Integrand size = 367, antiderivative size = 25 \[ \int \frac {196+168 x+400 x^2+268 x^3+329 x^4+152 x^5+120 x^6+32 x^7+16 x^8+\left (336 x+256 x^2+584 x^3+296 x^4+336 x^5+96 x^6+64 x^7\right ) \log (x)+\left (256 x^2+144 x^3+312 x^4+96 x^5+96 x^6\right ) \log ^2(x)+\left (96 x^3+32 x^4+64 x^5\right ) \log ^3(x)+16 x^4 \log ^4(x)+e^{\frac {5}{14+6 x+13 x^2+4 x^3+4 x^4+\left (12 x+4 x^2+8 x^3\right ) \log (x)+4 x^2 \log ^2(x)}} \left (-90-150 x-100 x^2-80 x^3+\left (-60-80 x-120 x^2\right ) \log (x)-40 x \log ^2(x)\right )}{196+168 x+400 x^2+268 x^3+329 x^4+152 x^5+120 x^6+32 x^7+16 x^8+\left (336 x+256 x^2+584 x^3+296 x^4+336 x^5+96 x^6+64 x^7\right ) \log (x)+\left (256 x^2+144 x^3+312 x^4+96 x^5+96 x^6\right ) \log ^2(x)+\left (96 x^3+32 x^4+64 x^5\right ) \log ^3(x)+16 x^4 \log ^4(x)} \, dx=e^{\frac {5}{5+(3-x+2 x (1+x+\log (x)))^2}}+x \]
Time = 0.40 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.00 \[ \int \frac {196+168 x+400 x^2+268 x^3+329 x^4+152 x^5+120 x^6+32 x^7+16 x^8+\left (336 x+256 x^2+584 x^3+296 x^4+336 x^5+96 x^6+64 x^7\right ) \log (x)+\left (256 x^2+144 x^3+312 x^4+96 x^5+96 x^6\right ) \log ^2(x)+\left (96 x^3+32 x^4+64 x^5\right ) \log ^3(x)+16 x^4 \log ^4(x)+e^{\frac {5}{14+6 x+13 x^2+4 x^3+4 x^4+\left (12 x+4 x^2+8 x^3\right ) \log (x)+4 x^2 \log ^2(x)}} \left (-90-150 x-100 x^2-80 x^3+\left (-60-80 x-120 x^2\right ) \log (x)-40 x \log ^2(x)\right )}{196+168 x+400 x^2+268 x^3+329 x^4+152 x^5+120 x^6+32 x^7+16 x^8+\left (336 x+256 x^2+584 x^3+296 x^4+336 x^5+96 x^6+64 x^7\right ) \log (x)+\left (256 x^2+144 x^3+312 x^4+96 x^5+96 x^6\right ) \log ^2(x)+\left (96 x^3+32 x^4+64 x^5\right ) \log ^3(x)+16 x^4 \log ^4(x)} \, dx=e^{\frac {5}{14+6 x+13 x^2+4 x^3+4 x^4+4 x \left (3+x+2 x^2\right ) \log (x)+4 x^2 \log ^2(x)}}+x \]
Integrate[(196 + 168*x + 400*x^2 + 268*x^3 + 329*x^4 + 152*x^5 + 120*x^6 + 32*x^7 + 16*x^8 + (336*x + 256*x^2 + 584*x^3 + 296*x^4 + 336*x^5 + 96*x^6 + 64*x^7)*Log[x] + (256*x^2 + 144*x^3 + 312*x^4 + 96*x^5 + 96*x^6)*Log[x] ^2 + (96*x^3 + 32*x^4 + 64*x^5)*Log[x]^3 + 16*x^4*Log[x]^4 + E^(5/(14 + 6* x + 13*x^2 + 4*x^3 + 4*x^4 + (12*x + 4*x^2 + 8*x^3)*Log[x] + 4*x^2*Log[x]^ 2))*(-90 - 150*x - 100*x^2 - 80*x^3 + (-60 - 80*x - 120*x^2)*Log[x] - 40*x *Log[x]^2))/(196 + 168*x + 400*x^2 + 268*x^3 + 329*x^4 + 152*x^5 + 120*x^6 + 32*x^7 + 16*x^8 + (336*x + 256*x^2 + 584*x^3 + 296*x^4 + 336*x^5 + 96*x ^6 + 64*x^7)*Log[x] + (256*x^2 + 144*x^3 + 312*x^4 + 96*x^5 + 96*x^6)*Log[ x]^2 + (96*x^3 + 32*x^4 + 64*x^5)*Log[x]^3 + 16*x^4*Log[x]^4),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.
| \(\displaystyle \int \frac {\left (-80 x^3-100 x^2+\left (-120 x^2-80 x-60\right ) \log (x)-150 x-40 x \log ^2(x)-90\right ) \exp \left (\frac {5}{4 x^4+4 x^3+13 x^2+4 x^2 \log ^2(x)+\left (8 x^3+4 x^2+12 x\right ) \log (x)+6 x+14}\right )+16 x^8+32 x^7+120 x^6+152 x^5+329 x^4+16 x^4 \log ^4(x)+268 x^3+400 x^2+\left (64 x^5+32 x^4+96 x^3\right ) \log ^3(x)+\left (96 x^6+96 x^5+312 x^4+144 x^3+256 x^2\right ) \log ^2(x)+\left (64 x^7+96 x^6+336 x^5+296 x^4+584 x^3+256 x^2+336 x\right ) \log (x)+168 x+196}{16 x^8+32 x^7+120 x^6+152 x^5+329 x^4+16 x^4 \log ^4(x)+268 x^3+400 x^2+\left (64 x^5+32 x^4+96 x^3\right ) \log ^3(x)+\left (96 x^6+96 x^5+312 x^4+144 x^3+256 x^2\right ) \log ^2(x)+\left (64 x^7+96 x^6+336 x^5+296 x^4+584 x^3+256 x^2+336 x\right ) \log (x)+168 x+196} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (-80 x^3-100 x^2+\left (-120 x^2-80 x-60\right ) \log (x)-150 x-40 x \log ^2(x)-90\right ) \exp \left (\frac {5}{4 x^4+4 x^3+13 x^2+4 x^2 \log ^2(x)+\left (8 x^3+4 x^2+12 x\right ) \log (x)+6 x+14}\right )+16 x^8+32 x^7+120 x^6+152 x^5+329 x^4+16 x^4 \log ^4(x)+268 x^3+400 x^2+\left (64 x^5+32 x^4+96 x^3\right ) \log ^3(x)+\left (96 x^6+96 x^5+312 x^4+144 x^3+256 x^2\right ) \log ^2(x)+\left (64 x^7+96 x^6+336 x^5+296 x^4+584 x^3+256 x^2+336 x\right ) \log (x)+168 x+196}{\left (4 x^4+4 x^3+8 x^3 \log (x)+13 x^2+4 x^2 \log ^2(x)+4 x^2 \log (x)+6 x+12 x \log (x)+14\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {16 x^8}{\left (4 x^4+8 \log (x) x^3+4 x^3+4 \log ^2(x) x^2+4 \log (x) x^2+13 x^2+12 \log (x) x+6 x+14\right )^2}+\frac {32 x^7}{\left (4 x^4+8 \log (x) x^3+4 x^3+4 \log ^2(x) x^2+4 \log (x) x^2+13 x^2+12 \log (x) x+6 x+14\right )^2}+\frac {120 x^6}{\left (4 x^4+8 \log (x) x^3+4 x^3+4 \log ^2(x) x^2+4 \log (x) x^2+13 x^2+12 \log (x) x+6 x+14\right )^2}+\frac {152 x^5}{\left (4 x^4+8 \log (x) x^3+4 x^3+4 \log ^2(x) x^2+4 \log (x) x^2+13 x^2+12 \log (x) x+6 x+14\right )^2}+\frac {16 \log ^4(x) x^4}{\left (4 x^4+8 \log (x) x^3+4 x^3+4 \log ^2(x) x^2+4 \log (x) x^2+13 x^2+12 \log (x) x+6 x+14\right )^2}+\frac {329 x^4}{\left (4 x^4+8 \log (x) x^3+4 x^3+4 \log ^2(x) x^2+4 \log (x) x^2+13 x^2+12 \log (x) x+6 x+14\right )^2}+\frac {32 \left (2 x^2+x+3\right ) \log ^3(x) x^3}{\left (4 x^4+8 \log (x) x^3+4 x^3+4 \log ^2(x) x^2+4 \log (x) x^2+13 x^2+12 \log (x) x+6 x+14\right )^2}+\frac {268 x^3}{\left (4 x^4+8 \log (x) x^3+4 x^3+4 \log ^2(x) x^2+4 \log (x) x^2+13 x^2+12 \log (x) x+6 x+14\right )^2}+\frac {8 \left (12 x^4+12 x^3+39 x^2+18 x+32\right ) \log ^2(x) x^2}{\left (4 x^4+8 \log (x) x^3+4 x^3+4 \log ^2(x) x^2+4 \log (x) x^2+13 x^2+12 \log (x) x+6 x+14\right )^2}+\frac {400 x^2}{\left (4 x^4+8 \log (x) x^3+4 x^3+4 \log ^2(x) x^2+4 \log (x) x^2+13 x^2+12 \log (x) x+6 x+14\right )^2}+\frac {8 \left (2 x^2+x+3\right ) \left (4 x^4+4 x^3+13 x^2+6 x+14\right ) \log (x) x}{\left (4 x^4+8 \log (x) x^3+4 x^3+4 \log ^2(x) x^2+4 \log (x) x^2+13 x^2+12 \log (x) x+6 x+14\right )^2}+\frac {168 x}{\left (4 x^4+8 \log (x) x^3+4 x^3+4 \log ^2(x) x^2+4 \log (x) x^2+13 x^2+12 \log (x) x+6 x+14\right )^2}-\frac {10 e^{\frac {5}{4 x^4+4 x^3+4 \log ^2(x) x^2+13 x^2+4 \left (2 x^2+x+3\right ) \log (x) x+6 x+14}} (4 x+2 \log (x)+3) \left (2 x^2+2 \log (x) x+x+3\right )}{\left (4 x^4+8 \log (x) x^3+4 x^3+4 \log ^2(x) x^2+4 \log (x) x^2+13 x^2+12 \log (x) x+6 x+14\right )^2}+\frac {196}{\left (4 x^4+8 \log (x) x^3+4 x^3+4 \log ^2(x) x^2+4 \log (x) x^2+13 x^2+12 \log (x) x+6 x+14\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {8 x \log ^2(x) \left (x \left (12 x^4+12 x^3+39 x^2+18 x+32\right )-5 \exp \left (\frac {5}{4 x^4+4 x^3+13 x^2+4 x^2 \log ^2(x)+4 \left (2 x^2+x+3\right ) x \log (x)+6 x+14}\right )\right )-10 \left (8 x^3+10 x^2+15 x+9\right ) \exp \left (\frac {5}{4 x^4+4 x^3+13 x^2+4 x^2 \log ^2(x)+4 \left (2 x^2+x+3\right ) x \log (x)+6 x+14}\right )+\log (x) \left (8 x \left (8 x^6+12 x^5+42 x^4+37 x^3+73 x^2+32 x+42\right )-20 \left (6 x^2+4 x+3\right ) \exp \left (\frac {5}{4 x^4+4 x^3+13 x^2+4 x^2 \log ^2(x)+4 \left (2 x^2+x+3\right ) x \log (x)+6 x+14}\right )\right )+16 x^4 \log ^4(x)+32 x^3 \left (2 x^2+x+3\right ) \log ^3(x)+\left (4 x^4+4 x^3+13 x^2+6 x+14\right )^2}{\left (4 x^4+4 x^3+13 x^2+4 x^2 \log ^2(x)+4 \left (2 x^2+x+3\right ) x \log (x)+6 x+14\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {10 \left (8 x^3+10 x^2+12 x^2 \log (x)+15 x+4 x \log ^2(x)+8 x \log (x)+6 \log (x)+9\right ) \exp \left (\frac {5}{4 x^4+4 x^3+13 x^2+4 x^2 \log ^2(x)+4 \left (2 x^2+x+3\right ) x \log (x)+6 x+14}\right )}{\left (4 x^4+4 x^3+8 x^3 \log (x)+13 x^2+4 x^2 \log ^2(x)+4 x^2 \log (x)+6 x+12 x \log (x)+14\right )^2}+\frac {8 x^2 \left (12 x^4+12 x^3+39 x^2+18 x+32\right ) \log ^2(x)}{\left (4 x^4+4 x^3+8 x^3 \log (x)+13 x^2+4 x^2 \log ^2(x)+4 x^2 \log (x)+6 x+12 x \log (x)+14\right )^2}+\frac {8 x \left (2 x^2+x+3\right ) \left (4 x^4+4 x^3+13 x^2+6 x+14\right ) \log (x)}{\left (4 x^4+4 x^3+8 x^3 \log (x)+13 x^2+4 x^2 \log ^2(x)+4 x^2 \log (x)+6 x+12 x \log (x)+14\right )^2}+\frac {\left (4 x^4+4 x^3+13 x^2+6 x+14\right )^2}{\left (4 x^4+4 x^3+8 x^3 \log (x)+13 x^2+4 x^2 \log ^2(x)+4 x^2 \log (x)+6 x+12 x \log (x)+14\right )^2}+\frac {16 x^4 \log ^4(x)}{\left (4 x^4+4 x^3+8 x^3 \log (x)+13 x^2+4 x^2 \log ^2(x)+4 x^2 \log (x)+6 x+12 x \log (x)+14\right )^2}+\frac {32 x^3 \left (2 x^2+x+3\right ) \log ^3(x)}{\left (4 x^4+4 x^3+8 x^3 \log (x)+13 x^2+4 x^2 \log ^2(x)+4 x^2 \log (x)+6 x+12 x \log (x)+14\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (-\frac {10 \left (8 x^3+10 x^2+12 x^2 \log (x)+15 x+4 x \log ^2(x)+8 x \log (x)+6 \log (x)+9\right ) \exp \left (\frac {5}{4 x^4+4 x^3+13 x^2+4 x^2 \log ^2(x)+4 \left (2 x^2+x+3\right ) x \log (x)+6 x+14}\right )}{\left (4 x^4+4 x^3+8 x^3 \log (x)+13 x^2+4 x^2 \log ^2(x)+4 x^2 \log (x)+6 x+12 x \log (x)+14\right )^2}+\frac {8 x^2 \left (12 x^4+12 x^3+39 x^2+18 x+32\right ) \log ^2(x)}{\left (4 x^4+4 x^3+8 x^3 \log (x)+13 x^2+4 x^2 \log ^2(x)+4 x^2 \log (x)+6 x+12 x \log (x)+14\right )^2}+\frac {8 x \left (2 x^2+x+3\right ) \left (4 x^4+4 x^3+13 x^2+6 x+14\right ) \log (x)}{\left (4 x^4+4 x^3+8 x^3 \log (x)+13 x^2+4 x^2 \log ^2(x)+4 x^2 \log (x)+6 x+12 x \log (x)+14\right )^2}+\frac {\left (4 x^4+4 x^3+13 x^2+6 x+14\right )^2}{\left (4 x^4+4 x^3+8 x^3 \log (x)+13 x^2+4 x^2 \log ^2(x)+4 x^2 \log (x)+6 x+12 x \log (x)+14\right )^2}+\frac {16 x^4 \log ^4(x)}{\left (4 x^4+4 x^3+8 x^3 \log (x)+13 x^2+4 x^2 \log ^2(x)+4 x^2 \log (x)+6 x+12 x \log (x)+14\right )^2}+\frac {32 x^3 \left (2 x^2+x+3\right ) \log ^3(x)}{\left (4 x^4+4 x^3+8 x^3 \log (x)+13 x^2+4 x^2 \log ^2(x)+4 x^2 \log (x)+6 x+12 x \log (x)+14\right )^2}\right )dx\) |
Int[(196 + 168*x + 400*x^2 + 268*x^3 + 329*x^4 + 152*x^5 + 120*x^6 + 32*x^ 7 + 16*x^8 + (336*x + 256*x^2 + 584*x^3 + 296*x^4 + 336*x^5 + 96*x^6 + 64* x^7)*Log[x] + (256*x^2 + 144*x^3 + 312*x^4 + 96*x^5 + 96*x^6)*Log[x]^2 + ( 96*x^3 + 32*x^4 + 64*x^5)*Log[x]^3 + 16*x^4*Log[x]^4 + E^(5/(14 + 6*x + 13 *x^2 + 4*x^3 + 4*x^4 + (12*x + 4*x^2 + 8*x^3)*Log[x] + 4*x^2*Log[x]^2))*(- 90 - 150*x - 100*x^2 - 80*x^3 + (-60 - 80*x - 120*x^2)*Log[x] - 40*x*Log[x ]^2))/(196 + 168*x + 400*x^2 + 268*x^3 + 329*x^4 + 152*x^5 + 120*x^6 + 32* x^7 + 16*x^8 + (336*x + 256*x^2 + 584*x^3 + 296*x^4 + 336*x^5 + 96*x^6 + 6 4*x^7)*Log[x] + (256*x^2 + 144*x^3 + 312*x^4 + 96*x^5 + 96*x^6)*Log[x]^2 + (96*x^3 + 32*x^4 + 64*x^5)*Log[x]^3 + 16*x^4*Log[x]^4),x]
3.1.61.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs. \(2(24)=48\).
Time = 41.21 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.24
| method | result | size |
| risch | \(x +{\mathrm e}^{\frac {5}{4 x^{2} \ln \left (x \right )^{2}+8 x^{3} \ln \left (x \right )+4 x^{4}+4 x^{2} \ln \left (x \right )+4 x^{3}+12 x \ln \left (x \right )+13 x^{2}+6 x +14}}\) | \(56\) |
| parallelrisch | \(x +{\mathrm e}^{\frac {5}{4 x^{2} \ln \left (x \right )^{2}+8 x^{3} \ln \left (x \right )+4 x^{4}+4 x^{2} \ln \left (x \right )+4 x^{3}+12 x \ln \left (x \right )+13 x^{2}+6 x +14}}-\frac {2707}{40}\) | \(57\) |
int(((-40*x*ln(x)^2+(-120*x^2-80*x-60)*ln(x)-80*x^3-100*x^2-150*x-90)*exp( 5/(4*x^2*ln(x)^2+(8*x^3+4*x^2+12*x)*ln(x)+4*x^4+4*x^3+13*x^2+6*x+14))+16*x ^4*ln(x)^4+(64*x^5+32*x^4+96*x^3)*ln(x)^3+(96*x^6+96*x^5+312*x^4+144*x^3+2 56*x^2)*ln(x)^2+(64*x^7+96*x^6+336*x^5+296*x^4+584*x^3+256*x^2+336*x)*ln(x )+16*x^8+32*x^7+120*x^6+152*x^5+329*x^4+268*x^3+400*x^2+168*x+196)/(16*x^4 *ln(x)^4+(64*x^5+32*x^4+96*x^3)*ln(x)^3+(96*x^6+96*x^5+312*x^4+144*x^3+256 *x^2)*ln(x)^2+(64*x^7+96*x^6+336*x^5+296*x^4+584*x^3+256*x^2+336*x)*ln(x)+ 16*x^8+32*x^7+120*x^6+152*x^5+329*x^4+268*x^3+400*x^2+168*x+196),x,method= _RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (24) = 48\).
Time = 0.27 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.08 \[ \int \frac {196+168 x+400 x^2+268 x^3+329 x^4+152 x^5+120 x^6+32 x^7+16 x^8+\left (336 x+256 x^2+584 x^3+296 x^4+336 x^5+96 x^6+64 x^7\right ) \log (x)+\left (256 x^2+144 x^3+312 x^4+96 x^5+96 x^6\right ) \log ^2(x)+\left (96 x^3+32 x^4+64 x^5\right ) \log ^3(x)+16 x^4 \log ^4(x)+e^{\frac {5}{14+6 x+13 x^2+4 x^3+4 x^4+\left (12 x+4 x^2+8 x^3\right ) \log (x)+4 x^2 \log ^2(x)}} \left (-90-150 x-100 x^2-80 x^3+\left (-60-80 x-120 x^2\right ) \log (x)-40 x \log ^2(x)\right )}{196+168 x+400 x^2+268 x^3+329 x^4+152 x^5+120 x^6+32 x^7+16 x^8+\left (336 x+256 x^2+584 x^3+296 x^4+336 x^5+96 x^6+64 x^7\right ) \log (x)+\left (256 x^2+144 x^3+312 x^4+96 x^5+96 x^6\right ) \log ^2(x)+\left (96 x^3+32 x^4+64 x^5\right ) \log ^3(x)+16 x^4 \log ^4(x)} \, dx=x + e^{\left (\frac {5}{4 \, x^{4} + 4 \, x^{2} \log \left (x\right )^{2} + 4 \, x^{3} + 13 \, x^{2} + 4 \, {\left (2 \, x^{3} + x^{2} + 3 \, x\right )} \log \left (x\right ) + 6 \, x + 14}\right )} \]
integrate(((-40*x*log(x)^2+(-120*x^2-80*x-60)*log(x)-80*x^3-100*x^2-150*x- 90)*exp(5/(4*x^2*log(x)^2+(8*x^3+4*x^2+12*x)*log(x)+4*x^4+4*x^3+13*x^2+6*x +14))+16*x^4*log(x)^4+(64*x^5+32*x^4+96*x^3)*log(x)^3+(96*x^6+96*x^5+312*x ^4+144*x^3+256*x^2)*log(x)^2+(64*x^7+96*x^6+336*x^5+296*x^4+584*x^3+256*x^ 2+336*x)*log(x)+16*x^8+32*x^7+120*x^6+152*x^5+329*x^4+268*x^3+400*x^2+168* x+196)/(16*x^4*log(x)^4+(64*x^5+32*x^4+96*x^3)*log(x)^3+(96*x^6+96*x^5+312 *x^4+144*x^3+256*x^2)*log(x)^2+(64*x^7+96*x^6+336*x^5+296*x^4+584*x^3+256* x^2+336*x)*log(x)+16*x^8+32*x^7+120*x^6+152*x^5+329*x^4+268*x^3+400*x^2+16 8*x+196),x, algorithm=\
x + e^(5/(4*x^4 + 4*x^2*log(x)^2 + 4*x^3 + 13*x^2 + 4*(2*x^3 + x^2 + 3*x)* log(x) + 6*x + 14))
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (22) = 44\).
Time = 0.66 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.04 \[ \int \frac {196+168 x+400 x^2+268 x^3+329 x^4+152 x^5+120 x^6+32 x^7+16 x^8+\left (336 x+256 x^2+584 x^3+296 x^4+336 x^5+96 x^6+64 x^7\right ) \log (x)+\left (256 x^2+144 x^3+312 x^4+96 x^5+96 x^6\right ) \log ^2(x)+\left (96 x^3+32 x^4+64 x^5\right ) \log ^3(x)+16 x^4 \log ^4(x)+e^{\frac {5}{14+6 x+13 x^2+4 x^3+4 x^4+\left (12 x+4 x^2+8 x^3\right ) \log (x)+4 x^2 \log ^2(x)}} \left (-90-150 x-100 x^2-80 x^3+\left (-60-80 x-120 x^2\right ) \log (x)-40 x \log ^2(x)\right )}{196+168 x+400 x^2+268 x^3+329 x^4+152 x^5+120 x^6+32 x^7+16 x^8+\left (336 x+256 x^2+584 x^3+296 x^4+336 x^5+96 x^6+64 x^7\right ) \log (x)+\left (256 x^2+144 x^3+312 x^4+96 x^5+96 x^6\right ) \log ^2(x)+\left (96 x^3+32 x^4+64 x^5\right ) \log ^3(x)+16 x^4 \log ^4(x)} \, dx=x + e^{\frac {5}{4 x^{4} + 4 x^{3} + 4 x^{2} \log {\left (x \right )}^{2} + 13 x^{2} + 6 x + \left (8 x^{3} + 4 x^{2} + 12 x\right ) \log {\left (x \right )} + 14}} \]
integrate(((-40*x*ln(x)**2+(-120*x**2-80*x-60)*ln(x)-80*x**3-100*x**2-150* x-90)*exp(5/(4*x**2*ln(x)**2+(8*x**3+4*x**2+12*x)*ln(x)+4*x**4+4*x**3+13*x **2+6*x+14))+16*x**4*ln(x)**4+(64*x**5+32*x**4+96*x**3)*ln(x)**3+(96*x**6+ 96*x**5+312*x**4+144*x**3+256*x**2)*ln(x)**2+(64*x**7+96*x**6+336*x**5+296 *x**4+584*x**3+256*x**2+336*x)*ln(x)+16*x**8+32*x**7+120*x**6+152*x**5+329 *x**4+268*x**3+400*x**2+168*x+196)/(16*x**4*ln(x)**4+(64*x**5+32*x**4+96*x **3)*ln(x)**3+(96*x**6+96*x**5+312*x**4+144*x**3+256*x**2)*ln(x)**2+(64*x* *7+96*x**6+336*x**5+296*x**4+584*x**3+256*x**2+336*x)*ln(x)+16*x**8+32*x** 7+120*x**6+152*x**5+329*x**4+268*x**3+400*x**2+168*x+196),x)
x + exp(5/(4*x**4 + 4*x**3 + 4*x**2*log(x)**2 + 13*x**2 + 6*x + (8*x**3 + 4*x**2 + 12*x)*log(x) + 14))
Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (24) = 48\).
Time = 0.26 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.08 \[ \int \frac {196+168 x+400 x^2+268 x^3+329 x^4+152 x^5+120 x^6+32 x^7+16 x^8+\left (336 x+256 x^2+584 x^3+296 x^4+336 x^5+96 x^6+64 x^7\right ) \log (x)+\left (256 x^2+144 x^3+312 x^4+96 x^5+96 x^6\right ) \log ^2(x)+\left (96 x^3+32 x^4+64 x^5\right ) \log ^3(x)+16 x^4 \log ^4(x)+e^{\frac {5}{14+6 x+13 x^2+4 x^3+4 x^4+\left (12 x+4 x^2+8 x^3\right ) \log (x)+4 x^2 \log ^2(x)}} \left (-90-150 x-100 x^2-80 x^3+\left (-60-80 x-120 x^2\right ) \log (x)-40 x \log ^2(x)\right )}{196+168 x+400 x^2+268 x^3+329 x^4+152 x^5+120 x^6+32 x^7+16 x^8+\left (336 x+256 x^2+584 x^3+296 x^4+336 x^5+96 x^6+64 x^7\right ) \log (x)+\left (256 x^2+144 x^3+312 x^4+96 x^5+96 x^6\right ) \log ^2(x)+\left (96 x^3+32 x^4+64 x^5\right ) \log ^3(x)+16 x^4 \log ^4(x)} \, dx=x + e^{\left (\frac {5}{4 \, x^{4} + 4 \, x^{2} \log \left (x\right )^{2} + 4 \, x^{3} + 13 \, x^{2} + 4 \, {\left (2 \, x^{3} + x^{2} + 3 \, x\right )} \log \left (x\right ) + 6 \, x + 14}\right )} \]
integrate(((-40*x*log(x)^2+(-120*x^2-80*x-60)*log(x)-80*x^3-100*x^2-150*x- 90)*exp(5/(4*x^2*log(x)^2+(8*x^3+4*x^2+12*x)*log(x)+4*x^4+4*x^3+13*x^2+6*x +14))+16*x^4*log(x)^4+(64*x^5+32*x^4+96*x^3)*log(x)^3+(96*x^6+96*x^5+312*x ^4+144*x^3+256*x^2)*log(x)^2+(64*x^7+96*x^6+336*x^5+296*x^4+584*x^3+256*x^ 2+336*x)*log(x)+16*x^8+32*x^7+120*x^6+152*x^5+329*x^4+268*x^3+400*x^2+168* x+196)/(16*x^4*log(x)^4+(64*x^5+32*x^4+96*x^3)*log(x)^3+(96*x^6+96*x^5+312 *x^4+144*x^3+256*x^2)*log(x)^2+(64*x^7+96*x^6+336*x^5+296*x^4+584*x^3+256* x^2+336*x)*log(x)+16*x^8+32*x^7+120*x^6+152*x^5+329*x^4+268*x^3+400*x^2+16 8*x+196),x, algorithm=\
x + e^(5/(4*x^4 + 4*x^2*log(x)^2 + 4*x^3 + 13*x^2 + 4*(2*x^3 + x^2 + 3*x)* log(x) + 6*x + 14))
Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (24) = 48\).
Time = 0.82 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.20 \[ \int \frac {196+168 x+400 x^2+268 x^3+329 x^4+152 x^5+120 x^6+32 x^7+16 x^8+\left (336 x+256 x^2+584 x^3+296 x^4+336 x^5+96 x^6+64 x^7\right ) \log (x)+\left (256 x^2+144 x^3+312 x^4+96 x^5+96 x^6\right ) \log ^2(x)+\left (96 x^3+32 x^4+64 x^5\right ) \log ^3(x)+16 x^4 \log ^4(x)+e^{\frac {5}{14+6 x+13 x^2+4 x^3+4 x^4+\left (12 x+4 x^2+8 x^3\right ) \log (x)+4 x^2 \log ^2(x)}} \left (-90-150 x-100 x^2-80 x^3+\left (-60-80 x-120 x^2\right ) \log (x)-40 x \log ^2(x)\right )}{196+168 x+400 x^2+268 x^3+329 x^4+152 x^5+120 x^6+32 x^7+16 x^8+\left (336 x+256 x^2+584 x^3+296 x^4+336 x^5+96 x^6+64 x^7\right ) \log (x)+\left (256 x^2+144 x^3+312 x^4+96 x^5+96 x^6\right ) \log ^2(x)+\left (96 x^3+32 x^4+64 x^5\right ) \log ^3(x)+16 x^4 \log ^4(x)} \, dx=x + e^{\left (\frac {5}{4 \, x^{4} + 8 \, x^{3} \log \left (x\right ) + 4 \, x^{2} \log \left (x\right )^{2} + 4 \, x^{3} + 4 \, x^{2} \log \left (x\right ) + 13 \, x^{2} + 12 \, x \log \left (x\right ) + 6 \, x + 14}\right )} \]
integrate(((-40*x*log(x)^2+(-120*x^2-80*x-60)*log(x)-80*x^3-100*x^2-150*x- 90)*exp(5/(4*x^2*log(x)^2+(8*x^3+4*x^2+12*x)*log(x)+4*x^4+4*x^3+13*x^2+6*x +14))+16*x^4*log(x)^4+(64*x^5+32*x^4+96*x^3)*log(x)^3+(96*x^6+96*x^5+312*x ^4+144*x^3+256*x^2)*log(x)^2+(64*x^7+96*x^6+336*x^5+296*x^4+584*x^3+256*x^ 2+336*x)*log(x)+16*x^8+32*x^7+120*x^6+152*x^5+329*x^4+268*x^3+400*x^2+168* x+196)/(16*x^4*log(x)^4+(64*x^5+32*x^4+96*x^3)*log(x)^3+(96*x^6+96*x^5+312 *x^4+144*x^3+256*x^2)*log(x)^2+(64*x^7+96*x^6+336*x^5+296*x^4+584*x^3+256* x^2+336*x)*log(x)+16*x^8+32*x^7+120*x^6+152*x^5+329*x^4+268*x^3+400*x^2+16 8*x+196),x, algorithm=\
x + e^(5/(4*x^4 + 8*x^3*log(x) + 4*x^2*log(x)^2 + 4*x^3 + 4*x^2*log(x) + 1 3*x^2 + 12*x*log(x) + 6*x + 14))
Time = 10.36 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.20 \[ \int \frac {196+168 x+400 x^2+268 x^3+329 x^4+152 x^5+120 x^6+32 x^7+16 x^8+\left (336 x+256 x^2+584 x^3+296 x^4+336 x^5+96 x^6+64 x^7\right ) \log (x)+\left (256 x^2+144 x^3+312 x^4+96 x^5+96 x^6\right ) \log ^2(x)+\left (96 x^3+32 x^4+64 x^5\right ) \log ^3(x)+16 x^4 \log ^4(x)+e^{\frac {5}{14+6 x+13 x^2+4 x^3+4 x^4+\left (12 x+4 x^2+8 x^3\right ) \log (x)+4 x^2 \log ^2(x)}} \left (-90-150 x-100 x^2-80 x^3+\left (-60-80 x-120 x^2\right ) \log (x)-40 x \log ^2(x)\right )}{196+168 x+400 x^2+268 x^3+329 x^4+152 x^5+120 x^6+32 x^7+16 x^8+\left (336 x+256 x^2+584 x^3+296 x^4+336 x^5+96 x^6+64 x^7\right ) \log (x)+\left (256 x^2+144 x^3+312 x^4+96 x^5+96 x^6\right ) \log ^2(x)+\left (96 x^3+32 x^4+64 x^5\right ) \log ^3(x)+16 x^4 \log ^4(x)} \, dx=x+{\mathrm {e}}^{\frac {5}{4\,x^4+8\,x^3\,\ln \left (x\right )+4\,x^3+4\,x^2\,{\ln \left (x\right )}^2+4\,x^2\,\ln \left (x\right )+13\,x^2+12\,x\,\ln \left (x\right )+6\,x+14}} \]
int((168*x + log(x)^2*(256*x^2 + 144*x^3 + 312*x^4 + 96*x^5 + 96*x^6) - ex p(5/(6*x + 4*x^2*log(x)^2 + 13*x^2 + 4*x^3 + 4*x^4 + log(x)*(12*x + 4*x^2 + 8*x^3) + 14))*(150*x + 40*x*log(x)^2 + log(x)*(80*x + 120*x^2 + 60) + 10 0*x^2 + 80*x^3 + 90) + 16*x^4*log(x)^4 + log(x)^3*(96*x^3 + 32*x^4 + 64*x^ 5) + log(x)*(336*x + 256*x^2 + 584*x^3 + 296*x^4 + 336*x^5 + 96*x^6 + 64*x ^7) + 400*x^2 + 268*x^3 + 329*x^4 + 152*x^5 + 120*x^6 + 32*x^7 + 16*x^8 + 196)/(168*x + log(x)^2*(256*x^2 + 144*x^3 + 312*x^4 + 96*x^5 + 96*x^6) + 1 6*x^4*log(x)^4 + log(x)^3*(96*x^3 + 32*x^4 + 64*x^5) + log(x)*(336*x + 256 *x^2 + 584*x^3 + 296*x^4 + 336*x^5 + 96*x^6 + 64*x^7) + 400*x^2 + 268*x^3 + 329*x^4 + 152*x^5 + 120*x^6 + 32*x^7 + 16*x^8 + 196),x)