Integrand size = 121, antiderivative size = 30 \[ \int \frac {-18 e^6 x^2+\left (6 e^3 x^2+e^6 \left (54 x+18 x^2\right ) \log (3+x)\right ) \log (\log (3+x))+e^3 \left (-36 x-12 x^2\right ) \log (3+x) \log ^2(\log (3+x))+\left (27+e^{16-x} (-27-9 x)+15 x+2 x^2\right ) \log (3+x) \log ^3(\log (3+x))}{(27+9 x) \log (3+x) \log ^3(\log (3+x))} \, dx=e^{16-x}+x+\left (\frac {x}{3}-\frac {e^3 x}{\log (\log (3+x))}\right )^2 \] Output:
exp(16-x)+x+(1/3*x-x/ln(ln(3+x))*exp(3))^2
Time = 0.16 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.67 \[ \int \frac {-18 e^6 x^2+\left (6 e^3 x^2+e^6 \left (54 x+18 x^2\right ) \log (3+x)\right ) \log (\log (3+x))+e^3 \left (-36 x-12 x^2\right ) \log (3+x) \log ^2(\log (3+x))+\left (27+e^{16-x} (-27-9 x)+15 x+2 x^2\right ) \log (3+x) \log ^3(\log (3+x))}{(27+9 x) \log (3+x) \log ^3(\log (3+x))} \, dx=\frac {1}{9} \left (9 e^{16-x}+9 x+x^2+\frac {9 e^6 x^2}{\log ^2(\log (3+x))}-\frac {6 e^3 x^2}{\log (\log (3+x))}\right ) \] Input:
Integrate[(-18*E^6*x^2 + (6*E^3*x^2 + E^6*(54*x + 18*x^2)*Log[3 + x])*Log[ Log[3 + x]] + E^3*(-36*x - 12*x^2)*Log[3 + x]*Log[Log[3 + x]]^2 + (27 + E^ (16 - x)*(-27 - 9*x) + 15*x + 2*x^2)*Log[3 + x]*Log[Log[3 + x]]^3)/((27 + 9*x)*Log[3 + x]*Log[Log[3 + x]]^3),x]
Output:
(9*E^(16 - x) + 9*x + x^2 + (9*E^6*x^2)/Log[Log[3 + x]]^2 - (6*E^3*x^2)/Lo g[Log[3 + x]])/9
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.
| \(\displaystyle \int \frac {-18 e^6 x^2+\left (2 x^2+15 x+e^{16-x} (-9 x-27)+27\right ) \log (x+3) \log ^3(\log (x+3))+e^3 \left (-12 x^2-36 x\right ) \log (x+3) \log ^2(\log (x+3))+\left (6 e^3 x^2+e^6 \left (18 x^2+54 x\right ) \log (x+3)\right ) \log (\log (x+3))}{(9 x+27) \log (x+3) \log ^3(\log (x+3))} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {-18 e^6 x^2+2 x^2 \log (x+3) \log ^3(\log (x+3))-12 e^3 x^2 \log (x+3) \log ^2(\log (x+3))+6 e^3 x^2 \log (\log (x+3))+18 e^6 x^2 \log (x+3) \log (\log (x+3))+15 x \log (x+3) \log ^3(\log (x+3))+27 \log (x+3) \log ^3(\log (x+3))-36 e^3 x \log (x+3) \log ^2(\log (x+3))+54 e^6 x \log (x+3) \log (\log (x+3))}{9 (x+3) \log (x+3) \log ^3(\log (x+3))}-e^{16-x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 6 e^6 \text {Subst}\left (\int \frac {1}{\log (x) \log ^3(\log (x))}dx,x,x+3\right )-2 e^3 \text {Subst}\left (\int \frac {1}{\log (x) \log ^2(\log (x))}dx,x,x+3\right )-2 e^6 \int \frac {x}{\log (x+3) \log ^3(\log (x+3))}dx+2 e^6 \int \frac {x}{\log ^2(\log (x+3))}dx+\frac {2}{3} e^3 \int \frac {x}{\log (x+3) \log ^2(\log (x+3))}dx-\frac {4}{3} e^3 \int \frac {x}{\log (\log (x+3))}dx+\frac {x^2}{9}+x+e^{16-x}+\frac {9 e^6}{\log ^2(\log (x+3))}-\frac {6 e^3}{\log (\log (x+3))}\) |
Input:
Int[(-18*E^6*x^2 + (6*E^3*x^2 + E^6*(54*x + 18*x^2)*Log[3 + x])*Log[Log[3 + x]] + E^3*(-36*x - 12*x^2)*Log[3 + x]*Log[Log[3 + x]]^2 + (27 + E^(16 - x)*(-27 - 9*x) + 15*x + 2*x^2)*Log[3 + x]*Log[Log[3 + x]]^3)/((27 + 9*x)*L og[3 + x]*Log[Log[3 + x]]^3),x]
Output:
$Aborted
Time = 9.46 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.33
| method | result | size |
| risch | \(\frac {x^{2}}{9}+x +{\mathrm e}^{16-x}+\frac {x^{2} {\mathrm e}^{3} \left (3 \,{\mathrm e}^{3}-2 \ln \left (\ln \left (3+x \right )\right )\right )}{3 \ln \left (\ln \left (3+x \right )\right )^{2}}\) | \(40\) |
| parallelrisch | \(\frac {9 x^{2} {\mathrm e}^{6}-6 \ln \left (\ln \left (3+x \right )\right ) {\mathrm e}^{3} x^{2}+x^{2} \ln \left (\ln \left (3+x \right )\right )^{2}+9 x \ln \left (\ln \left (3+x \right )\right )^{2}+9 \,{\mathrm e}^{16-x} \ln \left (\ln \left (3+x \right )\right )^{2}-63 \ln \left (\ln \left (3+x \right )\right )^{2}}{9 \ln \left (\ln \left (3+x \right )\right )^{2}}\) | \(77\) |
Input:
int((((-9*x-27)*exp(16-x)+2*x^2+15*x+27)*ln(3+x)*ln(ln(3+x))^3+(-12*x^2-36 *x)*exp(3)*ln(3+x)*ln(ln(3+x))^2+((18*x^2+54*x)*exp(3)^2*ln(3+x)+6*x^2*exp (3))*ln(ln(3+x))-18*x^2*exp(3)^2)/(9*x+27)/ln(3+x)/ln(ln(3+x))^3,x,method= _RETURNVERBOSE)
Output:
1/9*x^2+x+exp(16-x)+1/3*x^2*exp(3)*(3*exp(3)-2*ln(ln(3+x)))/ln(ln(3+x))^2
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (26) = 52\).
Time = 0.08 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.77 \[ \int \frac {-18 e^6 x^2+\left (6 e^3 x^2+e^6 \left (54 x+18 x^2\right ) \log (3+x)\right ) \log (\log (3+x))+e^3 \left (-36 x-12 x^2\right ) \log (3+x) \log ^2(\log (3+x))+\left (27+e^{16-x} (-27-9 x)+15 x+2 x^2\right ) \log (3+x) \log ^3(\log (3+x))}{(27+9 x) \log (3+x) \log ^3(\log (3+x))} \, dx=-\frac {6 \, x^{2} e^{3} \log \left (\log \left (x + 3\right )\right ) - 9 \, x^{2} e^{6} - {\left (x^{2} + 9 \, x + 9 \, e^{\left (-x + 16\right )}\right )} \log \left (\log \left (x + 3\right )\right )^{2}}{9 \, \log \left (\log \left (x + 3\right )\right )^{2}} \] Input:
integrate((((-9*x-27)*exp(16-x)+2*x^2+15*x+27)*log(3+x)*log(log(3+x))^3+(- 12*x^2-36*x)*exp(3)*log(3+x)*log(log(3+x))^2+((18*x^2+54*x)*exp(3)^2*log(3 +x)+6*x^2*exp(3))*log(log(3+x))-18*x^2*exp(3)^2)/(9*x+27)/log(3+x)/log(log (3+x))^3,x, algorithm="fricas")
Output:
-1/9*(6*x^2*e^3*log(log(x + 3)) - 9*x^2*e^6 - (x^2 + 9*x + 9*e^(-x + 16))* log(log(x + 3))^2)/log(log(x + 3))^2
Time = 0.21 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.47 \[ \int \frac {-18 e^6 x^2+\left (6 e^3 x^2+e^6 \left (54 x+18 x^2\right ) \log (3+x)\right ) \log (\log (3+x))+e^3 \left (-36 x-12 x^2\right ) \log (3+x) \log ^2(\log (3+x))+\left (27+e^{16-x} (-27-9 x)+15 x+2 x^2\right ) \log (3+x) \log ^3(\log (3+x))}{(27+9 x) \log (3+x) \log ^3(\log (3+x))} \, dx=\frac {x^{2}}{9} + x + \frac {- 2 x^{2} e^{3} \log {\left (\log {\left (x + 3 \right )} \right )} + 3 x^{2} e^{6}}{3 \log {\left (\log {\left (x + 3 \right )} \right )}^{2}} + e^{16 - x} \] Input:
integrate((((-9*x-27)*exp(16-x)+2*x**2+15*x+27)*ln(3+x)*ln(ln(3+x))**3+(-1 2*x**2-36*x)*exp(3)*ln(3+x)*ln(ln(3+x))**2+((18*x**2+54*x)*exp(3)**2*ln(3+ x)+6*x**2*exp(3))*ln(ln(3+x))-18*x**2*exp(3)**2)/(9*x+27)/ln(3+x)/ln(ln(3+ x))**3,x)
Output:
x**2/9 + x + (-2*x**2*exp(3)*log(log(x + 3)) + 3*x**2*exp(6))/(3*log(log(x + 3))**2) + exp(16 - x)
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (26) = 52\).
Time = 0.08 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.03 \[ \int \frac {-18 e^6 x^2+\left (6 e^3 x^2+e^6 \left (54 x+18 x^2\right ) \log (3+x)\right ) \log (\log (3+x))+e^3 \left (-36 x-12 x^2\right ) \log (3+x) \log ^2(\log (3+x))+\left (27+e^{16-x} (-27-9 x)+15 x+2 x^2\right ) \log (3+x) \log ^3(\log (3+x))}{(27+9 x) \log (3+x) \log ^3(\log (3+x))} \, dx=-\frac {{\left (6 \, x^{2} e^{\left (x + 3\right )} \log \left (\log \left (x + 3\right )\right ) - 9 \, x^{2} e^{\left (x + 6\right )} - {\left ({\left (x^{2} + 9 \, x\right )} e^{x} + 9 \, e^{16}\right )} \log \left (\log \left (x + 3\right )\right )^{2}\right )} e^{\left (-x\right )}}{9 \, \log \left (\log \left (x + 3\right )\right )^{2}} \] Input:
integrate((((-9*x-27)*exp(16-x)+2*x^2+15*x+27)*log(3+x)*log(log(3+x))^3+(- 12*x^2-36*x)*exp(3)*log(3+x)*log(log(3+x))^2+((18*x^2+54*x)*exp(3)^2*log(3 +x)+6*x^2*exp(3))*log(log(3+x))-18*x^2*exp(3)^2)/(9*x+27)/log(3+x)/log(log (3+x))^3,x, algorithm="maxima")
Output:
-1/9*(6*x^2*e^(x + 3)*log(log(x + 3)) - 9*x^2*e^(x + 6) - ((x^2 + 9*x)*e^x + 9*e^16)*log(log(x + 3))^2)*e^(-x)/log(log(x + 3))^2
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (26) = 52\).
Time = 0.21 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.20 \[ \int \frac {-18 e^6 x^2+\left (6 e^3 x^2+e^6 \left (54 x+18 x^2\right ) \log (3+x)\right ) \log (\log (3+x))+e^3 \left (-36 x-12 x^2\right ) \log (3+x) \log ^2(\log (3+x))+\left (27+e^{16-x} (-27-9 x)+15 x+2 x^2\right ) \log (3+x) \log ^3(\log (3+x))}{(27+9 x) \log (3+x) \log ^3(\log (3+x))} \, dx=-\frac {6 \, x^{2} e^{3} \log \left (\log \left (x + 3\right )\right ) - x^{2} \log \left (\log \left (x + 3\right )\right )^{2} - 9 \, x^{2} e^{6} - 9 \, x \log \left (\log \left (x + 3\right )\right )^{2} - 9 \, e^{\left (-x + 16\right )} \log \left (\log \left (x + 3\right )\right )^{2}}{9 \, \log \left (\log \left (x + 3\right )\right )^{2}} \] Input:
integrate((((-9*x-27)*exp(16-x)+2*x^2+15*x+27)*log(3+x)*log(log(3+x))^3+(- 12*x^2-36*x)*exp(3)*log(3+x)*log(log(3+x))^2+((18*x^2+54*x)*exp(3)^2*log(3 +x)+6*x^2*exp(3))*log(log(3+x))-18*x^2*exp(3)^2)/(9*x+27)/log(3+x)/log(log (3+x))^3,x, algorithm="giac")
Output:
-1/9*(6*x^2*e^3*log(log(x + 3)) - x^2*log(log(x + 3))^2 - 9*x^2*e^6 - 9*x* log(log(x + 3))^2 - 9*e^(-x + 16)*log(log(x + 3))^2)/log(log(x + 3))^2
Time = 4.62 (sec) , antiderivative size = 281, normalized size of antiderivative = 9.37 \[ \int \frac {-18 e^6 x^2+\left (6 e^3 x^2+e^6 \left (54 x+18 x^2\right ) \log (3+x)\right ) \log (\log (3+x))+e^3 \left (-36 x-12 x^2\right ) \log (3+x) \log ^2(\log (3+x))+\left (27+e^{16-x} (-27-9 x)+15 x+2 x^2\right ) \log (3+x) \log ^3(\log (3+x))}{(27+9 x) \log (3+x) \log ^3(\log (3+x))} \, dx=x+{\mathrm {e}}^{16-x}+\frac {x^2}{9}+6\,{\ln \left (x+3\right )}^2\,{\mathrm {e}}^3\,\ln \left (\ln \left (x+3\right )\right )-\frac {2\,x^2\,{\mathrm {e}}^3}{3\,\ln \left (\ln \left (x+3\right )\right )}+\frac {x^2\,{\mathrm {e}}^6}{{\ln \left (\ln \left (x+3\right )\right )}^2}+2\,x\,\ln \left (x+3\right )\,{\mathrm {e}}^3\,\ln \left (\ln \left (x+3\right )\right )+6\,x\,{\ln \left (x+3\right )}^2\,{\mathrm {e}}^3\,\ln \left (\ln \left (x+3\right )\right )+\frac {2\,x^2\,\ln \left (x+3\right )\,{\mathrm {e}}^3\,\ln \left (\ln \left (x+3\right )\right )}{3}-\frac {54\,{\ln \left (x+3\right )}^2\,{\mathrm {e}}^3\,\ln \left (\ln \left (x+3\right )\right )}{3\,x+9}+\frac {4\,x^2\,{\ln \left (x+3\right )}^2\,{\mathrm {e}}^3\,\ln \left (\ln \left (x+3\right )\right )}{3}-\frac {72\,x\,{\ln \left (x+3\right )}^2\,{\mathrm {e}}^3\,\ln \left (\ln \left (x+3\right )\right )}{3\,x+9}-\frac {12\,x^2\,\ln \left (x+3\right )\,{\mathrm {e}}^3\,\ln \left (\ln \left (x+3\right )\right )}{3\,x+9}-\frac {2\,x^3\,\ln \left (x+3\right )\,{\mathrm {e}}^3\,\ln \left (\ln \left (x+3\right )\right )}{3\,x+9}-\frac {30\,x^2\,{\ln \left (x+3\right )}^2\,{\mathrm {e}}^3\,\ln \left (\ln \left (x+3\right )\right )}{3\,x+9}-\frac {4\,x^3\,{\ln \left (x+3\right )}^2\,{\mathrm {e}}^3\,\ln \left (\ln \left (x+3\right )\right )}{3\,x+9}-\frac {18\,x\,\ln \left (x+3\right )\,{\mathrm {e}}^3\,\ln \left (\ln \left (x+3\right )\right )}{3\,x+9} \] Input:
int((log(log(x + 3))*(6*x^2*exp(3) + log(x + 3)*exp(6)*(54*x + 18*x^2)) - 18*x^2*exp(6) + log(x + 3)*log(log(x + 3))^3*(15*x - exp(16 - x)*(9*x + 27 ) + 2*x^2 + 27) - log(x + 3)*exp(3)*log(log(x + 3))^2*(36*x + 12*x^2))/(lo g(x + 3)*log(log(x + 3))^3*(9*x + 27)),x)
Output:
x + exp(16 - x) + x^2/9 + 6*log(x + 3)^2*exp(3)*log(log(x + 3)) - (2*x^2*e xp(3))/(3*log(log(x + 3))) + (x^2*exp(6))/log(log(x + 3))^2 + 2*x*log(x + 3)*exp(3)*log(log(x + 3)) + 6*x*log(x + 3)^2*exp(3)*log(log(x + 3)) + (2*x ^2*log(x + 3)*exp(3)*log(log(x + 3)))/3 - (54*log(x + 3)^2*exp(3)*log(log( x + 3)))/(3*x + 9) + (4*x^2*log(x + 3)^2*exp(3)*log(log(x + 3)))/3 - (72*x *log(x + 3)^2*exp(3)*log(log(x + 3)))/(3*x + 9) - (12*x^2*log(x + 3)*exp(3 )*log(log(x + 3)))/(3*x + 9) - (2*x^3*log(x + 3)*exp(3)*log(log(x + 3)))/( 3*x + 9) - (30*x^2*log(x + 3)^2*exp(3)*log(log(x + 3)))/(3*x + 9) - (4*x^3 *log(x + 3)^2*exp(3)*log(log(x + 3)))/(3*x + 9) - (18*x*log(x + 3)*exp(3)* log(log(x + 3)))/(3*x + 9)
Time = 0.16 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.70 \[ \int \frac {-18 e^6 x^2+\left (6 e^3 x^2+e^6 \left (54 x+18 x^2\right ) \log (3+x)\right ) \log (\log (3+x))+e^3 \left (-36 x-12 x^2\right ) \log (3+x) \log ^2(\log (3+x))+\left (27+e^{16-x} (-27-9 x)+15 x+2 x^2\right ) \log (3+x) \log ^3(\log (3+x))}{(27+9 x) \log (3+x) \log ^3(\log (3+x))} \, dx=\frac {e^{x} \mathrm {log}\left (\mathrm {log}\left (x +3\right )\right )^{2} x^{2}+9 e^{x} \mathrm {log}\left (\mathrm {log}\left (x +3\right )\right )^{2} x -6 e^{x} \mathrm {log}\left (\mathrm {log}\left (x +3\right )\right ) e^{3} x^{2}+9 e^{x} e^{6} x^{2}+9 \mathrm {log}\left (\mathrm {log}\left (x +3\right )\right )^{2} e^{16}}{9 e^{x} \mathrm {log}\left (\mathrm {log}\left (x +3\right )\right )^{2}} \] Input:
int((((-9*x-27)*exp(16-x)+2*x^2+15*x+27)*log(3+x)*log(log(3+x))^3+(-12*x^2 -36*x)*exp(3)*log(3+x)*log(log(3+x))^2+((18*x^2+54*x)*exp(3)^2*log(3+x)+6* x^2*exp(3))*log(log(3+x))-18*x^2*exp(3)^2)/(9*x+27)/log(3+x)/log(log(3+x)) ^3,x)
Output:
(e**x*log(log(x + 3))**2*x**2 + 9*e**x*log(log(x + 3))**2*x - 6*e**x*log(l og(x + 3))*e**3*x**2 + 9*e**x*e**6*x**2 + 9*log(log(x + 3))**2*e**16)/(9*e **x*log(log(x + 3))**2)