Integrand size = 122, antiderivative size = 30 \[ \int \frac {e^{\frac {-2 e^{16} x+2 e^{16} \log (2)}{-100-5 x-19 x^2+(5+19 x) \log (2)}} \left (e^{16} \left (600-114 x^2\right )+228 e^{16} x \log (2)-114 e^{16} \log ^2(2)\right )}{10000+1000 x+3825 x^2+190 x^3+361 x^4+\left (-1000-3850 x-380 x^2-722 x^3\right ) \log (2)+\left (25+190 x+361 x^2\right ) \log ^2(2)} \, dx=3 e^{\frac {2 e^{16}}{5-x+20 \left (x+\frac {5}{x-\log (2)}\right )}} \]
\[ \int \frac {e^{\frac {-2 e^{16} x+2 e^{16} \log (2)}{-100-5 x-19 x^2+(5+19 x) \log (2)}} \left (e^{16} \left (600-114 x^2\right )+228 e^{16} x \log (2)-114 e^{16} \log ^2(2)\right )}{10000+1000 x+3825 x^2+190 x^3+361 x^4+\left (-1000-3850 x-380 x^2-722 x^3\right ) \log (2)+\left (25+190 x+361 x^2\right ) \log ^2(2)} \, dx=\int \frac {e^{\frac {-2 e^{16} x+2 e^{16} \log (2)}{-100-5 x-19 x^2+(5+19 x) \log (2)}} \left (e^{16} \left (600-114 x^2\right )+228 e^{16} x \log (2)-114 e^{16} \log ^2(2)\right )}{10000+1000 x+3825 x^2+190 x^3+361 x^4+\left (-1000-3850 x-380 x^2-722 x^3\right ) \log (2)+\left (25+190 x+361 x^2\right ) \log ^2(2)} \, dx \]
Integrate[(E^((-2*E^16*x + 2*E^16*Log[2])/(-100 - 5*x - 19*x^2 + (5 + 19*x )*Log[2]))*(E^16*(600 - 114*x^2) + 228*E^16*x*Log[2] - 114*E^16*Log[2]^2)) /(10000 + 1000*x + 3825*x^2 + 190*x^3 + 361*x^4 + (-1000 - 3850*x - 380*x^ 2 - 722*x^3)*Log[2] + (25 + 190*x + 361*x^2)*Log[2]^2),x]
Integrate[(E^((-2*E^16*x + 2*E^16*Log[2])/(-100 - 5*x - 19*x^2 + (5 + 19*x )*Log[2]))*(E^16*(600 - 114*x^2) + 228*E^16*x*Log[2] - 114*E^16*Log[2]^2)) /(10000 + 1000*x + 3825*x^2 + 190*x^3 + 361*x^4 + (-1000 - 3850*x - 380*x^ 2 - 722*x^3)*Log[2] + (25 + 190*x + 361*x^2)*Log[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.
| \(\displaystyle \int \frac {\left (e^{16} \left (600-114 x^2\right )+228 e^{16} x \log (2)-114 e^{16} \log ^2(2)\right ) \exp \left (\frac {2 e^{16} \log (2)-2 e^{16} x}{-19 x^2-5 x+(19 x+5) \log (2)-100}\right )}{361 x^4+190 x^3+3825 x^2+\left (361 x^2+190 x+25\right ) \log ^2(2)+\left (-722 x^3-380 x^2-3850 x-1000\right ) \log (2)+1000 x+10000} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \frac {\left (e^{16} \left (600-114 x^2\right )+228 e^{16} x \log (2)-114 e^{16} \log ^2(2)\right ) \exp \left (\frac {2 e^{16} \log (2)-2 e^{16} x}{-19 x^2-5 x+(19 x+5) \log (2)-100}\right )}{\left (19 x^2+5 x-19 x \log (2)+100-5 \log (2)\right )^2}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {\left (e^{16} \left (600-114 x^2\right )+228 e^{16} x \log (2)-114 e^{16} \log ^2(2)\right ) \exp \left (\frac {2 e^{16} \log (2)-2 e^{16} x}{-19 x^2-5 x+(19 x+5) \log (2)-100}\right )}{\left (19 x^2+x (5-19 \log (2))+100-5 \log (2)\right )^2}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (-114 e^{16} x^2+228 e^{16} x \log (2)+6 e^{16} \left (100-19 \log ^2(2)\right )\right ) \exp \left (\frac {2 e^{16} \log (2)-2 e^{16} x}{-19 x^2-x (5-19 \log (2))-100+\log (32)}\right )}{\left (19 x^2+x (5-19 \log (2))+5 (20-\log (2))\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {6 \left (x (5+19 \log (2))+200-19 \log ^2(2)-5 \log (2)\right ) \exp \left (\frac {2 e^{16} \log (2)-2 e^{16} x}{-19 x^2-x (5-19 \log (2))-100+\log (32)}+16\right )}{\left (19 x^2+x (5-19 \log (2))+5 (20-\log (2))\right )^2}+\frac {6 \exp \left (\frac {2 e^{16} \log (2)-2 e^{16} x}{-19 x^2-x (5-19 \log (2))-100+\log (32)}+16\right )}{-19 x^2-x (5-19 \log (2))-5 (20-\log (2))}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {8664 (200-\log (32)-\log (2) \log (524288)) \int \frac {e^{\frac {2 e^{16} \log (2)-2 e^{16} x}{-19 x^2-(5-19 \log (2)) x+\log (32)-100}+16}}{\left (38 x+i \sqrt {7575-190 \log (2)-361 \log ^2(2)}-19 \log (2)+5\right )^2}dx}{7575-190 \log (2)-361 \log ^2(2)}+\frac {228 (5+19 \log (2)) \left (5-19 \log (2)+i \sqrt {7575-190 \log (2)-361 \log ^2(2)}\right ) \int \frac {e^{\frac {2 e^{16} \log (2)-2 e^{16} x}{-19 x^2-(5-19 \log (2)) x+\log (32)-100}+16}}{\left (38 x+i \sqrt {7575-190 \log (2)-361 \log ^2(2)}-19 \log (2)+5\right )^2}dx}{7575-190 \log (2)-361 \log ^2(2)}+\frac {8664 i (200-\log (2) (5+\log (524288))) \int \frac {e^{\frac {2 e^{16} \log (2)-2 e^{16} x}{-19 x^2-(5-19 \log (2)) x+\log (32)-100}+16}}{38 x+i \sqrt {7575-190 \log (2)-361 \log ^2(2)}-19 \log (2)+5}dx}{\left (7575-190 \log (2)-361 \log ^2(2)\right )^{3/2}}-\frac {228 i \int \frac {e^{\frac {2 e^{16} \log (2)-2 e^{16} x}{-19 x^2-(5-19 \log (2)) x+\log (32)-100}+16}}{38 x+i \sqrt {7575-190 \log (2)-361 \log ^2(2)}-19 \log (2)+5}dx}{\sqrt {7575-190 \log (2)-361 \log ^2(2)}}-\frac {228 i (5-19 \log (2)) (5+19 \log (2)) \int \frac {e^{\frac {2 e^{16} \log (2)-2 e^{16} x}{-19 x^2-(5-19 \log (2)) x+\log (32)-100}+16}}{38 x+i \sqrt {7575-190 \log (2)-361 \log ^2(2)}-19 \log (2)+5}dx}{\left (7575-190 \log (2)-361 \log ^2(2)\right )^{3/2}}-\frac {8664 (200-\log (32)-\log (2) \log (524288)) \int \frac {e^{\frac {2 e^{16} \log (2)-2 e^{16} x}{-19 x^2-(5-19 \log (2)) x+\log (32)-100}+16}}{\left (-38 x+i \sqrt {7575-190 \log (2)-361 \log ^2(2)}+19 \log (2)-5\right )^2}dx}{7575-190 \log (2)-361 \log ^2(2)}+\frac {228 (5+19 \log (2)) \left (5-19 \log (2)-i \sqrt {7575-190 \log (2)-361 \log ^2(2)}\right ) \int \frac {e^{\frac {2 e^{16} \log (2)-2 e^{16} x}{-19 x^2-(5-19 \log (2)) x+\log (32)-100}+16}}{\left (-38 x+i \sqrt {7575-190 \log (2)-361 \log ^2(2)}+19 \log (2)-5\right )^2}dx}{7575-190 \log (2)-361 \log ^2(2)}+\frac {8664 i (200-\log (2) (5+\log (524288))) \int \frac {e^{\frac {2 e^{16} \log (2)-2 e^{16} x}{-19 x^2-(5-19 \log (2)) x+\log (32)-100}+16}}{-38 x+i \sqrt {7575-190 \log (2)-361 \log ^2(2)}+19 \log (2)-5}dx}{\left (7575-190 \log (2)-361 \log ^2(2)\right )^{3/2}}-\frac {228 i \int \frac {e^{\frac {2 e^{16} \log (2)-2 e^{16} x}{-19 x^2-(5-19 \log (2)) x+\log (32)-100}+16}}{-38 x+i \sqrt {7575-190 \log (2)-361 \log ^2(2)}+19 \log (2)-5}dx}{\sqrt {7575-190 \log (2)-361 \log ^2(2)}}-\frac {228 i (5-19 \log (2)) (5+19 \log (2)) \int \frac {e^{\frac {2 e^{16} \log (2)-2 e^{16} x}{-19 x^2-(5-19 \log (2)) x+\log (32)-100}+16}}{-38 x+i \sqrt {7575-190 \log (2)-361 \log ^2(2)}+19 \log (2)-5}dx}{\left (7575-190 \log (2)-361 \log ^2(2)\right )^{3/2}}\) |
Int[(E^((-2*E^16*x + 2*E^16*Log[2])/(-100 - 5*x - 19*x^2 + (5 + 19*x)*Log[ 2]))*(E^16*(600 - 114*x^2) + 228*E^16*x*Log[2] - 114*E^16*Log[2]^2))/(1000 0 + 1000*x + 3825*x^2 + 190*x^3 + 361*x^4 + (-1000 - 3850*x - 380*x^2 - 72 2*x^3)*Log[2] + (25 + 190*x + 361*x^2)*Log[2]^2),x]
3.13.28.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 0.49 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17
| method | result | size |
| gosper | \(3 \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{16} \left (\ln \left (2\right )-x \right )}{19 x \ln \left (2\right )-19 x^{2}+5 \ln \left (2\right )-5 x -100}}\) | \(35\) |
| risch | \(3 \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{16} \left (\ln \left (2\right )-x \right )}{19 x \ln \left (2\right )-19 x^{2}+5 \ln \left (2\right )-5 x -100}}\) | \(35\) |
| parallelrisch | \(3 \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{16} \left (\ln \left (2\right )-x \right )}{19 x \ln \left (2\right )-19 x^{2}+5 \ln \left (2\right )-5 x -100}}\) | \(35\) |
| norman | \(\frac {\left (15 \ln \left (2\right )-300\right ) {\mathrm e}^{\frac {2 \,{\mathrm e}^{16} \ln \left (2\right )-2 x \,{\mathrm e}^{16}}{\left (19 x +5\right ) \ln \left (2\right )-19 x^{2}-5 x -100}}+\left (-15+57 \ln \left (2\right )\right ) x \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{16} \ln \left (2\right )-2 x \,{\mathrm e}^{16}}{\left (19 x +5\right ) \ln \left (2\right )-19 x^{2}-5 x -100}}-57 x^{2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{16} \ln \left (2\right )-2 x \,{\mathrm e}^{16}}{\left (19 x +5\right ) \ln \left (2\right )-19 x^{2}-5 x -100}}}{19 x \ln \left (2\right )-19 x^{2}+5 \ln \left (2\right )-5 x -100}\) | \(146\) |
int((-114*exp(16)*ln(2)^2+228*x*exp(16)*ln(2)+(-114*x^2+600)*exp(16))*exp( (2*exp(16)*ln(2)-2*x*exp(16))/((19*x+5)*ln(2)-19*x^2-5*x-100))/((361*x^2+1 90*x+25)*ln(2)^2+(-722*x^3-380*x^2-3850*x-1000)*ln(2)+361*x^4+190*x^3+3825 *x^2+1000*x+10000),x,method=_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {e^{\frac {-2 e^{16} x+2 e^{16} \log (2)}{-100-5 x-19 x^2+(5+19 x) \log (2)}} \left (e^{16} \left (600-114 x^2\right )+228 e^{16} x \log (2)-114 e^{16} \log ^2(2)\right )}{10000+1000 x+3825 x^2+190 x^3+361 x^4+\left (-1000-3850 x-380 x^2-722 x^3\right ) \log (2)+\left (25+190 x+361 x^2\right ) \log ^2(2)} \, dx=3 \, e^{\left (\frac {2 \, {\left (x e^{16} - e^{16} \log \left (2\right )\right )}}{19 \, x^{2} - {\left (19 \, x + 5\right )} \log \left (2\right ) + 5 \, x + 100}\right )} \]
integrate((-114*exp(16)*log(2)^2+228*x*exp(16)*log(2)+(-114*x^2+600)*exp(1 6))*exp((2*exp(16)*log(2)-2*x*exp(16))/((19*x+5)*log(2)-19*x^2-5*x-100))/( (361*x^2+190*x+25)*log(2)^2+(-722*x^3-380*x^2-3850*x-1000)*log(2)+361*x^4+ 190*x^3+3825*x^2+1000*x+10000),x, algorithm=\
Time = 0.51 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20 \[ \int \frac {e^{\frac {-2 e^{16} x+2 e^{16} \log (2)}{-100-5 x-19 x^2+(5+19 x) \log (2)}} \left (e^{16} \left (600-114 x^2\right )+228 e^{16} x \log (2)-114 e^{16} \log ^2(2)\right )}{10000+1000 x+3825 x^2+190 x^3+361 x^4+\left (-1000-3850 x-380 x^2-722 x^3\right ) \log (2)+\left (25+190 x+361 x^2\right ) \log ^2(2)} \, dx=3 e^{\frac {- 2 x e^{16} + 2 e^{16} \log {\left (2 \right )}}{- 19 x^{2} - 5 x + \left (19 x + 5\right ) \log {\left (2 \right )} - 100}} \]
integrate((-114*exp(16)*ln(2)**2+228*x*exp(16)*ln(2)+(-114*x**2+600)*exp(1 6))*exp((2*exp(16)*ln(2)-2*x*exp(16))/((19*x+5)*ln(2)-19*x**2-5*x-100))/(( 361*x**2+190*x+25)*ln(2)**2+(-722*x**3-380*x**2-3850*x-1000)*ln(2)+361*x** 4+190*x**3+3825*x**2+1000*x+10000),x)
Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (24) = 48\).
Time = 0.43 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.97 \[ \int \frac {e^{\frac {-2 e^{16} x+2 e^{16} \log (2)}{-100-5 x-19 x^2+(5+19 x) \log (2)}} \left (e^{16} \left (600-114 x^2\right )+228 e^{16} x \log (2)-114 e^{16} \log ^2(2)\right )}{10000+1000 x+3825 x^2+190 x^3+361 x^4+\left (-1000-3850 x-380 x^2-722 x^3\right ) \log (2)+\left (25+190 x+361 x^2\right ) \log ^2(2)} \, dx=3 \, e^{\left (\frac {2 \, x e^{16}}{19 \, x^{2} - x {\left (19 \, \log \left (2\right ) - 5\right )} - 5 \, \log \left (2\right ) + 100} - \frac {2 \, e^{16} \log \left (2\right )}{19 \, x^{2} - x {\left (19 \, \log \left (2\right ) - 5\right )} - 5 \, \log \left (2\right ) + 100}\right )} \]
integrate((-114*exp(16)*log(2)^2+228*x*exp(16)*log(2)+(-114*x^2+600)*exp(1 6))*exp((2*exp(16)*log(2)-2*x*exp(16))/((19*x+5)*log(2)-19*x^2-5*x-100))/( (361*x^2+190*x+25)*log(2)^2+(-722*x^3-380*x^2-3850*x-1000)*log(2)+361*x^4+ 190*x^3+3825*x^2+1000*x+10000),x, algorithm=\
3*e^(2*x*e^16/(19*x^2 - x*(19*log(2) - 5) - 5*log(2) + 100) - 2*e^16*log(2 )/(19*x^2 - x*(19*log(2) - 5) - 5*log(2) + 100))
Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (24) = 48\).
Time = 0.27 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.90 \[ \int \frac {e^{\frac {-2 e^{16} x+2 e^{16} \log (2)}{-100-5 x-19 x^2+(5+19 x) \log (2)}} \left (e^{16} \left (600-114 x^2\right )+228 e^{16} x \log (2)-114 e^{16} \log ^2(2)\right )}{10000+1000 x+3825 x^2+190 x^3+361 x^4+\left (-1000-3850 x-380 x^2-722 x^3\right ) \log (2)+\left (25+190 x+361 x^2\right ) \log ^2(2)} \, dx=3 \, e^{\left (\frac {2 \, x e^{16}}{19 \, x^{2} - 19 \, x \log \left (2\right ) + 5 \, x - 5 \, \log \left (2\right ) + 100} - \frac {2 \, e^{16} \log \left (2\right )}{19 \, x^{2} - 19 \, x \log \left (2\right ) + 5 \, x - 5 \, \log \left (2\right ) + 100}\right )} \]
integrate((-114*exp(16)*log(2)^2+228*x*exp(16)*log(2)+(-114*x^2+600)*exp(1 6))*exp((2*exp(16)*log(2)-2*x*exp(16))/((19*x+5)*log(2)-19*x^2-5*x-100))/( (361*x^2+190*x+25)*log(2)^2+(-722*x^3-380*x^2-3850*x-1000)*log(2)+361*x^4+ 190*x^3+3825*x^2+1000*x+10000),x, algorithm=\
3*e^(2*x*e^16/(19*x^2 - 19*x*log(2) + 5*x - 5*log(2) + 100) - 2*e^16*log(2 )/(19*x^2 - 19*x*log(2) + 5*x - 5*log(2) + 100))
Time = 11.13 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.93 \[ \int \frac {e^{\frac {-2 e^{16} x+2 e^{16} \log (2)}{-100-5 x-19 x^2+(5+19 x) \log (2)}} \left (e^{16} \left (600-114 x^2\right )+228 e^{16} x \log (2)-114 e^{16} \log ^2(2)\right )}{10000+1000 x+3825 x^2+190 x^3+361 x^4+\left (-1000-3850 x-380 x^2-722 x^3\right ) \log (2)+\left (25+190 x+361 x^2\right ) \log ^2(2)} \, dx=\frac {3\,{\mathrm {e}}^{\frac {2\,x\,{\mathrm {e}}^{16}}{5\,x-5\,\ln \left (2\right )-19\,x\,\ln \left (2\right )+19\,x^2+100}}}{2^{\frac {2\,{\mathrm {e}}^{16}}{5\,x-5\,\ln \left (2\right )-19\,x\,\ln \left (2\right )+19\,x^2+100}}} \]
int(-(exp(-(2*exp(16)*log(2) - 2*x*exp(16))/(5*x - log(2)*(19*x + 5) + 19* x^2 + 100))*(exp(16)*(114*x^2 - 600) + 114*exp(16)*log(2)^2 - 228*x*exp(16 )*log(2)))/(1000*x - log(2)*(3850*x + 380*x^2 + 722*x^3 + 1000) + log(2)^2 *(190*x + 361*x^2 + 25) + 3825*x^2 + 190*x^3 + 361*x^4 + 10000),x)