Integrand size = 109, antiderivative size = 34 \[ \int \frac {-12+84 x-40 x^2+5 x^3+\left (-160 x+80 x^2-10 x^3\right ) \log (x)+\left (4-x+\left (80 x-40 x^2+5 x^3\right ) \log (x)\right ) \log \left (\frac {-1+\left (-20 x+5 x^2\right ) \log (x)}{-12 x^2+3 x^3}\right )}{4-x+\left (80 x-40 x^2+5 x^3\right ) \log (x)} \, dx=e^5-x+x \log \left (\frac {5 \left (\frac {1}{5 (4-x) x}+\log (x)\right )}{3 x}\right ) \]
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Time = 1.68 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.97, number of steps used = 17, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {6873, 6820, 6874, 2629} \[ \int \frac {-12+84 x-40 x^2+5 x^3+\left (-160 x+80 x^2-10 x^3\right ) \log (x)+\left (4-x+\left (80 x-40 x^2+5 x^3\right ) \log (x)\right ) \log \left (\frac {-1+\left (-20 x+5 x^2\right ) \log (x)}{-12 x^2+3 x^3}\right )}{4-x+\left (80 x-40 x^2+5 x^3\right ) \log (x)} \, dx=x \log \left (\frac {5 (4-x) x \log (x)+1}{3 (4-x) x^2}\right )-x \]
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Rule 2629
Rule 6820
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {-12+84 x-40 x^2+5 x^3+\left (-160 x+80 x^2-10 x^3\right ) \log (x)+\left (4-x+\left (80 x-40 x^2+5 x^3\right ) \log (x)\right ) \log \left (\frac {-1+\left (-20 x+5 x^2\right ) \log (x)}{-12 x^2+3 x^3}\right )}{(4-x) \left (1+20 x \log (x)-5 x^2 \log (x)\right )} \, dx \\ & = \int \frac {-12+84 x-40 x^2+5 x^3-10 (-4+x)^2 x \log (x)+(-4+x) (-1+5 (-4+x) x \log (x)) \log \left (\frac {-1+5 (-4+x) x \log (x)}{3 (-4+x) x^2}\right )}{(4-x) (1-5 (-4+x) x \log (x))} \, dx \\ & = \int \left (-\frac {12}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )}+\frac {84 x}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )}-\frac {40 x^2}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )}+\frac {5 x^3}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )}-\frac {10 (-4+x) x \log (x)}{-1-20 x \log (x)+5 x^2 \log (x)}+\log \left (\frac {-1+5 (-4+x) x \log (x)}{3 (-4+x) x^2}\right )\right ) \, dx \\ & = 5 \int \frac {x^3}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx-10 \int \frac {(-4+x) x \log (x)}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx-12 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx-40 \int \frac {x^2}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx+84 \int \frac {x}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx+\int \log \left (\frac {-1+5 (-4+x) x \log (x)}{3 (-4+x) x^2}\right ) \, dx \\ & = x \log \left (\frac {1+5 (4-x) x \log (x)}{3 (4-x) x^2}\right )+5 \int \left (\frac {16}{-1-20 x \log (x)+5 x^2 \log (x)}+\frac {64}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )}+\frac {4 x}{-1-20 x \log (x)+5 x^2 \log (x)}+\frac {x^2}{-1-20 x \log (x)+5 x^2 \log (x)}\right ) \, dx-10 \int \left (\frac {1}{5}+\frac {1}{5 \left (-1-20 x \log (x)+5 x^2 \log (x)\right )}\right ) \, dx-12 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx-40 \int \left (\frac {4}{-1-20 x \log (x)+5 x^2 \log (x)}+\frac {16}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )}+\frac {x}{-1-20 x \log (x)+5 x^2 \log (x)}\right ) \, dx+84 \int \left (\frac {1}{-1-20 x \log (x)+5 x^2 \log (x)}+\frac {4}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )}\right ) \, dx-\int \frac {-8+83 x-40 x^2+5 x^3-5 (-4+x)^2 x \log (x)}{(-4+x) (-1+5 (-4+x) x \log (x))} \, dx \\ & = -2 x+x \log \left (\frac {1+5 (4-x) x \log (x)}{3 (4-x) x^2}\right )-2 \int \frac {1}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx+5 \int \frac {x^2}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx-12 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx+20 \int \frac {x}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx-40 \int \frac {x}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx+80 \int \frac {1}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx+84 \int \frac {1}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx-160 \int \frac {1}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx+320 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx+336 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx-640 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx-\int \left (-1+\frac {-4+82 x-40 x^2+5 x^3}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )}\right ) \, dx \\ & = -x+x \log \left (\frac {1+5 (4-x) x \log (x)}{3 (4-x) x^2}\right )-2 \int \frac {1}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx+5 \int \frac {x^2}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx-12 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx+20 \int \frac {x}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx-40 \int \frac {x}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx+80 \int \frac {1}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx+84 \int \frac {1}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx-160 \int \frac {1}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx+320 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx+336 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx-640 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx-\int \frac {-4+82 x-40 x^2+5 x^3}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx \\ & = -x+x \log \left (\frac {1+5 (4-x) x \log (x)}{3 (4-x) x^2}\right )-2 \int \frac {1}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx+5 \int \frac {x^2}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx-12 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx+20 \int \frac {x}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx-40 \int \frac {x}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx+80 \int \frac {1}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx+84 \int \frac {1}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx-160 \int \frac {1}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx+320 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx+336 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx-640 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx-\int \left (\frac {2}{-1-20 x \log (x)+5 x^2 \log (x)}+\frac {4}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )}-\frac {20 x}{-1-20 x \log (x)+5 x^2 \log (x)}+\frac {5 x^2}{-1-20 x \log (x)+5 x^2 \log (x)}\right ) \, dx \\ & = -x+x \log \left (\frac {1+5 (4-x) x \log (x)}{3 (4-x) x^2}\right )-2 \left (2 \int \frac {1}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx\right )-4 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx-12 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx+2 \left (20 \int \frac {x}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx\right )-40 \int \frac {x}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx+80 \int \frac {1}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx+84 \int \frac {1}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx-160 \int \frac {1}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx+320 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx+336 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx-640 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85 \[ \int \frac {-12+84 x-40 x^2+5 x^3+\left (-160 x+80 x^2-10 x^3\right ) \log (x)+\left (4-x+\left (80 x-40 x^2+5 x^3\right ) \log (x)\right ) \log \left (\frac {-1+\left (-20 x+5 x^2\right ) \log (x)}{-12 x^2+3 x^3}\right )}{4-x+\left (80 x-40 x^2+5 x^3\right ) \log (x)} \, dx=-x+x \log \left (\frac {-1+5 (-4+x) x \log (x)}{3 (-4+x) x^2}\right ) \]
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Time = 1.77 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.97
method | result | size |
parallelrisch | \(-12+\ln \left (\frac {\left (5 x^{2}-20 x \right ) \ln \left (x \right )-1}{3 \left (x -4\right ) x^{2}}\right ) x -x\) | \(33\) |
risch | \(x \ln \left (x^{2} \ln \left (x \right )-4 x \ln \left (x \right )-\frac {1}{5}\right )-x \ln \left (x -4\right )-2 x \ln \left (x \right )-\frac {i \pi x {\operatorname {csgn}\left (\frac {i \left (x^{2} \ln \left (x \right )-4 x \ln \left (x \right )-\frac {1}{5}\right )}{x^{2} \left (x -4\right )}\right )}^{3}}{2}+\frac {i \pi x \,\operatorname {csgn}\left (\frac {i \left (x^{2} \ln \left (x \right )-4 x \ln \left (x \right )-\frac {1}{5}\right )}{x -4}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2} \ln \left (x \right )-4 x \ln \left (x \right )-\frac {1}{5}\right )}{x^{2} \left (x -4\right )}\right )}^{2}}{2}-i x \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )+\frac {i \pi x {\operatorname {csgn}\left (\frac {i \left (x^{2} \ln \left (x \right )-4 x \ln \left (x \right )-\frac {1}{5}\right )}{x -4}\right )}^{2} \operatorname {csgn}\left (\frac {i}{x -4}\right )}{2}+\frac {i x \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}}{2}-\frac {i \pi x \,\operatorname {csgn}\left (i \left (x^{2} \ln \left (x \right )-4 x \ln \left (x \right )-\frac {1}{5}\right )\right ) \operatorname {csgn}\left (\frac {i \left (x^{2} \ln \left (x \right )-4 x \ln \left (x \right )-\frac {1}{5}\right )}{x -4}\right ) \operatorname {csgn}\left (\frac {i}{x -4}\right )}{2}-\frac {i \pi x {\operatorname {csgn}\left (\frac {i \left (x^{2} \ln \left (x \right )-4 x \ln \left (x \right )-\frac {1}{5}\right )}{x -4}\right )}^{3}}{2}-\frac {i \pi x \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (x^{2} \ln \left (x \right )-4 x \ln \left (x \right )-\frac {1}{5}\right )}{x -4}\right ) \operatorname {csgn}\left (\frac {i \left (x^{2} \ln \left (x \right )-4 x \ln \left (x \right )-\frac {1}{5}\right )}{x^{2} \left (x -4\right )}\right )}{2}+\frac {i \pi x \,\operatorname {csgn}\left (i \left (x^{2} \ln \left (x \right )-4 x \ln \left (x \right )-\frac {1}{5}\right )\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2} \ln \left (x \right )-4 x \ln \left (x \right )-\frac {1}{5}\right )}{x -4}\right )}^{2}}{2}+\frac {i x \pi \operatorname {csgn}\left (i x^{2}\right )^{3}}{2}+\frac {i \pi x \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2} \ln \left (x \right )-4 x \ln \left (x \right )-\frac {1}{5}\right )}{x^{2} \left (x -4\right )}\right )}^{2}}{2}+x \ln \left (5\right )-x \ln \left (3\right )-x\) | \(444\) |
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Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.97 \[ \int \frac {-12+84 x-40 x^2+5 x^3+\left (-160 x+80 x^2-10 x^3\right ) \log (x)+\left (4-x+\left (80 x-40 x^2+5 x^3\right ) \log (x)\right ) \log \left (\frac {-1+\left (-20 x+5 x^2\right ) \log (x)}{-12 x^2+3 x^3}\right )}{4-x+\left (80 x-40 x^2+5 x^3\right ) \log (x)} \, dx=x \log \left (\frac {5 \, {\left (x^{2} - 4 \, x\right )} \log \left (x\right ) - 1}{3 \, {\left (x^{3} - 4 \, x^{2}\right )}}\right ) - x \]
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Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (24) = 48\).
Time = 0.52 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.65 \[ \int \frac {-12+84 x-40 x^2+5 x^3+\left (-160 x+80 x^2-10 x^3\right ) \log (x)+\left (4-x+\left (80 x-40 x^2+5 x^3\right ) \log (x)\right ) \log \left (\frac {-1+\left (-20 x+5 x^2\right ) \log (x)}{-12 x^2+3 x^3}\right )}{4-x+\left (80 x-40 x^2+5 x^3\right ) \log (x)} \, dx=- x + \left (x - \frac {2}{3}\right ) \log {\left (\frac {\left (5 x^{2} - 20 x\right ) \log {\left (x \right )} - 1}{3 x^{3} - 12 x^{2}} \right )} - \frac {2 \log {\left (x \right )}}{3} + \frac {2 \log {\left (\log {\left (x \right )} - \frac {1}{5 x^{2} - 20 x} \right )}}{3} \]
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Time = 0.31 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {-12+84 x-40 x^2+5 x^3+\left (-160 x+80 x^2-10 x^3\right ) \log (x)+\left (4-x+\left (80 x-40 x^2+5 x^3\right ) \log (x)\right ) \log \left (\frac {-1+\left (-20 x+5 x^2\right ) \log (x)}{-12 x^2+3 x^3}\right )}{4-x+\left (80 x-40 x^2+5 x^3\right ) \log (x)} \, dx=-x {\left (\log \left (3\right ) + 1\right )} + x \log \left (5 \, {\left (x^{2} - 4 \, x\right )} \log \left (x\right ) - 1\right ) - x \log \left (x - 4\right ) - 2 \, x \log \left (x\right ) \]
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Time = 0.31 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.03 \[ \int \frac {-12+84 x-40 x^2+5 x^3+\left (-160 x+80 x^2-10 x^3\right ) \log (x)+\left (4-x+\left (80 x-40 x^2+5 x^3\right ) \log (x)\right ) \log \left (\frac {-1+\left (-20 x+5 x^2\right ) \log (x)}{-12 x^2+3 x^3}\right )}{4-x+\left (80 x-40 x^2+5 x^3\right ) \log (x)} \, dx=x \log \left (5 \, x^{2} \log \left (x\right ) - 20 \, x \log \left (x\right ) - 1\right ) - x \log \left (3 \, x - 12\right ) - 2 \, x \log \left (x\right ) - x \]
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Time = 8.60 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.97 \[ \int \frac {-12+84 x-40 x^2+5 x^3+\left (-160 x+80 x^2-10 x^3\right ) \log (x)+\left (4-x+\left (80 x-40 x^2+5 x^3\right ) \log (x)\right ) \log \left (\frac {-1+\left (-20 x+5 x^2\right ) \log (x)}{-12 x^2+3 x^3}\right )}{4-x+\left (80 x-40 x^2+5 x^3\right ) \log (x)} \, dx=x\,\left (\ln \left (\frac {\ln \left (x\right )\,\left (20\,x-5\,x^2\right )+1}{12\,x^2-3\,x^3}\right )-1\right ) \]
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