Integrand size = 134, antiderivative size = 26 \[ \int \frac {25 x^6+10 x^7+x^8+e^x \left (-25 x^6-10 x^7-x^8\right )+\left (-150 x^6-70 x^7-8 x^8+e^3 \left (-150 x^5-70 x^6-8 x^7\right )+e^x \left (150 x^5+70 x^6+8 x^7\right )\right ) \log \left (-e^3+e^x-x\right )}{\left (-e^3+e^x-x\right ) \log ^2\left (-e^3+e^x-x\right )} \, dx=\frac {(-5-x)^2 x^6}{\log \left (-e^3+e^x-x\right )} \]
Time = 0.77 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {25 x^6+10 x^7+x^8+e^x \left (-25 x^6-10 x^7-x^8\right )+\left (-150 x^6-70 x^7-8 x^8+e^3 \left (-150 x^5-70 x^6-8 x^7\right )+e^x \left (150 x^5+70 x^6+8 x^7\right )\right ) \log \left (-e^3+e^x-x\right )}{\left (-e^3+e^x-x\right ) \log ^2\left (-e^3+e^x-x\right )} \, dx=\frac {x^6 (5+x)^2}{\log \left (-e^3+e^x-x\right )} \]
Integrate[(25*x^6 + 10*x^7 + x^8 + E^x*(-25*x^6 - 10*x^7 - x^8) + (-150*x^ 6 - 70*x^7 - 8*x^8 + E^3*(-150*x^5 - 70*x^6 - 8*x^7) + E^x*(150*x^5 + 70*x ^6 + 8*x^7))*Log[-E^3 + E^x - x])/((-E^3 + E^x - x)*Log[-E^3 + E^x - x]^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.
\(\displaystyle \int \frac {x^8+10 x^7+25 x^6+e^x \left (-x^8-10 x^7-25 x^6\right )+\left (-8 x^8-70 x^7-150 x^6+e^3 \left (-8 x^7-70 x^6-150 x^5\right )+e^x \left (8 x^7+70 x^6+150 x^5\right )\right ) \log \left (-x+e^x-e^3\right )}{\left (-x+e^x-e^3\right ) \log ^2\left (-x+e^x-e^3\right )} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {x^5 (x+5) \left (\left (e^x-1\right ) x (x+5)-2 \left (-x+e^x-e^3\right ) (4 x+15) \log \left (-x+e^x-e^3\right )\right )}{\left (x-e^x+e^3\right ) \log ^2\left (-x+e^x-e^3\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {(x+5)^2 \left (x+e^3-1\right ) x^6}{\left (-x+e^x-e^3\right ) \log ^2\left (-x+e^x-e^3\right )}-\frac {(x+5) x^5 \left (x^2+5 x-8 x \log \left (-x+e^x-e^3\right )-30 \log \left (-x+e^x-e^3\right )\right )}{\log ^2\left (-x+e^x-e^3\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\int \frac {x^9}{\left (-x+e^x-e^3\right ) \log ^2\left (-x+e^x-e^3\right )}dx-\int \frac {x^8}{\log ^2\left (-x+e^x-e^3\right )}dx+\left (9+e^3\right ) \int \frac {x^8}{\left (x-e^x+e^3\right ) \log ^2\left (-x+e^x-e^3\right )}dx-10 \int \frac {x^7}{\log ^2\left (-x+e^x-e^3\right )}dx+5 \left (3+2 e^3\right ) \int \frac {x^7}{\left (x-e^x+e^3\right ) \log ^2\left (-x+e^x-e^3\right )}dx+8 \int \frac {x^7}{\log \left (-x+e^x-e^3\right )}dx-25 \int \frac {x^6}{\log ^2\left (-x+e^x-e^3\right )}dx-25 \left (1-e^3\right ) \int \frac {x^6}{\left (x-e^x+e^3\right ) \log ^2\left (-x+e^x-e^3\right )}dx+70 \int \frac {x^6}{\log \left (-x+e^x-e^3\right )}dx+150 \int \frac {x^5}{\log \left (-x+e^x-e^3\right )}dx\) |
Int[(25*x^6 + 10*x^7 + x^8 + E^x*(-25*x^6 - 10*x^7 - x^8) + (-150*x^6 - 70 *x^7 - 8*x^8 + E^3*(-150*x^5 - 70*x^6 - 8*x^7) + E^x*(150*x^5 + 70*x^6 + 8 *x^7))*Log[-E^3 + E^x - x])/((-E^3 + E^x - x)*Log[-E^3 + E^x - x]^2),x]
3.28.16.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 1.75 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00
method | result | size |
risch | \(\frac {x^{6} \left (x^{2}+10 x +25\right )}{\ln \left ({\mathrm e}^{x}-{\mathrm e}^{3}-x \right )}\) | \(26\) |
parallelrisch | \(\frac {x^{8}+10 x^{7}+25 x^{6}}{\ln \left ({\mathrm e}^{x}-{\mathrm e}^{3}-x \right )}\) | \(29\) |
int((((8*x^7+70*x^6+150*x^5)*exp(x)+(-8*x^7-70*x^6-150*x^5)*exp(3)-8*x^8-7 0*x^7-150*x^6)*ln(exp(x)-exp(3)-x)+(-x^8-10*x^7-25*x^6)*exp(x)+x^8+10*x^7+ 25*x^6)/(exp(x)-exp(3)-x)/ln(exp(x)-exp(3)-x)^2,x,method=_RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {25 x^6+10 x^7+x^8+e^x \left (-25 x^6-10 x^7-x^8\right )+\left (-150 x^6-70 x^7-8 x^8+e^3 \left (-150 x^5-70 x^6-8 x^7\right )+e^x \left (150 x^5+70 x^6+8 x^7\right )\right ) \log \left (-e^3+e^x-x\right )}{\left (-e^3+e^x-x\right ) \log ^2\left (-e^3+e^x-x\right )} \, dx=\frac {x^{8} + 10 \, x^{7} + 25 \, x^{6}}{\log \left (-x - e^{3} + e^{x}\right )} \]
integrate((((8*x^7+70*x^6+150*x^5)*exp(x)+(-8*x^7-70*x^6-150*x^5)*exp(3)-8 *x^8-70*x^7-150*x^6)*log(exp(x)-exp(3)-x)+(-x^8-10*x^7-25*x^6)*exp(x)+x^8+ 10*x^7+25*x^6)/(exp(x)-exp(3)-x)/log(exp(x)-exp(3)-x)^2,x, algorithm=\
Time = 0.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {25 x^6+10 x^7+x^8+e^x \left (-25 x^6-10 x^7-x^8\right )+\left (-150 x^6-70 x^7-8 x^8+e^3 \left (-150 x^5-70 x^6-8 x^7\right )+e^x \left (150 x^5+70 x^6+8 x^7\right )\right ) \log \left (-e^3+e^x-x\right )}{\left (-e^3+e^x-x\right ) \log ^2\left (-e^3+e^x-x\right )} \, dx=\frac {x^{8} + 10 x^{7} + 25 x^{6}}{\log {\left (- x + e^{x} - e^{3} \right )}} \]
integrate((((8*x**7+70*x**6+150*x**5)*exp(x)+(-8*x**7-70*x**6-150*x**5)*ex p(3)-8*x**8-70*x**7-150*x**6)*ln(exp(x)-exp(3)-x)+(-x**8-10*x**7-25*x**6)* exp(x)+x**8+10*x**7+25*x**6)/(exp(x)-exp(3)-x)/ln(exp(x)-exp(3)-x)**2,x)
Time = 0.23 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {25 x^6+10 x^7+x^8+e^x \left (-25 x^6-10 x^7-x^8\right )+\left (-150 x^6-70 x^7-8 x^8+e^3 \left (-150 x^5-70 x^6-8 x^7\right )+e^x \left (150 x^5+70 x^6+8 x^7\right )\right ) \log \left (-e^3+e^x-x\right )}{\left (-e^3+e^x-x\right ) \log ^2\left (-e^3+e^x-x\right )} \, dx=\frac {x^{8} + 10 \, x^{7} + 25 \, x^{6}}{\log \left (-x - e^{3} + e^{x}\right )} \]
integrate((((8*x^7+70*x^6+150*x^5)*exp(x)+(-8*x^7-70*x^6-150*x^5)*exp(3)-8 *x^8-70*x^7-150*x^6)*log(exp(x)-exp(3)-x)+(-x^8-10*x^7-25*x^6)*exp(x)+x^8+ 10*x^7+25*x^6)/(exp(x)-exp(3)-x)/log(exp(x)-exp(3)-x)^2,x, algorithm=\
Time = 0.31 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {25 x^6+10 x^7+x^8+e^x \left (-25 x^6-10 x^7-x^8\right )+\left (-150 x^6-70 x^7-8 x^8+e^3 \left (-150 x^5-70 x^6-8 x^7\right )+e^x \left (150 x^5+70 x^6+8 x^7\right )\right ) \log \left (-e^3+e^x-x\right )}{\left (-e^3+e^x-x\right ) \log ^2\left (-e^3+e^x-x\right )} \, dx=\frac {x^{8} + 10 \, x^{7} + 25 \, x^{6}}{\log \left (-x - e^{3} + e^{x}\right )} \]
integrate((((8*x^7+70*x^6+150*x^5)*exp(x)+(-8*x^7-70*x^6-150*x^5)*exp(3)-8 *x^8-70*x^7-150*x^6)*log(exp(x)-exp(3)-x)+(-x^8-10*x^7-25*x^6)*exp(x)+x^8+ 10*x^7+25*x^6)/(exp(x)-exp(3)-x)/log(exp(x)-exp(3)-x)^2,x, algorithm=\
Time = 0.44 (sec) , antiderivative size = 130, normalized size of antiderivative = 5.00 \[ \int \frac {25 x^6+10 x^7+x^8+e^x \left (-25 x^6-10 x^7-x^8\right )+\left (-150 x^6-70 x^7-8 x^8+e^3 \left (-150 x^5-70 x^6-8 x^7\right )+e^x \left (150 x^5+70 x^6+8 x^7\right )\right ) \log \left (-e^3+e^x-x\right )}{\left (-e^3+e^x-x\right ) \log ^2\left (-e^3+e^x-x\right )} \, dx=\frac {x^6\,{\left (x+5\right )}^2+\frac {2\,x^5\,\ln \left ({\mathrm {e}}^x-{\mathrm {e}}^3-x\right )\,\left (4\,x^2+35\,x+75\right )\,\left (x+{\mathrm {e}}^3-{\mathrm {e}}^x\right )}{{\mathrm {e}}^x-1}}{\ln \left ({\mathrm {e}}^x-{\mathrm {e}}^3-x\right )}-\frac {150\,x^5\,{\mathrm {e}}^3+70\,x^6\,{\mathrm {e}}^3+8\,x^7\,{\mathrm {e}}^3-150\,x^5+80\,x^6+62\,x^7+8\,x^8}{{\mathrm {e}}^x-1}+150\,x^5+70\,x^6+8\,x^7 \]
int(-(25*x^6 - log(exp(x) - exp(3) - x)*(exp(3)*(150*x^5 + 70*x^6 + 8*x^7) - exp(x)*(150*x^5 + 70*x^6 + 8*x^7) + 150*x^6 + 70*x^7 + 8*x^8) + 10*x^7 + x^8 - exp(x)*(25*x^6 + 10*x^7 + x^8))/(log(exp(x) - exp(3) - x)^2*(x + e xp(3) - exp(x))),x)