Integrand size = 201, antiderivative size = 38 \[ \int \frac {1853 x-7252 x^2+10252 x^3-5952 x^4+1508 x^5+784 x^6-1856 x^7+768 x^8+\left (2890-7752 x+10312 x^2-10688 x^3+8296 x^4-4448 x^5+1920 x^6-512 x^7\right ) \log (x)+\left (408 x-3784 x^2+8880 x^3-9488 x^4+6848 x^5-3840 x^6+1024 x^7\right ) \log ^2(x)}{-500 x+200 x^2+940 x^3-464 x^4-448 x^5+256 x^6+\left (-1000 x+1400 x^2+480 x^3-1408 x^4+512 x^5\right ) \log ^2(x)+\left (-500 x+1200 x^2-960 x^3+256 x^4\right ) \log ^4(x)} \, dx=\frac {\left (-2+\frac {1}{2 \left (\frac {5-x}{3}-x\right )}-x^2\right )^2}{1+x+\log ^2(x)} \]
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\[ \int \frac {1853 x-7252 x^2+10252 x^3-5952 x^4+1508 x^5+784 x^6-1856 x^7+768 x^8+\left (2890-7752 x+10312 x^2-10688 x^3+8296 x^4-4448 x^5+1920 x^6-512 x^7\right ) \log (x)+\left (408 x-3784 x^2+8880 x^3-9488 x^4+6848 x^5-3840 x^6+1024 x^7\right ) \log ^2(x)}{-500 x+200 x^2+940 x^3-464 x^4-448 x^5+256 x^6+\left (-1000 x+1400 x^2+480 x^3-1408 x^4+512 x^5\right ) \log ^2(x)+\left (-500 x+1200 x^2-960 x^3+256 x^4\right ) \log ^4(x)} \, dx=\int \frac {1853 x-7252 x^2+10252 x^3-5952 x^4+1508 x^5+784 x^6-1856 x^7+768 x^8+\left (2890-7752 x+10312 x^2-10688 x^3+8296 x^4-4448 x^5+1920 x^6-512 x^7\right ) \log (x)+\left (408 x-3784 x^2+8880 x^3-9488 x^4+6848 x^5-3840 x^6+1024 x^7\right ) \log ^2(x)}{-500 x+200 x^2+940 x^3-464 x^4-448 x^5+256 x^6+\left (-1000 x+1400 x^2+480 x^3-1408 x^4+512 x^5\right ) \log ^2(x)+\left (-500 x+1200 x^2-960 x^3+256 x^4\right ) \log ^4(x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (17-16 x+10 x^2-8 x^3\right ) \left (x \left (-109+324 x-234 x^2-112 x^3+96 x^4\right )-2 \left (85-148 x+114 x^2-80 x^3+32 x^4\right ) \log (x)+8 x \left (-3+25 x-40 x^2+16 x^3\right ) \log ^2(x)\right )}{4 (5-4 x)^3 x \left (1+x+\log ^2(x)\right )^2} \, dx \\ & = \frac {1}{4} \int \frac {\left (17-16 x+10 x^2-8 x^3\right ) \left (x \left (-109+324 x-234 x^2-112 x^3+96 x^4\right )-2 \left (85-148 x+114 x^2-80 x^3+32 x^4\right ) \log (x)+8 x \left (-3+25 x-40 x^2+16 x^3\right ) \log ^2(x)\right )}{(5-4 x)^3 x \left (1+x+\log ^2(x)\right )^2} \, dx \\ & = \frac {1}{4} \int \left (-\frac {\left (-17+16 x-10 x^2+8 x^3\right )^2 (x+2 \log (x))}{x (-5+4 x)^2 \left (1+x+\log ^2(x)\right )^2}+\frac {8 \left (51-473 x+1110 x^2-1186 x^3+856 x^4-480 x^5+128 x^6\right )}{(-5+4 x)^3 \left (1+x+\log ^2(x)\right )}\right ) \, dx \\ & = -\left (\frac {1}{4} \int \frac {\left (-17+16 x-10 x^2+8 x^3\right )^2 (x+2 \log (x))}{x (-5+4 x)^2 \left (1+x+\log ^2(x)\right )^2} \, dx\right )+2 \int \frac {51-473 x+1110 x^2-1186 x^3+856 x^4-480 x^5+128 x^6}{(-5+4 x)^3 \left (1+x+\log ^2(x)\right )} \, dx \\ & = -\left (\frac {1}{4} \int \left (\frac {3 (x+2 \log (x))}{\left (1+x+\log ^2(x)\right )^2}+\frac {289 (x+2 \log (x))}{25 x \left (1+x+\log ^2(x)\right )^2}+\frac {16 x (x+2 \log (x))}{\left (1+x+\log ^2(x)\right )^2}+\frac {4 x^3 (x+2 \log (x))}{\left (1+x+\log ^2(x)\right )^2}+\frac {36 (x+2 \log (x))}{5 (-5+4 x)^2 \left (1+x+\log ^2(x)\right )^2}+\frac {819 (x+2 \log (x))}{25 (-5+4 x) \left (1+x+\log ^2(x)\right )^2}\right ) \, dx\right )+2 \int \left (\frac {3}{8 \left (1+x+\log ^2(x)\right )}+\frac {4 x}{1+x+\log ^2(x)}+\frac {2 x^3}{1+x+\log ^2(x)}-\frac {9}{(-5+4 x)^3 \left (1+x+\log ^2(x)\right )}-\frac {171}{8 (-5+4 x)^2 \left (1+x+\log ^2(x)\right )}\right ) \, dx \\ & = -\left (\frac {3}{4} \int \frac {x+2 \log (x)}{\left (1+x+\log ^2(x)\right )^2} \, dx\right )+\frac {3}{4} \int \frac {1}{1+x+\log ^2(x)} \, dx-\frac {9}{5} \int \frac {x+2 \log (x)}{(-5+4 x)^2 \left (1+x+\log ^2(x)\right )^2} \, dx-\frac {289}{100} \int \frac {x+2 \log (x)}{x \left (1+x+\log ^2(x)\right )^2} \, dx-4 \int \frac {x (x+2 \log (x))}{\left (1+x+\log ^2(x)\right )^2} \, dx+4 \int \frac {x^3}{1+x+\log ^2(x)} \, dx+8 \int \frac {x}{1+x+\log ^2(x)} \, dx-\frac {819}{100} \int \frac {x+2 \log (x)}{(-5+4 x) \left (1+x+\log ^2(x)\right )^2} \, dx-18 \int \frac {1}{(-5+4 x)^3 \left (1+x+\log ^2(x)\right )} \, dx-\frac {171}{4} \int \frac {1}{(-5+4 x)^2 \left (1+x+\log ^2(x)\right )} \, dx-\int \frac {x^3 (x+2 \log (x))}{\left (1+x+\log ^2(x)\right )^2} \, dx \\ & = \frac {289}{100 \left (1+x+\log ^2(x)\right )}+\frac {3}{4} \int \frac {1}{1+x+\log ^2(x)} \, dx-\frac {3}{4} \int \left (\frac {x}{\left (1+x+\log ^2(x)\right )^2}+\frac {2 \log (x)}{\left (1+x+\log ^2(x)\right )^2}\right ) \, dx-\frac {9}{5} \int \left (\frac {x}{(-5+4 x)^2 \left (1+x+\log ^2(x)\right )^2}+\frac {2 \log (x)}{(-5+4 x)^2 \left (1+x+\log ^2(x)\right )^2}\right ) \, dx+4 \int \frac {x^3}{1+x+\log ^2(x)} \, dx-4 \int \left (\frac {x^2}{\left (1+x+\log ^2(x)\right )^2}+\frac {2 x \log (x)}{\left (1+x+\log ^2(x)\right )^2}\right ) \, dx+8 \int \frac {x}{1+x+\log ^2(x)} \, dx-\frac {819}{100} \int \left (\frac {x}{(-5+4 x) \left (1+x+\log ^2(x)\right )^2}+\frac {2 \log (x)}{(-5+4 x) \left (1+x+\log ^2(x)\right )^2}\right ) \, dx-18 \int \frac {1}{(-5+4 x)^3 \left (1+x+\log ^2(x)\right )} \, dx-\frac {171}{4} \int \frac {1}{(-5+4 x)^2 \left (1+x+\log ^2(x)\right )} \, dx-\int \left (\frac {x^4}{\left (1+x+\log ^2(x)\right )^2}+\frac {2 x^3 \log (x)}{\left (1+x+\log ^2(x)\right )^2}\right ) \, dx \\ & = \frac {289}{100 \left (1+x+\log ^2(x)\right )}-\frac {3}{4} \int \frac {x}{\left (1+x+\log ^2(x)\right )^2} \, dx+\frac {3}{4} \int \frac {1}{1+x+\log ^2(x)} \, dx-\frac {3}{2} \int \frac {\log (x)}{\left (1+x+\log ^2(x)\right )^2} \, dx-\frac {9}{5} \int \frac {x}{(-5+4 x)^2 \left (1+x+\log ^2(x)\right )^2} \, dx-2 \int \frac {x^3 \log (x)}{\left (1+x+\log ^2(x)\right )^2} \, dx-\frac {18}{5} \int \frac {\log (x)}{(-5+4 x)^2 \left (1+x+\log ^2(x)\right )^2} \, dx-4 \int \frac {x^2}{\left (1+x+\log ^2(x)\right )^2} \, dx+4 \int \frac {x^3}{1+x+\log ^2(x)} \, dx-8 \int \frac {x \log (x)}{\left (1+x+\log ^2(x)\right )^2} \, dx+8 \int \frac {x}{1+x+\log ^2(x)} \, dx-\frac {819}{100} \int \frac {x}{(-5+4 x) \left (1+x+\log ^2(x)\right )^2} \, dx-\frac {819}{50} \int \frac {\log (x)}{(-5+4 x) \left (1+x+\log ^2(x)\right )^2} \, dx-18 \int \frac {1}{(-5+4 x)^3 \left (1+x+\log ^2(x)\right )} \, dx-\frac {171}{4} \int \frac {1}{(-5+4 x)^2 \left (1+x+\log ^2(x)\right )} \, dx-\int \frac {x^4}{\left (1+x+\log ^2(x)\right )^2} \, dx \\ & = \frac {289}{100 \left (1+x+\log ^2(x)\right )}-\frac {3}{4} \int \frac {x}{\left (1+x+\log ^2(x)\right )^2} \, dx+\frac {3}{4} \int \frac {1}{1+x+\log ^2(x)} \, dx-\frac {3}{2} \int \frac {\log (x)}{\left (1+x+\log ^2(x)\right )^2} \, dx-\frac {9}{5} \int \left (\frac {5}{4 (-5+4 x)^2 \left (1+x+\log ^2(x)\right )^2}+\frac {1}{4 (-5+4 x) \left (1+x+\log ^2(x)\right )^2}\right ) \, dx-2 \int \frac {x^3 \log (x)}{\left (1+x+\log ^2(x)\right )^2} \, dx-\frac {18}{5} \int \frac {\log (x)}{(-5+4 x)^2 \left (1+x+\log ^2(x)\right )^2} \, dx-4 \int \frac {x^2}{\left (1+x+\log ^2(x)\right )^2} \, dx+4 \int \frac {x^3}{1+x+\log ^2(x)} \, dx-8 \int \frac {x \log (x)}{\left (1+x+\log ^2(x)\right )^2} \, dx+8 \int \frac {x}{1+x+\log ^2(x)} \, dx-\frac {819}{100} \int \left (\frac {1}{4 \left (1+x+\log ^2(x)\right )^2}+\frac {5}{4 (-5+4 x) \left (1+x+\log ^2(x)\right )^2}\right ) \, dx-\frac {819}{50} \int \frac {\log (x)}{(-5+4 x) \left (1+x+\log ^2(x)\right )^2} \, dx-18 \int \frac {1}{(-5+4 x)^3 \left (1+x+\log ^2(x)\right )} \, dx-\frac {171}{4} \int \frac {1}{(-5+4 x)^2 \left (1+x+\log ^2(x)\right )} \, dx-\int \frac {x^4}{\left (1+x+\log ^2(x)\right )^2} \, dx \\ & = \frac {289}{100 \left (1+x+\log ^2(x)\right )}-\frac {9}{20} \int \frac {1}{(-5+4 x) \left (1+x+\log ^2(x)\right )^2} \, dx-\frac {3}{4} \int \frac {x}{\left (1+x+\log ^2(x)\right )^2} \, dx+\frac {3}{4} \int \frac {1}{1+x+\log ^2(x)} \, dx-\frac {3}{2} \int \frac {\log (x)}{\left (1+x+\log ^2(x)\right )^2} \, dx-2 \int \frac {x^3 \log (x)}{\left (1+x+\log ^2(x)\right )^2} \, dx-\frac {819}{400} \int \frac {1}{\left (1+x+\log ^2(x)\right )^2} \, dx-\frac {9}{4} \int \frac {1}{(-5+4 x)^2 \left (1+x+\log ^2(x)\right )^2} \, dx-\frac {18}{5} \int \frac {\log (x)}{(-5+4 x)^2 \left (1+x+\log ^2(x)\right )^2} \, dx-4 \int \frac {x^2}{\left (1+x+\log ^2(x)\right )^2} \, dx+4 \int \frac {x^3}{1+x+\log ^2(x)} \, dx-8 \int \frac {x \log (x)}{\left (1+x+\log ^2(x)\right )^2} \, dx+8 \int \frac {x}{1+x+\log ^2(x)} \, dx-\frac {819}{80} \int \frac {1}{(-5+4 x) \left (1+x+\log ^2(x)\right )^2} \, dx-\frac {819}{50} \int \frac {\log (x)}{(-5+4 x) \left (1+x+\log ^2(x)\right )^2} \, dx-18 \int \frac {1}{(-5+4 x)^3 \left (1+x+\log ^2(x)\right )} \, dx-\frac {171}{4} \int \frac {1}{(-5+4 x)^2 \left (1+x+\log ^2(x)\right )} \, dx-\int \frac {x^4}{\left (1+x+\log ^2(x)\right )^2} \, dx \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.97 \[ \int \frac {1853 x-7252 x^2+10252 x^3-5952 x^4+1508 x^5+784 x^6-1856 x^7+768 x^8+\left (2890-7752 x+10312 x^2-10688 x^3+8296 x^4-4448 x^5+1920 x^6-512 x^7\right ) \log (x)+\left (408 x-3784 x^2+8880 x^3-9488 x^4+6848 x^5-3840 x^6+1024 x^7\right ) \log ^2(x)}{-500 x+200 x^2+940 x^3-464 x^4-448 x^5+256 x^6+\left (-1000 x+1400 x^2+480 x^3-1408 x^4+512 x^5\right ) \log ^2(x)+\left (-500 x+1200 x^2-960 x^3+256 x^4\right ) \log ^4(x)} \, dx=\frac {\left (17-16 x+10 x^2-8 x^3\right )^2}{4 (5-4 x)^2 \left (1+x+\log ^2(x)\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(53\) vs. \(2(26)=52\).
Time = 2.05 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.42
method | result | size |
risch | \(\frac {64 x^{6}-160 x^{5}+356 x^{4}-592 x^{3}+596 x^{2}-544 x +289}{4 \left (16 x^{2}-40 x +25\right ) \left (x +\ln \left (x \right )^{2}+1\right )}\) | \(54\) |
default | \(\frac {714-799 x +425 \ln \left (x \right )^{2}+64 x^{6}-320 x^{3}+272 x^{2} \ln \left (x \right )^{2}-680 x \ln \left (x \right )^{2}+188 x^{2}+356 x^{4}-160 x^{5}}{4 \left (-5+4 x \right )^{2} \left (x +\ln \left (x \right )^{2}+1\right )}\) | \(71\) |
parallelrisch | \(\frac {1024 x^{6}-2560 x^{5}+5696 x^{4}-9472 x^{3}+9536 x^{2}-8704 x +4624}{1024 x^{2} \ln \left (x \right )^{2}+1024 x^{3}-2560 x \ln \left (x \right )^{2}-1536 x^{2}+1600 \ln \left (x \right )^{2}-960 x +1600}\) | \(72\) |
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Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (29) = 58\).
Time = 0.25 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.68 \[ \int \frac {1853 x-7252 x^2+10252 x^3-5952 x^4+1508 x^5+784 x^6-1856 x^7+768 x^8+\left (2890-7752 x+10312 x^2-10688 x^3+8296 x^4-4448 x^5+1920 x^6-512 x^7\right ) \log (x)+\left (408 x-3784 x^2+8880 x^3-9488 x^4+6848 x^5-3840 x^6+1024 x^7\right ) \log ^2(x)}{-500 x+200 x^2+940 x^3-464 x^4-448 x^5+256 x^6+\left (-1000 x+1400 x^2+480 x^3-1408 x^4+512 x^5\right ) \log ^2(x)+\left (-500 x+1200 x^2-960 x^3+256 x^4\right ) \log ^4(x)} \, dx=\frac {64 \, x^{6} - 160 \, x^{5} + 356 \, x^{4} - 592 \, x^{3} + 596 \, x^{2} - 544 \, x + 289}{4 \, {\left (16 \, x^{3} + {\left (16 \, x^{2} - 40 \, x + 25\right )} \log \left (x\right )^{2} - 24 \, x^{2} - 15 \, x + 25\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (24) = 48\).
Time = 0.14 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.58 \[ \int \frac {1853 x-7252 x^2+10252 x^3-5952 x^4+1508 x^5+784 x^6-1856 x^7+768 x^8+\left (2890-7752 x+10312 x^2-10688 x^3+8296 x^4-4448 x^5+1920 x^6-512 x^7\right ) \log (x)+\left (408 x-3784 x^2+8880 x^3-9488 x^4+6848 x^5-3840 x^6+1024 x^7\right ) \log ^2(x)}{-500 x+200 x^2+940 x^3-464 x^4-448 x^5+256 x^6+\left (-1000 x+1400 x^2+480 x^3-1408 x^4+512 x^5\right ) \log ^2(x)+\left (-500 x+1200 x^2-960 x^3+256 x^4\right ) \log ^4(x)} \, dx=\frac {64 x^{6} - 160 x^{5} + 356 x^{4} - 592 x^{3} + 596 x^{2} - 544 x + 289}{64 x^{3} - 96 x^{2} - 60 x + \left (64 x^{2} - 160 x + 100\right ) \log {\left (x \right )}^{2} + 100} \]
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Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (29) = 58\).
Time = 0.26 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.68 \[ \int \frac {1853 x-7252 x^2+10252 x^3-5952 x^4+1508 x^5+784 x^6-1856 x^7+768 x^8+\left (2890-7752 x+10312 x^2-10688 x^3+8296 x^4-4448 x^5+1920 x^6-512 x^7\right ) \log (x)+\left (408 x-3784 x^2+8880 x^3-9488 x^4+6848 x^5-3840 x^6+1024 x^7\right ) \log ^2(x)}{-500 x+200 x^2+940 x^3-464 x^4-448 x^5+256 x^6+\left (-1000 x+1400 x^2+480 x^3-1408 x^4+512 x^5\right ) \log ^2(x)+\left (-500 x+1200 x^2-960 x^3+256 x^4\right ) \log ^4(x)} \, dx=\frac {64 \, x^{6} - 160 \, x^{5} + 356 \, x^{4} - 592 \, x^{3} + 596 \, x^{2} - 544 \, x + 289}{4 \, {\left (16 \, x^{3} + {\left (16 \, x^{2} - 40 \, x + 25\right )} \log \left (x\right )^{2} - 24 \, x^{2} - 15 \, x + 25\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (29) = 58\).
Time = 0.35 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.87 \[ \int \frac {1853 x-7252 x^2+10252 x^3-5952 x^4+1508 x^5+784 x^6-1856 x^7+768 x^8+\left (2890-7752 x+10312 x^2-10688 x^3+8296 x^4-4448 x^5+1920 x^6-512 x^7\right ) \log (x)+\left (408 x-3784 x^2+8880 x^3-9488 x^4+6848 x^5-3840 x^6+1024 x^7\right ) \log ^2(x)}{-500 x+200 x^2+940 x^3-464 x^4-448 x^5+256 x^6+\left (-1000 x+1400 x^2+480 x^3-1408 x^4+512 x^5\right ) \log ^2(x)+\left (-500 x+1200 x^2-960 x^3+256 x^4\right ) \log ^4(x)} \, dx=\frac {64 \, x^{6} - 160 \, x^{5} + 356 \, x^{4} - 592 \, x^{3} + 596 \, x^{2} - 544 \, x + 289}{4 \, {\left (16 \, x^{2} \log \left (x\right )^{2} + 16 \, x^{3} - 40 \, x \log \left (x\right )^{2} - 24 \, x^{2} + 25 \, \log \left (x\right )^{2} - 15 \, x + 25\right )}} \]
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Time = 14.34 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.13 \[ \int \frac {1853 x-7252 x^2+10252 x^3-5952 x^4+1508 x^5+784 x^6-1856 x^7+768 x^8+\left (2890-7752 x+10312 x^2-10688 x^3+8296 x^4-4448 x^5+1920 x^6-512 x^7\right ) \log (x)+\left (408 x-3784 x^2+8880 x^3-9488 x^4+6848 x^5-3840 x^6+1024 x^7\right ) \log ^2(x)}{-500 x+200 x^2+940 x^3-464 x^4-448 x^5+256 x^6+\left (-1000 x+1400 x^2+480 x^3-1408 x^4+512 x^5\right ) \log ^2(x)+\left (-500 x+1200 x^2-960 x^3+256 x^4\right ) \log ^4(x)} \, dx=-\frac {-64\,x^{10}-16\,x^9+148\,x^8-227\,x^7+588\,x^6+93\,x^5-1157\,x^4+\frac {6565\,x^3}{4}-2431\,x^2+1445\,x}{{\left (4\,x-5\right )}^3\,\left ({\ln \left (x\right )}^2+x+1\right )\,\left (x^3+4\,x^2+4\,x\right )} \]
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