Integrand size = 263, antiderivative size = 30 \[ \int \frac {-54 x+81 x^3+e^8 \left (-6 x+9 x^3\right )+e^{2 x} \left (-5 x-6 x^2+9 x^3\right )+e^4 \left (-36 x+54 x^3\right )+e^x \left (-2+30 x+18 x^2-54 x^3+e^4 \left (10 x+6 x^2-18 x^3\right )\right )+\left (36 x-54 x^3+e^4 \left (12 x-18 x^3\right )+e^x \left (-10 x-6 x^2+18 x^3\right )\right ) \log (x)+\left (-6 x+9 x^3\right ) \log ^2(x)}{81 x^3+54 e^4 x^3+9 e^8 x^3+e^{2 x} \left (x-6 x^2+9 x^3\right )+e^x \left (18 x^2-54 x^3+e^4 \left (6 x^2-18 x^3\right )\right )+\left (-54 x^3-18 e^4 x^3+e^x \left (-6 x^2+18 x^3\right )\right ) \log (x)+9 x^3 \log ^2(x)} \, dx=5+x+\frac {2}{3 x-\frac {e^x}{-3-e^4+e^x+\log (x)}} \] Output:
2/(3*x-exp(x)/(exp(x)-exp(4)+ln(x)-3))+5+x
Time = 0.12 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.57 \[ \int \frac {-54 x+81 x^3+e^8 \left (-6 x+9 x^3\right )+e^{2 x} \left (-5 x-6 x^2+9 x^3\right )+e^4 \left (-36 x+54 x^3\right )+e^x \left (-2+30 x+18 x^2-54 x^3+e^4 \left (10 x+6 x^2-18 x^3\right )\right )+\left (36 x-54 x^3+e^4 \left (12 x-18 x^3\right )+e^x \left (-10 x-6 x^2+18 x^3\right )\right ) \log (x)+\left (-6 x+9 x^3\right ) \log ^2(x)}{81 x^3+54 e^4 x^3+9 e^8 x^3+e^{2 x} \left (x-6 x^2+9 x^3\right )+e^x \left (18 x^2-54 x^3+e^4 \left (6 x^2-18 x^3\right )\right )+\left (-54 x^3-18 e^4 x^3+e^x \left (-6 x^2+18 x^3\right )\right ) \log (x)+9 x^3 \log ^2(x)} \, dx=\frac {2}{3 x}+x+\frac {2 e^x}{3 x \left (-e^x-9 x-3 e^4 x+3 e^x x+3 x \log (x)\right )} \] Input:
Integrate[(-54*x + 81*x^3 + E^8*(-6*x + 9*x^3) + E^(2*x)*(-5*x - 6*x^2 + 9 *x^3) + E^4*(-36*x + 54*x^3) + E^x*(-2 + 30*x + 18*x^2 - 54*x^3 + E^4*(10* x + 6*x^2 - 18*x^3)) + (36*x - 54*x^3 + E^4*(12*x - 18*x^3) + E^x*(-10*x - 6*x^2 + 18*x^3))*Log[x] + (-6*x + 9*x^3)*Log[x]^2)/(81*x^3 + 54*E^4*x^3 + 9*E^8*x^3 + E^(2*x)*(x - 6*x^2 + 9*x^3) + E^x*(18*x^2 - 54*x^3 + E^4*(6*x ^2 - 18*x^3)) + (-54*x^3 - 18*E^4*x^3 + E^x*(-6*x^2 + 18*x^3))*Log[x] + 9* x^3*Log[x]^2),x]
Output:
2/(3*x) + x + (2*E^x)/(3*x*(-E^x - 9*x - 3*E^4*x + 3*E^x*x + 3*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.
| \(\displaystyle \int \frac {81 x^3+e^8 \left (9 x^3-6 x\right )+e^4 \left (54 x^3-36 x\right )+\left (9 x^3-6 x\right ) \log ^2(x)+e^{2 x} \left (9 x^3-6 x^2-5 x\right )+e^x \left (-54 x^3+18 x^2+e^4 \left (-18 x^3+6 x^2+10 x\right )+30 x-2\right )+\left (-54 x^3+e^4 \left (12 x-18 x^3\right )+e^x \left (18 x^3-6 x^2-10 x\right )+36 x\right ) \log (x)-54 x}{9 e^8 x^3+54 e^4 x^3+81 x^3+9 x^3 \log ^2(x)+e^{2 x} \left (9 x^3-6 x^2+x\right )+e^x \left (-54 x^3+18 x^2+e^4 \left (6 x^2-18 x^3\right )\right )+\left (-18 e^4 x^3-54 x^3+e^x \left (18 x^3-6 x^2\right )\right ) \log (x)} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {81 x^3+e^8 \left (9 x^3-6 x\right )+e^4 \left (54 x^3-36 x\right )+\left (9 x^3-6 x\right ) \log ^2(x)+e^{2 x} \left (9 x^3-6 x^2-5 x\right )+e^x \left (-54 x^3+18 x^2+e^4 \left (-18 x^3+6 x^2+10 x\right )+30 x-2\right )+\left (-54 x^3+e^4 \left (12 x-18 x^3\right )+e^x \left (18 x^3-6 x^2-10 x\right )+36 x\right ) \log (x)-54 x}{\left (81+54 e^4\right ) x^3+9 e^8 x^3+9 x^3 \log ^2(x)+e^{2 x} \left (9 x^3-6 x^2+x\right )+e^x \left (-54 x^3+18 x^2+e^4 \left (6 x^2-18 x^3\right )\right )+\left (-18 e^4 x^3-54 x^3+e^x \left (18 x^3-6 x^2\right )\right ) \log (x)}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {81 x^3+e^8 \left (9 x^3-6 x\right )+e^4 \left (54 x^3-36 x\right )+\left (9 x^3-6 x\right ) \log ^2(x)+e^{2 x} \left (9 x^3-6 x^2-5 x\right )+e^x \left (-54 x^3+18 x^2+e^4 \left (-18 x^3+6 x^2+10 x\right )+30 x-2\right )+\left (-54 x^3+e^4 \left (12 x-18 x^3\right )+e^x \left (18 x^3-6 x^2-10 x\right )+36 x\right ) \log (x)-54 x}{\left (81+54 e^4+9 e^8\right ) x^3+9 x^3 \log ^2(x)+e^{2 x} \left (9 x^3-6 x^2+x\right )+e^x \left (-54 x^3+18 x^2+e^4 \left (6 x^2-18 x^3\right )\right )+\left (-18 e^4 x^3-54 x^3+e^x \left (18 x^3-6 x^2\right )\right ) \log (x)}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {81 x^3+\left (1+\frac {6}{e^4}\right ) e^8 \left (9 x^3-6 x\right )+\left (9 x^3-6 x\right ) \log ^2(x)+e^{2 x} \left (9 x^3-6 x^2-5 x\right )+e^x \left (-54 x^3+18 x^2+e^4 \left (-18 x^3+6 x^2+10 x\right )+30 x-2\right )+\left (-54 x^3+e^4 \left (12 x-18 x^3\right )+e^x \left (18 x^3-6 x^2-10 x\right )+36 x\right ) \log (x)-54 x}{x \left (-3 e^x x+9 \left (1+\frac {e^4}{3}\right ) x+e^x-3 x \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {9 x^2-6 x-5}{(3 x-1)^2}+\frac {2 \left (9 \left (1+\frac {e^4}{3}\right ) x^2-3 x^2 \log (x)+18 \left (1+\frac {5 e^4}{18}\right ) x-5 x \log (x)-1\right )}{(1-3 x)^2 x \left (-3 e^x x+9 \left (1+\frac {e^4}{3}\right ) x+e^x-3 x \log (x)\right )}+\frac {6 \left (3 \left (1+\frac {e^4}{3}\right )-\log (x)\right ) \left (-9 \left (1+\frac {e^4}{3}\right ) x^2+3 x^2 \log (x)+e^4 x-x \log (x)+\log (x)-2 \left (1+\frac {e^4}{2}\right )\right )}{(1-3 x)^2 \left (-3 e^x x+9 \left (1+\frac {e^4}{3}\right ) x+e^x-3 x \log (x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {9 x^2-6 x-5}{(3 x-1)^2}+\frac {2 \left (9 \left (1+\frac {e^4}{3}\right ) x^2-3 x^2 \log (x)+18 \left (1+\frac {5 e^4}{18}\right ) x-5 x \log (x)-1\right )}{(1-3 x)^2 x \left (-3 e^x x+9 \left (1+\frac {e^4}{3}\right ) x+e^x-3 x \log (x)\right )}+\frac {6 \left (3 \left (1+\frac {e^4}{3}\right )-\log (x)\right ) \left (-9 \left (1+\frac {e^4}{3}\right ) x^2+3 x^2 \log (x)+e^4 x-x \log (x)+\log (x)-2 \left (1+\frac {e^4}{2}\right )\right )}{(1-3 x)^2 \left (-3 e^x x+9 \left (1+\frac {e^4}{3}\right ) x+e^x-3 x \log (x)\right )^2}\right )dx\) |
Input:
Int[(-54*x + 81*x^3 + E^8*(-6*x + 9*x^3) + E^(2*x)*(-5*x - 6*x^2 + 9*x^3) + E^4*(-36*x + 54*x^3) + E^x*(-2 + 30*x + 18*x^2 - 54*x^3 + E^4*(10*x + 6* x^2 - 18*x^3)) + (36*x - 54*x^3 + E^4*(12*x - 18*x^3) + E^x*(-10*x - 6*x^2 + 18*x^3))*Log[x] + (-6*x + 9*x^3)*Log[x]^2)/(81*x^3 + 54*E^4*x^3 + 9*E^8 *x^3 + E^(2*x)*(x - 6*x^2 + 9*x^3) + E^x*(18*x^2 - 54*x^3 + E^4*(6*x^2 - 1 8*x^3)) + (-54*x^3 - 18*E^4*x^3 + E^x*(-6*x^2 + 18*x^3))*Log[x] + 9*x^3*Lo g[x]^2),x]
Output:
$Aborted
Time = 55.96 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.47
| method | result | size |
| risch | \(\frac {3 x^{2}+2}{3 x}-\frac {2 \,{\mathrm e}^{x}}{3 x \left (3 x \,{\mathrm e}^{4}-3 \,{\mathrm e}^{x} x -3 x \ln \left (x \right )+9 x +{\mathrm e}^{x}\right )}\) | \(44\) |
| parallelrisch | \(\frac {18+9 x -9 \,{\mathrm e}^{x} x^{2}+9 x^{2} {\mathrm e}^{4}+3 x \,{\mathrm e}^{4}-9 x^{2} \ln \left (x \right )-3 x \ln \left (x \right )+6 \,{\mathrm e}^{4}-6 \ln \left (x \right )+27 x^{2}-5 \,{\mathrm e}^{x}}{9 x \,{\mathrm e}^{4}-9 \,{\mathrm e}^{x} x -9 x \ln \left (x \right )+27 x +3 \,{\mathrm e}^{x}}\) | \(79\) |
Input:
int(((9*x^3-6*x)*ln(x)^2+((18*x^3-6*x^2-10*x)*exp(x)+(-18*x^3+12*x)*exp(4) -54*x^3+36*x)*ln(x)+(9*x^3-6*x^2-5*x)*exp(x)^2+((-18*x^3+6*x^2+10*x)*exp(4 )-54*x^3+18*x^2+30*x-2)*exp(x)+(9*x^3-6*x)*exp(4)^2+(54*x^3-36*x)*exp(4)+8 1*x^3-54*x)/(9*x^3*ln(x)^2+((18*x^3-6*x^2)*exp(x)-18*x^3*exp(4)-54*x^3)*ln (x)+(9*x^3-6*x^2+x)*exp(x)^2+((-18*x^3+6*x^2)*exp(4)-54*x^3+18*x^2)*exp(x) +9*x^3*exp(4)^2+54*x^3*exp(4)+81*x^3),x,method=_RETURNVERBOSE)
Output:
1/3*(3*x^2+2)/x-2/3/x*exp(x)/(3*x*exp(4)-3*exp(x)*x-3*x*ln(x)+9*x+exp(x))
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (28) = 56\).
Time = 0.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.27 \[ \int \frac {-54 x+81 x^3+e^8 \left (-6 x+9 x^3\right )+e^{2 x} \left (-5 x-6 x^2+9 x^3\right )+e^4 \left (-36 x+54 x^3\right )+e^x \left (-2+30 x+18 x^2-54 x^3+e^4 \left (10 x+6 x^2-18 x^3\right )\right )+\left (36 x-54 x^3+e^4 \left (12 x-18 x^3\right )+e^x \left (-10 x-6 x^2+18 x^3\right )\right ) \log (x)+\left (-6 x+9 x^3\right ) \log ^2(x)}{81 x^3+54 e^4 x^3+9 e^8 x^3+e^{2 x} \left (x-6 x^2+9 x^3\right )+e^x \left (18 x^2-54 x^3+e^4 \left (6 x^2-18 x^3\right )\right )+\left (-54 x^3-18 e^4 x^3+e^x \left (-6 x^2+18 x^3\right )\right ) \log (x)+9 x^3 \log ^2(x)} \, dx=\frac {9 \, x^{2} + {\left (3 \, x^{2} + 2\right )} e^{4} - {\left (3 \, x^{2} - x + 2\right )} e^{x} - {\left (3 \, x^{2} + 2\right )} \log \left (x\right ) + 6}{3 \, x e^{4} - {\left (3 \, x - 1\right )} e^{x} - 3 \, x \log \left (x\right ) + 9 \, x} \] Input:
integrate(((9*x^3-6*x)*log(x)^2+((18*x^3-6*x^2-10*x)*exp(x)+(-18*x^3+12*x) *exp(4)-54*x^3+36*x)*log(x)+(9*x^3-6*x^2-5*x)*exp(x)^2+((-18*x^3+6*x^2+10* x)*exp(4)-54*x^3+18*x^2+30*x-2)*exp(x)+(9*x^3-6*x)*exp(4)^2+(54*x^3-36*x)* exp(4)+81*x^3-54*x)/(9*x^3*log(x)^2+((18*x^3-6*x^2)*exp(x)-18*x^3*exp(4)-5 4*x^3)*log(x)+(9*x^3-6*x^2+x)*exp(x)^2+((-18*x^3+6*x^2)*exp(4)-54*x^3+18*x ^2)*exp(x)+9*x^3*exp(4)^2+54*x^3*exp(4)+81*x^3),x, algorithm="fricas")
Output:
(9*x^2 + (3*x^2 + 2)*e^4 - (3*x^2 - x + 2)*e^x - (3*x^2 + 2)*log(x) + 6)/( 3*x*e^4 - (3*x - 1)*e^x - 3*x*log(x) + 9*x)
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (22) = 44\).
Time = 0.64 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.37 \[ \int \frac {-54 x+81 x^3+e^8 \left (-6 x+9 x^3\right )+e^{2 x} \left (-5 x-6 x^2+9 x^3\right )+e^4 \left (-36 x+54 x^3\right )+e^x \left (-2+30 x+18 x^2-54 x^3+e^4 \left (10 x+6 x^2-18 x^3\right )\right )+\left (36 x-54 x^3+e^4 \left (12 x-18 x^3\right )+e^x \left (-10 x-6 x^2+18 x^3\right )\right ) \log (x)+\left (-6 x+9 x^3\right ) \log ^2(x)}{81 x^3+54 e^4 x^3+9 e^8 x^3+e^{2 x} \left (x-6 x^2+9 x^3\right )+e^x \left (18 x^2-54 x^3+e^4 \left (6 x^2-18 x^3\right )\right )+\left (-54 x^3-18 e^4 x^3+e^x \left (-6 x^2+18 x^3\right )\right ) \log (x)+9 x^3 \log ^2(x)} \, dx=x + \frac {- 2 \log {\left (x \right )} + 6 + 2 e^{4}}{9 x^{2} \log {\left (x \right )} - 9 x^{2} e^{4} - 27 x^{2} - 3 x \log {\left (x \right )} + 9 x + 3 x e^{4} + \left (9 x^{2} - 6 x + 1\right ) e^{x}} + \frac {2}{3 x - 1} \] Input:
integrate(((9*x**3-6*x)*ln(x)**2+((18*x**3-6*x**2-10*x)*exp(x)+(-18*x**3+1 2*x)*exp(4)-54*x**3+36*x)*ln(x)+(9*x**3-6*x**2-5*x)*exp(x)**2+((-18*x**3+6 *x**2+10*x)*exp(4)-54*x**3+18*x**2+30*x-2)*exp(x)+(9*x**3-6*x)*exp(4)**2+( 54*x**3-36*x)*exp(4)+81*x**3-54*x)/(9*x**3*ln(x)**2+((18*x**3-6*x**2)*exp( x)-18*x**3*exp(4)-54*x**3)*ln(x)+(9*x**3-6*x**2+x)*exp(x)**2+((-18*x**3+6* x**2)*exp(4)-54*x**3+18*x**2)*exp(x)+9*x**3*exp(4)**2+54*x**3*exp(4)+81*x* *3),x)
Output:
x + (-2*log(x) + 6 + 2*exp(4))/(9*x**2*log(x) - 9*x**2*exp(4) - 27*x**2 - 3*x*log(x) + 9*x + 3*x*exp(4) + (9*x**2 - 6*x + 1)*exp(x)) + 2/(3*x - 1)
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (28) = 56\).
Time = 0.15 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.17 \[ \int \frac {-54 x+81 x^3+e^8 \left (-6 x+9 x^3\right )+e^{2 x} \left (-5 x-6 x^2+9 x^3\right )+e^4 \left (-36 x+54 x^3\right )+e^x \left (-2+30 x+18 x^2-54 x^3+e^4 \left (10 x+6 x^2-18 x^3\right )\right )+\left (36 x-54 x^3+e^4 \left (12 x-18 x^3\right )+e^x \left (-10 x-6 x^2+18 x^3\right )\right ) \log (x)+\left (-6 x+9 x^3\right ) \log ^2(x)}{81 x^3+54 e^4 x^3+9 e^8 x^3+e^{2 x} \left (x-6 x^2+9 x^3\right )+e^x \left (18 x^2-54 x^3+e^4 \left (6 x^2-18 x^3\right )\right )+\left (-54 x^3-18 e^4 x^3+e^x \left (-6 x^2+18 x^3\right )\right ) \log (x)+9 x^3 \log ^2(x)} \, dx=\frac {3 \, x^{2} {\left (e^{4} + 3\right )} - {\left (3 \, x^{2} - x + 2\right )} e^{x} - {\left (3 \, x^{2} + 2\right )} \log \left (x\right ) + 2 \, e^{4} + 6}{3 \, x {\left (e^{4} + 3\right )} - {\left (3 \, x - 1\right )} e^{x} - 3 \, x \log \left (x\right )} \] Input:
integrate(((9*x^3-6*x)*log(x)^2+((18*x^3-6*x^2-10*x)*exp(x)+(-18*x^3+12*x) *exp(4)-54*x^3+36*x)*log(x)+(9*x^3-6*x^2-5*x)*exp(x)^2+((-18*x^3+6*x^2+10* x)*exp(4)-54*x^3+18*x^2+30*x-2)*exp(x)+(9*x^3-6*x)*exp(4)^2+(54*x^3-36*x)* exp(4)+81*x^3-54*x)/(9*x^3*log(x)^2+((18*x^3-6*x^2)*exp(x)-18*x^3*exp(4)-5 4*x^3)*log(x)+(9*x^3-6*x^2+x)*exp(x)^2+((-18*x^3+6*x^2)*exp(4)-54*x^3+18*x ^2)*exp(x)+9*x^3*exp(4)^2+54*x^3*exp(4)+81*x^3),x, algorithm="maxima")
Output:
(3*x^2*(e^4 + 3) - (3*x^2 - x + 2)*e^x - (3*x^2 + 2)*log(x) + 2*e^4 + 6)/( 3*x*(e^4 + 3) - (3*x - 1)*e^x - 3*x*log(x))
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (28) = 56\).
Time = 0.23 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.73 \[ \int \frac {-54 x+81 x^3+e^8 \left (-6 x+9 x^3\right )+e^{2 x} \left (-5 x-6 x^2+9 x^3\right )+e^4 \left (-36 x+54 x^3\right )+e^x \left (-2+30 x+18 x^2-54 x^3+e^4 \left (10 x+6 x^2-18 x^3\right )\right )+\left (36 x-54 x^3+e^4 \left (12 x-18 x^3\right )+e^x \left (-10 x-6 x^2+18 x^3\right )\right ) \log (x)+\left (-6 x+9 x^3\right ) \log ^2(x)}{81 x^3+54 e^4 x^3+9 e^8 x^3+e^{2 x} \left (x-6 x^2+9 x^3\right )+e^x \left (18 x^2-54 x^3+e^4 \left (6 x^2-18 x^3\right )\right )+\left (-54 x^3-18 e^4 x^3+e^x \left (-6 x^2+18 x^3\right )\right ) \log (x)+9 x^3 \log ^2(x)} \, dx=\frac {3 \, x^{2} e^{4} - 3 \, x^{2} e^{x} - 3 \, x^{2} \log \left (x\right ) + 9 \, x^{2} + 3 \, x e^{4} - 2 \, x e^{x} - 3 \, x \log \left (x\right ) + 9 \, x + 2 \, e^{4} - e^{x} - 2 \, \log \left (x\right ) + 6}{3 \, x e^{4} - 3 \, x e^{x} - 3 \, x \log \left (x\right ) + 9 \, x + e^{x}} \] Input:
integrate(((9*x^3-6*x)*log(x)^2+((18*x^3-6*x^2-10*x)*exp(x)+(-18*x^3+12*x) *exp(4)-54*x^3+36*x)*log(x)+(9*x^3-6*x^2-5*x)*exp(x)^2+((-18*x^3+6*x^2+10* x)*exp(4)-54*x^3+18*x^2+30*x-2)*exp(x)+(9*x^3-6*x)*exp(4)^2+(54*x^3-36*x)* exp(4)+81*x^3-54*x)/(9*x^3*log(x)^2+((18*x^3-6*x^2)*exp(x)-18*x^3*exp(4)-5 4*x^3)*log(x)+(9*x^3-6*x^2+x)*exp(x)^2+((-18*x^3+6*x^2)*exp(4)-54*x^3+18*x ^2)*exp(x)+9*x^3*exp(4)^2+54*x^3*exp(4)+81*x^3),x, algorithm="giac")
Output:
(3*x^2*e^4 - 3*x^2*e^x - 3*x^2*log(x) + 9*x^2 + 3*x*e^4 - 2*x*e^x - 3*x*lo g(x) + 9*x + 2*e^4 - e^x - 2*log(x) + 6)/(3*x*e^4 - 3*x*e^x - 3*x*log(x) + 9*x + e^x)
Timed out. \[ \int \frac {-54 x+81 x^3+e^8 \left (-6 x+9 x^3\right )+e^{2 x} \left (-5 x-6 x^2+9 x^3\right )+e^4 \left (-36 x+54 x^3\right )+e^x \left (-2+30 x+18 x^2-54 x^3+e^4 \left (10 x+6 x^2-18 x^3\right )\right )+\left (36 x-54 x^3+e^4 \left (12 x-18 x^3\right )+e^x \left (-10 x-6 x^2+18 x^3\right )\right ) \log (x)+\left (-6 x+9 x^3\right ) \log ^2(x)}{81 x^3+54 e^4 x^3+9 e^8 x^3+e^{2 x} \left (x-6 x^2+9 x^3\right )+e^x \left (18 x^2-54 x^3+e^4 \left (6 x^2-18 x^3\right )\right )+\left (-54 x^3-18 e^4 x^3+e^x \left (-6 x^2+18 x^3\right )\right ) \log (x)+9 x^3 \log ^2(x)} \, dx=\int -\frac {54\,x+{\ln \left (x\right )}^2\,\left (6\,x-9\,x^3\right )+{\mathrm {e}}^8\,\left (6\,x-9\,x^3\right )+{\mathrm {e}}^4\,\left (36\,x-54\,x^3\right )+{\mathrm {e}}^{2\,x}\,\left (-9\,x^3+6\,x^2+5\,x\right )-{\mathrm {e}}^x\,\left (30\,x+{\mathrm {e}}^4\,\left (-18\,x^3+6\,x^2+10\,x\right )+18\,x^2-54\,x^3-2\right )-\ln \left (x\right )\,\left (36\,x+{\mathrm {e}}^4\,\left (12\,x-18\,x^3\right )-54\,x^3-{\mathrm {e}}^x\,\left (-18\,x^3+6\,x^2+10\,x\right )\right )-81\,x^3}{9\,x^3\,{\ln \left (x\right )}^2-\ln \left (x\right )\,\left ({\mathrm {e}}^x\,\left (6\,x^2-18\,x^3\right )+18\,x^3\,{\mathrm {e}}^4+54\,x^3\right )+{\mathrm {e}}^x\,\left ({\mathrm {e}}^4\,\left (6\,x^2-18\,x^3\right )+18\,x^2-54\,x^3\right )+54\,x^3\,{\mathrm {e}}^4+9\,x^3\,{\mathrm {e}}^8+81\,x^3+{\mathrm {e}}^{2\,x}\,\left (9\,x^3-6\,x^2+x\right )} \,d x \] Input:
int(-(54*x + log(x)^2*(6*x - 9*x^3) + exp(8)*(6*x - 9*x^3) + exp(4)*(36*x - 54*x^3) + exp(2*x)*(5*x + 6*x^2 - 9*x^3) - exp(x)*(30*x + exp(4)*(10*x + 6*x^2 - 18*x^3) + 18*x^2 - 54*x^3 - 2) - log(x)*(36*x + exp(4)*(12*x - 18 *x^3) - 54*x^3 - exp(x)*(10*x + 6*x^2 - 18*x^3)) - 81*x^3)/(9*x^3*log(x)^2 - log(x)*(exp(x)*(6*x^2 - 18*x^3) + 18*x^3*exp(4) + 54*x^3) + exp(x)*(exp (4)*(6*x^2 - 18*x^3) + 18*x^2 - 54*x^3) + 54*x^3*exp(4) + 9*x^3*exp(8) + 8 1*x^3 + exp(2*x)*(x - 6*x^2 + 9*x^3)),x)
Output:
int(-(54*x + log(x)^2*(6*x - 9*x^3) + exp(8)*(6*x - 9*x^3) + exp(4)*(36*x - 54*x^3) + exp(2*x)*(5*x + 6*x^2 - 9*x^3) - exp(x)*(30*x + exp(4)*(10*x + 6*x^2 - 18*x^3) + 18*x^2 - 54*x^3 - 2) - log(x)*(36*x + exp(4)*(12*x - 18 *x^3) - 54*x^3 - exp(x)*(10*x + 6*x^2 - 18*x^3)) - 81*x^3)/(9*x^3*log(x)^2 - log(x)*(exp(x)*(6*x^2 - 18*x^3) + 18*x^3*exp(4) + 54*x^3) + exp(x)*(exp (4)*(6*x^2 - 18*x^3) + 18*x^2 - 54*x^3) + 54*x^3*exp(4) + 9*x^3*exp(8) + 8 1*x^3 + exp(2*x)*(x - 6*x^2 + 9*x^3)), x)
Time = 0.26 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.93 \[ \int \frac {-54 x+81 x^3+e^8 \left (-6 x+9 x^3\right )+e^{2 x} \left (-5 x-6 x^2+9 x^3\right )+e^4 \left (-36 x+54 x^3\right )+e^x \left (-2+30 x+18 x^2-54 x^3+e^4 \left (10 x+6 x^2-18 x^3\right )\right )+\left (36 x-54 x^3+e^4 \left (12 x-18 x^3\right )+e^x \left (-10 x-6 x^2+18 x^3\right )\right ) \log (x)+\left (-6 x+9 x^3\right ) \log ^2(x)}{81 x^3+54 e^4 x^3+9 e^8 x^3+e^{2 x} \left (x-6 x^2+9 x^3\right )+e^x \left (18 x^2-54 x^3+e^4 \left (6 x^2-18 x^3\right )\right )+\left (-54 x^3-18 e^4 x^3+e^x \left (-6 x^2+18 x^3\right )\right ) \log (x)+9 x^3 \log ^2(x)} \, dx=\frac {3 e^{x} x^{2}+5 e^{x} x +3 \,\mathrm {log}\left (x \right ) x^{2}+6 \,\mathrm {log}\left (x \right ) x +2 \,\mathrm {log}\left (x \right )-3 e^{4} x^{2}-6 e^{4} x -2 e^{4}-9 x^{2}-18 x -6}{3 e^{x} x -e^{x}+3 \,\mathrm {log}\left (x \right ) x -3 e^{4} x -9 x} \] Input:
int(((9*x^3-6*x)*log(x)^2+((18*x^3-6*x^2-10*x)*exp(x)+(-18*x^3+12*x)*exp(4 )-54*x^3+36*x)*log(x)+(9*x^3-6*x^2-5*x)*exp(x)^2+((-18*x^3+6*x^2+10*x)*exp (4)-54*x^3+18*x^2+30*x-2)*exp(x)+(9*x^3-6*x)*exp(4)^2+(54*x^3-36*x)*exp(4) +81*x^3-54*x)/(9*x^3*log(x)^2+((18*x^3-6*x^2)*exp(x)-18*x^3*exp(4)-54*x^3) *log(x)+(9*x^3-6*x^2+x)*exp(x)^2+((-18*x^3+6*x^2)*exp(4)-54*x^3+18*x^2)*ex p(x)+9*x^3*exp(4)^2+54*x^3*exp(4)+81*x^3),x)
Output:
(3*e**x*x**2 + 5*e**x*x + 3*log(x)*x**2 + 6*log(x)*x + 2*log(x) - 3*e**4*x **2 - 6*e**4*x - 2*e**4 - 9*x**2 - 18*x - 6)/(3*e**x*x - e**x + 3*log(x)*x - 3*e**4*x - 9*x)