Integrand size = 111, antiderivative size = 26 \[ \int \frac {300 x^2-440 x^3-20 x^4+160 x^5+e^3 \left (-25 x-85 x^2+35 x^3+40 x^4\right )+\left (100-120 x-60 x^2+80 x^3+e^3 \left (-25+5 x+20 x^2\right )\right ) \log \left (\frac {4-e^3-4 x}{-1+x}\right )}{4+e^3 (-1+x)-8 x+4 x^2} \, dx=5 x (5+2 x) \left (x^2+\log \left (-4+\frac {e^3}{1-x}\right )\right ) \]
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Leaf count is larger than twice the leaf count of optimal. \(101\) vs. \(2(26)=52\).
Time = 1.22 (sec) , antiderivative size = 101, normalized size of antiderivative = 3.88, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {6820, 12, 6874, 1626, 2548, 84} \[ \int \frac {300 x^2-440 x^3-20 x^4+160 x^5+e^3 \left (-25 x-85 x^2+35 x^3+40 x^4\right )+\left (100-120 x-60 x^2+80 x^3+e^3 \left (-25+5 x+20 x^2\right )\right ) \log \left (\frac {4-e^3-4 x}{-1+x}\right )}{4+e^3 (-1+x)-8 x+4 x^2} \, dx=10 x^4+25 x^3+\frac {5}{8} \left (56-18 e^3+e^6\right ) \log \left (-4 x-e^3+4\right )-\frac {5}{8} \left (9-e^3\right )^2 \log \left (-4 x-e^3+4\right )+\frac {5}{8} (4 x+5)^2 \log \left (-\frac {-4 x-e^3+4}{1-x}\right )+\frac {125}{8} \log (1-x) \]
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Rule 12
Rule 84
Rule 1626
Rule 2548
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {5 \left (-\frac {x \left (4 (-1+x)^2 x (15+8 x)+e^3 \left (-5-17 x+7 x^2+8 x^3\right )\right )}{-4+e^3+4 x}-\left (-5+x+4 x^2\right ) \log \left (-\frac {-4+e^3+4 x}{-1+x}\right )\right )}{1-x} \, dx \\ & = 5 \int \frac {-\frac {x \left (4 (-1+x)^2 x (15+8 x)+e^3 \left (-5-17 x+7 x^2+8 x^3\right )\right )}{-4+e^3+4 x}-\left (-5+x+4 x^2\right ) \log \left (-\frac {-4+e^3+4 x}{-1+x}\right )}{1-x} \, dx \\ & = 5 \int \left (\frac {x \left (-5 e^3+\left (60-17 e^3\right ) x-\left (88-7 e^3\right ) x^2-4 \left (1-2 e^3\right ) x^3+32 x^4\right )}{\left (4-e^3-4 x\right ) (1-x)}+(5+4 x) \log \left (-\frac {-4+e^3+4 x}{-1+x}\right )\right ) \, dx \\ & = 5 \int \frac {x \left (-5 e^3+\left (60-17 e^3\right ) x-\left (88-7 e^3\right ) x^2-4 \left (1-2 e^3\right ) x^3+32 x^4\right )}{\left (4-e^3-4 x\right ) (1-x)} \, dx+5 \int (5+4 x) \log \left (-\frac {-4+e^3+4 x}{-1+x}\right ) \, dx \\ & = \frac {5}{8} (5+4 x)^2 \log \left (-\frac {4-e^3-4 x}{1-x}\right )+5 \int \left (-\frac {e^3}{2}-\frac {7}{-1+x}+15 x^2+8 x^3+\frac {56-18 e^3+e^6}{2 \left (-4+e^3+4 x\right )}\right ) \, dx+\frac {1}{8} \left (5 e^3\right ) \int \frac {(5+4 x)^2}{(-1+x) \left (-4+e^3+4 x\right )} \, dx \\ & = -\frac {5 e^3 x}{2}+25 x^3+10 x^4+\frac {5}{8} \left (56-18 e^3+e^6\right ) \log \left (4-e^3-4 x\right )+\frac {5}{8} (5+4 x)^2 \log \left (-\frac {4-e^3-4 x}{1-x}\right )-35 \log (1-x)+\frac {1}{8} \left (5 e^3\right ) \int \left (4+\frac {81}{e^3 (-1+x)}-\frac {4 \left (-9+e^3\right )^2}{e^3 \left (-4+e^3+4 x\right )}\right ) \, dx \\ & = 25 x^3+10 x^4-\frac {5}{8} \left (9-e^3\right )^2 \log \left (4-e^3-4 x\right )+\frac {5}{8} \left (56-18 e^3+e^6\right ) \log \left (4-e^3-4 x\right )+\frac {5}{8} (5+4 x)^2 \log \left (-\frac {4-e^3-4 x}{1-x}\right )+\frac {125}{8} \log (1-x) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(162\) vs. \(2(26)=52\).
Time = 0.14 (sec) , antiderivative size = 162, normalized size of antiderivative = 6.23 \[ \int \frac {300 x^2-440 x^3-20 x^4+160 x^5+e^3 \left (-25 x-85 x^2+35 x^3+40 x^4\right )+\left (100-120 x-60 x^2+80 x^3+e^3 \left (-25+5 x+20 x^2\right )\right ) \log \left (\frac {4-e^3-4 x}{-1+x}\right )}{4+e^3 (-1+x)-8 x+4 x^2} \, dx=5 \left (5 x^3+2 x^4+5 \log \left (4-e^3-4 x\right )-\frac {5}{4} e^3 \log \left (4-e^3-4 x\right )-5 \log \left (-\frac {4-e^3-4 x}{1-x}\right )+\frac {5}{4} e^3 \log \left (-\frac {4-e^3-4 x}{1-x}\right )+5 x \log \left (-\frac {4-e^3-4 x}{1-x}\right )+2 x^2 \log \left (-\frac {4-e^3-4 x}{1-x}\right )-5 \log (1-x)+\frac {5}{4} e^3 \log (1-x)\right ) \]
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Time = 0.32 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.46
method | result | size |
risch | \(\left (10 x^{2}+25 x \right ) \ln \left (\frac {-{\mathrm e}^{3}-4 x +4}{-1+x}\right )+10 x^{4}+25 x^{3}\) | \(38\) |
norman | \(25 x^{3}+10 x^{4}+25 x \ln \left (\frac {-{\mathrm e}^{3}-4 x +4}{-1+x}\right )+10 x^{2} \ln \left (\frac {-{\mathrm e}^{3}-4 x +4}{-1+x}\right )\) | \(52\) |
parallelrisch | \(10 x^{4}-\frac {5 \,{\mathrm e}^{6} \ln \left (-1+x \right )}{8}+\frac {5 \,{\mathrm e}^{6} \ln \left (x +\frac {{\mathrm e}^{3}}{4}-1\right )}{8}-\frac {5 \,{\mathrm e}^{6} \ln \left (-\frac {{\mathrm e}^{3}+4 x -4}{-1+x}\right )}{8}+25 x^{3}+10 \ln \left (-\frac {{\mathrm e}^{3}+4 x -4}{-1+x}\right ) x^{2}+\frac {5 \,{\mathrm e}^{6}}{2}+\frac {5 \,{\mathrm e}^{3} \ln \left (-1+x \right )}{2}-\frac {5 \,{\mathrm e}^{3} \ln \left (x +\frac {{\mathrm e}^{3}}{4}-1\right )}{2}+\frac {5 \,{\mathrm e}^{3} \ln \left (-\frac {{\mathrm e}^{3}+4 x -4}{-1+x}\right )}{2}+90+25 \ln \left (-\frac {{\mathrm e}^{3}+4 x -4}{-1+x}\right ) x -\frac {65 \,{\mathrm e}^{3}}{2}+40 \ln \left (-1+x \right )-40 \ln \left (x +\frac {{\mathrm e}^{3}}{4}-1\right )+40 \ln \left (-\frac {{\mathrm e}^{3}+4 x -4}{-1+x}\right )\) | \(178\) |
parts | \(5 \,{\mathrm e}^{3} \left (-4 \,{\mathrm e}^{3} \left (-\frac {\left (-1+x \right ) {\mathrm e}^{-3}}{8}-\frac {\ln \left (-\frac {{\mathrm e}^{3}}{-1+x}\right )}{32}+\frac {\ln \left (-4-\frac {{\mathrm e}^{3}}{-1+x}\right ) \left (-4-\frac {{\mathrm e}^{3}}{-1+x}\right ) \left (-\frac {{\mathrm e}^{3}}{-1+x}+4\right ) \left (-1+x \right )^{2} {\mathrm e}^{-6}}{32}\right )-\frac {9 \ln \left (-\frac {{\mathrm e}^{3}}{-1+x}\right )}{4}-\frac {9 \ln \left (-4-\frac {{\mathrm e}^{3}}{-1+x}\right ) \left (-4-\frac {{\mathrm e}^{3}}{-1+x}\right ) \left (-1+x \right ) {\mathrm e}^{-3}}{4}\right )+10 x^{4}+25 x^{3}-\frac {5 x \,{\mathrm e}^{3}}{2}-35 \,{\mathrm e}^{3} {\mathrm e}^{-3} \ln \left (-1+x \right )+\frac {5 \left (-18 \,{\mathrm e}^{6}+{\mathrm e}^{6} {\mathrm e}^{3}+56 \,{\mathrm e}^{3}\right ) {\mathrm e}^{-3} \ln \left ({\mathrm e}^{3}+4 x -4\right )}{8}\) | \(194\) |
derivativedivides | \({\mathrm e}^{3} \left (-\frac {45 \ln \left (-\frac {{\mathrm e}^{3}}{-1+x}\right )}{4}-\frac {45 \ln \left (-4-\frac {{\mathrm e}^{3}}{-1+x}\right ) \left (-4-\frac {{\mathrm e}^{3}}{-1+x}\right ) \left (-1+x \right ) {\mathrm e}^{-3}}{4}-20 \,{\mathrm e}^{3} \left (-\frac {\left (-1+x \right ) {\mathrm e}^{-3}}{8}-\frac {\ln \left (-\frac {{\mathrm e}^{3}}{-1+x}\right )}{32}+\frac {\ln \left (-4-\frac {{\mathrm e}^{3}}{-1+x}\right ) \left (-4-\frac {{\mathrm e}^{3}}{-1+x}\right ) \left (-\frac {{\mathrm e}^{3}}{-1+x}+4\right ) \left (-1+x \right )^{2} {\mathrm e}^{-6}}{32}\right )-5 \,{\mathrm e}^{-3} \left (\left (-\frac {9 \,{\mathrm e}^{3}}{4}+\frac {{\mathrm e}^{6}}{8}\right ) \ln \left (-\frac {{\mathrm e}^{3}}{-1+x}\right )+\left (\frac {{\mathrm e}^{6}}{2}-23 \,{\mathrm e}^{3}\right ) \left (-1+x \right ) {\mathrm e}^{-3}-27 \,{\mathrm e}^{6} \left (-1+x \right )^{2} {\mathrm e}^{-6}-13 \,{\mathrm e}^{9} \left (-1+x \right )^{3} {\mathrm e}^{-9}-2 \,{\mathrm e}^{12} \left (-1+x \right )^{4} {\mathrm e}^{-12}+\left (\frac {9 \,{\mathrm e}^{3}}{4}-\frac {{\mathrm e}^{6}}{8}-7\right ) \ln \left (-4-\frac {{\mathrm e}^{3}}{-1+x}\right )\right )\right )\) | \(226\) |
default | \({\mathrm e}^{3} \left (-\frac {45 \ln \left (-\frac {{\mathrm e}^{3}}{-1+x}\right )}{4}-\frac {45 \ln \left (-4-\frac {{\mathrm e}^{3}}{-1+x}\right ) \left (-4-\frac {{\mathrm e}^{3}}{-1+x}\right ) \left (-1+x \right ) {\mathrm e}^{-3}}{4}-20 \,{\mathrm e}^{3} \left (-\frac {\left (-1+x \right ) {\mathrm e}^{-3}}{8}-\frac {\ln \left (-\frac {{\mathrm e}^{3}}{-1+x}\right )}{32}+\frac {\ln \left (-4-\frac {{\mathrm e}^{3}}{-1+x}\right ) \left (-4-\frac {{\mathrm e}^{3}}{-1+x}\right ) \left (-\frac {{\mathrm e}^{3}}{-1+x}+4\right ) \left (-1+x \right )^{2} {\mathrm e}^{-6}}{32}\right )-5 \,{\mathrm e}^{-3} \left (\left (-\frac {9 \,{\mathrm e}^{3}}{4}+\frac {{\mathrm e}^{6}}{8}\right ) \ln \left (-\frac {{\mathrm e}^{3}}{-1+x}\right )+\left (\frac {{\mathrm e}^{6}}{2}-23 \,{\mathrm e}^{3}\right ) \left (-1+x \right ) {\mathrm e}^{-3}-27 \,{\mathrm e}^{6} \left (-1+x \right )^{2} {\mathrm e}^{-6}-13 \,{\mathrm e}^{9} \left (-1+x \right )^{3} {\mathrm e}^{-9}-2 \,{\mathrm e}^{12} \left (-1+x \right )^{4} {\mathrm e}^{-12}+\left (\frac {9 \,{\mathrm e}^{3}}{4}-\frac {{\mathrm e}^{6}}{8}-7\right ) \ln \left (-4-\frac {{\mathrm e}^{3}}{-1+x}\right )\right )\right )\) | \(226\) |
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Time = 0.23 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.42 \[ \int \frac {300 x^2-440 x^3-20 x^4+160 x^5+e^3 \left (-25 x-85 x^2+35 x^3+40 x^4\right )+\left (100-120 x-60 x^2+80 x^3+e^3 \left (-25+5 x+20 x^2\right )\right ) \log \left (\frac {4-e^3-4 x}{-1+x}\right )}{4+e^3 (-1+x)-8 x+4 x^2} \, dx=10 \, x^{4} + 25 \, x^{3} + 5 \, {\left (2 \, x^{2} + 5 \, x\right )} \log \left (-\frac {4 \, x + e^{3} - 4}{x - 1}\right ) \]
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Time = 0.12 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {300 x^2-440 x^3-20 x^4+160 x^5+e^3 \left (-25 x-85 x^2+35 x^3+40 x^4\right )+\left (100-120 x-60 x^2+80 x^3+e^3 \left (-25+5 x+20 x^2\right )\right ) \log \left (\frac {4-e^3-4 x}{-1+x}\right )}{4+e^3 (-1+x)-8 x+4 x^2} \, dx=10 x^{4} + 25 x^{3} + \left (10 x^{2} + 25 x\right ) \log {\left (\frac {- 4 x - e^{3} + 4}{x - 1} \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 641 vs. \(2 (24) = 48\).
Time = 0.26 (sec) , antiderivative size = 641, normalized size of antiderivative = 24.65 \[ \int \frac {300 x^2-440 x^3-20 x^4+160 x^5+e^3 \left (-25 x-85 x^2+35 x^3+40 x^4\right )+\left (100-120 x-60 x^2+80 x^3+e^3 \left (-25+5 x+20 x^2\right )\right ) \log \left (\frac {4-e^3-4 x}{-1+x}\right )}{4+e^3 (-1+x)-8 x+4 x^2} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 546 vs. \(2 (24) = 48\).
Time = 0.39 (sec) , antiderivative size = 546, normalized size of antiderivative = 21.00 \[ \int \frac {300 x^2-440 x^3-20 x^4+160 x^5+e^3 \left (-25 x-85 x^2+35 x^3+40 x^4\right )+\left (100-120 x-60 x^2+80 x^3+e^3 \left (-25+5 x+20 x^2\right )\right ) \log \left (\frac {4-e^3-4 x}{-1+x}\right )}{4+e^3 (-1+x)-8 x+4 x^2} \, dx=\frac {5 \, {\left ({\left (e^{3} - 4\right )} e^{\left (-6\right )} + 4 \, e^{\left (-6\right )}\right )} {\left (\frac {2 \, {\left (4 \, x + e^{3} - 4\right )}^{2} e^{9} \log \left (-\frac {4 \, x + e^{3} - 4}{x - 1}\right )}{{\left (x - 1\right )}^{2}} - \frac {16 \, {\left (4 \, x + e^{3} - 4\right )} e^{9} \log \left (-\frac {4 \, x + e^{3} - 4}{x - 1}\right )}{x - 1} + 32 \, e^{9} \log \left (-\frac {4 \, x + e^{3} - 4}{x - 1}\right ) + \frac {9 \, {\left (4 \, x + e^{3} - 4\right )}^{3} e^{6} \log \left (-\frac {4 \, x + e^{3} - 4}{x - 1}\right )}{{\left (x - 1\right )}^{3}} - \frac {108 \, {\left (4 \, x + e^{3} - 4\right )}^{2} e^{6} \log \left (-\frac {4 \, x + e^{3} - 4}{x - 1}\right )}{{\left (x - 1\right )}^{2}} + \frac {432 \, {\left (4 \, x + e^{3} - 4\right )} e^{6} \log \left (-\frac {4 \, x + e^{3} - 4}{x - 1}\right )}{x - 1} - 576 \, e^{6} \log \left (-\frac {4 \, x + e^{3} - 4}{x - 1}\right ) + \frac {7 \, {\left (4 \, x + e^{3} - 4\right )}^{4} e^{3} \log \left (-\frac {4 \, x + e^{3} - 4}{x - 1}\right )}{{\left (x - 1\right )}^{4}} - \frac {112 \, {\left (4 \, x + e^{3} - 4\right )}^{3} e^{3} \log \left (-\frac {4 \, x + e^{3} - 4}{x - 1}\right )}{{\left (x - 1\right )}^{3}} + \frac {672 \, {\left (4 \, x + e^{3} - 4\right )}^{2} e^{3} \log \left (-\frac {4 \, x + e^{3} - 4}{x - 1}\right )}{{\left (x - 1\right )}^{2}} - \frac {1792 \, {\left (4 \, x + e^{3} - 4\right )} e^{3} \log \left (-\frac {4 \, x + e^{3} - 4}{x - 1}\right )}{x - 1} + 1792 \, e^{3} \log \left (-\frac {4 \, x + e^{3} - 4}{x - 1}\right ) + \frac {13 \, {\left (4 \, x + e^{3} - 4\right )} e^{12}}{x - 1} + \frac {27 \, {\left (4 \, x + e^{3} - 4\right )}^{2} e^{9}}{{\left (x - 1\right )}^{2}} - \frac {216 \, {\left (4 \, x + e^{3} - 4\right )} e^{9}}{x - 1} + \frac {23 \, {\left (4 \, x + e^{3} - 4\right )}^{3} e^{6}}{{\left (x - 1\right )}^{3}} - \frac {276 \, {\left (4 \, x + e^{3} - 4\right )}^{2} e^{6}}{{\left (x - 1\right )}^{2}} + \frac {1104 \, {\left (4 \, x + e^{3} - 4\right )} e^{6}}{x - 1} + 2 \, e^{15} - 52 \, e^{12} + 432 \, e^{9} - 1472 \, e^{6}\right )}}{\frac {{\left (4 \, x + e^{3} - 4\right )}^{4}}{{\left (x - 1\right )}^{4}} - \frac {16 \, {\left (4 \, x + e^{3} - 4\right )}^{3}}{{\left (x - 1\right )}^{3}} + \frac {96 \, {\left (4 \, x + e^{3} - 4\right )}^{2}}{{\left (x - 1\right )}^{2}} - \frac {256 \, {\left (4 \, x + e^{3} - 4\right )}}{x - 1} + 256} \]
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Time = 1.37 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {300 x^2-440 x^3-20 x^4+160 x^5+e^3 \left (-25 x-85 x^2+35 x^3+40 x^4\right )+\left (100-120 x-60 x^2+80 x^3+e^3 \left (-25+5 x+20 x^2\right )\right ) \log \left (\frac {4-e^3-4 x}{-1+x}\right )}{4+e^3 (-1+x)-8 x+4 x^2} \, dx=5\,x\,\left (2\,x+5\right )\,\left (\ln \left (-\frac {4\,x+{\mathrm {e}}^3-4}{x-1}\right )+x^2\right ) \]
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