Integrand size = 124, antiderivative size = 27 \[ \int \frac {-100 x^2+40 x^3-4 x^4+\left (200 x^2-120 x^3+16 x^4\right ) \log (x)+\left (-1-76 x^2+24 x^3\right ) \log ^2(x)-\log \left (3 e^{\frac {100 x^2-40 x^3+4 x^4+\left (-40 x^2+8 x^3\right ) \log (x)+4 x^2 \log ^2(x)}{\log (x)}}\right ) \log ^2(x)+8 x^2 \log ^3(x)}{x^2 \log ^2(x)} \, dx=\frac {1+\log \left (3 e^{\frac {4 x^2 (-5+x+\log (x))^2}{\log (x)}}\right )}{x} \] Output:
(1+ln(3*exp(4/ln(x)*(ln(x)+x-5)^2*x^2)))/x
Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {-100 x^2+40 x^3-4 x^4+\left (200 x^2-120 x^3+16 x^4\right ) \log (x)+\left (-1-76 x^2+24 x^3\right ) \log ^2(x)-\log \left (3 e^{\frac {100 x^2-40 x^3+4 x^4+\left (-40 x^2+8 x^3\right ) \log (x)+4 x^2 \log ^2(x)}{\log (x)}}\right ) \log ^2(x)+8 x^2 \log ^3(x)}{x^2 \log ^2(x)} \, dx=\frac {1+\log \left (3 e^{\frac {4 (-5+x) x^2 (-5+x+2 \log (x))}{\log (x)}} x^{4 x^2}\right )}{x} \] Input:
Integrate[(-100*x^2 + 40*x^3 - 4*x^4 + (200*x^2 - 120*x^3 + 16*x^4)*Log[x] + (-1 - 76*x^2 + 24*x^3)*Log[x]^2 - Log[3*E^((100*x^2 - 40*x^3 + 4*x^4 + (-40*x^2 + 8*x^3)*Log[x] + 4*x^2*Log[x]^2)/Log[x])]*Log[x]^2 + 8*x^2*Log[x ]^3)/(x^2*Log[x]^2),x]
Output:
(1 + Log[3*E^((4*(-5 + x)*x^2*(-5 + x + 2*Log[x]))/Log[x])*x^(4*x^2)])/x
Time = 0.89 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.89, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {7293, 2009}
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.
| \(\displaystyle \int \frac {-\log ^2(x) \log \left (3 \exp \left (\frac {4 x^4-40 x^3+100 x^2+4 x^2 \log ^2(x)+\left (8 x^3-40 x^2\right ) \log (x)}{\log (x)}\right )\right )-4 x^4+40 x^3-100 x^2+8 x^2 \log ^3(x)+\left (24 x^3-76 x^2-1\right ) \log ^2(x)+\left (16 x^4-120 x^3+200 x^2\right ) \log (x)}{x^2 \log ^2(x)} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {24 x^3-76 x^2-\log \left (3 x^{4 x^2} e^{\frac {4 (x-5) x^2 (x+2 \log (x)-5)}{\log (x)}}\right )-1}{x^2}-\frac {4 (x-5)^2}{\log ^2(x)}+\frac {8 (2 x-5) (x-5)}{\log (x)}+8 \log (x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\log \left (3 e^{\frac {4 (5-x)^2 x^2}{\log (x)}} x^{4 x^2-\frac {8 (5-x) x^2}{\log (x)}}\right )}{x}+\frac {1}{x}\) |
Input:
Int[(-100*x^2 + 40*x^3 - 4*x^4 + (200*x^2 - 120*x^3 + 16*x^4)*Log[x] + (-1 - 76*x^2 + 24*x^3)*Log[x]^2 - Log[3*E^((100*x^2 - 40*x^3 + 4*x^4 + (-40*x ^2 + 8*x^3)*Log[x] + 4*x^2*Log[x]^2)/Log[x])]*Log[x]^2 + 8*x^2*Log[x]^3)/( x^2*Log[x]^2),x]
Output:
x^(-1) + Log[3*E^((4*(5 - x)^2*x^2)/Log[x])*x^(4*x^2 - (8*(5 - x)*x^2)/Log [x])]/x
Time = 1.12 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22
| method | result | size |
| risch | \(\frac {\ln \left ({\mathrm e}^{\frac {4 \left (\ln \left (x \right )+x -5\right )^{2} x^{2}}{\ln \left (x \right )}}\right )}{x}+\frac {\ln \left (3\right )}{x}+\frac {1}{x}\) | \(33\) |
| parallelrisch | \(-\frac {-1-\ln \left (3 \,{\mathrm e}^{\frac {4 x^{2} \left (\ln \left (x \right )^{2}+2 x \ln \left (x \right )+x^{2}-10 \ln \left (x \right )-10 x +25\right )}{\ln \left (x \right )}}\right )}{x}\) | \(44\) |
| default | \(\frac {4 x^{3}}{\ln \left (x \right )}-\frac {40 x^{2}}{\ln \left (x \right )}+\frac {100 x}{\ln \left (x \right )}+8 x^{2}-40 x +\frac {1}{x}+4 x \ln \left (x \right )+\frac {\ln \left (3 \,{\mathrm e}^{\frac {4 x^{2} \ln \left (x \right )^{2}+\left (8 x^{3}-40 x^{2}\right ) \ln \left (x \right )+4 x^{4}-40 x^{3}+100 x^{2}}{\ln \left (x \right )}}\right )-\frac {4 x^{2} \ln \left (x \right )^{2}+\left (8 x^{3}-40 x^{2}\right ) \ln \left (x \right )+4 x^{4}-40 x^{3}+100 x^{2}}{\ln \left (x \right )}}{x}\) | \(141\) |
| parts | \(\frac {4 x^{3}}{\ln \left (x \right )}-\frac {40 x^{2}}{\ln \left (x \right )}+\frac {100 x}{\ln \left (x \right )}+8 x^{2}-40 x +\frac {1}{x}+4 x \ln \left (x \right )+\frac {\ln \left (3 \,{\mathrm e}^{\frac {4 x^{2} \ln \left (x \right )^{2}+\left (8 x^{3}-40 x^{2}\right ) \ln \left (x \right )+4 x^{4}-40 x^{3}+100 x^{2}}{\ln \left (x \right )}}\right )-\frac {4 x^{2} \ln \left (x \right )^{2}+\left (8 x^{3}-40 x^{2}\right ) \ln \left (x \right )+4 x^{4}-40 x^{3}+100 x^{2}}{\ln \left (x \right )}}{x}\) | \(141\) |
Input:
int((-ln(x)^2*ln(3*exp((4*x^2*ln(x)^2+(8*x^3-40*x^2)*ln(x)+4*x^4-40*x^3+10 0*x^2)/ln(x)))+8*x^2*ln(x)^3+(24*x^3-76*x^2-1)*ln(x)^2+(16*x^4-120*x^3+200 *x^2)*ln(x)-4*x^4+40*x^3-100*x^2)/x^2/ln(x)^2,x,method=_RETURNVERBOSE)
Output:
1/x*ln(exp(4/ln(x)*(ln(x)+x-5)^2*x^2))+ln(3)/x+1/x
Time = 0.09 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.85 \[ \int \frac {-100 x^2+40 x^3-4 x^4+\left (200 x^2-120 x^3+16 x^4\right ) \log (x)+\left (-1-76 x^2+24 x^3\right ) \log ^2(x)-\log \left (3 e^{\frac {100 x^2-40 x^3+4 x^4+\left (-40 x^2+8 x^3\right ) \log (x)+4 x^2 \log ^2(x)}{\log (x)}}\right ) \log ^2(x)+8 x^2 \log ^3(x)}{x^2 \log ^2(x)} \, dx=\frac {4 \, x^{4} + 4 \, x^{2} \log \left (x\right )^{2} - 40 \, x^{3} + 100 \, x^{2} + {\left (8 \, x^{3} - 40 \, x^{2} + \log \left (3\right ) + 1\right )} \log \left (x\right )}{x \log \left (x\right )} \] Input:
integrate((-log(x)^2*log(3*exp((4*x^2*log(x)^2+(8*x^3-40*x^2)*log(x)+4*x^4 -40*x^3+100*x^2)/log(x)))+8*x^2*log(x)^3+(24*x^3-76*x^2-1)*log(x)^2+(16*x^ 4-120*x^3+200*x^2)*log(x)-4*x^4+40*x^3-100*x^2)/x^2/log(x)^2,x, algorithm= "fricas")
Output:
(4*x^4 + 4*x^2*log(x)^2 - 40*x^3 + 100*x^2 + (8*x^3 - 40*x^2 + log(3) + 1) *log(x))/(x*log(x))
Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {-100 x^2+40 x^3-4 x^4+\left (200 x^2-120 x^3+16 x^4\right ) \log (x)+\left (-1-76 x^2+24 x^3\right ) \log ^2(x)-\log \left (3 e^{\frac {100 x^2-40 x^3+4 x^4+\left (-40 x^2+8 x^3\right ) \log (x)+4 x^2 \log ^2(x)}{\log (x)}}\right ) \log ^2(x)+8 x^2 \log ^3(x)}{x^2 \log ^2(x)} \, dx=8 x^{2} + 4 x \log {\left (x \right )} - 40 x + \frac {4 x^{3} - 40 x^{2} + 100 x}{\log {\left (x \right )}} + \frac {1 + \log {\left (3 \right )}}{x} \] Input:
integrate((-ln(x)**2*ln(3*exp((4*x**2*ln(x)**2+(8*x**3-40*x**2)*ln(x)+4*x* *4-40*x**3+100*x**2)/ln(x)))+8*x**2*ln(x)**3+(24*x**3-76*x**2-1)*ln(x)**2+ (16*x**4-120*x**3+200*x**2)*ln(x)-4*x**4+40*x**3-100*x**2)/x**2/ln(x)**2,x )
Output:
8*x**2 + 4*x*log(x) - 40*x + (4*x**3 - 40*x**2 + 100*x)/log(x) + (1 + log( 3))/x
\[ \int \frac {-100 x^2+40 x^3-4 x^4+\left (200 x^2-120 x^3+16 x^4\right ) \log (x)+\left (-1-76 x^2+24 x^3\right ) \log ^2(x)-\log \left (3 e^{\frac {100 x^2-40 x^3+4 x^4+\left (-40 x^2+8 x^3\right ) \log (x)+4 x^2 \log ^2(x)}{\log (x)}}\right ) \log ^2(x)+8 x^2 \log ^3(x)}{x^2 \log ^2(x)} \, dx=\int { \frac {8 \, x^{2} \log \left (x\right )^{3} - 4 \, x^{4} + 40 \, x^{3} + {\left (24 \, x^{3} - 76 \, x^{2} - 1\right )} \log \left (x\right )^{2} - \log \left (x\right )^{2} \log \left (3 \, e^{\left (\frac {4 \, {\left (x^{4} + x^{2} \log \left (x\right )^{2} - 10 \, x^{3} + 25 \, x^{2} + 2 \, {\left (x^{3} - 5 \, x^{2}\right )} \log \left (x\right )\right )}}{\log \left (x\right )}\right )}\right ) - 100 \, x^{2} + 8 \, {\left (2 \, x^{4} - 15 \, x^{3} + 25 \, x^{2}\right )} \log \left (x\right )}{x^{2} \log \left (x\right )^{2}} \,d x } \] Input:
integrate((-log(x)^2*log(3*exp((4*x^2*log(x)^2+(8*x^3-40*x^2)*log(x)+4*x^4 -40*x^3+100*x^2)/log(x)))+8*x^2*log(x)^3+(24*x^3-76*x^2-1)*log(x)^2+(16*x^ 4-120*x^3+200*x^2)*log(x)-4*x^4+40*x^3-100*x^2)/x^2/log(x)^2,x, algorithm= "maxima")
Output:
12*x^2 + 8*x*log(x) - 84*x + 1/x - (4*x^4 + 8*x^2*log(x)^2 - 40*x^3 + 100* x^2 + (4*x^3 - 44*x^2 - log(3))*log(x) - 4*log(x)*log(x^(x^2)) - 4*log(x)* log(e^(x^4/log(x))) + 40*log(x)*log(e^(x^3/log(x))) - 100*log(x)*log(e^(x^ 2/log(x))))/(x*log(x)) + 16*Ei(3*log(x)) - 120*Ei(2*log(x)) + 200*Ei(log(x )) - 100*gamma(-1, -log(x)) + 80*gamma(-1, -2*log(x)) - 12*gamma(-1, -3*lo g(x)) - integrate(4*(x^2 - 10*x + 25)/log(x), x)
Time = 0.30 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.48 \[ \int \frac {-100 x^2+40 x^3-4 x^4+\left (200 x^2-120 x^3+16 x^4\right ) \log (x)+\left (-1-76 x^2+24 x^3\right ) \log ^2(x)-\log \left (3 e^{\frac {100 x^2-40 x^3+4 x^4+\left (-40 x^2+8 x^3\right ) \log (x)+4 x^2 \log ^2(x)}{\log (x)}}\right ) \log ^2(x)+8 x^2 \log ^3(x)}{x^2 \log ^2(x)} \, dx=8 \, x^{2} + 4 \, x \log \left (x\right ) - 40 \, x + \frac {\log \left (3\right ) + 1}{x} + \frac {4 \, {\left (x^{3} - 10 \, x^{2} + 25 \, x\right )}}{\log \left (x\right )} \] Input:
integrate((-log(x)^2*log(3*exp((4*x^2*log(x)^2+(8*x^3-40*x^2)*log(x)+4*x^4 -40*x^3+100*x^2)/log(x)))+8*x^2*log(x)^3+(24*x^3-76*x^2-1)*log(x)^2+(16*x^ 4-120*x^3+200*x^2)*log(x)-4*x^4+40*x^3-100*x^2)/x^2/log(x)^2,x, algorithm= "giac")
Output:
8*x^2 + 4*x*log(x) - 40*x + (log(3) + 1)/x + 4*(x^3 - 10*x^2 + 25*x)/log(x )
Time = 3.26 (sec) , antiderivative size = 122, normalized size of antiderivative = 4.52 \[ \int \frac {-100 x^2+40 x^3-4 x^4+\left (200 x^2-120 x^3+16 x^4\right ) \log (x)+\left (-1-76 x^2+24 x^3\right ) \log ^2(x)-\log \left (3 e^{\frac {100 x^2-40 x^3+4 x^4+\left (-40 x^2+8 x^3\right ) \log (x)+4 x^2 \log ^2(x)}{\log (x)}}\right ) \log ^2(x)+8 x^2 \log ^3(x)}{x^2 \log ^2(x)} \, dx=60\,x+\frac {4\,x\,{\left (x-5\right )}^2-4\,x\,\ln \left (x\right )\,\left (3\,x^2-20\,x+25\right )}{\ln \left (x\right )}+4\,x\,\ln \left (x\right )-72\,x^2+12\,x^3+\frac {\ln \left (3\,x^{4\,x^2}\,{\mathrm {e}}^{8\,x^3}\,{\mathrm {e}}^{-40\,x^2}\,{\mathrm {e}}^{\frac {4\,x^4}{\ln \left (x\right )}}\,{\mathrm {e}}^{-\frac {40\,x^3}{\ln \left (x\right )}}\,{\mathrm {e}}^{\frac {100\,x^2}{\ln \left (x\right )}}\right )-\frac {4\,x^2\,{\left (x+\ln \left (x\right )-5\right )}^2}{\ln \left (x\right )}+1}{x} \] Input:
int(-(log(3*exp((4*x^2*log(x)^2 - log(x)*(40*x^2 - 8*x^3) + 100*x^2 - 40*x ^3 + 4*x^4)/log(x)))*log(x)^2 - log(x)*(200*x^2 - 120*x^3 + 16*x^4) + log( x)^2*(76*x^2 - 24*x^3 + 1) - 8*x^2*log(x)^3 + 100*x^2 - 40*x^3 + 4*x^4)/(x ^2*log(x)^2),x)
Output:
60*x + (4*x*(x - 5)^2 - 4*x*log(x)*(3*x^2 - 20*x + 25))/log(x) + 4*x*log(x ) - 72*x^2 + 12*x^3 + (log(3*x^(4*x^2)*exp(8*x^3)*exp(-40*x^2)*exp((4*x^4) /log(x))*exp(-(40*x^3)/log(x))*exp((100*x^2)/log(x))) - (4*x^2*(x + log(x) - 5)^2)/log(x) + 1)/x
\[ \int \frac {-100 x^2+40 x^3-4 x^4+\left (200 x^2-120 x^3+16 x^4\right ) \log (x)+\left (-1-76 x^2+24 x^3\right ) \log ^2(x)-\log \left (3 e^{\frac {100 x^2-40 x^3+4 x^4+\left (-40 x^2+8 x^3\right ) \log (x)+4 x^2 \log ^2(x)}{\log (x)}}\right ) \log ^2(x)+8 x^2 \log ^3(x)}{x^2 \log ^2(x)} \, dx=\frac {100 \mathit {ei} \left (\mathrm {log}\left (x \right )\right ) \mathrm {log}\left (x \right ) x +4 \mathit {ei} \left (3 \,\mathrm {log}\left (x \right )\right ) \mathrm {log}\left (x \right ) x -40 \mathit {ei} \left (2 \,\mathrm {log}\left (x \right )\right ) \mathrm {log}\left (x \right ) x -\left (\int \frac {\mathrm {log}\left (\frac {3 x^{4 x^{2}} e^{\frac {8 \,\mathrm {log}\left (x \right ) x^{3}+4 x^{4}+100 x^{2}}{\mathrm {log}\left (x \right )}}}{e^{\frac {40 \,\mathrm {log}\left (x \right ) x^{2}+40 x^{3}}{\mathrm {log}\left (x \right )}}}\right )}{x^{2}}d x \right ) \mathrm {log}\left (x \right ) x +8 \mathrm {log}\left (x \right )^{2} x^{2}+12 \,\mathrm {log}\left (x \right ) x^{3}-84 \,\mathrm {log}\left (x \right ) x^{2}+\mathrm {log}\left (x \right )+4 x^{4}-40 x^{3}+100 x^{2}}{\mathrm {log}\left (x \right ) x} \] Input:
int((-log(x)^2*log(3*exp((4*x^2*log(x)^2+(8*x^3-40*x^2)*log(x)+4*x^4-40*x^ 3+100*x^2)/log(x)))+8*x^2*log(x)^3+(24*x^3-76*x^2-1)*log(x)^2+(16*x^4-120* x^3+200*x^2)*log(x)-4*x^4+40*x^3-100*x^2)/x^2/log(x)^2,x)
Output:
(100*ei(log(x))*log(x)*x + 4*ei(3*log(x))*log(x)*x - 40*ei(2*log(x))*log(x )*x - int(log((3*x**(4*x**2)*e**((8*log(x)*x**3 + 4*x**4 + 100*x**2)/log(x )))/e**((40*log(x)*x**2 + 40*x**3)/log(x)))/x**2,x)*log(x)*x + 8*log(x)**2 *x**2 + 12*log(x)*x**3 - 84*log(x)*x**2 + log(x) + 4*x**4 - 40*x**3 + 100* x**2)/(log(x)*x)