Optimal. Leaf size=92 \[ \frac {x}{4374 \left (9-6 x+4 x^2\right )}-\frac {\tan ^{-1}\left (\frac {3-4 x}{3 \sqrt {3}}\right )}{4374 \sqrt {3}}-\frac {\log (3-2 x)}{26244}+\frac {\log (3+2 x)}{78732}-\frac {\log \left (9-6 x+4 x^2\right )}{157464}+\frac {\log \left (9+6 x+4 x^2\right )}{52488} \]
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Rubi [A]
time = 0.08, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {1600, 2099,
652, 632, 210, 648, 642} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {3-4 x}{3 \sqrt {3}}\right )}{4374 \sqrt {3}}+\frac {x}{4374 \left (4 x^2-6 x+9\right )}-\frac {\log \left (4 x^2-6 x+9\right )}{157464}+\frac {\log \left (4 x^2+6 x+9\right )}{52488}-\frac {\log (3-2 x)}{26244}+\frac {\log (2 x+3)}{78732} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 652
Rule 1600
Rule 2099
Rubi steps
\begin {align*} \int \frac {81+54 x-24 x^3-16 x^4}{\left (729-64 x^6\right )^2} \, dx &=\int \frac {1}{\left (9-6 x+4 x^2\right )^2 \left (81+54 x-24 x^3-16 x^4\right )} \, dx\\ &=\int \left (-\frac {1}{13122 (-3+2 x)}+\frac {1}{39366 (3+2 x)}+\frac {3-x}{729 \left (9-6 x+4 x^2\right )^2}+\frac {39-4 x}{78732 \left (9-6 x+4 x^2\right )}+\frac {3+4 x}{26244 \left (9+6 x+4 x^2\right )}\right ) \, dx\\ &=-\frac {\log (3-2 x)}{26244}+\frac {\log (3+2 x)}{78732}+\frac {\int \frac {39-4 x}{9-6 x+4 x^2} \, dx}{78732}+\frac {\int \frac {3+4 x}{9+6 x+4 x^2} \, dx}{26244}+\frac {1}{729} \int \frac {3-x}{\left (9-6 x+4 x^2\right )^2} \, dx\\ &=\frac {x}{4374 \left (9-6 x+4 x^2\right )}-\frac {\log (3-2 x)}{26244}+\frac {\log (3+2 x)}{78732}+\frac {\log \left (9+6 x+4 x^2\right )}{52488}-\frac {\int \frac {-6+8 x}{9-6 x+4 x^2} \, dx}{157464}+\frac {\int \frac {1}{9-6 x+4 x^2} \, dx}{4374}+\frac {\int \frac {1}{9-6 x+4 x^2} \, dx}{2187}\\ &=\frac {x}{4374 \left (9-6 x+4 x^2\right )}-\frac {\log (3-2 x)}{26244}+\frac {\log (3+2 x)}{78732}-\frac {\log \left (9-6 x+4 x^2\right )}{157464}+\frac {\log \left (9+6 x+4 x^2\right )}{52488}-\frac {\text {Subst}\left (\int \frac {1}{-108-x^2} \, dx,x,-6+8 x\right )}{2187}-\frac {2 \text {Subst}\left (\int \frac {1}{-108-x^2} \, dx,x,-6+8 x\right )}{2187}\\ &=\frac {x}{4374 \left (9-6 x+4 x^2\right )}-\frac {\tan ^{-1}\left (\frac {3-4 x}{3 \sqrt {3}}\right )}{4374 \sqrt {3}}-\frac {\log (3-2 x)}{26244}+\frac {\log (3+2 x)}{78732}-\frac {\log \left (9-6 x+4 x^2\right )}{157464}+\frac {\log \left (9+6 x+4 x^2\right )}{52488}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 84, normalized size = 0.91 \begin {gather*} \frac {\frac {36 x}{9-6 x+4 x^2}+12 \sqrt {3} \tan ^{-1}\left (\frac {-3+4 x}{3 \sqrt {3}}\right )-6 \log (3-2 x)+2 \log (3+2 x)-\log \left (9-6 x+4 x^2\right )+3 \log \left (9+6 x+4 x^2\right )}{157464} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 73, normalized size = 0.79
method | result | size |
default | \(\frac {x}{17496 x^{2}-26244 x +39366}-\frac {\ln \left (4 x^{2}-6 x +9\right )}{157464}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (8 x -6\right ) \sqrt {3}}{18}\right )}{13122}+\frac {\ln \left (4 x^{2}+6 x +9\right )}{52488}+\frac {\ln \left (2 x +3\right )}{78732}-\frac {\ln \left (-3+2 x \right )}{26244}\) | \(73\) |
risch | \(\frac {x}{17496 x^{2}-26244 x +39366}+\frac {\ln \left (2 x +3\right )}{78732}-\frac {\ln \left (64 x^{2}-96 x +144\right )}{157464}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (8 x -6\right ) \sqrt {3}}{18}\right )}{13122}-\frac {\ln \left (-3+2 x \right )}{26244}+\frac {\ln \left (4 x^{2}+6 x +9\right )}{52488}\) | \(73\) |
meijerg | \(-\frac {\left (-1\right )^{\frac {5}{6}} \left (\frac {4 x \left (-1\right )^{\frac {1}{6}}}{6-\frac {128 x^{6}}{243}}-\frac {5 x \left (-1\right )^{\frac {1}{6}} \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 )}{6 \left (x^{6}\right )^{\frac {1}{6}}}\right )}{26244}+\frac {\left (-1\right )^{\frac {1}{6}} \left (\frac {64 x^{5} \left (-1\right )^{\frac {5}{6}}}{81 \left (6-\frac {128 x^{6}}{243}\right )}-\frac {x^{5} \left (-1\right )^{\frac {5}{6}} \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 )}{6 \left (x^{6}\right )^{\frac {5}{6}}}\right )}{26244}+\frac {\left (-1\right )^{\frac {1}{3}} \left (\frac {16 x^{4} \left (-1\right )^{\frac {2}{3}}}{27 \left (3-\frac {64 x^{6}}{243}\right )}-\frac {x^{4} \left (-1\right )^{\frac {2}{3}} \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 )}{3 \left (x^{6}\right )^{\frac {2}{3}}}\right )}{26244}-\frac {\left (-1\right )^{\frac {2}{3}} \left (\frac {4 x^{2} \left (-1\right )^{\frac {1}{3}}}{3 \left (3-\frac {64 x^{6}}{243}\right )}-\frac {2 x^{2} \left (-1\right )^{\frac {1}{3}} \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 )}{3 \left (x^{6}\right )^{\frac {1}{3}}}\right )}{26244}\) | \(483\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 74, normalized size = 0.80 \begin {gather*} \frac {1}{13122} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) + \frac {x}{4374 \, {\left (4 \, x^{2} - 6 \, x + 9\right )}} + \frac {1}{52488} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac {1}{157464} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac {1}{78732} \, \log \left (2 \, x + 3\right ) - \frac {1}{26244} \, \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 = 126, normalized size = 1.37 \begin {gather*} \frac {12 \, \sqrt {3} {\left (4 \, x^{2} - 6 \, x + 9\right )} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) + 3 \, {\left (4 \, x^{2} - 6 \, x + 9\right )} \log \left (4 \, x^{2} + 6 \, x + 9\right ) - {\left (4 \, x^{2} - 6 \, x + 9\right )} \log \left (4 \, x^{2} - 6 \, x + 9\right ) + 2 \, {\left (4 \, x^{2} - 6 \, x + 9\right )} \log \left (2 \, x + 3\right ) - 6 \, {\left (4 \, x^{2} - 6 \, x + 9\right )} \log \left (2 \, x - 3\right ) + 36 \, x}{157464 \, {\left (4 \, x^{2} - 6 \, x + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.14, size = 82, normalized size = 0.89 \begin {gather*} \frac {x}{17496 x^{2} - 26244 x + 39366} - \frac {\log {\left (x - \frac {3}{2} \right )}}{26244} + \frac {\log {\left (x + \frac {3}{2} \right )}}{78732} - \frac {\log {\left (x^{2} - \frac {3 x}{2} + \frac {9}{4} \right )}}{157464} + \frac {\log {\left (4 x^{2} + 6 x + 9 \right )}}{52488} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {4 \sqrt {3} x}{9} - \frac {\sqrt {3}}{3} \right )}}{13122} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.79, size = 76, normalized size = 0.83 \begin {gather*} \frac {1}{13122} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) + \frac {x}{4374 \, {\left (4 \, x^{2} - 6 \, x + 9\right )}} + \frac {1}{52488} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac {1}{157464} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac {1}{78732} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - \frac {1}{26244} \, \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 = 0.12, size = 77, normalized size = 0.84 \begin {gather*} \frac {\ln \left (x+\frac {3}{2}\right )}{78732}-\frac {\ln \left (x-\frac {3}{2}\right )}{26244}+\frac {\ln \left (x^2+\frac {3\,x}{2}+\frac {9}{4}\right )}{52488}+\frac {x}{17496\,\left (x^2-\frac {3\,x}{2}+\frac {9}{4}\right )}-\ln \left (x-\frac {3}{4}-\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (\frac {1}{157464}+\frac {\sqrt {3}\,1{}\mathrm {i}}{26244}\right )+\ln \left (x-\frac {3}{4}+\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (-\frac {1}{157464}+\frac {\sqrt {3}\,1{}\mathrm {i}}{26244}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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