Optimal. Leaf size=50 \[ -\frac {\tan ^{-1}\left (\frac {3-4 x}{3 \sqrt {3}}\right )}{18 \sqrt {3}}-\frac {1}{18} \log (3-2 x)+\frac {1}{36} \log \left (9-6 x+4 x^2\right ) \]
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Rubi [A]
time = 0.03, 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 = {1600, 2083,
648, 632, 210, 642} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {3-4 x}{3 \sqrt {3}}\right )}{18 \sqrt {3}}+\frac {1}{36} \log \left (4 x^2-6 x+9\right )-\frac {1}{18} \log (3-2 x) \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 {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} \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 )}{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 \begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.50, size = 39, normalized size = 0.78
method | result | size |
default | \(\frac {\ln \left (4 x^{2}-6 x +9\right )}{36}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (8 x -6\right ) \sqrt {3}}{18}\right )}{54}-\frac {\ln \left (-3+2 x \right )}{18}\) | \(39\) |
risch | \(-\frac {\ln \left (-3+2 x \right )}{18}+\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (2 x -\frac {3}{2}\right ) \sqrt {3}}{9}\right )}{54}+\frac {\ln \left (4 x^{2}-6 x +9\right )}{36}\) | \(39\) |
meijerg | \(-\frac {x \left (\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )-\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )+\frac {\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3-\left (x^{6}\right )^{\frac {1}{6}}}\right )-\frac {\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3+\left (x^{6}\right )^{\frac {1}{6}}}\right )\right )}{108 \left (x^{6}\right )^{\frac {1}{6}}}-\frac {x^{4} \left (\ln \left (1-\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )-\frac {\ln \left (1+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}+\frac {16 \left (x^{6}\right )^{\frac {2}{3}}}{81}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, \left (x^{6}\right )^{\frac {1}{3}}}{9 \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}\right )\right )}{108 \left (x^{6}\right )^{\frac {2}{3}}}+\frac {\arctanh \left (\frac {8 x^{3}}{27}\right )}{27}-\frac {x^{2} \left (\ln \left (1-\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )-\frac {\ln \left (1+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}+\frac {16 \left (x^{6}\right )^{\frac {2}{3}}}{81}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, \left (x^{6}\right )^{\frac {1}{3}}}{9 \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}\right )\right )}{54 \left (x^{6}\right )^{\frac {1}{3}}}\) | \(266\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 38, normalized size = 0.76 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 38, normalized size = 0.76 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 48, normalized size = 0.96 \begin {gather*} - \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.81, size = 39, normalized size = 0.78 \begin {gather*} \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 ) \end {gather*}
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 \begin {gather*} -\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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