Optimal. Leaf size=60 \[ \frac {\tan ^{-1}\left (\frac {3+4 x}{3 \sqrt {3}}\right )}{54 \sqrt {3}}-\frac {1}{324} \log (3-2 x)+\frac {1}{108} \log (3+2 x)-\frac {1}{324} \log \left (9+6 x+4 x^2\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1600, 2083,
648, 632, 210, 642} \begin {gather*} \frac {\text {ArcTan}\left (\frac {4 x+3}{3 \sqrt {3}}\right )}{54 \sqrt {3}}-\frac {1}{324} \log \left (4 x^2+6 x+9\right )-\frac {1}{324} \log (3-2 x)+\frac {1}{108} \log (2 x+3) \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 1600
Rule 2083
Rubi steps
\begin {align*} \int \frac {9-6 x+4 x^2}{729-64 x^6} \, dx &=\int \frac {1}{81+54 x-24 x^3-16 x^4} \, dx\\ &=\int \left (-\frac {1}{162 (-3+2 x)}+\frac {1}{54 (3+2 x)}+\frac {3-2 x}{81 \left (9+6 x+4 x^2\right )}\right ) \, dx\\ &=-\frac {1}{324} \log (3-2 x)+\frac {1}{108} \log (3+2 x)+\frac {1}{81} \int \frac {3-2 x}{9+6 x+4 x^2} \, dx\\ &=-\frac {1}{324} \log (3-2 x)+\frac {1}{108} \log (3+2 x)-\frac {1}{324} \int \frac {6+8 x}{9+6 x+4 x^2} \, dx+\frac {1}{18} \int \frac {1}{9+6 x+4 x^2} \, dx\\ &=-\frac {1}{324} \log (3-2 x)+\frac {1}{108} \log (3+2 x)-\frac {1}{324} \log \left (9+6 x+4 x^2\right )-\frac {1}{9} \text {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 )}{54 \sqrt {3}}-\frac {1}{324} \log (3-2 x)+\frac {1}{108} \log (3+2 x)-\frac {1}{324} \log \left (9+6 x+4 x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 56, normalized size = 0.93 \begin {gather*} \frac {1}{324} \left (2 \sqrt {3} \tan ^{-1}\left (\frac {3+4 x}{3 \sqrt {3}}\right )-\log (3-2 x)+3 \log (3+2 x)-\log \left (9+6 x+4 x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.37, size = 47, normalized size = 0.78
method | result | size |
default | \(-\frac {\ln \left (4 x^{2}+6 x +9\right )}{324}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (8 x +6\right ) \sqrt {3}}{18}\right )}{162}+\frac {\ln \left (2 x +3\right )}{108}-\frac {\ln \left (-3+2 x \right )}{324}\) | \(47\) |
risch | \(-\frac {\ln \left (-3+2 x \right )}{324}+\frac {\ln \left (2 x +3\right )}{108}-\frac {\ln \left (4 x^{2}+6 x +9\right )}{324}+\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (2 x +\frac {3}{2}\right ) \sqrt {3}}{9}\right )}{162}\) | \(47\) |
meijerg | \(-\frac {x \left (\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )-\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )+\frac {\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3-\left (x^{6}\right )^{\frac {1}{6}}}\right )-\frac {\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3+\left (x^{6}\right )^{\frac {1}{6}}}\right )\right )}{324 \left (x^{6}\right )^{\frac {1}{6}}}+\frac {\arctanh \left (\frac {8 x^{3}}{27}\right )}{162}+\frac {x^{2} \left (\ln \left (1-\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )-\frac {\ln \left (1+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}+\frac {16 \left (x^{6}\right )^{\frac {2}{3}}}{81}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, \left (x^{6}\right )^{\frac {1}{3}}}{9 \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}\right )\right )}{324 \left (x^{6}\right )^{\frac {1}{3}}}\) | \(200\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 46, normalized size = 0.77 \begin {gather*} \frac {1}{162} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) - \frac {1}{324} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) + \frac {1}{108} \, \log \left (2 \, x + 3\right ) - \frac {1}{324} \, \log \left (2 \, x - 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 46, normalized size = 0.77 \begin {gather*} \frac {1}{162} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) - \frac {1}{324} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) + \frac {1}{108} \, \log \left (2 \, x + 3\right ) - \frac {1}{324} \, \log \left (2 \, x - 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 56, normalized size = 0.93 \begin {gather*} - \frac {\log {\left (x - \frac {3}{2} \right )}}{324} + \frac {\log {\left (x + \frac {3}{2} \right )}}{108} - \frac {\log {\left (x^{2} + \frac {3 x}{2} + \frac {9}{4} \right )}}{324} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {4 \sqrt {3} x}{9} + \frac {\sqrt {3}}{3} \right )}}{162} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.88, size = 48, normalized size = 0.80 \begin {gather*} \frac {1}{162} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) - \frac {1}{324} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) + \frac {1}{108} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - \frac {1}{324} \, \log \left ({\left | 2 \, x - 3 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.01, size = 52, normalized size = 0.87 \begin {gather*} \frac {\ln \left (x+\frac {3}{2}\right )}{108}-\frac {\ln \left (x-\frac {3}{2}\right )}{324}-\ln \left (x+\frac {3}{4}-\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (\frac {1}{324}+\frac {\sqrt {3}\,1{}\mathrm {i}}{324}\right )+\ln \left (x+\frac {3}{4}+\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (-\frac {1}{324}+\frac {\sqrt {3}\,1{}\mathrm {i}}{324}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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