Integrand size = 191, antiderivative size = 26 \[ \int \frac {4-10 x+4 x^2+e^x \left (-1-2 x+7 x^2-4 x^3\right )+\left (-8+9 x+e^x \left (2+5 x-6 x^2-x^3\right )\right ) \log (x)+\left (4+e^x \left (-1-2 x-2 x^2\right )\right ) \log ^2(x)-e^x x \log ^3(x)}{x^2-x^3+e^x \left (4 x-8 x^2+4 x^3\right )+\left (-4 x+7 x^2-4 x^3+e^x \left (-7 x+6 x^2+x^3\right )\right ) \log (x)+\left (8 x-8 x^2+e^x \left (2 x+2 x^2\right )\right ) \log ^2(x)+\left (-4 x+e^x x\right ) \log ^3(x)} \, dx=\log \left (\frac {1}{-4 \log (x)+e^x (4+\log (x))-\frac {x}{-1+x+\log (x)}}\right ) \] Output:
ln(1/((ln(x)+4)*exp(x)-4*ln(x)-x/(-1+ln(x)+x)))
Leaf count is larger than twice the leaf count of optimal. \(66\) vs. \(2(26)=52\).
Time = 0.13 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.54 \[ \int \frac {4-10 x+4 x^2+e^x \left (-1-2 x+7 x^2-4 x^3\right )+\left (-8+9 x+e^x \left (2+5 x-6 x^2-x^3\right )\right ) \log (x)+\left (4+e^x \left (-1-2 x-2 x^2\right )\right ) \log ^2(x)-e^x x \log ^3(x)}{x^2-x^3+e^x \left (4 x-8 x^2+4 x^3\right )+\left (-4 x+7 x^2-4 x^3+e^x \left (-7 x+6 x^2+x^3\right )\right ) \log (x)+\left (8 x-8 x^2+e^x \left (2 x+2 x^2\right )\right ) \log ^2(x)+\left (-4 x+e^x x\right ) \log ^3(x)} \, dx=\log (1-x-\log (x))-\log \left (-4 e^x-x+4 e^x x+4 \log (x)+3 e^x \log (x)-4 x \log (x)+e^x x \log (x)-4 \log ^2(x)+e^x \log ^2(x)\right ) \] Input:
Integrate[(4 - 10*x + 4*x^2 + E^x*(-1 - 2*x + 7*x^2 - 4*x^3) + (-8 + 9*x + E^x*(2 + 5*x - 6*x^2 - x^3))*Log[x] + (4 + E^x*(-1 - 2*x - 2*x^2))*Log[x] ^2 - E^x*x*Log[x]^3)/(x^2 - x^3 + E^x*(4*x - 8*x^2 + 4*x^3) + (-4*x + 7*x^ 2 - 4*x^3 + E^x*(-7*x + 6*x^2 + x^3))*Log[x] + (8*x - 8*x^2 + E^x*(2*x + 2 *x^2))*Log[x]^2 + (-4*x + E^x*x)*Log[x]^3),x]
Output:
Log[1 - x - Log[x]] - Log[-4*E^x - x + 4*E^x*x + 4*Log[x] + 3*E^x*Log[x] - 4*x*Log[x] + E^x*x*Log[x] - 4*Log[x]^2 + E^x*Log[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.
| \(\displaystyle \int \frac {4 x^2+\left (e^x \left (-2 x^2-2 x-1\right )+4\right ) \log ^2(x)+e^x \left (-4 x^3+7 x^2-2 x-1\right )+\left (e^x \left (-x^3-6 x^2+5 x+2\right )+9 x-8\right ) \log (x)-10 x-e^x x \log ^3(x)+4}{-x^3+x^2+\left (-8 x^2+e^x \left (2 x^2+2 x\right )+8 x\right ) \log ^2(x)+e^x \left (4 x^3-8 x^2+4 x\right )+\left (-4 x^3+7 x^2+e^x \left (x^3+6 x^2-7 x\right )-4 x\right ) \log (x)+\left (e^x x-4 x\right ) \log ^3(x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {4 x^2+\left (e^x \left (-2 x^2-2 x-1\right )+4\right ) \log ^2(x)+e^x \left (-4 x^3+7 x^2-2 x-1\right )+\left (e^x \left (-x^3-6 x^2+5 x+2\right )+9 x-8\right ) \log (x)-10 x-e^x x \log ^3(x)+4}{x (-x-\log (x)+1) \left (4 e^x-4 e^x x+x-e^x \log ^2(x)+4 \log ^2(x)-3 e^x \log (x)-e^x x \log (x)+4 x \log (x)-4 \log (x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {-4 x+x (-\log (x))-1}{x (\log (x)+4)}-\frac {4 x^3+4 x^3 \log ^2(x)+17 x^3 \log (x)-19 x^2+8 x^2 \log ^3(x)+25 x^2 \log ^2(x)-29 x^2 \log (x)+39 x+4 x \log ^4(x)+8 x \log ^3(x)-29 x \log ^2(x)-16 \log ^2(x)-17 x \log (x)+32 \log (x)-16}{x (\log (x)+4) (x+\log (x)-1) \left (-4 e^x+4 e^x x-x+e^x \log ^2(x)-4 \log ^2(x)+3 e^x \log (x)+e^x x \log (x)-4 x \log (x)+4 \log (x)\right )}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {-4 x+x (-\log (x))-1}{x (\log (x)+4)}-\frac {4 x^3+4 x^3 \log ^2(x)+17 x^3 \log (x)-19 x^2+8 x^2 \log ^3(x)+25 x^2 \log ^2(x)-29 x^2 \log (x)+39 x+4 x \log ^4(x)+8 x \log ^3(x)-29 x \log ^2(x)-16 \log ^2(x)-17 x \log (x)+32 \log (x)-16}{x (\log (x)+4) (x+\log (x)-1) \left (-4 e^x+4 e^x x-x+e^x \log ^2(x)-4 \log ^2(x)+3 e^x \log (x)+e^x x \log (x)-4 x \log (x)+4 \log (x)\right )}\right )dx\) |
Input:
Int[(4 - 10*x + 4*x^2 + E^x*(-1 - 2*x + 7*x^2 - 4*x^3) + (-8 + 9*x + E^x*( 2 + 5*x - 6*x^2 - x^3))*Log[x] + (4 + E^x*(-1 - 2*x - 2*x^2))*Log[x]^2 - E ^x*x*Log[x]^3)/(x^2 - x^3 + E^x*(4*x - 8*x^2 + 4*x^3) + (-4*x + 7*x^2 - 4* x^3 + E^x*(-7*x + 6*x^2 + x^3))*Log[x] + (8*x - 8*x^2 + E^x*(2*x + 2*x^2)) *Log[x]^2 + (-4*x + E^x*x)*Log[x]^3),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(64\) vs. \(2(25)=50\).
Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.50
\[-\ln \left ({\mathrm e}^{x}-4\right )+\ln \left (-1+\ln \left (x \right )+x \right )-\ln \left (\ln \left (x \right )^{2}+\frac {\left ({\mathrm e}^{x} x -4 x +3 \,{\mathrm e}^{x}+4\right ) \ln \left (x \right )}{{\mathrm e}^{x}-4}+\frac {4 \,{\mathrm e}^{x} x -x -4 \,{\mathrm e}^{x}}{{\mathrm e}^{x}-4}\right )\]
Input:
int((-x*exp(x)*ln(x)^3+((-2*x^2-2*x-1)*exp(x)+4)*ln(x)^2+((-x^3-6*x^2+5*x+ 2)*exp(x)+9*x-8)*ln(x)+(-4*x^3+7*x^2-2*x-1)*exp(x)+4*x^2-10*x+4)/((exp(x)* x-4*x)*ln(x)^3+((2*x^2+2*x)*exp(x)-8*x^2+8*x)*ln(x)^2+((x^3+6*x^2-7*x)*exp (x)-4*x^3+7*x^2-4*x)*ln(x)+(4*x^3-8*x^2+4*x)*exp(x)-x^3+x^2),x)
Output:
-ln(exp(x)-4)+ln(-1+ln(x)+x)-ln(ln(x)^2+(exp(x)*x-4*x+3*exp(x)+4)/(exp(x)- 4)*ln(x)+(4*exp(x)*x-x-4*exp(x))/(exp(x)-4))
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (25) = 50\).
Time = 0.10 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.23 \[ \int \frac {4-10 x+4 x^2+e^x \left (-1-2 x+7 x^2-4 x^3\right )+\left (-8+9 x+e^x \left (2+5 x-6 x^2-x^3\right )\right ) \log (x)+\left (4+e^x \left (-1-2 x-2 x^2\right )\right ) \log ^2(x)-e^x x \log ^3(x)}{x^2-x^3+e^x \left (4 x-8 x^2+4 x^3\right )+\left (-4 x+7 x^2-4 x^3+e^x \left (-7 x+6 x^2+x^3\right )\right ) \log (x)+\left (8 x-8 x^2+e^x \left (2 x+2 x^2\right )\right ) \log ^2(x)+\left (-4 x+e^x x\right ) \log ^3(x)} \, dx=\log \left (x + \log \left (x\right ) - 1\right ) - \log \left (\frac {{\left (e^{x} - 4\right )} \log \left (x\right )^{2} + 4 \, {\left (x - 1\right )} e^{x} + {\left ({\left (x + 3\right )} e^{x} - 4 \, x + 4\right )} \log \left (x\right ) - x}{e^{x} - 4}\right ) - \log \left (e^{x} - 4\right ) \] Input:
integrate((-x*exp(x)*log(x)^3+((-2*x^2-2*x-1)*exp(x)+4)*log(x)^2+((-x^3-6* x^2+5*x+2)*exp(x)+9*x-8)*log(x)+(-4*x^3+7*x^2-2*x-1)*exp(x)+4*x^2-10*x+4)/ ((exp(x)*x-4*x)*log(x)^3+((2*x^2+2*x)*exp(x)-8*x^2+8*x)*log(x)^2+((x^3+6*x ^2-7*x)*exp(x)-4*x^3+7*x^2-4*x)*log(x)+(4*x^3-8*x^2+4*x)*exp(x)-x^3+x^2),x , algorithm="fricas")
Output:
log(x + log(x) - 1) - log(((e^x - 4)*log(x)^2 + 4*(x - 1)*e^x + ((x + 3)*e ^x - 4*x + 4)*log(x) - x)/(e^x - 4)) - log(e^x - 4)
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (24) = 48\).
Time = 1.77 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.04 \[ \int \frac {4-10 x+4 x^2+e^x \left (-1-2 x+7 x^2-4 x^3\right )+\left (-8+9 x+e^x \left (2+5 x-6 x^2-x^3\right )\right ) \log (x)+\left (4+e^x \left (-1-2 x-2 x^2\right )\right ) \log ^2(x)-e^x x \log ^3(x)}{x^2-x^3+e^x \left (4 x-8 x^2+4 x^3\right )+\left (-4 x+7 x^2-4 x^3+e^x \left (-7 x+6 x^2+x^3\right )\right ) \log (x)+\left (8 x-8 x^2+e^x \left (2 x+2 x^2\right )\right ) \log ^2(x)+\left (-4 x+e^x x\right ) \log ^3(x)} \, dx=- \log {\left (\frac {- 4 x \log {\left (x \right )} - x - 4 \log {\left (x \right )}^{2} + 4 \log {\left (x \right )}}{x \log {\left (x \right )} + 4 x + \log {\left (x \right )}^{2} + 3 \log {\left (x \right )} - 4} + e^{x} \right )} - \log {\left (\log {\left (x \right )} + 4 \right )} \] Input:
integrate((-x*exp(x)*ln(x)**3+((-2*x**2-2*x-1)*exp(x)+4)*ln(x)**2+((-x**3- 6*x**2+5*x+2)*exp(x)+9*x-8)*ln(x)+(-4*x**3+7*x**2-2*x-1)*exp(x)+4*x**2-10* x+4)/((exp(x)*x-4*x)*ln(x)**3+((2*x**2+2*x)*exp(x)-8*x**2+8*x)*ln(x)**2+(( x**3+6*x**2-7*x)*exp(x)-4*x**3+7*x**2-4*x)*ln(x)+(4*x**3-8*x**2+4*x)*exp(x )-x**3+x**2),x)
Output:
-log((-4*x*log(x) - x - 4*log(x)**2 + 4*log(x))/(x*log(x) + 4*x + log(x)** 2 + 3*log(x) - 4) + exp(x)) - log(log(x) + 4)
Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (25) = 50\).
Time = 0.13 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.46 \[ \int \frac {4-10 x+4 x^2+e^x \left (-1-2 x+7 x^2-4 x^3\right )+\left (-8+9 x+e^x \left (2+5 x-6 x^2-x^3\right )\right ) \log (x)+\left (4+e^x \left (-1-2 x-2 x^2\right )\right ) \log ^2(x)-e^x x \log ^3(x)}{x^2-x^3+e^x \left (4 x-8 x^2+4 x^3\right )+\left (-4 x+7 x^2-4 x^3+e^x \left (-7 x+6 x^2+x^3\right )\right ) \log (x)+\left (8 x-8 x^2+e^x \left (2 x+2 x^2\right )\right ) \log ^2(x)+\left (-4 x+e^x x\right ) \log ^3(x)} \, dx=-\log \left (\frac {{\left ({\left (x + 3\right )} \log \left (x\right ) + \log \left (x\right )^{2} + 4 \, x - 4\right )} e^{x} - 4 \, {\left (x - 1\right )} \log \left (x\right ) - 4 \, \log \left (x\right )^{2} - x}{{\left (x + 3\right )} \log \left (x\right ) + \log \left (x\right )^{2} + 4 \, x - 4}\right ) - \log \left (\log \left (x\right ) + 4\right ) \] Input:
integrate((-x*exp(x)*log(x)^3+((-2*x^2-2*x-1)*exp(x)+4)*log(x)^2+((-x^3-6* x^2+5*x+2)*exp(x)+9*x-8)*log(x)+(-4*x^3+7*x^2-2*x-1)*exp(x)+4*x^2-10*x+4)/ ((exp(x)*x-4*x)*log(x)^3+((2*x^2+2*x)*exp(x)-8*x^2+8*x)*log(x)^2+((x^3+6*x ^2-7*x)*exp(x)-4*x^3+7*x^2-4*x)*log(x)+(4*x^3-8*x^2+4*x)*exp(x)-x^3+x^2),x , algorithm="maxima")
Output:
-log((((x + 3)*log(x) + log(x)^2 + 4*x - 4)*e^x - 4*(x - 1)*log(x) - 4*log (x)^2 - x)/((x + 3)*log(x) + log(x)^2 + 4*x - 4)) - log(log(x) + 4)
Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (25) = 50\).
Time = 0.34 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.19 \[ \int \frac {4-10 x+4 x^2+e^x \left (-1-2 x+7 x^2-4 x^3\right )+\left (-8+9 x+e^x \left (2+5 x-6 x^2-x^3\right )\right ) \log (x)+\left (4+e^x \left (-1-2 x-2 x^2\right )\right ) \log ^2(x)-e^x x \log ^3(x)}{x^2-x^3+e^x \left (4 x-8 x^2+4 x^3\right )+\left (-4 x+7 x^2-4 x^3+e^x \left (-7 x+6 x^2+x^3\right )\right ) \log (x)+\left (8 x-8 x^2+e^x \left (2 x+2 x^2\right )\right ) \log ^2(x)+\left (-4 x+e^x x\right ) \log ^3(x)} \, dx=-\log \left (x e^{x} \log \left (x\right ) + e^{x} \log \left (x\right )^{2} + 4 \, x e^{x} - 4 \, x \log \left (x\right ) + 3 \, e^{x} \log \left (x\right ) - 4 \, \log \left (x\right )^{2} - x - 4 \, e^{x} + 4 \, \log \left (x\right )\right ) + \log \left (x + \log \left (x\right ) - 1\right ) \] Input:
integrate((-x*exp(x)*log(x)^3+((-2*x^2-2*x-1)*exp(x)+4)*log(x)^2+((-x^3-6* x^2+5*x+2)*exp(x)+9*x-8)*log(x)+(-4*x^3+7*x^2-2*x-1)*exp(x)+4*x^2-10*x+4)/ ((exp(x)*x-4*x)*log(x)^3+((2*x^2+2*x)*exp(x)-8*x^2+8*x)*log(x)^2+((x^3+6*x ^2-7*x)*exp(x)-4*x^3+7*x^2-4*x)*log(x)+(4*x^3-8*x^2+4*x)*exp(x)-x^3+x^2),x , algorithm="giac")
Output:
-log(x*e^x*log(x) + e^x*log(x)^2 + 4*x*e^x - 4*x*log(x) + 3*e^x*log(x) - 4 *log(x)^2 - x - 4*e^x + 4*log(x)) + log(x + log(x) - 1)
Timed out. \[ \int \frac {4-10 x+4 x^2+e^x \left (-1-2 x+7 x^2-4 x^3\right )+\left (-8+9 x+e^x \left (2+5 x-6 x^2-x^3\right )\right ) \log (x)+\left (4+e^x \left (-1-2 x-2 x^2\right )\right ) \log ^2(x)-e^x x \log ^3(x)}{x^2-x^3+e^x \left (4 x-8 x^2+4 x^3\right )+\left (-4 x+7 x^2-4 x^3+e^x \left (-7 x+6 x^2+x^3\right )\right ) \log (x)+\left (8 x-8 x^2+e^x \left (2 x+2 x^2\right )\right ) \log ^2(x)+\left (-4 x+e^x x\right ) \log ^3(x)} \, dx=\int \frac {10\,x-\ln \left (x\right )\,\left (9\,x+{\mathrm {e}}^x\,\left (-x^3-6\,x^2+5\,x+2\right )-8\right )+{\ln \left (x\right )}^2\,\left ({\mathrm {e}}^x\,\left (2\,x^2+2\,x+1\right )-4\right )-4\,x^2+{\mathrm {e}}^x\,\left (4\,x^3-7\,x^2+2\,x+1\right )+x\,{\mathrm {e}}^x\,{\ln \left (x\right )}^3-4}{\ln \left (x\right )\,\left (4\,x-{\mathrm {e}}^x\,\left (x^3+6\,x^2-7\,x\right )-7\,x^2+4\,x^3\right )+{\ln \left (x\right )}^3\,\left (4\,x-x\,{\mathrm {e}}^x\right )-{\ln \left (x\right )}^2\,\left (8\,x+{\mathrm {e}}^x\,\left (2\,x^2+2\,x\right )-8\,x^2\right )-x^2+x^3-{\mathrm {e}}^x\,\left (4\,x^3-8\,x^2+4\,x\right )} \,d x \] Input:
int((10*x - log(x)*(9*x + exp(x)*(5*x - 6*x^2 - x^3 + 2) - 8) + log(x)^2*( exp(x)*(2*x + 2*x^2 + 1) - 4) - 4*x^2 + exp(x)*(2*x - 7*x^2 + 4*x^3 + 1) + x*exp(x)*log(x)^3 - 4)/(log(x)*(4*x - exp(x)*(6*x^2 - 7*x + x^3) - 7*x^2 + 4*x^3) + log(x)^3*(4*x - x*exp(x)) - log(x)^2*(8*x + exp(x)*(2*x + 2*x^2 ) - 8*x^2) - x^2 + x^3 - exp(x)*(4*x - 8*x^2 + 4*x^3)),x)
Output:
int((10*x - log(x)*(9*x + exp(x)*(5*x - 6*x^2 - x^3 + 2) - 8) + log(x)^2*( exp(x)*(2*x + 2*x^2 + 1) - 4) - 4*x^2 + exp(x)*(2*x - 7*x^2 + 4*x^3 + 1) + x*exp(x)*log(x)^3 - 4)/(log(x)*(4*x - exp(x)*(6*x^2 - 7*x + x^3) - 7*x^2 + 4*x^3) + log(x)^3*(4*x - x*exp(x)) - log(x)^2*(8*x + exp(x)*(2*x + 2*x^2 ) - 8*x^2) - x^2 + x^3 - exp(x)*(4*x - 8*x^2 + 4*x^3)), x)
Time = 0.18 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.38 \[ \int \frac {4-10 x+4 x^2+e^x \left (-1-2 x+7 x^2-4 x^3\right )+\left (-8+9 x+e^x \left (2+5 x-6 x^2-x^3\right )\right ) \log (x)+\left (4+e^x \left (-1-2 x-2 x^2\right )\right ) \log ^2(x)-e^x x \log ^3(x)}{x^2-x^3+e^x \left (4 x-8 x^2+4 x^3\right )+\left (-4 x+7 x^2-4 x^3+e^x \left (-7 x+6 x^2+x^3\right )\right ) \log (x)+\left (8 x-8 x^2+e^x \left (2 x+2 x^2\right )\right ) \log ^2(x)+\left (-4 x+e^x x\right ) \log ^3(x)} \, dx=\mathrm {log}\left (\mathrm {log}\left (x \right )+x -1\right )-\mathrm {log}\left (e^{x} \mathrm {log}\left (x \right )^{2}+e^{x} \mathrm {log}\left (x \right ) x +3 e^{x} \mathrm {log}\left (x \right )+4 e^{x} x -4 e^{x}-4 \mathrm {log}\left (x \right )^{2}-4 \,\mathrm {log}\left (x \right ) x +4 \,\mathrm {log}\left (x \right )-x \right ) \] Input:
int((-x*exp(x)*log(x)^3+((-2*x^2-2*x-1)*exp(x)+4)*log(x)^2+((-x^3-6*x^2+5* x+2)*exp(x)+9*x-8)*log(x)+(-4*x^3+7*x^2-2*x-1)*exp(x)+4*x^2-10*x+4)/((exp( x)*x-4*x)*log(x)^3+((2*x^2+2*x)*exp(x)-8*x^2+8*x)*log(x)^2+((x^3+6*x^2-7*x )*exp(x)-4*x^3+7*x^2-4*x)*log(x)+(4*x^3-8*x^2+4*x)*exp(x)-x^3+x^2),x)
Output:
log(log(x) + x - 1) - log(e**x*log(x)**2 + e**x*log(x)*x + 3*e**x*log(x) + 4*e**x*x - 4*e**x - 4*log(x)**2 - 4*log(x)*x + 4*log(x) - x)