Integrand size = 90, antiderivative size = 32 \[ \int \frac {e^4 x+e^4 \left (175-251 x+90 x^2\right ) \log \left (\frac {175-126 x}{-105+75 x}\right )+\left (875-1255 x+450 x^2\right ) \log ^2\left (\frac {175-126 x}{-105+75 x}\right )}{\left (175-251 x+90 x^2\right ) \log ^2\left (\frac {175-126 x}{-105+75 x}\right )} \, dx=x \left (5+\frac {e^4}{\log \left (\frac {1}{3} \left (-5-\frac {1}{5 \left (4+\frac {-7+x}{x}\right )}\right )\right )}\right ) \] Output:
(5+exp(4)/ln(-1/15/(1/x*(-7+x)+4)-5/3))*x
Time = 0.20 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \frac {e^4 x+e^4 \left (175-251 x+90 x^2\right ) \log \left (\frac {175-126 x}{-105+75 x}\right )+\left (875-1255 x+450 x^2\right ) \log ^2\left (\frac {175-126 x}{-105+75 x}\right )}{\left (175-251 x+90 x^2\right ) \log ^2\left (\frac {175-126 x}{-105+75 x}\right )} \, dx=5 x+\frac {e^4 x}{\log \left (-\frac {7 (-25+18 x)}{15 (-7+5 x)}\right )} \] Input:
Integrate[(E^4*x + E^4*(175 - 251*x + 90*x^2)*Log[(175 - 126*x)/(-105 + 75 *x)] + (875 - 1255*x + 450*x^2)*Log[(175 - 126*x)/(-105 + 75*x)]^2)/((175 - 251*x + 90*x^2)*Log[(175 - 126*x)/(-105 + 75*x)]^2),x]
Output:
5*x + (E^4*x)/Log[(-7*(-25 + 18*x))/(15*(-7 + 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.
\(\displaystyle \int \frac {\left (450 x^2-1255 x+875\right ) \log ^2\left (\frac {175-126 x}{75 x-105}\right )+e^4 \left (90 x^2-251 x+175\right ) \log \left (\frac {175-126 x}{75 x-105}\right )+e^4 x}{\left (90 x^2-251 x+175\right ) \log ^2\left (\frac {175-126 x}{75 x-105}\right )} \, dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (\frac {e^4 x}{(5 x-7) (18 x-25) \log ^2\left (-\frac {7 (18 x-25)}{15 (5 x-7)}\right )}+\frac {e^4}{\log \left (-\frac {7 (18 x-25)}{15 (5 x-7)}\right )}+5\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle e^4 \int \frac {x}{(5 x-7) (18 x-25) \log ^2\left (-\frac {7 (18 x-25)}{15 (5 x-7)}\right )}dx+e^4 \int \frac {1}{\log \left (-\frac {7 (18 x-25)}{15 (5 x-7)}\right )}dx+5 x\) |
Input:
Int[(E^4*x + E^4*(175 - 251*x + 90*x^2)*Log[(175 - 126*x)/(-105 + 75*x)] + (875 - 1255*x + 450*x^2)*Log[(175 - 126*x)/(-105 + 75*x)]^2)/((175 - 251* x + 90*x^2)*Log[(175 - 126*x)/(-105 + 75*x)]^2),x]
Output:
$Aborted
Time = 4.39 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78
method | result | size |
risch | \(5 x +\frac {{\mathrm e}^{4} x}{\ln \left (\frac {-126 x +175}{75 x -105}\right )}\) | \(25\) |
norman | \(\frac {x \,{\mathrm e}^{4}+5 x \ln \left (\frac {-126 x +175}{75 x -105}\right )}{\ln \left (\frac {-126 x +175}{75 x -105}\right )}\) | \(40\) |
derivativedivides | \(-\frac {25 \left (-\frac {\ln \left (-\frac {42}{25}-\frac {7}{75 \left (5 x -7\right )}\right )}{25}+\frac {3 \,{\mathrm e}^{4} \left (-\frac {42}{25}-\frac {7}{75 \left (5 x -7\right )}\right )}{5}+{\mathrm e}^{4}\right ) \left (5 x -7\right )}{\ln \left (-\frac {42}{25}-\frac {7}{75 \left (5 x -7\right )}\right )}\) | \(54\) |
default | \(-\frac {25 \left (-\frac {\ln \left (-\frac {42}{25}-\frac {7}{75 \left (5 x -7\right )}\right )}{25}+\frac {3 \,{\mathrm e}^{4} \left (-\frac {42}{25}-\frac {7}{75 \left (5 x -7\right )}\right )}{5}+{\mathrm e}^{4}\right ) \left (5 x -7\right )}{\ln \left (-\frac {42}{25}-\frac {7}{75 \left (5 x -7\right )}\right )}\) | \(54\) |
parallelrisch | \(\frac {8100 x \,{\mathrm e}^{4}+40500 x \ln \left (-\frac {7 \left (18 x -25\right )}{15 \left (5 x -7\right )}\right )+225900 \ln \left (-\frac {7 \left (18 x -25\right )}{15 \left (5 x -7\right )}\right )}{8100 \ln \left (-\frac {7 \left (18 x -25\right )}{15 \left (5 x -7\right )}\right )}\) | \(61\) |
Input:
int(((450*x^2-1255*x+875)*ln((-126*x+175)/(75*x-105))^2+(90*x^2-251*x+175) *exp(4)*ln((-126*x+175)/(75*x-105))+x*exp(4))/(90*x^2-251*x+175)/ln((-126* x+175)/(75*x-105))^2,x,method=_RETURNVERBOSE)
Output:
5*x+exp(4)*x/ln((-126*x+175)/(75*x-105))
Time = 0.11 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.28 \[ \int \frac {e^4 x+e^4 \left (175-251 x+90 x^2\right ) \log \left (\frac {175-126 x}{-105+75 x}\right )+\left (875-1255 x+450 x^2\right ) \log ^2\left (\frac {175-126 x}{-105+75 x}\right )}{\left (175-251 x+90 x^2\right ) \log ^2\left (\frac {175-126 x}{-105+75 x}\right )} \, dx=\frac {x e^{4} + 5 \, x \log \left (-\frac {7 \, {\left (18 \, x - 25\right )}}{15 \, {\left (5 \, x - 7\right )}}\right )}{\log \left (-\frac {7 \, {\left (18 \, x - 25\right )}}{15 \, {\left (5 \, x - 7\right )}}\right )} \] Input:
integrate(((450*x^2-1255*x+875)*log((-126*x+175)/(75*x-105))^2+(90*x^2-251 *x+175)*exp(4)*log((-126*x+175)/(75*x-105))+x*exp(4))/(90*x^2-251*x+175)/l og((-126*x+175)/(75*x-105))^2,x, algorithm="fricas")
Output:
(x*e^4 + 5*x*log(-7/15*(18*x - 25)/(5*x - 7)))/log(-7/15*(18*x - 25)/(5*x - 7))
Time = 0.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.59 \[ \int \frac {e^4 x+e^4 \left (175-251 x+90 x^2\right ) \log \left (\frac {175-126 x}{-105+75 x}\right )+\left (875-1255 x+450 x^2\right ) \log ^2\left (\frac {175-126 x}{-105+75 x}\right )}{\left (175-251 x+90 x^2\right ) \log ^2\left (\frac {175-126 x}{-105+75 x}\right )} \, dx=5 x + \frac {x e^{4}}{\log {\left (\frac {175 - 126 x}{75 x - 105} \right )}} \] Input:
integrate(((450*x**2-1255*x+875)*ln((-126*x+175)/(75*x-105))**2+(90*x**2-2 51*x+175)*exp(4)*ln((-126*x+175)/(75*x-105))+x*exp(4))/(90*x**2-251*x+175) /ln((-126*x+175)/(75*x-105))**2,x)
Output:
5*x + x*exp(4)/log((175 - 126*x)/(75*x - 105))
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.19 \[ \int \frac {e^4 x+e^4 \left (175-251 x+90 x^2\right ) \log \left (\frac {175-126 x}{-105+75 x}\right )+\left (875-1255 x+450 x^2\right ) \log ^2\left (\frac {175-126 x}{-105+75 x}\right )}{\left (175-251 x+90 x^2\right ) \log ^2\left (\frac {175-126 x}{-105+75 x}\right )} \, dx=\frac {{\left (5 i \, \pi + e^{4} + 5 \, \log \left (7\right ) - 5 \, \log \left (5\right ) - 5 \, \log \left (3\right )\right )} x + 5 \, x \log \left (18 \, x - 25\right ) - 5 \, x \log \left (5 \, x - 7\right )}{i \, \pi + \log \left (7\right ) - \log \left (5\right ) - \log \left (3\right ) + \log \left (18 \, x - 25\right ) - \log \left (5 \, x - 7\right )} \] Input:
integrate(((450*x^2-1255*x+875)*log((-126*x+175)/(75*x-105))^2+(90*x^2-251 *x+175)*exp(4)*log((-126*x+175)/(75*x-105))+x*exp(4))/(90*x^2-251*x+175)/l og((-126*x+175)/(75*x-105))^2,x, algorithm="maxima")
Output:
((5*I*pi + e^4 + 5*log(7) - 5*log(5) - 5*log(3))*x + 5*x*log(18*x - 25) - 5*x*log(5*x - 7))/(I*pi + log(7) - log(5) - log(3) + log(18*x - 25) - log( 5*x - 7))
Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (25) = 50\).
Time = 0.16 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.69 \[ \int \frac {e^4 x+e^4 \left (175-251 x+90 x^2\right ) \log \left (\frac {175-126 x}{-105+75 x}\right )+\left (875-1255 x+450 x^2\right ) \log ^2\left (\frac {175-126 x}{-105+75 x}\right )}{\left (175-251 x+90 x^2\right ) \log ^2\left (\frac {175-126 x}{-105+75 x}\right )} \, dx=\frac {\frac {7 \, {\left (18 \, x - 25\right )} e^{4}}{5 \, x - 7} - 25 \, e^{4} + \log \left (-\frac {7 \, {\left (18 \, x - 25\right )}}{15 \, {\left (5 \, x - 7\right )}}\right )}{\frac {5 \, {\left (18 \, x - 25\right )} \log \left (-\frac {7 \, {\left (18 \, x - 25\right )}}{15 \, {\left (5 \, x - 7\right )}}\right )}{5 \, x - 7} - 18 \, \log \left (-\frac {7 \, {\left (18 \, x - 25\right )}}{15 \, {\left (5 \, x - 7\right )}}\right )} \] Input:
integrate(((450*x^2-1255*x+875)*log((-126*x+175)/(75*x-105))^2+(90*x^2-251 *x+175)*exp(4)*log((-126*x+175)/(75*x-105))+x*exp(4))/(90*x^2-251*x+175)/l og((-126*x+175)/(75*x-105))^2,x, algorithm="giac")
Output:
(7*(18*x - 25)*e^4/(5*x - 7) - 25*e^4 + log(-7/15*(18*x - 25)/(5*x - 7)))/ (5*(18*x - 25)*log(-7/15*(18*x - 25)/(5*x - 7))/(5*x - 7) - 18*log(-7/15*( 18*x - 25)/(5*x - 7)))
Time = 3.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78 \[ \int \frac {e^4 x+e^4 \left (175-251 x+90 x^2\right ) \log \left (\frac {175-126 x}{-105+75 x}\right )+\left (875-1255 x+450 x^2\right ) \log ^2\left (\frac {175-126 x}{-105+75 x}\right )}{\left (175-251 x+90 x^2\right ) \log ^2\left (\frac {175-126 x}{-105+75 x}\right )} \, dx=5\,x+\frac {x\,{\mathrm {e}}^4}{\ln \left (-\frac {126\,x-175}{75\,x-105}\right )} \] Input:
int((x*exp(4) + log(-(126*x - 175)/(75*x - 105))^2*(450*x^2 - 1255*x + 875 ) + exp(4)*log(-(126*x - 175)/(75*x - 105))*(90*x^2 - 251*x + 175))/(log(- (126*x - 175)/(75*x - 105))^2*(90*x^2 - 251*x + 175)),x)
Output:
5*x + (x*exp(4))/log(-(126*x - 175)/(75*x - 105))
Time = 0.16 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.19 \[ \int \frac {e^4 x+e^4 \left (175-251 x+90 x^2\right ) \log \left (\frac {175-126 x}{-105+75 x}\right )+\left (875-1255 x+450 x^2\right ) \log ^2\left (\frac {175-126 x}{-105+75 x}\right )}{\left (175-251 x+90 x^2\right ) \log ^2\left (\frac {175-126 x}{-105+75 x}\right )} \, dx=\frac {x \left (5 \,\mathrm {log}\left (\frac {-126 x +175}{75 x -105}\right )+e^{4}\right )}{\mathrm {log}\left (\frac {-126 x +175}{75 x -105}\right )} \] Input:
int(((450*x^2-1255*x+875)*log((-126*x+175)/(75*x-105))^2+(90*x^2-251*x+175 )*exp(4)*log((-126*x+175)/(75*x-105))+x*exp(4))/(90*x^2-251*x+175)/log((-1 26*x+175)/(75*x-105))^2,x)
Output:
(x*(5*log(( - 126*x + 175)/(75*x - 105)) + e**4))/log(( - 126*x + 175)/(75 *x - 105))