Integrand size = 318, antiderivative size = 24 \[ \int \frac {-x^3-2 x^4+e^x \left (-8 x^3-2 e^6 x^3-2 x^4\right )+e^{2 x} \left (18+e^{12} (2-2 x)-12 x-12 x^2-2 x^3+e^6 \left (12-10 x-4 x^2\right )\right )+\left (x^3+x^4+e^x \left (6 x^2+2 e^6 x^2+2 x^3\right )+e^{2 x} \left (9+e^{12}+6 x+x^2+e^6 (6+2 x)\right )\right ) \log \left (\frac {x^3+x^4+e^x \left (6 x^2+2 e^6 x^2+2 x^3\right )+e^{2 x} \left (9+e^{12}+6 x+x^2+e^6 (6+2 x)\right )}{x^2}\right )}{\left (x^3+x^4+e^x \left (6 x^2+2 e^6 x^2+2 x^3\right )+e^{2 x} \left (9+e^{12}+6 x+x^2+e^6 (6+2 x)\right )\right ) \log ^2\left (\frac {x^3+x^4+e^x \left (6 x^2+2 e^6 x^2+2 x^3\right )+e^{2 x} \left (9+e^{12}+6 x+x^2+e^6 (6+2 x)\right )}{x^2}\right )} \, dx=\frac {x}{\log \left (x+\left (x+\frac {e^x \left (3+e^6+x\right )}{x}\right )^2\right )} \] Output:
x/ln((exp(x)*(exp(6)+3+x)/x+x)^2+x)
Time = 0.13 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.58 \[ \int \frac {-x^3-2 x^4+e^x \left (-8 x^3-2 e^6 x^3-2 x^4\right )+e^{2 x} \left (18+e^{12} (2-2 x)-12 x-12 x^2-2 x^3+e^6 \left (12-10 x-4 x^2\right )\right )+\left (x^3+x^4+e^x \left (6 x^2+2 e^6 x^2+2 x^3\right )+e^{2 x} \left (9+e^{12}+6 x+x^2+e^6 (6+2 x)\right )\right ) \log \left (\frac {x^3+x^4+e^x \left (6 x^2+2 e^6 x^2+2 x^3\right )+e^{2 x} \left (9+e^{12}+6 x+x^2+e^6 (6+2 x)\right )}{x^2}\right )}{\left (x^3+x^4+e^x \left (6 x^2+2 e^6 x^2+2 x^3\right )+e^{2 x} \left (9+e^{12}+6 x+x^2+e^6 (6+2 x)\right )\right ) \log ^2\left (\frac {x^3+x^4+e^x \left (6 x^2+2 e^6 x^2+2 x^3\right )+e^{2 x} \left (9+e^{12}+6 x+x^2+e^6 (6+2 x)\right )}{x^2}\right )} \, dx=\frac {x}{\log \left (x+x^2+2 e^x \left (3+e^6+x\right )+\frac {e^{2 x} \left (3+e^6+x\right )^2}{x^2}\right )} \] Input:
Integrate[(-x^3 - 2*x^4 + E^x*(-8*x^3 - 2*E^6*x^3 - 2*x^4) + E^(2*x)*(18 + E^12*(2 - 2*x) - 12*x - 12*x^2 - 2*x^3 + E^6*(12 - 10*x - 4*x^2)) + (x^3 + x^4 + E^x*(6*x^2 + 2*E^6*x^2 + 2*x^3) + E^(2*x)*(9 + E^12 + 6*x + x^2 + E^6*(6 + 2*x)))*Log[(x^3 + x^4 + E^x*(6*x^2 + 2*E^6*x^2 + 2*x^3) + E^(2*x) *(9 + E^12 + 6*x + x^2 + E^6*(6 + 2*x)))/x^2])/((x^3 + x^4 + E^x*(6*x^2 + 2*E^6*x^2 + 2*x^3) + E^(2*x)*(9 + E^12 + 6*x + x^2 + E^6*(6 + 2*x)))*Log[( x^3 + x^4 + E^x*(6*x^2 + 2*E^6*x^2 + 2*x^3) + E^(2*x)*(9 + E^12 + 6*x + x^ 2 + E^6*(6 + 2*x)))/x^2]^2),x]
Output:
x/Log[x + x^2 + 2*E^x*(3 + E^6 + x) + (E^(2*x)*(3 + E^6 + x)^2)/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.
| \(\displaystyle \int \frac {-2 x^4-x^3+e^x \left (-2 x^4-2 e^6 x^3-8 x^3\right )+e^{2 x} \left (-2 x^3-12 x^2+e^6 \left (-4 x^2-10 x+12\right )-12 x+e^{12} (2-2 x)+18\right )+\left (x^4+x^3+e^{2 x} \left (x^2+6 x+e^6 (2 x+6)+e^{12}+9\right )+e^x \left (2 x^3+2 e^6 x^2+6 x^2\right )\right ) \log \left (\frac {x^4+x^3+e^{2 x} \left (x^2+6 x+e^6 (2 x+6)+e^{12}+9\right )+e^x \left (2 x^3+2 e^6 x^2+6 x^2\right )}{x^2}\right )}{\left (x^4+x^3+e^{2 x} \left (x^2+6 x+e^6 (2 x+6)+e^{12}+9\right )+e^x \left (2 x^3+2 e^6 x^2+6 x^2\right )\right ) \log ^2\left (\frac {x^4+x^3+e^{2 x} \left (x^2+6 x+e^6 (2 x+6)+e^{12}+9\right )+e^x \left (2 x^3+2 e^6 x^2+6 x^2\right )}{x^2}\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-2 x^4-x^3+e^x \left (-2 x^4-2 e^6 x^3-8 x^3\right )+e^{2 x} \left (-2 x^3-12 x^2+e^6 \left (-4 x^2-10 x+12\right )-12 x+e^{12} (2-2 x)+18\right )+\left (x^4+x^3+e^{2 x} \left (x^2+6 x+e^6 (2 x+6)+e^{12}+9\right )+e^x \left (2 x^3+2 e^6 x^2+6 x^2\right )\right ) \log \left (\frac {x^4+x^3+e^{2 x} \left (x^2+6 x+e^6 (2 x+6)+e^{12}+9\right )+e^x \left (2 x^3+2 e^6 x^2+6 x^2\right )}{x^2}\right )}{\left (x^4+x^3+e^{2 x} \left (x^2+6 x+e^6 (2 x+6)+e^{12}+9\right )+e^x \left (2 x^3+2 e^6 x^2+6 x^2\right )\right ) \log ^2\left (x^2+\frac {e^{2 x} \left (x^2+6 x+e^6 (2 x+6)+e^{12}+9\right )}{x^2}+\frac {e^x \left (2 x^3+2 e^6 x^2+6 x^2\right )}{x^2}+x\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {-2 x^2+x \log \left (x^2+\frac {e^{2 x} \left (x+e^6+3\right )^2}{x^2}+x+2 e^x \left (x+e^6+3\right )\right )+3 \left (1+\frac {e^6}{3}\right ) \log \left (x^2+\frac {e^{2 x} \left (x+e^6+3\right )^2}{x^2}+x+2 e^x \left (x+e^6+3\right )\right )-6 \left (1+\frac {e^6}{3}\right ) x+6 \left (1+\frac {e^6}{3}\right )}{\left (x+e^6+3\right ) \log ^2\left (x^2+\frac {e^{2 x} \left (x+e^6+3\right )^2}{x^2}+x+2 e^x \left (x+e^6+3\right )\right )}+\frac {\left (2 x^4+2 e^x x^3+6 \left (1+\frac {e^6}{3}\right ) x^3+10 \left (1+\frac {2 e^6}{5}\right ) e^x x^2-7 \left (1+\frac {2 e^6}{7}\right ) x^2+6 \left (1+\frac {e^6}{3}\right ) e^{x+6} x-9 \left (1+\frac {e^6}{3}\right ) x-36 \left (1+\frac {1}{9} e^6 \left (6+e^6\right )\right ) e^x\right ) x^2}{\left (x+e^6+3\right ) \left (x^4+2 e^x x^3+x^3+e^{2 x} x^2+6 \left (1+\frac {e^6}{3}\right ) e^x x^2+6 \left (1+\frac {e^6}{3}\right ) e^{2 x} x+9 \left (1+\frac {1}{9} e^6 \left (6+e^6\right )\right ) e^{2 x}\right ) \log ^2\left (x^2+\frac {e^{2 x} \left (x+e^6+3\right )^2}{x^2}+x+2 e^x \left (x+e^6+3\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {-2 x^2+x \log \left (x^2+\frac {e^{2 x} \left (x+e^6+3\right )^2}{x^2}+x+2 e^x \left (x+e^6+3\right )\right )+3 \left (1+\frac {e^6}{3}\right ) \log \left (x^2+\frac {e^{2 x} \left (x+e^6+3\right )^2}{x^2}+x+2 e^x \left (x+e^6+3\right )\right )-6 \left (1+\frac {e^6}{3}\right ) x+6 \left (1+\frac {e^6}{3}\right )}{\left (x+e^6+3\right ) \log ^2\left (x^2+\frac {e^{2 x} \left (x+e^6+3\right )^2}{x^2}+x+2 e^x \left (x+e^6+3\right )\right )}+\frac {\left (2 x^4+2 e^x x^3+6 \left (1+\frac {e^6}{3}\right ) x^3+10 \left (1+\frac {2 e^6}{5}\right ) e^x x^2-7 \left (1+\frac {2 e^6}{7}\right ) x^2+6 \left (1+\frac {e^6}{3}\right ) e^{x+6} x-9 \left (1+\frac {e^6}{3}\right ) x-36 \left (1+\frac {1}{9} e^6 \left (6+e^6\right )\right ) e^x\right ) x^2}{\left (x+e^6+3\right ) \left (x^4+2 e^x x^3+x^3+e^{2 x} x^2+6 \left (1+\frac {e^6}{3}\right ) e^x x^2+6 \left (1+\frac {e^6}{3}\right ) e^{2 x} x+9 \left (1+\frac {1}{9} e^6 \left (6+e^6\right )\right ) e^{2 x}\right ) \log ^2\left (x^2+\frac {e^{2 x} \left (x+e^6+3\right )^2}{x^2}+x+2 e^x \left (x+e^6+3\right )\right )}\right )dx\) |
Input:
Int[(-x^3 - 2*x^4 + E^x*(-8*x^3 - 2*E^6*x^3 - 2*x^4) + E^(2*x)*(18 + E^12* (2 - 2*x) - 12*x - 12*x^2 - 2*x^3 + E^6*(12 - 10*x - 4*x^2)) + (x^3 + x^4 + E^x*(6*x^2 + 2*E^6*x^2 + 2*x^3) + E^(2*x)*(9 + E^12 + 6*x + x^2 + E^6*(6 + 2*x)))*Log[(x^3 + x^4 + E^x*(6*x^2 + 2*E^6*x^2 + 2*x^3) + E^(2*x)*(9 + E^12 + 6*x + x^2 + E^6*(6 + 2*x)))/x^2])/((x^3 + x^4 + E^x*(6*x^2 + 2*E^6* x^2 + 2*x^3) + E^(2*x)*(9 + E^12 + 6*x + x^2 + E^6*(6 + 2*x)))*Log[(x^3 + x^4 + E^x*(6*x^2 + 2*E^6*x^2 + 2*x^3) + E^(2*x)*(9 + E^12 + 6*x + x^2 + E^ 6*(6 + 2*x)))/x^2]^2),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(62\) vs. \(2(22)=44\).
Time = 14.55 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.62
| method | result | size |
| parallelrisch | \(\frac {x}{\ln \left (\frac {\left ({\mathrm e}^{12}+\left (2 x +6\right ) {\mathrm e}^{6}+x^{2}+6 x +9\right ) {\mathrm e}^{2 x}+\left (2 x^{2} {\mathrm e}^{6}+2 x^{3}+6 x^{2}\right ) {\mathrm e}^{x}+x^{4}+x^{3}}{x^{2}}\right )}\) | \(63\) |
| risch | \(-\frac {2 i x}{\pi \operatorname {csgn}\left (i x^{2}\right )^{3}-2 \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )+\pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}+\pi \,\operatorname {csgn}\left (i \left (2 x^{2} {\mathrm e}^{6+x}+2 \,{\mathrm e}^{x} x^{3}+x^{4}+6 \,{\mathrm e}^{x} x^{2}+2 x \,{\mathrm e}^{2 x +6}+{\mathrm e}^{2 x} x^{2}+x^{3}+6 \,{\mathrm e}^{2 x +6}+{\mathrm e}^{2 x +12}+6 x \,{\mathrm e}^{2 x}+9 \,{\mathrm e}^{2 x}\right )\right ) {\operatorname {csgn}\left (\frac {i \left (2 x^{2} {\mathrm e}^{6+x}+2 \,{\mathrm e}^{x} x^{3}+x^{4}+6 \,{\mathrm e}^{x} x^{2}+2 x \,{\mathrm e}^{2 x +6}+{\mathrm e}^{2 x} x^{2}+x^{3}+6 \,{\mathrm e}^{2 x +6}+{\mathrm e}^{2 x +12}+6 x \,{\mathrm e}^{2 x}+9 \,{\mathrm e}^{2 x}\right )}{x^{2}}\right )}^{2}-\pi \,\operatorname {csgn}\left (i \left (2 x^{2} {\mathrm e}^{6+x}+2 \,{\mathrm e}^{x} x^{3}+x^{4}+6 \,{\mathrm e}^{x} x^{2}+2 x \,{\mathrm e}^{2 x +6}+{\mathrm e}^{2 x} x^{2}+x^{3}+6 \,{\mathrm e}^{2 x +6}+{\mathrm e}^{2 x +12}+6 x \,{\mathrm e}^{2 x}+9 \,{\mathrm e}^{2 x}\right )\right ) \operatorname {csgn}\left (\frac {i \left (2 x^{2} {\mathrm e}^{6+x}+2 \,{\mathrm e}^{x} x^{3}+x^{4}+6 \,{\mathrm e}^{x} x^{2}+2 x \,{\mathrm e}^{2 x +6}+{\mathrm e}^{2 x} x^{2}+x^{3}+6 \,{\mathrm e}^{2 x +6}+{\mathrm e}^{2 x +12}+6 x \,{\mathrm e}^{2 x}+9 \,{\mathrm e}^{2 x}\right )}{x^{2}}\right ) \operatorname {csgn}\left (\frac {i}{x^{2}}\right )-\pi {\operatorname {csgn}\left (\frac {i \left (2 x^{2} {\mathrm e}^{6+x}+2 \,{\mathrm e}^{x} x^{3}+x^{4}+6 \,{\mathrm e}^{x} x^{2}+2 x \,{\mathrm e}^{2 x +6}+{\mathrm e}^{2 x} x^{2}+x^{3}+6 \,{\mathrm e}^{2 x +6}+{\mathrm e}^{2 x +12}+6 x \,{\mathrm e}^{2 x}+9 \,{\mathrm e}^{2 x}\right )}{x^{2}}\right )}^{3}+\pi {\operatorname {csgn}\left (\frac {i \left (2 x^{2} {\mathrm e}^{6+x}+2 \,{\mathrm e}^{x} x^{3}+x^{4}+6 \,{\mathrm e}^{x} x^{2}+2 x \,{\mathrm e}^{2 x +6}+{\mathrm e}^{2 x} x^{2}+x^{3}+6 \,{\mathrm e}^{2 x +6}+{\mathrm e}^{2 x +12}+6 x \,{\mathrm e}^{2 x}+9 \,{\mathrm e}^{2 x}\right )}{x^{2}}\right )}^{2} \operatorname {csgn}\left (\frac {i}{x^{2}}\right )+4 i \ln \left (x \right )-2 i \ln \left (2 x^{2} {\mathrm e}^{6+x}+2 \,{\mathrm e}^{x} x^{3}+x^{4}+6 \,{\mathrm e}^{x} x^{2}+2 x \,{\mathrm e}^{2 x +6}+{\mathrm e}^{2 x} x^{2}+x^{3}+6 \,{\mathrm e}^{2 x +6}+{\mathrm e}^{2 x +12}+6 x \,{\mathrm e}^{2 x}+9 \,{\mathrm e}^{2 x}\right )}\) | \(645\) |
Input:
int((((exp(6)^2+(2*x+6)*exp(6)+x^2+6*x+9)*exp(x)^2+(2*x^2*exp(6)+2*x^3+6*x ^2)*exp(x)+x^4+x^3)*ln(((exp(6)^2+(2*x+6)*exp(6)+x^2+6*x+9)*exp(x)^2+(2*x^ 2*exp(6)+2*x^3+6*x^2)*exp(x)+x^4+x^3)/x^2)+((2-2*x)*exp(6)^2+(-4*x^2-10*x+ 12)*exp(6)-2*x^3-12*x^2-12*x+18)*exp(x)^2+(-2*x^3*exp(6)-2*x^4-8*x^3)*exp( x)-2*x^4-x^3)/((exp(6)^2+(2*x+6)*exp(6)+x^2+6*x+9)*exp(x)^2+(2*x^2*exp(6)+ 2*x^3+6*x^2)*exp(x)+x^4+x^3)/ln(((exp(6)^2+(2*x+6)*exp(6)+x^2+6*x+9)*exp(x )^2+(2*x^2*exp(6)+2*x^3+6*x^2)*exp(x)+x^4+x^3)/x^2)^2,x,method=_RETURNVERB OSE)
Output:
x/ln(((exp(6)^2+(2*x+6)*exp(6)+x^2+6*x+9)*exp(x)^2+(2*x^2*exp(6)+2*x^3+6*x ^2)*exp(x)+x^4+x^3)/x^2)
Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (22) = 44\).
Time = 0.11 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.38 \[ \int \frac {-x^3-2 x^4+e^x \left (-8 x^3-2 e^6 x^3-2 x^4\right )+e^{2 x} \left (18+e^{12} (2-2 x)-12 x-12 x^2-2 x^3+e^6 \left (12-10 x-4 x^2\right )\right )+\left (x^3+x^4+e^x \left (6 x^2+2 e^6 x^2+2 x^3\right )+e^{2 x} \left (9+e^{12}+6 x+x^2+e^6 (6+2 x)\right )\right ) \log \left (\frac {x^3+x^4+e^x \left (6 x^2+2 e^6 x^2+2 x^3\right )+e^{2 x} \left (9+e^{12}+6 x+x^2+e^6 (6+2 x)\right )}{x^2}\right )}{\left (x^3+x^4+e^x \left (6 x^2+2 e^6 x^2+2 x^3\right )+e^{2 x} \left (9+e^{12}+6 x+x^2+e^6 (6+2 x)\right )\right ) \log ^2\left (\frac {x^3+x^4+e^x \left (6 x^2+2 e^6 x^2+2 x^3\right )+e^{2 x} \left (9+e^{12}+6 x+x^2+e^6 (6+2 x)\right )}{x^2}\right )} \, dx=\frac {x}{\log \left (\frac {x^{4} + x^{3} + {\left (x^{2} + 2 \, {\left (x + 3\right )} e^{6} + 6 \, x + e^{12} + 9\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x^{3} + x^{2} e^{6} + 3 \, x^{2}\right )} e^{x}}{x^{2}}\right )} \] Input:
integrate((((exp(6)^2+(2*x+6)*exp(6)+x^2+6*x+9)*exp(x)^2+(2*x^2*exp(6)+2*x ^3+6*x^2)*exp(x)+x^4+x^3)*log(((exp(6)^2+(2*x+6)*exp(6)+x^2+6*x+9)*exp(x)^ 2+(2*x^2*exp(6)+2*x^3+6*x^2)*exp(x)+x^4+x^3)/x^2)+((2-2*x)*exp(6)^2+(-4*x^ 2-10*x+12)*exp(6)-2*x^3-12*x^2-12*x+18)*exp(x)^2+(-2*x^3*exp(6)-2*x^4-8*x^ 3)*exp(x)-2*x^4-x^3)/((exp(6)^2+(2*x+6)*exp(6)+x^2+6*x+9)*exp(x)^2+(2*x^2* exp(6)+2*x^3+6*x^2)*exp(x)+x^4+x^3)/log(((exp(6)^2+(2*x+6)*exp(6)+x^2+6*x+ 9)*exp(x)^2+(2*x^2*exp(6)+2*x^3+6*x^2)*exp(x)+x^4+x^3)/x^2)^2,x, algorithm ="fricas")
Output:
x/log((x^4 + x^3 + (x^2 + 2*(x + 3)*e^6 + 6*x + e^12 + 9)*e^(2*x) + 2*(x^3 + x^2*e^6 + 3*x^2)*e^x)/x^2)
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (19) = 38\).
Time = 0.69 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.50 \[ \int \frac {-x^3-2 x^4+e^x \left (-8 x^3-2 e^6 x^3-2 x^4\right )+e^{2 x} \left (18+e^{12} (2-2 x)-12 x-12 x^2-2 x^3+e^6 \left (12-10 x-4 x^2\right )\right )+\left (x^3+x^4+e^x \left (6 x^2+2 e^6 x^2+2 x^3\right )+e^{2 x} \left (9+e^{12}+6 x+x^2+e^6 (6+2 x)\right )\right ) \log \left (\frac {x^3+x^4+e^x \left (6 x^2+2 e^6 x^2+2 x^3\right )+e^{2 x} \left (9+e^{12}+6 x+x^2+e^6 (6+2 x)\right )}{x^2}\right )}{\left (x^3+x^4+e^x \left (6 x^2+2 e^6 x^2+2 x^3\right )+e^{2 x} \left (9+e^{12}+6 x+x^2+e^6 (6+2 x)\right )\right ) \log ^2\left (\frac {x^3+x^4+e^x \left (6 x^2+2 e^6 x^2+2 x^3\right )+e^{2 x} \left (9+e^{12}+6 x+x^2+e^6 (6+2 x)\right )}{x^2}\right )} \, dx=\frac {x}{\log {\left (\frac {x^{4} + x^{3} + \left (2 x^{3} + 6 x^{2} + 2 x^{2} e^{6}\right ) e^{x} + \left (x^{2} + 6 x + \left (2 x + 6\right ) e^{6} + 9 + e^{12}\right ) e^{2 x}}{x^{2}} \right )}} \] Input:
integrate((((exp(6)**2+(2*x+6)*exp(6)+x**2+6*x+9)*exp(x)**2+(2*x**2*exp(6) +2*x**3+6*x**2)*exp(x)+x**4+x**3)*ln(((exp(6)**2+(2*x+6)*exp(6)+x**2+6*x+9 )*exp(x)**2+(2*x**2*exp(6)+2*x**3+6*x**2)*exp(x)+x**4+x**3)/x**2)+((2-2*x) *exp(6)**2+(-4*x**2-10*x+12)*exp(6)-2*x**3-12*x**2-12*x+18)*exp(x)**2+(-2* x**3*exp(6)-2*x**4-8*x**3)*exp(x)-2*x**4-x**3)/((exp(6)**2+(2*x+6)*exp(6)+ x**2+6*x+9)*exp(x)**2+(2*x**2*exp(6)+2*x**3+6*x**2)*exp(x)+x**4+x**3)/ln(( (exp(6)**2+(2*x+6)*exp(6)+x**2+6*x+9)*exp(x)**2+(2*x**2*exp(6)+2*x**3+6*x* *2)*exp(x)+x**4+x**3)/x**2)**2,x)
Output:
x/log((x**4 + x**3 + (2*x**3 + 6*x**2 + 2*x**2*exp(6))*exp(x) + (x**2 + 6* x + (2*x + 6)*exp(6) + 9 + exp(12))*exp(2*x))/x**2)
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (22) = 44\).
Time = 4.39 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.33 \[ \int \frac {-x^3-2 x^4+e^x \left (-8 x^3-2 e^6 x^3-2 x^4\right )+e^{2 x} \left (18+e^{12} (2-2 x)-12 x-12 x^2-2 x^3+e^6 \left (12-10 x-4 x^2\right )\right )+\left (x^3+x^4+e^x \left (6 x^2+2 e^6 x^2+2 x^3\right )+e^{2 x} \left (9+e^{12}+6 x+x^2+e^6 (6+2 x)\right )\right ) \log \left (\frac {x^3+x^4+e^x \left (6 x^2+2 e^6 x^2+2 x^3\right )+e^{2 x} \left (9+e^{12}+6 x+x^2+e^6 (6+2 x)\right )}{x^2}\right )}{\left (x^3+x^4+e^x \left (6 x^2+2 e^6 x^2+2 x^3\right )+e^{2 x} \left (9+e^{12}+6 x+x^2+e^6 (6+2 x)\right )\right ) \log ^2\left (\frac {x^3+x^4+e^x \left (6 x^2+2 e^6 x^2+2 x^3\right )+e^{2 x} \left (9+e^{12}+6 x+x^2+e^6 (6+2 x)\right )}{x^2}\right )} \, dx=\frac {x}{\log \left (x^{4} + x^{3} + {\left (x^{2} + 2 \, x {\left (e^{6} + 3\right )} + e^{12} + 6 \, e^{6} + 9\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x^{3} + x^{2} {\left (e^{6} + 3\right )}\right )} e^{x}\right ) - 2 \, \log \left (x\right )} \] Input:
integrate((((exp(6)^2+(2*x+6)*exp(6)+x^2+6*x+9)*exp(x)^2+(2*x^2*exp(6)+2*x ^3+6*x^2)*exp(x)+x^4+x^3)*log(((exp(6)^2+(2*x+6)*exp(6)+x^2+6*x+9)*exp(x)^ 2+(2*x^2*exp(6)+2*x^3+6*x^2)*exp(x)+x^4+x^3)/x^2)+((2-2*x)*exp(6)^2+(-4*x^ 2-10*x+12)*exp(6)-2*x^3-12*x^2-12*x+18)*exp(x)^2+(-2*x^3*exp(6)-2*x^4-8*x^ 3)*exp(x)-2*x^4-x^3)/((exp(6)^2+(2*x+6)*exp(6)+x^2+6*x+9)*exp(x)^2+(2*x^2* exp(6)+2*x^3+6*x^2)*exp(x)+x^4+x^3)/log(((exp(6)^2+(2*x+6)*exp(6)+x^2+6*x+ 9)*exp(x)^2+(2*x^2*exp(6)+2*x^3+6*x^2)*exp(x)+x^4+x^3)/x^2)^2,x, algorithm ="maxima")
Output:
x/(log(x^4 + x^3 + (x^2 + 2*x*(e^6 + 3) + e^12 + 6*e^6 + 9)*e^(2*x) + 2*(x ^3 + x^2*(e^6 + 3))*e^x) - 2*log(x))
Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (22) = 44\).
Time = 70.11 (sec) , antiderivative size = 86, normalized size of antiderivative = 3.58 \[ \int \frac {-x^3-2 x^4+e^x \left (-8 x^3-2 e^6 x^3-2 x^4\right )+e^{2 x} \left (18+e^{12} (2-2 x)-12 x-12 x^2-2 x^3+e^6 \left (12-10 x-4 x^2\right )\right )+\left (x^3+x^4+e^x \left (6 x^2+2 e^6 x^2+2 x^3\right )+e^{2 x} \left (9+e^{12}+6 x+x^2+e^6 (6+2 x)\right )\right ) \log \left (\frac {x^3+x^4+e^x \left (6 x^2+2 e^6 x^2+2 x^3\right )+e^{2 x} \left (9+e^{12}+6 x+x^2+e^6 (6+2 x)\right )}{x^2}\right )}{\left (x^3+x^4+e^x \left (6 x^2+2 e^6 x^2+2 x^3\right )+e^{2 x} \left (9+e^{12}+6 x+x^2+e^6 (6+2 x)\right )\right ) \log ^2\left (\frac {x^3+x^4+e^x \left (6 x^2+2 e^6 x^2+2 x^3\right )+e^{2 x} \left (9+e^{12}+6 x+x^2+e^6 (6+2 x)\right )}{x^2}\right )} \, dx=\frac {x}{\log \left (x^{4} + 2 \, x^{3} e^{x} + x^{3} + x^{2} e^{\left (2 \, x\right )} + 2 \, x^{2} e^{\left (x + 6\right )} + 6 \, x^{2} e^{x} + 6 \, x e^{\left (2 \, x\right )} + 2 \, x e^{\left (2 \, x + 6\right )} + 9 \, e^{\left (2 \, x\right )} + e^{\left (2 \, x + 12\right )} + 6 \, e^{\left (2 \, x + 6\right )}\right ) - \log \left (x^{2}\right )} \] Input:
integrate((((exp(6)^2+(2*x+6)*exp(6)+x^2+6*x+9)*exp(x)^2+(2*x^2*exp(6)+2*x ^3+6*x^2)*exp(x)+x^4+x^3)*log(((exp(6)^2+(2*x+6)*exp(6)+x^2+6*x+9)*exp(x)^ 2+(2*x^2*exp(6)+2*x^3+6*x^2)*exp(x)+x^4+x^3)/x^2)+((2-2*x)*exp(6)^2+(-4*x^ 2-10*x+12)*exp(6)-2*x^3-12*x^2-12*x+18)*exp(x)^2+(-2*x^3*exp(6)-2*x^4-8*x^ 3)*exp(x)-2*x^4-x^3)/((exp(6)^2+(2*x+6)*exp(6)+x^2+6*x+9)*exp(x)^2+(2*x^2* exp(6)+2*x^3+6*x^2)*exp(x)+x^4+x^3)/log(((exp(6)^2+(2*x+6)*exp(6)+x^2+6*x+ 9)*exp(x)^2+(2*x^2*exp(6)+2*x^3+6*x^2)*exp(x)+x^4+x^3)/x^2)^2,x, algorithm ="giac")
Output:
x/(log(x^4 + 2*x^3*e^x + x^3 + x^2*e^(2*x) + 2*x^2*e^(x + 6) + 6*x^2*e^x + 6*x*e^(2*x) + 2*x*e^(2*x + 6) + 9*e^(2*x) + e^(2*x + 12) + 6*e^(2*x + 6)) - log(x^2))
Timed out. \[ \int \frac {-x^3-2 x^4+e^x \left (-8 x^3-2 e^6 x^3-2 x^4\right )+e^{2 x} \left (18+e^{12} (2-2 x)-12 x-12 x^2-2 x^3+e^6 \left (12-10 x-4 x^2\right )\right )+\left (x^3+x^4+e^x \left (6 x^2+2 e^6 x^2+2 x^3\right )+e^{2 x} \left (9+e^{12}+6 x+x^2+e^6 (6+2 x)\right )\right ) \log \left (\frac {x^3+x^4+e^x \left (6 x^2+2 e^6 x^2+2 x^3\right )+e^{2 x} \left (9+e^{12}+6 x+x^2+e^6 (6+2 x)\right )}{x^2}\right )}{\left (x^3+x^4+e^x \left (6 x^2+2 e^6 x^2+2 x^3\right )+e^{2 x} \left (9+e^{12}+6 x+x^2+e^6 (6+2 x)\right )\right ) \log ^2\left (\frac {x^3+x^4+e^x \left (6 x^2+2 e^6 x^2+2 x^3\right )+e^{2 x} \left (9+e^{12}+6 x+x^2+e^6 (6+2 x)\right )}{x^2}\right )} \, dx=-\int \frac {{\mathrm {e}}^{2\,x}\,\left (12\,x+{\mathrm {e}}^6\,\left (4\,x^2+10\,x-12\right )+12\,x^2+2\,x^3+{\mathrm {e}}^{12}\,\left (2\,x-2\right )-18\right )-\ln \left (\frac {{\mathrm {e}}^{2\,x}\,\left (6\,x+{\mathrm {e}}^{12}+x^2+{\mathrm {e}}^6\,\left (2\,x+6\right )+9\right )+{\mathrm {e}}^x\,\left (2\,x^2\,{\mathrm {e}}^6+6\,x^2+2\,x^3\right )+x^3+x^4}{x^2}\right )\,\left ({\mathrm {e}}^{2\,x}\,\left (6\,x+{\mathrm {e}}^{12}+x^2+{\mathrm {e}}^6\,\left (2\,x+6\right )+9\right )+{\mathrm {e}}^x\,\left (2\,x^2\,{\mathrm {e}}^6+6\,x^2+2\,x^3\right )+x^3+x^4\right )+{\mathrm {e}}^x\,\left (2\,x^3\,{\mathrm {e}}^6+8\,x^3+2\,x^4\right )+x^3+2\,x^4}{{\ln \left (\frac {{\mathrm {e}}^{2\,x}\,\left (6\,x+{\mathrm {e}}^{12}+x^2+{\mathrm {e}}^6\,\left (2\,x+6\right )+9\right )+{\mathrm {e}}^x\,\left (2\,x^2\,{\mathrm {e}}^6+6\,x^2+2\,x^3\right )+x^3+x^4}{x^2}\right )}^2\,\left ({\mathrm {e}}^{2\,x}\,\left (6\,x+{\mathrm {e}}^{12}+x^2+{\mathrm {e}}^6\,\left (2\,x+6\right )+9\right )+{\mathrm {e}}^x\,\left (2\,x^2\,{\mathrm {e}}^6+6\,x^2+2\,x^3\right )+x^3+x^4\right )} \,d x \] Input:
int(-(exp(2*x)*(12*x + exp(6)*(10*x + 4*x^2 - 12) + 12*x^2 + 2*x^3 + exp(1 2)*(2*x - 2) - 18) - log((exp(2*x)*(6*x + exp(12) + x^2 + exp(6)*(2*x + 6) + 9) + exp(x)*(2*x^2*exp(6) + 6*x^2 + 2*x^3) + x^3 + x^4)/x^2)*(exp(2*x)* (6*x + exp(12) + x^2 + exp(6)*(2*x + 6) + 9) + exp(x)*(2*x^2*exp(6) + 6*x^ 2 + 2*x^3) + x^3 + x^4) + exp(x)*(2*x^3*exp(6) + 8*x^3 + 2*x^4) + x^3 + 2* x^4)/(log((exp(2*x)*(6*x + exp(12) + x^2 + exp(6)*(2*x + 6) + 9) + exp(x)* (2*x^2*exp(6) + 6*x^2 + 2*x^3) + x^3 + x^4)/x^2)^2*(exp(2*x)*(6*x + exp(12 ) + x^2 + exp(6)*(2*x + 6) + 9) + exp(x)*(2*x^2*exp(6) + 6*x^2 + 2*x^3) + x^3 + x^4)),x)
Output:
-int((exp(2*x)*(12*x + exp(6)*(10*x + 4*x^2 - 12) + 12*x^2 + 2*x^3 + exp(1 2)*(2*x - 2) - 18) - log((exp(2*x)*(6*x + exp(12) + x^2 + exp(6)*(2*x + 6) + 9) + exp(x)*(2*x^2*exp(6) + 6*x^2 + 2*x^3) + x^3 + x^4)/x^2)*(exp(2*x)* (6*x + exp(12) + x^2 + exp(6)*(2*x + 6) + 9) + exp(x)*(2*x^2*exp(6) + 6*x^ 2 + 2*x^3) + x^3 + x^4) + exp(x)*(2*x^3*exp(6) + 8*x^3 + 2*x^4) + x^3 + 2* x^4)/(log((exp(2*x)*(6*x + exp(12) + x^2 + exp(6)*(2*x + 6) + 9) + exp(x)* (2*x^2*exp(6) + 6*x^2 + 2*x^3) + x^3 + x^4)/x^2)^2*(exp(2*x)*(6*x + exp(12 ) + x^2 + exp(6)*(2*x + 6) + 9) + exp(x)*(2*x^2*exp(6) + 6*x^2 + 2*x^3) + x^3 + x^4)), x)
Time = 0.18 (sec) , antiderivative size = 97, normalized size of antiderivative = 4.04 \[ \int \frac {-x^3-2 x^4+e^x \left (-8 x^3-2 e^6 x^3-2 x^4\right )+e^{2 x} \left (18+e^{12} (2-2 x)-12 x-12 x^2-2 x^3+e^6 \left (12-10 x-4 x^2\right )\right )+\left (x^3+x^4+e^x \left (6 x^2+2 e^6 x^2+2 x^3\right )+e^{2 x} \left (9+e^{12}+6 x+x^2+e^6 (6+2 x)\right )\right ) \log \left (\frac {x^3+x^4+e^x \left (6 x^2+2 e^6 x^2+2 x^3\right )+e^{2 x} \left (9+e^{12}+6 x+x^2+e^6 (6+2 x)\right )}{x^2}\right )}{\left (x^3+x^4+e^x \left (6 x^2+2 e^6 x^2+2 x^3\right )+e^{2 x} \left (9+e^{12}+6 x+x^2+e^6 (6+2 x)\right )\right ) \log ^2\left (\frac {x^3+x^4+e^x \left (6 x^2+2 e^6 x^2+2 x^3\right )+e^{2 x} \left (9+e^{12}+6 x+x^2+e^6 (6+2 x)\right )}{x^2}\right )} \, dx=\frac {x}{\mathrm {log}\left (\frac {e^{2 x} e^{12}+2 e^{2 x} e^{6} x +6 e^{2 x} e^{6}+e^{2 x} x^{2}+6 e^{2 x} x +9 e^{2 x}+2 e^{x} e^{6} x^{2}+2 e^{x} x^{3}+6 e^{x} x^{2}+x^{4}+x^{3}}{x^{2}}\right )} \] Input:
int((((exp(6)^2+(2*x+6)*exp(6)+x^2+6*x+9)*exp(x)^2+(2*x^2*exp(6)+2*x^3+6*x ^2)*exp(x)+x^4+x^3)*log(((exp(6)^2+(2*x+6)*exp(6)+x^2+6*x+9)*exp(x)^2+(2*x ^2*exp(6)+2*x^3+6*x^2)*exp(x)+x^4+x^3)/x^2)+((2-2*x)*exp(6)^2+(-4*x^2-10*x +12)*exp(6)-2*x^3-12*x^2-12*x+18)*exp(x)^2+(-2*x^3*exp(6)-2*x^4-8*x^3)*exp (x)-2*x^4-x^3)/((exp(6)^2+(2*x+6)*exp(6)+x^2+6*x+9)*exp(x)^2+(2*x^2*exp(6) +2*x^3+6*x^2)*exp(x)+x^4+x^3)/log(((exp(6)^2+(2*x+6)*exp(6)+x^2+6*x+9)*exp (x)^2+(2*x^2*exp(6)+2*x^3+6*x^2)*exp(x)+x^4+x^3)/x^2)^2,x)
Output:
x/log((e**(2*x)*e**12 + 2*e**(2*x)*e**6*x + 6*e**(2*x)*e**6 + e**(2*x)*x** 2 + 6*e**(2*x)*x + 9*e**(2*x) + 2*e**x*e**6*x**2 + 2*e**x*x**3 + 6*e**x*x* *2 + x**4 + x**3)/x**2)