Integrand size = 400, antiderivative size = 31 \[ \int \frac {-18 x+72 x^2+e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )} \left (-72 x+e^{x/2} x+36 x^2+e^{x/4} \left (-12 x-3 x^2\right )\right )+\left (36 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )} x-36 x^2\right ) \log \left (-e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+x\right )+\left (-36 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+36 x+\left (18 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}-18 x\right ) \log \left (-e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+x\right )\right ) \log \left (2-\log \left (-e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+x\right )\right )}{-36 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+36 x+\left (18 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}-18 x\right ) \log \left (-e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+x\right )} \, dx=x \left (x+\log \left (2-\log \left (-e^{\left (-\frac {e^{x/4}}{3}+x\right )^2}+x\right )\right )\right ) \] Output:
(ln(2-ln(x-exp((x-1/3*exp(1/4*x))^2)))+x)*x
Time = 0.60 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.68 \[ \int \frac {-18 x+72 x^2+e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )} \left (-72 x+e^{x/2} x+36 x^2+e^{x/4} \left (-12 x-3 x^2\right )\right )+\left (36 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )} x-36 x^2\right ) \log \left (-e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+x\right )+\left (-36 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+36 x+\left (18 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}-18 x\right ) \log \left (-e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+x\right )\right ) \log \left (2-\log \left (-e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+x\right )\right )}{-36 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+36 x+\left (18 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}-18 x\right ) \log \left (-e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+x\right )} \, dx=\frac {1}{18} \left (18 x^2+18 x \log \left (2-\log \left (-e^{\frac {e^{x/2}}{9}-\frac {2}{3} e^{x/4} x+x^2}+x\right )\right )\right ) \] Input:
Integrate[(-18*x + 72*x^2 + E^((E^(x/2) - 6*E^(x/4)*x + 9*x^2)/9)*(-72*x + E^(x/2)*x + 36*x^2 + E^(x/4)*(-12*x - 3*x^2)) + (36*E^((E^(x/2) - 6*E^(x/ 4)*x + 9*x^2)/9)*x - 36*x^2)*Log[-E^((E^(x/2) - 6*E^(x/4)*x + 9*x^2)/9) + x] + (-36*E^((E^(x/2) - 6*E^(x/4)*x + 9*x^2)/9) + 36*x + (18*E^((E^(x/2) - 6*E^(x/4)*x + 9*x^2)/9) - 18*x)*Log[-E^((E^(x/2) - 6*E^(x/4)*x + 9*x^2)/9 ) + x])*Log[2 - Log[-E^((E^(x/2) - 6*E^(x/4)*x + 9*x^2)/9) + x]])/(-36*E^( (E^(x/2) - 6*E^(x/4)*x + 9*x^2)/9) + 36*x + (18*E^((E^(x/2) - 6*E^(x/4)*x + 9*x^2)/9) - 18*x)*Log[-E^((E^(x/2) - 6*E^(x/4)*x + 9*x^2)/9) + x]),x]
Output:
(18*x^2 + 18*x*Log[2 - Log[-E^(E^(x/2)/9 - (2*E^(x/4)*x)/3 + x^2) + x]])/1 8
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.
| \(\displaystyle \int \frac {72 x^2+e^{\frac {1}{9} \left (9 x^2-6 e^{x/4} x+e^{x/2}\right )} \left (36 x^2+e^{x/4} \left (-3 x^2-12 x\right )+e^{x/2} x-72 x\right )+\left (36 e^{\frac {1}{9} \left (9 x^2-6 e^{x/4} x+e^{x/2}\right )} x-36 x^2\right ) \log \left (x-e^{\frac {1}{9} \left (9 x^2-6 e^{x/4} x+e^{x/2}\right )}\right )+\left (-36 e^{\frac {1}{9} \left (9 x^2-6 e^{x/4} x+e^{x/2}\right )}+\left (18 e^{\frac {1}{9} \left (9 x^2-6 e^{x/4} x+e^{x/2}\right )}-18 x\right ) \log \left (x-e^{\frac {1}{9} \left (9 x^2-6 e^{x/4} x+e^{x/2}\right )}\right )+36 x\right ) \log \left (2-\log \left (x-e^{\frac {1}{9} \left (9 x^2-6 e^{x/4} x+e^{x/2}\right )}\right )\right )-18 x}{-36 e^{\frac {1}{9} \left (9 x^2-6 e^{x/4} x+e^{x/2}\right )}+\left (18 e^{\frac {1}{9} \left (9 x^2-6 e^{x/4} x+e^{x/2}\right )}-18 x\right ) \log \left (x-e^{\frac {1}{9} \left (9 x^2-6 e^{x/4} x+e^{x/2}\right )}\right )+36 x} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{\frac {2}{3} e^{x/4} x} \left (-72 x^2-e^{\frac {1}{9} \left (9 x^2-6 e^{x/4} x+e^{x/2}\right )} \left (36 x^2+e^{x/4} \left (-3 x^2-12 x\right )+e^{x/2} x-72 x\right )-\left (36 e^{\frac {1}{9} \left (9 x^2-6 e^{x/4} x+e^{x/2}\right )} x-36 x^2\right ) \log \left (x-e^{\frac {1}{9} \left (9 x^2-6 e^{x/4} x+e^{x/2}\right )}\right )-\left (-36 e^{\frac {1}{9} \left (9 x^2-6 e^{x/4} x+e^{x/2}\right )}+\left (18 e^{\frac {1}{9} \left (9 x^2-6 e^{x/4} x+e^{x/2}\right )}-18 x\right ) \log \left (x-e^{\frac {1}{9} \left (9 x^2-6 e^{x/4} x+e^{x/2}\right )}\right )+36 x\right ) \log \left (2-\log \left (x-e^{\frac {1}{9} \left (9 x^2-6 e^{x/4} x+e^{x/2}\right )}\right )\right )+18 x\right )}{18 \left (e^{x^2+\frac {e^{x/2}}{9}}-e^{\frac {2}{3} e^{x/4} x} x\right ) \left (2-\log \left (x-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}\right )\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{18} \int \frac {e^{\frac {2}{3} e^{x/4} x} \left (-72 x^2+18 x+e^{\frac {1}{9} \left (9 x^2-6 e^{x/4} x+e^{x/2}\right )} \left (-36 x^2-e^{x/2} x+72 x+3 e^{x/4} \left (x^2+4 x\right )\right )-36 \left (e^{\frac {1}{9} \left (9 x^2-6 e^{x/4} x+e^{x/2}\right )} x-x^2\right ) \log \left (x-e^{\frac {1}{9} \left (9 x^2-6 e^{x/4} x+e^{x/2}\right )}\right )+18 \left (-2 x+2 e^{\frac {1}{9} \left (9 x^2-6 e^{x/4} x+e^{x/2}\right )}-\left (e^{\frac {1}{9} \left (9 x^2-6 e^{x/4} x+e^{x/2}\right )}-x\right ) \log \left (x-e^{\frac {1}{9} \left (9 x^2-6 e^{x/4} x+e^{x/2}\right )}\right )\right ) \log \left (2-\log \left (x-e^{\frac {1}{9} \left (9 x^2-6 e^{x/4} x+e^{x/2}\right )}\right )\right )\right )}{\left (e^{x^2+\frac {e^{x/2}}{9}}-e^{\frac {2}{3} e^{x/4} x} x\right ) \left (2-\log \left (x-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{18} \int \left (\frac {e^{\frac {2}{3} e^{x/4} x} x \left (3 e^{x/4} x^2-36 x^2+12 e^{x/4} x-e^{x/2} x+18\right )}{\left (e^{\frac {2}{3} e^{x/4} x} x-e^{x^2+\frac {e^{x/2}}{9}}\right ) \left (\log \left (x-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}\right )-2\right )}+\frac {-3 e^{x/4} x^2+36 x^2-12 e^{x/4} x+e^{x/2} x+36 \log \left (x-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}\right ) x-72 x+18 \log \left (x-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}\right ) \log \left (2-\log \left (x-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}\right )\right )-36 \log \left (2-\log \left (x-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}\right )\right )}{\log \left (x-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}\right )-2}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \frac {1}{18} \int \left (\frac {e^{\frac {2}{3} e^{x/4} x} x \left (3 e^{x/4} x^2-36 x^2+12 e^{x/4} x-e^{x/2} x+18\right )}{\left (e^{\frac {2}{3} e^{x/4} x} x-e^{x^2+\frac {e^{x/2}}{9}}\right ) \left (\log \left (x-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}\right )-2\right )}+\frac {-3 e^{x/4} x^2+36 x^2-12 e^{x/4} x+e^{x/2} x+36 \log \left (x-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}\right ) x-72 x+18 \log \left (x-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}\right ) \log \left (2-\log \left (x-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}\right )\right )-36 \log \left (2-\log \left (x-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}\right )\right )}{\log \left (x-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}\right )-2}\right )dx\) |
Input:
Int[(-18*x + 72*x^2 + E^((E^(x/2) - 6*E^(x/4)*x + 9*x^2)/9)*(-72*x + E^(x/ 2)*x + 36*x^2 + E^(x/4)*(-12*x - 3*x^2)) + (36*E^((E^(x/2) - 6*E^(x/4)*x + 9*x^2)/9)*x - 36*x^2)*Log[-E^((E^(x/2) - 6*E^(x/4)*x + 9*x^2)/9) + x] + ( -36*E^((E^(x/2) - 6*E^(x/4)*x + 9*x^2)/9) + 36*x + (18*E^((E^(x/2) - 6*E^( x/4)*x + 9*x^2)/9) - 18*x)*Log[-E^((E^(x/2) - 6*E^(x/4)*x + 9*x^2)/9) + x] )*Log[2 - Log[-E^((E^(x/2) - 6*E^(x/4)*x + 9*x^2)/9) + x]])/(-36*E^((E^(x/ 2) - 6*E^(x/4)*x + 9*x^2)/9) + 36*x + (18*E^((E^(x/2) - 6*E^(x/4)*x + 9*x^ 2)/9) - 18*x)*Log[-E^((E^(x/2) - 6*E^(x/4)*x + 9*x^2)/9) + x]),x]
Output:
$Aborted
Time = 7.22 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13
| method | result | size |
| risch | \(x^{2}+\ln \left (-\ln \left (-{\mathrm e}^{\frac {{\mathrm e}^{\frac {x}{2}}}{9}-\frac {2 x \,{\mathrm e}^{\frac {x}{4}}}{3}+x^{2}}+x \right )+2\right ) x\) | \(35\) |
| parallelrisch | \(x^{2}+\ln \left (-\ln \left (-{\mathrm e}^{\frac {{\mathrm e}^{\frac {x}{2}}}{9}-\frac {2 x \,{\mathrm e}^{\frac {x}{4}}}{3}+x^{2}}+x \right )+2\right ) x\) | \(37\) |
Input:
int((((18*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)-18*x)*ln(-exp(1/9*exp (1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+x)-36*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x )+x^2)+36*x)*ln(-ln(-exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+x)+2)+(36* x*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)-36*x^2)*ln(-exp(1/9*exp(1/4*x )^2-2/3*x*exp(1/4*x)+x^2)+x)+(x*exp(1/4*x)^2+(-3*x^2-12*x)*exp(1/4*x)+36*x ^2-72*x)*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+72*x^2-18*x)/((18*exp( 1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)-18*x)*ln(-exp(1/9*exp(1/4*x)^2-2/3* x*exp(1/4*x)+x^2)+x)-36*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+36*x),x ,method=_RETURNVERBOSE)
Output:
x^2+ln(-ln(-exp(1/9*exp(1/2*x)-2/3*x*exp(1/4*x)+x^2)+x)+2)*x
Time = 0.13 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {-18 x+72 x^2+e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )} \left (-72 x+e^{x/2} x+36 x^2+e^{x/4} \left (-12 x-3 x^2\right )\right )+\left (36 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )} x-36 x^2\right ) \log \left (-e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+x\right )+\left (-36 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+36 x+\left (18 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}-18 x\right ) \log \left (-e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+x\right )\right ) \log \left (2-\log \left (-e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+x\right )\right )}{-36 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+36 x+\left (18 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}-18 x\right ) \log \left (-e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+x\right )} \, dx=x^{2} + x \log \left (-\log \left (x - e^{\left (x^{2} - \frac {2}{3} \, x e^{\left (\frac {1}{4} \, x\right )} + \frac {1}{9} \, e^{\left (\frac {1}{2} \, x\right )}\right )}\right ) + 2\right ) \] Input:
integrate((((18*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)-18*x)*log(-exp( 1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+x)-36*exp(1/9*exp(1/4*x)^2-2/3*x*ex p(1/4*x)+x^2)+36*x)*log(-log(-exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+x )+2)+(36*x*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)-36*x^2)*log(-exp(1/9 *exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+x)+(x*exp(1/4*x)^2+(-3*x^2-12*x)*exp(1 /4*x)+36*x^2-72*x)*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+72*x^2-18*x) /((18*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)-18*x)*log(-exp(1/9*exp(1/ 4*x)^2-2/3*x*exp(1/4*x)+x^2)+x)-36*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x ^2)+36*x),x, algorithm="fricas")
Output:
x^2 + x*log(-log(x - e^(x^2 - 2/3*x*e^(1/4*x) + 1/9*e^(1/2*x))) + 2)
Timed out. \[ \int \frac {-18 x+72 x^2+e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )} \left (-72 x+e^{x/2} x+36 x^2+e^{x/4} \left (-12 x-3 x^2\right )\right )+\left (36 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )} x-36 x^2\right ) \log \left (-e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+x\right )+\left (-36 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+36 x+\left (18 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}-18 x\right ) \log \left (-e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+x\right )\right ) \log \left (2-\log \left (-e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+x\right )\right )}{-36 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+36 x+\left (18 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}-18 x\right ) \log \left (-e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+x\right )} \, dx=\text {Timed out} \] Input:
integrate((((18*exp(1/9*exp(1/4*x)**2-2/3*x*exp(1/4*x)+x**2)-18*x)*ln(-exp (1/9*exp(1/4*x)**2-2/3*x*exp(1/4*x)+x**2)+x)-36*exp(1/9*exp(1/4*x)**2-2/3* x*exp(1/4*x)+x**2)+36*x)*ln(-ln(-exp(1/9*exp(1/4*x)**2-2/3*x*exp(1/4*x)+x* *2)+x)+2)+(36*x*exp(1/9*exp(1/4*x)**2-2/3*x*exp(1/4*x)+x**2)-36*x**2)*ln(- exp(1/9*exp(1/4*x)**2-2/3*x*exp(1/4*x)+x**2)+x)+(x*exp(1/4*x)**2+(-3*x**2- 12*x)*exp(1/4*x)+36*x**2-72*x)*exp(1/9*exp(1/4*x)**2-2/3*x*exp(1/4*x)+x**2 )+72*x**2-18*x)/((18*exp(1/9*exp(1/4*x)**2-2/3*x*exp(1/4*x)+x**2)-18*x)*ln (-exp(1/9*exp(1/4*x)**2-2/3*x*exp(1/4*x)+x**2)+x)-36*exp(1/9*exp(1/4*x)**2 -2/3*x*exp(1/4*x)+x**2)+36*x),x)
Output:
Timed out
Time = 0.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.55 \[ \int \frac {-18 x+72 x^2+e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )} \left (-72 x+e^{x/2} x+36 x^2+e^{x/4} \left (-12 x-3 x^2\right )\right )+\left (36 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )} x-36 x^2\right ) \log \left (-e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+x\right )+\left (-36 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+36 x+\left (18 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}-18 x\right ) \log \left (-e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+x\right )\right ) \log \left (2-\log \left (-e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+x\right )\right )}{-36 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+36 x+\left (18 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}-18 x\right ) \log \left (-e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+x\right )} \, dx=x^{2} - x \log \left (3\right ) + x \log \left (2 \, x e^{\left (\frac {1}{4} \, x\right )} - 3 \, \log \left (x e^{\left (\frac {2}{3} \, x e^{\left (\frac {1}{4} \, x\right )}\right )} - e^{\left (x^{2} + \frac {1}{9} \, e^{\left (\frac {1}{2} \, x\right )}\right )}\right ) + 6\right ) \] Input:
integrate((((18*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)-18*x)*log(-exp( 1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+x)-36*exp(1/9*exp(1/4*x)^2-2/3*x*ex p(1/4*x)+x^2)+36*x)*log(-log(-exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+x )+2)+(36*x*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)-36*x^2)*log(-exp(1/9 *exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+x)+(x*exp(1/4*x)^2+(-3*x^2-12*x)*exp(1 /4*x)+36*x^2-72*x)*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+72*x^2-18*x) /((18*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)-18*x)*log(-exp(1/9*exp(1/ 4*x)^2-2/3*x*exp(1/4*x)+x^2)+x)-36*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x ^2)+36*x),x, algorithm="maxima")
Output:
x^2 - x*log(3) + x*log(2*x*e^(1/4*x) - 3*log(x*e^(2/3*x*e^(1/4*x)) - e^(x^ 2 + 1/9*e^(1/2*x))) + 6)
\[ \int \frac {-18 x+72 x^2+e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )} \left (-72 x+e^{x/2} x+36 x^2+e^{x/4} \left (-12 x-3 x^2\right )\right )+\left (36 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )} x-36 x^2\right ) \log \left (-e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+x\right )+\left (-36 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+36 x+\left (18 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}-18 x\right ) \log \left (-e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+x\right )\right ) \log \left (2-\log \left (-e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+x\right )\right )}{-36 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+36 x+\left (18 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}-18 x\right ) \log \left (-e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+x\right )} \, dx=\int { -\frac {72 \, x^{2} + {\left (36 \, x^{2} + x e^{\left (\frac {1}{2} \, x\right )} - 3 \, {\left (x^{2} + 4 \, x\right )} e^{\left (\frac {1}{4} \, x\right )} - 72 \, x\right )} e^{\left (x^{2} - \frac {2}{3} \, x e^{\left (\frac {1}{4} \, x\right )} + \frac {1}{9} \, e^{\left (\frac {1}{2} \, x\right )}\right )} - 36 \, {\left (x^{2} - x e^{\left (x^{2} - \frac {2}{3} \, x e^{\left (\frac {1}{4} \, x\right )} + \frac {1}{9} \, e^{\left (\frac {1}{2} \, x\right )}\right )}\right )} \log \left (x - e^{\left (x^{2} - \frac {2}{3} \, x e^{\left (\frac {1}{4} \, x\right )} + \frac {1}{9} \, e^{\left (\frac {1}{2} \, x\right )}\right )}\right ) - 18 \, {\left ({\left (x - e^{\left (x^{2} - \frac {2}{3} \, x e^{\left (\frac {1}{4} \, x\right )} + \frac {1}{9} \, e^{\left (\frac {1}{2} \, x\right )}\right )}\right )} \log \left (x - e^{\left (x^{2} - \frac {2}{3} \, x e^{\left (\frac {1}{4} \, x\right )} + \frac {1}{9} \, e^{\left (\frac {1}{2} \, x\right )}\right )}\right ) - 2 \, x + 2 \, e^{\left (x^{2} - \frac {2}{3} \, x e^{\left (\frac {1}{4} \, x\right )} + \frac {1}{9} \, e^{\left (\frac {1}{2} \, x\right )}\right )}\right )} \log \left (-\log \left (x - e^{\left (x^{2} - \frac {2}{3} \, x e^{\left (\frac {1}{4} \, x\right )} + \frac {1}{9} \, e^{\left (\frac {1}{2} \, x\right )}\right )}\right ) + 2\right ) - 18 \, x}{18 \, {\left ({\left (x - e^{\left (x^{2} - \frac {2}{3} \, x e^{\left (\frac {1}{4} \, x\right )} + \frac {1}{9} \, e^{\left (\frac {1}{2} \, x\right )}\right )}\right )} \log \left (x - e^{\left (x^{2} - \frac {2}{3} \, x e^{\left (\frac {1}{4} \, x\right )} + \frac {1}{9} \, e^{\left (\frac {1}{2} \, x\right )}\right )}\right ) - 2 \, x + 2 \, e^{\left (x^{2} - \frac {2}{3} \, x e^{\left (\frac {1}{4} \, x\right )} + \frac {1}{9} \, e^{\left (\frac {1}{2} \, x\right )}\right )}\right )}} \,d x } \] Input:
integrate((((18*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)-18*x)*log(-exp( 1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+x)-36*exp(1/9*exp(1/4*x)^2-2/3*x*ex p(1/4*x)+x^2)+36*x)*log(-log(-exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+x )+2)+(36*x*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)-36*x^2)*log(-exp(1/9 *exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+x)+(x*exp(1/4*x)^2+(-3*x^2-12*x)*exp(1 /4*x)+36*x^2-72*x)*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+72*x^2-18*x) /((18*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)-18*x)*log(-exp(1/9*exp(1/ 4*x)^2-2/3*x*exp(1/4*x)+x^2)+x)-36*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x ^2)+36*x),x, algorithm="giac")
Output:
integrate(-1/18*(72*x^2 + (36*x^2 + x*e^(1/2*x) - 3*(x^2 + 4*x)*e^(1/4*x) - 72*x)*e^(x^2 - 2/3*x*e^(1/4*x) + 1/9*e^(1/2*x)) - 36*(x^2 - x*e^(x^2 - 2 /3*x*e^(1/4*x) + 1/9*e^(1/2*x)))*log(x - e^(x^2 - 2/3*x*e^(1/4*x) + 1/9*e^ (1/2*x))) - 18*((x - e^(x^2 - 2/3*x*e^(1/4*x) + 1/9*e^(1/2*x)))*log(x - e^ (x^2 - 2/3*x*e^(1/4*x) + 1/9*e^(1/2*x))) - 2*x + 2*e^(x^2 - 2/3*x*e^(1/4*x ) + 1/9*e^(1/2*x)))*log(-log(x - e^(x^2 - 2/3*x*e^(1/4*x) + 1/9*e^(1/2*x)) ) + 2) - 18*x)/((x - e^(x^2 - 2/3*x*e^(1/4*x) + 1/9*e^(1/2*x)))*log(x - e^ (x^2 - 2/3*x*e^(1/4*x) + 1/9*e^(1/2*x))) - 2*x + 2*e^(x^2 - 2/3*x*e^(1/4*x ) + 1/9*e^(1/2*x))), x)
Time = 3.90 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {-18 x+72 x^2+e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )} \left (-72 x+e^{x/2} x+36 x^2+e^{x/4} \left (-12 x-3 x^2\right )\right )+\left (36 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )} x-36 x^2\right ) \log \left (-e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+x\right )+\left (-36 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+36 x+\left (18 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}-18 x\right ) \log \left (-e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+x\right )\right ) \log \left (2-\log \left (-e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+x\right )\right )}{-36 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+36 x+\left (18 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}-18 x\right ) \log \left (-e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+x\right )} \, dx=x\,\left (x+\ln \left (2-\ln \left (x-{\mathrm {e}}^{\frac {{\mathrm {e}}^{x/2}}{9}}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-\frac {2\,x\,{\mathrm {e}}^{x/4}}{3}}\right )\right )\right ) \] Input:
int((18*x - log(x - exp(exp(x/2)/9 - (2*x*exp(x/4))/3 + x^2))*(36*x*exp(ex p(x/2)/9 - (2*x*exp(x/4))/3 + x^2) - 36*x^2) + exp(exp(x/2)/9 - (2*x*exp(x /4))/3 + x^2)*(72*x + exp(x/4)*(12*x + 3*x^2) - x*exp(x/2) - 36*x^2) + log (2 - log(x - exp(exp(x/2)/9 - (2*x*exp(x/4))/3 + x^2)))*(36*exp(exp(x/2)/9 - (2*x*exp(x/4))/3 + x^2) - 36*x + log(x - exp(exp(x/2)/9 - (2*x*exp(x/4) )/3 + x^2))*(18*x - 18*exp(exp(x/2)/9 - (2*x*exp(x/4))/3 + x^2))) - 72*x^2 )/(36*exp(exp(x/2)/9 - (2*x*exp(x/4))/3 + x^2) - 36*x + log(x - exp(exp(x/ 2)/9 - (2*x*exp(x/4))/3 + x^2))*(18*x - 18*exp(exp(x/2)/9 - (2*x*exp(x/4)) /3 + x^2))),x)
Output:
x*(x + log(2 - log(x - exp(exp(x/2)/9)*exp(x^2)*exp(-(2*x*exp(x/4))/3))))
Time = 0.22 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.65 \[ \int \frac {-18 x+72 x^2+e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )} \left (-72 x+e^{x/2} x+36 x^2+e^{x/4} \left (-12 x-3 x^2\right )\right )+\left (36 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )} x-36 x^2\right ) \log \left (-e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+x\right )+\left (-36 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+36 x+\left (18 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}-18 x\right ) \log \left (-e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+x\right )\right ) \log \left (2-\log \left (-e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+x\right )\right )}{-36 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+36 x+\left (18 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}-18 x\right ) \log \left (-e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+x\right )} \, dx=x \left (\mathrm {log}\left (-\mathrm {log}\left (\frac {-e^{\frac {e^{\frac {x}{2}}}{9}+x^{2}}+e^{\frac {2 e^{\frac {x}{4}} x}{3}} x}{e^{\frac {2 e^{\frac {x}{4}} x}{3}}}\right )+2\right )+x \right ) \] Input:
int((((18*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)-18*x)*log(-exp(1/9*ex p(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+x)-36*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4* x)+x^2)+36*x)*log(-log(-exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+x)+2)+( 36*x*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)-36*x^2)*log(-exp(1/9*exp(1 /4*x)^2-2/3*x*exp(1/4*x)+x^2)+x)+(x*exp(1/4*x)^2+(-3*x^2-12*x)*exp(1/4*x)+ 36*x^2-72*x)*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+72*x^2-18*x)/((18* exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)-18*x)*log(-exp(1/9*exp(1/4*x)^2 -2/3*x*exp(1/4*x)+x^2)+x)-36*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+36 *x),x)
Output:
x*(log( - log(( - e**((e**(x/2) + 9*x**2)/9) + e**((2*e**(x/4)*x)/3)*x)/e* *((2*e**(x/4)*x)/3)) + 2) + x)