Integrand size = 458, antiderivative size = 35 \[ \int \frac {4 x^3+4 x^4-12 x^5-12 x^6+e^3 \left (-4 x^3+12 x^5\right )+\left (8 x^2+8 x^3-24 x^4-24 x^5+e^3 \left (-8 x^2+24 x^4\right )\right ) \log \left (1-e^3+x\right )+\left (4 x+4 x^2-12 x^3-12 x^4+e^3 \left (-4 x+12 x^3\right )\right ) \log ^2\left (1-e^3+x\right )+\log ^2(x) \left (3 x+2 x^2-12 x^3-6 x^4+9 x^5+e^3 \left (-2 x+6 x^3\right )+\left (1+x-9 x^4-9 x^5+e^3 \left (-1+9 x^4\right )\right ) \log \left (1-e^3+x\right )\right )+\log (x) \left (-x-x^2-6 x^3-2 x^4+3 x^5-9 x^6+e^3 \left (x+2 x^3+9 x^5\right )+\left (-1-x-10 x^2-6 x^3-9 x^4-21 x^5+e^3 \left (1+6 x^2+21 x^4\right )\right ) \log \left (1-e^3+x\right )+\left (-4 x-4 x^2-12 x^3-12 x^4+e^3 \left (4 x+12 x^3\right )\right ) \log ^2\left (1-e^3+x\right )\right )}{-2 x^6+2 e^3 x^6-2 x^7+\left (-6 x^5+6 e^3 x^5-6 x^6\right ) \log \left (1-e^3+x\right )+\left (-6 x^4+6 e^3 x^4-6 x^5\right ) \log ^2\left (1-e^3+x\right )+\left (-2 x^3+2 e^3 x^3-2 x^4\right ) \log ^3\left (1-e^3+x\right )} \, dx=\left (-2+\frac {\left (\frac {1}{2} \left (\frac {1}{x}-x\right )-x\right ) \log (x)}{x+\log \left (1-e^3+x\right )}\right )^2 \] Output:
((1/2/x-3/2*x)/(ln(-exp(3)+x+1)+x)*ln(x)-2)^2
Time = 0.09 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.54 \[ \int \frac {4 x^3+4 x^4-12 x^5-12 x^6+e^3 \left (-4 x^3+12 x^5\right )+\left (8 x^2+8 x^3-24 x^4-24 x^5+e^3 \left (-8 x^2+24 x^4\right )\right ) \log \left (1-e^3+x\right )+\left (4 x+4 x^2-12 x^3-12 x^4+e^3 \left (-4 x+12 x^3\right )\right ) \log ^2\left (1-e^3+x\right )+\log ^2(x) \left (3 x+2 x^2-12 x^3-6 x^4+9 x^5+e^3 \left (-2 x+6 x^3\right )+\left (1+x-9 x^4-9 x^5+e^3 \left (-1+9 x^4\right )\right ) \log \left (1-e^3+x\right )\right )+\log (x) \left (-x-x^2-6 x^3-2 x^4+3 x^5-9 x^6+e^3 \left (x+2 x^3+9 x^5\right )+\left (-1-x-10 x^2-6 x^3-9 x^4-21 x^5+e^3 \left (1+6 x^2+21 x^4\right )\right ) \log \left (1-e^3+x\right )+\left (-4 x-4 x^2-12 x^3-12 x^4+e^3 \left (4 x+12 x^3\right )\right ) \log ^2\left (1-e^3+x\right )\right )}{-2 x^6+2 e^3 x^6-2 x^7+\left (-6 x^5+6 e^3 x^5-6 x^6\right ) \log \left (1-e^3+x\right )+\left (-6 x^4+6 e^3 x^4-6 x^5\right ) \log ^2\left (1-e^3+x\right )+\left (-2 x^3+2 e^3 x^3-2 x^4\right ) \log ^3\left (1-e^3+x\right )} \, dx=\frac {\left (-1+3 x^2\right ) \log (x) \left (\left (-1+3 x^2\right ) \log (x)+8 x \left (x+\log \left (1-e^3+x\right )\right )\right )}{4 x^2 \left (x+\log \left (1-e^3+x\right )\right )^2} \] Input:
Integrate[(4*x^3 + 4*x^4 - 12*x^5 - 12*x^6 + E^3*(-4*x^3 + 12*x^5) + (8*x^ 2 + 8*x^3 - 24*x^4 - 24*x^5 + E^3*(-8*x^2 + 24*x^4))*Log[1 - E^3 + x] + (4 *x + 4*x^2 - 12*x^3 - 12*x^4 + E^3*(-4*x + 12*x^3))*Log[1 - E^3 + x]^2 + L og[x]^2*(3*x + 2*x^2 - 12*x^3 - 6*x^4 + 9*x^5 + E^3*(-2*x + 6*x^3) + (1 + x - 9*x^4 - 9*x^5 + E^3*(-1 + 9*x^4))*Log[1 - E^3 + x]) + Log[x]*(-x - x^2 - 6*x^3 - 2*x^4 + 3*x^5 - 9*x^6 + E^3*(x + 2*x^3 + 9*x^5) + (-1 - x - 10* x^2 - 6*x^3 - 9*x^4 - 21*x^5 + E^3*(1 + 6*x^2 + 21*x^4))*Log[1 - E^3 + x] + (-4*x - 4*x^2 - 12*x^3 - 12*x^4 + E^3*(4*x + 12*x^3))*Log[1 - E^3 + x]^2 ))/(-2*x^6 + 2*E^3*x^6 - 2*x^7 + (-6*x^5 + 6*E^3*x^5 - 6*x^6)*Log[1 - E^3 + x] + (-6*x^4 + 6*E^3*x^4 - 6*x^5)*Log[1 - E^3 + x]^2 + (-2*x^3 + 2*E^3*x ^3 - 2*x^4)*Log[1 - E^3 + x]^3),x]
Output:
((-1 + 3*x^2)*Log[x]*((-1 + 3*x^2)*Log[x] + 8*x*(x + Log[1 - E^3 + x])))/( 4*x^2*(x + Log[1 - E^3 + x])^2)
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.
| \(\displaystyle \int \frac {-12 x^6-12 x^5+4 x^4+4 x^3+e^3 \left (12 x^5-4 x^3\right )+\left (-12 x^4-12 x^3+e^3 \left (12 x^3-4 x\right )+4 x^2+4 x\right ) \log ^2\left (x-e^3+1\right )+\log ^2(x) \left (9 x^5-6 x^4-12 x^3+e^3 \left (6 x^3-2 x\right )+2 x^2+\left (-9 x^5-9 x^4+e^3 \left (9 x^4-1\right )+x+1\right ) \log \left (x-e^3+1\right )+3 x\right )+\left (-24 x^5-24 x^4+8 x^3+8 x^2+e^3 \left (24 x^4-8 x^2\right )\right ) \log \left (x-e^3+1\right )+\log (x) \left (-9 x^6+3 x^5-2 x^4-6 x^3-x^2+e^3 \left (9 x^5+2 x^3+x\right )+\left (-12 x^4-12 x^3+e^3 \left (12 x^3+4 x\right )-4 x^2-4 x\right ) \log ^2\left (x-e^3+1\right )+\left (-21 x^5-9 x^4-6 x^3-10 x^2+e^3 \left (21 x^4+6 x^2+1\right )-x-1\right ) \log \left (x-e^3+1\right )-x\right )}{-2 x^7+2 e^3 x^6-2 x^6+\left (-6 x^6+6 e^3 x^5-6 x^5\right ) \log \left (x-e^3+1\right )+\left (-6 x^5+6 e^3 x^4-6 x^4\right ) \log ^2\left (x-e^3+1\right )+\left (-2 x^4+2 e^3 x^3-2 x^3\right ) \log ^3\left (x-e^3+1\right )} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {-12 x^6-12 x^5+4 x^4+4 x^3+e^3 \left (12 x^5-4 x^3\right )+\left (-12 x^4-12 x^3+e^3 \left (12 x^3-4 x\right )+4 x^2+4 x\right ) \log ^2\left (x-e^3+1\right )+\log ^2(x) \left (9 x^5-6 x^4-12 x^3+e^3 \left (6 x^3-2 x\right )+2 x^2+\left (-9 x^5-9 x^4+e^3 \left (9 x^4-1\right )+x+1\right ) \log \left (x-e^3+1\right )+3 x\right )+\left (-24 x^5-24 x^4+8 x^3+8 x^2+e^3 \left (24 x^4-8 x^2\right )\right ) \log \left (x-e^3+1\right )+\log (x) \left (-9 x^6+3 x^5-2 x^4-6 x^3-x^2+e^3 \left (9 x^5+2 x^3+x\right )+\left (-12 x^4-12 x^3+e^3 \left (12 x^3+4 x\right )-4 x^2-4 x\right ) \log ^2\left (x-e^3+1\right )+\left (-21 x^5-9 x^4-6 x^3-10 x^2+e^3 \left (21 x^4+6 x^2+1\right )-x-1\right ) \log \left (x-e^3+1\right )-x\right )}{-2 x^7+\left (2 e^3-2\right ) x^6+\left (-6 x^6+6 e^3 x^5-6 x^5\right ) \log \left (x-e^3+1\right )+\left (-6 x^5+6 e^3 x^4-6 x^4\right ) \log ^2\left (x-e^3+1\right )+\left (-2 x^4+2 e^3 x^3-2 x^3\right ) \log ^3\left (x-e^3+1\right )}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (\left (3 x^2-1\right ) \log (x)+4 x \left (x+\log \left (x-e^3+1\right )\right )\right ) \left (\left (x-e^3+1\right ) \left (3 x^2-1\right ) \left (x+\log \left (x-e^3+1\right )\right )+\log (x) \left (x \left (-3 x^2+2 x-2 e^3+3\right )-\left (-x+e^3-1\right ) \left (3 x^2+1\right ) \log \left (x-e^3+1\right )\right )\right )}{2 x^3 \left (x-e^3+1\right ) \left (x+\log \left (x-e^3+1\right )\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int \frac {\left (\left (1-3 x^2\right ) \log (x)-4 x \left (x+\log \left (x-e^3+1\right )\right )\right ) \left (\left (x-e^3+1\right ) \left (1-3 x^2\right ) \left (x+\log \left (x-e^3+1\right )\right )-\log (x) \left (x \left (-3 x^2+2 x-2 e^3+3\right )+\left (x-e^3+1\right ) \left (3 x^2+1\right ) \log \left (x-e^3+1\right )\right )\right )}{x^3 \left (x-e^3+1\right ) \left (x+\log \left (x-e^3+1\right )\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (-\frac {\left (-x+e^3-2\right ) \left (3 x^2-1\right )^2 \log ^2(x)}{\left (-x+e^3-1\right ) x^2 \left (x+\log \left (x-e^3+1\right )\right )^3}+\frac {\left (1-3 x^2\right ) \left (-3 \log (x) x^3+x^3-3 \left (1-e^3\right ) \log (x) x^2+5 \left (1-\frac {e^3}{5}\right ) x^2-\log (x) x+x-\left (1-e^3\right ) \log (x)-e^3+1\right ) \log (x)}{x^3 \left (x-e^3+1\right ) \left (x+\log \left (x-e^3+1\right )\right )^2}+\frac {4 \left (3 \log (x) x^2+3 x^2+\log (x)-1\right )}{x^2 \left (x+\log \left (x-e^3+1\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \frac {1}{2} \int \left (-\frac {\left (-x+e^3-2\right ) \left (3 x^2-1\right )^2 \log ^2(x)}{\left (-x+e^3-1\right ) x^2 \left (x+\log \left (x-e^3+1\right )\right )^3}+\frac {\left (1-3 x^2\right ) \left (-3 \log (x) x^3+x^3-3 \left (1-e^3\right ) \log (x) x^2+5 \left (1-\frac {e^3}{5}\right ) x^2-\log (x) x+x-\left (1-e^3\right ) \log (x)-e^3+1\right ) \log (x)}{x^3 \left (x-e^3+1\right ) \left (x+\log \left (x-e^3+1\right )\right )^2}+\frac {4 \left (3 \log (x) x^2+3 x^2+\log (x)-1\right )}{x^2 \left (x+\log \left (x-e^3+1\right )\right )}\right )dx\) |
Input:
Int[(4*x^3 + 4*x^4 - 12*x^5 - 12*x^6 + E^3*(-4*x^3 + 12*x^5) + (8*x^2 + 8* x^3 - 24*x^4 - 24*x^5 + E^3*(-8*x^2 + 24*x^4))*Log[1 - E^3 + x] + (4*x + 4 *x^2 - 12*x^3 - 12*x^4 + E^3*(-4*x + 12*x^3))*Log[1 - E^3 + x]^2 + Log[x]^ 2*(3*x + 2*x^2 - 12*x^3 - 6*x^4 + 9*x^5 + E^3*(-2*x + 6*x^3) + (1 + x - 9* x^4 - 9*x^5 + E^3*(-1 + 9*x^4))*Log[1 - E^3 + x]) + Log[x]*(-x - x^2 - 6*x ^3 - 2*x^4 + 3*x^5 - 9*x^6 + E^3*(x + 2*x^3 + 9*x^5) + (-1 - x - 10*x^2 - 6*x^3 - 9*x^4 - 21*x^5 + E^3*(1 + 6*x^2 + 21*x^4))*Log[1 - E^3 + x] + (-4* x - 4*x^2 - 12*x^3 - 12*x^4 + E^3*(4*x + 12*x^3))*Log[1 - E^3 + x]^2))/(-2 *x^6 + 2*E^3*x^6 - 2*x^7 + (-6*x^5 + 6*E^3*x^5 - 6*x^6)*Log[1 - E^3 + x] + (-6*x^4 + 6*E^3*x^4 - 6*x^5)*Log[1 - E^3 + x]^2 + (-2*x^3 + 2*E^3*x^3 - 2 *x^4)*Log[1 - E^3 + x]^3),x]
Output:
$Aborted
\[\int \frac {\left (\left (\left (9 x^{4}-1\right ) {\mathrm e}^{3}-9 x^{5}-9 x^{4}+x +1\right ) \ln \left (-{\mathrm e}^{3}+x +1\right )+\left (6 x^{3}-2 x \right ) {\mathrm e}^{3}+9 x^{5}-6 x^{4}-12 x^{3}+2 x^{2}+3 x \right ) \ln \left (x \right )^{2}+\left (\left (\left (12 x^{3}+4 x \right ) {\mathrm e}^{3}-12 x^{4}-12 x^{3}-4 x^{2}-4 x \right ) \ln \left (-{\mathrm e}^{3}+x +1\right )^{2}+\left (\left (21 x^{4}+6 x^{2}+1\right ) {\mathrm e}^{3}-21 x^{5}-9 x^{4}-6 x^{3}-10 x^{2}-x -1\right ) \ln \left (-{\mathrm e}^{3}+x +1\right )+\left (9 x^{5}+2 x^{3}+x \right ) {\mathrm e}^{3}-9 x^{6}+3 x^{5}-2 x^{4}-6 x^{3}-x^{2}-x \right ) \ln \left (x \right )+\left (\left (12 x^{3}-4 x \right ) {\mathrm e}^{3}-12 x^{4}-12 x^{3}+4 x^{2}+4 x \right ) \ln \left (-{\mathrm e}^{3}+x +1\right )^{2}+\left (\left (24 x^{4}-8 x^{2}\right ) {\mathrm e}^{3}-24 x^{5}-24 x^{4}+8 x^{3}+8 x^{2}\right ) \ln \left (-{\mathrm e}^{3}+x +1\right )+\left (12 x^{5}-4 x^{3}\right ) {\mathrm e}^{3}-12 x^{6}-12 x^{5}+4 x^{4}+4 x^{3}}{\left (2 x^{3} {\mathrm e}^{3}-2 x^{4}-2 x^{3}\right ) \ln \left (-{\mathrm e}^{3}+x +1\right )^{3}+\left (6 x^{4} {\mathrm e}^{3}-6 x^{5}-6 x^{4}\right ) \ln \left (-{\mathrm e}^{3}+x +1\right )^{2}+\left (6 x^{5} {\mathrm e}^{3}-6 x^{6}-6 x^{5}\right ) \ln \left (-{\mathrm e}^{3}+x +1\right )+2 x^{6} {\mathrm e}^{3}-2 x^{7}-2 x^{6}}d x\]
Input:
int(((((9*x^4-1)*exp(3)-9*x^5-9*x^4+x+1)*ln(-exp(3)+x+1)+(6*x^3-2*x)*exp(3 )+9*x^5-6*x^4-12*x^3+2*x^2+3*x)*ln(x)^2+(((12*x^3+4*x)*exp(3)-12*x^4-12*x^ 3-4*x^2-4*x)*ln(-exp(3)+x+1)^2+((21*x^4+6*x^2+1)*exp(3)-21*x^5-9*x^4-6*x^3 -10*x^2-x-1)*ln(-exp(3)+x+1)+(9*x^5+2*x^3+x)*exp(3)-9*x^6+3*x^5-2*x^4-6*x^ 3-x^2-x)*ln(x)+((12*x^3-4*x)*exp(3)-12*x^4-12*x^3+4*x^2+4*x)*ln(-exp(3)+x+ 1)^2+((24*x^4-8*x^2)*exp(3)-24*x^5-24*x^4+8*x^3+8*x^2)*ln(-exp(3)+x+1)+(12 *x^5-4*x^3)*exp(3)-12*x^6-12*x^5+4*x^4+4*x^3)/((2*x^3*exp(3)-2*x^4-2*x^3)* ln(-exp(3)+x+1)^3+(6*x^4*exp(3)-6*x^5-6*x^4)*ln(-exp(3)+x+1)^2+(6*x^5*exp( 3)-6*x^6-6*x^5)*ln(-exp(3)+x+1)+2*x^6*exp(3)-2*x^7-2*x^6),x)
Output:
int(((((9*x^4-1)*exp(3)-9*x^5-9*x^4+x+1)*ln(-exp(3)+x+1)+(6*x^3-2*x)*exp(3 )+9*x^5-6*x^4-12*x^3+2*x^2+3*x)*ln(x)^2+(((12*x^3+4*x)*exp(3)-12*x^4-12*x^ 3-4*x^2-4*x)*ln(-exp(3)+x+1)^2+((21*x^4+6*x^2+1)*exp(3)-21*x^5-9*x^4-6*x^3 -10*x^2-x-1)*ln(-exp(3)+x+1)+(9*x^5+2*x^3+x)*exp(3)-9*x^6+3*x^5-2*x^4-6*x^ 3-x^2-x)*ln(x)+((12*x^3-4*x)*exp(3)-12*x^4-12*x^3+4*x^2+4*x)*ln(-exp(3)+x+ 1)^2+((24*x^4-8*x^2)*exp(3)-24*x^5-24*x^4+8*x^3+8*x^2)*ln(-exp(3)+x+1)+(12 *x^5-4*x^3)*exp(3)-12*x^6-12*x^5+4*x^4+4*x^3)/((2*x^3*exp(3)-2*x^4-2*x^3)* ln(-exp(3)+x+1)^3+(6*x^4*exp(3)-6*x^5-6*x^4)*ln(-exp(3)+x+1)^2+(6*x^5*exp( 3)-6*x^6-6*x^5)*ln(-exp(3)+x+1)+2*x^6*exp(3)-2*x^7-2*x^6),x)
Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (30) = 60\).
Time = 0.10 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.46 \[ \int \frac {4 x^3+4 x^4-12 x^5-12 x^6+e^3 \left (-4 x^3+12 x^5\right )+\left (8 x^2+8 x^3-24 x^4-24 x^5+e^3 \left (-8 x^2+24 x^4\right )\right ) \log \left (1-e^3+x\right )+\left (4 x+4 x^2-12 x^3-12 x^4+e^3 \left (-4 x+12 x^3\right )\right ) \log ^2\left (1-e^3+x\right )+\log ^2(x) \left (3 x+2 x^2-12 x^3-6 x^4+9 x^5+e^3 \left (-2 x+6 x^3\right )+\left (1+x-9 x^4-9 x^5+e^3 \left (-1+9 x^4\right )\right ) \log \left (1-e^3+x\right )\right )+\log (x) \left (-x-x^2-6 x^3-2 x^4+3 x^5-9 x^6+e^3 \left (x+2 x^3+9 x^5\right )+\left (-1-x-10 x^2-6 x^3-9 x^4-21 x^5+e^3 \left (1+6 x^2+21 x^4\right )\right ) \log \left (1-e^3+x\right )+\left (-4 x-4 x^2-12 x^3-12 x^4+e^3 \left (4 x+12 x^3\right )\right ) \log ^2\left (1-e^3+x\right )\right )}{-2 x^6+2 e^3 x^6-2 x^7+\left (-6 x^5+6 e^3 x^5-6 x^6\right ) \log \left (1-e^3+x\right )+\left (-6 x^4+6 e^3 x^4-6 x^5\right ) \log ^2\left (1-e^3+x\right )+\left (-2 x^3+2 e^3 x^3-2 x^4\right ) \log ^3\left (1-e^3+x\right )} \, dx=\frac {{\left (9 \, x^{4} - 6 \, x^{2} + 1\right )} \log \left (x\right )^{2} + 8 \, {\left (3 \, x^{4} - x^{2} + {\left (3 \, x^{3} - x\right )} \log \left (x - e^{3} + 1\right )\right )} \log \left (x\right )}{4 \, {\left (x^{4} + 2 \, x^{3} \log \left (x - e^{3} + 1\right ) + x^{2} \log \left (x - e^{3} + 1\right )^{2}\right )}} \] Input:
integrate(((((9*x^4-1)*exp(3)-9*x^5-9*x^4+x+1)*log(-exp(3)+x+1)+(6*x^3-2*x )*exp(3)+9*x^5-6*x^4-12*x^3+2*x^2+3*x)*log(x)^2+(((12*x^3+4*x)*exp(3)-12*x ^4-12*x^3-4*x^2-4*x)*log(-exp(3)+x+1)^2+((21*x^4+6*x^2+1)*exp(3)-21*x^5-9* x^4-6*x^3-10*x^2-x-1)*log(-exp(3)+x+1)+(9*x^5+2*x^3+x)*exp(3)-9*x^6+3*x^5- 2*x^4-6*x^3-x^2-x)*log(x)+((12*x^3-4*x)*exp(3)-12*x^4-12*x^3+4*x^2+4*x)*lo g(-exp(3)+x+1)^2+((24*x^4-8*x^2)*exp(3)-24*x^5-24*x^4+8*x^3+8*x^2)*log(-ex p(3)+x+1)+(12*x^5-4*x^3)*exp(3)-12*x^6-12*x^5+4*x^4+4*x^3)/((2*x^3*exp(3)- 2*x^4-2*x^3)*log(-exp(3)+x+1)^3+(6*x^4*exp(3)-6*x^5-6*x^4)*log(-exp(3)+x+1 )^2+(6*x^5*exp(3)-6*x^6-6*x^5)*log(-exp(3)+x+1)+2*x^6*exp(3)-2*x^7-2*x^6), x, algorithm="fricas")
Output:
1/4*((9*x^4 - 6*x^2 + 1)*log(x)^2 + 8*(3*x^4 - x^2 + (3*x^3 - x)*log(x - e ^3 + 1))*log(x))/(x^4 + 2*x^3*log(x - e^3 + 1) + x^2*log(x - e^3 + 1)^2)
Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (26) = 52\).
Time = 0.28 (sec) , antiderivative size = 99, normalized size of antiderivative = 2.83 \[ \int \frac {4 x^3+4 x^4-12 x^5-12 x^6+e^3 \left (-4 x^3+12 x^5\right )+\left (8 x^2+8 x^3-24 x^4-24 x^5+e^3 \left (-8 x^2+24 x^4\right )\right ) \log \left (1-e^3+x\right )+\left (4 x+4 x^2-12 x^3-12 x^4+e^3 \left (-4 x+12 x^3\right )\right ) \log ^2\left (1-e^3+x\right )+\log ^2(x) \left (3 x+2 x^2-12 x^3-6 x^4+9 x^5+e^3 \left (-2 x+6 x^3\right )+\left (1+x-9 x^4-9 x^5+e^3 \left (-1+9 x^4\right )\right ) \log \left (1-e^3+x\right )\right )+\log (x) \left (-x-x^2-6 x^3-2 x^4+3 x^5-9 x^6+e^3 \left (x+2 x^3+9 x^5\right )+\left (-1-x-10 x^2-6 x^3-9 x^4-21 x^5+e^3 \left (1+6 x^2+21 x^4\right )\right ) \log \left (1-e^3+x\right )+\left (-4 x-4 x^2-12 x^3-12 x^4+e^3 \left (4 x+12 x^3\right )\right ) \log ^2\left (1-e^3+x\right )\right )}{-2 x^6+2 e^3 x^6-2 x^7+\left (-6 x^5+6 e^3 x^5-6 x^6\right ) \log \left (1-e^3+x\right )+\left (-6 x^4+6 e^3 x^4-6 x^5\right ) \log ^2\left (1-e^3+x\right )+\left (-2 x^3+2 e^3 x^3-2 x^4\right ) \log ^3\left (1-e^3+x\right )} \, dx=\frac {9 x^{4} \log {\left (x \right )}^{2} + 24 x^{4} \log {\left (x \right )} - 6 x^{2} \log {\left (x \right )}^{2} - 8 x^{2} \log {\left (x \right )} + \left (24 x^{3} \log {\left (x \right )} - 8 x \log {\left (x \right )}\right ) \log {\left (x - e^{3} + 1 \right )} + \log {\left (x \right )}^{2}}{4 x^{4} + 8 x^{3} \log {\left (x - e^{3} + 1 \right )} + 4 x^{2} \log {\left (x - e^{3} + 1 \right )}^{2}} \] Input:
integrate(((((9*x**4-1)*exp(3)-9*x**5-9*x**4+x+1)*ln(-exp(3)+x+1)+(6*x**3- 2*x)*exp(3)+9*x**5-6*x**4-12*x**3+2*x**2+3*x)*ln(x)**2+(((12*x**3+4*x)*exp (3)-12*x**4-12*x**3-4*x**2-4*x)*ln(-exp(3)+x+1)**2+((21*x**4+6*x**2+1)*exp (3)-21*x**5-9*x**4-6*x**3-10*x**2-x-1)*ln(-exp(3)+x+1)+(9*x**5+2*x**3+x)*e xp(3)-9*x**6+3*x**5-2*x**4-6*x**3-x**2-x)*ln(x)+((12*x**3-4*x)*exp(3)-12*x **4-12*x**3+4*x**2+4*x)*ln(-exp(3)+x+1)**2+((24*x**4-8*x**2)*exp(3)-24*x** 5-24*x**4+8*x**3+8*x**2)*ln(-exp(3)+x+1)+(12*x**5-4*x**3)*exp(3)-12*x**6-1 2*x**5+4*x**4+4*x**3)/((2*x**3*exp(3)-2*x**4-2*x**3)*ln(-exp(3)+x+1)**3+(6 *x**4*exp(3)-6*x**5-6*x**4)*ln(-exp(3)+x+1)**2+(6*x**5*exp(3)-6*x**6-6*x** 5)*ln(-exp(3)+x+1)+2*x**6*exp(3)-2*x**7-2*x**6),x)
Output:
(9*x**4*log(x)**2 + 24*x**4*log(x) - 6*x**2*log(x)**2 - 8*x**2*log(x) + (2 4*x**3*log(x) - 8*x*log(x))*log(x - exp(3) + 1) + log(x)**2)/(4*x**4 + 8*x **3*log(x - exp(3) + 1) + 4*x**2*log(x - exp(3) + 1)**2)
Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (30) = 60\).
Time = 0.12 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.54 \[ \int \frac {4 x^3+4 x^4-12 x^5-12 x^6+e^3 \left (-4 x^3+12 x^5\right )+\left (8 x^2+8 x^3-24 x^4-24 x^5+e^3 \left (-8 x^2+24 x^4\right )\right ) \log \left (1-e^3+x\right )+\left (4 x+4 x^2-12 x^3-12 x^4+e^3 \left (-4 x+12 x^3\right )\right ) \log ^2\left (1-e^3+x\right )+\log ^2(x) \left (3 x+2 x^2-12 x^3-6 x^4+9 x^5+e^3 \left (-2 x+6 x^3\right )+\left (1+x-9 x^4-9 x^5+e^3 \left (-1+9 x^4\right )\right ) \log \left (1-e^3+x\right )\right )+\log (x) \left (-x-x^2-6 x^3-2 x^4+3 x^5-9 x^6+e^3 \left (x+2 x^3+9 x^5\right )+\left (-1-x-10 x^2-6 x^3-9 x^4-21 x^5+e^3 \left (1+6 x^2+21 x^4\right )\right ) \log \left (1-e^3+x\right )+\left (-4 x-4 x^2-12 x^3-12 x^4+e^3 \left (4 x+12 x^3\right )\right ) \log ^2\left (1-e^3+x\right )\right )}{-2 x^6+2 e^3 x^6-2 x^7+\left (-6 x^5+6 e^3 x^5-6 x^6\right ) \log \left (1-e^3+x\right )+\left (-6 x^4+6 e^3 x^4-6 x^5\right ) \log ^2\left (1-e^3+x\right )+\left (-2 x^3+2 e^3 x^3-2 x^4\right ) \log ^3\left (1-e^3+x\right )} \, dx=\frac {8 \, {\left (3 \, x^{3} - x\right )} \log \left (x - e^{3} + 1\right ) \log \left (x\right ) + {\left (9 \, x^{4} - 6 \, x^{2} + 1\right )} \log \left (x\right )^{2} + 8 \, {\left (3 \, x^{4} - x^{2}\right )} \log \left (x\right )}{4 \, {\left (x^{4} + 2 \, x^{3} \log \left (x - e^{3} + 1\right ) + x^{2} \log \left (x - e^{3} + 1\right )^{2}\right )}} \] Input:
integrate(((((9*x^4-1)*exp(3)-9*x^5-9*x^4+x+1)*log(-exp(3)+x+1)+(6*x^3-2*x )*exp(3)+9*x^5-6*x^4-12*x^3+2*x^2+3*x)*log(x)^2+(((12*x^3+4*x)*exp(3)-12*x ^4-12*x^3-4*x^2-4*x)*log(-exp(3)+x+1)^2+((21*x^4+6*x^2+1)*exp(3)-21*x^5-9* x^4-6*x^3-10*x^2-x-1)*log(-exp(3)+x+1)+(9*x^5+2*x^3+x)*exp(3)-9*x^6+3*x^5- 2*x^4-6*x^3-x^2-x)*log(x)+((12*x^3-4*x)*exp(3)-12*x^4-12*x^3+4*x^2+4*x)*lo g(-exp(3)+x+1)^2+((24*x^4-8*x^2)*exp(3)-24*x^5-24*x^4+8*x^3+8*x^2)*log(-ex p(3)+x+1)+(12*x^5-4*x^3)*exp(3)-12*x^6-12*x^5+4*x^4+4*x^3)/((2*x^3*exp(3)- 2*x^4-2*x^3)*log(-exp(3)+x+1)^3+(6*x^4*exp(3)-6*x^5-6*x^4)*log(-exp(3)+x+1 )^2+(6*x^5*exp(3)-6*x^6-6*x^5)*log(-exp(3)+x+1)+2*x^6*exp(3)-2*x^7-2*x^6), x, algorithm="maxima")
Output:
1/4*(8*(3*x^3 - x)*log(x - e^3 + 1)*log(x) + (9*x^4 - 6*x^2 + 1)*log(x)^2 + 8*(3*x^4 - x^2)*log(x))/(x^4 + 2*x^3*log(x - e^3 + 1) + x^2*log(x - e^3 + 1)^2)
Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (30) = 60\).
Time = 0.29 (sec) , antiderivative size = 100, normalized size of antiderivative = 2.86 \[ \int \frac {4 x^3+4 x^4-12 x^5-12 x^6+e^3 \left (-4 x^3+12 x^5\right )+\left (8 x^2+8 x^3-24 x^4-24 x^5+e^3 \left (-8 x^2+24 x^4\right )\right ) \log \left (1-e^3+x\right )+\left (4 x+4 x^2-12 x^3-12 x^4+e^3 \left (-4 x+12 x^3\right )\right ) \log ^2\left (1-e^3+x\right )+\log ^2(x) \left (3 x+2 x^2-12 x^3-6 x^4+9 x^5+e^3 \left (-2 x+6 x^3\right )+\left (1+x-9 x^4-9 x^5+e^3 \left (-1+9 x^4\right )\right ) \log \left (1-e^3+x\right )\right )+\log (x) \left (-x-x^2-6 x^3-2 x^4+3 x^5-9 x^6+e^3 \left (x+2 x^3+9 x^5\right )+\left (-1-x-10 x^2-6 x^3-9 x^4-21 x^5+e^3 \left (1+6 x^2+21 x^4\right )\right ) \log \left (1-e^3+x\right )+\left (-4 x-4 x^2-12 x^3-12 x^4+e^3 \left (4 x+12 x^3\right )\right ) \log ^2\left (1-e^3+x\right )\right )}{-2 x^6+2 e^3 x^6-2 x^7+\left (-6 x^5+6 e^3 x^5-6 x^6\right ) \log \left (1-e^3+x\right )+\left (-6 x^4+6 e^3 x^4-6 x^5\right ) \log ^2\left (1-e^3+x\right )+\left (-2 x^3+2 e^3 x^3-2 x^4\right ) \log ^3\left (1-e^3+x\right )} \, dx=\frac {9 \, x^{4} \log \left (x\right )^{2} + 24 \, x^{4} \log \left (x\right ) + 24 \, x^{3} \log \left (x - e^{3} + 1\right ) \log \left (x\right ) - 6 \, x^{2} \log \left (x\right )^{2} - 8 \, x^{2} \log \left (x\right ) - 8 \, x \log \left (x - e^{3} + 1\right ) \log \left (x\right ) + \log \left (x\right )^{2}}{4 \, {\left (x^{4} + 2 \, x^{3} \log \left (x - e^{3} + 1\right ) + x^{2} \log \left (x - e^{3} + 1\right )^{2}\right )}} \] Input:
integrate(((((9*x^4-1)*exp(3)-9*x^5-9*x^4+x+1)*log(-exp(3)+x+1)+(6*x^3-2*x )*exp(3)+9*x^5-6*x^4-12*x^3+2*x^2+3*x)*log(x)^2+(((12*x^3+4*x)*exp(3)-12*x ^4-12*x^3-4*x^2-4*x)*log(-exp(3)+x+1)^2+((21*x^4+6*x^2+1)*exp(3)-21*x^5-9* x^4-6*x^3-10*x^2-x-1)*log(-exp(3)+x+1)+(9*x^5+2*x^3+x)*exp(3)-9*x^6+3*x^5- 2*x^4-6*x^3-x^2-x)*log(x)+((12*x^3-4*x)*exp(3)-12*x^4-12*x^3+4*x^2+4*x)*lo g(-exp(3)+x+1)^2+((24*x^4-8*x^2)*exp(3)-24*x^5-24*x^4+8*x^3+8*x^2)*log(-ex p(3)+x+1)+(12*x^5-4*x^3)*exp(3)-12*x^6-12*x^5+4*x^4+4*x^3)/((2*x^3*exp(3)- 2*x^4-2*x^3)*log(-exp(3)+x+1)^3+(6*x^4*exp(3)-6*x^5-6*x^4)*log(-exp(3)+x+1 )^2+(6*x^5*exp(3)-6*x^6-6*x^5)*log(-exp(3)+x+1)+2*x^6*exp(3)-2*x^7-2*x^6), x, algorithm="giac")
Output:
1/4*(9*x^4*log(x)^2 + 24*x^4*log(x) + 24*x^3*log(x - e^3 + 1)*log(x) - 6*x ^2*log(x)^2 - 8*x^2*log(x) - 8*x*log(x - e^3 + 1)*log(x) + log(x)^2)/(x^4 + 2*x^3*log(x - e^3 + 1) + x^2*log(x - e^3 + 1)^2)
Timed out. \[ \int \frac {4 x^3+4 x^4-12 x^5-12 x^6+e^3 \left (-4 x^3+12 x^5\right )+\left (8 x^2+8 x^3-24 x^4-24 x^5+e^3 \left (-8 x^2+24 x^4\right )\right ) \log \left (1-e^3+x\right )+\left (4 x+4 x^2-12 x^3-12 x^4+e^3 \left (-4 x+12 x^3\right )\right ) \log ^2\left (1-e^3+x\right )+\log ^2(x) \left (3 x+2 x^2-12 x^3-6 x^4+9 x^5+e^3 \left (-2 x+6 x^3\right )+\left (1+x-9 x^4-9 x^5+e^3 \left (-1+9 x^4\right )\right ) \log \left (1-e^3+x\right )\right )+\log (x) \left (-x-x^2-6 x^3-2 x^4+3 x^5-9 x^6+e^3 \left (x+2 x^3+9 x^5\right )+\left (-1-x-10 x^2-6 x^3-9 x^4-21 x^5+e^3 \left (1+6 x^2+21 x^4\right )\right ) \log \left (1-e^3+x\right )+\left (-4 x-4 x^2-12 x^3-12 x^4+e^3 \left (4 x+12 x^3\right )\right ) \log ^2\left (1-e^3+x\right )\right )}{-2 x^6+2 e^3 x^6-2 x^7+\left (-6 x^5+6 e^3 x^5-6 x^6\right ) \log \left (1-e^3+x\right )+\left (-6 x^4+6 e^3 x^4-6 x^5\right ) \log ^2\left (1-e^3+x\right )+\left (-2 x^3+2 e^3 x^3-2 x^4\right ) \log ^3\left (1-e^3+x\right )} \, dx=\int \frac {\ln \left (x-{\mathrm {e}}^3+1\right )\,\left ({\mathrm {e}}^3\,\left (8\,x^2-24\,x^4\right )-8\,x^2-8\,x^3+24\,x^4+24\,x^5\right )+\ln \left (x\right )\,\left (x-{\mathrm {e}}^3\,\left (9\,x^5+2\,x^3+x\right )+\ln \left (x-{\mathrm {e}}^3+1\right )\,\left (x-{\mathrm {e}}^3\,\left (21\,x^4+6\,x^2+1\right )+10\,x^2+6\,x^3+9\,x^4+21\,x^5+1\right )+{\ln \left (x-{\mathrm {e}}^3+1\right )}^2\,\left (4\,x-{\mathrm {e}}^3\,\left (12\,x^3+4\,x\right )+4\,x^2+12\,x^3+12\,x^4\right )+x^2+6\,x^3+2\,x^4-3\,x^5+9\,x^6\right )+{\mathrm {e}}^3\,\left (4\,x^3-12\,x^5\right )+{\ln \left (x-{\mathrm {e}}^3+1\right )}^2\,\left ({\mathrm {e}}^3\,\left (4\,x-12\,x^3\right )-4\,x-4\,x^2+12\,x^3+12\,x^4\right )-4\,x^3-4\,x^4+12\,x^5+12\,x^6-{\ln \left (x\right )}^2\,\left (3\,x+\ln \left (x-{\mathrm {e}}^3+1\right )\,\left (x+{\mathrm {e}}^3\,\left (9\,x^4-1\right )-9\,x^4-9\,x^5+1\right )-{\mathrm {e}}^3\,\left (2\,x-6\,x^3\right )+2\,x^2-12\,x^3-6\,x^4+9\,x^5\right )}{{\ln \left (x-{\mathrm {e}}^3+1\right )}^3\,\left (2\,x^3-2\,x^3\,{\mathrm {e}}^3+2\,x^4\right )+{\ln \left (x-{\mathrm {e}}^3+1\right )}^2\,\left (6\,x^4-6\,x^4\,{\mathrm {e}}^3+6\,x^5\right )-2\,x^6\,{\mathrm {e}}^3+\ln \left (x-{\mathrm {e}}^3+1\right )\,\left (6\,x^5-6\,x^5\,{\mathrm {e}}^3+6\,x^6\right )+2\,x^6+2\,x^7} \,d x \] Input:
int((log(x - exp(3) + 1)*(exp(3)*(8*x^2 - 24*x^4) - 8*x^2 - 8*x^3 + 24*x^4 + 24*x^5) + log(x)*(x - exp(3)*(x + 2*x^3 + 9*x^5) + log(x - exp(3) + 1)* (x - exp(3)*(6*x^2 + 21*x^4 + 1) + 10*x^2 + 6*x^3 + 9*x^4 + 21*x^5 + 1) + log(x - exp(3) + 1)^2*(4*x - exp(3)*(4*x + 12*x^3) + 4*x^2 + 12*x^3 + 12*x ^4) + x^2 + 6*x^3 + 2*x^4 - 3*x^5 + 9*x^6) + exp(3)*(4*x^3 - 12*x^5) + log (x - exp(3) + 1)^2*(exp(3)*(4*x - 12*x^3) - 4*x - 4*x^2 + 12*x^3 + 12*x^4) - 4*x^3 - 4*x^4 + 12*x^5 + 12*x^6 - log(x)^2*(3*x + log(x - exp(3) + 1)*( x + exp(3)*(9*x^4 - 1) - 9*x^4 - 9*x^5 + 1) - exp(3)*(2*x - 6*x^3) + 2*x^2 - 12*x^3 - 6*x^4 + 9*x^5))/(log(x - exp(3) + 1)^3*(2*x^3 - 2*x^3*exp(3) + 2*x^4) + log(x - exp(3) + 1)^2*(6*x^4 - 6*x^4*exp(3) + 6*x^5) - 2*x^6*exp (3) + log(x - exp(3) + 1)*(6*x^5 - 6*x^5*exp(3) + 6*x^6) + 2*x^6 + 2*x^7), x)
Output:
int((log(x - exp(3) + 1)*(exp(3)*(8*x^2 - 24*x^4) - 8*x^2 - 8*x^3 + 24*x^4 + 24*x^5) + log(x)*(x - exp(3)*(x + 2*x^3 + 9*x^5) + log(x - exp(3) + 1)* (x - exp(3)*(6*x^2 + 21*x^4 + 1) + 10*x^2 + 6*x^3 + 9*x^4 + 21*x^5 + 1) + log(x - exp(3) + 1)^2*(4*x - exp(3)*(4*x + 12*x^3) + 4*x^2 + 12*x^3 + 12*x ^4) + x^2 + 6*x^3 + 2*x^4 - 3*x^5 + 9*x^6) + exp(3)*(4*x^3 - 12*x^5) + log (x - exp(3) + 1)^2*(exp(3)*(4*x - 12*x^3) - 4*x - 4*x^2 + 12*x^3 + 12*x^4) - 4*x^3 - 4*x^4 + 12*x^5 + 12*x^6 - log(x)^2*(3*x + log(x - exp(3) + 1)*( x + exp(3)*(9*x^4 - 1) - 9*x^4 - 9*x^5 + 1) - exp(3)*(2*x - 6*x^3) + 2*x^2 - 12*x^3 - 6*x^4 + 9*x^5))/(log(x - exp(3) + 1)^3*(2*x^3 - 2*x^3*exp(3) + 2*x^4) + log(x - exp(3) + 1)^2*(6*x^4 - 6*x^4*exp(3) + 6*x^5) - 2*x^6*exp (3) + log(x - exp(3) + 1)*(6*x^5 - 6*x^5*exp(3) + 6*x^6) + 2*x^6 + 2*x^7), x)
Time = 0.19 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.54 \[ \int \frac {4 x^3+4 x^4-12 x^5-12 x^6+e^3 \left (-4 x^3+12 x^5\right )+\left (8 x^2+8 x^3-24 x^4-24 x^5+e^3 \left (-8 x^2+24 x^4\right )\right ) \log \left (1-e^3+x\right )+\left (4 x+4 x^2-12 x^3-12 x^4+e^3 \left (-4 x+12 x^3\right )\right ) \log ^2\left (1-e^3+x\right )+\log ^2(x) \left (3 x+2 x^2-12 x^3-6 x^4+9 x^5+e^3 \left (-2 x+6 x^3\right )+\left (1+x-9 x^4-9 x^5+e^3 \left (-1+9 x^4\right )\right ) \log \left (1-e^3+x\right )\right )+\log (x) \left (-x-x^2-6 x^3-2 x^4+3 x^5-9 x^6+e^3 \left (x+2 x^3+9 x^5\right )+\left (-1-x-10 x^2-6 x^3-9 x^4-21 x^5+e^3 \left (1+6 x^2+21 x^4\right )\right ) \log \left (1-e^3+x\right )+\left (-4 x-4 x^2-12 x^3-12 x^4+e^3 \left (4 x+12 x^3\right )\right ) \log ^2\left (1-e^3+x\right )\right )}{-2 x^6+2 e^3 x^6-2 x^7+\left (-6 x^5+6 e^3 x^5-6 x^6\right ) \log \left (1-e^3+x\right )+\left (-6 x^4+6 e^3 x^4-6 x^5\right ) \log ^2\left (1-e^3+x\right )+\left (-2 x^3+2 e^3 x^3-2 x^4\right ) \log ^3\left (1-e^3+x\right )} \, dx=\frac {\mathrm {log}\left (x \right ) \left (24 \,\mathrm {log}\left (-e^{3}+x +1\right ) x^{3}-8 \,\mathrm {log}\left (-e^{3}+x +1\right ) x +9 \,\mathrm {log}\left (x \right ) x^{4}-6 \,\mathrm {log}\left (x \right ) x^{2}+\mathrm {log}\left (x \right )+24 x^{4}-8 x^{2}\right )}{4 x^{2} \left (\mathrm {log}\left (-e^{3}+x +1\right )^{2}+2 \,\mathrm {log}\left (-e^{3}+x +1\right ) x +x^{2}\right )} \] Input:
int(((((9*x^4-1)*exp(3)-9*x^5-9*x^4+x+1)*log(-exp(3)+x+1)+(6*x^3-2*x)*exp( 3)+9*x^5-6*x^4-12*x^3+2*x^2+3*x)*log(x)^2+(((12*x^3+4*x)*exp(3)-12*x^4-12* x^3-4*x^2-4*x)*log(-exp(3)+x+1)^2+((21*x^4+6*x^2+1)*exp(3)-21*x^5-9*x^4-6* x^3-10*x^2-x-1)*log(-exp(3)+x+1)+(9*x^5+2*x^3+x)*exp(3)-9*x^6+3*x^5-2*x^4- 6*x^3-x^2-x)*log(x)+((12*x^3-4*x)*exp(3)-12*x^4-12*x^3+4*x^2+4*x)*log(-exp (3)+x+1)^2+((24*x^4-8*x^2)*exp(3)-24*x^5-24*x^4+8*x^3+8*x^2)*log(-exp(3)+x +1)+(12*x^5-4*x^3)*exp(3)-12*x^6-12*x^5+4*x^4+4*x^3)/((2*x^3*exp(3)-2*x^4- 2*x^3)*log(-exp(3)+x+1)^3+(6*x^4*exp(3)-6*x^5-6*x^4)*log(-exp(3)+x+1)^2+(6 *x^5*exp(3)-6*x^6-6*x^5)*log(-exp(3)+x+1)+2*x^6*exp(3)-2*x^7-2*x^6),x)
Output:
(log(x)*(24*log( - e**3 + x + 1)*x**3 - 8*log( - e**3 + x + 1)*x + 9*log(x )*x**4 - 6*log(x)*x**2 + log(x) + 24*x**4 - 8*x**2))/(4*x**2*(log( - e**3 + x + 1)**2 + 2*log( - e**3 + x + 1)*x + x**2))