Integrand size = 245, antiderivative size = 31 \[ \int \frac {e^6 \left (-162+198 x-90 x^2-18 x^3-6 x^4-10 x^5-18 x^6-4 x^7\right )+e^3 \left (18 x-18 x^2+6 x^4+4 x^5\right ) \log (2)+\left (e^6 \left (-36+90 x+72 x^2+18 x^3+30 x^4+54 x^5+12 x^6\right )+e^3 \left (-18+18 x-18 x^3-12 x^4\right ) \log (2)\right ) \log (x)+\left (e^6 \left (-54 x-30 x^3-54 x^4-12 x^5\right )+e^3 \left (18 x^2+12 x^3\right ) \log (2)\right ) \log ^2(x)+\left (e^6 \left (-12 x+10 x^2+18 x^3+4 x^4\right )+e^3 \left (-6 x-4 x^2\right ) \log (2)\right ) \log ^3(x)}{-x^4+3 x^3 \log (x)-3 x^2 \log ^2(x)+x \log ^3(x)} \, dx=1+\left (\log (2)+e^3 \left (2-3 x-x^2-\frac {9}{x-\log (x)}\right )\right )^2 \] Output:
1+(ln(2)+(2-3*x-9/(x-ln(x))-x^2)*exp(3))^2
Leaf count is larger than twice the leaf count of optimal. \(85\) vs. \(2(31)=62\).
Time = 0.09 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.74 \[ \int \frac {e^6 \left (-162+198 x-90 x^2-18 x^3-6 x^4-10 x^5-18 x^6-4 x^7\right )+e^3 \left (18 x-18 x^2+6 x^4+4 x^5\right ) \log (2)+\left (e^6 \left (-36+90 x+72 x^2+18 x^3+30 x^4+54 x^5+12 x^6\right )+e^3 \left (-18+18 x-18 x^3-12 x^4\right ) \log (2)\right ) \log (x)+\left (e^6 \left (-54 x-30 x^3-54 x^4-12 x^5\right )+e^3 \left (18 x^2+12 x^3\right ) \log (2)\right ) \log ^2(x)+\left (e^6 \left (-12 x+10 x^2+18 x^3+4 x^4\right )+e^3 \left (-6 x-4 x^2\right ) \log (2)\right ) \log ^3(x)}{-x^4+3 x^3 \log (x)-3 x^2 \log ^2(x)+x \log ^3(x)} \, dx=e^3 \left (6 e^3 x^3+e^3 x^4-6 x \left (2 e^3+\log (2)\right )+x^2 \left (5 e^3-\log (4)\right )+\frac {81 e^3}{(x-\log (x))^2}+\frac {18 \left (e^3 \left (-2+3 x+x^2\right )-\log (2)\right )}{x-\log (x)}\right ) \] Input:
Integrate[(E^6*(-162 + 198*x - 90*x^2 - 18*x^3 - 6*x^4 - 10*x^5 - 18*x^6 - 4*x^7) + E^3*(18*x - 18*x^2 + 6*x^4 + 4*x^5)*Log[2] + (E^6*(-36 + 90*x + 72*x^2 + 18*x^3 + 30*x^4 + 54*x^5 + 12*x^6) + E^3*(-18 + 18*x - 18*x^3 - 1 2*x^4)*Log[2])*Log[x] + (E^6*(-54*x - 30*x^3 - 54*x^4 - 12*x^5) + E^3*(18* x^2 + 12*x^3)*Log[2])*Log[x]^2 + (E^6*(-12*x + 10*x^2 + 18*x^3 + 4*x^4) + E^3*(-6*x - 4*x^2)*Log[2])*Log[x]^3)/(-x^4 + 3*x^3*Log[x] - 3*x^2*Log[x]^2 + x*Log[x]^3),x]
Output:
E^3*(6*E^3*x^3 + E^3*x^4 - 6*x*(2*E^3 + Log[2]) + x^2*(5*E^3 - Log[4]) + ( 81*E^3)/(x - Log[x])^2 + (18*(E^3*(-2 + 3*x + x^2) - Log[2]))/(x - Log[x]) )
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.
| \(\displaystyle \int \frac {e^3 \left (4 x^5+6 x^4-18 x^2+18 x\right ) \log (2)+\left (e^3 \left (-4 x^2-6 x\right ) \log (2)+e^6 \left (4 x^4+18 x^3+10 x^2-12 x\right )\right ) \log ^3(x)+\left (e^3 \left (12 x^3+18 x^2\right ) \log (2)+e^6 \left (-12 x^5-54 x^4-30 x^3-54 x\right )\right ) \log ^2(x)+\left (e^3 \left (-12 x^4-18 x^3+18 x-18\right ) \log (2)+e^6 \left (12 x^6+54 x^5+30 x^4+18 x^3+72 x^2+90 x-36\right )\right ) \log (x)+e^6 \left (-4 x^7-18 x^6-10 x^5-6 x^4-18 x^3-90 x^2+198 x-162\right )}{-x^4+3 x^3 \log (x)-3 x^2 \log ^2(x)+x \log ^3(x)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 e^3 \left (\left (\log (2)-e^3 \left (x^2+3 x-2\right )\right ) \log (x)+e^3 \left (x^3+3 x^2-2 x+9\right )+x (-\log (2))\right ) \left (2 x^4+3 x^3-2 (2 x+3) x^2 \log (x)-9 x+(2 x+3) x \log ^2(x)+9\right )}{x (x-\log (x))^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \int \frac {\left (-\log (2) x+e^3 \left (x^3+3 x^2-2 x+9\right )+\left (e^3 \left (-x^2-3 x+2\right )+\log (2)\right ) \log (x)\right ) \left (2 x^4+3 x^3-2 (2 x+3) \log (x) x^2+(2 x+3) \log ^2(x) x-9 x+9\right )}{x (x-\log (x))^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 e^3 \int \left (-\frac {9 \left (e^3 x^2+3 e^3 x-\log (2)-2 e^3\right ) (x-1)}{x (x-\log (x))^2}-\frac {81 e^3 (x-1)}{x (x-\log (x))^3}+(2 x+3) \left (e^3 x^2+3 e^3 x-\log (2)-2 e^3\right )+\frac {9 e^3 (2 x+3)}{x-\log (x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 e^3 \left (-9 e^3 \int \frac {x^2}{(x-\log (x))^2}dx+9 \left (5 e^3+\log (2)\right ) \int \frac {1}{(x-\log (x))^2}dx-9 \left (2 e^3+\log (2)\right ) \int \frac {1}{x (x-\log (x))^2}dx-18 e^3 \int \frac {x}{(x-\log (x))^2}dx+27 e^3 \int \frac {1}{x-\log (x)}dx+18 e^3 \int \frac {x}{x-\log (x)}dx+\frac {\left (-e^3 x^2-3 e^3 x+2 e^3+\log (2)\right )^2}{2 e^3}+\frac {81 e^3}{2 (x-\log (x))^2}\right )\) |
Input:
Int[(E^6*(-162 + 198*x - 90*x^2 - 18*x^3 - 6*x^4 - 10*x^5 - 18*x^6 - 4*x^7 ) + E^3*(18*x - 18*x^2 + 6*x^4 + 4*x^5)*Log[2] + (E^6*(-36 + 90*x + 72*x^2 + 18*x^3 + 30*x^4 + 54*x^5 + 12*x^6) + E^3*(-18 + 18*x - 18*x^3 - 12*x^4) *Log[2])*Log[x] + (E^6*(-54*x - 30*x^3 - 54*x^4 - 12*x^5) + E^3*(18*x^2 + 12*x^3)*Log[2])*Log[x]^2 + (E^6*(-12*x + 10*x^2 + 18*x^3 + 4*x^4) + E^3*(- 6*x - 4*x^2)*Log[2])*Log[x]^3)/(-x^4 + 3*x^3*Log[x] - 3*x^2*Log[x]^2 + x*L og[x]^3),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(92\) vs. \(2(30)=60\).
Time = 264.12 (sec) , antiderivative size = 93, normalized size of antiderivative = 3.00
| method | result | size |
| risch | \(\left (x^{2} {\mathrm e}^{3}+3 x \,{\mathrm e}^{3}-2 \,{\mathrm e}^{3}-\ln \left (2\right )\right )^{2}+\frac {9 \,{\mathrm e}^{3} \left (2 x^{3} {\mathrm e}^{3}-2 \,{\mathrm e}^{3} \ln \left (x \right ) x^{2}+6 x^{2} {\mathrm e}^{3}-6 \,{\mathrm e}^{3} \ln \left (x \right ) x -4 x \,{\mathrm e}^{3}+4 \,{\mathrm e}^{3} \ln \left (x \right )-2 x \ln \left (2\right )+2 \ln \left (2\right ) \ln \left (x \right )+9 \,{\mathrm e}^{3}\right )}{\left (x -\ln \left (x \right )\right )^{2}}\) | \(93\) |
| default | \(2 \,{\mathrm e}^{6} \left (9 \ln \left (x \right )+\frac {x^{4}}{2}+3 x^{3}+\frac {5 x^{2}}{2}+3 x -\frac {9 \left (2 \ln \left (x \right )^{3}-2 x \ln \left (x \right )^{2}+6 \ln \left (x \right )^{2}-6 x \ln \left (x \right )-4 \ln \left (x \right )+4 x -9\right )}{2 \left (\ln \left (x \right )-x \right )^{2}}\right )-\frac {2 \,{\mathrm e}^{3} \ln \left (2\right ) \left (-9+x^{2} \ln \left (x \right )-3 x^{2}-x^{3}+3 x \ln \left (x \right )\right )}{\ln \left (x \right )-x}\) | \(111\) |
| parallelrisch | \(\frac {-2 \,{\mathrm e}^{3} x^{4} \ln \left (2\right )-6 \,{\mathrm e}^{3} x^{3} \ln \left (2\right )+18 \,{\mathrm e}^{3} \ln \left (2\right ) \ln \left (x \right )-18 \,{\mathrm e}^{3} x \ln \left (2\right )+6 x^{5} {\mathrm e}^{6}+81 \,{\mathrm e}^{6}+6 x^{3} {\mathrm e}^{6}+5 x^{4} {\mathrm e}^{6}-36 x \,{\mathrm e}^{6}-2 \,{\mathrm e}^{3} x^{2} \ln \left (2\right ) \ln \left (x \right )^{2}+4 \,{\mathrm e}^{3} x^{3} \ln \left (2\right ) \ln \left (x \right )-6 \,{\mathrm e}^{3} x \ln \left (2\right ) \ln \left (x \right )^{2}+12 \,{\mathrm e}^{3} x^{2} \ln \left (2\right ) \ln \left (x \right )+54 x^{2} {\mathrm e}^{6}+{\mathrm e}^{6} x^{4} \ln \left (x \right )^{2}-2 \,{\mathrm e}^{6} x^{5} \ln \left (x \right )+6 \,{\mathrm e}^{6} x^{3} \ln \left (x \right )^{2}-12 \,{\mathrm e}^{6} x^{4} \ln \left (x \right )-10 \,{\mathrm e}^{6} x^{3} \ln \left (x \right )-12 \,{\mathrm e}^{6} \ln \left (x \right )^{2} x +6 \,{\mathrm e}^{6} x^{2} \ln \left (x \right )-54 \,{\mathrm e}^{6} x \ln \left (x \right )+5 x^{2} {\mathrm e}^{6} \ln \left (x \right )^{2}+36 \,{\mathrm e}^{6} \ln \left (x \right )+x^{6} {\mathrm e}^{6}}{\ln \left (x \right )^{2}-2 x \ln \left (x \right )+x^{2}}\) | \(264\) |
Input:
int((((-4*x^2-6*x)*exp(3)*ln(2)+(4*x^4+18*x^3+10*x^2-12*x)*exp(3)^2)*ln(x) ^3+((12*x^3+18*x^2)*exp(3)*ln(2)+(-12*x^5-54*x^4-30*x^3-54*x)*exp(3)^2)*ln (x)^2+((-12*x^4-18*x^3+18*x-18)*exp(3)*ln(2)+(12*x^6+54*x^5+30*x^4+18*x^3+ 72*x^2+90*x-36)*exp(3)^2)*ln(x)+(4*x^5+6*x^4-18*x^2+18*x)*exp(3)*ln(2)+(-4 *x^7-18*x^6-10*x^5-6*x^4-18*x^3-90*x^2+198*x-162)*exp(3)^2)/(x*ln(x)^3-3*x ^2*ln(x)^2+3*x^3*ln(x)-x^4),x,method=_RETURNVERBOSE)
Output:
(x^2*exp(3)+3*x*exp(3)-2*exp(3)-ln(2))^2+9*exp(3)*(2*x^3*exp(3)-2*exp(3)*l n(x)*x^2+6*x^2*exp(3)-6*exp(3)*ln(x)*x-4*x*exp(3)+4*exp(3)*ln(x)-2*x*ln(2) +2*ln(2)*ln(x)+9*exp(3))/(x-ln(x))^2
Leaf count of result is larger than twice the leaf count of optimal. 157 vs. \(2 (30) = 60\).
Time = 0.08 (sec) , antiderivative size = 157, normalized size of antiderivative = 5.06 \[ \int \frac {e^6 \left (-162+198 x-90 x^2-18 x^3-6 x^4-10 x^5-18 x^6-4 x^7\right )+e^3 \left (18 x-18 x^2+6 x^4+4 x^5\right ) \log (2)+\left (e^6 \left (-36+90 x+72 x^2+18 x^3+30 x^4+54 x^5+12 x^6\right )+e^3 \left (-18+18 x-18 x^3-12 x^4\right ) \log (2)\right ) \log (x)+\left (e^6 \left (-54 x-30 x^3-54 x^4-12 x^5\right )+e^3 \left (18 x^2+12 x^3\right ) \log (2)\right ) \log ^2(x)+\left (e^6 \left (-12 x+10 x^2+18 x^3+4 x^4\right )+e^3 \left (-6 x-4 x^2\right ) \log (2)\right ) \log ^3(x)}{-x^4+3 x^3 \log (x)-3 x^2 \log ^2(x)+x \log ^3(x)} \, dx=-\frac {2 \, {\left (x^{4} + 3 \, x^{3} + 9 \, x\right )} e^{3} \log \left (2\right ) + {\left (2 \, {\left (x^{2} + 3 \, x\right )} e^{3} \log \left (2\right ) - {\left (x^{4} + 6 \, x^{3} + 5 \, x^{2} - 12 \, x\right )} e^{6}\right )} \log \left (x\right )^{2} - {\left (x^{6} + 6 \, x^{5} + 5 \, x^{4} + 6 \, x^{3} + 54 \, x^{2} - 36 \, x + 81\right )} e^{6} - 2 \, {\left ({\left (2 \, x^{3} + 6 \, x^{2} + 9\right )} e^{3} \log \left (2\right ) - {\left (x^{5} + 6 \, x^{4} + 5 \, x^{3} - 3 \, x^{2} + 27 \, x - 18\right )} e^{6}\right )} \log \left (x\right )}{x^{2} - 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}} \] Input:
integrate((((-4*x^2-6*x)*exp(3)*log(2)+(4*x^4+18*x^3+10*x^2-12*x)*exp(3)^2 )*log(x)^3+((12*x^3+18*x^2)*exp(3)*log(2)+(-12*x^5-54*x^4-30*x^3-54*x)*exp (3)^2)*log(x)^2+((-12*x^4-18*x^3+18*x-18)*exp(3)*log(2)+(12*x^6+54*x^5+30* x^4+18*x^3+72*x^2+90*x-36)*exp(3)^2)*log(x)+(4*x^5+6*x^4-18*x^2+18*x)*exp( 3)*log(2)+(-4*x^7-18*x^6-10*x^5-6*x^4-18*x^3-90*x^2+198*x-162)*exp(3)^2)/( x*log(x)^3-3*x^2*log(x)^2+3*x^3*log(x)-x^4),x, algorithm="fricas")
Output:
-(2*(x^4 + 3*x^3 + 9*x)*e^3*log(2) + (2*(x^2 + 3*x)*e^3*log(2) - (x^4 + 6* x^3 + 5*x^2 - 12*x)*e^6)*log(x)^2 - (x^6 + 6*x^5 + 5*x^4 + 6*x^3 + 54*x^2 - 36*x + 81)*e^6 - 2*((2*x^3 + 6*x^2 + 9)*e^3*log(2) - (x^5 + 6*x^4 + 5*x^ 3 - 3*x^2 + 27*x - 18)*e^6)*log(x))/(x^2 - 2*x*log(x) + log(x)^2)
Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (24) = 48\).
Time = 0.12 (sec) , antiderivative size = 134, normalized size of antiderivative = 4.32 \[ \int \frac {e^6 \left (-162+198 x-90 x^2-18 x^3-6 x^4-10 x^5-18 x^6-4 x^7\right )+e^3 \left (18 x-18 x^2+6 x^4+4 x^5\right ) \log (2)+\left (e^6 \left (-36+90 x+72 x^2+18 x^3+30 x^4+54 x^5+12 x^6\right )+e^3 \left (-18+18 x-18 x^3-12 x^4\right ) \log (2)\right ) \log (x)+\left (e^6 \left (-54 x-30 x^3-54 x^4-12 x^5\right )+e^3 \left (18 x^2+12 x^3\right ) \log (2)\right ) \log ^2(x)+\left (e^6 \left (-12 x+10 x^2+18 x^3+4 x^4\right )+e^3 \left (-6 x-4 x^2\right ) \log (2)\right ) \log ^3(x)}{-x^4+3 x^3 \log (x)-3 x^2 \log ^2(x)+x \log ^3(x)} \, dx=x^{4} e^{6} + 6 x^{3} e^{6} + x^{2} \left (- 2 e^{3} \log {\left (2 \right )} + 5 e^{6}\right ) + x \left (- 12 e^{6} - 6 e^{3} \log {\left (2 \right )}\right ) + \frac {18 x^{3} e^{6} + 54 x^{2} e^{6} - 36 x e^{6} - 18 x e^{3} \log {\left (2 \right )} + \left (- 18 x^{2} e^{6} - 54 x e^{6} + 18 e^{3} \log {\left (2 \right )} + 36 e^{6}\right ) \log {\left (x \right )} + 81 e^{6}}{x^{2} - 2 x \log {\left (x \right )} + \log {\left (x \right )}^{2}} \] Input:
integrate((((-4*x**2-6*x)*exp(3)*ln(2)+(4*x**4+18*x**3+10*x**2-12*x)*exp(3 )**2)*ln(x)**3+((12*x**3+18*x**2)*exp(3)*ln(2)+(-12*x**5-54*x**4-30*x**3-5 4*x)*exp(3)**2)*ln(x)**2+((-12*x**4-18*x**3+18*x-18)*exp(3)*ln(2)+(12*x**6 +54*x**5+30*x**4+18*x**3+72*x**2+90*x-36)*exp(3)**2)*ln(x)+(4*x**5+6*x**4- 18*x**2+18*x)*exp(3)*ln(2)+(-4*x**7-18*x**6-10*x**5-6*x**4-18*x**3-90*x**2 +198*x-162)*exp(3)**2)/(x*ln(x)**3-3*x**2*ln(x)**2+3*x**3*ln(x)-x**4),x)
Output:
x**4*exp(6) + 6*x**3*exp(6) + x**2*(-2*exp(3)*log(2) + 5*exp(6)) + x*(-12* exp(6) - 6*exp(3)*log(2)) + (18*x**3*exp(6) + 54*x**2*exp(6) - 36*x*exp(6) - 18*x*exp(3)*log(2) + (-18*x**2*exp(6) - 54*x*exp(6) + 18*exp(3)*log(2) + 36*exp(6))*log(x) + 81*exp(6))/(x**2 - 2*x*log(x) + log(x)**2)
Leaf count of result is larger than twice the leaf count of optimal. 196 vs. \(2 (30) = 60\).
Time = 0.17 (sec) , antiderivative size = 196, normalized size of antiderivative = 6.32 \[ \int \frac {e^6 \left (-162+198 x-90 x^2-18 x^3-6 x^4-10 x^5-18 x^6-4 x^7\right )+e^3 \left (18 x-18 x^2+6 x^4+4 x^5\right ) \log (2)+\left (e^6 \left (-36+90 x+72 x^2+18 x^3+30 x^4+54 x^5+12 x^6\right )+e^3 \left (-18+18 x-18 x^3-12 x^4\right ) \log (2)\right ) \log (x)+\left (e^6 \left (-54 x-30 x^3-54 x^4-12 x^5\right )+e^3 \left (18 x^2+12 x^3\right ) \log (2)\right ) \log ^2(x)+\left (e^6 \left (-12 x+10 x^2+18 x^3+4 x^4\right )+e^3 \left (-6 x-4 x^2\right ) \log (2)\right ) \log ^3(x)}{-x^4+3 x^3 \log (x)-3 x^2 \log ^2(x)+x \log ^3(x)} \, dx=\frac {x^{6} e^{6} + 6 \, x^{5} e^{6} - {\left (2 \, e^{3} \log \left (2\right ) - 5 \, e^{6}\right )} x^{4} - 6 \, {\left (e^{3} \log \left (2\right ) - e^{6}\right )} x^{3} + 54 \, x^{2} e^{6} + {\left (x^{4} e^{6} + 6 \, x^{3} e^{6} - {\left (2 \, e^{3} \log \left (2\right ) - 5 \, e^{6}\right )} x^{2} - 6 \, {\left (e^{3} \log \left (2\right ) + 2 \, e^{6}\right )} x\right )} \log \left (x\right )^{2} - 18 \, {\left (e^{3} \log \left (2\right ) + 2 \, e^{6}\right )} x - 2 \, {\left (x^{5} e^{6} + 6 \, x^{4} e^{6} - {\left (2 \, e^{3} \log \left (2\right ) - 5 \, e^{6}\right )} x^{3} - 3 \, {\left (2 \, e^{3} \log \left (2\right ) + e^{6}\right )} x^{2} + 27 \, x e^{6} - 9 \, e^{3} \log \left (2\right ) - 18 \, e^{6}\right )} \log \left (x\right ) + 81 \, e^{6}}{x^{2} - 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}} \] Input:
integrate((((-4*x^2-6*x)*exp(3)*log(2)+(4*x^4+18*x^3+10*x^2-12*x)*exp(3)^2 )*log(x)^3+((12*x^3+18*x^2)*exp(3)*log(2)+(-12*x^5-54*x^4-30*x^3-54*x)*exp (3)^2)*log(x)^2+((-12*x^4-18*x^3+18*x-18)*exp(3)*log(2)+(12*x^6+54*x^5+30* x^4+18*x^3+72*x^2+90*x-36)*exp(3)^2)*log(x)+(4*x^5+6*x^4-18*x^2+18*x)*exp( 3)*log(2)+(-4*x^7-18*x^6-10*x^5-6*x^4-18*x^3-90*x^2+198*x-162)*exp(3)^2)/( x*log(x)^3-3*x^2*log(x)^2+3*x^3*log(x)-x^4),x, algorithm="maxima")
Output:
(x^6*e^6 + 6*x^5*e^6 - (2*e^3*log(2) - 5*e^6)*x^4 - 6*(e^3*log(2) - e^6)*x ^3 + 54*x^2*e^6 + (x^4*e^6 + 6*x^3*e^6 - (2*e^3*log(2) - 5*e^6)*x^2 - 6*(e ^3*log(2) + 2*e^6)*x)*log(x)^2 - 18*(e^3*log(2) + 2*e^6)*x - 2*(x^5*e^6 + 6*x^4*e^6 - (2*e^3*log(2) - 5*e^6)*x^3 - 3*(2*e^3*log(2) + e^6)*x^2 + 27*x *e^6 - 9*e^3*log(2) - 18*e^6)*log(x) + 81*e^6)/(x^2 - 2*x*log(x) + log(x)^ 2)
Leaf count of result is larger than twice the leaf count of optimal. 229 vs. \(2 (30) = 60\).
Time = 0.13 (sec) , antiderivative size = 229, normalized size of antiderivative = 7.39 \[ \int \frac {e^6 \left (-162+198 x-90 x^2-18 x^3-6 x^4-10 x^5-18 x^6-4 x^7\right )+e^3 \left (18 x-18 x^2+6 x^4+4 x^5\right ) \log (2)+\left (e^6 \left (-36+90 x+72 x^2+18 x^3+30 x^4+54 x^5+12 x^6\right )+e^3 \left (-18+18 x-18 x^3-12 x^4\right ) \log (2)\right ) \log (x)+\left (e^6 \left (-54 x-30 x^3-54 x^4-12 x^5\right )+e^3 \left (18 x^2+12 x^3\right ) \log (2)\right ) \log ^2(x)+\left (e^6 \left (-12 x+10 x^2+18 x^3+4 x^4\right )+e^3 \left (-6 x-4 x^2\right ) \log (2)\right ) \log ^3(x)}{-x^4+3 x^3 \log (x)-3 x^2 \log ^2(x)+x \log ^3(x)} \, dx=\frac {x^{6} e^{6} - 2 \, x^{5} e^{6} \log \left (x\right ) + x^{4} e^{6} \log \left (x\right )^{2} + 6 \, x^{5} e^{6} - 2 \, x^{4} e^{3} \log \left (2\right ) - 12 \, x^{4} e^{6} \log \left (x\right ) + 4 \, x^{3} e^{3} \log \left (2\right ) \log \left (x\right ) + 6 \, x^{3} e^{6} \log \left (x\right )^{2} - 2 \, x^{2} e^{3} \log \left (2\right ) \log \left (x\right )^{2} + 5 \, x^{4} e^{6} - 6 \, x^{3} e^{3} \log \left (2\right ) - 10 \, x^{3} e^{6} \log \left (x\right ) + 12 \, x^{2} e^{3} \log \left (2\right ) \log \left (x\right ) + 5 \, x^{2} e^{6} \log \left (x\right )^{2} - 6 \, x e^{3} \log \left (2\right ) \log \left (x\right )^{2} + 6 \, x^{3} e^{6} + 6 \, x^{2} e^{6} \log \left (x\right ) - 12 \, x e^{6} \log \left (x\right )^{2} + 54 \, x^{2} e^{6} - 18 \, x e^{3} \log \left (2\right ) - 54 \, x e^{6} \log \left (x\right ) + 18 \, e^{3} \log \left (2\right ) \log \left (x\right ) - 36 \, x e^{6} + 36 \, e^{6} \log \left (x\right ) + 81 \, e^{6}}{x^{2} - 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}} \] Input:
integrate((((-4*x^2-6*x)*exp(3)*log(2)+(4*x^4+18*x^3+10*x^2-12*x)*exp(3)^2 )*log(x)^3+((12*x^3+18*x^2)*exp(3)*log(2)+(-12*x^5-54*x^4-30*x^3-54*x)*exp (3)^2)*log(x)^2+((-12*x^4-18*x^3+18*x-18)*exp(3)*log(2)+(12*x^6+54*x^5+30* x^4+18*x^3+72*x^2+90*x-36)*exp(3)^2)*log(x)+(4*x^5+6*x^4-18*x^2+18*x)*exp( 3)*log(2)+(-4*x^7-18*x^6-10*x^5-6*x^4-18*x^3-90*x^2+198*x-162)*exp(3)^2)/( x*log(x)^3-3*x^2*log(x)^2+3*x^3*log(x)-x^4),x, algorithm="giac")
Output:
(x^6*e^6 - 2*x^5*e^6*log(x) + x^4*e^6*log(x)^2 + 6*x^5*e^6 - 2*x^4*e^3*log (2) - 12*x^4*e^6*log(x) + 4*x^3*e^3*log(2)*log(x) + 6*x^3*e^6*log(x)^2 - 2 *x^2*e^3*log(2)*log(x)^2 + 5*x^4*e^6 - 6*x^3*e^3*log(2) - 10*x^3*e^6*log(x ) + 12*x^2*e^3*log(2)*log(x) + 5*x^2*e^6*log(x)^2 - 6*x*e^3*log(2)*log(x)^ 2 + 6*x^3*e^6 + 6*x^2*e^6*log(x) - 12*x*e^6*log(x)^2 + 54*x^2*e^6 - 18*x*e ^3*log(2) - 54*x*e^6*log(x) + 18*e^3*log(2)*log(x) - 36*x*e^6 + 36*e^6*log (x) + 81*e^6)/(x^2 - 2*x*log(x) + log(x)^2)
Time = 0.74 (sec) , antiderivative size = 231, normalized size of antiderivative = 7.45 \[ \int \frac {e^6 \left (-162+198 x-90 x^2-18 x^3-6 x^4-10 x^5-18 x^6-4 x^7\right )+e^3 \left (18 x-18 x^2+6 x^4+4 x^5\right ) \log (2)+\left (e^6 \left (-36+90 x+72 x^2+18 x^3+30 x^4+54 x^5+12 x^6\right )+e^3 \left (-18+18 x-18 x^3-12 x^4\right ) \log (2)\right ) \log (x)+\left (e^6 \left (-54 x-30 x^3-54 x^4-12 x^5\right )+e^3 \left (18 x^2+12 x^3\right ) \log (2)\right ) \log ^2(x)+\left (e^6 \left (-12 x+10 x^2+18 x^3+4 x^4\right )+e^3 \left (-6 x-4 x^2\right ) \log (2)\right ) \log ^3(x)}{-x^4+3 x^3 \log (x)-3 x^2 \log ^2(x)+x \log ^3(x)} \, dx=\frac {{\mathrm {e}}^6\,x^8-2\,{\mathrm {e}}^6\,x^7\,\ln \left (x\right )+6\,{\mathrm {e}}^6\,x^7+{\mathrm {e}}^6\,x^6\,{\ln \left (x\right )}^2-12\,{\mathrm {e}}^6\,x^6\,\ln \left (x\right )+\left (5\,{\mathrm {e}}^6-2\,{\mathrm {e}}^3\,\ln \left (2\right )\right )\,x^6+6\,{\mathrm {e}}^6\,x^5\,{\ln \left (x\right )}^2+\left (4\,{\mathrm {e}}^3\,\ln \left (2\right )-10\,{\mathrm {e}}^6\right )\,x^5\,\ln \left (x\right )+6\,{\mathrm {e}}^3\,\left ({\mathrm {e}}^3-\ln \left (2\right )\right )\,x^5+\left (5\,{\mathrm {e}}^6-2\,{\mathrm {e}}^3\,\ln \left (2\right )\right )\,x^4\,{\ln \left (x\right )}^2+\left (6\,{\mathrm {e}}^6+12\,{\mathrm {e}}^3\,\ln \left (2\right )\right )\,x^4\,\ln \left (x\right )+54\,{\mathrm {e}}^6\,x^4+\left (-12\,{\mathrm {e}}^6-6\,{\mathrm {e}}^3\,\ln \left (2\right )\right )\,x^3\,{\ln \left (x\right )}^2-54\,{\mathrm {e}}^6\,x^3\,\ln \left (x\right )-18\,{\mathrm {e}}^3\,\left (2\,{\mathrm {e}}^3+\ln \left (2\right )\right )\,x^3+18\,{\mathrm {e}}^3\,\left (2\,{\mathrm {e}}^3+\ln \left (2\right )\right )\,x^2\,\ln \left (x\right )+81\,{\mathrm {e}}^6\,x^2}{x^4-2\,x^3\,\ln \left (x\right )+x^2\,{\ln \left (x\right )}^2} \] Input:
int((log(x)^3*(exp(6)*(10*x^2 - 12*x + 18*x^3 + 4*x^4) - exp(3)*log(2)*(6* x + 4*x^2)) - exp(6)*(90*x^2 - 198*x + 18*x^3 + 6*x^4 + 10*x^5 + 18*x^6 + 4*x^7 + 162) + log(x)*(exp(6)*(90*x + 72*x^2 + 18*x^3 + 30*x^4 + 54*x^5 + 12*x^6 - 36) - exp(3)*log(2)*(18*x^3 - 18*x + 12*x^4 + 18)) - log(x)^2*(ex p(6)*(54*x + 30*x^3 + 54*x^4 + 12*x^5) - exp(3)*log(2)*(18*x^2 + 12*x^3)) + exp(3)*log(2)*(18*x - 18*x^2 + 6*x^4 + 4*x^5))/(x*log(x)^3 + 3*x^3*log(x ) - 3*x^2*log(x)^2 - x^4),x)
Output:
(x^6*(5*exp(6) - 2*exp(3)*log(2)) + 81*x^2*exp(6) + 54*x^4*exp(6) + 6*x^7* exp(6) + x^8*exp(6) - 18*x^3*exp(3)*(2*exp(3) + log(2)) + 6*x^5*exp(3)*(ex p(3) - log(2)) - x^5*log(x)*(10*exp(6) - 4*exp(3)*log(2)) + x^4*log(x)*(6* exp(6) + 12*exp(3)*log(2)) - 54*x^3*exp(6)*log(x) - 12*x^6*exp(6)*log(x) - 2*x^7*exp(6)*log(x) + x^4*log(x)^2*(5*exp(6) - 2*exp(3)*log(2)) - x^3*log (x)^2*(12*exp(6) + 6*exp(3)*log(2)) + 6*x^5*exp(6)*log(x)^2 + x^6*exp(6)*l og(x)^2 + 18*x^2*exp(3)*log(x)*(2*exp(3) + log(2)))/(x^2*log(x)^2 - 2*x^3* log(x) + x^4)
Time = 0.20 (sec) , antiderivative size = 234, normalized size of antiderivative = 7.55 \[ \int \frac {e^6 \left (-162+198 x-90 x^2-18 x^3-6 x^4-10 x^5-18 x^6-4 x^7\right )+e^3 \left (18 x-18 x^2+6 x^4+4 x^5\right ) \log (2)+\left (e^6 \left (-36+90 x+72 x^2+18 x^3+30 x^4+54 x^5+12 x^6\right )+e^3 \left (-18+18 x-18 x^3-12 x^4\right ) \log (2)\right ) \log (x)+\left (e^6 \left (-54 x-30 x^3-54 x^4-12 x^5\right )+e^3 \left (18 x^2+12 x^3\right ) \log (2)\right ) \log ^2(x)+\left (e^6 \left (-12 x+10 x^2+18 x^3+4 x^4\right )+e^3 \left (-6 x-4 x^2\right ) \log (2)\right ) \log ^3(x)}{-x^4+3 x^3 \log (x)-3 x^2 \log ^2(x)+x \log ^3(x)} \, dx=\frac {e^{3} \left (-2 \mathrm {log}\left (x \right )^{2} \mathrm {log}\left (2\right ) x^{2}-6 \mathrm {log}\left (x \right )^{2} \mathrm {log}\left (2\right ) x +6 \mathrm {log}\left (x \right )^{2} e^{3} x^{3}+5 \mathrm {log}\left (x \right )^{2} e^{3} x^{2}-12 \mathrm {log}\left (x \right )^{2} e^{3} x +4 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (2\right ) x^{3}+12 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (2\right ) x^{2}-2 \,\mathrm {log}\left (x \right ) e^{3} x^{5}-12 \,\mathrm {log}\left (x \right ) e^{3} x^{4}-10 \,\mathrm {log}\left (x \right ) e^{3} x^{3}+6 \,\mathrm {log}\left (x \right ) e^{3} x^{2}-18 \,\mathrm {log}\left (2\right ) x +6 e^{3} x^{5}+e^{3} x^{6}-27 \mathrm {log}\left (x \right )^{2} e^{3}+18 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (2\right )+36 \,\mathrm {log}\left (x \right ) e^{3}+5 e^{3} x^{4}+6 e^{3} x^{3}+27 e^{3} x^{2}-36 e^{3} x -2 \,\mathrm {log}\left (2\right ) x^{4}-6 \,\mathrm {log}\left (2\right ) x^{3}+\mathrm {log}\left (x \right )^{2} e^{3} x^{4}+81 e^{3}\right )}{\mathrm {log}\left (x \right )^{2}-2 \,\mathrm {log}\left (x \right ) x +x^{2}} \] Input:
int((((-4*x^2-6*x)*exp(3)*log(2)+(4*x^4+18*x^3+10*x^2-12*x)*exp(3)^2)*log( x)^3+((12*x^3+18*x^2)*exp(3)*log(2)+(-12*x^5-54*x^4-30*x^3-54*x)*exp(3)^2) *log(x)^2+((-12*x^4-18*x^3+18*x-18)*exp(3)*log(2)+(12*x^6+54*x^5+30*x^4+18 *x^3+72*x^2+90*x-36)*exp(3)^2)*log(x)+(4*x^5+6*x^4-18*x^2+18*x)*exp(3)*log (2)+(-4*x^7-18*x^6-10*x^5-6*x^4-18*x^3-90*x^2+198*x-162)*exp(3)^2)/(x*log( x)^3-3*x^2*log(x)^2+3*x^3*log(x)-x^4),x)
Output:
(e**3*( - 2*log(x)**2*log(2)*x**2 - 6*log(x)**2*log(2)*x + log(x)**2*e**3* x**4 + 6*log(x)**2*e**3*x**3 + 5*log(x)**2*e**3*x**2 - 12*log(x)**2*e**3*x - 27*log(x)**2*e**3 + 4*log(x)*log(2)*x**3 + 12*log(x)*log(2)*x**2 + 18*l og(x)*log(2) - 2*log(x)*e**3*x**5 - 12*log(x)*e**3*x**4 - 10*log(x)*e**3*x **3 + 6*log(x)*e**3*x**2 + 36*log(x)*e**3 - 2*log(2)*x**4 - 6*log(2)*x**3 - 18*log(2)*x + e**3*x**6 + 6*e**3*x**5 + 5*e**3*x**4 + 6*e**3*x**3 + 27*e **3*x**2 - 36*e**3*x + 81*e**3))/(log(x)**2 - 2*log(x)*x + x**2)