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 = 0.01, antiderivative size = 34, 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 = {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 206
Rule 217
Rule 277
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.41, size = 35, normalized size = 1.03 \[ -\frac {3 \, x \log \left (-3 \, x + \sqrt {9 \, x^{2} - 1}\right ) + 3 \, x + \sqrt {9 \, x^{2} - 1}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.03, size = 44, normalized size = 1.29 \[ -\frac {6}{{\left (3 \, x - \sqrt {9 \, x^{2} - 1}\right )}^{2} + 1} - \frac {3}{2} \, \log \left ({\left (3 \, x - \sqrt {9 \, x^{2} - 1}\right )}^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 47, normalized size = 1.38 \[ -9 \sqrt {9 x^{2}-1}\, x +\sqrt {9}\, \ln \left (\sqrt {9}\, x +\sqrt {9 x^{2}-1}\right )+\frac {\left (9 x^{2}-1\right )^{\frac {3}{2}}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.24, size = 33, normalized size = 0.97 \[ -\frac {\sqrt {9 \, x^{2} - 1}}{x} + 3 \, \log \left (18 \, x + 6 \, \sqrt {9 \, x^{2} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.52, size = 32, normalized size = 0.94 \[ -\frac {\left (\frac {3\,x\,\mathrm {asin}\left (3\,x\right )}{\sqrt {1-9\,x^2}}+1\right )\,\sqrt {9\,x^2-1}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.25, size = 17, normalized size = 0.50 \[ 3 \operatorname {acosh}{\left (3 x \right )} - \frac {\sqrt {9 x^{2} - 1}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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