Integrand size = 131, antiderivative size = 27 \[ \int \frac {-400 x+\left (-2800+e^5 (1200-300 x)+700 x\right ) \log (-4+x)+(-1600+400 x) \log (-4+x) \log (\log (-4+x))}{\left (-4900-5495 x-624 x^2+576 x^3+e^{10} (-900+225 x)+e^5 \left (4200+1830 x-720 x^2\right )\right ) \log (-4+x)+\left (-5600+e^5 (2400-600 x)-2440 x+960 x^2\right ) \log (-4+x) \log (\log (-4+x))+(-1600+400 x) \log (-4+x) \log ^2(\log (-4+x))} \, dx=\frac {x}{1+\frac {3}{4} \left (1-e^5\right )+\frac {6 x}{5}+\log (\log (-4+x))} \]
Time = 0.69 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {-400 x+\left (-2800+e^5 (1200-300 x)+700 x\right ) \log (-4+x)+(-1600+400 x) \log (-4+x) \log (\log (-4+x))}{\left (-4900-5495 x-624 x^2+576 x^3+e^{10} (-900+225 x)+e^5 \left (4200+1830 x-720 x^2\right )\right ) \log (-4+x)+\left (-5600+e^5 (2400-600 x)-2440 x+960 x^2\right ) \log (-4+x) \log (\log (-4+x))+(-1600+400 x) \log (-4+x) \log ^2(\log (-4+x))} \, dx=\frac {100 x}{175-75 e^5+120 x+100 \log (\log (-4+x))} \]
Integrate[(-400*x + (-2800 + E^5*(1200 - 300*x) + 700*x)*Log[-4 + x] + (-1 600 + 400*x)*Log[-4 + x]*Log[Log[-4 + x]])/((-4900 - 5495*x - 624*x^2 + 57 6*x^3 + E^10*(-900 + 225*x) + E^5*(4200 + 1830*x - 720*x^2))*Log[-4 + x] + (-5600 + E^5*(2400 - 600*x) - 2440*x + 960*x^2)*Log[-4 + x]*Log[Log[-4 + x]] + (-1600 + 400*x)*Log[-4 + x]*Log[Log[-4 + 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.
| \(\displaystyle \int \frac {-400 x+\left (e^5 (1200-300 x)+700 x-2800\right ) \log (x-4)+(400 x-1600) \log (x-4) \log (\log (x-4))}{\left (960 x^2-2440 x+e^5 (2400-600 x)-5600\right ) \log (x-4) \log (\log (x-4))+\left (576 x^3-624 x^2+e^5 \left (-720 x^2+1830 x+4200\right )-5495 x+e^{10} (225 x-900)-4900\right ) \log (x-4)+(400 x-1600) \log (x-4) \log ^2(\log (x-4))} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {100 \left (4 x+(x-4) \log (x-4) \left (-4 \log (\log (x-4))+3 e^5-7\right )\right )}{(4-x) \log (x-4) \left (24 x+20 \log (\log (x-4))+35 \left (1-\frac {3 e^5}{7}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 100 \int \frac {4 x+(4-x) \log (x-4) \left (4 \log (\log (x-4))-3 e^5+7\right )}{(4-x) \log (x-4) \left (24 x+20 \log (\log (x-4))+5 \left (7-3 e^5\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 100 \int \left (\frac {4 x (6 x \log (x-4)-24 \log (x-4)+5)}{5 (4-x) \log (x-4) \left (24 x+20 \log (\log (x-4))+35 \left (1-\frac {3 e^5}{7}\right )\right )^2}+\frac {1}{5 \left (24 x+20 \log (\log (x-4))+35 \left (1-\frac {3 e^5}{7}\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 100 \left (\frac {96}{5} \int \frac {1}{\left (24 x+20 \log (\log (x-4))+35 \left (1-\frac {3 e^5}{7}\right )\right )^2}dx-\frac {24}{5} \int \frac {x}{\left (24 x+20 \log (\log (x-4))+35 \left (1-\frac {3 e^5}{7}\right )\right )^2}dx-4 \int \frac {1}{\log (x-4) \left (24 x+20 \log (\log (x-4))+35 \left (1-\frac {3 e^5}{7}\right )\right )^2}dx+\frac {1}{5} \int \frac {1}{24 x+20 \log (\log (x-4))+35 \left (1-\frac {3 e^5}{7}\right )}dx+\frac {4}{5 \left (24 x+20 \log (\log (x-4))+5 \left (7-3 e^5\right )\right )}\right )\) |
Int[(-400*x + (-2800 + E^5*(1200 - 300*x) + 700*x)*Log[-4 + x] + (-1600 + 400*x)*Log[-4 + x]*Log[Log[-4 + x]])/((-4900 - 5495*x - 624*x^2 + 576*x^3 + E^10*(-900 + 225*x) + E^5*(4200 + 1830*x - 720*x^2))*Log[-4 + x] + (-560 0 + E^5*(2400 - 600*x) - 2440*x + 960*x^2)*Log[-4 + x]*Log[Log[-4 + x]] + (-1600 + 400*x)*Log[-4 + x]*Log[Log[-4 + x]]^2),x]
3.13.87.3.1 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 = 2.14 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81
| method | result | size |
| risch | \(-\frac {20 x}{15 \,{\mathrm e}^{5}-20 \ln \left (\ln \left (x -4\right )\right )-24 x -35}\) | \(22\) |
| parallelrisch | \(-\frac {20 x}{15 \,{\mathrm e}^{5}-20 \ln \left (\ln \left (x -4\right )\right )-24 x -35}\) | \(22\) |
| default | \(-\frac {120 \ln \left (x -4\right ) \left (x -4\right ) x}{\left (6 \left (x -4\right ) \ln \left (x -4\right )+5\right ) \left (15 \,{\mathrm e}^{5}-20 \ln \left (\ln \left (x -4\right )\right )-24 x -35\right )}-\frac {100 x}{\left (6 \left (x -4\right ) \ln \left (x -4\right )+5\right ) \left (15 \,{\mathrm e}^{5}-20 \ln \left (\ln \left (x -4\right )\right )-24 x -35\right )}\) | \(77\) |
| parts | \(-\frac {120 \ln \left (x -4\right ) \left (x -4\right ) x}{\left (6 \left (x -4\right ) \ln \left (x -4\right )+5\right ) \left (15 \,{\mathrm e}^{5}-20 \ln \left (\ln \left (x -4\right )\right )-24 x -35\right )}-\frac {100 x}{\left (6 \left (x -4\right ) \ln \left (x -4\right )+5\right ) \left (15 \,{\mathrm e}^{5}-20 \ln \left (\ln \left (x -4\right )\right )-24 x -35\right )}\) | \(77\) |
int(((400*x-1600)*ln(x-4)*ln(ln(x-4))+((-300*x+1200)*exp(5)+700*x-2800)*ln (x-4)-400*x)/((400*x-1600)*ln(x-4)*ln(ln(x-4))^2+((-600*x+2400)*exp(5)+960 *x^2-2440*x-5600)*ln(x-4)*ln(ln(x-4))+((225*x-900)*exp(5)^2+(-720*x^2+1830 *x+4200)*exp(5)+576*x^3-624*x^2-5495*x-4900)*ln(x-4)),x,method=_RETURNVERB OSE)
Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {-400 x+\left (-2800+e^5 (1200-300 x)+700 x\right ) \log (-4+x)+(-1600+400 x) \log (-4+x) \log (\log (-4+x))}{\left (-4900-5495 x-624 x^2+576 x^3+e^{10} (-900+225 x)+e^5 \left (4200+1830 x-720 x^2\right )\right ) \log (-4+x)+\left (-5600+e^5 (2400-600 x)-2440 x+960 x^2\right ) \log (-4+x) \log (\log (-4+x))+(-1600+400 x) \log (-4+x) \log ^2(\log (-4+x))} \, dx=\frac {20 \, x}{24 \, x - 15 \, e^{5} + 20 \, \log \left (\log \left (x - 4\right )\right ) + 35} \]
integrate(((400*x-1600)*log(x-4)*log(log(x-4))+((-300*x+1200)*exp(5)+700*x -2800)*log(x-4)-400*x)/((400*x-1600)*log(x-4)*log(log(x-4))^2+((-600*x+240 0)*exp(5)+960*x^2-2440*x-5600)*log(x-4)*log(log(x-4))+((225*x-900)*exp(5)^ 2+(-720*x^2+1830*x+4200)*exp(5)+576*x^3-624*x^2-5495*x-4900)*log(x-4)),x, algorithm=\
Time = 0.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {-400 x+\left (-2800+e^5 (1200-300 x)+700 x\right ) \log (-4+x)+(-1600+400 x) \log (-4+x) \log (\log (-4+x))}{\left (-4900-5495 x-624 x^2+576 x^3+e^{10} (-900+225 x)+e^5 \left (4200+1830 x-720 x^2\right )\right ) \log (-4+x)+\left (-5600+e^5 (2400-600 x)-2440 x+960 x^2\right ) \log (-4+x) \log (\log (-4+x))+(-1600+400 x) \log (-4+x) \log ^2(\log (-4+x))} \, dx=\frac {x}{\frac {6 x}{5} + \log {\left (\log {\left (x - 4 \right )} \right )} - \frac {3 e^{5}}{4} + \frac {7}{4}} \]
integrate(((400*x-1600)*ln(x-4)*ln(ln(x-4))+((-300*x+1200)*exp(5)+700*x-28 00)*ln(x-4)-400*x)/((400*x-1600)*ln(x-4)*ln(ln(x-4))**2+((-600*x+2400)*exp (5)+960*x**2-2440*x-5600)*ln(x-4)*ln(ln(x-4))+((225*x-900)*exp(5)**2+(-720 *x**2+1830*x+4200)*exp(5)+576*x**3-624*x**2-5495*x-4900)*ln(x-4)),x)
Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {-400 x+\left (-2800+e^5 (1200-300 x)+700 x\right ) \log (-4+x)+(-1600+400 x) \log (-4+x) \log (\log (-4+x))}{\left (-4900-5495 x-624 x^2+576 x^3+e^{10} (-900+225 x)+e^5 \left (4200+1830 x-720 x^2\right )\right ) \log (-4+x)+\left (-5600+e^5 (2400-600 x)-2440 x+960 x^2\right ) \log (-4+x) \log (\log (-4+x))+(-1600+400 x) \log (-4+x) \log ^2(\log (-4+x))} \, dx=\frac {20 \, x}{24 \, x - 15 \, e^{5} + 20 \, \log \left (\log \left (x - 4\right )\right ) + 35} \]
integrate(((400*x-1600)*log(x-4)*log(log(x-4))+((-300*x+1200)*exp(5)+700*x -2800)*log(x-4)-400*x)/((400*x-1600)*log(x-4)*log(log(x-4))^2+((-600*x+240 0)*exp(5)+960*x^2-2440*x-5600)*log(x-4)*log(log(x-4))+((225*x-900)*exp(5)^ 2+(-720*x^2+1830*x+4200)*exp(5)+576*x^3-624*x^2-5495*x-4900)*log(x-4)),x, algorithm=\
Time = 0.32 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {-400 x+\left (-2800+e^5 (1200-300 x)+700 x\right ) \log (-4+x)+(-1600+400 x) \log (-4+x) \log (\log (-4+x))}{\left (-4900-5495 x-624 x^2+576 x^3+e^{10} (-900+225 x)+e^5 \left (4200+1830 x-720 x^2\right )\right ) \log (-4+x)+\left (-5600+e^5 (2400-600 x)-2440 x+960 x^2\right ) \log (-4+x) \log (\log (-4+x))+(-1600+400 x) \log (-4+x) \log ^2(\log (-4+x))} \, dx=\frac {20 \, x}{24 \, x - 15 \, e^{5} + 20 \, \log \left (\log \left (x - 4\right )\right ) + 35} \]
integrate(((400*x-1600)*log(x-4)*log(log(x-4))+((-300*x+1200)*exp(5)+700*x -2800)*log(x-4)-400*x)/((400*x-1600)*log(x-4)*log(log(x-4))^2+((-600*x+240 0)*exp(5)+960*x^2-2440*x-5600)*log(x-4)*log(log(x-4))+((225*x-900)*exp(5)^ 2+(-720*x^2+1830*x+4200)*exp(5)+576*x^3-624*x^2-5495*x-4900)*log(x-4)),x, algorithm=\
Timed out. \[ \int \frac {-400 x+\left (-2800+e^5 (1200-300 x)+700 x\right ) \log (-4+x)+(-1600+400 x) \log (-4+x) \log (\log (-4+x))}{\left (-4900-5495 x-624 x^2+576 x^3+e^{10} (-900+225 x)+e^5 \left (4200+1830 x-720 x^2\right )\right ) \log (-4+x)+\left (-5600+e^5 (2400-600 x)-2440 x+960 x^2\right ) \log (-4+x) \log (\log (-4+x))+(-1600+400 x) \log (-4+x) \log ^2(\log (-4+x))} \, dx=\int \frac {400\,x+\ln \left (x-4\right )\,\left ({\mathrm {e}}^5\,\left (300\,x-1200\right )-700\,x+2800\right )-\ln \left (x-4\right )\,\ln \left (\ln \left (x-4\right )\right )\,\left (400\,x-1600\right )}{-\ln \left (x-4\right )\,\left (400\,x-1600\right )\,{\ln \left (\ln \left (x-4\right )\right )}^2+\ln \left (x-4\right )\,\left (2440\,x-960\,x^2+{\mathrm {e}}^5\,\left (600\,x-2400\right )+5600\right )\,\ln \left (\ln \left (x-4\right )\right )+\ln \left (x-4\right )\,\left (5495\,x-{\mathrm {e}}^5\,\left (-720\,x^2+1830\,x+4200\right )+624\,x^2-576\,x^3-{\mathrm {e}}^{10}\,\left (225\,x-900\right )+4900\right )} \,d x \]
int((400*x + log(x - 4)*(exp(5)*(300*x - 1200) - 700*x + 2800) - log(x - 4 )*log(log(x - 4))*(400*x - 1600))/(log(x - 4)*(5495*x - exp(5)*(1830*x - 7 20*x^2 + 4200) + 624*x^2 - 576*x^3 - exp(10)*(225*x - 900) + 4900) + log(x - 4)*log(log(x - 4))*(2440*x - 960*x^2 + exp(5)*(600*x - 2400) + 5600) - log(x - 4)*log(log(x - 4))^2*(400*x - 1600)),x)
int((400*x + log(x - 4)*(exp(5)*(300*x - 1200) - 700*x + 2800) - log(x - 4 )*log(log(x - 4))*(400*x - 1600))/(log(x - 4)*(5495*x - exp(5)*(1830*x - 7 20*x^2 + 4200) + 624*x^2 - 576*x^3 - exp(10)*(225*x - 900) + 4900) + log(x - 4)*log(log(x - 4))*(2440*x - 960*x^2 + exp(5)*(600*x - 2400) + 5600) - log(x - 4)*log(log(x - 4))^2*(400*x - 1600)), x)