Integrand size = 189, antiderivative size = 34 \[ \int \frac {\left (e^3 \left (-4 x^3+4 x^4\right )+e^6 \left (2 x^3-6 x^4+4 x^5\right )\right ) \log \left (\frac {-1-2 e^3 x+e^6 \left (x-x^2\right )}{e^6}\right )+\left (-3 x^2+2 x^3+e^3 \left (-6 x^3+4 x^4\right )+e^6 \left (3 x^3-5 x^4+2 x^5\right )\right ) \log ^2\left (\frac {-1-2 e^3 x+e^6 \left (x-x^2\right )}{e^6}\right )}{20-40 x+20 x^2+e^3 \left (40 x-80 x^2+40 x^3\right )+e^6 \left (-20 x+60 x^2-60 x^3+20 x^4\right )} \, dx=\frac {1}{4} \left (2-\frac {x^2 \log ^2\left (x-\left (\frac {1}{e^3}+x\right )^2\right )}{-5+\frac {5}{x}}\right ) \] Output:
1/2-1/4*x^2*ln(x-(exp(-3)+x)^2)^2/(5/x-5)
Time = 0.07 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.97 \[ \int \frac {\left (e^3 \left (-4 x^3+4 x^4\right )+e^6 \left (2 x^3-6 x^4+4 x^5\right )\right ) \log \left (\frac {-1-2 e^3 x+e^6 \left (x-x^2\right )}{e^6}\right )+\left (-3 x^2+2 x^3+e^3 \left (-6 x^3+4 x^4\right )+e^6 \left (3 x^3-5 x^4+2 x^5\right )\right ) \log ^2\left (\frac {-1-2 e^3 x+e^6 \left (x-x^2\right )}{e^6}\right )}{20-40 x+20 x^2+e^3 \left (40 x-80 x^2+40 x^3\right )+e^6 \left (-20 x+60 x^2-60 x^3+20 x^4\right )} \, dx=\frac {x^3 \log ^2\left (-\frac {1}{e^6}+x-\frac {2 x}{e^3}-x^2\right )}{20 (-1+x)} \] Input:
Integrate[((E^3*(-4*x^3 + 4*x^4) + E^6*(2*x^3 - 6*x^4 + 4*x^5))*Log[(-1 - 2*E^3*x + E^6*(x - x^2))/E^6] + (-3*x^2 + 2*x^3 + E^3*(-6*x^3 + 4*x^4) + E ^6*(3*x^3 - 5*x^4 + 2*x^5))*Log[(-1 - 2*E^3*x + E^6*(x - x^2))/E^6]^2)/(20 - 40*x + 20*x^2 + E^3*(40*x - 80*x^2 + 40*x^3) + E^6*(-20*x + 60*x^2 - 60 *x^3 + 20*x^4)),x]
Output:
(x^3*Log[-E^(-6) + x - (2*x)/E^3 - x^2]^2)/(20*(-1 + x))
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 20.96 (sec) , antiderivative size = 3616, normalized size of antiderivative = 106.35, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {2463, 7239, 27, 25, 7293, 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.
| \(\displaystyle \int \frac {\left (2 x^3-3 x^2+e^3 \left (4 x^4-6 x^3\right )+e^6 \left (2 x^5-5 x^4+3 x^3\right )\right ) \log ^2\left (\frac {e^6 \left (x-x^2\right )-2 e^3 x-1}{e^6}\right )+\left (e^3 \left (4 x^4-4 x^3\right )+e^6 \left (4 x^5-6 x^4+2 x^3\right )\right ) \log \left (\frac {e^6 \left (x-x^2\right )-2 e^3 x-1}{e^6}\right )}{20 x^2+e^3 \left (40 x^3-80 x^2+40 x\right )+e^6 \left (20 x^4-60 x^3+60 x^2-20 x\right )-40 x+20} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (-\frac {e^3 \left (2+e^3\right ) \left (\left (2 x^3-3 x^2+e^3 \left (4 x^4-6 x^3\right )+e^6 \left (2 x^5-5 x^4+3 x^3\right )\right ) \log ^2\left (\frac {e^6 \left (x-x^2\right )-2 e^3 x-1}{e^6}\right )+\left (e^3 \left (4 x^4-4 x^3\right )+e^6 \left (4 x^5-6 x^4+2 x^3\right )\right ) \log \left (\frac {e^6 \left (x-x^2\right )-2 e^3 x-1}{e^6}\right )\right )}{20 \left (1+2 e^3\right )^2 (x-1)}+\frac {e^6 \left (e^3 \left (2+e^3\right ) x+3\right ) \left (\left (2 x^3-3 x^2+e^3 \left (4 x^4-6 x^3\right )+e^6 \left (2 x^5-5 x^4+3 x^3\right )\right ) \log ^2\left (\frac {e^6 \left (x-x^2\right )-2 e^3 x-1}{e^6}\right )+\left (e^3 \left (4 x^4-4 x^3\right )+e^6 \left (4 x^5-6 x^4+2 x^3\right )\right ) \log \left (\frac {e^6 \left (x-x^2\right )-2 e^3 x-1}{e^6}\right )\right )}{20 \left (1+2 e^3\right )^2 \left (e^6 x^2+e^3 \left (2-e^3\right ) x+1\right )}+\frac {\left (2 x^3-3 x^2+e^3 \left (4 x^4-6 x^3\right )+e^6 \left (2 x^5-5 x^4+3 x^3\right )\right ) \log ^2\left (\frac {e^6 \left (x-x^2\right )-2 e^3 x-1}{e^6}\right )+\left (e^3 \left (4 x^4-4 x^3\right )+e^6 \left (4 x^5-6 x^4+2 x^3\right )\right ) \log \left (\frac {e^6 \left (x-x^2\right )-2 e^3 x-1}{e^6}\right )}{20 \left (1+2 e^3\right ) (x-1)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^2 \log \left (-x^2+\left (1-\frac {2}{e^3}\right ) x-\frac {1}{e^6}\right ) \left ((2 x-3) \log \left (-x^2-\frac {2 x}{e^3}+x-\frac {1}{e^6}\right )+\frac {2 e^3 (x-1) x \left (e^3 (2 x-1)+2\right )}{e^6 (x-1) x+2 e^3 x+1}\right )}{20 (1-x)^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{20} \int -\frac {x^2 \log \left (-x^2+\left (1-\frac {2}{e^3}\right ) x-\frac {1}{e^6}\right ) \left (\frac {2 e^3 \left (2-e^3 (1-2 x)\right ) (1-x) x}{-e^6 (1-x) x+2 e^3 x+1}+(3-2 x) \log \left (-x^2-\frac {2 x}{e^3}+x-\frac {1}{e^6}\right )\right )}{(1-x)^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{20} \int \frac {x^2 \log \left (-x^2+\left (1-\frac {2}{e^3}\right ) x-\frac {1}{e^6}\right ) \left (\frac {2 e^3 \left (2-e^3 (1-2 x)\right ) (1-x) x}{-e^6 (1-x) x+2 e^3 x+1}+(3-2 x) \log \left (-x^2-\frac {2 x}{e^3}+x-\frac {1}{e^6}\right )\right )}{(1-x)^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{20} \int \left (\frac {2 e^3 \left (2 e^3 x-e^3+2\right ) \log \left (-x^2+\left (1-\frac {2}{e^3}\right ) x-\frac {1}{e^6}\right ) x^3}{(1-x) \left (e^6 x^2+e^3 \left (2-e^3\right ) x+1\right )}+\frac {(3-2 x) \log ^2\left (-x^2+\left (1-\frac {2}{e^3}\right ) x-\frac {1}{e^6}\right ) x^2}{(1-x)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{20} \left (-\frac {\left (2-2 e^3-8 e^6+6 e^9-e^{12}+\frac {e^{3/2} \left (8+6 e^3-18 e^6+8 e^9-e^{12}\right )}{\sqrt {-4+e^3}}\right ) \log ^2\left (e^3 \left (-e^3 (1-2 x)-e^{3/2} \sqrt {-4+e^3}+2\right )\right )}{2 e^6 \left (1+2 e^3\right )}+\frac {\left (2-4 e^3+e^6+\frac {e^{3/2} \left (8-6 e^3+e^6\right )}{\sqrt {-4+e^3}}\right ) \log ^2\left (e^3 \left (-e^3 (1-2 x)-e^{3/2} \sqrt {-4+e^3}+2\right )\right )}{2 e^6}-\frac {e^3 \left (2+e^3+e^{3/2} \sqrt {-4+e^3}\right ) \log ^2\left (e^3 \left (-e^3 (1-2 x)-e^{3/2} \sqrt {-4+e^3}+2\right )\right )}{2 \left (1+2 e^3\right )}-\frac {\left (2-e^3-e^{3/2} \sqrt {-4+e^3}\right ) \log ^2\left (e^3 \left (-e^3 (1-2 x)-e^{3/2} \sqrt {-4+e^3}+2\right )\right )}{2 e^3}-\frac {\left (2-2 e^3-8 e^6+6 e^9-e^{12}-\frac {e^{3/2} \left (8+6 e^3-18 e^6+8 e^9-e^{12}\right )}{\sqrt {-4+e^3}}\right ) \log ^2\left (e^3 \left (-e^3 (1-2 x)+e^{3/2} \sqrt {-4+e^3}+2\right )\right )}{2 e^6 \left (1+2 e^3\right )}+\frac {\left (2-4 e^3+e^6-\frac {e^{3/2} \left (8-6 e^3+e^6\right )}{\sqrt {-4+e^3}}\right ) \log ^2\left (e^3 \left (-e^3 (1-2 x)+e^{3/2} \sqrt {-4+e^3}+2\right )\right )}{2 e^6}-\frac {\left (2-e^3+e^{3/2} \sqrt {-4+e^3}\right ) \log ^2\left (e^3 \left (-e^3 (1-2 x)+e^{3/2} \sqrt {-4+e^3}+2\right )\right )}{2 e^3}-\frac {e^3 \left (2+e^3-e^{3/2} \sqrt {-4+e^3}\right ) \log ^2\left (e^3 \left (-e^3 (1-2 x)+e^{3/2} \sqrt {-4+e^3}+2\right )\right )}{2 \left (1+2 e^3\right )}+x^2 \log ^2\left (-x^2+\left (1-\frac {2}{e^3}\right ) x-\frac {1}{e^6}\right )+x \log ^2\left (-x^2+\left (1-\frac {2}{e^3}\right ) x-\frac {1}{e^6}\right )-\frac {\log ^2\left (-x^2+\left (1-\frac {2}{e^3}\right ) x-\frac {1}{e^6}\right )}{1-x}-\frac {4 \left (2-e^3\right ) x}{e^3}+\frac {4 \left (2-3 e^3\right ) x}{e^3}+8 x+\frac {2 \left (2-e^3\right ) \sqrt {-4+e^3} \text {arctanh}\left (\frac {2-e^3 (1-2 x)}{e^{3/2} \sqrt {-4+e^3}}\right )}{e^{9/2}}-\frac {2 \left (2-3 e^3\right ) \sqrt {-4+e^3} \text {arctanh}\left (\frac {2-e^3 (1-2 x)}{e^{3/2} \sqrt {-4+e^3}}\right )}{e^{9/2}}-\frac {4 \sqrt {-4+e^3} \text {arctanh}\left (\frac {2-e^3 (1-2 x)}{e^{3/2} \sqrt {-4+e^3}}\right )}{e^{3/2}}-\frac {\left (2-2 e^3-8 e^6+6 e^9-e^{12}+\frac {e^{3/2} \left (8+6 e^3-18 e^6+8 e^9-e^{12}\right )}{\sqrt {-4+e^3}}\right ) \log \left (-\frac {e^{3/2} \left (-2 x-\frac {\sqrt {-4+e^3}}{e^{3/2}}-\frac {2}{e^3}+1\right )}{2 \sqrt {-4+e^3}}\right ) \log \left (2 e^6 x+e^3 \left (2-e^3-e^{3/2} \sqrt {-4+e^3}\right )\right )}{e^6 \left (1+2 e^3\right )}+\frac {\left (2-4 e^3+e^6+\frac {e^{3/2} \left (8-6 e^3+e^6\right )}{\sqrt {-4+e^3}}\right ) \log \left (-\frac {e^{3/2} \left (-2 x-\frac {\sqrt {-4+e^3}}{e^{3/2}}-\frac {2}{e^3}+1\right )}{2 \sqrt {-4+e^3}}\right ) \log \left (2 e^6 x+e^3 \left (2-e^3-e^{3/2} \sqrt {-4+e^3}\right )\right )}{e^6}-\frac {e^3 \left (2+e^3+e^{3/2} \sqrt {-4+e^3}\right ) \log \left (-\frac {e^{3/2} \left (-2 x-\frac {\sqrt {-4+e^3}}{e^{3/2}}-\frac {2}{e^3}+1\right )}{2 \sqrt {-4+e^3}}\right ) \log \left (2 e^6 x+e^3 \left (2-e^3-e^{3/2} \sqrt {-4+e^3}\right )\right )}{1+2 e^3}-\frac {\left (2-e^3-e^{3/2} \sqrt {-4+e^3}\right ) \log \left (-\frac {e^{3/2} \left (-2 x-\frac {\sqrt {-4+e^3}}{e^{3/2}}-\frac {2}{e^3}+1\right )}{2 \sqrt {-4+e^3}}\right ) \log \left (2 e^6 x+e^3 \left (2-e^3-e^{3/2} \sqrt {-4+e^3}\right )\right )}{e^3}-\frac {\left (2-2 e^3-8 e^6+6 e^9-e^{12}-\frac {e^{3/2} \left (8+6 e^3-18 e^6+8 e^9-e^{12}\right )}{\sqrt {-4+e^3}}\right ) \log \left (\frac {e^{3/2} \left (-2 x+\frac {\sqrt {-4+e^3}}{e^{3/2}}-\frac {2}{e^3}+1\right )}{2 \sqrt {-4+e^3}}\right ) \log \left (2 e^6 x+e^3 \left (2-e^3+e^{3/2} \sqrt {-4+e^3}\right )\right )}{e^6 \left (1+2 e^3\right )}+\frac {\left (2-4 e^3+e^6-\frac {e^{3/2} \left (8-6 e^3+e^6\right )}{\sqrt {-4+e^3}}\right ) \log \left (\frac {e^{3/2} \left (-2 x+\frac {\sqrt {-4+e^3}}{e^{3/2}}-\frac {2}{e^3}+1\right )}{2 \sqrt {-4+e^3}}\right ) \log \left (2 e^6 x+e^3 \left (2-e^3+e^{3/2} \sqrt {-4+e^3}\right )\right )}{e^6}-\frac {\left (2-e^3+e^{3/2} \sqrt {-4+e^3}\right ) \log \left (\frac {e^{3/2} \left (-2 x+\frac {\sqrt {-4+e^3}}{e^{3/2}}-\frac {2}{e^3}+1\right )}{2 \sqrt {-4+e^3}}\right ) \log \left (2 e^6 x+e^3 \left (2-e^3+e^{3/2} \sqrt {-4+e^3}\right )\right )}{e^3}-\frac {e^3 \left (2+e^3-e^{3/2} \sqrt {-4+e^3}\right ) \log \left (\frac {e^{3/2} \left (-2 x+\frac {\sqrt {-4+e^3}}{e^{3/2}}-\frac {2}{e^3}+1\right )}{2 \sqrt {-4+e^3}}\right ) \log \left (2 e^6 x+e^3 \left (2-e^3+e^{3/2} \sqrt {-4+e^3}\right )\right )}{1+2 e^3}+\frac {2 \left (2-e^3\right ) x \log \left (-x^2+\left (1-\frac {2}{e^3}\right ) x-\frac {1}{e^6}\right )}{e^3}-\frac {2 \left (2-3 e^3\right ) x \log \left (-x^2+\left (1-\frac {2}{e^3}\right ) x-\frac {1}{e^6}\right )}{e^3}-4 x \log \left (-x^2+\left (1-\frac {2}{e^3}\right ) x-\frac {1}{e^6}\right )+\frac {\left (2-2 e^3-8 e^6+6 e^9-e^{12}+\frac {e^{3/2} \left (8+6 e^3-18 e^6+8 e^9-e^{12}\right )}{\sqrt {-4+e^3}}\right ) \log \left (2 e^6 x+e^3 \left (2-e^3-e^{3/2} \sqrt {-4+e^3}\right )\right ) \log \left (-x^2+\left (1-\frac {2}{e^3}\right ) x-\frac {1}{e^6}\right )}{e^6 \left (1+2 e^3\right )}-\frac {\left (2-4 e^3+e^6+\frac {e^{3/2} \left (8-6 e^3+e^6\right )}{\sqrt {-4+e^3}}\right ) \log \left (2 e^6 x+e^3 \left (2-e^3-e^{3/2} \sqrt {-4+e^3}\right )\right ) \log \left (-x^2+\left (1-\frac {2}{e^3}\right ) x-\frac {1}{e^6}\right )}{e^6}+\frac {e^3 \left (2+e^3+e^{3/2} \sqrt {-4+e^3}\right ) \log \left (2 e^6 x+e^3 \left (2-e^3-e^{3/2} \sqrt {-4+e^3}\right )\right ) \log \left (-x^2+\left (1-\frac {2}{e^3}\right ) x-\frac {1}{e^6}\right )}{1+2 e^3}+\frac {\left (2-e^3-e^{3/2} \sqrt {-4+e^3}\right ) \log \left (2 e^6 x+e^3 \left (2-e^3-e^{3/2} \sqrt {-4+e^3}\right )\right ) \log \left (-x^2+\left (1-\frac {2}{e^3}\right ) x-\frac {1}{e^6}\right )}{e^3}+\frac {\left (2-2 e^3-8 e^6+6 e^9-e^{12}-\frac {e^{3/2} \left (8+6 e^3-18 e^6+8 e^9-e^{12}\right )}{\sqrt {-4+e^3}}\right ) \log \left (2 e^6 x+e^3 \left (2-e^3+e^{3/2} \sqrt {-4+e^3}\right )\right ) \log \left (-x^2+\left (1-\frac {2}{e^3}\right ) x-\frac {1}{e^6}\right )}{e^6 \left (1+2 e^3\right )}-\frac {\left (2-4 e^3+e^6-\frac {e^{3/2} \left (8-6 e^3+e^6\right )}{\sqrt {-4+e^3}}\right ) \log \left (2 e^6 x+e^3 \left (2-e^3+e^{3/2} \sqrt {-4+e^3}\right )\right ) \log \left (-x^2+\left (1-\frac {2}{e^3}\right ) x-\frac {1}{e^6}\right )}{e^6}+\frac {\left (2-e^3+e^{3/2} \sqrt {-4+e^3}\right ) \log \left (2 e^6 x+e^3 \left (2-e^3+e^{3/2} \sqrt {-4+e^3}\right )\right ) \log \left (-x^2+\left (1-\frac {2}{e^3}\right ) x-\frac {1}{e^6}\right )}{e^3}+\frac {e^3 \left (2+e^3-e^{3/2} \sqrt {-4+e^3}\right ) \log \left (2 e^6 x+e^3 \left (2-e^3+e^{3/2} \sqrt {-4+e^3}\right )\right ) \log \left (-x^2+\left (1-\frac {2}{e^3}\right ) x-\frac {1}{e^6}\right )}{1+2 e^3}+\frac {\left (2-e^3\right )^2 \log \left (e^6 x^2+e^3 \left (2-e^3\right ) x+1\right )}{e^6}-\frac {\left (2-3 e^3\right ) \left (2-e^3\right ) \log \left (e^6 x^2+e^3 \left (2-e^3\right ) x+1\right )}{e^6}-\frac {2 \left (2-e^3\right ) \log \left (e^6 x^2+e^3 \left (2-e^3\right ) x+1\right )}{e^3}-\frac {\left (2-2 e^3-8 e^6+6 e^9-e^{12}+\frac {e^{3/2} \left (8+6 e^3-18 e^6+8 e^9-e^{12}\right )}{\sqrt {-4+e^3}}\right ) \operatorname {PolyLog}\left (2,-\frac {-e^3 (1-2 x)-e^{3/2} \sqrt {-4+e^3}+2}{2 e^{3/2} \sqrt {-4+e^3}}\right )}{e^6 \left (1+2 e^3\right )}+\frac {\left (2-4 e^3+e^6+\frac {e^{3/2} \left (8-6 e^3+e^6\right )}{\sqrt {-4+e^3}}\right ) \operatorname {PolyLog}\left (2,-\frac {-e^3 (1-2 x)-e^{3/2} \sqrt {-4+e^3}+2}{2 e^{3/2} \sqrt {-4+e^3}}\right )}{e^6}-\frac {e^3 \left (2+e^3+e^{3/2} \sqrt {-4+e^3}\right ) \operatorname {PolyLog}\left (2,-\frac {-e^3 (1-2 x)-e^{3/2} \sqrt {-4+e^3}+2}{2 e^{3/2} \sqrt {-4+e^3}}\right )}{1+2 e^3}-\frac {\left (2-e^3-e^{3/2} \sqrt {-4+e^3}\right ) \operatorname {PolyLog}\left (2,-\frac {-e^3 (1-2 x)-e^{3/2} \sqrt {-4+e^3}+2}{2 e^{3/2} \sqrt {-4+e^3}}\right )}{e^3}-\frac {\left (2-2 e^3-8 e^6+6 e^9-e^{12}-\frac {e^{3/2} \left (8+6 e^3-18 e^6+8 e^9-e^{12}\right )}{\sqrt {-4+e^3}}\right ) \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {2}{e^{3/2} \sqrt {-4+e^3}}\right )-\frac {e^{3/2} (1-2 x)}{2 \sqrt {-4+e^3}}\right )}{e^6 \left (1+2 e^3\right )}+\frac {\left (2-4 e^3+e^6-\frac {e^{3/2} \left (8-6 e^3+e^6\right )}{\sqrt {-4+e^3}}\right ) \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {2}{e^{3/2} \sqrt {-4+e^3}}\right )-\frac {e^{3/2} (1-2 x)}{2 \sqrt {-4+e^3}}\right )}{e^6}-\frac {\left (2-e^3+e^{3/2} \sqrt {-4+e^3}\right ) \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {2}{e^{3/2} \sqrt {-4+e^3}}\right )-\frac {e^{3/2} (1-2 x)}{2 \sqrt {-4+e^3}}\right )}{e^3}-\frac {e^3 \left (2+e^3-e^{3/2} \sqrt {-4+e^3}\right ) \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {2}{e^{3/2} \sqrt {-4+e^3}}\right )-\frac {e^{3/2} (1-2 x)}{2 \sqrt {-4+e^3}}\right )}{1+2 e^3}\right )\) |
Input:
Int[((E^3*(-4*x^3 + 4*x^4) + E^6*(2*x^3 - 6*x^4 + 4*x^5))*Log[(-1 - 2*E^3* x + E^6*(x - x^2))/E^6] + (-3*x^2 + 2*x^3 + E^3*(-6*x^3 + 4*x^4) + E^6*(3* x^3 - 5*x^4 + 2*x^5))*Log[(-1 - 2*E^3*x + E^6*(x - x^2))/E^6]^2)/(20 - 40* x + 20*x^2 + E^3*(40*x - 80*x^2 + 40*x^3) + E^6*(-20*x + 60*x^2 - 60*x^3 + 20*x^4)),x]
Output:
(8*x + (4*(2 - 3*E^3)*x)/E^3 - (4*(2 - E^3)*x)/E^3 - (4*Sqrt[-4 + E^3]*Arc Tanh[(2 - E^3*(1 - 2*x))/(E^(3/2)*Sqrt[-4 + E^3])])/E^(3/2) - (2*(2 - 3*E^ 3)*Sqrt[-4 + E^3]*ArcTanh[(2 - E^3*(1 - 2*x))/(E^(3/2)*Sqrt[-4 + E^3])])/E ^(9/2) + (2*(2 - E^3)*Sqrt[-4 + E^3]*ArcTanh[(2 - E^3*(1 - 2*x))/(E^(3/2)* Sqrt[-4 + E^3])])/E^(9/2) - ((2 - E^3 - E^(3/2)*Sqrt[-4 + E^3])*Log[E^3*(2 - E^(3/2)*Sqrt[-4 + E^3] - E^3*(1 - 2*x))]^2)/(2*E^3) - (E^3*(2 + E^3 + E ^(3/2)*Sqrt[-4 + E^3])*Log[E^3*(2 - E^(3/2)*Sqrt[-4 + E^3] - E^3*(1 - 2*x) )]^2)/(2*(1 + 2*E^3)) + ((2 - 4*E^3 + E^6 + (E^(3/2)*(8 - 6*E^3 + E^6))/Sq rt[-4 + E^3])*Log[E^3*(2 - E^(3/2)*Sqrt[-4 + E^3] - E^3*(1 - 2*x))]^2)/(2* E^6) - ((2 - 2*E^3 - 8*E^6 + 6*E^9 - E^12 + (E^(3/2)*(8 + 6*E^3 - 18*E^6 + 8*E^9 - E^12))/Sqrt[-4 + E^3])*Log[E^3*(2 - E^(3/2)*Sqrt[-4 + E^3] - E^3* (1 - 2*x))]^2)/(2*E^6*(1 + 2*E^3)) - (E^3*(2 + E^3 - E^(3/2)*Sqrt[-4 + E^3 ])*Log[E^3*(2 + E^(3/2)*Sqrt[-4 + E^3] - E^3*(1 - 2*x))]^2)/(2*(1 + 2*E^3) ) - ((2 - E^3 + E^(3/2)*Sqrt[-4 + E^3])*Log[E^3*(2 + E^(3/2)*Sqrt[-4 + E^3 ] - E^3*(1 - 2*x))]^2)/(2*E^3) + ((2 - 4*E^3 + E^6 - (E^(3/2)*(8 - 6*E^3 + E^6))/Sqrt[-4 + E^3])*Log[E^3*(2 + E^(3/2)*Sqrt[-4 + E^3] - E^3*(1 - 2*x) )]^2)/(2*E^6) - ((2 - 2*E^3 - 8*E^6 + 6*E^9 - E^12 - (E^(3/2)*(8 + 6*E^3 - 18*E^6 + 8*E^9 - E^12))/Sqrt[-4 + E^3])*Log[E^3*(2 + E^(3/2)*Sqrt[-4 + E^ 3] - E^3*(1 - 2*x))]^2)/(2*E^6*(1 + 2*E^3)) - ((2 - E^3 - E^(3/2)*Sqrt[-4 + E^3])*Log[-1/2*(E^(3/2)*(1 - 2/E^3 - Sqrt[-4 + E^3]/E^(3/2) - 2*x))/S...
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]]
Time = 65.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00
| method | result | size |
| risch | \(\frac {x^{3} {\ln \left (\left (\left (-x^{2}+x \right ) {\mathrm e}^{6}-2 x \,{\mathrm e}^{3}-1\right ) {\mathrm e}^{-6}\right )}^{2}}{-20+20 x}\) | \(34\) |
| norman | \(\frac {x^{3} {\ln \left (\left (\left (-x^{2}+x \right ) {\mathrm e}^{6}-2 x \,{\mathrm e}^{3}-1\right ) {\mathrm e}^{-6}\right )}^{2}}{-20+20 x}\) | \(38\) |
| parallelrisch | \(\frac {x^{3} {\ln \left (\left (\left (-x^{2}+x \right ) {\mathrm e}^{6}-2 x \,{\mathrm e}^{3}-1\right ) {\mathrm e}^{-6}\right )}^{2}}{-20+20 x}\) | \(38\) |
Input:
int((((2*x^5-5*x^4+3*x^3)*exp(3)^2+(4*x^4-6*x^3)*exp(3)+2*x^3-3*x^2)*ln((( -x^2+x)*exp(3)^2-2*x*exp(3)-1)/exp(3)^2)^2+((4*x^5-6*x^4+2*x^3)*exp(3)^2+( 4*x^4-4*x^3)*exp(3))*ln(((-x^2+x)*exp(3)^2-2*x*exp(3)-1)/exp(3)^2))/((20*x ^4-60*x^3+60*x^2-20*x)*exp(3)^2+(40*x^3-80*x^2+40*x)*exp(3)+20*x^2-40*x+20 ),x,method=_RETURNVERBOSE)
Output:
1/20*x^3*ln(((-x^2+x)*exp(6)-2*x*exp(3)-1)*exp(-6))^2/(-1+x)
Time = 0.09 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {\left (e^3 \left (-4 x^3+4 x^4\right )+e^6 \left (2 x^3-6 x^4+4 x^5\right )\right ) \log \left (\frac {-1-2 e^3 x+e^6 \left (x-x^2\right )}{e^6}\right )+\left (-3 x^2+2 x^3+e^3 \left (-6 x^3+4 x^4\right )+e^6 \left (3 x^3-5 x^4+2 x^5\right )\right ) \log ^2\left (\frac {-1-2 e^3 x+e^6 \left (x-x^2\right )}{e^6}\right )}{20-40 x+20 x^2+e^3 \left (40 x-80 x^2+40 x^3\right )+e^6 \left (-20 x+60 x^2-60 x^3+20 x^4\right )} \, dx=\frac {x^{3} \log \left (-{\left ({\left (x^{2} - x\right )} e^{6} + 2 \, x e^{3} + 1\right )} e^{\left (-6\right )}\right )^{2}}{20 \, {\left (x - 1\right )}} \] Input:
integrate((((2*x^5-5*x^4+3*x^3)*exp(3)^2+(4*x^4-6*x^3)*exp(3)+2*x^3-3*x^2) *log(((-x^2+x)*exp(3)^2-2*x*exp(3)-1)/exp(3)^2)^2+((4*x^5-6*x^4+2*x^3)*exp (3)^2+(4*x^4-4*x^3)*exp(3))*log(((-x^2+x)*exp(3)^2-2*x*exp(3)-1)/exp(3)^2) )/((20*x^4-60*x^3+60*x^2-20*x)*exp(3)^2+(40*x^3-80*x^2+40*x)*exp(3)+20*x^2 -40*x+20),x, algorithm="fricas")
Output:
1/20*x^3*log(-((x^2 - x)*e^6 + 2*x*e^3 + 1)*e^(-6))^2/(x - 1)
Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91 \[ \int \frac {\left (e^3 \left (-4 x^3+4 x^4\right )+e^6 \left (2 x^3-6 x^4+4 x^5\right )\right ) \log \left (\frac {-1-2 e^3 x+e^6 \left (x-x^2\right )}{e^6}\right )+\left (-3 x^2+2 x^3+e^3 \left (-6 x^3+4 x^4\right )+e^6 \left (3 x^3-5 x^4+2 x^5\right )\right ) \log ^2\left (\frac {-1-2 e^3 x+e^6 \left (x-x^2\right )}{e^6}\right )}{20-40 x+20 x^2+e^3 \left (40 x-80 x^2+40 x^3\right )+e^6 \left (-20 x+60 x^2-60 x^3+20 x^4\right )} \, dx=\frac {x^{3} \log {\left (\frac {- 2 x e^{3} + \left (- x^{2} + x\right ) e^{6} - 1}{e^{6}} \right )}^{2}}{20 x - 20} \] Input:
integrate((((2*x**5-5*x**4+3*x**3)*exp(3)**2+(4*x**4-6*x**3)*exp(3)+2*x**3 -3*x**2)*ln(((-x**2+x)*exp(3)**2-2*x*exp(3)-1)/exp(3)**2)**2+((4*x**5-6*x* *4+2*x**3)*exp(3)**2+(4*x**4-4*x**3)*exp(3))*ln(((-x**2+x)*exp(3)**2-2*x*e xp(3)-1)/exp(3)**2))/((20*x**4-60*x**3+60*x**2-20*x)*exp(3)**2+(40*x**3-80 *x**2+40*x)*exp(3)+20*x**2-40*x+20),x)
Output:
x**3*log((-2*x*exp(3) + (-x**2 + x)*exp(6) - 1)*exp(-6))**2/(20*x - 20)
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (27) = 54\).
Time = 0.28 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.94 \[ \int \frac {\left (e^3 \left (-4 x^3+4 x^4\right )+e^6 \left (2 x^3-6 x^4+4 x^5\right )\right ) \log \left (\frac {-1-2 e^3 x+e^6 \left (x-x^2\right )}{e^6}\right )+\left (-3 x^2+2 x^3+e^3 \left (-6 x^3+4 x^4\right )+e^6 \left (3 x^3-5 x^4+2 x^5\right )\right ) \log ^2\left (\frac {-1-2 e^3 x+e^6 \left (x-x^2\right )}{e^6}\right )}{20-40 x+20 x^2+e^3 \left (40 x-80 x^2+40 x^3\right )+e^6 \left (-20 x+60 x^2-60 x^3+20 x^4\right )} \, dx=\frac {x^{3} \log \left (-x^{2} e^{6} + x {\left (e^{6} - 2 \, e^{3}\right )} - 1\right )^{2} - 12 \, x^{3} \log \left (-x^{2} e^{6} + x {\left (e^{6} - 2 \, e^{3}\right )} - 1\right ) + 36 \, x^{3} - 36 \, x + 36}{20 \, {\left (x - 1\right )}} \] Input:
integrate((((2*x^5-5*x^4+3*x^3)*exp(3)^2+(4*x^4-6*x^3)*exp(3)+2*x^3-3*x^2) *log(((-x^2+x)*exp(3)^2-2*x*exp(3)-1)/exp(3)^2)^2+((4*x^5-6*x^4+2*x^3)*exp (3)^2+(4*x^4-4*x^3)*exp(3))*log(((-x^2+x)*exp(3)^2-2*x*exp(3)-1)/exp(3)^2) )/((20*x^4-60*x^3+60*x^2-20*x)*exp(3)^2+(40*x^3-80*x^2+40*x)*exp(3)+20*x^2 -40*x+20),x, algorithm="maxima")
Output:
1/20*(x^3*log(-x^2*e^6 + x*(e^6 - 2*e^3) - 1)^2 - 12*x^3*log(-x^2*e^6 + x* (e^6 - 2*e^3) - 1) + 36*x^3 - 36*x + 36)/(x - 1)
\[ \int \frac {\left (e^3 \left (-4 x^3+4 x^4\right )+e^6 \left (2 x^3-6 x^4+4 x^5\right )\right ) \log \left (\frac {-1-2 e^3 x+e^6 \left (x-x^2\right )}{e^6}\right )+\left (-3 x^2+2 x^3+e^3 \left (-6 x^3+4 x^4\right )+e^6 \left (3 x^3-5 x^4+2 x^5\right )\right ) \log ^2\left (\frac {-1-2 e^3 x+e^6 \left (x-x^2\right )}{e^6}\right )}{20-40 x+20 x^2+e^3 \left (40 x-80 x^2+40 x^3\right )+e^6 \left (-20 x+60 x^2-60 x^3+20 x^4\right )} \, dx=\int { \frac {{\left (2 \, x^{3} - 3 \, x^{2} + {\left (2 \, x^{5} - 5 \, x^{4} + 3 \, x^{3}\right )} e^{6} + 2 \, {\left (2 \, x^{4} - 3 \, x^{3}\right )} e^{3}\right )} \log \left (-{\left ({\left (x^{2} - x\right )} e^{6} + 2 \, x e^{3} + 1\right )} e^{\left (-6\right )}\right )^{2} + 2 \, {\left ({\left (2 \, x^{5} - 3 \, x^{4} + x^{3}\right )} e^{6} + 2 \, {\left (x^{4} - x^{3}\right )} e^{3}\right )} \log \left (-{\left ({\left (x^{2} - x\right )} e^{6} + 2 \, x e^{3} + 1\right )} e^{\left (-6\right )}\right )}{20 \, {\left (x^{2} + {\left (x^{4} - 3 \, x^{3} + 3 \, x^{2} - x\right )} e^{6} + 2 \, {\left (x^{3} - 2 \, x^{2} + x\right )} e^{3} - 2 \, x + 1\right )}} \,d x } \] Input:
integrate((((2*x^5-5*x^4+3*x^3)*exp(3)^2+(4*x^4-6*x^3)*exp(3)+2*x^3-3*x^2) *log(((-x^2+x)*exp(3)^2-2*x*exp(3)-1)/exp(3)^2)^2+((4*x^5-6*x^4+2*x^3)*exp (3)^2+(4*x^4-4*x^3)*exp(3))*log(((-x^2+x)*exp(3)^2-2*x*exp(3)-1)/exp(3)^2) )/((20*x^4-60*x^3+60*x^2-20*x)*exp(3)^2+(40*x^3-80*x^2+40*x)*exp(3)+20*x^2 -40*x+20),x, algorithm="giac")
Output:
integrate(1/20*((2*x^3 - 3*x^2 + (2*x^5 - 5*x^4 + 3*x^3)*e^6 + 2*(2*x^4 - 3*x^3)*e^3)*log(-((x^2 - x)*e^6 + 2*x*e^3 + 1)*e^(-6))^2 + 2*((2*x^5 - 3*x ^4 + x^3)*e^6 + 2*(x^4 - x^3)*e^3)*log(-((x^2 - x)*e^6 + 2*x*e^3 + 1)*e^(- 6)))/(x^2 + (x^4 - 3*x^3 + 3*x^2 - x)*e^6 + 2*(x^3 - 2*x^2 + x)*e^3 - 2*x + 1), x)
Time = 4.78 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88 \[ \int \frac {\left (e^3 \left (-4 x^3+4 x^4\right )+e^6 \left (2 x^3-6 x^4+4 x^5\right )\right ) \log \left (\frac {-1-2 e^3 x+e^6 \left (x-x^2\right )}{e^6}\right )+\left (-3 x^2+2 x^3+e^3 \left (-6 x^3+4 x^4\right )+e^6 \left (3 x^3-5 x^4+2 x^5\right )\right ) \log ^2\left (\frac {-1-2 e^3 x+e^6 \left (x-x^2\right )}{e^6}\right )}{20-40 x+20 x^2+e^3 \left (40 x-80 x^2+40 x^3\right )+e^6 \left (-20 x+60 x^2-60 x^3+20 x^4\right )} \, dx=\frac {x^3\,{\ln \left (x-{\mathrm {e}}^{-6}-2\,x\,{\mathrm {e}}^{-3}-x^2\right )}^2}{20\,\left (x-1\right )} \] Input:
int(-(log(-exp(-6)*(2*x*exp(3) - exp(6)*(x - x^2) + 1))^2*(exp(3)*(6*x^3 - 4*x^4) - exp(6)*(3*x^3 - 5*x^4 + 2*x^5) + 3*x^2 - 2*x^3) + log(-exp(-6)*( 2*x*exp(3) - exp(6)*(x - x^2) + 1))*(exp(3)*(4*x^3 - 4*x^4) - exp(6)*(2*x^ 3 - 6*x^4 + 4*x^5)))/(exp(3)*(40*x - 80*x^2 + 40*x^3) - 40*x - exp(6)*(20* x - 60*x^2 + 60*x^3 - 20*x^4) + 20*x^2 + 20),x)
Output:
(x^3*log(x - exp(-6) - 2*x*exp(-3) - x^2)^2)/(20*(x - 1))
Time = 0.17 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.15 \[ \int \frac {\left (e^3 \left (-4 x^3+4 x^4\right )+e^6 \left (2 x^3-6 x^4+4 x^5\right )\right ) \log \left (\frac {-1-2 e^3 x+e^6 \left (x-x^2\right )}{e^6}\right )+\left (-3 x^2+2 x^3+e^3 \left (-6 x^3+4 x^4\right )+e^6 \left (3 x^3-5 x^4+2 x^5\right )\right ) \log ^2\left (\frac {-1-2 e^3 x+e^6 \left (x-x^2\right )}{e^6}\right )}{20-40 x+20 x^2+e^3 \left (40 x-80 x^2+40 x^3\right )+e^6 \left (-20 x+60 x^2-60 x^3+20 x^4\right )} \, dx=\frac {\mathrm {log}\left (\frac {-e^{6} x^{2}+e^{6} x -2 e^{3} x -1}{e^{6}}\right )^{2} x^{3}}{20 x -20} \] Input:
int((((2*x^5-5*x^4+3*x^3)*exp(3)^2+(4*x^4-6*x^3)*exp(3)+2*x^3-3*x^2)*log(( (-x^2+x)*exp(3)^2-2*x*exp(3)-1)/exp(3)^2)^2+((4*x^5-6*x^4+2*x^3)*exp(3)^2+ (4*x^4-4*x^3)*exp(3))*log(((-x^2+x)*exp(3)^2-2*x*exp(3)-1)/exp(3)^2))/((20 *x^4-60*x^3+60*x^2-20*x)*exp(3)^2+(40*x^3-80*x^2+40*x)*exp(3)+20*x^2-40*x+ 20),x)
Output:
(log(( - e**6*x**2 + e**6*x - 2*e**3*x - 1)/e**6)**2*x**3)/(20*(x - 1))