Integrand size = 25, antiderivative size = 24 \[ \int \frac {81+54 x-24 x^3-16 x^4}{729-64 x^6} \, dx=-\frac {\arctan \left (\frac {3-4 x}{3 \sqrt {3}}\right )}{3 \sqrt {3}} \]
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Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {1600, 632, 210} \[ \int \frac {81+54 x-24 x^3-16 x^4}{729-64 x^6} \, dx=-\frac {\arctan \left (\frac {3-4 x}{3 \sqrt {3}}\right )}{3 \sqrt {3}} \]
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Rule 210
Rule 632
Rule 1600
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{9-6 x+4 x^2} \, dx \\ & = -\left (2 \text {Subst}\left (\int \frac {1}{-108-x^2} \, dx,x,-6+8 x\right )\right ) \\ & = -\frac {\tan ^{-1}\left (\frac {3-4 x}{3 \sqrt {3}}\right )}{3 \sqrt {3}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {81+54 x-24 x^3-16 x^4}{729-64 x^6} \, dx=\frac {\arctan \left (\frac {-3+4 x}{3 \sqrt {3}}\right )}{3 \sqrt {3}} \]
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Time = 1.53 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.71
method | result | size |
default | \(\frac {\sqrt {3}\, \arctan \left (\frac {\left (8 x -6\right ) \sqrt {3}}{18}\right )}{9}\) | \(17\) |
risch | \(\frac {\sqrt {3}\, \arctan \left (\frac {\left (4 x -3\right ) \sqrt {3}}{9}\right )}{9}\) | \(17\) |
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 )}{36 \left (x^{6}\right )^{\frac {1}{6}}}+\frac {x^{5} \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 )}{36 \left (x^{6}\right )^{\frac {5}{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 )}{36 \left (x^{6}\right )^{\frac {2}{3}}}-\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 )}{36 \left (x^{6}\right )^{\frac {1}{3}}}\) | \(381\) |
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Time = 0.28 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.67 \[ \int \frac {81+54 x-24 x^3-16 x^4}{729-64 x^6} \, dx=\frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {81+54 x-24 x^3-16 x^4}{729-64 x^6} \, dx=\frac {\sqrt {3} \operatorname {atan}{\left (\frac {4 \sqrt {3} x}{9} - \frac {\sqrt {3}}{3} \right )}}{9} \]
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Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.67 \[ \int \frac {81+54 x-24 x^3-16 x^4}{729-64 x^6} \, dx=\frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) \]
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Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.67 \[ \int \frac {81+54 x-24 x^3-16 x^4}{729-64 x^6} \, dx=\frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.67 \[ \int \frac {81+54 x-24 x^3-16 x^4}{729-64 x^6} \, dx=\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,\left (4\,x-3\right )}{9}\right )}{9} \]
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