Integrand size = 105, antiderivative size = 19 \[ \int \frac {e^{179 x/3} (-6-349 x)+e^x \left (-12-6 x+3 x^2\right )}{24 x^3+3 e^{176 x} x^3+36 x^4+18 x^5+3 x^6+e^{352 x/3} \left (18 x^3+9 x^4\right )+e^{176 x/3} \left (36 x^3+36 x^4+9 x^5\right )} \, dx=\frac {e^x}{x^2 \left (2+e^{176 x/3}+x\right )^2} \]
Time = 1.79 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {e^{179 x/3} (-6-349 x)+e^x \left (-12-6 x+3 x^2\right )}{24 x^3+3 e^{176 x} x^3+36 x^4+18 x^5+3 x^6+e^{352 x/3} \left (18 x^3+9 x^4\right )+e^{176 x/3} \left (36 x^3+36 x^4+9 x^5\right )} \, dx=\frac {e^x}{x^2 \left (2+e^{176 x/3}+x\right )^2} \]
Integrate[(E^((179*x)/3)*(-6 - 349*x) + E^x*(-12 - 6*x + 3*x^2))/(24*x^3 + 3*E^(176*x)*x^3 + 36*x^4 + 18*x^5 + 3*x^6 + E^((352*x)/3)*(18*x^3 + 9*x^4 ) + E^((176*x)/3)*(36*x^3 + 36*x^4 + 9*x^5)),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.
\(\displaystyle \int \frac {e^x \left (3 x^2-6 x-12\right )+e^{179 x/3} (-349 x-6)}{3 x^6+18 x^5+36 x^4+3 e^{176 x} x^3+24 x^3+e^{352 x/3} \left (9 x^4+18 x^3\right )+e^{176 x/3} \left (9 x^5+36 x^4+36 x^3\right )} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {e^x \left (3 x^2-6 x-e^{176 x/3} (349 x+6)-12\right )}{3 x^3 \left (x+e^{176 x/3}+2\right )^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \int -\frac {e^x \left (-3 x^2+6 x+e^{176 x/3} (349 x+6)+12\right )}{x^3 \left (x+e^{176 x/3}+2\right )^3}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{3} \int \frac {e^x \left (-3 x^2+6 x+e^{176 x/3} (349 x+6)+12\right )}{x^3 \left (x+e^{176 x/3}+2\right )^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {1}{3} \int \left (\frac {e^x (349 x+6)}{x^3 \left (x+e^{176 x/3}+2\right )^2}-\frac {2 e^x (176 x+349)}{x^2 \left (x+e^{176 x/3}+2\right )^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (-6 \int \frac {e^x}{x^3 \left (x+e^{176 x/3}+2\right )^2}dx+698 \int \frac {e^x}{x^2 \left (x+e^{176 x/3}+2\right )^3}dx-349 \int \frac {e^x}{x^2 \left (x+e^{176 x/3}+2\right )^2}dx+352 \int \frac {e^x}{x \left (x+e^{176 x/3}+2\right )^3}dx\right )\) |
Int[(E^((179*x)/3)*(-6 - 349*x) + E^x*(-12 - 6*x + 3*x^2))/(24*x^3 + 3*E^( 176*x)*x^3 + 36*x^4 + 18*x^5 + 3*x^6 + E^((352*x)/3)*(18*x^3 + 9*x^4) + E^ ((176*x)/3)*(36*x^3 + 36*x^4 + 9*x^5)),x]
3.8.2.3.1 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 = 0.56 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84
method | result | size |
risch | \(\frac {{\mathrm e}^{x}}{x^{2} \left ({\mathrm e}^{\frac {176 x}{3}}+x +2\right )^{2}}\) | \(16\) |
parallelrisch | \(\frac {{\mathrm e}^{x}}{x^{2} \left ({\mathrm e}^{\frac {352 x}{3}}+2 \,{\mathrm e}^{\frac {176 x}{3}} x +x^{2}+4 \,{\mathrm e}^{\frac {176 x}{3}}+4 x +4\right )}\) | \(40\) |
int(((-349*x-6)*exp(x)*exp(88/3*x)^2+(3*x^2-6*x-12)*exp(x))/(3*x^3*exp(88/ 3*x)^6+(9*x^4+18*x^3)*exp(88/3*x)^4+(9*x^5+36*x^4+36*x^3)*exp(88/3*x)^2+3* x^6+18*x^5+36*x^4+24*x^3),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (15) = 30\).
Time = 0.38 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.21 \[ \int \frac {e^{179 x/3} (-6-349 x)+e^x \left (-12-6 x+3 x^2\right )}{24 x^3+3 e^{176 x} x^3+36 x^4+18 x^5+3 x^6+e^{352 x/3} \left (18 x^3+9 x^4\right )+e^{176 x/3} \left (36 x^3+36 x^4+9 x^5\right )} \, dx=\frac {e^{x}}{x^{4} + 4 \, x^{3} + x^{2} e^{\left (\frac {352}{3} \, x\right )} + 4 \, x^{2} + 2 \, {\left (x^{3} + 2 \, x^{2}\right )} e^{\left (\frac {176}{3} \, x\right )}} \]
integrate(((-349*x-6)*exp(x)*exp(88/3*x)^2+(3*x^2-6*x-12)*exp(x))/(3*x^3*e xp(88/3*x)^6+(9*x^4+18*x^3)*exp(88/3*x)^4+(9*x^5+36*x^4+36*x^3)*exp(88/3*x )^2+3*x^6+18*x^5+36*x^4+24*x^3),x, algorithm=\
Timed out. \[ \int \frac {e^{179 x/3} (-6-349 x)+e^x \left (-12-6 x+3 x^2\right )}{24 x^3+3 e^{176 x} x^3+36 x^4+18 x^5+3 x^6+e^{352 x/3} \left (18 x^3+9 x^4\right )+e^{176 x/3} \left (36 x^3+36 x^4+9 x^5\right )} \, dx=\text {Timed out} \]
integrate(((-349*x-6)*exp(x)*exp(88/3*x)**2+(3*x**2-6*x-12)*exp(x))/(3*x** 3*exp(88/3*x)**6+(9*x**4+18*x**3)*exp(88/3*x)**4+(9*x**5+36*x**4+36*x**3)* exp(88/3*x)**2+3*x**6+18*x**5+36*x**4+24*x**3),x)
Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (15) = 30\).
Time = 1.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.21 \[ \int \frac {e^{179 x/3} (-6-349 x)+e^x \left (-12-6 x+3 x^2\right )}{24 x^3+3 e^{176 x} x^3+36 x^4+18 x^5+3 x^6+e^{352 x/3} \left (18 x^3+9 x^4\right )+e^{176 x/3} \left (36 x^3+36 x^4+9 x^5\right )} \, dx=\frac {e^{x}}{x^{4} + 4 \, x^{3} + x^{2} e^{\left (\frac {352}{3} \, x\right )} + 4 \, x^{2} + 2 \, {\left (x^{3} + 2 \, x^{2}\right )} e^{\left (\frac {176}{3} \, x\right )}} \]
integrate(((-349*x-6)*exp(x)*exp(88/3*x)^2+(3*x^2-6*x-12)*exp(x))/(3*x^3*e xp(88/3*x)^6+(9*x^4+18*x^3)*exp(88/3*x)^4+(9*x^5+36*x^4+36*x^3)*exp(88/3*x )^2+3*x^6+18*x^5+36*x^4+24*x^3),x, algorithm=\
\[ \int \frac {e^{179 x/3} (-6-349 x)+e^x \left (-12-6 x+3 x^2\right )}{24 x^3+3 e^{176 x} x^3+36 x^4+18 x^5+3 x^6+e^{352 x/3} \left (18 x^3+9 x^4\right )+e^{176 x/3} \left (36 x^3+36 x^4+9 x^5\right )} \, dx=\int { -\frac {{\left (349 \, x + 6\right )} e^{\left (\frac {179}{3} \, x\right )} - 3 \, {\left (x^{2} - 2 \, x - 4\right )} e^{x}}{3 \, {\left (x^{6} + 6 \, x^{5} + 12 \, x^{4} + x^{3} e^{\left (176 \, x\right )} + 8 \, x^{3} + 3 \, {\left (x^{4} + 2 \, x^{3}\right )} e^{\left (\frac {352}{3} \, x\right )} + 3 \, {\left (x^{5} + 4 \, x^{4} + 4 \, x^{3}\right )} e^{\left (\frac {176}{3} \, x\right )}\right )}} \,d x } \]
integrate(((-349*x-6)*exp(x)*exp(88/3*x)^2+(3*x^2-6*x-12)*exp(x))/(3*x^3*e xp(88/3*x)^6+(9*x^4+18*x^3)*exp(88/3*x)^4+(9*x^5+36*x^4+36*x^3)*exp(88/3*x )^2+3*x^6+18*x^5+36*x^4+24*x^3),x, algorithm=\
integrate(-1/3*((349*x + 6)*e^(179/3*x) - 3*(x^2 - 2*x - 4)*e^x)/(x^6 + 6* x^5 + 12*x^4 + x^3*e^(176*x) + 8*x^3 + 3*(x^4 + 2*x^3)*e^(352/3*x) + 3*(x^ 5 + 4*x^4 + 4*x^3)*e^(176/3*x)), x)
Timed out. \[ \int \frac {e^{179 x/3} (-6-349 x)+e^x \left (-12-6 x+3 x^2\right )}{24 x^3+3 e^{176 x} x^3+36 x^4+18 x^5+3 x^6+e^{352 x/3} \left (18 x^3+9 x^4\right )+e^{176 x/3} \left (36 x^3+36 x^4+9 x^5\right )} \, dx=\int -\frac {{\mathrm {e}}^x\,\left (-3\,x^2+6\,x+12\right )+{\mathrm {e}}^{\frac {176\,x}{3}}\,{\mathrm {e}}^x\,\left (349\,x+6\right )}{{\mathrm {e}}^{\frac {352\,x}{3}}\,\left (9\,x^4+18\,x^3\right )+3\,x^3\,{\mathrm {e}}^{176\,x}+{\mathrm {e}}^{\frac {176\,x}{3}}\,\left (9\,x^5+36\,x^4+36\,x^3\right )+24\,x^3+36\,x^4+18\,x^5+3\,x^6} \,d x \]
int(-(exp(x)*(6*x - 3*x^2 + 12) + exp((176*x)/3)*exp(x)*(349*x + 6))/(exp( (352*x)/3)*(18*x^3 + 9*x^4) + 3*x^3*exp(176*x) + exp((176*x)/3)*(36*x^3 + 36*x^4 + 9*x^5) + 24*x^3 + 36*x^4 + 18*x^5 + 3*x^6),x)