Integrand size = 131, antiderivative size = 27 \[ \int \frac {-400 x+\left (-2800+e^5 (1200-300 x)+700 x\right ) \log (-4+x)+(-1600+400 x) \log (-4+x) \log (\log (-4+x))}{\left (-4900-5495 x-624 x^2+576 x^3+e^{10} (-900+225 x)+e^5 \left (4200+1830 x-720 x^2\right )\right ) \log (-4+x)+\left (-5600+e^5 (2400-600 x)-2440 x+960 x^2\right ) \log (-4+x) \log (\log (-4+x))+(-1600+400 x) \log (-4+x) \log ^2(\log (-4+x))} \, dx=\frac {x}{1+\frac {3}{4} \left (1-e^5\right )+\frac {6 x}{5}+\log (\log (-4+x))} \] Output:
x/(ln(ln(-4+x))+7/4-3/4*exp(5)+6/5*x)
Time = 0.41 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {-400 x+\left (-2800+e^5 (1200-300 x)+700 x\right ) \log (-4+x)+(-1600+400 x) \log (-4+x) \log (\log (-4+x))}{\left (-4900-5495 x-624 x^2+576 x^3+e^{10} (-900+225 x)+e^5 \left (4200+1830 x-720 x^2\right )\right ) \log (-4+x)+\left (-5600+e^5 (2400-600 x)-2440 x+960 x^2\right ) \log (-4+x) \log (\log (-4+x))+(-1600+400 x) \log (-4+x) \log ^2(\log (-4+x))} \, dx=\frac {100 x}{175-75 e^5+120 x+100 \log (\log (-4+x))} \] Input:
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]
Output:
(100*x)/(175 - 75*E^5 + 120*x + 100*Log[Log[-4 + 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 |
| \(\displaystyle 100 \left (\frac {96}{5} \int \frac {1}{\left (24 x+20 \log (\log (x-4))+35 \left (1-\frac {3 e^5}{7}\right )\right )^2}dx-\frac {24}{5} \int \frac {x}{\left (24 x+20 \log (\log (x-4))+35 \left (1-\frac {3 e^5}{7}\right )\right )^2}dx-4 \int \frac {1}{\log (x-4) \left (24 x+20 \log (\log (x-4))+35 \left (1-\frac {3 e^5}{7}\right )\right )^2}dx+\frac {1}{5} \int \frac {1}{24 x+20 \log (\log (x-4))+35 \left (1-\frac {3 e^5}{7}\right )}dx+\frac {4}{5 \left (24 x+20 \log (\log (x-4))+5 \left (7-3 e^5\right )\right )}\right )\) |
Input:
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]
Output:
$Aborted
Time = 0.96 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81
| method | result | size |
| risch | \(-\frac {20 x}{-20 \ln \left (\ln \left (x -4\right )\right )-35+15 \,{\mathrm e}^{5}-24 x}\) | \(22\) |
| parallelrisch | \(-\frac {20 x}{-20 \ln \left (\ln \left (x -4\right )\right )-35+15 \,{\mathrm e}^{5}-24 x}\) | \(22\) |
| default | \(-\frac {120 \ln \left (x -4\right ) \left (x -4\right ) x}{\left (6 \left (x -4\right ) \ln \left (x -4\right )+5\right ) \left (-20 \ln \left (\ln \left (x -4\right )\right )-35+15 \,{\mathrm e}^{5}-24 x \right )}-\frac {100 x}{\left (6 \left (x -4\right ) \ln \left (x -4\right )+5\right ) \left (-20 \ln \left (\ln \left (x -4\right )\right )-35+15 \,{\mathrm e}^{5}-24 x \right )}\) | \(77\) |
| parts | \(-\frac {120 \ln \left (x -4\right ) \left (x -4\right ) x}{\left (6 \left (x -4\right ) \ln \left (x -4\right )+5\right ) \left (-20 \ln \left (\ln \left (x -4\right )\right )-35+15 \,{\mathrm e}^{5}-24 x \right )}-\frac {100 x}{\left (6 \left (x -4\right ) \ln \left (x -4\right )+5\right ) \left (-20 \ln \left (\ln \left (x -4\right )\right )-35+15 \,{\mathrm e}^{5}-24 x \right )}\) | \(77\) |
Input:
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)
Output:
-20*x/(-20*ln(ln(x-4))-35+15*exp(5)-24*x)
Time = 0.10 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {-400 x+\left (-2800+e^5 (1200-300 x)+700 x\right ) \log (-4+x)+(-1600+400 x) \log (-4+x) \log (\log (-4+x))}{\left (-4900-5495 x-624 x^2+576 x^3+e^{10} (-900+225 x)+e^5 \left (4200+1830 x-720 x^2\right )\right ) \log (-4+x)+\left (-5600+e^5 (2400-600 x)-2440 x+960 x^2\right ) \log (-4+x) \log (\log (-4+x))+(-1600+400 x) \log (-4+x) \log ^2(\log (-4+x))} \, dx=\frac {20 \, x}{24 \, x - 15 \, e^{5} + 20 \, \log \left (\log \left (x - 4\right )\right ) + 35} \] Input:
integrate(((400*x-1600)*log(-4+x)*log(log(-4+x))+((-300*x+1200)*exp(5)+700 *x-2800)*log(-4+x)-400*x)/((400*x-1600)*log(-4+x)*log(log(-4+x))^2+((-600* x+2400)*exp(5)+960*x^2-2440*x-5600)*log(-4+x)*log(log(-4+x))+((225*x-900)* exp(5)^2+(-720*x^2+1830*x+4200)*exp(5)+576*x^3-624*x^2-5495*x-4900)*log(-4 +x)),x, algorithm="fricas")
Output:
20*x/(24*x - 15*e^5 + 20*log(log(x - 4)) + 35)
Time = 0.14 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {-400 x+\left (-2800+e^5 (1200-300 x)+700 x\right ) \log (-4+x)+(-1600+400 x) \log (-4+x) \log (\log (-4+x))}{\left (-4900-5495 x-624 x^2+576 x^3+e^{10} (-900+225 x)+e^5 \left (4200+1830 x-720 x^2\right )\right ) \log (-4+x)+\left (-5600+e^5 (2400-600 x)-2440 x+960 x^2\right ) \log (-4+x) \log (\log (-4+x))+(-1600+400 x) \log (-4+x) \log ^2(\log (-4+x))} \, dx=\frac {x}{\frac {6 x}{5} + \log {\left (\log {\left (x - 4 \right )} \right )} - \frac {3 e^{5}}{4} + \frac {7}{4}} \] Input:
integrate(((400*x-1600)*ln(-4+x)*ln(ln(-4+x))+((-300*x+1200)*exp(5)+700*x- 2800)*ln(-4+x)-400*x)/((400*x-1600)*ln(-4+x)*ln(ln(-4+x))**2+((-600*x+2400 )*exp(5)+960*x**2-2440*x-5600)*ln(-4+x)*ln(ln(-4+x))+((225*x-900)*exp(5)** 2+(-720*x**2+1830*x+4200)*exp(5)+576*x**3-624*x**2-5495*x-4900)*ln(-4+x)), x)
Output:
x/(6*x/5 + log(log(x - 4)) - 3*exp(5)/4 + 7/4)
Time = 0.10 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {-400 x+\left (-2800+e^5 (1200-300 x)+700 x\right ) \log (-4+x)+(-1600+400 x) \log (-4+x) \log (\log (-4+x))}{\left (-4900-5495 x-624 x^2+576 x^3+e^{10} (-900+225 x)+e^5 \left (4200+1830 x-720 x^2\right )\right ) \log (-4+x)+\left (-5600+e^5 (2400-600 x)-2440 x+960 x^2\right ) \log (-4+x) \log (\log (-4+x))+(-1600+400 x) \log (-4+x) \log ^2(\log (-4+x))} \, dx=\frac {20 \, x}{24 \, x - 15 \, e^{5} + 20 \, \log \left (\log \left (x - 4\right )\right ) + 35} \] Input:
integrate(((400*x-1600)*log(-4+x)*log(log(-4+x))+((-300*x+1200)*exp(5)+700 *x-2800)*log(-4+x)-400*x)/((400*x-1600)*log(-4+x)*log(log(-4+x))^2+((-600* x+2400)*exp(5)+960*x^2-2440*x-5600)*log(-4+x)*log(log(-4+x))+((225*x-900)* exp(5)^2+(-720*x^2+1830*x+4200)*exp(5)+576*x^3-624*x^2-5495*x-4900)*log(-4 +x)),x, algorithm="maxima")
Output:
20*x/(24*x - 15*e^5 + 20*log(log(x - 4)) + 35)
Time = 0.16 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {-400 x+\left (-2800+e^5 (1200-300 x)+700 x\right ) \log (-4+x)+(-1600+400 x) \log (-4+x) \log (\log (-4+x))}{\left (-4900-5495 x-624 x^2+576 x^3+e^{10} (-900+225 x)+e^5 \left (4200+1830 x-720 x^2\right )\right ) \log (-4+x)+\left (-5600+e^5 (2400-600 x)-2440 x+960 x^2\right ) \log (-4+x) \log (\log (-4+x))+(-1600+400 x) \log (-4+x) \log ^2(\log (-4+x))} \, dx=\frac {20 \, x}{24 \, x - 15 \, e^{5} + 20 \, \log \left (\log \left (x - 4\right )\right ) + 35} \] Input:
integrate(((400*x-1600)*log(-4+x)*log(log(-4+x))+((-300*x+1200)*exp(5)+700 *x-2800)*log(-4+x)-400*x)/((400*x-1600)*log(-4+x)*log(log(-4+x))^2+((-600* x+2400)*exp(5)+960*x^2-2440*x-5600)*log(-4+x)*log(log(-4+x))+((225*x-900)* exp(5)^2+(-720*x^2+1830*x+4200)*exp(5)+576*x^3-624*x^2-5495*x-4900)*log(-4 +x)),x, algorithm="giac")
Output:
20*x/(24*x - 15*e^5 + 20*log(log(x - 4)) + 35)
Timed out. \[ \int \frac {-400 x+\left (-2800+e^5 (1200-300 x)+700 x\right ) \log (-4+x)+(-1600+400 x) \log (-4+x) \log (\log (-4+x))}{\left (-4900-5495 x-624 x^2+576 x^3+e^{10} (-900+225 x)+e^5 \left (4200+1830 x-720 x^2\right )\right ) \log (-4+x)+\left (-5600+e^5 (2400-600 x)-2440 x+960 x^2\right ) \log (-4+x) \log (\log (-4+x))+(-1600+400 x) \log (-4+x) \log ^2(\log (-4+x))} \, dx=\int \frac {400\,x+\ln \left (x-4\right )\,\left ({\mathrm {e}}^5\,\left (300\,x-1200\right )-700\,x+2800\right )-\ln \left (x-4\right )\,\ln \left (\ln \left (x-4\right )\right )\,\left (400\,x-1600\right )}{-\ln \left (x-4\right )\,\left (400\,x-1600\right )\,{\ln \left (\ln \left (x-4\right )\right )}^2+\ln \left (x-4\right )\,\left (2440\,x-960\,x^2+{\mathrm {e}}^5\,\left (600\,x-2400\right )+5600\right )\,\ln \left (\ln \left (x-4\right )\right )+\ln \left (x-4\right )\,\left (5495\,x-{\mathrm {e}}^5\,\left (-720\,x^2+1830\,x+4200\right )+624\,x^2-576\,x^3-{\mathrm {e}}^{10}\,\left (225\,x-900\right )+4900\right )} \,d x \] Input:
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)
Output:
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)
Time = 0.18 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.30 \[ \int \frac {-400 x+\left (-2800+e^5 (1200-300 x)+700 x\right ) \log (-4+x)+(-1600+400 x) \log (-4+x) \log (\log (-4+x))}{\left (-4900-5495 x-624 x^2+576 x^3+e^{10} (-900+225 x)+e^5 \left (4200+1830 x-720 x^2\right )\right ) \log (-4+x)+\left (-5600+e^5 (2400-600 x)-2440 x+960 x^2\right ) \log (-4+x) \log (\log (-4+x))+(-1600+400 x) \log (-4+x) \log ^2(\log (-4+x))} \, dx=\frac {-100 \,\mathrm {log}\left (\mathrm {log}\left (x -4\right )\right )+75 e^{5}-175}{120 \,\mathrm {log}\left (\mathrm {log}\left (x -4\right )\right )-90 e^{5}+144 x +210} \] Input:
int(((400*x-1600)*log(-4+x)*log(log(-4+x))+((-300*x+1200)*exp(5)+700*x-280 0)*log(-4+x)-400*x)/((400*x-1600)*log(-4+x)*log(log(-4+x))^2+((-600*x+2400 )*exp(5)+960*x^2-2440*x-5600)*log(-4+x)*log(log(-4+x))+((225*x-900)*exp(5) ^2+(-720*x^2+1830*x+4200)*exp(5)+576*x^3-624*x^2-5495*x-4900)*log(-4+x)),x )
Output:
(25*( - 4*log(log(x - 4)) + 3*e**5 - 7))/(6*(20*log(log(x - 4)) - 15*e**5 + 24*x + 35))