Integrand size = 218, antiderivative size = 28 \[ \int \frac {2^{\frac {5}{4 x+e^{\frac {9+6 x^2+x^4}{1-2 x+x^2}} x}} \left (\left (20-60 x+60 x^2-20 x^3\right ) \log (2)+e^{\frac {9+6 x^2+x^4}{1-2 x+x^2}} \left (5+75 x+75 x^2-5 x^3+20 x^4-10 x^5\right ) \log (2)\right )}{-16 x^2+48 x^3-48 x^4+16 x^5+e^{\frac {2 \left (9+6 x^2+x^4\right )}{1-2 x+x^2}} \left (-x^2+3 x^3-3 x^4+x^5\right )+e^{\frac {9+6 x^2+x^4}{1-2 x+x^2}} \left (-8 x^2+24 x^3-24 x^4+8 x^5\right )} \, dx=2^{\frac {5}{\left (4+e^{\left (3-x-\frac {4 x}{-1+x}\right )^2}\right ) x}} \]
Time = 0.27 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {2^{\frac {5}{4 x+e^{\frac {9+6 x^2+x^4}{1-2 x+x^2}} x}} \left (\left (20-60 x+60 x^2-20 x^3\right ) \log (2)+e^{\frac {9+6 x^2+x^4}{1-2 x+x^2}} \left (5+75 x+75 x^2-5 x^3+20 x^4-10 x^5\right ) \log (2)\right )}{-16 x^2+48 x^3-48 x^4+16 x^5+e^{\frac {2 \left (9+6 x^2+x^4\right )}{1-2 x+x^2}} \left (-x^2+3 x^3-3 x^4+x^5\right )+e^{\frac {9+6 x^2+x^4}{1-2 x+x^2}} \left (-8 x^2+24 x^3-24 x^4+8 x^5\right )} \, dx=\frac {5\ 2^{\frac {5}{\left (4+e^{\frac {\left (3+x^2\right )^2}{(-1+x)^2}}\right ) x}} \log (2)}{\log (32)} \]
Integrate[(2^(5/(4*x + E^((9 + 6*x^2 + x^4)/(1 - 2*x + x^2))*x))*((20 - 60 *x + 60*x^2 - 20*x^3)*Log[2] + E^((9 + 6*x^2 + x^4)/(1 - 2*x + x^2))*(5 + 75*x + 75*x^2 - 5*x^3 + 20*x^4 - 10*x^5)*Log[2]))/(-16*x^2 + 48*x^3 - 48*x ^4 + 16*x^5 + E^((2*(9 + 6*x^2 + x^4))/(1 - 2*x + x^2))*(-x^2 + 3*x^3 - 3* x^4 + x^5) + E^((9 + 6*x^2 + x^4)/(1 - 2*x + x^2))*(-8*x^2 + 24*x^3 - 24*x ^4 + 8*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.
| \(\displaystyle \int \frac {2^{\frac {5}{e^{\frac {x^4+6 x^2+9}{x^2-2 x+1}} x+4 x}} \left (\left (-20 x^3+60 x^2-60 x+20\right ) \log (2)+e^{\frac {x^4+6 x^2+9}{x^2-2 x+1}} \left (-10 x^5+20 x^4-5 x^3+75 x^2+75 x+5\right ) \log (2)\right )}{16 x^5-48 x^4+48 x^3-16 x^2+e^{\frac {2 \left (x^4+6 x^2+9\right )}{x^2-2 x+1}} \left (x^5-3 x^4+3 x^3-x^2\right )+e^{\frac {x^4+6 x^2+9}{x^2-2 x+1}} \left (8 x^5-24 x^4+24 x^3-8 x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {5\ 2^{\frac {5}{\left (e^{\frac {\left (x^2+3\right )^2}{(x-1)^2}}+4\right ) x}} \left (e^{\frac {\left (x^2+3\right )^2}{(x-1)^2}} \left (2 x^5-4 x^4+x^3-15 x^2-15 x-1\right )+4 (x-1)^3\right ) \log (2)}{\left (e^{\frac {\left (x^2+3\right )^2}{(x-1)^2}}+4\right )^2 (1-x)^3 x^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 5 \log (2) \int -\frac {2^{\frac {5}{\left (4+e^{\frac {\left (x^2+3\right )^2}{(1-x)^2}}\right ) x}} \left (4 (1-x)^3+e^{\frac {\left (x^2+3\right )^2}{(1-x)^2}} \left (-2 x^5+4 x^4-x^3+15 x^2+15 x+1\right )\right )}{\left (4+e^{\frac {\left (x^2+3\right )^2}{(1-x)^2}}\right )^2 (1-x)^3 x^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -5 \log (2) \int \frac {2^{\frac {5}{\left (4+e^{\frac {\left (x^2+3\right )^2}{(1-x)^2}}\right ) x}} \left (4 (1-x)^3+e^{\frac {\left (x^2+3\right )^2}{(1-x)^2}} \left (-2 x^5+4 x^4-x^3+15 x^2+15 x+1\right )\right )}{\left (4+e^{\frac {\left (x^2+3\right )^2}{(1-x)^2}}\right )^2 (1-x)^3 x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -5 \log (2) \int \left (\frac {2^{\frac {5}{\left (4+e^{\frac {\left (x^2+3\right )^2}{(1-x)^2}}\right ) x}} \left (2 x^5-4 x^4+x^3-15 x^2-15 x-1\right )}{\left (4+e^{\frac {\left (x^2+3\right )^2}{(x-1)^2}}\right ) (x-1)^3 x^2}-\frac {2^{3+\frac {5}{\left (4+e^{\frac {\left (x^2+3\right )^2}{(1-x)^2}}\right ) x}} \left (x^4-2 x^3-6 x-9\right )}{\left (4+e^{\frac {\left (x^2+3\right )^2}{(x-1)^2}}\right )^2 (x-1)^3 x}\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -5 \log (2) \int \left (\frac {2^{\frac {5}{\left (4+e^{\frac {\left (x^2+3\right )^2}{(x-1)^2}}\right ) x}} \left (2 x^5-4 x^4+x^3-15 x^2-15 x-1\right )}{\left (4+e^{\frac {\left (x^2+3\right )^2}{(x-1)^2}}\right ) (x-1)^3 x^2}-\frac {2^{3+\frac {5}{\left (4+e^{\frac {\left (x^2+3\right )^2}{(x-1)^2}}\right ) x}} \left (x^4-2 x^3-6 x-9\right )}{\left (4+e^{\frac {\left (x^2+3\right )^2}{(x-1)^2}}\right )^2 (x-1)^3 x}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle -5 \log (2) \int \left (\frac {2^{\frac {5}{\left (4+e^{\frac {\left (x^2+3\right )^2}{(x-1)^2}}\right ) x}} \left (2 x^5-4 x^4+x^3-15 x^2-15 x-1\right )}{\left (4+e^{\frac {\left (x^2+3\right )^2}{(x-1)^2}}\right ) (x-1)^3 x^2}-\frac {2^{3+\frac {5}{\left (4+e^{\frac {\left (x^2+3\right )^2}{(x-1)^2}}\right ) x}} \left (x^4-2 x^3-6 x-9\right )}{\left (4+e^{\frac {\left (x^2+3\right )^2}{(x-1)^2}}\right )^2 (x-1)^3 x}\right )dx\) |
Int[(2^(5/(4*x + E^((9 + 6*x^2 + x^4)/(1 - 2*x + x^2))*x))*((20 - 60*x + 6 0*x^2 - 20*x^3)*Log[2] + E^((9 + 6*x^2 + x^4)/(1 - 2*x + x^2))*(5 + 75*x + 75*x^2 - 5*x^3 + 20*x^4 - 10*x^5)*Log[2]))/(-16*x^2 + 48*x^3 - 48*x^4 + 1 6*x^5 + E^((2*(9 + 6*x^2 + x^4))/(1 - 2*x + x^2))*(-x^2 + 3*x^3 - 3*x^4 + x^5) + E^((9 + 6*x^2 + x^4)/(1 - 2*x + x^2))*(-8*x^2 + 24*x^3 - 24*x^4 + 8 *x^5)),x]
3.10.42.3.1 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 = 21.83 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89
| method | result | size |
| risch | \(32^{\frac {1}{x \left ({\mathrm e}^{\frac {\left (x^{2}+3\right )^{2}}{\left (-1+x \right )^{2}}}+4\right )}}\) | \(25\) |
| parallelrisch | \({\mathrm e}^{\frac {5 \ln \left (2\right )}{x \left ({\mathrm e}^{\frac {x^{4}+6 x^{2}+9}{x^{2}-2 x +1}}+4\right )}}\) | \(35\) |
int(((-10*x^5+20*x^4-5*x^3+75*x^2+75*x+5)*ln(2)*exp((x^4+6*x^2+9)/(x^2-2*x +1))+(-20*x^3+60*x^2-60*x+20)*ln(2))*exp(5*ln(2)/(x*exp((x^4+6*x^2+9)/(x^2 -2*x+1))+4*x))/((x^5-3*x^4+3*x^3-x^2)*exp((x^4+6*x^2+9)/(x^2-2*x+1))^2+(8* x^5-24*x^4+24*x^3-8*x^2)*exp((x^4+6*x^2+9)/(x^2-2*x+1))+16*x^5-48*x^4+48*x ^3-16*x^2),x,method=_RETURNVERBOSE)
Time = 0.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {2^{\frac {5}{4 x+e^{\frac {9+6 x^2+x^4}{1-2 x+x^2}} x}} \left (\left (20-60 x+60 x^2-20 x^3\right ) \log (2)+e^{\frac {9+6 x^2+x^4}{1-2 x+x^2}} \left (5+75 x+75 x^2-5 x^3+20 x^4-10 x^5\right ) \log (2)\right )}{-16 x^2+48 x^3-48 x^4+16 x^5+e^{\frac {2 \left (9+6 x^2+x^4\right )}{1-2 x+x^2}} \left (-x^2+3 x^3-3 x^4+x^5\right )+e^{\frac {9+6 x^2+x^4}{1-2 x+x^2}} \left (-8 x^2+24 x^3-24 x^4+8 x^5\right )} \, dx=2^{\frac {5}{x e^{\left (\frac {x^{4} + 6 \, x^{2} + 9}{x^{2} - 2 \, x + 1}\right )} + 4 \, x}} \]
integrate(((-10*x^5+20*x^4-5*x^3+75*x^2+75*x+5)*log(2)*exp((x^4+6*x^2+9)/( x^2-2*x+1))+(-20*x^3+60*x^2-60*x+20)*log(2))*exp(5*log(2)/(x*exp((x^4+6*x^ 2+9)/(x^2-2*x+1))+4*x))/((x^5-3*x^4+3*x^3-x^2)*exp((x^4+6*x^2+9)/(x^2-2*x+ 1))^2+(8*x^5-24*x^4+24*x^3-8*x^2)*exp((x^4+6*x^2+9)/(x^2-2*x+1))+16*x^5-48 *x^4+48*x^3-16*x^2),x, algorithm=\
Time = 0.75 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {2^{\frac {5}{4 x+e^{\frac {9+6 x^2+x^4}{1-2 x+x^2}} x}} \left (\left (20-60 x+60 x^2-20 x^3\right ) \log (2)+e^{\frac {9+6 x^2+x^4}{1-2 x+x^2}} \left (5+75 x+75 x^2-5 x^3+20 x^4-10 x^5\right ) \log (2)\right )}{-16 x^2+48 x^3-48 x^4+16 x^5+e^{\frac {2 \left (9+6 x^2+x^4\right )}{1-2 x+x^2}} \left (-x^2+3 x^3-3 x^4+x^5\right )+e^{\frac {9+6 x^2+x^4}{1-2 x+x^2}} \left (-8 x^2+24 x^3-24 x^4+8 x^5\right )} \, dx=e^{\frac {5 \log {\left (2 \right )}}{x e^{\frac {x^{4} + 6 x^{2} + 9}{x^{2} - 2 x + 1}} + 4 x}} \]
integrate(((-10*x**5+20*x**4-5*x**3+75*x**2+75*x+5)*ln(2)*exp((x**4+6*x**2 +9)/(x**2-2*x+1))+(-20*x**3+60*x**2-60*x+20)*ln(2))*exp(5*ln(2)/(x*exp((x* *4+6*x**2+9)/(x**2-2*x+1))+4*x))/((x**5-3*x**4+3*x**3-x**2)*exp((x**4+6*x* *2+9)/(x**2-2*x+1))**2+(8*x**5-24*x**4+24*x**3-8*x**2)*exp((x**4+6*x**2+9) /(x**2-2*x+1))+16*x**5-48*x**4+48*x**3-16*x**2),x)
Time = 0.47 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43 \[ \int \frac {2^{\frac {5}{4 x+e^{\frac {9+6 x^2+x^4}{1-2 x+x^2}} x}} \left (\left (20-60 x+60 x^2-20 x^3\right ) \log (2)+e^{\frac {9+6 x^2+x^4}{1-2 x+x^2}} \left (5+75 x+75 x^2-5 x^3+20 x^4-10 x^5\right ) \log (2)\right )}{-16 x^2+48 x^3-48 x^4+16 x^5+e^{\frac {2 \left (9+6 x^2+x^4\right )}{1-2 x+x^2}} \left (-x^2+3 x^3-3 x^4+x^5\right )+e^{\frac {9+6 x^2+x^4}{1-2 x+x^2}} \left (-8 x^2+24 x^3-24 x^4+8 x^5\right )} \, dx=2^{\frac {5}{x e^{\left (x^{2} + 2 \, x + \frac {16}{x^{2} - 2 \, x + 1} + \frac {16}{x - 1} + 9\right )} + 4 \, x}} \]
integrate(((-10*x^5+20*x^4-5*x^3+75*x^2+75*x+5)*log(2)*exp((x^4+6*x^2+9)/( x^2-2*x+1))+(-20*x^3+60*x^2-60*x+20)*log(2))*exp(5*log(2)/(x*exp((x^4+6*x^ 2+9)/(x^2-2*x+1))+4*x))/((x^5-3*x^4+3*x^3-x^2)*exp((x^4+6*x^2+9)/(x^2-2*x+ 1))^2+(8*x^5-24*x^4+24*x^3-8*x^2)*exp((x^4+6*x^2+9)/(x^2-2*x+1))+16*x^5-48 *x^4+48*x^3-16*x^2),x, algorithm=\
\[ \int \frac {2^{\frac {5}{4 x+e^{\frac {9+6 x^2+x^4}{1-2 x+x^2}} x}} \left (\left (20-60 x+60 x^2-20 x^3\right ) \log (2)+e^{\frac {9+6 x^2+x^4}{1-2 x+x^2}} \left (5+75 x+75 x^2-5 x^3+20 x^4-10 x^5\right ) \log (2)\right )}{-16 x^2+48 x^3-48 x^4+16 x^5+e^{\frac {2 \left (9+6 x^2+x^4\right )}{1-2 x+x^2}} \left (-x^2+3 x^3-3 x^4+x^5\right )+e^{\frac {9+6 x^2+x^4}{1-2 x+x^2}} \left (-8 x^2+24 x^3-24 x^4+8 x^5\right )} \, dx=\int { -\frac {5 \, {\left ({\left (2 \, x^{5} - 4 \, x^{4} + x^{3} - 15 \, x^{2} - 15 \, x - 1\right )} e^{\left (\frac {x^{4} + 6 \, x^{2} + 9}{x^{2} - 2 \, x + 1}\right )} \log \left (2\right ) + 4 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )} \log \left (2\right )\right )} 2^{\frac {5}{x e^{\left (\frac {x^{4} + 6 \, x^{2} + 9}{x^{2} - 2 \, x + 1}\right )} + 4 \, x}}}{16 \, x^{5} - 48 \, x^{4} + 48 \, x^{3} - 16 \, x^{2} + {\left (x^{5} - 3 \, x^{4} + 3 \, x^{3} - x^{2}\right )} e^{\left (\frac {2 \, {\left (x^{4} + 6 \, x^{2} + 9\right )}}{x^{2} - 2 \, x + 1}\right )} + 8 \, {\left (x^{5} - 3 \, x^{4} + 3 \, x^{3} - x^{2}\right )} e^{\left (\frac {x^{4} + 6 \, x^{2} + 9}{x^{2} - 2 \, x + 1}\right )}} \,d x } \]
integrate(((-10*x^5+20*x^4-5*x^3+75*x^2+75*x+5)*log(2)*exp((x^4+6*x^2+9)/( x^2-2*x+1))+(-20*x^3+60*x^2-60*x+20)*log(2))*exp(5*log(2)/(x*exp((x^4+6*x^ 2+9)/(x^2-2*x+1))+4*x))/((x^5-3*x^4+3*x^3-x^2)*exp((x^4+6*x^2+9)/(x^2-2*x+ 1))^2+(8*x^5-24*x^4+24*x^3-8*x^2)*exp((x^4+6*x^2+9)/(x^2-2*x+1))+16*x^5-48 *x^4+48*x^3-16*x^2),x, algorithm=\
integrate(-5*((2*x^5 - 4*x^4 + x^3 - 15*x^2 - 15*x - 1)*e^((x^4 + 6*x^2 + 9)/(x^2 - 2*x + 1))*log(2) + 4*(x^3 - 3*x^2 + 3*x - 1)*log(2))*2^(5/(x*e^( (x^4 + 6*x^2 + 9)/(x^2 - 2*x + 1)) + 4*x))/(16*x^5 - 48*x^4 + 48*x^3 - 16* x^2 + (x^5 - 3*x^4 + 3*x^3 - x^2)*e^(2*(x^4 + 6*x^2 + 9)/(x^2 - 2*x + 1)) + 8*(x^5 - 3*x^4 + 3*x^3 - x^2)*e^((x^4 + 6*x^2 + 9)/(x^2 - 2*x + 1))), x)
Time = 10.14 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.00 \[ \int \frac {2^{\frac {5}{4 x+e^{\frac {9+6 x^2+x^4}{1-2 x+x^2}} x}} \left (\left (20-60 x+60 x^2-20 x^3\right ) \log (2)+e^{\frac {9+6 x^2+x^4}{1-2 x+x^2}} \left (5+75 x+75 x^2-5 x^3+20 x^4-10 x^5\right ) \log (2)\right )}{-16 x^2+48 x^3-48 x^4+16 x^5+e^{\frac {2 \left (9+6 x^2+x^4\right )}{1-2 x+x^2}} \left (-x^2+3 x^3-3 x^4+x^5\right )+e^{\frac {9+6 x^2+x^4}{1-2 x+x^2}} \left (-8 x^2+24 x^3-24 x^4+8 x^5\right )} \, dx=2^{\frac {5}{4\,x+x\,{\mathrm {e}}^{\frac {x^4}{x^2-2\,x+1}}\,{\mathrm {e}}^{\frac {6\,x^2}{x^2-2\,x+1}}\,{\mathrm {e}}^{\frac {9}{x^2-2\,x+1}}}} \]
int((exp((5*log(2))/(4*x + x*exp((6*x^2 + x^4 + 9)/(x^2 - 2*x + 1))))*(log (2)*(60*x - 60*x^2 + 20*x^3 - 20) - exp((6*x^2 + x^4 + 9)/(x^2 - 2*x + 1)) *log(2)*(75*x + 75*x^2 - 5*x^3 + 20*x^4 - 10*x^5 + 5)))/(exp((6*x^2 + x^4 + 9)/(x^2 - 2*x + 1))*(8*x^2 - 24*x^3 + 24*x^4 - 8*x^5) + 16*x^2 - 48*x^3 + 48*x^4 - 16*x^5 + exp((2*(6*x^2 + x^4 + 9))/(x^2 - 2*x + 1))*(x^2 - 3*x^ 3 + 3*x^4 - x^5)),x)