Integrand size = 146, antiderivative size = 27 \[ \int \frac {e^{20-4 e^x} \left (-144 x^2-576 x^3-864 x^4-576 x^5-144 x^6\right )+e^{10-2 e^x} \left (12 x+36 x^2+24 x^3+e^x \left (-12 x^2-24 x^3-12 x^4\right )\right )}{1+e^{10-2 e^x} \left (-48 x-96 x^2-48 x^3\right )+e^{20-4 e^x} \left (576 x^2+2304 x^3+3456 x^4+2304 x^5+576 x^6\right )} \, dx=\frac {x}{-4+\frac {e^{-10+2 e^x}}{6 x (1+x)^2}} \]
Time = 2.99 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {e^{20-4 e^x} \left (-144 x^2-576 x^3-864 x^4-576 x^5-144 x^6\right )+e^{10-2 e^x} \left (12 x+36 x^2+24 x^3+e^x \left (-12 x^2-24 x^3-12 x^4\right )\right )}{1+e^{10-2 e^x} \left (-48 x-96 x^2-48 x^3\right )+e^{20-4 e^x} \left (576 x^2+2304 x^3+3456 x^4+2304 x^5+576 x^6\right )} \, dx=-\frac {12 e^{10} x^2 (1+x)^2}{-2 e^{2 e^x}+48 e^{10} x (1+x)^2} \]
Integrate[(E^(20 - 4*E^x)*(-144*x^2 - 576*x^3 - 864*x^4 - 576*x^5 - 144*x^ 6) + E^(10 - 2*E^x)*(12*x + 36*x^2 + 24*x^3 + E^x*(-12*x^2 - 24*x^3 - 12*x ^4)))/(1 + E^(10 - 2*E^x)*(-48*x - 96*x^2 - 48*x^3) + E^(20 - 4*E^x)*(576* x^2 + 2304*x^3 + 3456*x^4 + 2304*x^5 + 576*x^6)),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.
\(\displaystyle \int \frac {e^{10-2 e^x} \left (24 x^3+36 x^2+e^x \left (-12 x^4-24 x^3-12 x^2\right )+12 x\right )+e^{20-4 e^x} \left (-144 x^6-576 x^5-864 x^4-576 x^3-144 x^2\right )}{e^{10-2 e^x} \left (-48 x^3-96 x^2-48 x\right )+e^{20-4 e^x} \left (576 x^6+2304 x^5+3456 x^4+2304 x^3+576 x^2\right )+1} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {12 e^{10} x (x+1) \left (-12 e^{10} x (x+1)^3-e^{x+2 e^x} x (x+1)+e^{2 e^x} (2 x+1)\right )}{\left (e^{2 e^x}-24 e^{10} x (x+1)^2\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 12 e^{10} \int -\frac {x (x+1) \left (12 e^{10} x (x+1)^3+e^{x+2 e^x} x (x+1)-e^{2 e^x} (2 x+1)\right )}{\left (e^{2 e^x}-24 e^{10} x (x+1)^2\right )^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -12 e^{10} \int \frac {x (x+1) \left (12 e^{10} x (x+1)^3+e^{x+2 e^x} x (x+1)-e^{2 e^x} (2 x+1)\right )}{\left (e^{2 e^x}-24 e^{10} x (x+1)^2\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -12 e^{10} \int \left (\frac {e^{x+2 e^x} x^2 (x+1)^2}{\left (-24 e^{10} x^3-48 e^{10} x^2-24 e^{10} x+e^{2 e^x}\right )^2}+\frac {x \left (12 e^{10} x^4+36 e^{10} x^3+36 e^{10} x^2-2 e^{2 e^x} x+12 e^{10} x-e^{2 e^x}\right ) (x+1)}{\left (24 e^{10} x^3+48 e^{10} x^2+24 e^{10} x-e^{2 e^x}\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -12 e^{10} \left (-12 e^{10} \int \frac {x^2}{\left (24 e^{10} x^3+48 e^{10} x^2+24 e^{10} x-e^{2 e^x}\right )^2}dx+\int \frac {e^{x+2 e^x} x^2}{\left (24 e^{10} x^3+48 e^{10} x^2+24 e^{10} x-e^{2 e^x}\right )^2}dx-72 e^{10} \int \frac {x^3}{\left (24 e^{10} x^3+48 e^{10} x^2+24 e^{10} x-e^{2 e^x}\right )^2}dx+2 \int \frac {e^{x+2 e^x} x^3}{\left (24 e^{10} x^3+48 e^{10} x^2+24 e^{10} x-e^{2 e^x}\right )^2}dx+\int \frac {x}{24 e^{10} x^3+48 e^{10} x^2+24 e^{10} x-e^{2 e^x}}dx+3 \int \frac {x^2}{24 e^{10} x^3+48 e^{10} x^2+24 e^{10} x-e^{2 e^x}}dx+2 \int \frac {x^3}{24 e^{10} x^3+48 e^{10} x^2+24 e^{10} x-e^{2 e^x}}dx-36 e^{10} \int \frac {x^6}{\left (24 e^{10} x^3+48 e^{10} x^2+24 e^{10} x-e^{2 e^x}\right )^2}dx-120 e^{10} \int \frac {x^5}{\left (24 e^{10} x^3+48 e^{10} x^2+24 e^{10} x-e^{2 e^x}\right )^2}dx-144 e^{10} \int \frac {x^4}{\left (24 e^{10} x^3+48 e^{10} x^2+24 e^{10} x-e^{2 e^x}\right )^2}dx+\int \frac {e^{x+2 e^x} x^4}{\left (24 e^{10} x^3+48 e^{10} x^2+24 e^{10} x-e^{2 e^x}\right )^2}dx\right )\) |
Int[(E^(20 - 4*E^x)*(-144*x^2 - 576*x^3 - 864*x^4 - 576*x^5 - 144*x^6) + E ^(10 - 2*E^x)*(12*x + 36*x^2 + 24*x^3 + E^x*(-12*x^2 - 24*x^3 - 12*x^4)))/ (1 + E^(10 - 2*E^x)*(-48*x - 96*x^2 - 48*x^3) + E^(20 - 4*E^x)*(576*x^2 + 2304*x^3 + 3456*x^4 + 2304*x^5 + 576*x^6)),x]
3.7.78.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 1.86 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.70
method | result | size |
risch | \(-\frac {x}{4}-\frac {x}{4 \left (24 \,{\mathrm e}^{-2 \,{\mathrm e}^{x}+10} x^{3}+48 \,{\mathrm e}^{-2 \,{\mathrm e}^{x}+10} x^{2}+24 \,{\mathrm e}^{-2 \,{\mathrm e}^{x}+10} x -1\right )}\) | \(46\) |
parallelrisch | \(-\frac {6 \,{\mathrm e}^{-2 \,{\mathrm e}^{x}+10} x^{4}+12 \,{\mathrm e}^{-2 \,{\mathrm e}^{x}+10} x^{3}+6 \,{\mathrm e}^{-2 \,{\mathrm e}^{x}+10} x^{2}}{24 \,{\mathrm e}^{-2 \,{\mathrm e}^{x}+10} x^{3}+48 \,{\mathrm e}^{-2 \,{\mathrm e}^{x}+10} x^{2}+24 \,{\mathrm e}^{-2 \,{\mathrm e}^{x}+10} x -1}\) | \(90\) |
int(((-144*x^6-576*x^5-864*x^4-576*x^3-144*x^2)*exp(5-exp(x))^4+((-12*x^4- 24*x^3-12*x^2)*exp(x)+24*x^3+36*x^2+12*x)*exp(5-exp(x))^2)/((576*x^6+2304* x^5+3456*x^4+2304*x^3+576*x^2)*exp(5-exp(x))^4+(-48*x^3-96*x^2-48*x)*exp(5 -exp(x))^2+1),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.63 \[ \int \frac {e^{20-4 e^x} \left (-144 x^2-576 x^3-864 x^4-576 x^5-144 x^6\right )+e^{10-2 e^x} \left (12 x+36 x^2+24 x^3+e^x \left (-12 x^2-24 x^3-12 x^4\right )\right )}{1+e^{10-2 e^x} \left (-48 x-96 x^2-48 x^3\right )+e^{20-4 e^x} \left (576 x^2+2304 x^3+3456 x^4+2304 x^5+576 x^6\right )} \, dx=-\frac {6 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} e^{\left (-2 \, e^{x} + 10\right )}}{24 \, {\left (x^{3} + 2 \, x^{2} + x\right )} e^{\left (-2 \, e^{x} + 10\right )} - 1} \]
integrate(((-144*x^6-576*x^5-864*x^4-576*x^3-144*x^2)*exp(5-exp(x))^4+((-1 2*x^4-24*x^3-12*x^2)*exp(x)+24*x^3+36*x^2+12*x)*exp(5-exp(x))^2)/((576*x^6 +2304*x^5+3456*x^4+2304*x^3+576*x^2)*exp(5-exp(x))^4+(-48*x^3-96*x^2-48*x) *exp(5-exp(x))^2+1),x, algorithm=\
Time = 0.21 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {e^{20-4 e^x} \left (-144 x^2-576 x^3-864 x^4-576 x^5-144 x^6\right )+e^{10-2 e^x} \left (12 x+36 x^2+24 x^3+e^x \left (-12 x^2-24 x^3-12 x^4\right )\right )}{1+e^{10-2 e^x} \left (-48 x-96 x^2-48 x^3\right )+e^{20-4 e^x} \left (576 x^2+2304 x^3+3456 x^4+2304 x^5+576 x^6\right )} \, dx=- \frac {x}{4} - \frac {x}{\left (96 x^{3} + 192 x^{2} + 96 x\right ) e^{10 - 2 e^{x}} - 4} \]
integrate(((-144*x**6-576*x**5-864*x**4-576*x**3-144*x**2)*exp(5-exp(x))** 4+((-12*x**4-24*x**3-12*x**2)*exp(x)+24*x**3+36*x**2+12*x)*exp(5-exp(x))** 2)/((576*x**6+2304*x**5+3456*x**4+2304*x**3+576*x**2)*exp(5-exp(x))**4+(-4 8*x**3-96*x**2-48*x)*exp(5-exp(x))**2+1),x)
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (23) = 46\).
Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.89 \[ \int \frac {e^{20-4 e^x} \left (-144 x^2-576 x^3-864 x^4-576 x^5-144 x^6\right )+e^{10-2 e^x} \left (12 x+36 x^2+24 x^3+e^x \left (-12 x^2-24 x^3-12 x^4\right )\right )}{1+e^{10-2 e^x} \left (-48 x-96 x^2-48 x^3\right )+e^{20-4 e^x} \left (576 x^2+2304 x^3+3456 x^4+2304 x^5+576 x^6\right )} \, dx=-\frac {6 \, {\left (x^{4} e^{10} + 2 \, x^{3} e^{10} + x^{2} e^{10}\right )}}{24 \, x^{3} e^{10} + 48 \, x^{2} e^{10} + 24 \, x e^{10} - e^{\left (2 \, e^{x}\right )}} \]
integrate(((-144*x^6-576*x^5-864*x^4-576*x^3-144*x^2)*exp(5-exp(x))^4+((-1 2*x^4-24*x^3-12*x^2)*exp(x)+24*x^3+36*x^2+12*x)*exp(5-exp(x))^2)/((576*x^6 +2304*x^5+3456*x^4+2304*x^3+576*x^2)*exp(5-exp(x))^4+(-48*x^3-96*x^2-48*x) *exp(5-exp(x))^2+1),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (23) = 46\).
Time = 0.29 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.89 \[ \int \frac {e^{20-4 e^x} \left (-144 x^2-576 x^3-864 x^4-576 x^5-144 x^6\right )+e^{10-2 e^x} \left (12 x+36 x^2+24 x^3+e^x \left (-12 x^2-24 x^3-12 x^4\right )\right )}{1+e^{10-2 e^x} \left (-48 x-96 x^2-48 x^3\right )+e^{20-4 e^x} \left (576 x^2+2304 x^3+3456 x^4+2304 x^5+576 x^6\right )} \, dx=-\frac {6 \, {\left (x^{4} e^{10} + 2 \, x^{3} e^{10} + x^{2} e^{10}\right )}}{24 \, x^{3} e^{10} + 48 \, x^{2} e^{10} + 24 \, x e^{10} - e^{\left (2 \, e^{x}\right )}} \]
integrate(((-144*x^6-576*x^5-864*x^4-576*x^3-144*x^2)*exp(5-exp(x))^4+((-1 2*x^4-24*x^3-12*x^2)*exp(x)+24*x^3+36*x^2+12*x)*exp(5-exp(x))^2)/((576*x^6 +2304*x^5+3456*x^4+2304*x^3+576*x^2)*exp(5-exp(x))^4+(-48*x^3-96*x^2-48*x) *exp(5-exp(x))^2+1),x, algorithm=\
Timed out. \[ \int \frac {e^{20-4 e^x} \left (-144 x^2-576 x^3-864 x^4-576 x^5-144 x^6\right )+e^{10-2 e^x} \left (12 x+36 x^2+24 x^3+e^x \left (-12 x^2-24 x^3-12 x^4\right )\right )}{1+e^{10-2 e^x} \left (-48 x-96 x^2-48 x^3\right )+e^{20-4 e^x} \left (576 x^2+2304 x^3+3456 x^4+2304 x^5+576 x^6\right )} \, dx=\int \frac {{\mathrm {e}}^{10-2\,{\mathrm {e}}^x}\,\left (12\,x-{\mathrm {e}}^x\,\left (12\,x^4+24\,x^3+12\,x^2\right )+36\,x^2+24\,x^3\right )-{\mathrm {e}}^{20-4\,{\mathrm {e}}^x}\,\left (144\,x^6+576\,x^5+864\,x^4+576\,x^3+144\,x^2\right )}{{\mathrm {e}}^{20-4\,{\mathrm {e}}^x}\,\left (576\,x^6+2304\,x^5+3456\,x^4+2304\,x^3+576\,x^2\right )-{\mathrm {e}}^{10-2\,{\mathrm {e}}^x}\,\left (48\,x^3+96\,x^2+48\,x\right )+1} \,d x \]
int((exp(10 - 2*exp(x))*(12*x - exp(x)*(12*x^2 + 24*x^3 + 12*x^4) + 36*x^2 + 24*x^3) - exp(20 - 4*exp(x))*(144*x^2 + 576*x^3 + 864*x^4 + 576*x^5 + 1 44*x^6))/(exp(20 - 4*exp(x))*(576*x^2 + 2304*x^3 + 3456*x^4 + 2304*x^5 + 5 76*x^6) - exp(10 - 2*exp(x))*(48*x + 96*x^2 + 48*x^3) + 1),x)