Optimal. Leaf size=50 \[ \frac {1}{36} \log \left (4 x^2-6 x+9\right )-\frac {1}{18} \log (3-2 x)-\frac {\tan ^{-1}\left (\frac {3-4 x}{3 \sqrt {3}}\right )}{18 \sqrt {3}} \]
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Rubi [A] time = 0.04, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1586, 2058, 634, 618, 204, 628} \[ \frac {1}{36} \log \left (4 x^2-6 x+9\right )-\frac {1}{18} \log (3-2 x)-\frac {\tan ^{-1}\left (\frac {3-4 x}{3 \sqrt {3}}\right )}{18 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 1586
Rule 2058
Rubi steps
\begin {align*} \int \frac {27+36 x+24 x^2+8 x^3}{729-64 x^6} \, dx &=\int \frac {1}{27-36 x+24 x^2-8 x^3} \, dx\\ &=\int \left (-\frac {1}{9 (-3+2 x)}+\frac {2 x}{9 \left (9-6 x+4 x^2\right )}\right ) \, dx\\ &=-\frac {1}{18} \log (3-2 x)+\frac {2}{9} \int \frac {x}{9-6 x+4 x^2} \, dx\\ &=-\frac {1}{18} \log (3-2 x)+\frac {1}{36} \int \frac {-6+8 x}{9-6 x+4 x^2} \, dx+\frac {1}{6} \int \frac {1}{9-6 x+4 x^2} \, dx\\ &=-\frac {1}{18} \log (3-2 x)+\frac {1}{36} \log \left (9-6 x+4 x^2\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{-108-x^2} \, dx,x,-6+8 x\right )\\ &=-\frac {\tan ^{-1}\left (\frac {3-4 x}{3 \sqrt {3}}\right )}{18 \sqrt {3}}-\frac {1}{18} \log (3-2 x)+\frac {1}{36} \log \left (9-6 x+4 x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 50, normalized size = 1.00 \[ \frac {1}{36} \log \left (4 x^2-6 x+9\right )-\frac {1}{18} \log (3-2 x)+\frac {\tan ^{-1}\left (\frac {4 x-3}{3 \sqrt {3}}\right )}{18 \sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 38, normalized size = 0.76 \[ \frac {1}{54} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) + \frac {1}{36} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) - \frac {1}{18} \, \log \left (2 \, x - 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 39, normalized size = 0.78 \[ \frac {1}{54} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) + \frac {1}{36} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) - \frac {1}{18} \, \log \left ({\left | 2 \, x - 3 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 39, normalized size = 0.78 \[ \frac {\sqrt {3}\, \arctan \left (\frac {\left (8 x -6\right ) \sqrt {3}}{18}\right )}{54}-\frac {\ln \left (2 x -3\right )}{18}+\frac {\ln \left (4 x^{2}-6 x +9\right )}{36} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.95, size = 38, normalized size = 0.76 \[ \frac {1}{54} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) + \frac {1}{36} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) - \frac {1}{18} \, \log \left (2 \, x - 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 46, normalized size = 0.92 \[ -\frac {\ln \left (x-\frac {3}{2}\right )}{18}-\ln \left (x-\frac {3}{4}-\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (-\frac {1}{36}+\frac {\sqrt {3}\,1{}\mathrm {i}}{108}\right )+\ln \left (x-\frac {3}{4}+\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (\frac {1}{36}+\frac {\sqrt {3}\,1{}\mathrm {i}}{108}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.20, size = 48, normalized size = 0.96 \[ - \frac {\log {\left (x - \frac {3}{2} \right )}}{18} + \frac {\log {\left (x^{2} - \frac {3 x}{2} + \frac {9}{4} \right )}}{36} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {4 \sqrt {3} x}{9} - \frac {\sqrt {3}}{3} \right )}}{54} \]
Verification of antiderivative is not currently implemented for this CAS.
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