Integrand size = 165, antiderivative size = 22 \[ \int \frac {-12 x^2+3 x^3+e \left (-12 x^2+3 x^3\right )+\left (24 x-6 x^2+e \left (12 x-3 x^2\right )\right ) \log (-4+x)+(-12+3 x) \log ^2(-4+x)+\left (12 x^2-3 x^3+e \left (9 x^2-3 x^3\right )+\left (-24 x+6 x^2+e \left (-24 x+6 x^2\right )\right ) \log (-4+x)+(12-3 x) \log ^2(-4+x)\right ) \log (x)}{\left (-4 x^2+x^3+\left (8 x-2 x^2\right ) \log (-4+x)+(-4+x) \log ^2(-4+x)\right ) \log ^2(x)} \, dx=\frac {3 x \left (-1+\frac {e x}{-x+\log (-4+x)}\right )}{\log (x)} \]
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\[ \int \frac {-12 x^2+3 x^3+e \left (-12 x^2+3 x^3\right )+\left (24 x-6 x^2+e \left (12 x-3 x^2\right )\right ) \log (-4+x)+(-12+3 x) \log ^2(-4+x)+\left (12 x^2-3 x^3+e \left (9 x^2-3 x^3\right )+\left (-24 x+6 x^2+e \left (-24 x+6 x^2\right )\right ) \log (-4+x)+(12-3 x) \log ^2(-4+x)\right ) \log (x)}{\left (-4 x^2+x^3+\left (8 x-2 x^2\right ) \log (-4+x)+(-4+x) \log ^2(-4+x)\right ) \log ^2(x)} \, dx=\int \frac {-12 x^2+3 x^3+e \left (-12 x^2+3 x^3\right )+\left (24 x-6 x^2+e \left (12 x-3 x^2\right )\right ) \log (-4+x)+(-12+3 x) \log ^2(-4+x)+\left (12 x^2-3 x^3+e \left (9 x^2-3 x^3\right )+\left (-24 x+6 x^2+e \left (-24 x+6 x^2\right )\right ) \log (-4+x)+(12-3 x) \log ^2(-4+x)\right ) \log (x)}{\left (-4 x^2+x^3+\left (8 x-2 x^2\right ) \log (-4+x)+(-4+x) \log ^2(-4+x)\right ) \log ^2(x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {3 \left ((-4+x) \log ^2(-4+x) (-1+\log (x))-(-4+x) x \log (-4+x) (-2-e+2 (1+e) \log (x))+x^2 (-((1+e) (-4+x))+(-4+e (-3+x)+x) \log (x))\right )}{(4-x) (x-\log (-4+x))^2 \log ^2(x)} \, dx \\ & = 3 \int \frac {(-4+x) \log ^2(-4+x) (-1+\log (x))-(-4+x) x \log (-4+x) (-2-e+2 (1+e) \log (x))+x^2 (-((1+e) (-4+x))+(-4+e (-3+x)+x) \log (x))}{(4-x) (x-\log (-4+x))^2 \log ^2(x)} \, dx \\ & = 3 \int \left (\frac {(1+e) x-\log (-4+x)}{(x-\log (-4+x)) \log ^2(x)}+\frac {-4 \left (1+\frac {3 e}{4}\right ) x^2+(1+e) x^3+8 (1+e) x \log (-4+x)-2 (1+e) x^2 \log (-4+x)-4 \log ^2(-4+x)+x \log ^2(-4+x)}{(4-x) (x-\log (-4+x))^2 \log (x)}\right ) \, dx \\ & = 3 \int \frac {(1+e) x-\log (-4+x)}{(x-\log (-4+x)) \log ^2(x)} \, dx+3 \int \frac {-4 \left (1+\frac {3 e}{4}\right ) x^2+(1+e) x^3+8 (1+e) x \log (-4+x)-2 (1+e) x^2 \log (-4+x)-4 \log ^2(-4+x)+x \log ^2(-4+x)}{(4-x) (x-\log (-4+x))^2 \log (x)} \, dx \\ & = 3 \int \left (\frac {(1+e) x}{(x-\log (-4+x)) \log ^2(x)}-\frac {\log (-4+x)}{(x-\log (-4+x)) \log ^2(x)}\right ) \, dx+3 \int \frac {x^2 (-4+e (-3+x)+x)-2 (1+e) (-4+x) x \log (-4+x)+(-4+x) \log ^2(-4+x)}{(4-x) (x-\log (-4+x))^2 \log (x)} \, dx \\ & = 3 \int \left (\frac {4 \left (1+\frac {3 e}{4}\right ) x^2}{(-4+x) (x-\log (-4+x))^2 \log (x)}-\frac {(1+e) x^3}{(-4+x) (x-\log (-4+x))^2 \log (x)}-\frac {8 (1+e) x \log (-4+x)}{(-4+x) (x-\log (-4+x))^2 \log (x)}+\frac {2 (1+e) x^2 \log (-4+x)}{(-4+x) (x-\log (-4+x))^2 \log (x)}+\frac {4 \log ^2(-4+x)}{(-4+x) (x-\log (-4+x))^2 \log (x)}-\frac {x \log ^2(-4+x)}{(-4+x) (x-\log (-4+x))^2 \log (x)}\right ) \, dx-3 \int \frac {\log (-4+x)}{(x-\log (-4+x)) \log ^2(x)} \, dx+(3 (1+e)) \int \frac {x}{(x-\log (-4+x)) \log ^2(x)} \, dx \\ & = -\left (3 \int \frac {\log (-4+x)}{(x-\log (-4+x)) \log ^2(x)} \, dx\right )-3 \int \frac {x \log ^2(-4+x)}{(-4+x) (x-\log (-4+x))^2 \log (x)} \, dx+12 \int \frac {\log ^2(-4+x)}{(-4+x) (x-\log (-4+x))^2 \log (x)} \, dx+(3 (1+e)) \int \frac {x}{(x-\log (-4+x)) \log ^2(x)} \, dx-(3 (1+e)) \int \frac {x^3}{(-4+x) (x-\log (-4+x))^2 \log (x)} \, dx+(6 (1+e)) \int \frac {x^2 \log (-4+x)}{(-4+x) (x-\log (-4+x))^2 \log (x)} \, dx-(24 (1+e)) \int \frac {x \log (-4+x)}{(-4+x) (x-\log (-4+x))^2 \log (x)} \, dx+(3 (4+3 e)) \int \frac {x^2}{(-4+x) (x-\log (-4+x))^2 \log (x)} \, dx \\ & = -\left (3 \int \left (\frac {\log ^2(-4+x)}{(x-\log (-4+x))^2 \log (x)}+\frac {4 \log ^2(-4+x)}{(-4+x) (x-\log (-4+x))^2 \log (x)}\right ) \, dx\right )-3 \int \frac {\log (-4+x)}{(x-\log (-4+x)) \log ^2(x)} \, dx+12 \int \frac {\log ^2(-4+x)}{(-4+x) (x-\log (-4+x))^2 \log (x)} \, dx-(3 (1+e)) \int \left (\frac {16}{(x-\log (-4+x))^2 \log (x)}+\frac {64}{(-4+x) (x-\log (-4+x))^2 \log (x)}+\frac {4 x}{(x-\log (-4+x))^2 \log (x)}+\frac {x^2}{(x-\log (-4+x))^2 \log (x)}\right ) \, dx+(3 (1+e)) \int \frac {x}{(x-\log (-4+x)) \log ^2(x)} \, dx+(6 (1+e)) \int \left (\frac {4 \log (-4+x)}{(x-\log (-4+x))^2 \log (x)}+\frac {16 \log (-4+x)}{(-4+x) (x-\log (-4+x))^2 \log (x)}+\frac {x \log (-4+x)}{(x-\log (-4+x))^2 \log (x)}\right ) \, dx-(24 (1+e)) \int \left (\frac {\log (-4+x)}{(x-\log (-4+x))^2 \log (x)}+\frac {4 \log (-4+x)}{(-4+x) (x-\log (-4+x))^2 \log (x)}\right ) \, dx+(3 (4+3 e)) \int \left (\frac {4}{(x-\log (-4+x))^2 \log (x)}+\frac {16}{(-4+x) (x-\log (-4+x))^2 \log (x)}+\frac {x}{(x-\log (-4+x))^2 \log (x)}\right ) \, dx \\ & = -\left (3 \int \frac {\log (-4+x)}{(x-\log (-4+x)) \log ^2(x)} \, dx\right )-3 \int \frac {\log ^2(-4+x)}{(x-\log (-4+x))^2 \log (x)} \, dx+(3 (1+e)) \int \frac {x}{(x-\log (-4+x)) \log ^2(x)} \, dx-(3 (1+e)) \int \frac {x^2}{(x-\log (-4+x))^2 \log (x)} \, dx+(6 (1+e)) \int \frac {x \log (-4+x)}{(x-\log (-4+x))^2 \log (x)} \, dx-(12 (1+e)) \int \frac {x}{(x-\log (-4+x))^2 \log (x)} \, dx-(48 (1+e)) \int \frac {1}{(x-\log (-4+x))^2 \log (x)} \, dx-(192 (1+e)) \int \frac {1}{(-4+x) (x-\log (-4+x))^2 \log (x)} \, dx+(3 (4+3 e)) \int \frac {x}{(x-\log (-4+x))^2 \log (x)} \, dx+(12 (4+3 e)) \int \frac {1}{(x-\log (-4+x))^2 \log (x)} \, dx+(48 (4+3 e)) \int \frac {1}{(-4+x) (x-\log (-4+x))^2 \log (x)} \, dx \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {-12 x^2+3 x^3+e \left (-12 x^2+3 x^3\right )+\left (24 x-6 x^2+e \left (12 x-3 x^2\right )\right ) \log (-4+x)+(-12+3 x) \log ^2(-4+x)+\left (12 x^2-3 x^3+e \left (9 x^2-3 x^3\right )+\left (-24 x+6 x^2+e \left (-24 x+6 x^2\right )\right ) \log (-4+x)+(12-3 x) \log ^2(-4+x)\right ) \log (x)}{\left (-4 x^2+x^3+\left (8 x-2 x^2\right ) \log (-4+x)+(-4+x) \log ^2(-4+x)\right ) \log ^2(x)} \, dx=-\frac {3 x (x+e x-\log (-4+x))}{(x-\log (-4+x)) \log (x)} \]
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Time = 1.40 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36
method | result | size |
default | \(-\frac {3 x}{\ln \left (x \right )}-\frac {3 \,{\mathrm e} x^{2}}{\ln \left (x \right ) \left (-\ln \left (x -4\right )+x \right )}\) | \(30\) |
risch | \(-\frac {3 x}{\ln \left (x \right )}-\frac {3 \,{\mathrm e} x^{2}}{\ln \left (x \right ) \left (-\ln \left (x -4\right )+x \right )}\) | \(30\) |
parts | \(-\frac {3 x}{\ln \left (x \right )}-\frac {3 \,{\mathrm e} x^{2}}{\ln \left (x \right ) \left (-\ln \left (x -4\right )+x \right )}\) | \(30\) |
parallelrisch | \(-\frac {3 x^{2} {\mathrm e}+3 x^{2}-3 x \ln \left (x -4\right )}{\left (-\ln \left (x -4\right )+x \right ) \ln \left (x \right )}\) | \(37\) |
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Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {-12 x^2+3 x^3+e \left (-12 x^2+3 x^3\right )+\left (24 x-6 x^2+e \left (12 x-3 x^2\right )\right ) \log (-4+x)+(-12+3 x) \log ^2(-4+x)+\left (12 x^2-3 x^3+e \left (9 x^2-3 x^3\right )+\left (-24 x+6 x^2+e \left (-24 x+6 x^2\right )\right ) \log (-4+x)+(12-3 x) \log ^2(-4+x)\right ) \log (x)}{\left (-4 x^2+x^3+\left (8 x-2 x^2\right ) \log (-4+x)+(-4+x) \log ^2(-4+x)\right ) \log ^2(x)} \, dx=-\frac {3 \, {\left (x^{2} e + x^{2} - x \log \left (x - 4\right )\right )}}{{\left (x - \log \left (x - 4\right )\right )} \log \left (x\right )} \]
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Time = 0.11 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.23 \[ \int \frac {-12 x^2+3 x^3+e \left (-12 x^2+3 x^3\right )+\left (24 x-6 x^2+e \left (12 x-3 x^2\right )\right ) \log (-4+x)+(-12+3 x) \log ^2(-4+x)+\left (12 x^2-3 x^3+e \left (9 x^2-3 x^3\right )+\left (-24 x+6 x^2+e \left (-24 x+6 x^2\right )\right ) \log (-4+x)+(12-3 x) \log ^2(-4+x)\right ) \log (x)}{\left (-4 x^2+x^3+\left (8 x-2 x^2\right ) \log (-4+x)+(-4+x) \log ^2(-4+x)\right ) \log ^2(x)} \, dx=\frac {3 e x^{2}}{- x \log {\left (x \right )} + \log {\left (x \right )} \log {\left (x - 4 \right )}} - \frac {3 x}{\log {\left (x \right )}} \]
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Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {-12 x^2+3 x^3+e \left (-12 x^2+3 x^3\right )+\left (24 x-6 x^2+e \left (12 x-3 x^2\right )\right ) \log (-4+x)+(-12+3 x) \log ^2(-4+x)+\left (12 x^2-3 x^3+e \left (9 x^2-3 x^3\right )+\left (-24 x+6 x^2+e \left (-24 x+6 x^2\right )\right ) \log (-4+x)+(12-3 x) \log ^2(-4+x)\right ) \log (x)}{\left (-4 x^2+x^3+\left (8 x-2 x^2\right ) \log (-4+x)+(-4+x) \log ^2(-4+x)\right ) \log ^2(x)} \, dx=-\frac {3 \, {\left (x^{2} {\left (e + 1\right )} - x \log \left (x - 4\right )\right )}}{x \log \left (x\right ) - \log \left (x - 4\right ) \log \left (x\right )} \]
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Time = 0.27 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.55 \[ \int \frac {-12 x^2+3 x^3+e \left (-12 x^2+3 x^3\right )+\left (24 x-6 x^2+e \left (12 x-3 x^2\right )\right ) \log (-4+x)+(-12+3 x) \log ^2(-4+x)+\left (12 x^2-3 x^3+e \left (9 x^2-3 x^3\right )+\left (-24 x+6 x^2+e \left (-24 x+6 x^2\right )\right ) \log (-4+x)+(12-3 x) \log ^2(-4+x)\right ) \log (x)}{\left (-4 x^2+x^3+\left (8 x-2 x^2\right ) \log (-4+x)+(-4+x) \log ^2(-4+x)\right ) \log ^2(x)} \, dx=-\frac {3 \, {\left (x^{2} e + x^{2} - x \log \left (x - 4\right )\right )}}{x \log \left (x\right ) - \log \left (x - 4\right ) \log \left (x\right )} \]
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Time = 10.22 (sec) , antiderivative size = 402, normalized size of antiderivative = 18.27 \[ \int \frac {-12 x^2+3 x^3+e \left (-12 x^2+3 x^3\right )+\left (24 x-6 x^2+e \left (12 x-3 x^2\right )\right ) \log (-4+x)+(-12+3 x) \log ^2(-4+x)+\left (12 x^2-3 x^3+e \left (9 x^2-3 x^3\right )+\left (-24 x+6 x^2+e \left (-24 x+6 x^2\right )\right ) \log (-4+x)+(12-3 x) \log ^2(-4+x)\right ) \log (x)}{\left (-4 x^2+x^3+\left (8 x-2 x^2\right ) \log (-4+x)+(-4+x) \log ^2(-4+x)\right ) \log ^2(x)} \, dx=\frac {\frac {3\,x\,\ln \left (x\right )\,\left (75\,x-100\,\mathrm {e}+60\,x\,\mathrm {e}-15\,x^2\,\mathrm {e}+x^3\,\mathrm {e}-15\,x^2+x^3-125\right )}{2\,{\left (x-5\right )}^3}-\frac {3\,x\,\left (20\,\mathrm {e}-10\,x-8\,x\,\mathrm {e}+x^2\,\mathrm {e}+x^2+25\right )}{2\,{\left (x-5\right )}^2}+\frac {3\,x\,{\ln \left (x\right )}^2\,\left (75\,x-200\,\mathrm {e}+160\,x\,\mathrm {e}-30\,x^2\,\mathrm {e}+2\,x^3\,\mathrm {e}-15\,x^2+x^3-125\right )}{2\,{\left (x-5\right )}^3}}{\ln \left (x\right )}+\frac {\frac {3\,x^2\,\left (4\,\mathrm {e}-x\,\mathrm {e}-3\,\mathrm {e}\,\ln \left (x\right )+x\,\mathrm {e}\,\ln \left (x\right )\right )}{{\ln \left (x\right )}^2\,\left (x-5\right )}+\frac {3\,\ln \left (x-4\right )\,\left (x\,\mathrm {e}-2\,x\,\mathrm {e}\,\ln \left (x\right )\right )\,\left (x-4\right )}{{\ln \left (x\right )}^2\,\left (x-5\right )}}{x-\ln \left (x-4\right )}+\ln \left (x\right )\,\left (45\,\mathrm {e}+\frac {45}{2}\right )-\frac {225\,x\,\mathrm {e}-75\,x^2\,\mathrm {e}}{2\,x^3-30\,x^2+150\,x-250}+\frac {\frac {3\,x\,\mathrm {e}\,\left (x-4\right )}{x-5}-\frac {3\,x\,\ln \left (x\right )\,\left (60\,\mathrm {e}-10\,x-28\,x\,\mathrm {e}+3\,x^2\,\mathrm {e}+x^2+25\right )}{2\,{\left (x-5\right )}^2}+\frac {3\,x\,{\ln \left (x\right )}^2\,\left (40\,\mathrm {e}-10\,x-20\,x\,\mathrm {e}+2\,x^2\,\mathrm {e}+x^2+25\right )}{2\,{\left (x-5\right )}^2}}{{\ln \left (x\right )}^2}-x\,\left (\frac {9\,\mathrm {e}}{2}+3\right )+\frac {\ln \left (x\right )\,\left (\left (-3\,\mathrm {e}-\frac {3}{2}\right )\,x^4+\left (435\,\mathrm {e}+225\right )\,x^2+\left (-3075\,\mathrm {e}-1500\right )\,x+5625\,\mathrm {e}+\frac {5625}{2}\right )}{x^3-15\,x^2+75\,x-125} \]
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