Integrand size = 93, antiderivative size = 30 \[ \int \frac {\left (64+32 x+12 x^2+x^3\right ) \log (x) \log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )+\left (32+16 x+2 x^2\right ) \log ^2\left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )}{16 x+8 x^2+x^3} \, dx=2 \log (x) \log ^2\left (\frac {1}{3} e^{\frac {1}{4} (-2+x)-\frac {x}{4+x}} x\right ) \] Output:
2*ln(1/3*exp(1/4*x-1/2-x/(4+x))*x)^2*ln(x)
Leaf count is larger than twice the leaf count of optimal. \(81\) vs. \(2(30)=60\).
Time = 0.09 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.70 \[ \int \frac {\left (64+32 x+12 x^2+x^3\right ) \log (x) \log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )+\left (32+16 x+2 x^2\right ) \log ^2\left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )}{16 x+8 x^2+x^3} \, dx=\frac {18+\frac {9 x}{2}-\frac {9 x^2}{4}+9 (4+x) \log \left (\frac {1}{3} e^{\frac {1}{4} \left (-6+x+\frac {16}{4+x}\right )} x\right )+(4+x) \log (x) \left (-9+2 \log ^2\left (\frac {1}{3} e^{\frac {1}{4} \left (-6+x+\frac {16}{4+x}\right )} x\right )\right )}{4+x} \] Input:
Integrate[((64 + 32*x + 12*x^2 + x^3)*Log[x]*Log[(E^((-8 - 2*x + x^2)/(16 + 4*x))*x)/3] + (32 + 16*x + 2*x^2)*Log[(E^((-8 - 2*x + x^2)/(16 + 4*x))*x )/3]^2)/(16*x + 8*x^2 + x^3),x]
Output:
(18 + (9*x)/2 - (9*x^2)/4 + 9*(4 + x)*Log[(E^((-6 + x + 16/(4 + x))/4)*x)/ 3] + (4 + x)*Log[x]*(-9 + 2*Log[(E^((-6 + x + 16/(4 + x))/4)*x)/3]^2))/(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.
| \(\displaystyle \int \frac {\left (2 x^2+16 x+32\right ) \log ^2\left (\frac {1}{3} e^{\frac {x^2-2 x-8}{4 x+16}} x\right )+\left (x^3+12 x^2+32 x+64\right ) \log (x) \log \left (\frac {1}{3} e^{\frac {x^2-2 x-8}{4 x+16}} x\right )}{x^3+8 x^2+16 x} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {\left (2 x^2+16 x+32\right ) \log ^2\left (\frac {1}{3} e^{\frac {x^2-2 x-8}{4 x+16}} x\right )+\left (x^3+12 x^2+32 x+64\right ) \log (x) \log \left (\frac {1}{3} e^{\frac {x^2-2 x-8}{4 x+16}} x\right )}{x \left (x^2+8 x+16\right )}dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {\left (2 x^2+16 x+32\right ) \log ^2\left (\frac {1}{3} e^{\frac {x^2-2 x-8}{4 x+16}} x\right )+\left (x^3+12 x^2+32 x+64\right ) \log (x) \log \left (\frac {1}{3} e^{\frac {x^2-2 x-8}{4 x+16}} x\right )}{x (x+4)^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 \log ^2\left (\frac {1}{3} e^{\frac {x^2-2 x-8}{4 (x+4)}} x\right )}{x}+\frac {\left (x^3+12 x^2+32 x+64\right ) \log (x) \log \left (\frac {1}{3} e^{\frac {x^2-2 x-8}{4 (x+4)}} x\right )}{x (x+4)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \int \frac {\log ^2\left (\frac {1}{3} e^{\frac {x^2-2 x-8}{4 (x+4)}} x\right )}{x}dx-4 \int \frac {\log \left (\frac {1}{3} e^{\frac {x^2-2 x-8}{4 (x+4)}} x\right )}{x}dx+4 \int \frac {\log \left (\frac {1}{3} e^{\frac {x^2-2 x-8}{4 (x+4)}} x\right )}{x+4}dx+4 \int \frac {\log (x) \log \left (\frac {1}{3} e^{\frac {x^2-2 x-8}{4 (x+4)}} x\right )}{x}dx+4 \operatorname {PolyLog}\left (2,-\frac {x}{4}\right )+\frac {3 x^2}{16}-\frac {1}{8} x^2 \log (x)+x \log (x) \log \left (\frac {1}{3} e^{-\frac {-x^2+2 x+8}{4 (x+4)}} x\right )-x \log \left (\frac {1}{3} e^{-\frac {-x^2+2 x+8}{4 (x+4)}} x\right )+\frac {16 \log (x) \log \left (\frac {1}{3} e^{-\frac {-x^2+2 x+8}{4 (x+4)}} x\right )}{x+4}+2 x-\frac {8}{x+4}-2 \log ^2(x)-\frac {4 x \log (x)}{x+4}-x \log (x)+4 \log \left (\frac {x}{4}+1\right ) \log (x)-\frac {32 \log (x)}{(x+4)^2}+2 \log (x)-2 \log (x+4)\) |
Input:
Int[((64 + 32*x + 12*x^2 + x^3)*Log[x]*Log[(E^((-8 - 2*x + x^2)/(16 + 4*x) )*x)/3] + (32 + 16*x + 2*x^2)*Log[(E^((-8 - 2*x + x^2)/(16 + 4*x))*x)/3]^2 )/(16*x + 8*x^2 + x^3),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(511\) vs. \(2(24)=48\).
Time = 1.86 (sec) , antiderivative size = 512, normalized size of antiderivative = 17.07
| method | result | size |
| parallelrisch | \(-\frac {-6291456 \ln \left (\frac {x \,{\mathrm e}^{\frac {x^{2}-2 x -8}{4 x +16}}}{3}\right ) \ln \left (x \right )^{2}-2097152 \ln \left (\frac {x \,{\mathrm e}^{\frac {x^{2}-2 x -8}{4 x +16}}}{3}\right )^{3}+24576 x^{4} \ln \left (x \right )^{2}+6144 x^{5} \ln \left (x \right )+24576 x^{4} \ln \left (x \right )+196608 x^{3} \ln \left (x \right )^{2}+32768 x^{3} \ln \left (x \right )^{3}+393216 x^{2} \ln \left (x \right )^{2}+1572864 x \ln \left (x \right )^{3}+393216 x^{2} \ln \left (x \right )^{3}+2097152 \ln \left (x \right )^{3}+512 x^{6}+393216 \ln \left (\frac {x \,{\mathrm e}^{\frac {x^{2}-2 x -8}{4 x +16}}}{3}\right )^{2} x^{2}+196608 \ln \left (\frac {x \,{\mathrm e}^{\frac {x^{2}-2 x -8}{4 x +16}}}{3}\right )^{2} x^{3}-24576 \ln \left (\frac {x \,{\mathrm e}^{\frac {x^{2}-2 x -8}{4 x +16}}}{3}\right ) x^{4}-393216 \ln \left (x \right ) \ln \left (\frac {x \,{\mathrm e}^{\frac {x^{2}-2 x -8}{4 x +16}}}{3}\right ) x^{3}-49152 \ln \left (x \right ) \ln \left (\frac {x \,{\mathrm e}^{\frac {x^{2}-2 x -8}{4 x +16}}}{3}\right ) x^{4}-4718592 \ln \left (\frac {x \,{\mathrm e}^{\frac {x^{2}-2 x -8}{4 x +16}}}{3}\right ) \ln \left (x \right )^{2} x -1179648 \ln \left (\frac {x \,{\mathrm e}^{\frac {x^{2}-2 x -8}{4 x +16}}}{3}\right ) x^{2} \ln \left (x \right )^{2}-98304 \ln \left (\frac {x \,{\mathrm e}^{\frac {x^{2}-2 x -8}{4 x +16}}}{3}\right ) \ln \left (x \right )^{2} x^{3}+24576 \ln \left (\frac {x \,{\mathrm e}^{\frac {x^{2}-2 x -8}{4 x +16}}}{3}\right )^{2} x^{4}-6144 \ln \left (\frac {x \,{\mathrm e}^{\frac {x^{2}-2 x -8}{4 x +16}}}{3}\right ) x^{5}-1572864 \ln \left (\frac {x \,{\mathrm e}^{\frac {x^{2}-2 x -8}{4 x +16}}}{3}\right )^{3} x -393216 \ln \left (\frac {x \,{\mathrm e}^{\frac {x^{2}-2 x -8}{4 x +16}}}{3}\right )^{3} x^{2}-32768 \ln \left (\frac {x \,{\mathrm e}^{\frac {x^{2}-2 x -8}{4 x +16}}}{3}\right )^{3} x^{3}-786432 \ln \left (x \right ) \ln \left (\frac {x \,{\mathrm e}^{\frac {x^{2}-2 x -8}{4 x +16}}}{3}\right ) x^{2}}{49152 \left (4+x \right )^{3}}\) | \(512\) |
| risch | \(\text {Expression too large to display}\) | \(944\) |
Input:
int(((2*x^2+16*x+32)*ln(1/3*x*exp((x^2-2*x-8)/(4*x+16)))^2+(x^3+12*x^2+32* x+64)*ln(x)*ln(1/3*x*exp((x^2-2*x-8)/(4*x+16))))/(x^3+8*x^2+16*x),x,method =_RETURNVERBOSE)
Output:
-1/49152*(-2097152*ln(1/3*x*exp(1/4*(x^2-2*x-8)/(4+x)))^3-6291456*ln(1/3*x *exp(1/4*(x^2-2*x-8)/(4+x)))*ln(x)^2-1572864*ln(1/3*x*exp(1/4*(x^2-2*x-8)/ (4+x)))^3*x-393216*ln(1/3*x*exp(1/4*(x^2-2*x-8)/(4+x)))^3*x^2-32768*ln(1/3 *x*exp(1/4*(x^2-2*x-8)/(4+x)))^3*x^3+24576*x^4*ln(x)^2+6144*x^5*ln(x)+2457 6*x^4*ln(x)+196608*x^3*ln(x)^2+32768*x^3*ln(x)^3+393216*x^2*ln(x)^2+157286 4*x*ln(x)^3+393216*x^2*ln(x)^3+2097152*ln(x)^3+512*x^6+24576*ln(1/3*x*exp( 1/4*(x^2-2*x-8)/(4+x)))^2*x^4-6144*ln(1/3*x*exp(1/4*(x^2-2*x-8)/(4+x)))*x^ 5+196608*ln(1/3*x*exp(1/4*(x^2-2*x-8)/(4+x)))^2*x^3-24576*ln(1/3*x*exp(1/4 *(x^2-2*x-8)/(4+x)))*x^4+393216*ln(1/3*x*exp(1/4*(x^2-2*x-8)/(4+x)))^2*x^2 -786432*ln(x)*ln(1/3*x*exp(1/4*(x^2-2*x-8)/(4+x)))*x^2-393216*ln(x)*ln(1/3 *x*exp(1/4*(x^2-2*x-8)/(4+x)))*x^3-49152*ln(x)*ln(1/3*x*exp(1/4*(x^2-2*x-8 )/(4+x)))*x^4-4718592*ln(1/3*x*exp(1/4*(x^2-2*x-8)/(4+x)))*ln(x)^2*x-11796 48*ln(1/3*x*exp(1/4*(x^2-2*x-8)/(4+x)))*x^2*ln(x)^2-98304*ln(1/3*x*exp(1/4 *(x^2-2*x-8)/(4+x)))*ln(x)^2*x^3)/(4+x)^3
Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (24) = 48\).
Time = 0.08 (sec) , antiderivative size = 110, normalized size of antiderivative = 3.67 \[ \int \frac {\left (64+32 x+12 x^2+x^3\right ) \log (x) \log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )+\left (32+16 x+2 x^2\right ) \log ^2\left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )}{16 x+8 x^2+x^3} \, dx=\frac {16 \, {\left (x^{2} + 8 \, x + 16\right )} \log \left (x\right )^{3} + 8 \, {\left (x^{3} + 2 \, x^{2} - 4 \, {\left (x^{2} + 8 \, x + 16\right )} \log \left (3\right ) - 16 \, x - 32\right )} \log \left (x\right )^{2} + {\left (x^{4} - 4 \, x^{3} + 16 \, {\left (x^{2} + 8 \, x + 16\right )} \log \left (3\right )^{2} - 12 \, x^{2} - 8 \, {\left (x^{3} + 2 \, x^{2} - 16 \, x - 32\right )} \log \left (3\right ) + 32 \, x + 64\right )} \log \left (x\right )}{8 \, {\left (x^{2} + 8 \, x + 16\right )}} \] Input:
integrate(((2*x^2+16*x+32)*log(1/3*x*exp((x^2-2*x-8)/(4*x+16)))^2+(x^3+12* x^2+32*x+64)*log(x)*log(1/3*x*exp((x^2-2*x-8)/(4*x+16))))/(x^3+8*x^2+16*x) ,x, algorithm="fricas")
Output:
1/8*(16*(x^2 + 8*x + 16)*log(x)^3 + 8*(x^3 + 2*x^2 - 4*(x^2 + 8*x + 16)*lo g(3) - 16*x - 32)*log(x)^2 + (x^4 - 4*x^3 + 16*(x^2 + 8*x + 16)*log(3)^2 - 12*x^2 - 8*(x^3 + 2*x^2 - 16*x - 32)*log(3) + 32*x + 64)*log(x))/(x^2 + 8 *x + 16)
Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (24) = 48\).
Time = 0.47 (sec) , antiderivative size = 114, normalized size of antiderivative = 3.80 \[ \int \frac {\left (64+32 x+12 x^2+x^3\right ) \log (x) \log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )+\left (32+16 x+2 x^2\right ) \log ^2\left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )}{16 x+8 x^2+x^3} \, dx=2 \log {\left (x \right )}^{3} + \left (2 \log {\left (3 \right )}^{2} + 6 \log {\left (3 \right )} + \frac {17}{2}\right ) \log {\left (x \right )} + \frac {\left (x^{4} - 8 x^{3} \log {\left (3 \right )} - 4 x^{3} - 80 x^{2} - 64 x^{2} \log {\left (3 \right )} - 512 x - 256 x \log {\left (3 \right )} - 1024 - 512 \log {\left (3 \right )}\right ) \log {\left (x \right )}}{8 x^{2} + 64 x + 128} + \frac {\left (x^{2} - 4 x \log {\left (3 \right )} - 2 x - 16 \log {\left (3 \right )} - 8\right ) \log {\left (x \right )}^{2}}{x + 4} \] Input:
integrate(((2*x**2+16*x+32)*ln(1/3*x*exp((x**2-2*x-8)/(4*x+16)))**2+(x**3+ 12*x**2+32*x+64)*ln(x)*ln(1/3*x*exp((x**2-2*x-8)/(4*x+16))))/(x**3+8*x**2+ 16*x),x)
Output:
2*log(x)**3 + (2*log(3)**2 + 6*log(3) + 17/2)*log(x) + (x**4 - 8*x**3*log( 3) - 4*x**3 - 80*x**2 - 64*x**2*log(3) - 512*x - 256*x*log(3) - 1024 - 512 *log(3))*log(x)/(8*x**2 + 64*x + 128) + (x**2 - 4*x*log(3) - 2*x - 16*log( 3) - 8)*log(x)**2/(x + 4)
Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (24) = 48\).
Time = 0.21 (sec) , antiderivative size = 123, normalized size of antiderivative = 4.10 \[ \int \frac {\left (64+32 x+12 x^2+x^3\right ) \log (x) \log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )+\left (32+16 x+2 x^2\right ) \log ^2\left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )}{16 x+8 x^2+x^3} \, dx=\frac {16 \, {\left (x^{2} + 8 \, x + 16\right )} \log \left (x\right )^{3} + 8 \, {\left (x^{3} - 2 \, x^{2} {\left (2 \, \log \left (3\right ) - 1\right )} - 16 \, x {\left (2 \, \log \left (3\right ) + 1\right )} - 64 \, \log \left (3\right ) - 32\right )} \log \left (x\right )^{2} + {\left (x^{4} - 4 \, x^{3} {\left (2 \, \log \left (3\right ) + 1\right )} + 4 \, {\left (4 \, \log \left (3\right )^{2} - 4 \, \log \left (3\right ) - 3\right )} x^{2} + 32 \, {\left (4 \, \log \left (3\right )^{2} + 4 \, \log \left (3\right ) + 1\right )} x + 256 \, \log \left (3\right )^{2} + 256 \, \log \left (3\right ) + 64\right )} \log \left (x\right )}{8 \, {\left (x^{2} + 8 \, x + 16\right )}} \] Input:
integrate(((2*x^2+16*x+32)*log(1/3*x*exp((x^2-2*x-8)/(4*x+16)))^2+(x^3+12* x^2+32*x+64)*log(x)*log(1/3*x*exp((x^2-2*x-8)/(4*x+16))))/(x^3+8*x^2+16*x) ,x, algorithm="maxima")
Output:
1/8*(16*(x^2 + 8*x + 16)*log(x)^3 + 8*(x^3 - 2*x^2*(2*log(3) - 1) - 16*x*( 2*log(3) + 1) - 64*log(3) - 32)*log(x)^2 + (x^4 - 4*x^3*(2*log(3) + 1) + 4 *(4*log(3)^2 - 4*log(3) - 3)*x^2 + 32*(4*log(3)^2 + 4*log(3) + 1)*x + 256* log(3)^2 + 256*log(3) + 64)*log(x))/(x^2 + 8*x + 16)
\[ \int \frac {\left (64+32 x+12 x^2+x^3\right ) \log (x) \log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )+\left (32+16 x+2 x^2\right ) \log ^2\left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )}{16 x+8 x^2+x^3} \, dx=\int { \frac {2 \, {\left (x^{2} + 8 \, x + 16\right )} \log \left (\frac {1}{3} \, x e^{\left (\frac {x^{2} - 2 \, x - 8}{4 \, {\left (x + 4\right )}}\right )}\right )^{2} + {\left (x^{3} + 12 \, x^{2} + 32 \, x + 64\right )} \log \left (\frac {1}{3} \, x e^{\left (\frac {x^{2} - 2 \, x - 8}{4 \, {\left (x + 4\right )}}\right )}\right ) \log \left (x\right )}{x^{3} + 8 \, x^{2} + 16 \, x} \,d x } \] Input:
integrate(((2*x^2+16*x+32)*log(1/3*x*exp((x^2-2*x-8)/(4*x+16)))^2+(x^3+12* x^2+32*x+64)*log(x)*log(1/3*x*exp((x^2-2*x-8)/(4*x+16))))/(x^3+8*x^2+16*x) ,x, algorithm="giac")
Output:
integrate((2*(x^2 + 8*x + 16)*log(1/3*x*e^(1/4*(x^2 - 2*x - 8)/(x + 4)))^2 + (x^3 + 12*x^2 + 32*x + 64)*log(1/3*x*e^(1/4*(x^2 - 2*x - 8)/(x + 4)))*l og(x))/(x^3 + 8*x^2 + 16*x), x)
Timed out. \[ \int \frac {\left (64+32 x+12 x^2+x^3\right ) \log (x) \log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )+\left (32+16 x+2 x^2\right ) \log ^2\left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )}{16 x+8 x^2+x^3} \, dx=\int \frac {\left (2\,x^2+16\,x+32\right )\,{\ln \left (\frac {x\,{\mathrm {e}}^{-\frac {-x^2+2\,x+8}{4\,x+16}}}{3}\right )}^2+\ln \left (x\right )\,\left (x^3+12\,x^2+32\,x+64\right )\,\ln \left (\frac {x\,{\mathrm {e}}^{-\frac {-x^2+2\,x+8}{4\,x+16}}}{3}\right )}{x^3+8\,x^2+16\,x} \,d x \] Input:
int((log((x*exp(-(2*x - x^2 + 8)/(4*x + 16)))/3)^2*(16*x + 2*x^2 + 32) + l og((x*exp(-(2*x - x^2 + 8)/(4*x + 16)))/3)*log(x)*(32*x + 12*x^2 + x^3 + 6 4))/(16*x + 8*x^2 + x^3),x)
Output:
int((log((x*exp(-(2*x - x^2 + 8)/(4*x + 16)))/3)^2*(16*x + 2*x^2 + 32) + l og((x*exp(-(2*x - x^2 + 8)/(4*x + 16)))/3)*log(x)*(32*x + 12*x^2 + x^3 + 6 4))/(16*x + 8*x^2 + x^3), x)
Time = 0.18 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90 \[ \int \frac {\left (64+32 x+12 x^2+x^3\right ) \log (x) \log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )+\left (32+16 x+2 x^2\right ) \log ^2\left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )}{16 x+8 x^2+x^3} \, dx=2 \mathrm {log}\left (\frac {e^{\frac {x^{2}}{4 x +16}} x}{3 \sqrt {e}}\right )^{2} \mathrm {log}\left (x \right ) \] Input:
int(((2*x^2+16*x+32)*log(1/3*x*exp((x^2-2*x-8)/(4*x+16)))^2+(x^3+12*x^2+32 *x+64)*log(x)*log(1/3*x*exp((x^2-2*x-8)/(4*x+16))))/(x^3+8*x^2+16*x),x)
Output:
2*log((e**(x**2/(4*x + 16))*x)/(3*sqrt(e)))**2*log(x)