Optimal. Leaf size=34 \[ 3 \tanh ^{-1}\left (\frac{3 x}{\sqrt{9 x^2-1}}\right )-\frac{\sqrt{9 x^2-1}}{x} \]
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Rubi [A] time = 0.007611, antiderivative size = 34, 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 = {277, 217, 206} \[ 3 \tanh ^{-1}\left (\frac{3 x}{\sqrt{9 x^2-1}}\right )-\frac{\sqrt{9 x^2-1}}{x} \]
Antiderivative was successfully verified.
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Rule 277
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{-1+9 x^2}}{x^2} \, dx &=-\frac{\sqrt{-1+9 x^2}}{x}+9 \int \frac{1}{\sqrt{-1+9 x^2}} \, dx\\ &=-\frac{\sqrt{-1+9 x^2}}{x}+9 \operatorname{Subst}\left (\int \frac{1}{1-9 x^2} \, dx,x,\frac{x}{\sqrt{-1+9 x^2}}\right )\\ &=-\frac{\sqrt{-1+9 x^2}}{x}+3 \tanh ^{-1}\left (\frac{3 x}{\sqrt{-1+9 x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0152595, size = 35, normalized size = 1.03 \[ \sqrt{9 x^2-1} \left (-\frac{3 \sin ^{-1}(3 x)}{\sqrt{1-9 x^2}}-\frac{1}{x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 47, normalized size = 1.4 \begin{align*}{\frac{1}{x} \left ( 9\,{x}^{2}-1 \right ) ^{{\frac{3}{2}}}}-9\,x\sqrt{9\,{x}^{2}-1}+\ln \left ( x\sqrt{9}+\sqrt{9\,{x}^{2}-1} \right ) \sqrt{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.4237, size = 45, normalized size = 1.32 \begin{align*} -\frac{\sqrt{9 \, x^{2} - 1}}{x} + 3 \, \log \left (18 \, x + 6 \, \sqrt{9 \, x^{2} - 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.89075, size = 84, normalized size = 2.47 \begin{align*} -\frac{3 \, x \log \left (-3 \, x + \sqrt{9 \, x^{2} - 1}\right ) + 3 \, x + \sqrt{9 \, x^{2} - 1}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.224803, size = 17, normalized size = 0.5 \begin{align*} 3 \operatorname{acosh}{\left (3 x \right )} - \frac{\sqrt{9 x^{2} - 1}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.073, size = 59, normalized size = 1.74 \begin{align*} -\frac{6}{{\left (3 \, x - \sqrt{9 \, x^{2} - 1}\right )}^{2} + 1} - \frac{3}{2} \, \log \left ({\left (3 \, x - \sqrt{9 \, x^{2} - 1}\right )}^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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