\right )\\ &=\operatorname {Subst}\left (\int \frac {1}{-3+x^2} \, dx,x,\sqrt {3+x^2}\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {3+x^2}}{\sqrt {3}}\right )}{\sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 23, normalized size = 1.00 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {x^2+3}}{\sqrt {3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 24, normalized size = 1.04 \[ \frac {1}{3} \, \sqrt {3} \log \left (-\frac {\sqrt {3} - \sqrt {x^{2} + 3}}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.91, size = 37, normalized size = 1.61 \[ -\frac {1}{6} \, \sqrt {3} \log \left (\sqrt {3} + \sqrt {x^{2} + 3}\right ) + \frac {1}{6} \, \sqrt {3} \log \left (-\sqrt {3} + \sqrt {x^{2} + 3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 18, normalized size = 0.78 \[ -\frac {\sqrt {3}\, \arctanh \left (\frac {\sqrt {3}}{\sqrt {x^{2}+3}}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 14, normalized size = 0.61 \[ -\frac {1}{3} \, \sqrt {3} \operatorname {arsinh}\left (\frac {\sqrt {3}}{{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 18, normalized size = 0.78 \[ -\frac {\sqrt {3}\,\mathrm {atanh}\left (\frac {\sqrt {3}\,\sqrt {x^2+3}}{3}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.06, size = 15, normalized size = 0.65 \[ - \frac {\sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {3}}{x} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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