Integrand size = 106, antiderivative size = 24 \[ \int \frac {6 x^2-6 x^2 \log (x)+4 e^6 x \log ^3(x)+\left (3 x^2 \log (x)+e^6 (-64-4 x) \log ^3(x)\right ) \log \left (\frac {-45 x^2+e^6 (960+60 x) \log ^2(x)}{4 e^6 \log ^2(x)}\right )}{-3 x^4 \log (x)+e^6 \left (64 x^2+4 x^3\right ) \log ^3(x)} \, dx=\frac {\log \left (15 \left (16+x-\frac {3 x^2}{4 e^6 \log ^2(x)}\right )\right )}{x} \] Output:
ln(240-45/4/exp(3)^2/ln(x)^2*x^2+15*x)/x
Time = 0.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04 \[ \int \frac {6 x^2-6 x^2 \log (x)+4 e^6 x \log ^3(x)+\left (3 x^2 \log (x)+e^6 (-64-4 x) \log ^3(x)\right ) \log \left (\frac {-45 x^2+e^6 (960+60 x) \log ^2(x)}{4 e^6 \log ^2(x)}\right )}{-3 x^4 \log (x)+e^6 \left (64 x^2+4 x^3\right ) \log ^3(x)} \, dx=\frac {\log \left (15 (16+x)-\frac {45 x^2}{4 e^6 \log ^2(x)}\right )}{x} \] Input:
Integrate[(6*x^2 - 6*x^2*Log[x] + 4*E^6*x*Log[x]^3 + (3*x^2*Log[x] + E^6*( -64 - 4*x)*Log[x]^3)*Log[(-45*x^2 + E^6*(960 + 60*x)*Log[x]^2)/(4*E^6*Log[ x]^2)])/(-3*x^4*Log[x] + E^6*(64*x^2 + 4*x^3)*Log[x]^3),x]
Output:
Log[15*(16 + x) - (45*x^2)/(4*E^6*Log[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.
| \(\displaystyle \int \frac {6 x^2+\left (3 x^2 \log (x)+e^6 (-4 x-64) \log ^3(x)\right ) \log \left (\frac {e^6 (60 x+960) \log ^2(x)-45 x^2}{4 e^6 \log ^2(x)}\right )-6 x^2 \log (x)+4 e^6 x \log ^3(x)}{e^6 \left (4 x^3+64 x^2\right ) \log ^3(x)-3 x^4 \log (x)} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 \left (-3 x-2 e^6 \log ^3(x)+3 x \log (x)\right )}{x \log (x) \left (3 x^2-4 e^6 x \log ^2(x)-64 e^6 \log ^2(x)\right )}-\frac {\log \left (15 (x+16)-\frac {45 x^2}{4 e^6 \log ^2(x)}\right )}{x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 \int \frac {1}{3 x^2-4 e^6 \log ^2(x) x-64 e^6 \log ^2(x)}dx+48 \int \frac {1}{(x+16) \left (3 x^2-4 e^6 \log ^2(x) x-64 e^6 \log ^2(x)\right )}dx-\frac {1}{32} e^6 \int \frac {x \log (x)}{3 x^2-4 e^6 \log ^2(x) x-64 e^6 \log ^2(x)}dx+128 e^6 \int \frac {\log (x)}{x^2 \left (-3 x^2+4 e^6 \log ^2(x) x+64 e^6 \log ^2(x)\right )}dx+8 e^6 \int \frac {\log (x)}{x \left (-3 x^2+4 e^6 \log ^2(x) x+64 e^6 \log ^2(x)\right )}dx-\frac {1}{32} e^6 \int \frac {x \log (x)}{-3 x^2+4 e^6 \log ^2(x) x+64 e^6 \log ^2(x)}dx-\int \frac {\log \left (15 (x+16)-\frac {45 x^2}{4 e^6 \log ^2(x)}\right )}{x^2}dx-2 \operatorname {ExpIntegralEi}(-\log (x))+\frac {\log (x)}{16}-\frac {1}{16} \log (x+16)\) |
Input:
Int[(6*x^2 - 6*x^2*Log[x] + 4*E^6*x*Log[x]^3 + (3*x^2*Log[x] + E^6*(-64 - 4*x)*Log[x]^3)*Log[(-45*x^2 + E^6*(960 + 60*x)*Log[x]^2)/(4*E^6*Log[x]^2)] )/(-3*x^4*Log[x] + E^6*(64*x^2 + 4*x^3)*Log[x]^3),x]
Output:
$Aborted
Time = 6.05 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50
| method | result | size |
| parallelrisch | \(\frac {\ln \left (\frac {\left (\left (60 x +960\right ) {\mathrm e}^{6} \ln \left (x \right )^{2}-45 x^{2}\right ) {\mathrm e}^{-6}}{4 \ln \left (x \right )^{2}}\right )}{x}\) | \(36\) |
| risch | \(\frac {\ln \left (\left (x \ln \left (x \right )^{2}+16 \ln \left (x \right )^{2}\right ) {\mathrm e}^{6}-\frac {3 x^{2}}{4}\right )}{x}+\frac {-i \pi \,\operatorname {csgn}\left (i \left (\left (x \ln \left (x \right )^{2}+16 \ln \left (x \right )^{2}\right ) {\mathrm e}^{6}-\frac {3 x^{2}}{4}\right )\right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x \right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (\left (x \ln \left (x \right )^{2}+16 \ln \left (x \right )^{2}\right ) {\mathrm e}^{6}-\frac {3 x^{2}}{4}\right )}{\ln \left (x \right )^{2}}\right )+i \pi \,\operatorname {csgn}\left (i \left (\left (x \ln \left (x \right )^{2}+16 \ln \left (x \right )^{2}\right ) {\mathrm e}^{6}-\frac {3 x^{2}}{4}\right )\right ) {\operatorname {csgn}\left (\frac {i \left (\left (x \ln \left (x \right )^{2}+16 \ln \left (x \right )^{2}\right ) {\mathrm e}^{6}-\frac {3 x^{2}}{4}\right )}{\ln \left (x \right )^{2}}\right )}^{2}+i \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )^{2}}\right ) {\operatorname {csgn}\left (\frac {i \left (\left (x \ln \left (x \right )^{2}+16 \ln \left (x \right )^{2}\right ) {\mathrm e}^{6}-\frac {3 x^{2}}{4}\right )}{\ln \left (x \right )^{2}}\right )}^{2}+i \pi \operatorname {csgn}\left (i \ln \left (x \right )\right )^{2} \operatorname {csgn}\left (i \ln \left (x \right )^{2}\right )-2 i \pi \,\operatorname {csgn}\left (i \ln \left (x \right )\right ) \operatorname {csgn}\left (i \ln \left (x \right )^{2}\right )^{2}+i \pi \operatorname {csgn}\left (i \ln \left (x \right )^{2}\right )^{3}-12-i \pi {\operatorname {csgn}\left (\frac {i \left (\left (x \ln \left (x \right )^{2}+16 \ln \left (x \right )^{2}\right ) {\mathrm e}^{6}-\frac {3 x^{2}}{4}\right )}{\ln \left (x \right )^{2}}\right )}^{3}+2 \ln \left (3\right )+2 \ln \left (5\right )-4 \ln \left (\ln \left (x \right )\right )}{2 x}\) | \(313\) |
Input:
int((((-4*x-64)*exp(3)^2*ln(x)^3+3*x^2*ln(x))*ln(1/4*((60*x+960)*exp(3)^2* ln(x)^2-45*x^2)/exp(3)^2/ln(x)^2)+4*x*exp(3)^2*ln(x)^3-6*x^2*ln(x)+6*x^2)/ ((4*x^3+64*x^2)*exp(3)^2*ln(x)^3-3*x^4*ln(x)),x,method=_RETURNVERBOSE)
Output:
ln(1/4*((60*x+960)*exp(3)^2*ln(x)^2-45*x^2)/exp(3)^2/ln(x)^2)/x
Time = 0.10 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.25 \[ \int \frac {6 x^2-6 x^2 \log (x)+4 e^6 x \log ^3(x)+\left (3 x^2 \log (x)+e^6 (-64-4 x) \log ^3(x)\right ) \log \left (\frac {-45 x^2+e^6 (960+60 x) \log ^2(x)}{4 e^6 \log ^2(x)}\right )}{-3 x^4 \log (x)+e^6 \left (64 x^2+4 x^3\right ) \log ^3(x)} \, dx=\frac {\log \left (\frac {15 \, {\left (4 \, {\left (x + 16\right )} e^{6} \log \left (x\right )^{2} - 3 \, x^{2}\right )} e^{\left (-6\right )}}{4 \, \log \left (x\right )^{2}}\right )}{x} \] Input:
integrate((((-4*x-64)*exp(3)^2*log(x)^3+3*x^2*log(x))*log(1/4*((60*x+960)* exp(3)^2*log(x)^2-45*x^2)/exp(3)^2/log(x)^2)+4*x*exp(3)^2*log(x)^3-6*x^2*l og(x)+6*x^2)/((4*x^3+64*x^2)*exp(3)^2*log(x)^3-3*x^4*log(x)),x, algorithm= "fricas")
Output:
log(15/4*(4*(x + 16)*e^6*log(x)^2 - 3*x^2)*e^(-6)/log(x)^2)/x
Time = 0.24 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int \frac {6 x^2-6 x^2 \log (x)+4 e^6 x \log ^3(x)+\left (3 x^2 \log (x)+e^6 (-64-4 x) \log ^3(x)\right ) \log \left (\frac {-45 x^2+e^6 (960+60 x) \log ^2(x)}{4 e^6 \log ^2(x)}\right )}{-3 x^4 \log (x)+e^6 \left (64 x^2+4 x^3\right ) \log ^3(x)} \, dx=\frac {\log {\left (\frac {- \frac {45 x^{2}}{4} + \frac {\left (60 x + 960\right ) e^{6} \log {\left (x \right )}^{2}}{4}}{e^{6} \log {\left (x \right )}^{2}} \right )}}{x} \] Input:
integrate((((-4*x-64)*exp(3)**2*ln(x)**3+3*x**2*ln(x))*ln(1/4*((60*x+960)* exp(3)**2*ln(x)**2-45*x**2)/exp(3)**2/ln(x)**2)+4*x*exp(3)**2*ln(x)**3-6*x **2*ln(x)+6*x**2)/((4*x**3+64*x**2)*exp(3)**2*ln(x)**3-3*x**4*ln(x)),x)
Output:
log((-45*x**2/4 + (60*x + 960)*exp(6)*log(x)**2/4)*exp(-6)/log(x)**2)/x
Time = 0.17 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.71 \[ \int \frac {6 x^2-6 x^2 \log (x)+4 e^6 x \log ^3(x)+\left (3 x^2 \log (x)+e^6 (-64-4 x) \log ^3(x)\right ) \log \left (\frac {-45 x^2+e^6 (960+60 x) \log ^2(x)}{4 e^6 \log ^2(x)}\right )}{-3 x^4 \log (x)+e^6 \left (64 x^2+4 x^3\right ) \log ^3(x)} \, dx=\frac {\log \left (5\right ) + \log \left (3\right ) - 2 \, \log \left (2\right ) + \log \left (4 \, {\left (x e^{6} + 16 \, e^{6}\right )} \log \left (x\right )^{2} - 3 \, x^{2}\right ) - 2 \, \log \left (\log \left (x\right )\right ) - 6}{x} \] Input:
integrate((((-4*x-64)*exp(3)^2*log(x)^3+3*x^2*log(x))*log(1/4*((60*x+960)* exp(3)^2*log(x)^2-45*x^2)/exp(3)^2/log(x)^2)+4*x*exp(3)^2*log(x)^3-6*x^2*l og(x)+6*x^2)/((4*x^3+64*x^2)*exp(3)^2*log(x)^3-3*x^4*log(x)),x, algorithm= "maxima")
Output:
(log(5) + log(3) - 2*log(2) + log(4*(x*e^6 + 16*e^6)*log(x)^2 - 3*x^2) - 2 *log(log(x)) - 6)/x
Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62 \[ \int \frac {6 x^2-6 x^2 \log (x)+4 e^6 x \log ^3(x)+\left (3 x^2 \log (x)+e^6 (-64-4 x) \log ^3(x)\right ) \log \left (\frac {-45 x^2+e^6 (960+60 x) \log ^2(x)}{4 e^6 \log ^2(x)}\right )}{-3 x^4 \log (x)+e^6 \left (64 x^2+4 x^3\right ) \log ^3(x)} \, dx=\frac {\log \left (60 \, x e^{6} \log \left (x\right )^{2} + 960 \, e^{6} \log \left (x\right )^{2} - 45 \, x^{2}\right ) - \log \left (4 \, \log \left (x\right )^{2}\right ) - 6}{x} \] Input:
integrate((((-4*x-64)*exp(3)^2*log(x)^3+3*x^2*log(x))*log(1/4*((60*x+960)* exp(3)^2*log(x)^2-45*x^2)/exp(3)^2/log(x)^2)+4*x*exp(3)^2*log(x)^3-6*x^2*l og(x)+6*x^2)/((4*x^3+64*x^2)*exp(3)^2*log(x)^3-3*x^4*log(x)),x, algorithm= "giac")
Output:
(log(60*x*e^6*log(x)^2 + 960*e^6*log(x)^2 - 45*x^2) - log(4*log(x)^2) - 6) /x
Time = 3.78 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int \frac {6 x^2-6 x^2 \log (x)+4 e^6 x \log ^3(x)+\left (3 x^2 \log (x)+e^6 (-64-4 x) \log ^3(x)\right ) \log \left (\frac {-45 x^2+e^6 (960+60 x) \log ^2(x)}{4 e^6 \log ^2(x)}\right )}{-3 x^4 \log (x)+e^6 \left (64 x^2+4 x^3\right ) \log ^3(x)} \, dx=\frac {\ln \left (-\frac {\frac {45\,x^2}{4}-\frac {{\mathrm {e}}^6\,{\ln \left (x\right )}^2\,\left (60\,x+960\right )}{4}}{{\ln \left (x\right )}^2}\right )-6}{x} \] Input:
int(-(log(-(exp(-6)*((45*x^2)/4 - (exp(6)*log(x)^2*(60*x + 960))/4))/log(x )^2)*(3*x^2*log(x) - exp(6)*log(x)^3*(4*x + 64)) - 6*x^2*log(x) + 6*x^2 + 4*x*exp(6)*log(x)^3)/(3*x^4*log(x) - exp(6)*log(x)^3*(64*x^2 + 4*x^3)),x)
Output:
(log(-((45*x^2)/4 - (exp(6)*log(x)^2*(60*x + 960))/4)/log(x)^2) - 6)/x
\[ \int \frac {6 x^2-6 x^2 \log (x)+4 e^6 x \log ^3(x)+\left (3 x^2 \log (x)+e^6 (-64-4 x) \log ^3(x)\right ) \log \left (\frac {-45 x^2+e^6 (960+60 x) \log ^2(x)}{4 e^6 \log ^2(x)}\right )}{-3 x^4 \log (x)+e^6 \left (64 x^2+4 x^3\right ) \log ^3(x)} \, dx=4 \left (\int \frac {\mathrm {log}\left (x \right )^{2}}{4 \mathrm {log}\left (x \right )^{2} e^{6} x^{2}+64 \mathrm {log}\left (x \right )^{2} e^{6} x -3 x^{3}}d x \right ) e^{6}+3 \left (\int \frac {\mathrm {log}\left (\frac {60 \mathrm {log}\left (x \right )^{2} e^{6} x +960 \mathrm {log}\left (x \right )^{2} e^{6}-45 x^{2}}{4 \mathrm {log}\left (x \right )^{2} e^{6}}\right )}{4 \mathrm {log}\left (x \right )^{2} e^{6} x +64 \mathrm {log}\left (x \right )^{2} e^{6}-3 x^{2}}d x \right )-64 \left (\int \frac {\mathrm {log}\left (\frac {60 \mathrm {log}\left (x \right )^{2} e^{6} x +960 \mathrm {log}\left (x \right )^{2} e^{6}-45 x^{2}}{4 \mathrm {log}\left (x \right )^{2} e^{6}}\right ) \mathrm {log}\left (x \right )^{2}}{4 \mathrm {log}\left (x \right )^{2} e^{6} x^{3}+64 \mathrm {log}\left (x \right )^{2} e^{6} x^{2}-3 x^{4}}d x \right ) e^{6}-4 \left (\int \frac {\mathrm {log}\left (\frac {60 \mathrm {log}\left (x \right )^{2} e^{6} x +960 \mathrm {log}\left (x \right )^{2} e^{6}-45 x^{2}}{4 \mathrm {log}\left (x \right )^{2} e^{6}}\right ) \mathrm {log}\left (x \right )^{2}}{4 \mathrm {log}\left (x \right )^{2} e^{6} x^{2}+64 \mathrm {log}\left (x \right )^{2} e^{6} x -3 x^{3}}d x \right ) e^{6}+6 \left (\int \frac {1}{4 \mathrm {log}\left (x \right )^{3} e^{6} x +64 \mathrm {log}\left (x \right )^{3} e^{6}-3 \,\mathrm {log}\left (x \right ) x^{2}}d x \right )-6 \left (\int \frac {1}{4 \mathrm {log}\left (x \right )^{2} e^{6} x +64 \mathrm {log}\left (x \right )^{2} e^{6}-3 x^{2}}d x \right ) \] Input:
int((((-4*x-64)*exp(3)^2*log(x)^3+3*x^2*log(x))*log(1/4*((60*x+960)*exp(3) ^2*log(x)^2-45*x^2)/exp(3)^2/log(x)^2)+4*x*exp(3)^2*log(x)^3-6*x^2*log(x)+ 6*x^2)/((4*x^3+64*x^2)*exp(3)^2*log(x)^3-3*x^4*log(x)),x)
Output:
4*int(log(x)**2/(4*log(x)**2*e**6*x**2 + 64*log(x)**2*e**6*x - 3*x**3),x)* e**6 + 3*int(log((60*log(x)**2*e**6*x + 960*log(x)**2*e**6 - 45*x**2)/(4*l og(x)**2*e**6))/(4*log(x)**2*e**6*x + 64*log(x)**2*e**6 - 3*x**2),x) - 64* int((log((60*log(x)**2*e**6*x + 960*log(x)**2*e**6 - 45*x**2)/(4*log(x)**2 *e**6))*log(x)**2)/(4*log(x)**2*e**6*x**3 + 64*log(x)**2*e**6*x**2 - 3*x** 4),x)*e**6 - 4*int((log((60*log(x)**2*e**6*x + 960*log(x)**2*e**6 - 45*x** 2)/(4*log(x)**2*e**6))*log(x)**2)/(4*log(x)**2*e**6*x**2 + 64*log(x)**2*e* *6*x - 3*x**3),x)*e**6 + 6*int(1/(4*log(x)**3*e**6*x + 64*log(x)**3*e**6 - 3*log(x)*x**2),x) - 6*int(1/(4*log(x)**2*e**6*x + 64*log(x)**2*e**6 - 3*x **2),x)