Integrand size = 179, antiderivative size = 25 \[ \int \frac {-2352-532 x+292 x^2-24 x^3+\left (-392-84 x+48 x^2-4 x^3\right ) \log (9 x)+(-56-28 x) \log (2+x)}{2450 x^2+525 x^3-300 x^4+25 x^5+\left (98 x^2+21 x^3-12 x^4+x^5\right ) \log ^2(9 x)+\left (140 x^2+50 x^3-10 x^4\right ) \log (2+x)+\left (2 x^2+x^3\right ) \log ^2(2+x)+\log (9 x) \left (980 x^2+210 x^3-120 x^4+10 x^5+\left (28 x^2+10 x^3-2 x^4\right ) \log (2+x)\right )} \, dx=\frac {4}{x \left (5+\log (9 x)+\frac {\log (2+x)}{7-x}\right )} \]
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\[ \int \frac {-2352-532 x+292 x^2-24 x^3+\left (-392-84 x+48 x^2-4 x^3\right ) \log (9 x)+(-56-28 x) \log (2+x)}{2450 x^2+525 x^3-300 x^4+25 x^5+\left (98 x^2+21 x^3-12 x^4+x^5\right ) \log ^2(9 x)+\left (140 x^2+50 x^3-10 x^4\right ) \log (2+x)+\left (2 x^2+x^3\right ) \log ^2(2+x)+\log (9 x) \left (980 x^2+210 x^3-120 x^4+10 x^5+\left (28 x^2+10 x^3-2 x^4\right ) \log (2+x)\right )} \, dx=\int \frac {-2352-532 x+292 x^2-24 x^3+\left (-392-84 x+48 x^2-4 x^3\right ) \log (9 x)+(-56-28 x) \log (2+x)}{2450 x^2+525 x^3-300 x^4+25 x^5+\left (98 x^2+21 x^3-12 x^4+x^5\right ) \log ^2(9 x)+\left (140 x^2+50 x^3-10 x^4\right ) \log (2+x)+\left (2 x^2+x^3\right ) \log ^2(2+x)+\log (9 x) \left (980 x^2+210 x^3-120 x^4+10 x^5+\left (28 x^2+10 x^3-2 x^4\right ) \log (2+x)\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {4 \left (-588-133 x+73 x^2-6 x^3-(-7+x)^2 (2+x) \log (9 x)-7 (2+x) \log (2+x)\right )}{x^2 (2+x) (5 (-7+x)+(-7+x) \log (9 x)-\log (2+x))^2} \, dx \\ & = 4 \int \frac {-588-133 x+73 x^2-6 x^3-(-7+x)^2 (2+x) \log (9 x)-7 (2+x) \log (2+x)}{x^2 (2+x) (5 (-7+x)+(-7+x) \log (9 x)-\log (2+x))^2} \, dx \\ & = 4 \int \left (-\frac {(-7+x) \left (-14+4 x+6 x^2+2 x \log (9 x)+x^2 \log (9 x)\right )}{x^2 (2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}+\frac {7}{x^2 (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))}\right ) \, dx \\ & = -\left (4 \int \frac {(-7+x) \left (-14+4 x+6 x^2+2 x \log (9 x)+x^2 \log (9 x)\right )}{x^2 (2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx\right )+28 \int \frac {1}{x^2 (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))} \, dx \\ & = -\left (4 \int \left (-\frac {7 \left (-14+4 x+6 x^2+2 x \log (9 x)+x^2 \log (9 x)\right )}{2 x^2 (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}+\frac {9 \left (-14+4 x+6 x^2+2 x \log (9 x)+x^2 \log (9 x)\right )}{4 x (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}-\frac {9 \left (-14+4 x+6 x^2+2 x \log (9 x)+x^2 \log (9 x)\right )}{4 (2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}\right ) \, dx\right )+28 \int \frac {1}{x^2 (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))} \, dx \\ & = -\left (9 \int \frac {-14+4 x+6 x^2+2 x \log (9 x)+x^2 \log (9 x)}{x (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx\right )+9 \int \frac {-14+4 x+6 x^2+2 x \log (9 x)+x^2 \log (9 x)}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+14 \int \frac {-14+4 x+6 x^2+2 x \log (9 x)+x^2 \log (9 x)}{x^2 (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+28 \int \frac {1}{x^2 (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))} \, dx \\ & = -\left (9 \int \left (\frac {4}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}-\frac {14}{x (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}+\frac {6 x}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}+\frac {2 \log (9 x)}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}+\frac {x \log (9 x)}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}\right ) \, dx\right )+9 \int \left (-\frac {14}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}+\frac {4 x}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}+\frac {6 x^2}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}+\frac {2 x \log (9 x)}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}+\frac {x^2 \log (9 x)}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}\right ) \, dx+14 \int \left (\frac {6}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}-\frac {14}{x^2 (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}+\frac {4}{x (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}+\frac {\log (9 x)}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}+\frac {2 \log (9 x)}{x (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}\right ) \, dx+28 \int \frac {1}{x^2 (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))} \, dx \\ & = -\left (9 \int \frac {x \log (9 x)}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx\right )+9 \int \frac {x^2 \log (9 x)}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+14 \int \frac {\log (9 x)}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx-18 \int \frac {\log (9 x)}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+18 \int \frac {x \log (9 x)}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+28 \int \frac {\log (9 x)}{x (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+28 \int \frac {1}{x^2 (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))} \, dx-36 \int \frac {1}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+36 \int \frac {x}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx-54 \int \frac {x}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+54 \int \frac {x^2}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+56 \int \frac {1}{x (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+84 \int \frac {1}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+126 \int \frac {1}{x (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx-126 \int \frac {1}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx-196 \int \frac {1}{x^2 (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx \\ & = 9 \int \left (-\frac {2 \log (9 x)}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}+\frac {x \log (9 x)}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}+\frac {4 \log (9 x)}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}\right ) \, dx-9 \int \frac {x \log (9 x)}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+14 \int \frac {\log (9 x)}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+18 \int \left (\frac {\log (9 x)}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}-\frac {2 \log (9 x)}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}\right ) \, dx-18 \int \frac {\log (9 x)}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+28 \int \frac {\log (9 x)}{x (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+28 \int \frac {1}{x^2 (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))} \, dx+36 \int \left (\frac {1}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}-\frac {2}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}\right ) \, dx-36 \int \frac {1}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+54 \int \left (-\frac {2}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}+\frac {x}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}+\frac {4}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}\right ) \, dx-54 \int \frac {x}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+56 \int \frac {1}{x (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+84 \int \frac {1}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+126 \int \frac {1}{x (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx-126 \int \frac {1}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx-196 \int \frac {1}{x^2 (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx \\ & = 14 \int \frac {\log (9 x)}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx-18 \int \frac {\log (9 x)}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+28 \int \frac {\log (9 x)}{x (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+28 \int \frac {1}{x^2 (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))} \, dx+56 \int \frac {1}{x (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx-72 \int \frac {1}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+84 \int \frac {1}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx-108 \int \frac {1}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+126 \int \frac {1}{x (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx-126 \int \frac {1}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx-196 \int \frac {1}{x^2 (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+216 \int \frac {1}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx \\ \end{align*}
Time = 0.86 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {-2352-532 x+292 x^2-24 x^3+\left (-392-84 x+48 x^2-4 x^3\right ) \log (9 x)+(-56-28 x) \log (2+x)}{2450 x^2+525 x^3-300 x^4+25 x^5+\left (98 x^2+21 x^3-12 x^4+x^5\right ) \log ^2(9 x)+\left (140 x^2+50 x^3-10 x^4\right ) \log (2+x)+\left (2 x^2+x^3\right ) \log ^2(2+x)+\log (9 x) \left (980 x^2+210 x^3-120 x^4+10 x^5+\left (28 x^2+10 x^3-2 x^4\right ) \log (2+x)\right )} \, dx=-\frac {4 (-7+x)}{x (35-5 x-(-7+x) \log (9 x)+\log (2+x))} \]
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Time = 1.40 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36
method | result | size |
risch | \(\frac {-28+4 x}{x \left (x \ln \left (9 x \right )+5 x -7 \ln \left (9 x \right )-\ln \left (2+x \right )-35\right )}\) | \(34\) |
parallelrisch | \(\frac {-28+4 x}{x \left (x \ln \left (9 x \right )+5 x -7 \ln \left (9 x \right )-\ln \left (2+x \right )-35\right )}\) | \(35\) |
default | \(\frac {-28+4 x}{x \left (2 x \ln \left (3\right )+x \ln \left (x \right )-14 \ln \left (3\right )+5 x -\ln \left (2+x \right )-7 \ln \left (x \right )-35\right )}\) | \(39\) |
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Time = 0.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40 \[ \int \frac {-2352-532 x+292 x^2-24 x^3+\left (-392-84 x+48 x^2-4 x^3\right ) \log (9 x)+(-56-28 x) \log (2+x)}{2450 x^2+525 x^3-300 x^4+25 x^5+\left (98 x^2+21 x^3-12 x^4+x^5\right ) \log ^2(9 x)+\left (140 x^2+50 x^3-10 x^4\right ) \log (2+x)+\left (2 x^2+x^3\right ) \log ^2(2+x)+\log (9 x) \left (980 x^2+210 x^3-120 x^4+10 x^5+\left (28 x^2+10 x^3-2 x^4\right ) \log (2+x)\right )} \, dx=\frac {4 \, {\left (x - 7\right )}}{5 \, x^{2} + {\left (x^{2} - 7 \, x\right )} \log \left (9 \, x\right ) - x \log \left (x + 2\right ) - 35 \, x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (17) = 34\).
Time = 0.14 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {-2352-532 x+292 x^2-24 x^3+\left (-392-84 x+48 x^2-4 x^3\right ) \log (9 x)+(-56-28 x) \log (2+x)}{2450 x^2+525 x^3-300 x^4+25 x^5+\left (98 x^2+21 x^3-12 x^4+x^5\right ) \log ^2(9 x)+\left (140 x^2+50 x^3-10 x^4\right ) \log (2+x)+\left (2 x^2+x^3\right ) \log ^2(2+x)+\log (9 x) \left (980 x^2+210 x^3-120 x^4+10 x^5+\left (28 x^2+10 x^3-2 x^4\right ) \log (2+x)\right )} \, dx=\frac {28 - 4 x}{- x^{2} \log {\left (9 x \right )} - 5 x^{2} + 7 x \log {\left (9 x \right )} + x \log {\left (x + 2 \right )} + 35 x} \]
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Time = 0.32 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.76 \[ \int \frac {-2352-532 x+292 x^2-24 x^3+\left (-392-84 x+48 x^2-4 x^3\right ) \log (9 x)+(-56-28 x) \log (2+x)}{2450 x^2+525 x^3-300 x^4+25 x^5+\left (98 x^2+21 x^3-12 x^4+x^5\right ) \log ^2(9 x)+\left (140 x^2+50 x^3-10 x^4\right ) \log (2+x)+\left (2 x^2+x^3\right ) \log ^2(2+x)+\log (9 x) \left (980 x^2+210 x^3-120 x^4+10 x^5+\left (28 x^2+10 x^3-2 x^4\right ) \log (2+x)\right )} \, dx=\frac {4 \, {\left (x - 7\right )}}{x^{2} {\left (2 \, \log \left (3\right ) + 5\right )} - 7 \, x {\left (2 \, \log \left (3\right ) + 5\right )} - x \log \left (x + 2\right ) + {\left (x^{2} - 7 \, x\right )} \log \left (x\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (25) = 50\).
Time = 0.35 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.36 \[ \int \frac {-2352-532 x+292 x^2-24 x^3+\left (-392-84 x+48 x^2-4 x^3\right ) \log (9 x)+(-56-28 x) \log (2+x)}{2450 x^2+525 x^3-300 x^4+25 x^5+\left (98 x^2+21 x^3-12 x^4+x^5\right ) \log ^2(9 x)+\left (140 x^2+50 x^3-10 x^4\right ) \log (2+x)+\left (2 x^2+x^3\right ) \log ^2(2+x)+\log (9 x) \left (980 x^2+210 x^3-120 x^4+10 x^5+\left (28 x^2+10 x^3-2 x^4\right ) \log (2+x)\right )} \, dx=\frac {4 \, {\left (x - 7\right )}}{{\left (x + 2\right )}^{2} \log \left (9 \, x\right ) + 5 \, {\left (x + 2\right )}^{2} - 11 \, {\left (x + 2\right )} \log \left (9 \, x\right ) - {\left (x + 2\right )} \log \left (x + 2\right ) - 55 \, x + 18 \, \log \left (9 \, x\right ) + 2 \, \log \left (x + 2\right ) - 20} \]
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Time = 11.58 (sec) , antiderivative size = 142, normalized size of antiderivative = 5.68 \[ \int \frac {-2352-532 x+292 x^2-24 x^3+\left (-392-84 x+48 x^2-4 x^3\right ) \log (9 x)+(-56-28 x) \log (2+x)}{2450 x^2+525 x^3-300 x^4+25 x^5+\left (98 x^2+21 x^3-12 x^4+x^5\right ) \log ^2(9 x)+\left (140 x^2+50 x^3-10 x^4\right ) \log (2+x)+\left (2 x^2+x^3\right ) \log ^2(2+x)+\log (9 x) \left (980 x^2+210 x^3-120 x^4+10 x^5+\left (28 x^2+10 x^3-2 x^4\right ) \log (2+x)\right )} \, dx=-\frac {4\,{\left (x^2+2\,x\right )}^2\,\left (-x^4+20\,x^3-119\,x^2+98\,x+686\right )+4\,\ln \left (x+2\right )\,{\left (x^2+2\,x\right )}^2\,\left (-x^3+5\,x^2+14\,x\right )}{x^2\,\left (x+2\right )\,\left (5\,x-\ln \left (x+2\right )+\ln \left (9\,x\right )\,\left (x-7\right )-35\right )\,\left (196\,x+4\,x^2\,\ln \left (x+2\right )+4\,x^3\,\ln \left (x+2\right )+x^4\,\ln \left (x+2\right )+154\,x^2+2\,x^3-11\,x^4+x^5\right )} \]
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