Integrand size = 107, antiderivative size = 25 \begin {dmath*} \int \frac {e^{\frac {-27 x^4+x^5}{-x^4+(-108+4 x) \log (2)}} \left (x^8-x^9+\left (11880 x^4-872 x^5+16 x^6\right ) \log (2)+\left (11664-864 x+16 x^2\right ) \log ^2(2)\right )}{x^8+\left (216 x^4-8 x^5\right ) \log (2)+\left (11664-864 x+16 x^2\right ) \log ^2(2)} \, dx=e^{\frac {x}{\frac {x}{27-x}+\frac {4 \log (2)}{x^3}}} x \end {dmath*}
Time = 0.07 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \begin {dmath*} \int \frac {e^{\frac {-27 x^4+x^5}{-x^4+(-108+4 x) \log (2)}} \left (x^8-x^9+\left (11880 x^4-872 x^5+16 x^6\right ) \log (2)+\left (11664-864 x+16 x^2\right ) \log ^2(2)\right )}{x^8+\left (216 x^4-8 x^5\right ) \log (2)+\left (11664-864 x+16 x^2\right ) \log ^2(2)} \, dx=2^{-\frac {4 (-27+x)^2}{x^4+108 \log (2)-4 x \log (2)}} e^{27-x} x \end {dmath*}
Integrate[(E^((-27*x^4 + x^5)/(-x^4 + (-108 + 4*x)*Log[2]))*(x^8 - x^9 + ( 11880*x^4 - 872*x^5 + 16*x^6)*Log[2] + (11664 - 864*x + 16*x^2)*Log[2]^2)) /(x^8 + (216*x^4 - 8*x^5)*Log[2] + (11664 - 864*x + 16*x^2)*Log[2]^2),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.
\(\displaystyle \int \frac {e^{\frac {x^5-27 x^4}{(4 x-108) \log (2)-x^4}} \left (-x^9+x^8+\left (16 x^2-864 x+11664\right ) \log ^2(2)+\left (16 x^6-872 x^5+11880 x^4\right ) \log (2)\right )}{x^8+\left (16 x^2-864 x+11664\right ) \log ^2(2)+\left (216 x^4-8 x^5\right ) \log (2)} \, dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \frac {e^{\frac {x^5-27 x^4}{(4 x-108) \log (2)-x^4}} \left (-x^9+x^8+\left (16 x^2-864 x+11664\right ) \log ^2(2)+\left (16 x^6-872 x^5+11880 x^4\right ) \log (2)\right )}{\left (x^4-4 x \log (2)+108 \log (2)\right )^2}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{\frac {(x-27) x^4}{-x^4+4 x \log (2)-108 \log (2)}} \left (-x^9+x^8+\left (16 x^2-864 x+11664\right ) \log ^2(2)+\left (16 x^6-872 x^5+11880 x^4\right ) \log (2)\right )}{\left (x^4-4 x \log (2)+108 \log (2)\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {48 (x-36) (x-27)^2 \log ^2(2) e^{\frac {(x-27) x^4}{-x^4+4 x \log (2)-108 \log (2)}}}{\left (x^4-4 x \log (2)+108 \log (2)\right )^2}+e^{\frac {(x-27) x^4}{-x^4+4 x \log (2)-108 \log (2)}}-x e^{\frac {(x-27) x^4}{-x^4+4 x \log (2)-108 \log (2)}}+\frac {8 \left (x^2-81 x+1458\right ) \log (2) e^{\frac {(x-27) x^4}{-x^4+4 x \log (2)-108 \log (2)}}}{x^4-4 x \log (2)+108 \log (2)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -1259712 \log ^2(2) \int \frac {e^{\frac {(x-27) x^4}{-x^4+4 \log (2) x-108 \log (2)}}}{\left (x^4-4 \log (2) x+108 \log (2)\right )^2}dx+128304 \log ^2(2) \int \frac {e^{\frac {(x-27) x^4}{-x^4+4 \log (2) x-108 \log (2)}} x}{\left (x^4-4 \log (2) x+108 \log (2)\right )^2}dx+\int e^{\frac {(x-27) x^4}{-x^4+4 \log (2) x-108 \log (2)}}dx-\int e^{\frac {(x-27) x^4}{-x^4+4 \log (2) x-108 \log (2)}} xdx+11664 \log (2) \int \frac {e^{\frac {(x-27) x^4}{-x^4+4 \log (2) x-108 \log (2)}}}{x^4-4 \log (2) x+108 \log (2)}dx-648 \log (2) \int \frac {e^{\frac {(x-27) x^4}{-x^4+4 \log (2) x-108 \log (2)}} x}{x^4-4 \log (2) x+108 \log (2)}dx+48 \log ^2(2) \int \frac {e^{\frac {(x-27) x^4}{-x^4+4 \log (2) x-108 \log (2)}} x^3}{\left (x^4-4 \log (2) x+108 \log (2)\right )^2}dx-4320 \log ^2(2) \int \frac {e^{\frac {(x-27) x^4}{-x^4+4 \log (2) x-108 \log (2)}} x^2}{\left (x^4-4 \log (2) x+108 \log (2)\right )^2}dx+8 \log (2) \int \frac {e^{\frac {(x-27) x^4}{-x^4+4 \log (2) x-108 \log (2)}} x^2}{x^4-4 \log (2) x+108 \log (2)}dx\) |
Int[(E^((-27*x^4 + x^5)/(-x^4 + (-108 + 4*x)*Log[2]))*(x^8 - x^9 + (11880* x^4 - 872*x^5 + 16*x^6)*Log[2] + (11664 - 864*x + 16*x^2)*Log[2]^2))/(x^8 + (216*x^4 - 8*x^5)*Log[2] + (11664 - 864*x + 16*x^2)*Log[2]^2),x]
3.7.86.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 1.52 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12
method | result | size |
gosper | \(x \,{\mathrm e}^{\frac {x^{4} \left (x -27\right )}{-x^{4}+4 x \ln \left (2\right )-108 \ln \left (2\right )}}\) | \(28\) |
risch | \(x \,{\mathrm e}^{\frac {x^{4} \left (x -27\right )}{-x^{4}+4 x \ln \left (2\right )-108 \ln \left (2\right )}}\) | \(28\) |
parallelrisch | \(x \,{\mathrm e}^{\frac {x^{4} \left (x -27\right )}{-x^{4}+4 x \ln \left (2\right )-108 \ln \left (2\right )}}\) | \(28\) |
norman | \(\frac {-x^{5} {\mathrm e}^{\frac {x^{5}-27 x^{4}}{\left (4 x -108\right ) \ln \left (2\right )-x^{4}}}-108 x \ln \left (2\right ) {\mathrm e}^{\frac {x^{5}-27 x^{4}}{\left (4 x -108\right ) \ln \left (2\right )-x^{4}}}+4 x^{2} \ln \left (2\right ) {\mathrm e}^{\frac {x^{5}-27 x^{4}}{\left (4 x -108\right ) \ln \left (2\right )-x^{4}}}}{-x^{4}+4 x \ln \left (2\right )-108 \ln \left (2\right )}\) | \(118\) |
int(((16*x^2-864*x+11664)*ln(2)^2+(16*x^6-872*x^5+11880*x^4)*ln(2)-x^9+x^8 )*exp((x^5-27*x^4)/((4*x-108)*ln(2)-x^4))/((16*x^2-864*x+11664)*ln(2)^2+(- 8*x^5+216*x^4)*ln(2)+x^8),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \begin {dmath*} \int \frac {e^{\frac {-27 x^4+x^5}{-x^4+(-108+4 x) \log (2)}} \left (x^8-x^9+\left (11880 x^4-872 x^5+16 x^6\right ) \log (2)+\left (11664-864 x+16 x^2\right ) \log ^2(2)\right )}{x^8+\left (216 x^4-8 x^5\right ) \log (2)+\left (11664-864 x+16 x^2\right ) \log ^2(2)} \, dx=x e^{\left (-\frac {x^{5} - 27 \, x^{4}}{x^{4} - 4 \, {\left (x - 27\right )} \log \left (2\right )}\right )} \end {dmath*}
integrate(((16*x^2-864*x+11664)*log(2)^2+(16*x^6-872*x^5+11880*x^4)*log(2) -x^9+x^8)*exp((x^5-27*x^4)/((4*x-108)*log(2)-x^4))/((16*x^2-864*x+11664)*l og(2)^2+(-8*x^5+216*x^4)*log(2)+x^8),x, algorithm=\
Timed out. \begin {dmath*} \int \frac {e^{\frac {-27 x^4+x^5}{-x^4+(-108+4 x) \log (2)}} \left (x^8-x^9+\left (11880 x^4-872 x^5+16 x^6\right ) \log (2)+\left (11664-864 x+16 x^2\right ) \log ^2(2)\right )}{x^8+\left (216 x^4-8 x^5\right ) \log (2)+\left (11664-864 x+16 x^2\right ) \log ^2(2)} \, dx=\text {Timed out} \end {dmath*}
integrate(((16*x**2-864*x+11664)*ln(2)**2+(16*x**6-872*x**5+11880*x**4)*ln (2)-x**9+x**8)*exp((x**5-27*x**4)/((4*x-108)*ln(2)-x**4))/((16*x**2-864*x+ 11664)*ln(2)**2+(-8*x**5+216*x**4)*ln(2)+x**8),x)
Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (23) = 46\).
Time = 0.49 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.76 \begin {dmath*} \int \frac {e^{\frac {-27 x^4+x^5}{-x^4+(-108+4 x) \log (2)}} \left (x^8-x^9+\left (11880 x^4-872 x^5+16 x^6\right ) \log (2)+\left (11664-864 x+16 x^2\right ) \log ^2(2)\right )}{x^8+\left (216 x^4-8 x^5\right ) \log (2)+\left (11664-864 x+16 x^2\right ) \log ^2(2)} \, dx=x e^{\left (-\frac {4 \, x^{2} \log \left (2\right )}{x^{4} - 4 \, x \log \left (2\right ) + 108 \, \log \left (2\right )} - x + \frac {216 \, x \log \left (2\right )}{x^{4} - 4 \, x \log \left (2\right ) + 108 \, \log \left (2\right )} - \frac {2916 \, \log \left (2\right )}{x^{4} - 4 \, x \log \left (2\right ) + 108 \, \log \left (2\right )} + 27\right )} \end {dmath*}
integrate(((16*x^2-864*x+11664)*log(2)^2+(16*x^6-872*x^5+11880*x^4)*log(2) -x^9+x^8)*exp((x^5-27*x^4)/((4*x-108)*log(2)-x^4))/((16*x^2-864*x+11664)*l og(2)^2+(-8*x^5+216*x^4)*log(2)+x^8),x, algorithm=\
x*e^(-4*x^2*log(2)/(x^4 - 4*x*log(2) + 108*log(2)) - x + 216*x*log(2)/(x^4 - 4*x*log(2) + 108*log(2)) - 2916*log(2)/(x^4 - 4*x*log(2) + 108*log(2)) + 27)
Time = 0.36 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \begin {dmath*} \int \frac {e^{\frac {-27 x^4+x^5}{-x^4+(-108+4 x) \log (2)}} \left (x^8-x^9+\left (11880 x^4-872 x^5+16 x^6\right ) \log (2)+\left (11664-864 x+16 x^2\right ) \log ^2(2)\right )}{x^8+\left (216 x^4-8 x^5\right ) \log (2)+\left (11664-864 x+16 x^2\right ) \log ^2(2)} \, dx=x e^{\left (-\frac {x^{5} - 27 \, x^{4}}{x^{4} - 4 \, x \log \left (2\right ) + 108 \, \log \left (2\right )}\right )} \end {dmath*}
integrate(((16*x^2-864*x+11664)*log(2)^2+(16*x^6-872*x^5+11880*x^4)*log(2) -x^9+x^8)*exp((x^5-27*x^4)/((4*x-108)*log(2)-x^4))/((16*x^2-864*x+11664)*l og(2)^2+(-8*x^5+216*x^4)*log(2)+x^8),x, algorithm=\
Time = 15.29 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20 \begin {dmath*} \int \frac {e^{\frac {-27 x^4+x^5}{-x^4+(-108+4 x) \log (2)}} \left (x^8-x^9+\left (11880 x^4-872 x^5+16 x^6\right ) \log (2)+\left (11664-864 x+16 x^2\right ) \log ^2(2)\right )}{x^8+\left (216 x^4-8 x^5\right ) \log (2)+\left (11664-864 x+16 x^2\right ) \log ^2(2)} \, dx=x\,{\mathrm {e}}^{\frac {27\,x^4-x^5}{x^4-4\,\ln \left (2\right )\,x+108\,\ln \left (2\right )}} \end {dmath*}