Integrand size = 181, antiderivative size = 35 \[ \int \frac {\left (16 x+4 e x+e^{3 x} \left (-18 x+54 x^2+e^3 \left (-2 x+6 x^2\right )\right )\right ) \log \left (\frac {-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x}{18 x+2 e^3 x}\right )+\left (-16 x-4 e x+18 x^2+2 e^3 x^2+e^{3 x} \left (18 x+2 e^3 x\right )\right ) \log ^2\left (\frac {-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x}{18 x+2 e^3 x}\right )}{-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x} \, dx=x^2 \log ^2\left (\frac {-\frac {4+e}{9+e^3}+\frac {1}{2} \left (e^{3 x}+x\right )}{x}\right ) \] Output:
ln((1/2*exp(3*x)+1/2*x-(exp(1)+4)/(exp(3)+9))/x)^2*x^2
\[ \int \frac {\left (16 x+4 e x+e^{3 x} \left (-18 x+54 x^2+e^3 \left (-2 x+6 x^2\right )\right )\right ) \log \left (\frac {-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x}{18 x+2 e^3 x}\right )+\left (-16 x-4 e x+18 x^2+2 e^3 x^2+e^{3 x} \left (18 x+2 e^3 x\right )\right ) \log ^2\left (\frac {-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x}{18 x+2 e^3 x}\right )}{-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x} \, dx=\int \frac {\left (16 x+4 e x+e^{3 x} \left (-18 x+54 x^2+e^3 \left (-2 x+6 x^2\right )\right )\right ) \log \left (\frac {-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x}{18 x+2 e^3 x}\right )+\left (-16 x-4 e x+18 x^2+2 e^3 x^2+e^{3 x} \left (18 x+2 e^3 x\right )\right ) \log ^2\left (\frac {-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x}{18 x+2 e^3 x}\right )}{-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x} \, dx \] Input:
Integrate[((16*x + 4*E*x + E^(3*x)*(-18*x + 54*x^2 + E^3*(-2*x + 6*x^2)))* Log[(-8 - 2*E + E^(3*x)*(9 + E^3) + 9*x + E^3*x)/(18*x + 2*E^3*x)] + (-16* x - 4*E*x + 18*x^2 + 2*E^3*x^2 + E^(3*x)*(18*x + 2*E^3*x))*Log[(-8 - 2*E + E^(3*x)*(9 + E^3) + 9*x + E^3*x)/(18*x + 2*E^3*x)]^2)/(-8 - 2*E + E^(3*x) *(9 + E^3) + 9*x + E^3*x),x]
Output:
Integrate[((16*x + 4*E*x + E^(3*x)*(-18*x + 54*x^2 + E^3*(-2*x + 6*x^2)))* Log[(-8 - 2*E + E^(3*x)*(9 + E^3) + 9*x + E^3*x)/(18*x + 2*E^3*x)] + (-16* x - 4*E*x + 18*x^2 + 2*E^3*x^2 + E^(3*x)*(18*x + 2*E^3*x))*Log[(-8 - 2*E + E^(3*x)*(9 + E^3) + 9*x + E^3*x)/(18*x + 2*E^3*x)]^2)/(-8 - 2*E + E^(3*x) *(9 + E^3) + 9*x + E^3*x), 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.
| \(\displaystyle \int \frac {\left (2 e^3 x^2+18 x^2-4 e x-16 x+e^{3 x} \left (2 e^3 x+18 x\right )\right ) \log ^2\left (\frac {e^3 x+9 x+\left (9+e^3\right ) e^{3 x}-2 e-8}{2 e^3 x+18 x}\right )+\left (e^{3 x} \left (54 x^2+e^3 \left (6 x^2-2 x\right )-18 x\right )+4 e x+16 x\right ) \log \left (\frac {e^3 x+9 x+\left (9+e^3\right ) e^{3 x}-2 e-8}{2 e^3 x+18 x}\right )}{e^3 x+9 x+\left (9+e^3\right ) e^{3 x}-2 e-8} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {\left (2 e^3 x^2+18 x^2-4 e x-16 x+e^{3 x} \left (2 e^3 x+18 x\right )\right ) \log ^2\left (\frac {e^3 x+9 x+\left (9+e^3\right ) e^{3 x}-2 e-8}{2 e^3 x+18 x}\right )+\left (e^{3 x} \left (54 x^2+e^3 \left (6 x^2-2 x\right )-18 x\right )+4 e x+16 x\right ) \log \left (\frac {e^3 x+9 x+\left (9+e^3\right ) e^{3 x}-2 e-8}{2 e^3 x+18 x}\right )}{\left (9+e^3\right ) x+\left (9+e^3\right ) e^{3 x}-2 e-8}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-\left (\left (2 e^3 x^2+18 x^2-4 e x-16 x+e^{3 x} \left (2 e^3 x+18 x\right )\right ) \log ^2\left (\frac {e^3 x+9 x+\left (9+e^3\right ) e^{3 x}-2 e-8}{2 e^3 x+18 x}\right )\right )-\left (e^{3 x} \left (54 x^2+e^3 \left (6 x^2-2 x\right )-18 x\right )+4 e x+16 x\right ) \log \left (\frac {e^3 x+9 x+\left (9+e^3\right ) e^{3 x}-2 e-8}{2 e^3 x+18 x}\right )}{-\left (\left (9+e^3\right ) x\right )-\left (9+e^3\right ) e^{3 x}+8 \left (1+\frac {e}{4}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 \left (3 \left (9+e^3\right ) x-e^3-6 e-33\right ) x^2 \log \left (\frac {9 \left (1+\frac {e^3}{9}\right ) x+\left (9+e^3\right ) e^{3 x}-8 \left (1+\frac {e}{4}\right )}{2 \left (9+e^3\right ) x}\right )}{-9 \left (1+\frac {e^3}{9}\right ) x-9 \left (1+\frac {e^3}{9}\right ) e^{3 x}+8 \left (1+\frac {e}{4}\right )}+2 x \left (3 x+\log \left (\frac {e^3 x+9 x+\left (9+e^3\right ) e^{3 x}-2 e-8}{2 e^3 x+18 x}\right )-1\right ) \log \left (\frac {9 \left (1+\frac {e^3}{9}\right ) x+\left (9+e^3\right ) e^{3 x}-8 \left (1+\frac {e}{4}\right )}{2 \left (9+e^3\right ) x}\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 x^4}{2}+2 \log \left (-\frac {-\left (\left (9+e^3\right ) x\right )-e^{3 x} \left (9+e^3\right )+2 (4+e)}{2 \left (9+e^3\right ) x}\right ) x^3+2 \log \left (2 \left (9+e^3\right )\right ) x^3+\frac {5 x^3}{3}-2 \log \left (2 \left (9+e^3\right )\right ) \log \left (e^3+9+\frac {e^{3 x} \left (9+e^3\right )}{x}-\frac {2 (4+e)}{x}\right ) x^2-\log \left (e^3+9+\frac {e^{3 x} \left (9+e^3\right )}{x}-\frac {2 (4+e)}{x}\right ) x^2+\log ^2\left (2 \left (9+e^3\right )\right ) x^2-\frac {x^2}{2}-2 \left (33+6 e+e^3\right ) \log \left (-\frac {-\left (\left (9+e^3\right ) x\right )-e^{3 x} \left (9+e^3\right )+2 (4+e)}{2 \left (9+e^3\right ) x}\right ) \int \frac {x^2}{-9 \left (1+\frac {e^3}{9}\right ) x-9 e^{3 x} \left (1+\frac {e^3}{9}\right )+8 \left (1+\frac {e}{4}\right )}dx-2 \left (33+6 e+e^3\right ) \log \left (2 \left (9+e^3\right )\right ) \int \frac {x^2}{-9 \left (1+\frac {e^3}{9}\right ) x-9 e^{3 x} \left (1+\frac {e^3}{9}\right )+8 \left (1+\frac {e}{4}\right )}dx-\left (33+6 e+e^3\right ) \int \frac {x^2}{-9 \left (1+\frac {e^3}{9}\right ) x-9 e^{3 x} \left (1+\frac {e^3}{9}\right )+8 \left (1+\frac {e}{4}\right )}dx+6 \left (9+e^3\right ) \log \left (-\frac {-\left (\left (9+e^3\right ) x\right )-e^{3 x} \left (9+e^3\right )+2 (4+e)}{2 \left (9+e^3\right ) x}\right ) \int \frac {x^3}{-9 \left (1+\frac {e^3}{9}\right ) x-9 e^{3 x} \left (1+\frac {e^3}{9}\right )+8 \left (1+\frac {e}{4}\right )}dx+6 \left (9+e^3\right ) \log \left (2 \left (9+e^3\right )\right ) \int \frac {x^3}{-9 \left (1+\frac {e^3}{9}\right ) x-9 e^{3 x} \left (1+\frac {e^3}{9}\right )+8 \left (1+\frac {e}{4}\right )}dx+2 \left (33+6 e+e^3\right ) \int \frac {x^3}{-9 \left (1+\frac {e^3}{9}\right ) x-9 e^{3 x} \left (1+\frac {e^3}{9}\right )+8 \left (1+\frac {e}{4}\right )}dx+3 \left (9+e^3\right ) \int \frac {x^3}{-9 \left (1+\frac {e^3}{9}\right ) x-9 e^{3 x} \left (1+\frac {e^3}{9}\right )+8 \left (1+\frac {e}{4}\right )}dx-6 \left (9+e^3\right ) \int \frac {x^4}{-9 \left (1+\frac {e^3}{9}\right ) x-9 e^{3 x} \left (1+\frac {e^3}{9}\right )+8 \left (1+\frac {e}{4}\right )}dx+2 \int x \log ^2\left (9 \left (1+\frac {e^3}{9}\right )+\frac {9 e^{3 x} \left (1+\frac {e^3}{9}\right )}{x}-\frac {8 \left (1+\frac {e}{4}\right )}{x}\right )dx+6 \left (33+6 e+e^3\right ) \int \int -\frac {x^2}{9 \left (1+\frac {e^3}{9}\right ) x+9 e^{3 x} \left (1+\frac {e^3}{9}\right )-8 \left (1+\frac {e}{4}\right )}dxdx-2 \left (33+6 e+e^3\right ) \int \frac {\int -\frac {x^2}{9 \left (1+\frac {e^3}{9}\right ) x+9 e^{3 x} \left (1+\frac {e^3}{9}\right )-8 \left (1+\frac {e}{4}\right )}dx}{x}dx-2 \left (33+6 e+e^3\right )^2 \int \frac {\int -\frac {x^2}{9 \left (1+\frac {e^3}{9}\right ) x+9 e^{3 x} \left (1+\frac {e^3}{9}\right )-8 \left (1+\frac {e}{4}\right )}dx}{-9 \left (1+\frac {e^3}{9}\right ) x-9 e^{3 x} \left (1+\frac {e^3}{9}\right )+8 \left (1+\frac {e}{4}\right )}dx+6 \left (9+e^3\right ) \left (33+6 e+e^3\right ) \int \frac {x \int -\frac {x^2}{9 \left (1+\frac {e^3}{9}\right ) x+9 e^{3 x} \left (1+\frac {e^3}{9}\right )-8 \left (1+\frac {e}{4}\right )}dx}{-9 \left (1+\frac {e^3}{9}\right ) x-9 e^{3 x} \left (1+\frac {e^3}{9}\right )+8 \left (1+\frac {e}{4}\right )}dx-18 \left (9+e^3\right ) \int \int -\frac {x^3}{9 \left (1+\frac {e^3}{9}\right ) x+9 e^{3 x} \left (1+\frac {e^3}{9}\right )-8 \left (1+\frac {e}{4}\right )}dxdx+6 \left (9+e^3\right ) \int \frac {\int -\frac {x^3}{9 \left (1+\frac {e^3}{9}\right ) x+9 e^{3 x} \left (1+\frac {e^3}{9}\right )-8 \left (1+\frac {e}{4}\right )}dx}{x}dx+6 \left (9+e^3\right ) \left (33+6 e+e^3\right ) \int \frac {\int -\frac {x^3}{9 \left (1+\frac {e^3}{9}\right ) x+9 e^{3 x} \left (1+\frac {e^3}{9}\right )-8 \left (1+\frac {e}{4}\right )}dx}{-9 \left (1+\frac {e^3}{9}\right ) x-9 e^{3 x} \left (1+\frac {e^3}{9}\right )+8 \left (1+\frac {e}{4}\right )}dx-18 \left (9+e^3\right )^2 \int \frac {x \int -\frac {x^3}{9 \left (1+\frac {e^3}{9}\right ) x+9 e^{3 x} \left (1+\frac {e^3}{9}\right )-8 \left (1+\frac {e}{4}\right )}dx}{-9 \left (1+\frac {e^3}{9}\right ) x-9 e^{3 x} \left (1+\frac {e^3}{9}\right )+8 \left (1+\frac {e}{4}\right )}dx\) |
Input:
Int[((16*x + 4*E*x + E^(3*x)*(-18*x + 54*x^2 + E^3*(-2*x + 6*x^2)))*Log[(- 8 - 2*E + E^(3*x)*(9 + E^3) + 9*x + E^3*x)/(18*x + 2*E^3*x)] + (-16*x - 4* E*x + 18*x^2 + 2*E^3*x^2 + E^(3*x)*(18*x + 2*E^3*x))*Log[(-8 - 2*E + E^(3* x)*(9 + E^3) + 9*x + E^3*x)/(18*x + 2*E^3*x)]^2)/(-8 - 2*E + E^(3*x)*(9 + E^3) + 9*x + E^3*x),x]
Output:
$Aborted
Time = 2.76 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.20
| method | result | size |
| norman | \(x^{2} {\ln \left (\frac {\left ({\mathrm e}^{3}+9\right ) {\mathrm e}^{3 x}+x \,{\mathrm e}^{3}-2 \,{\mathrm e}+9 x -8}{2 x \,{\mathrm e}^{3}+18 x}\right )}^{2}\) | \(42\) |
| parallelrisch | \(-\frac {-4 \,{\mathrm e}^{2} x^{2} {\ln \left (\frac {\left ({\mathrm e}^{3}+9\right ) {\mathrm e}^{3 x}+x \,{\mathrm e}^{3}-2 \,{\mathrm e}+9 x -8}{2 x \left ({\mathrm e}^{3}+9\right )}\right )}^{2}-32 \,{\mathrm e} x^{2} {\ln \left (\frac {\left ({\mathrm e}^{3}+9\right ) {\mathrm e}^{3 x}+x \,{\mathrm e}^{3}-2 \,{\mathrm e}+9 x -8}{2 x \left ({\mathrm e}^{3}+9\right )}\right )}^{2}-64 {\ln \left (\frac {\left ({\mathrm e}^{3}+9\right ) {\mathrm e}^{3 x}+x \,{\mathrm e}^{3}-2 \,{\mathrm e}+9 x -8}{2 x \left ({\mathrm e}^{3}+9\right )}\right )}^{2} x^{2}}{4 \left ({\mathrm e}+4\right )^{2}}\) | \(139\) |
| risch | \(\text {Expression too large to display}\) | \(1840\) |
Input:
int((((2*x*exp(3)+18*x)*exp(3*x)+2*x^2*exp(3)-4*x*exp(1)+18*x^2-16*x)*ln(( (exp(3)+9)*exp(3*x)+x*exp(3)-2*exp(1)+9*x-8)/(2*x*exp(3)+18*x))^2+(((6*x^2 -2*x)*exp(3)+54*x^2-18*x)*exp(3*x)+4*x*exp(1)+16*x)*ln(((exp(3)+9)*exp(3*x )+x*exp(3)-2*exp(1)+9*x-8)/(2*x*exp(3)+18*x)))/((exp(3)+9)*exp(3*x)+x*exp( 3)-2*exp(1)+9*x-8),x,method=_RETURNVERBOSE)
Output:
x^2*ln(((exp(3)+9)*exp(3*x)+x*exp(3)-2*exp(1)+9*x-8)/(2*x*exp(3)+18*x))^2
Time = 0.11 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.17 \[ \int \frac {\left (16 x+4 e x+e^{3 x} \left (-18 x+54 x^2+e^3 \left (-2 x+6 x^2\right )\right )\right ) \log \left (\frac {-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x}{18 x+2 e^3 x}\right )+\left (-16 x-4 e x+18 x^2+2 e^3 x^2+e^{3 x} \left (18 x+2 e^3 x\right )\right ) \log ^2\left (\frac {-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x}{18 x+2 e^3 x}\right )}{-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x} \, dx=x^{2} \log \left (\frac {x e^{3} + {\left (e^{3} + 9\right )} e^{\left (3 \, x\right )} + 9 \, x - 2 \, e - 8}{2 \, {\left (x e^{3} + 9 \, x\right )}}\right )^{2} \] Input:
integrate((((2*x*exp(3)+18*x)*exp(3*x)+2*x^2*exp(3)-4*exp(1)*x+18*x^2-16*x )*log(((exp(3)+9)*exp(3*x)+x*exp(3)-2*exp(1)+9*x-8)/(2*x*exp(3)+18*x))^2+( ((6*x^2-2*x)*exp(3)+54*x^2-18*x)*exp(3*x)+4*exp(1)*x+16*x)*log(((exp(3)+9) *exp(3*x)+x*exp(3)-2*exp(1)+9*x-8)/(2*x*exp(3)+18*x)))/((exp(3)+9)*exp(3*x )+x*exp(3)-2*exp(1)+9*x-8),x, algorithm="fricas")
Output:
x^2*log(1/2*(x*e^3 + (e^3 + 9)*e^(3*x) + 9*x - 2*e - 8)/(x*e^3 + 9*x))^2
Time = 0.44 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.17 \[ \int \frac {\left (16 x+4 e x+e^{3 x} \left (-18 x+54 x^2+e^3 \left (-2 x+6 x^2\right )\right )\right ) \log \left (\frac {-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x}{18 x+2 e^3 x}\right )+\left (-16 x-4 e x+18 x^2+2 e^3 x^2+e^{3 x} \left (18 x+2 e^3 x\right )\right ) \log ^2\left (\frac {-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x}{18 x+2 e^3 x}\right )}{-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x} \, dx=x^{2} \log {\left (\frac {9 x + x e^{3} + \left (9 + e^{3}\right ) e^{3 x} - 8 - 2 e}{18 x + 2 x e^{3}} \right )}^{2} \] Input:
integrate((((2*x*exp(3)+18*x)*exp(3*x)+2*x**2*exp(3)-4*exp(1)*x+18*x**2-16 *x)*ln(((exp(3)+9)*exp(3*x)+x*exp(3)-2*exp(1)+9*x-8)/(2*x*exp(3)+18*x))**2 +(((6*x**2-2*x)*exp(3)+54*x**2-18*x)*exp(3*x)+4*exp(1)*x+16*x)*ln(((exp(3) +9)*exp(3*x)+x*exp(3)-2*exp(1)+9*x-8)/(2*x*exp(3)+18*x)))/((exp(3)+9)*exp( 3*x)+x*exp(3)-2*exp(1)+9*x-8),x)
Output:
x**2*log((9*x + x*exp(3) + (9 + exp(3))*exp(3*x) - 8 - 2*E)/(18*x + 2*x*ex p(3)))**2
Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (30) = 60\).
Time = 0.18 (sec) , antiderivative size = 120, normalized size of antiderivative = 3.43 \[ \int \frac {\left (16 x+4 e x+e^{3 x} \left (-18 x+54 x^2+e^3 \left (-2 x+6 x^2\right )\right )\right ) \log \left (\frac {-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x}{18 x+2 e^3 x}\right )+\left (-16 x-4 e x+18 x^2+2 e^3 x^2+e^{3 x} \left (18 x+2 e^3 x\right )\right ) \log ^2\left (\frac {-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x}{18 x+2 e^3 x}\right )}{-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x} \, dx=x^{2} \log \left (x {\left (e^{3} + 9\right )} + {\left (e^{3} + 9\right )} e^{\left (3 \, x\right )} - 2 \, e - 8\right )^{2} + 2 \, x^{2} {\left (\log \left (2\right ) + \log \left (e^{3} + 9\right )\right )} \log \left (x\right ) + x^{2} \log \left (x\right )^{2} + {\left (\log \left (2\right )^{2} + 2 \, \log \left (2\right ) \log \left (e^{3} + 9\right ) + \log \left (e^{3} + 9\right )^{2}\right )} x^{2} - 2 \, {\left (x^{2} {\left (\log \left (2\right ) + \log \left (e^{3} + 9\right )\right )} + x^{2} \log \left (x\right )\right )} \log \left (x {\left (e^{3} + 9\right )} + {\left (e^{3} + 9\right )} e^{\left (3 \, x\right )} - 2 \, e - 8\right ) \] Input:
integrate((((2*x*exp(3)+18*x)*exp(3*x)+2*x^2*exp(3)-4*exp(1)*x+18*x^2-16*x )*log(((exp(3)+9)*exp(3*x)+x*exp(3)-2*exp(1)+9*x-8)/(2*x*exp(3)+18*x))^2+( ((6*x^2-2*x)*exp(3)+54*x^2-18*x)*exp(3*x)+4*exp(1)*x+16*x)*log(((exp(3)+9) *exp(3*x)+x*exp(3)-2*exp(1)+9*x-8)/(2*x*exp(3)+18*x)))/((exp(3)+9)*exp(3*x )+x*exp(3)-2*exp(1)+9*x-8),x, algorithm="maxima")
Output:
x^2*log(x*(e^3 + 9) + (e^3 + 9)*e^(3*x) - 2*e - 8)^2 + 2*x^2*(log(2) + log (e^3 + 9))*log(x) + x^2*log(x)^2 + (log(2)^2 + 2*log(2)*log(e^3 + 9) + log (e^3 + 9)^2)*x^2 - 2*(x^2*(log(2) + log(e^3 + 9)) + x^2*log(x))*log(x*(e^3 + 9) + (e^3 + 9)*e^(3*x) - 2*e - 8)
Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (30) = 60\).
Time = 5.09 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.57 \[ \int \frac {\left (16 x+4 e x+e^{3 x} \left (-18 x+54 x^2+e^3 \left (-2 x+6 x^2\right )\right )\right ) \log \left (\frac {-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x}{18 x+2 e^3 x}\right )+\left (-16 x-4 e x+18 x^2+2 e^3 x^2+e^{3 x} \left (18 x+2 e^3 x\right )\right ) \log ^2\left (\frac {-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x}{18 x+2 e^3 x}\right )}{-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x} \, dx=x^{2} \log \left (2 \, x e^{3} + 18 \, x\right )^{2} - 2 \, x^{2} \log \left (2 \, x e^{3} + 18 \, x\right ) \log \left (x e^{3} + 9 \, x - 2 \, e + 9 \, e^{\left (3 \, x\right )} + e^{\left (3 \, x + 3\right )} - 8\right ) + x^{2} \log \left (x e^{3} + 9 \, x - 2 \, e + 9 \, e^{\left (3 \, x\right )} + e^{\left (3 \, x + 3\right )} - 8\right )^{2} \] Input:
integrate((((2*x*exp(3)+18*x)*exp(3*x)+2*x^2*exp(3)-4*exp(1)*x+18*x^2-16*x )*log(((exp(3)+9)*exp(3*x)+x*exp(3)-2*exp(1)+9*x-8)/(2*x*exp(3)+18*x))^2+( ((6*x^2-2*x)*exp(3)+54*x^2-18*x)*exp(3*x)+4*exp(1)*x+16*x)*log(((exp(3)+9) *exp(3*x)+x*exp(3)-2*exp(1)+9*x-8)/(2*x*exp(3)+18*x)))/((exp(3)+9)*exp(3*x )+x*exp(3)-2*exp(1)+9*x-8),x, algorithm="giac")
Output:
x^2*log(2*x*e^3 + 18*x)^2 - 2*x^2*log(2*x*e^3 + 18*x)*log(x*e^3 + 9*x - 2* e + 9*e^(3*x) + e^(3*x + 3) - 8) + x^2*log(x*e^3 + 9*x - 2*e + 9*e^(3*x) + e^(3*x + 3) - 8)^2
Time = 3.88 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.17 \[ \int \frac {\left (16 x+4 e x+e^{3 x} \left (-18 x+54 x^2+e^3 \left (-2 x+6 x^2\right )\right )\right ) \log \left (\frac {-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x}{18 x+2 e^3 x}\right )+\left (-16 x-4 e x+18 x^2+2 e^3 x^2+e^{3 x} \left (18 x+2 e^3 x\right )\right ) \log ^2\left (\frac {-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x}{18 x+2 e^3 x}\right )}{-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x} \, dx=x^2\,{\ln \left (\frac {9\,x-2\,\mathrm {e}+x\,{\mathrm {e}}^3+{\mathrm {e}}^{3\,x}\,\left ({\mathrm {e}}^3+9\right )-8}{18\,x+2\,x\,{\mathrm {e}}^3}\right )}^2 \] Input:
int((log((9*x - 2*exp(1) + x*exp(3) + exp(3*x)*(exp(3) + 9) - 8)/(18*x + 2 *x*exp(3)))*(16*x - exp(3*x)*(18*x + exp(3)*(2*x - 6*x^2) - 54*x^2) + 4*x* exp(1)) + log((9*x - 2*exp(1) + x*exp(3) + exp(3*x)*(exp(3) + 9) - 8)/(18* x + 2*x*exp(3)))^2*(exp(3*x)*(18*x + 2*x*exp(3)) - 16*x - 4*x*exp(1) + 2*x ^2*exp(3) + 18*x^2))/(9*x - 2*exp(1) + x*exp(3) + exp(3*x)*(exp(3) + 9) - 8),x)
Output:
x^2*log((9*x - 2*exp(1) + x*exp(3) + exp(3*x)*(exp(3) + 9) - 8)/(18*x + 2* x*exp(3)))^2
Time = 0.16 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.40 \[ \int \frac {\left (16 x+4 e x+e^{3 x} \left (-18 x+54 x^2+e^3 \left (-2 x+6 x^2\right )\right )\right ) \log \left (\frac {-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x}{18 x+2 e^3 x}\right )+\left (-16 x-4 e x+18 x^2+2 e^3 x^2+e^{3 x} \left (18 x+2 e^3 x\right )\right ) \log ^2\left (\frac {-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x}{18 x+2 e^3 x}\right )}{-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x} \, dx=\mathrm {log}\left (\frac {e^{3 x} e^{3}+9 e^{3 x}+e^{3} x -2 e +9 x -8}{2 e^{3} x +18 x}\right )^{2} x^{2} \] Input:
int((((2*x*exp(3)+18*x)*exp(3*x)+2*x^2*exp(3)-4*exp(1)*x+18*x^2-16*x)*log( ((exp(3)+9)*exp(3*x)+x*exp(3)-2*exp(1)+9*x-8)/(2*x*exp(3)+18*x))^2+(((6*x^ 2-2*x)*exp(3)+54*x^2-18*x)*exp(3*x)+4*exp(1)*x+16*x)*log(((exp(3)+9)*exp(3 *x)+x*exp(3)-2*exp(1)+9*x-8)/(2*x*exp(3)+18*x)))/((exp(3)+9)*exp(3*x)+x*ex p(3)-2*exp(1)+9*x-8),x)
Output:
log((e**(3*x)*e**3 + 9*e**(3*x) + e**3*x - 2*e + 9*x - 8)/(2*e**3*x + 18*x ))**2*x**2