Integrand size = 121, antiderivative size = 31 \[ \int \frac {-8750+5250 x-1050 x^2+1820 x^3-1050 x^4+210 x^5-14 x^6+\left (-50+10 x-20 x^3+4 x^4\right ) \log \left (\frac {5-x^3}{x}\right )+\left (25+5 x-5 x^3-x^4\right ) \log ^2\left (\frac {5-x^3}{x}\right )}{7500-4500 x+900 x^2-1560 x^3+900 x^4-180 x^5+12 x^6} \, dx=-x+\frac {1}{12} x \left (-2+\frac {\log ^2\left (\frac {5}{x}-x^2\right )}{(-5+x)^2}\right ) \] Output:
1/12*x*(ln(5/x-x^2)^2/(-5+x)^2-2)-x
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 40.00 (sec) , antiderivative size = 173773, normalized size of antiderivative = 5605.58 \[ \int \frac {-8750+5250 x-1050 x^2+1820 x^3-1050 x^4+210 x^5-14 x^6+\left (-50+10 x-20 x^3+4 x^4\right ) \log \left (\frac {5-x^3}{x}\right )+\left (25+5 x-5 x^3-x^4\right ) \log ^2\left (\frac {5-x^3}{x}\right )}{7500-4500 x+900 x^2-1560 x^3+900 x^4-180 x^5+12 x^6} \, dx=\text {Result too large to show} \] Input:
Integrate[(-8750 + 5250*x - 1050*x^2 + 1820*x^3 - 1050*x^4 + 210*x^5 - 14* x^6 + (-50 + 10*x - 20*x^3 + 4*x^4)*Log[(5 - x^3)/x] + (25 + 5*x - 5*x^3 - x^4)*Log[(5 - x^3)/x]^2)/(7500 - 4500*x + 900*x^2 - 1560*x^3 + 900*x^4 - 180*x^5 + 12*x^6),x]
Output:
Result too large to show
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 18.43 (sec) , antiderivative size = 3581, normalized size of antiderivative = 115.52, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {2463, 7239, 27, 25, 7276, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {-14 x^6+210 x^5-1050 x^4+1820 x^3-1050 x^2+\left (-x^4-5 x^3+5 x+25\right ) \log ^2\left (\frac {5-x^3}{x}\right )+\left (4 x^4-20 x^3+10 x-50\right ) \log \left (\frac {5-x^3}{x}\right )+5250 x-8750}{12 x^6-180 x^5+900 x^4-1560 x^3+900 x^2-4500 x+7500} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {17 \left (-14 x^6+210 x^5-1050 x^4+1820 x^3-1050 x^2+\left (-x^4-5 x^3+5 x+25\right ) \log ^2\left (\frac {5-x^3}{x}\right )+\left (4 x^4-20 x^3+10 x-50\right ) \log \left (\frac {5-x^3}{x}\right )+5250 x-8750\right )}{92160 (x-5)}+\frac {\left (-17 x^2-45 x-89\right ) \left (-14 x^6+210 x^5-1050 x^4+1820 x^3-1050 x^2+\left (-x^4-5 x^3+5 x+25\right ) \log ^2\left (\frac {5-x^3}{x}\right )+\left (4 x^4-20 x^3+10 x-50\right ) \log \left (\frac {5-x^3}{x}\right )+5250 x-8750\right )}{92160 \left (x^3-5\right )}-\frac {-14 x^6+210 x^5-1050 x^4+1820 x^3-1050 x^2+\left (-x^4-5 x^3+5 x+25\right ) \log ^2\left (\frac {5-x^3}{x}\right )+\left (4 x^4-20 x^3+10 x-50\right ) \log \left (\frac {5-x^3}{x}\right )+5250 x-8750}{2304 (x-5)^2}+\frac {-14 x^6+210 x^5-1050 x^4+1820 x^3-1050 x^2+\left (-x^4-5 x^3+5 x+25\right ) \log ^2\left (\frac {5-x^3}{x}\right )+\left (4 x^4-20 x^3+10 x-50\right ) \log \left (\frac {5-x^3}{x}\right )+5250 x-8750}{1440 (x-5)^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-14 \left (x^3-5\right ) (x-5)^3-\left (x^4+5 x^3-5 x-25\right ) \log ^2\left (\frac {5-x^3}{x}\right )+2 \left (2 x^4-10 x^3+5 x-25\right ) \log \left (\frac {5-x^3}{x}\right )}{12 (5-x)^3 \left (5-x^3\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{12} \int -\frac {14 \left (5-x^3\right ) (5-x)^3-\left (-x^4-5 x^3+5 x+25\right ) \log ^2\left (\frac {5-x^3}{x}\right )+2 \left (-2 x^4+10 x^3-5 x+25\right ) \log \left (\frac {5-x^3}{x}\right )}{(5-x)^3 \left (5-x^3\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{12} \int \frac {14 \left (5-x^3\right ) (5-x)^3-\left (-x^4-5 x^3+5 x+25\right ) \log ^2\left (\frac {5-x^3}{x}\right )+2 \left (-2 x^4+10 x^3-5 x+25\right ) \log \left (\frac {5-x^3}{x}\right )}{(5-x)^3 \left (5-x^3\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {1}{12} \int \left (\frac {(x+5) \log ^2\left (\frac {5-x^3}{x}\right )}{(x-5)^3}-\frac {2 \left (2 x^3+5\right ) \log \left (\frac {5-x^3}{x}\right )}{(x-5)^2 \left (x^3-5\right )}+14\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{12} \left (-\frac {1}{960} \left (25+5^{2/3} \left (17+9\ 5^{2/3}\right )\right ) \log ^2\left (\sqrt [3]{5}-x\right )-\frac {1}{120} \left (25+5^{2/3} \left (1+5^{2/3}\right )\right ) \log ^2\left (\sqrt [3]{5}-x\right )+\frac {1}{192} \left (45+\sqrt [3]{5} \left (17+5 \sqrt [3]{5}\right )\right ) \log ^2\left (\sqrt [3]{5}-x\right )-\frac {1}{480} \left (25+5^{2/3} \left (17+9\ 5^{2/3}\right )\right ) \log \left (\frac {x+\sqrt [3]{-5}}{\sqrt [3]{-5}+\sqrt [3]{5}}\right ) \log \left (\sqrt [3]{5}-x\right )-\frac {1}{60} \left (25+5^{2/3} \left (1+5^{2/3}\right )\right ) \log \left (\frac {x+\sqrt [3]{-5}}{\sqrt [3]{-5}+\sqrt [3]{5}}\right ) \log \left (\sqrt [3]{5}-x\right )+\frac {1}{96} \left (45+\sqrt [3]{5} \left (17+5 \sqrt [3]{5}\right )\right ) \log \left (\frac {x+\sqrt [3]{-5}}{\sqrt [3]{-5}+\sqrt [3]{5}}\right ) \log \left (\sqrt [3]{5}-x\right )-\frac {1}{480} \left (25+5^{2/3} \left (17+9\ 5^{2/3}\right )\right ) \log \left (-\frac {(-1)^{2/3} \left (\sqrt [3]{-1} x+\sqrt [3]{5}\right )}{\sqrt [3]{5} \left (1-(-1)^{2/3}\right )}\right ) \log \left (\sqrt [3]{5}-x\right )-\frac {1}{60} \left (25+5^{2/3} \left (1+5^{2/3}\right )\right ) \log \left (-\frac {(-1)^{2/3} \left (\sqrt [3]{-1} x+\sqrt [3]{5}\right )}{\sqrt [3]{5} \left (1-(-1)^{2/3}\right )}\right ) \log \left (\sqrt [3]{5}-x\right )+\frac {1}{96} \left (45+\sqrt [3]{5} \left (17+5 \sqrt [3]{5}\right )\right ) \log \left (-\frac {(-1)^{2/3} \left (\sqrt [3]{-1} x+\sqrt [3]{5}\right )}{\sqrt [3]{5} \left (1-(-1)^{2/3}\right )}\right ) \log \left (\sqrt [3]{5}-x\right )+\frac {1}{480} \left (25+5^{2/3} \left (17+9\ 5^{2/3}\right )\right ) \log \left (\frac {5-x^3}{x}\right ) \log \left (\sqrt [3]{5}-x\right )+\frac {1}{60} \left (25+5^{2/3} \left (1+5^{2/3}\right )\right ) \log \left (\frac {5-x^3}{x}\right ) \log \left (\sqrt [3]{5}-x\right )-\frac {1}{96} \left (45+\sqrt [3]{5} \left (17+5 \sqrt [3]{5}\right )\right ) \log \left (\frac {5-x^3}{x}\right ) \log \left (\sqrt [3]{5}-x\right )+\frac {1}{192} \left (45-\sqrt [3]{5} \left (5 \sqrt [3]{-5}-17 (-1)^{2/3}\right )\right ) \log ^2\left (\sqrt [3]{-1} x+\sqrt [3]{5}\right )-\frac {1}{960} (-1)^{2/3} \left (17 (-5)^{2/3}-25 \sqrt [3]{-1}+45 \sqrt [3]{5}\right ) \log ^2\left (\sqrt [3]{-1} x+\sqrt [3]{5}\right )-\frac {1}{120} (-1)^{2/3} \left ((-5)^{2/3}-25 \sqrt [3]{-1}+5 \sqrt [3]{5}\right ) \log ^2\left (\sqrt [3]{-1} x+\sqrt [3]{5}\right )-\frac {1}{960} \left (25-45 \sqrt [3]{-5}+17 (-5)^{2/3}\right ) \log ^2\left (\sqrt [3]{5}-(-1)^{2/3} x\right )-\frac {1}{192} \sqrt [3]{-5} \left (17-5 \sqrt [3]{-5}+9 (-5)^{2/3}\right ) \log ^2\left (\sqrt [3]{5}-(-1)^{2/3} x\right )-\frac {1}{120} \left (25-5 \sqrt [3]{-5}+(-5)^{2/3}\right ) \log ^2\left (\sqrt [3]{5}-(-1)^{2/3} x\right )-\frac {\log ^2\left (\frac {5-x^3}{x}\right )}{5-x}+\frac {5 \log ^2\left (\frac {5-x^3}{x}\right )}{(5-x)^2}-14 x+\frac {\left (25+5^{2/3} \left (17+9\ 5^{2/3}\right )\right ) \log (5) \log (x)}{1440}+\frac {1}{180} \left (25+5^{2/3} \left (1+5^{2/3}\right )\right ) \log (5) \log (x)+\frac {1}{288} (-1)^{2/3} \left (45 \sqrt [3]{-1}-\sqrt [3]{5} \left (17+5 (-1)^{2/3} \sqrt [3]{5}\right )\right ) \log (5) \log (x)-\frac {1}{288} \left (45+\sqrt [3]{5} \left (17+5 \sqrt [3]{5}\right )\right ) \log (5) \log (x)+\frac {(-1)^{2/3} \left (17 (-5)^{2/3}-25 \sqrt [3]{-1}+45 \sqrt [3]{5}\right ) \log (5) \log (x)}{1440}+\frac {1}{180} (-1)^{2/3} \left ((-5)^{2/3}-25 \sqrt [3]{-1}+5 \sqrt [3]{5}\right ) \log (5) \log (x)+\frac {\left (25-45 \sqrt [3]{-5}+17 (-5)^{2/3}\right ) \log (5) \log (x)}{1440}+\frac {1}{288} \sqrt [3]{-5} \left (17-5 \sqrt [3]{-5}+9 (-5)^{2/3}\right ) \log (5) \log (x)+\frac {1}{180} \left (25-5 \sqrt [3]{-5}+(-5)^{2/3}\right ) \log (5) \log (x)+\frac {1}{96} \left (45-\sqrt [3]{5} \left (5 \sqrt [3]{-5}-17 (-1)^{2/3}\right )\right ) \log \left (\frac {\sqrt [3]{-\frac {1}{5}} \left (\sqrt [3]{5}-x\right )}{1+\sqrt [3]{-1}}\right ) \log \left (\sqrt [3]{-1} x+\sqrt [3]{5}\right )-\frac {1}{480} (-1)^{2/3} \left (17 (-5)^{2/3}-25 \sqrt [3]{-1}+45 \sqrt [3]{5}\right ) \log \left (\frac {\sqrt [3]{-\frac {1}{5}} \left (\sqrt [3]{5}-x\right )}{1+\sqrt [3]{-1}}\right ) \log \left (\sqrt [3]{-1} x+\sqrt [3]{5}\right )-\frac {1}{60} (-1)^{2/3} \left ((-5)^{2/3}-25 \sqrt [3]{-1}+5 \sqrt [3]{5}\right ) \log \left (\frac {\sqrt [3]{-\frac {1}{5}} \left (\sqrt [3]{5}-x\right )}{1+\sqrt [3]{-1}}\right ) \log \left (\sqrt [3]{-1} x+\sqrt [3]{5}\right )+\frac {1}{96} \left (45-\sqrt [3]{5} \left (5 \sqrt [3]{-5}-17 (-1)^{2/3}\right )\right ) \log \left (-\frac {\sqrt [3]{-\frac {1}{5}} \left (x+\sqrt [3]{-5}\right )}{1-(-1)^{2/3}}\right ) \log \left (\sqrt [3]{-1} x+\sqrt [3]{5}\right )-\frac {1}{480} (-1)^{2/3} \left (17 (-5)^{2/3}-25 \sqrt [3]{-1}+45 \sqrt [3]{5}\right ) \log \left (-\frac {\sqrt [3]{-\frac {1}{5}} \left (x+\sqrt [3]{-5}\right )}{1-(-1)^{2/3}}\right ) \log \left (\sqrt [3]{-1} x+\sqrt [3]{5}\right )-\frac {1}{60} (-1)^{2/3} \left ((-5)^{2/3}-25 \sqrt [3]{-1}+5 \sqrt [3]{5}\right ) \log \left (-\frac {\sqrt [3]{-\frac {1}{5}} \left (x+\sqrt [3]{-5}\right )}{1-(-1)^{2/3}}\right ) \log \left (\sqrt [3]{-1} x+\sqrt [3]{5}\right )-\frac {1}{480} \left (25-45 \sqrt [3]{-5}+17 (-5)^{2/3}\right ) \log \left (-\frac {(-1)^{2/3} \left (\sqrt [3]{5}-x\right )}{\sqrt [3]{5} \left (1-(-1)^{2/3}\right )}\right ) \log \left (\sqrt [3]{5}-(-1)^{2/3} x\right )-\frac {1}{96} \sqrt [3]{-5} \left (17-5 \sqrt [3]{-5}+9 (-5)^{2/3}\right ) \log \left (-\frac {(-1)^{2/3} \left (\sqrt [3]{5}-x\right )}{\sqrt [3]{5} \left (1-(-1)^{2/3}\right )}\right ) \log \left (\sqrt [3]{5}-(-1)^{2/3} x\right )-\frac {1}{60} \left (25-5 \sqrt [3]{-5}+(-5)^{2/3}\right ) \log \left (-\frac {(-1)^{2/3} \left (\sqrt [3]{5}-x\right )}{\sqrt [3]{5} \left (1-(-1)^{2/3}\right )}\right ) \log \left (\sqrt [3]{5}-(-1)^{2/3} x\right )-\frac {1}{480} \left (25-45 \sqrt [3]{-5}+17 (-5)^{2/3}\right ) \log \left (-\frac {(-1)^{2/3} \left ((-1)^{2/3} \sqrt [3]{5}-x\right )}{\sqrt [3]{-5}+\sqrt [3]{5}}\right ) \log \left (\sqrt [3]{5}-(-1)^{2/3} x\right )-\frac {1}{96} \sqrt [3]{-5} \left (17-5 \sqrt [3]{-5}+9 (-5)^{2/3}\right ) \log \left (-\frac {(-1)^{2/3} \left ((-1)^{2/3} \sqrt [3]{5}-x\right )}{\sqrt [3]{-5}+\sqrt [3]{5}}\right ) \log \left (\sqrt [3]{5}-(-1)^{2/3} x\right )-\frac {1}{60} \left (25-5 \sqrt [3]{-5}+(-5)^{2/3}\right ) \log \left (-\frac {(-1)^{2/3} \left ((-1)^{2/3} \sqrt [3]{5}-x\right )}{\sqrt [3]{-5}+\sqrt [3]{5}}\right ) \log \left (\sqrt [3]{5}-(-1)^{2/3} x\right )-\frac {1}{96} \left (45-\sqrt [3]{5} \left (5 \sqrt [3]{-5}-17 (-1)^{2/3}\right )\right ) \log \left (\sqrt [3]{-1} x+\sqrt [3]{5}\right ) \log \left (\frac {5-x^3}{x}\right )+\frac {1}{480} (-1)^{2/3} \left (17 (-5)^{2/3}-25 \sqrt [3]{-1}+45 \sqrt [3]{5}\right ) \log \left (\sqrt [3]{-1} x+\sqrt [3]{5}\right ) \log \left (\frac {5-x^3}{x}\right )+\frac {1}{60} (-1)^{2/3} \left ((-5)^{2/3}-25 \sqrt [3]{-1}+5 \sqrt [3]{5}\right ) \log \left (\sqrt [3]{-1} x+\sqrt [3]{5}\right ) \log \left (\frac {5-x^3}{x}\right )+\frac {1}{480} \left (25-45 \sqrt [3]{-5}+17 (-5)^{2/3}\right ) \log \left (\sqrt [3]{5}-(-1)^{2/3} x\right ) \log \left (\frac {5-x^3}{x}\right )+\frac {1}{96} \sqrt [3]{-5} \left (17-5 \sqrt [3]{-5}+9 (-5)^{2/3}\right ) \log \left (\sqrt [3]{5}-(-1)^{2/3} x\right ) \log \left (\frac {5-x^3}{x}\right )+\frac {1}{60} \left (25-5 \sqrt [3]{-5}+(-5)^{2/3}\right ) \log \left (\sqrt [3]{5}-(-1)^{2/3} x\right ) \log \left (\frac {5-x^3}{x}\right )-\frac {1}{480} \left (25+5^{2/3} \left (17+9\ 5^{2/3}\right )\right ) \operatorname {PolyLog}\left (2,\frac {2 \left (\sqrt [3]{5}-x\right )}{\sqrt [3]{5} \left (3-i \sqrt {3}\right )}\right )-\frac {1}{60} \left (25+5^{2/3} \left (1+5^{2/3}\right )\right ) \operatorname {PolyLog}\left (2,\frac {2 \left (\sqrt [3]{5}-x\right )}{\sqrt [3]{5} \left (3-i \sqrt {3}\right )}\right )+\frac {1}{96} \left (45+\sqrt [3]{5} \left (17+5 \sqrt [3]{5}\right )\right ) \operatorname {PolyLog}\left (2,\frac {2 \left (\sqrt [3]{5}-x\right )}{\sqrt [3]{5} \left (3-i \sqrt {3}\right )}\right )-\frac {1}{480} \left (25+5^{2/3} \left (17+9\ 5^{2/3}\right )\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{5}-x}{\sqrt [3]{-5}+\sqrt [3]{5}}\right )-\frac {1}{60} \left (25+5^{2/3} \left (1+5^{2/3}\right )\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{5}-x}{\sqrt [3]{-5}+\sqrt [3]{5}}\right )+\frac {1}{96} \left (45+\sqrt [3]{5} \left (17+5 \sqrt [3]{5}\right )\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{5}-x}{\sqrt [3]{-5}+\sqrt [3]{5}}\right )+\frac {1}{96} \left (45-\sqrt [3]{5} \left (5 \sqrt [3]{-5}-17 (-1)^{2/3}\right )\right ) \operatorname {PolyLog}\left (2,-\sqrt [3]{-\frac {1}{5}} x\right )-\frac {1}{480} (-1)^{2/3} \left (17 (-5)^{2/3}-25 \sqrt [3]{-1}+45 \sqrt [3]{5}\right ) \operatorname {PolyLog}\left (2,-\sqrt [3]{-\frac {1}{5}} x\right )-\frac {1}{60} (-1)^{2/3} \left ((-5)^{2/3}-25 \sqrt [3]{-1}+5 \sqrt [3]{5}\right ) \operatorname {PolyLog}\left (2,-\sqrt [3]{-\frac {1}{5}} x\right )-\frac {1}{480} \left (25+5^{2/3} \left (17+9\ 5^{2/3}\right )\right ) \operatorname {PolyLog}\left (2,\frac {x}{\sqrt [3]{5}}\right )-\frac {1}{60} \left (25+5^{2/3} \left (1+5^{2/3}\right )\right ) \operatorname {PolyLog}\left (2,\frac {x}{\sqrt [3]{5}}\right )+\frac {1}{96} \left (45+\sqrt [3]{5} \left (17+5 \sqrt [3]{5}\right )\right ) \operatorname {PolyLog}\left (2,\frac {x}{\sqrt [3]{5}}\right )-\frac {1}{480} \left (25-45 \sqrt [3]{-5}+17 (-5)^{2/3}\right ) \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} x}{\sqrt [3]{5}}\right )-\frac {1}{96} \sqrt [3]{-5} \left (17-5 \sqrt [3]{-5}+9 (-5)^{2/3}\right ) \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} x}{\sqrt [3]{5}}\right )-\frac {1}{60} \left (25-5 \sqrt [3]{-5}+(-5)^{2/3}\right ) \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} x}{\sqrt [3]{5}}\right )-\frac {1}{480} \left (25-45 \sqrt [3]{-5}+17 (-5)^{2/3}\right ) \operatorname {PolyLog}\left (2,-\frac {(-1)^{2/3} \left (x+\sqrt [3]{-5}\right )}{\sqrt [3]{5} \left (1-(-1)^{2/3}\right )}\right )-\frac {1}{96} \sqrt [3]{-5} \left (17-5 \sqrt [3]{-5}+9 (-5)^{2/3}\right ) \operatorname {PolyLog}\left (2,-\frac {(-1)^{2/3} \left (x+\sqrt [3]{-5}\right )}{\sqrt [3]{5} \left (1-(-1)^{2/3}\right )}\right )-\frac {1}{60} \left (25-5 \sqrt [3]{-5}+(-5)^{2/3}\right ) \operatorname {PolyLog}\left (2,-\frac {(-1)^{2/3} \left (x+\sqrt [3]{-5}\right )}{\sqrt [3]{5} \left (1-(-1)^{2/3}\right )}\right )+\frac {1}{96} \left (45-\sqrt [3]{5} \left (5 \sqrt [3]{-5}-17 (-1)^{2/3}\right )\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} x+\sqrt [3]{5}}{\sqrt [3]{5} \left (1-(-1)^{2/3}\right )}\right )-\frac {1}{480} (-1)^{2/3} \left (17 (-5)^{2/3}-25 \sqrt [3]{-1}+45 \sqrt [3]{5}\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} x+\sqrt [3]{5}}{\sqrt [3]{5} \left (1-(-1)^{2/3}\right )}\right )-\frac {1}{60} (-1)^{2/3} \left ((-5)^{2/3}-25 \sqrt [3]{-1}+5 \sqrt [3]{5}\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} x+\sqrt [3]{5}}{\sqrt [3]{5} \left (1-(-1)^{2/3}\right )}\right )+\frac {1}{96} \left (45-\sqrt [3]{5} \left (5 \sqrt [3]{-5}-17 (-1)^{2/3}\right )\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} x+\sqrt [3]{5}}{\sqrt [3]{-5}+\sqrt [3]{5}}\right )-\frac {1}{480} (-1)^{2/3} \left (17 (-5)^{2/3}-25 \sqrt [3]{-1}+45 \sqrt [3]{5}\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} x+\sqrt [3]{5}}{\sqrt [3]{-5}+\sqrt [3]{5}}\right )-\frac {1}{60} (-1)^{2/3} \left ((-5)^{2/3}-25 \sqrt [3]{-1}+5 \sqrt [3]{5}\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} x+\sqrt [3]{5}}{\sqrt [3]{-5}+\sqrt [3]{5}}\right )-\frac {1}{480} \left (25-45 \sqrt [3]{-5}+17 (-5)^{2/3}\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{5}-(-1)^{2/3} x}{\sqrt [3]{-5}+\sqrt [3]{5}}\right )-\frac {1}{96} \sqrt [3]{-5} \left (17-5 \sqrt [3]{-5}+9 (-5)^{2/3}\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{5}-(-1)^{2/3} x}{\sqrt [3]{-5}+\sqrt [3]{5}}\right )-\frac {1}{60} \left (25-5 \sqrt [3]{-5}+(-5)^{2/3}\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{5}-(-1)^{2/3} x}{\sqrt [3]{-5}+\sqrt [3]{5}}\right )\right )\) |
Input:
Int[(-8750 + 5250*x - 1050*x^2 + 1820*x^3 - 1050*x^4 + 210*x^5 - 14*x^6 + (-50 + 10*x - 20*x^3 + 4*x^4)*Log[(5 - x^3)/x] + (25 + 5*x - 5*x^3 - x^4)* Log[(5 - x^3)/x]^2)/(7500 - 4500*x + 900*x^2 - 1560*x^3 + 900*x^4 - 180*x^ 5 + 12*x^6),x]
Output:
(-14*x + ((45 + 5^(1/3)*(17 + 5*5^(1/3)))*Log[5^(1/3) - x]^2)/192 - ((25 + 5^(2/3)*(1 + 5^(2/3)))*Log[5^(1/3) - x]^2)/120 - ((25 + 5^(2/3)*(17 + 9*5 ^(2/3)))*Log[5^(1/3) - x]^2)/960 + ((25 - 5*(-5)^(1/3) + (-5)^(2/3))*Log[5 ]*Log[x])/180 + ((-5)^(1/3)*(17 - 5*(-5)^(1/3) + 9*(-5)^(2/3))*Log[5]*Log[ x])/288 + ((25 - 45*(-5)^(1/3) + 17*(-5)^(2/3))*Log[5]*Log[x])/1440 + ((-1 )^(2/3)*((-5)^(2/3) - 25*(-1)^(1/3) + 5*5^(1/3))*Log[5]*Log[x])/180 + ((-1 )^(2/3)*(17*(-5)^(2/3) - 25*(-1)^(1/3) + 45*5^(1/3))*Log[5]*Log[x])/1440 - ((45 + 5^(1/3)*(17 + 5*5^(1/3)))*Log[5]*Log[x])/288 + ((-1)^(2/3)*(45*(-1 )^(1/3) - 5^(1/3)*(17 + 5*(-1)^(2/3)*5^(1/3)))*Log[5]*Log[x])/288 + ((25 + 5^(2/3)*(1 + 5^(2/3)))*Log[5]*Log[x])/180 + ((25 + 5^(2/3)*(17 + 9*5^(2/3 )))*Log[5]*Log[x])/1440 + ((45 + 5^(1/3)*(17 + 5*5^(1/3)))*Log[5^(1/3) - x ]*Log[((-5)^(1/3) + x)/((-5)^(1/3) + 5^(1/3))])/96 - ((25 + 5^(2/3)*(1 + 5 ^(2/3)))*Log[5^(1/3) - x]*Log[((-5)^(1/3) + x)/((-5)^(1/3) + 5^(1/3))])/60 - ((25 + 5^(2/3)*(17 + 9*5^(2/3)))*Log[5^(1/3) - x]*Log[((-5)^(1/3) + x)/ ((-5)^(1/3) + 5^(1/3))])/480 - ((-1)^(2/3)*((-5)^(2/3) - 25*(-1)^(1/3) + 5 *5^(1/3))*Log[((-1/5)^(1/3)*(5^(1/3) - x))/(1 + (-1)^(1/3))]*Log[5^(1/3) + (-1)^(1/3)*x])/60 - ((-1)^(2/3)*(17*(-5)^(2/3) - 25*(-1)^(1/3) + 45*5^(1/ 3))*Log[((-1/5)^(1/3)*(5^(1/3) - x))/(1 + (-1)^(1/3))]*Log[5^(1/3) + (-1)^ (1/3)*x])/480 + ((45 - 5^(1/3)*(5*(-5)^(1/3) - 17*(-1)^(2/3)))*Log[((-1/5) ^(1/3)*(5^(1/3) - x))/(1 + (-1)^(1/3))]*Log[5^(1/3) + (-1)^(1/3)*x])/96...
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 9.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03
| method | result | size |
| risch | \(\frac {\ln \left (\frac {-x^{3}+5}{x}\right )^{2} x}{12 x^{2}-120 x +300}-\frac {7 x}{6}\) | \(32\) |
| norman | \(\frac {\frac {175 x}{2}-\frac {7 x^{3}}{6}+\frac {\ln \left (\frac {-x^{3}+5}{x}\right )^{2} x}{12}-\frac {875}{3}}{\left (-5+x \right )^{2}}\) | \(34\) |
| parallelrisch | \(-\frac {3500+14 x^{3}-\ln \left (-\frac {x^{3}-5}{x}\right )^{2} x -1050 x}{12 \left (x^{2}-10 x +25\right )}\) | \(39\) |
| orering | \(\text {Expression too large to display}\) | \(2008\) |
Input:
int(((-x^4-5*x^3+5*x+25)*ln((-x^3+5)/x)^2+(4*x^4-20*x^3+10*x-50)*ln((-x^3+ 5)/x)-14*x^6+210*x^5-1050*x^4+1820*x^3-1050*x^2+5250*x-8750)/(12*x^6-180*x ^5+900*x^4-1560*x^3+900*x^2-4500*x+7500),x,method=_RETURNVERBOSE)
Output:
1/12*x/(x^2-10*x+25)*ln((-x^3+5)/x)^2-7/6*x
Time = 0.07 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35 \[ \int \frac {-8750+5250 x-1050 x^2+1820 x^3-1050 x^4+210 x^5-14 x^6+\left (-50+10 x-20 x^3+4 x^4\right ) \log \left (\frac {5-x^3}{x}\right )+\left (25+5 x-5 x^3-x^4\right ) \log ^2\left (\frac {5-x^3}{x}\right )}{7500-4500 x+900 x^2-1560 x^3+900 x^4-180 x^5+12 x^6} \, dx=-\frac {14 \, x^{3} - x \log \left (-\frac {x^{3} - 5}{x}\right )^{2} - 140 \, x^{2} + 350 \, x}{12 \, {\left (x^{2} - 10 \, x + 25\right )}} \] Input:
integrate(((-x^4-5*x^3+5*x+25)*log((-x^3+5)/x)^2+(4*x^4-20*x^3+10*x-50)*lo g((-x^3+5)/x)-14*x^6+210*x^5-1050*x^4+1820*x^3-1050*x^2+5250*x-8750)/(12*x ^6-180*x^5+900*x^4-1560*x^3+900*x^2-4500*x+7500),x, algorithm="fricas")
Output:
-1/12*(14*x^3 - x*log(-(x^3 - 5)/x)^2 - 140*x^2 + 350*x)/(x^2 - 10*x + 25)
Time = 0.18 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \frac {-8750+5250 x-1050 x^2+1820 x^3-1050 x^4+210 x^5-14 x^6+\left (-50+10 x-20 x^3+4 x^4\right ) \log \left (\frac {5-x^3}{x}\right )+\left (25+5 x-5 x^3-x^4\right ) \log ^2\left (\frac {5-x^3}{x}\right )}{7500-4500 x+900 x^2-1560 x^3+900 x^4-180 x^5+12 x^6} \, dx=- \frac {7 x}{6} + \frac {x \log {\left (\frac {5 - x^{3}}{x} \right )}^{2}}{12 x^{2} - 120 x + 300} \] Input:
integrate(((-x**4-5*x**3+5*x+25)*ln((-x**3+5)/x)**2+(4*x**4-20*x**3+10*x-5 0)*ln((-x**3+5)/x)-14*x**6+210*x**5-1050*x**4+1820*x**3-1050*x**2+5250*x-8 750)/(12*x**6-180*x**5+900*x**4-1560*x**3+900*x**2-4500*x+7500),x)
Output:
-7*x/6 + x*log((5 - x**3)/x)**2/(12*x**2 - 120*x + 300)
Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (29) = 58\).
Time = 3.57 (sec) , antiderivative size = 165, normalized size of antiderivative = 5.32 \[ \int \frac {-8750+5250 x-1050 x^2+1820 x^3-1050 x^4+210 x^5-14 x^6+\left (-50+10 x-20 x^3+4 x^4\right ) \log \left (\frac {5-x^3}{x}\right )+\left (25+5 x-5 x^3-x^4\right ) \log ^2\left (\frac {5-x^3}{x}\right )}{7500-4500 x+900 x^2-1560 x^3+900 x^4-180 x^5+12 x^6} \, dx=-\frac {7}{6} \, x + \frac {x \log \left (-x^{3} + 5\right )^{2} - 2 \, x \log \left (-x^{3} + 5\right ) \log \left (x\right ) + x \log \left (x\right )^{2}}{12 \, {\left (x^{2} - 10 \, x + 25\right )}} + \frac {4375 \, {\left (23 \, x - 95\right )}}{1152 \, {\left (x^{2} - 10 \, x + 25\right )}} + \frac {175 \, {\left (17 \, x - 105\right )}}{384 \, {\left (x^{2} - 10 \, x + 25\right )}} - \frac {175 \, {\left (9 \, x - 65\right )}}{384 \, {\left (x^{2} - 10 \, x + 25\right )}} + \frac {4375 \, {\left (7 \, x - 15\right )}}{384 \, {\left (x^{2} - 10 \, x + 25\right )}} - \frac {875 \, {\left (5 \, x - 29\right )}}{1152 \, {\left (x^{2} - 10 \, x + 25\right )}} - \frac {21875 \, {\left (3 \, x - 11\right )}}{384 \, {\left (x^{2} - 10 \, x + 25\right )}} + \frac {2275 \, {\left (x - 25\right )}}{576 \, {\left (x^{2} - 10 \, x + 25\right )}} \] Input:
integrate(((-x^4-5*x^3+5*x+25)*log((-x^3+5)/x)^2+(4*x^4-20*x^3+10*x-50)*lo g((-x^3+5)/x)-14*x^6+210*x^5-1050*x^4+1820*x^3-1050*x^2+5250*x-8750)/(12*x ^6-180*x^5+900*x^4-1560*x^3+900*x^2-4500*x+7500),x, algorithm="maxima")
Output:
-7/6*x + 1/12*(x*log(-x^3 + 5)^2 - 2*x*log(-x^3 + 5)*log(x) + x*log(x)^2)/ (x^2 - 10*x + 25) + 4375/1152*(23*x - 95)/(x^2 - 10*x + 25) + 175/384*(17* x - 105)/(x^2 - 10*x + 25) - 175/384*(9*x - 65)/(x^2 - 10*x + 25) + 4375/3 84*(7*x - 15)/(x^2 - 10*x + 25) - 875/1152*(5*x - 29)/(x^2 - 10*x + 25) - 21875/384*(3*x - 11)/(x^2 - 10*x + 25) + 2275/576*(x - 25)/(x^2 - 10*x + 2 5)
Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int \frac {-8750+5250 x-1050 x^2+1820 x^3-1050 x^4+210 x^5-14 x^6+\left (-50+10 x-20 x^3+4 x^4\right ) \log \left (\frac {5-x^3}{x}\right )+\left (25+5 x-5 x^3-x^4\right ) \log ^2\left (\frac {5-x^3}{x}\right )}{7500-4500 x+900 x^2-1560 x^3+900 x^4-180 x^5+12 x^6} \, dx=\frac {x \log \left (-\frac {x^{3} - 5}{x}\right )^{2}}{12 \, {\left (x^{2} - 10 \, x + 25\right )}} - \frac {7}{6} \, x \] Input:
integrate(((-x^4-5*x^3+5*x+25)*log((-x^3+5)/x)^2+(4*x^4-20*x^3+10*x-50)*lo g((-x^3+5)/x)-14*x^6+210*x^5-1050*x^4+1820*x^3-1050*x^2+5250*x-8750)/(12*x ^6-180*x^5+900*x^4-1560*x^3+900*x^2-4500*x+7500),x, algorithm="giac")
Output:
1/12*x*log(-(x^3 - 5)/x)^2/(x^2 - 10*x + 25) - 7/6*x
Time = 2.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81 \[ \int \frac {-8750+5250 x-1050 x^2+1820 x^3-1050 x^4+210 x^5-14 x^6+\left (-50+10 x-20 x^3+4 x^4\right ) \log \left (\frac {5-x^3}{x}\right )+\left (25+5 x-5 x^3-x^4\right ) \log ^2\left (\frac {5-x^3}{x}\right )}{7500-4500 x+900 x^2-1560 x^3+900 x^4-180 x^5+12 x^6} \, dx=\frac {x\,{\ln \left (-\frac {x^3-5}{x}\right )}^2}{12\,{\left (x-5\right )}^2}-\frac {7\,x}{6} \] Input:
int((5250*x + log(-(x^3 - 5)/x)^2*(5*x - 5*x^3 - x^4 + 25) - 1050*x^2 + 18 20*x^3 - 1050*x^4 + 210*x^5 - 14*x^6 + log(-(x^3 - 5)/x)*(10*x - 20*x^3 + 4*x^4 - 50) - 8750)/(900*x^2 - 4500*x - 1560*x^3 + 900*x^4 - 180*x^5 + 12* x^6 + 7500),x)
Output:
(x*log(-(x^3 - 5)/x)^2)/(12*(x - 5)^2) - (7*x)/6
Time = 0.19 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.23 \[ \int \frac {-8750+5250 x-1050 x^2+1820 x^3-1050 x^4+210 x^5-14 x^6+\left (-50+10 x-20 x^3+4 x^4\right ) \log \left (\frac {5-x^3}{x}\right )+\left (25+5 x-5 x^3-x^4\right ) \log ^2\left (\frac {5-x^3}{x}\right )}{7500-4500 x+900 x^2-1560 x^3+900 x^4-180 x^5+12 x^6} \, dx=\frac {x \left (\mathrm {log}\left (\frac {-x^{3}+5}{x}\right )^{2}-14 x^{2}+140 x -350\right )}{12 x^{2}-120 x +300} \] Input:
int(((-x^4-5*x^3+5*x+25)*log((-x^3+5)/x)^2+(4*x^4-20*x^3+10*x-50)*log((-x^ 3+5)/x)-14*x^6+210*x^5-1050*x^4+1820*x^3-1050*x^2+5250*x-8750)/(12*x^6-180 *x^5+900*x^4-1560*x^3+900*x^2-4500*x+7500),x)
Output:
(x*(log(( - x**3 + 5)/x)**2 - 14*x**2 + 140*x - 350))/(12*(x**2 - 10*x + 2 5))