Integrand size = 171, antiderivative size = 28 \[ \int \frac {e^x \left (64+16 x^4+x^8\right )+e^x \left (64+8 x^4\right ) \log \left (x^2\right )+16 e^x \log ^2\left (x^2\right )+e^{\frac {5 x^3+x^4}{e^x \left (8+x^4\right )+4 e^x \log \left (x^2\right )}} \left (80 x^2-16 x^3-8 x^4-5 x^6-5 x^7-x^8+\left (60 x^2-4 x^3-4 x^4\right ) \log \left (x^2\right )\right )}{e^x \left (64+16 x^4+x^8\right )+e^x \left (64+8 x^4\right ) \log \left (x^2\right )+16 e^x \log ^2\left (x^2\right )} \, dx=e^{\frac {e^{-x} (5+x)}{x+\frac {4 \left (2+\log \left (x^2\right )\right )}{x^3}}}+x \]
Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {e^x \left (64+16 x^4+x^8\right )+e^x \left (64+8 x^4\right ) \log \left (x^2\right )+16 e^x \log ^2\left (x^2\right )+e^{\frac {5 x^3+x^4}{e^x \left (8+x^4\right )+4 e^x \log \left (x^2\right )}} \left (80 x^2-16 x^3-8 x^4-5 x^6-5 x^7-x^8+\left (60 x^2-4 x^3-4 x^4\right ) \log \left (x^2\right )\right )}{e^x \left (64+16 x^4+x^8\right )+e^x \left (64+8 x^4\right ) \log \left (x^2\right )+16 e^x \log ^2\left (x^2\right )} \, dx=e^{\frac {e^{-x} x^3 (5+x)}{8+x^4+4 \log \left (x^2\right )}}+x \]
Integrate[(E^x*(64 + 16*x^4 + x^8) + E^x*(64 + 8*x^4)*Log[x^2] + 16*E^x*Lo g[x^2]^2 + E^((5*x^3 + x^4)/(E^x*(8 + x^4) + 4*E^x*Log[x^2]))*(80*x^2 - 16 *x^3 - 8*x^4 - 5*x^6 - 5*x^7 - x^8 + (60*x^2 - 4*x^3 - 4*x^4)*Log[x^2]))/( E^x*(64 + 16*x^4 + x^8) + E^x*(64 + 8*x^4)*Log[x^2] + 16*E^x*Log[x^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.
| \(\displaystyle \int \frac {16 e^x \log ^2\left (x^2\right )+e^x \left (x^8+16 x^4+64\right )+e^x \left (8 x^4+64\right ) \log \left (x^2\right )+e^{\frac {x^4+5 x^3}{e^x \left (x^4+8\right )+4 e^x \log \left (x^2\right )}} \left (-x^8-5 x^7-5 x^6-8 x^4-16 x^3+80 x^2+\left (-4 x^4-4 x^3+60 x^2\right ) \log \left (x^2\right )\right )}{16 e^x \log ^2\left (x^2\right )+e^x \left (x^8+16 x^4+64\right )+e^x \left (8 x^4+64\right ) \log \left (x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{-x} \left (16 e^x \log ^2\left (x^2\right )+e^x \left (x^8+16 x^4+64\right )+e^x \left (8 x^4+64\right ) \log \left (x^2\right )+e^{\frac {x^4+5 x^3}{e^x \left (x^4+8\right )+4 e^x \log \left (x^2\right )}} \left (-x^8-5 x^7-5 x^6-8 x^4-16 x^3+80 x^2+\left (-4 x^4-4 x^3+60 x^2\right ) \log \left (x^2\right )\right )\right )}{\left (x^4+4 \log \left (x^2\right )+8\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (1-\frac {x^2 e^{\frac {e^{-x} x^3 (x+5)}{x^4+4 \log \left (x^2\right )+8}-x} \left (x^6+5 x^5+5 x^4+8 x^2+4 x^2 \log \left (x^2\right )+4 x \log \left (x^2\right )-60 \log \left (x^2\right )+16 x-80\right )}{\left (x^4+4 \log \left (x^2\right )+8\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (1-\frac {x^2 e^{\frac {e^{-x} x^3 (x+5)}{x^4+4 \log \left (x^2\right )+8}-x} \left (x^6+5 x^5+5 x^4+8 x^2+4 x^2 \log \left (x^2\right )+4 x \log \left (x^2\right )-60 \log \left (x^2\right )+16 x-80\right )}{\left (x^4+4 \log \left (x^2\right )+8\right )^2}\right )dx\) |
Int[(E^x*(64 + 16*x^4 + x^8) + E^x*(64 + 8*x^4)*Log[x^2] + 16*E^x*Log[x^2] ^2 + E^((5*x^3 + x^4)/(E^x*(8 + x^4) + 4*E^x*Log[x^2]))*(80*x^2 - 16*x^3 - 8*x^4 - 5*x^6 - 5*x^7 - x^8 + (60*x^2 - 4*x^3 - 4*x^4)*Log[x^2]))/(E^x*(6 4 + 16*x^4 + x^8) + E^x*(64 + 8*x^4)*Log[x^2] + 16*E^x*Log[x^2]^2),x]
3.12.17.3.1 Defintions of rubi rules used
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.52 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.68
\[x +{\mathrm e}^{\frac {x^{3} \left (5+x \right ) {\mathrm e}^{-x}}{-2 i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+4 i \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )-2 i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}+x^{4}+8 \ln \left (x \right )+8}}\]
int((((-4*x^4-4*x^3+60*x^2)*ln(x^2)-x^8-5*x^7-5*x^6-8*x^4-16*x^3+80*x^2)*e xp((x^4+5*x^3)/(4*exp(x)*ln(x^2)+(x^4+8)*exp(x)))+16*exp(x)*ln(x^2)^2+(8*x ^4+64)*exp(x)*ln(x^2)+(x^8+16*x^4+64)*exp(x))/(16*exp(x)*ln(x^2)^2+(8*x^4+ 64)*exp(x)*ln(x^2)+(x^8+16*x^4+64)*exp(x)),x)
x+exp(x^3*(5+x)*exp(-x)/(-2*I*Pi*csgn(I*x^2)^3+4*I*Pi*csgn(I*x^2)^2*csgn(I *x)-2*I*Pi*csgn(I*x^2)*csgn(I*x)^2+x^4+8*ln(x)+8))
Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {e^x \left (64+16 x^4+x^8\right )+e^x \left (64+8 x^4\right ) \log \left (x^2\right )+16 e^x \log ^2\left (x^2\right )+e^{\frac {5 x^3+x^4}{e^x \left (8+x^4\right )+4 e^x \log \left (x^2\right )}} \left (80 x^2-16 x^3-8 x^4-5 x^6-5 x^7-x^8+\left (60 x^2-4 x^3-4 x^4\right ) \log \left (x^2\right )\right )}{e^x \left (64+16 x^4+x^8\right )+e^x \left (64+8 x^4\right ) \log \left (x^2\right )+16 e^x \log ^2\left (x^2\right )} \, dx=x + e^{\left (\frac {x^{4} + 5 \, x^{3}}{{\left (x^{4} + 8\right )} e^{x} + 4 \, e^{x} \log \left (x^{2}\right )}\right )} \]
integrate((((-4*x^4-4*x^3+60*x^2)*log(x^2)-x^8-5*x^7-5*x^6-8*x^4-16*x^3+80 *x^2)*exp((x^4+5*x^3)/(4*exp(x)*log(x^2)+(x^4+8)*exp(x)))+16*exp(x)*log(x^ 2)^2+(8*x^4+64)*exp(x)*log(x^2)+(x^8+16*x^4+64)*exp(x))/(16*exp(x)*log(x^2 )^2+(8*x^4+64)*exp(x)*log(x^2)+(x^8+16*x^4+64)*exp(x)),x, algorithm=\
Time = 0.60 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {e^x \left (64+16 x^4+x^8\right )+e^x \left (64+8 x^4\right ) \log \left (x^2\right )+16 e^x \log ^2\left (x^2\right )+e^{\frac {5 x^3+x^4}{e^x \left (8+x^4\right )+4 e^x \log \left (x^2\right )}} \left (80 x^2-16 x^3-8 x^4-5 x^6-5 x^7-x^8+\left (60 x^2-4 x^3-4 x^4\right ) \log \left (x^2\right )\right )}{e^x \left (64+16 x^4+x^8\right )+e^x \left (64+8 x^4\right ) \log \left (x^2\right )+16 e^x \log ^2\left (x^2\right )} \, dx=x + e^{\frac {x^{4} + 5 x^{3}}{\left (x^{4} + 8\right ) e^{x} + 4 e^{x} \log {\left (x^{2} \right )}}} \]
integrate((((-4*x**4-4*x**3+60*x**2)*ln(x**2)-x**8-5*x**7-5*x**6-8*x**4-16 *x**3+80*x**2)*exp((x**4+5*x**3)/(4*exp(x)*ln(x**2)+(x**4+8)*exp(x)))+16*e xp(x)*ln(x**2)**2+(8*x**4+64)*exp(x)*ln(x**2)+(x**8+16*x**4+64)*exp(x))/(1 6*exp(x)*ln(x**2)**2+(8*x**4+64)*exp(x)*ln(x**2)+(x**8+16*x**4+64)*exp(x)) ,x)
Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (26) = 52\).
Time = 0.25 (sec) , antiderivative size = 116, normalized size of antiderivative = 4.14 \[ \int \frac {e^x \left (64+16 x^4+x^8\right )+e^x \left (64+8 x^4\right ) \log \left (x^2\right )+16 e^x \log ^2\left (x^2\right )+e^{\frac {5 x^3+x^4}{e^x \left (8+x^4\right )+4 e^x \log \left (x^2\right )}} \left (80 x^2-16 x^3-8 x^4-5 x^6-5 x^7-x^8+\left (60 x^2-4 x^3-4 x^4\right ) \log \left (x^2\right )\right )}{e^x \left (64+16 x^4+x^8\right )+e^x \left (64+8 x^4\right ) \log \left (x^2\right )+16 e^x \log ^2\left (x^2\right )} \, dx={\left (x e^{\left (\frac {8 \, \log \left (x\right )}{{\left (x^{4} + 8\right )} e^{x} + 8 \, e^{x} \log \left (x\right )} + \frac {8}{{\left (x^{4} + 8\right )} e^{x} + 8 \, e^{x} \log \left (x\right )}\right )} + e^{\left (\frac {5 \, x^{3}}{{\left (x^{4} + 8\right )} e^{x} + 8 \, e^{x} \log \left (x\right )} + e^{\left (-x\right )}\right )}\right )} e^{\left (-\frac {8 \, \log \left (x\right )}{{\left (x^{4} + 8\right )} e^{x} + 8 \, e^{x} \log \left (x\right )} - \frac {8}{{\left (x^{4} + 8\right )} e^{x} + 8 \, e^{x} \log \left (x\right )}\right )} \]
integrate((((-4*x^4-4*x^3+60*x^2)*log(x^2)-x^8-5*x^7-5*x^6-8*x^4-16*x^3+80 *x^2)*exp((x^4+5*x^3)/(4*exp(x)*log(x^2)+(x^4+8)*exp(x)))+16*exp(x)*log(x^ 2)^2+(8*x^4+64)*exp(x)*log(x^2)+(x^8+16*x^4+64)*exp(x))/(16*exp(x)*log(x^2 )^2+(8*x^4+64)*exp(x)*log(x^2)+(x^8+16*x^4+64)*exp(x)),x, algorithm=\
(x*e^(8*log(x)/((x^4 + 8)*e^x + 8*e^x*log(x)) + 8/((x^4 + 8)*e^x + 8*e^x*l og(x))) + e^(5*x^3/((x^4 + 8)*e^x + 8*e^x*log(x)) + e^(-x)))*e^(-8*log(x)/ ((x^4 + 8)*e^x + 8*e^x*log(x)) - 8/((x^4 + 8)*e^x + 8*e^x*log(x)))
\[ \int \frac {e^x \left (64+16 x^4+x^8\right )+e^x \left (64+8 x^4\right ) \log \left (x^2\right )+16 e^x \log ^2\left (x^2\right )+e^{\frac {5 x^3+x^4}{e^x \left (8+x^4\right )+4 e^x \log \left (x^2\right )}} \left (80 x^2-16 x^3-8 x^4-5 x^6-5 x^7-x^8+\left (60 x^2-4 x^3-4 x^4\right ) \log \left (x^2\right )\right )}{e^x \left (64+16 x^4+x^8\right )+e^x \left (64+8 x^4\right ) \log \left (x^2\right )+16 e^x \log ^2\left (x^2\right )} \, dx=\int { \frac {8 \, {\left (x^{4} + 8\right )} e^{x} \log \left (x^{2}\right ) + 16 \, e^{x} \log \left (x^{2}\right )^{2} + {\left (x^{8} + 16 \, x^{4} + 64\right )} e^{x} - {\left (x^{8} + 5 \, x^{7} + 5 \, x^{6} + 8 \, x^{4} + 16 \, x^{3} - 80 \, x^{2} + 4 \, {\left (x^{4} + x^{3} - 15 \, x^{2}\right )} \log \left (x^{2}\right )\right )} e^{\left (\frac {x^{4} + 5 \, x^{3}}{{\left (x^{4} + 8\right )} e^{x} + 4 \, e^{x} \log \left (x^{2}\right )}\right )}}{8 \, {\left (x^{4} + 8\right )} e^{x} \log \left (x^{2}\right ) + 16 \, e^{x} \log \left (x^{2}\right )^{2} + {\left (x^{8} + 16 \, x^{4} + 64\right )} e^{x}} \,d x } \]
integrate((((-4*x^4-4*x^3+60*x^2)*log(x^2)-x^8-5*x^7-5*x^6-8*x^4-16*x^3+80 *x^2)*exp((x^4+5*x^3)/(4*exp(x)*log(x^2)+(x^4+8)*exp(x)))+16*exp(x)*log(x^ 2)^2+(8*x^4+64)*exp(x)*log(x^2)+(x^8+16*x^4+64)*exp(x))/(16*exp(x)*log(x^2 )^2+(8*x^4+64)*exp(x)*log(x^2)+(x^8+16*x^4+64)*exp(x)),x, algorithm=\
integrate((8*(x^4 + 8)*e^x*log(x^2) + 16*e^x*log(x^2)^2 + (x^8 + 16*x^4 + 64)*e^x - (x^8 + 5*x^7 + 5*x^6 + 8*x^4 + 16*x^3 - 80*x^2 + 4*(x^4 + x^3 - 15*x^2)*log(x^2))*e^((x^4 + 5*x^3)/((x^4 + 8)*e^x + 4*e^x*log(x^2))))/(8*( x^4 + 8)*e^x*log(x^2) + 16*e^x*log(x^2)^2 + (x^8 + 16*x^4 + 64)*e^x), x)
Time = 9.92 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.00 \[ \int \frac {e^x \left (64+16 x^4+x^8\right )+e^x \left (64+8 x^4\right ) \log \left (x^2\right )+16 e^x \log ^2\left (x^2\right )+e^{\frac {5 x^3+x^4}{e^x \left (8+x^4\right )+4 e^x \log \left (x^2\right )}} \left (80 x^2-16 x^3-8 x^4-5 x^6-5 x^7-x^8+\left (60 x^2-4 x^3-4 x^4\right ) \log \left (x^2\right )\right )}{e^x \left (64+16 x^4+x^8\right )+e^x \left (64+8 x^4\right ) \log \left (x^2\right )+16 e^x \log ^2\left (x^2\right )} \, dx=x+{\mathrm {e}}^{\frac {x^4}{8\,{\mathrm {e}}^x+x^4\,{\mathrm {e}}^x+4\,\ln \left (x^2\right )\,{\mathrm {e}}^x}}\,{\mathrm {e}}^{\frac {5\,x^3}{8\,{\mathrm {e}}^x+x^4\,{\mathrm {e}}^x+4\,\ln \left (x^2\right )\,{\mathrm {e}}^x}} \]
int((exp(x)*(16*x^4 + x^8 + 64) - exp((5*x^3 + x^4)/(exp(x)*(x^4 + 8) + 4* log(x^2)*exp(x)))*(log(x^2)*(4*x^3 - 60*x^2 + 4*x^4) - 80*x^2 + 16*x^3 + 8 *x^4 + 5*x^6 + 5*x^7 + x^8) + 16*log(x^2)^2*exp(x) + log(x^2)*exp(x)*(8*x^ 4 + 64))/(exp(x)*(16*x^4 + x^8 + 64) + 16*log(x^2)^2*exp(x) + log(x^2)*exp (x)*(8*x^4 + 64)),x)