Integrand size = 206, antiderivative size = 31 \[ \int \frac {-2 x^3+x^5+\left (2 x^3-x^5\right ) \log \left (\frac {x}{3}\right )+\left (320-240 x-160 x^2+62 x^3-20 x^4+29 x^5+10 x^6\right ) \log ^2\left (\frac {x}{3}\right )+\left (-2 x^5 \log \left (\frac {x}{3}\right )+\left (160 x^2-240 x^3+10 x^4+58 x^5+10 x^6\right ) \log ^2\left (\frac {x}{3}\right )\right ) \log \left (\frac {x^3+\left (-80+120 x-5 x^2-29 x^3-5 x^4\right ) \log \left (\frac {x}{3}\right )}{x^2 \log \left (\frac {x}{3}\right )}\right )}{-x^4 \log \left (\frac {x}{3}\right )+\left (80 x-120 x^2+5 x^3+29 x^4+5 x^5\right ) \log ^2\left (\frac {x}{3}\right )} \, dx=\left (-2+x^2\right ) \log \left (x-5 \left (3-\frac {4}{x}+x\right )^2+\frac {x}{\log \left (\frac {x}{3}\right )}\right ) \] Output:
ln(x+x/ln(1/3*x)-5*(x+3-4/x)^2)*(x^2-2)
Leaf count is larger than twice the leaf count of optimal. \(112\) vs. \(2(31)=62\).
Time = 0.10 (sec) , antiderivative size = 112, normalized size of antiderivative = 3.61 \[ \int \frac {-2 x^3+x^5+\left (2 x^3-x^5\right ) \log \left (\frac {x}{3}\right )+\left (320-240 x-160 x^2+62 x^3-20 x^4+29 x^5+10 x^6\right ) \log ^2\left (\frac {x}{3}\right )+\left (-2 x^5 \log \left (\frac {x}{3}\right )+\left (160 x^2-240 x^3+10 x^4+58 x^5+10 x^6\right ) \log ^2\left (\frac {x}{3}\right )\right ) \log \left (\frac {x^3+\left (-80+120 x-5 x^2-29 x^3-5 x^4\right ) \log \left (\frac {x}{3}\right )}{x^2 \log \left (\frac {x}{3}\right )}\right )}{-x^4 \log \left (\frac {x}{3}\right )+\left (80 x-120 x^2+5 x^3+29 x^4+5 x^5\right ) \log ^2\left (\frac {x}{3}\right )} \, dx=4 \log \left (\frac {x}{3}\right )+x^2 \log \left (-5-\frac {80}{x^2}+\frac {120}{x}-29 x-5 x^2+\frac {x}{\log \left (\frac {x}{3}\right )}\right )+2 \log \left (\log \left (\frac {x}{3}\right )\right )-2 \log \left (-x^3+80 \log \left (\frac {x}{3}\right )-120 x \log \left (\frac {x}{3}\right )+5 x^2 \log \left (\frac {x}{3}\right )+29 x^3 \log \left (\frac {x}{3}\right )+5 x^4 \log \left (\frac {x}{3}\right )\right ) \] Input:
Integrate[(-2*x^3 + x^5 + (2*x^3 - x^5)*Log[x/3] + (320 - 240*x - 160*x^2 + 62*x^3 - 20*x^4 + 29*x^5 + 10*x^6)*Log[x/3]^2 + (-2*x^5*Log[x/3] + (160* x^2 - 240*x^3 + 10*x^4 + 58*x^5 + 10*x^6)*Log[x/3]^2)*Log[(x^3 + (-80 + 12 0*x - 5*x^2 - 29*x^3 - 5*x^4)*Log[x/3])/(x^2*Log[x/3])])/(-(x^4*Log[x/3]) + (80*x - 120*x^2 + 5*x^3 + 29*x^4 + 5*x^5)*Log[x/3]^2),x]
Output:
4*Log[x/3] + x^2*Log[-5 - 80/x^2 + 120/x - 29*x - 5*x^2 + x/Log[x/3]] + 2* Log[Log[x/3]] - 2*Log[-x^3 + 80*Log[x/3] - 120*x*Log[x/3] + 5*x^2*Log[x/3] + 29*x^3*Log[x/3] + 5*x^4*Log[x/3]]
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.
| \(\displaystyle \int \frac {x^5-2 x^3+\left (2 x^3-x^5\right ) \log \left (\frac {x}{3}\right )+\left (10 x^6+29 x^5-20 x^4+62 x^3-160 x^2-240 x+320\right ) \log ^2\left (\frac {x}{3}\right )+\left (\left (10 x^6+58 x^5+10 x^4-240 x^3+160 x^2\right ) \log ^2\left (\frac {x}{3}\right )-2 x^5 \log \left (\frac {x}{3}\right )\right ) \log \left (\frac {x^3+\left (-5 x^4-29 x^3-5 x^2+120 x-80\right ) \log \left (\frac {x}{3}\right )}{x^2 \log \left (\frac {x}{3}\right )}\right )}{\left (5 x^5+29 x^4+5 x^3-120 x^2+80 x\right ) \log ^2\left (\frac {x}{3}\right )-x^4 \log \left (\frac {x}{3}\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2 x \log \left (-5 x^2-\frac {80}{x^2}-29 x+\frac {120}{x}+\frac {x}{\log \left (\frac {x}{3}\right )}-5\right )+\frac {\left (x^2-2\right ) \left (10 x^4 \log ^2\left (\frac {x}{3}\right )+x^3+29 x^3 \log ^2\left (\frac {x}{3}\right )-x^3 \log \left (\frac {x}{3}\right )+120 x \log ^2\left (\frac {x}{3}\right )-160 \log ^2\left (\frac {x}{3}\right )\right )}{x \log \left (\frac {x}{3}\right ) \left (5 x^4 \log \left (\frac {x}{3}\right )-x^3+29 x^3 \log \left (\frac {x}{3}\right )+5 x^2 \log \left (\frac {x}{3}\right )-120 x \log \left (\frac {x}{3}\right )+80 \log \left (\frac {x}{3}\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle x^2-\frac {29 x}{5}-9 \operatorname {ExpIntegralEi}\left (2 \log \left (\frac {x}{3}\right )\right )+4 \log (x)+\frac {591}{100} \log \left (5 x^4+29 x^3+5 x^2-120 x+80\right )+2 \log \left (\log \left (\frac {x}{3}\right )\right )+\frac {7066}{5} \int \frac {1}{5 x^4+29 x^3+5 x^2-120 x+80}dx-\frac {10951}{10} \int \frac {x}{5 x^4+29 x^3+5 x^2-120 x+80}dx-\frac {29917}{100} \int \frac {x^2}{5 x^4+29 x^3+5 x^2-120 x+80}dx+\frac {346881}{625} \int \frac {1}{5 \log \left (\frac {x}{3}\right ) x^4+29 \log \left (\frac {x}{3}\right ) x^3-x^3+5 \log \left (\frac {x}{3}\right ) x^2-120 \log \left (\frac {x}{3}\right ) x+80 \log \left (\frac {x}{3}\right )}dx-\frac {3014}{125} \int \frac {x}{5 \log \left (\frac {x}{3}\right ) x^4+29 \log \left (\frac {x}{3}\right ) x^3-x^3+5 \log \left (\frac {x}{3}\right ) x^2-120 \log \left (\frac {x}{3}\right ) x+80 \log \left (\frac {x}{3}\right )}dx-\frac {3709}{25} \int \frac {x^2}{5 \log \left (\frac {x}{3}\right ) x^4+29 \log \left (\frac {x}{3}\right ) x^3-x^3+5 \log \left (\frac {x}{3}\right ) x^2-120 \log \left (\frac {x}{3}\right ) x+80 \log \left (\frac {x}{3}\right )}dx-\frac {54}{5} \int \frac {x^3}{5 \log \left (\frac {x}{3}\right ) x^4+29 \log \left (\frac {x}{3}\right ) x^3-x^3+5 \log \left (\frac {x}{3}\right ) x^2-120 \log \left (\frac {x}{3}\right ) x+80 \log \left (\frac {x}{3}\right )}dx+30 \int \frac {x^4}{5 \log \left (\frac {x}{3}\right ) x^4+29 \log \left (\frac {x}{3}\right ) x^3-x^3+5 \log \left (\frac {x}{3}\right ) x^2-120 \log \left (\frac {x}{3}\right ) x+80 \log \left (\frac {x}{3}\right )}dx+5 \int \frac {x^5}{5 \log \left (\frac {x}{3}\right ) x^4+29 \log \left (\frac {x}{3}\right ) x^3-x^3+5 \log \left (\frac {x}{3}\right ) x^2-120 \log \left (\frac {x}{3}\right ) x+80 \log \left (\frac {x}{3}\right )}dx-\frac {3150096}{125} \int \frac {1}{\left (5 x^4+29 x^3+5 x^2-120 x+80\right ) \left (5 \log \left (\frac {x}{3}\right ) x^4+29 \log \left (\frac {x}{3}\right ) x^3-x^3+5 \log \left (\frac {x}{3}\right ) x^2-120 \log \left (\frac {x}{3}\right ) x+80 \log \left (\frac {x}{3}\right )\right )}dx+\frac {5666264}{125} \int \frac {x}{\left (5 x^4+29 x^3+5 x^2-120 x+80\right ) \left (5 \log \left (\frac {x}{3}\right ) x^4+29 \log \left (\frac {x}{3}\right ) x^3-x^3+5 \log \left (\frac {x}{3}\right ) x^2-120 \log \left (\frac {x}{3}\right ) x+80 \log \left (\frac {x}{3}\right )\right )}dx-\frac {1844961}{125} \int \frac {x^2}{\left (5 x^4+29 x^3+5 x^2-120 x+80\right ) \left (5 \log \left (\frac {x}{3}\right ) x^4+29 \log \left (\frac {x}{3}\right ) x^3-x^3+5 \log \left (\frac {x}{3}\right ) x^2-120 \log \left (\frac {x}{3}\right ) x+80 \log \left (\frac {x}{3}\right )\right )}dx-\frac {3202449}{625} \int \frac {x^3}{\left (5 x^4+29 x^3+5 x^2-120 x+80\right ) \left (5 \log \left (\frac {x}{3}\right ) x^4+29 \log \left (\frac {x}{3}\right ) x^3-x^3+5 \log \left (\frac {x}{3}\right ) x^2-120 \log \left (\frac {x}{3}\right ) x+80 \log \left (\frac {x}{3}\right )\right )}dx-160 \int \frac {1}{x \left (5 x^4+29 x^3+5 x^2-120 x+80\right ) \log \left (\frac {x}{3}\right )-x^4}dx+2 \int x \log \left (-5 x^2+\frac {x}{\log \left (\frac {x}{3}\right )}-29 x-5+\frac {120}{x}-\frac {80}{x^2}\right )dx\) |
Input:
Int[(-2*x^3 + x^5 + (2*x^3 - x^5)*Log[x/3] + (320 - 240*x - 160*x^2 + 62*x ^3 - 20*x^4 + 29*x^5 + 10*x^6)*Log[x/3]^2 + (-2*x^5*Log[x/3] + (160*x^2 - 240*x^3 + 10*x^4 + 58*x^5 + 10*x^6)*Log[x/3]^2)*Log[(x^3 + (-80 + 120*x - 5*x^2 - 29*x^3 - 5*x^4)*Log[x/3])/(x^2*Log[x/3])])/(-(x^4*Log[x/3]) + (80* x - 120*x^2 + 5*x^3 + 29*x^4 + 5*x^5)*Log[x/3]^2),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(87\) vs. \(2(29)=58\).
Time = 13.19 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.84
| method | result | size |
| parallelrisch | \(\ln \left (\frac {\left (-5 x^{4}-29 x^{3}-5 x^{2}+120 x -80\right ) \ln \left (\frac {x}{3}\right )+x^{3}}{x^{2} \ln \left (\frac {x}{3}\right )}\right ) x^{2}-2 \ln \left (\frac {\left (-5 x^{4}-29 x^{3}-5 x^{2}+120 x -80\right ) \ln \left (\frac {x}{3}\right )+x^{3}}{x^{2} \ln \left (\frac {x}{3}\right )}\right )\) | \(88\) |
Input:
int((((10*x^6+58*x^5+10*x^4-240*x^3+160*x^2)*ln(1/3*x)^2-2*x^5*ln(1/3*x))* ln(((-5*x^4-29*x^3-5*x^2+120*x-80)*ln(1/3*x)+x^3)/x^2/ln(1/3*x))+(10*x^6+2 9*x^5-20*x^4+62*x^3-160*x^2-240*x+320)*ln(1/3*x)^2+(-x^5+2*x^3)*ln(1/3*x)+ x^5-2*x^3)/((5*x^5+29*x^4+5*x^3-120*x^2+80*x)*ln(1/3*x)^2-x^4*ln(1/3*x)),x ,method=_RETURNVERBOSE)
Output:
ln(((-5*x^4-29*x^3-5*x^2+120*x-80)*ln(1/3*x)+x^3)/x^2/ln(1/3*x))*x^2-2*ln( ((-5*x^4-29*x^3-5*x^2+120*x-80)*ln(1/3*x)+x^3)/x^2/ln(1/3*x))
Time = 0.09 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.52 \[ \int \frac {-2 x^3+x^5+\left (2 x^3-x^5\right ) \log \left (\frac {x}{3}\right )+\left (320-240 x-160 x^2+62 x^3-20 x^4+29 x^5+10 x^6\right ) \log ^2\left (\frac {x}{3}\right )+\left (-2 x^5 \log \left (\frac {x}{3}\right )+\left (160 x^2-240 x^3+10 x^4+58 x^5+10 x^6\right ) \log ^2\left (\frac {x}{3}\right )\right ) \log \left (\frac {x^3+\left (-80+120 x-5 x^2-29 x^3-5 x^4\right ) \log \left (\frac {x}{3}\right )}{x^2 \log \left (\frac {x}{3}\right )}\right )}{-x^4 \log \left (\frac {x}{3}\right )+\left (80 x-120 x^2+5 x^3+29 x^4+5 x^5\right ) \log ^2\left (\frac {x}{3}\right )} \, dx={\left (x^{2} - 2\right )} \log \left (\frac {x^{3} - {\left (5 \, x^{4} + 29 \, x^{3} + 5 \, x^{2} - 120 \, x + 80\right )} \log \left (\frac {1}{3} \, x\right )}{x^{2} \log \left (\frac {1}{3} \, x\right )}\right ) \] Input:
integrate((((10*x^6+58*x^5+10*x^4-240*x^3+160*x^2)*log(1/3*x)^2-2*x^5*log( 1/3*x))*log(((-5*x^4-29*x^3-5*x^2+120*x-80)*log(1/3*x)+x^3)/x^2/log(1/3*x) )+(10*x^6+29*x^5-20*x^4+62*x^3-160*x^2-240*x+320)*log(1/3*x)^2+(-x^5+2*x^3 )*log(1/3*x)+x^5-2*x^3)/((5*x^5+29*x^4+5*x^3-120*x^2+80*x)*log(1/3*x)^2-x^ 4*log(1/3*x)),x, algorithm="fricas")
Output:
(x^2 - 2)*log((x^3 - (5*x^4 + 29*x^3 + 5*x^2 - 120*x + 80)*log(1/3*x))/(x^ 2*log(1/3*x)))
Exception generated. \[ \int \frac {-2 x^3+x^5+\left (2 x^3-x^5\right ) \log \left (\frac {x}{3}\right )+\left (320-240 x-160 x^2+62 x^3-20 x^4+29 x^5+10 x^6\right ) \log ^2\left (\frac {x}{3}\right )+\left (-2 x^5 \log \left (\frac {x}{3}\right )+\left (160 x^2-240 x^3+10 x^4+58 x^5+10 x^6\right ) \log ^2\left (\frac {x}{3}\right )\right ) \log \left (\frac {x^3+\left (-80+120 x-5 x^2-29 x^3-5 x^4\right ) \log \left (\frac {x}{3}\right )}{x^2 \log \left (\frac {x}{3}\right )}\right )}{-x^4 \log \left (\frac {x}{3}\right )+\left (80 x-120 x^2+5 x^3+29 x^4+5 x^5\right ) \log ^2\left (\frac {x}{3}\right )} \, dx=\text {Exception raised: PolynomialError} \] Input:
integrate((((10*x**6+58*x**5+10*x**4-240*x**3+160*x**2)*ln(1/3*x)**2-2*x** 5*ln(1/3*x))*ln(((-5*x**4-29*x**3-5*x**2+120*x-80)*ln(1/3*x)+x**3)/x**2/ln (1/3*x))+(10*x**6+29*x**5-20*x**4+62*x**3-160*x**2-240*x+320)*ln(1/3*x)**2 +(-x**5+2*x**3)*ln(1/3*x)+x**5-2*x**3)/((5*x**5+29*x**4+5*x**3-120*x**2+80 *x)*ln(1/3*x)**2-x**4*ln(1/3*x)),x)
Output:
Exception raised: PolynomialError >> 1/(25*x**9 + 290*x**8 + 891*x**7 - 91 0*x**6 - 6135*x**5 + 3440*x**4 + 15200*x**3 - 19200*x**2 + 6400*x) contain s an element of the set of generators.
Leaf count of result is larger than twice the leaf count of optimal. 198 vs. \(2 (29) = 58\).
Time = 0.19 (sec) , antiderivative size = 198, normalized size of antiderivative = 6.39 \[ \int \frac {-2 x^3+x^5+\left (2 x^3-x^5\right ) \log \left (\frac {x}{3}\right )+\left (320-240 x-160 x^2+62 x^3-20 x^4+29 x^5+10 x^6\right ) \log ^2\left (\frac {x}{3}\right )+\left (-2 x^5 \log \left (\frac {x}{3}\right )+\left (160 x^2-240 x^3+10 x^4+58 x^5+10 x^6\right ) \log ^2\left (\frac {x}{3}\right )\right ) \log \left (\frac {x^3+\left (-80+120 x-5 x^2-29 x^3-5 x^4\right ) \log \left (\frac {x}{3}\right )}{x^2 \log \left (\frac {x}{3}\right )}\right )}{-x^4 \log \left (\frac {x}{3}\right )+\left (80 x-120 x^2+5 x^3+29 x^4+5 x^5\right ) \log ^2\left (\frac {x}{3}\right )} \, dx=x^{2} \log \left (5 \, x^{4} \log \left (3\right ) + x^{3} {\left (29 \, \log \left (3\right ) + 1\right )} + 5 \, x^{2} \log \left (3\right ) - 120 \, x \log \left (3\right ) - {\left (5 \, x^{4} + 29 \, x^{3} + 5 \, x^{2} - 120 \, x + 80\right )} \log \left (x\right ) + 80 \, \log \left (3\right )\right ) - 2 \, x^{2} \log \left (x\right ) - {\left (x^{2} - 2\right )} \log \left (-\log \left (3\right ) + \log \left (x\right )\right ) - 2 \, \log \left (5 \, x^{4} + 29 \, x^{3} + 5 \, x^{2} - 120 \, x + 80\right ) + 4 \, \log \left (x\right ) - 2 \, \log \left (-\frac {5 \, x^{4} \log \left (3\right ) + x^{3} {\left (29 \, \log \left (3\right ) + 1\right )} + 5 \, x^{2} \log \left (3\right ) - 120 \, x \log \left (3\right ) - {\left (5 \, x^{4} + 29 \, x^{3} + 5 \, x^{2} - 120 \, x + 80\right )} \log \left (x\right ) + 80 \, \log \left (3\right )}{5 \, x^{4} + 29 \, x^{3} + 5 \, x^{2} - 120 \, x + 80}\right ) \] Input:
integrate((((10*x^6+58*x^5+10*x^4-240*x^3+160*x^2)*log(1/3*x)^2-2*x^5*log( 1/3*x))*log(((-5*x^4-29*x^3-5*x^2+120*x-80)*log(1/3*x)+x^3)/x^2/log(1/3*x) )+(10*x^6+29*x^5-20*x^4+62*x^3-160*x^2-240*x+320)*log(1/3*x)^2+(-x^5+2*x^3 )*log(1/3*x)+x^5-2*x^3)/((5*x^5+29*x^4+5*x^3-120*x^2+80*x)*log(1/3*x)^2-x^ 4*log(1/3*x)),x, algorithm="maxima")
Output:
x^2*log(5*x^4*log(3) + x^3*(29*log(3) + 1) + 5*x^2*log(3) - 120*x*log(3) - (5*x^4 + 29*x^3 + 5*x^2 - 120*x + 80)*log(x) + 80*log(3)) - 2*x^2*log(x) - (x^2 - 2)*log(-log(3) + log(x)) - 2*log(5*x^4 + 29*x^3 + 5*x^2 - 120*x + 80) + 4*log(x) - 2*log(-(5*x^4*log(3) + x^3*(29*log(3) + 1) + 5*x^2*log(3 ) - 120*x*log(3) - (5*x^4 + 29*x^3 + 5*x^2 - 120*x + 80)*log(x) + 80*log(3 ))/(5*x^4 + 29*x^3 + 5*x^2 - 120*x + 80))
Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (29) = 58\).
Time = 0.63 (sec) , antiderivative size = 148, normalized size of antiderivative = 4.77 \[ \int \frac {-2 x^3+x^5+\left (2 x^3-x^5\right ) \log \left (\frac {x}{3}\right )+\left (320-240 x-160 x^2+62 x^3-20 x^4+29 x^5+10 x^6\right ) \log ^2\left (\frac {x}{3}\right )+\left (-2 x^5 \log \left (\frac {x}{3}\right )+\left (160 x^2-240 x^3+10 x^4+58 x^5+10 x^6\right ) \log ^2\left (\frac {x}{3}\right )\right ) \log \left (\frac {x^3+\left (-80+120 x-5 x^2-29 x^3-5 x^4\right ) \log \left (\frac {x}{3}\right )}{x^2 \log \left (\frac {x}{3}\right )}\right )}{-x^4 \log \left (\frac {x}{3}\right )+\left (80 x-120 x^2+5 x^3+29 x^4+5 x^5\right ) \log ^2\left (\frac {x}{3}\right )} \, dx=x^{2} \log \left (-5 \, x^{4} \log \left (\frac {1}{3} \, x\right ) - 29 \, x^{3} \log \left (\frac {1}{3} \, x\right ) + x^{3} - 5 \, x^{2} \log \left (\frac {1}{3} \, x\right ) + 120 \, x \log \left (\frac {1}{3} \, x\right ) - 80 \, \log \left (\frac {1}{3} \, x\right )\right ) - 2 \, x^{2} \log \left (x\right ) - x^{2} \log \left (\log \left (\frac {1}{3} \, x\right )\right ) - 2 \, \log \left (5 \, x^{4} \log \left (3\right ) - 5 \, x^{4} \log \left (x\right ) + 29 \, x^{3} \log \left (3\right ) - 29 \, x^{3} \log \left (x\right ) + x^{3} + 5 \, x^{2} \log \left (3\right ) - 5 \, x^{2} \log \left (x\right ) - 120 \, x \log \left (3\right ) + 120 \, x \log \left (x\right ) + 80 \, \log \left (3\right ) - 80 \, \log \left (x\right )\right ) + 4 \, \log \left (x\right ) + 2 \, \log \left (\log \left (3\right ) - \log \left (x\right )\right ) \] Input:
integrate((((10*x^6+58*x^5+10*x^4-240*x^3+160*x^2)*log(1/3*x)^2-2*x^5*log( 1/3*x))*log(((-5*x^4-29*x^3-5*x^2+120*x-80)*log(1/3*x)+x^3)/x^2/log(1/3*x) )+(10*x^6+29*x^5-20*x^4+62*x^3-160*x^2-240*x+320)*log(1/3*x)^2+(-x^5+2*x^3 )*log(1/3*x)+x^5-2*x^3)/((5*x^5+29*x^4+5*x^3-120*x^2+80*x)*log(1/3*x)^2-x^ 4*log(1/3*x)),x, algorithm="giac")
Output:
x^2*log(-5*x^4*log(1/3*x) - 29*x^3*log(1/3*x) + x^3 - 5*x^2*log(1/3*x) + 1 20*x*log(1/3*x) - 80*log(1/3*x)) - 2*x^2*log(x) - x^2*log(log(1/3*x)) - 2* log(5*x^4*log(3) - 5*x^4*log(x) + 29*x^3*log(3) - 29*x^3*log(x) + x^3 + 5* x^2*log(3) - 5*x^2*log(x) - 120*x*log(3) + 120*x*log(x) + 80*log(3) - 80*l og(x)) + 4*log(x) + 2*log(log(3) - log(x))
Time = 3.33 (sec) , antiderivative size = 357, normalized size of antiderivative = 11.52 \[ \int \frac {-2 x^3+x^5+\left (2 x^3-x^5\right ) \log \left (\frac {x}{3}\right )+\left (320-240 x-160 x^2+62 x^3-20 x^4+29 x^5+10 x^6\right ) \log ^2\left (\frac {x}{3}\right )+\left (-2 x^5 \log \left (\frac {x}{3}\right )+\left (160 x^2-240 x^3+10 x^4+58 x^5+10 x^6\right ) \log ^2\left (\frac {x}{3}\right )\right ) \log \left (\frac {x^3+\left (-80+120 x-5 x^2-29 x^3-5 x^4\right ) \log \left (\frac {x}{3}\right )}{x^2 \log \left (\frac {x}{3}\right )}\right )}{-x^4 \log \left (\frac {x}{3}\right )+\left (80 x-120 x^2+5 x^3+29 x^4+5 x^5\right ) \log ^2\left (\frac {x}{3}\right )} \, dx=2\,\ln \left (x^4+\frac {29\,x^3}{5}+x^2-24\,x+16\right )+2\,\ln \left (\frac {25600\,\ln \left (\frac {x}{3}\right )-76800\,x\,\ln \left (\frac {x}{3}\right )+100\,x^8\,\ln \left (x\right )+60800\,x^2\,\ln \left (\frac {x}{3}\right )+12800\,x^3\,\ln \left (\frac {x}{3}\right )-23580\,x^4\,\ln \left (\frac {x}{3}\right )-3660\,x^5\,\ln \left (\frac {x}{3}\right )+3564\,x^6\,\ln \left (\frac {x}{3}\right )+1180\,x^7\,\ln \left (\frac {x}{3}\right )-100\,x^8\,\ln \left (3\right )}{x\,{\left (5\,x^4+29\,x^3+5\,x^2-120\,x+80\right )}^2}\right )-2\,\ln \left (\frac {320\,\ln \left (3\right )-320\,\ln \left (x\right )-20\,x^2\,\ln \left (x\right )-116\,x^3\,\ln \left (x\right )-20\,x^4\,\ln \left (x\right )-480\,x\,\ln \left (3\right )+20\,x^2\,\ln \left (3\right )+116\,x^3\,\ln \left (3\right )+20\,x^4\,\ln \left (3\right )+480\,x\,\ln \left (x\right )+4\,x^3}{x\,\left (5\,x^4+29\,x^3+5\,x^2-120\,x+80\right )}\right )-2\,\ln \left (x^8+\frac {59\,x^7}{5}+\frac {891\,x^6}{25}-\frac {183\,x^5}{5}-\frac {1179\,x^4}{5}+128\,x^3+608\,x^2-768\,x+256\right )+4\,\ln \left (x\right )+x^2\,\ln \left (\frac {80\,\ln \left (3\right )-80\,\ln \left (x\right )-5\,x^2\,\ln \left (x\right )-29\,x^3\,\ln \left (x\right )-5\,x^4\,\ln \left (x\right )-120\,x\,\ln \left (3\right )+5\,x^2\,\ln \left (3\right )+29\,x^3\,\ln \left (3\right )+5\,x^4\,\ln \left (3\right )+120\,x\,\ln \left (x\right )+x^3}{x^2\,\ln \left (x\right )-x^2\,\ln \left (3\right )}\right ) \] Input:
int((log(-(log(x/3)*(5*x^2 - 120*x + 29*x^3 + 5*x^4 + 80) - x^3)/(x^2*log( x/3)))*(log(x/3)^2*(160*x^2 - 240*x^3 + 10*x^4 + 58*x^5 + 10*x^6) - 2*x^5* log(x/3)) + log(x/3)^2*(62*x^3 - 160*x^2 - 240*x - 20*x^4 + 29*x^5 + 10*x^ 6 + 320) + log(x/3)*(2*x^3 - x^5) - 2*x^3 + x^5)/(log(x/3)^2*(80*x - 120*x ^2 + 5*x^3 + 29*x^4 + 5*x^5) - x^4*log(x/3)),x)
Output:
2*log(x^2 - 24*x + (29*x^3)/5 + x^4 + 16) + 2*log((25600*log(x/3) - 76800* x*log(x/3) + 100*x^8*log(x) + 60800*x^2*log(x/3) + 12800*x^3*log(x/3) - 23 580*x^4*log(x/3) - 3660*x^5*log(x/3) + 3564*x^6*log(x/3) + 1180*x^7*log(x/ 3) - 100*x^8*log(3))/(x*(5*x^2 - 120*x + 29*x^3 + 5*x^4 + 80)^2)) - 2*log( (320*log(3) - 320*log(x) - 20*x^2*log(x) - 116*x^3*log(x) - 20*x^4*log(x) - 480*x*log(3) + 20*x^2*log(3) + 116*x^3*log(3) + 20*x^4*log(3) + 480*x*lo g(x) + 4*x^3)/(x*(5*x^2 - 120*x + 29*x^3 + 5*x^4 + 80))) - 2*log(608*x^2 - 768*x + 128*x^3 - (1179*x^4)/5 - (183*x^5)/5 + (891*x^6)/25 + (59*x^7)/5 + x^8 + 256) + 4*log(x) + x^2*log((80*log(3) - 80*log(x) - 5*x^2*log(x) - 29*x^3*log(x) - 5*x^4*log(x) - 120*x*log(3) + 5*x^2*log(3) + 29*x^3*log(3) + 5*x^4*log(3) + 120*x*log(x) + x^3)/(x^2*log(x) - x^2*log(3)))
Time = 0.18 (sec) , antiderivative size = 183, normalized size of antiderivative = 5.90 \[ \int \frac {-2 x^3+x^5+\left (2 x^3-x^5\right ) \log \left (\frac {x}{3}\right )+\left (320-240 x-160 x^2+62 x^3-20 x^4+29 x^5+10 x^6\right ) \log ^2\left (\frac {x}{3}\right )+\left (-2 x^5 \log \left (\frac {x}{3}\right )+\left (160 x^2-240 x^3+10 x^4+58 x^5+10 x^6\right ) \log ^2\left (\frac {x}{3}\right )\right ) \log \left (\frac {x^3+\left (-80+120 x-5 x^2-29 x^3-5 x^4\right ) \log \left (\frac {x}{3}\right )}{x^2 \log \left (\frac {x}{3}\right )}\right )}{-x^4 \log \left (\frac {x}{3}\right )+\left (80 x-120 x^2+5 x^3+29 x^4+5 x^5\right ) \log ^2\left (\frac {x}{3}\right )} \, dx=-\frac {11764 \,\mathrm {log}\left (\mathrm {log}\left (\frac {x}{3}\right )\right )}{725}+\frac {11764 \,\mathrm {log}\left (5 \,\mathrm {log}\left (\frac {x}{3}\right ) x^{4}+29 \,\mathrm {log}\left (\frac {x}{3}\right ) x^{3}+5 \,\mathrm {log}\left (\frac {x}{3}\right ) x^{2}-120 \,\mathrm {log}\left (\frac {x}{3}\right ) x +80 \,\mathrm {log}\left (\frac {x}{3}\right )-x^{3}\right )}{725}+\mathrm {log}\left (\frac {-5 \,\mathrm {log}\left (\frac {x}{3}\right ) x^{4}-29 \,\mathrm {log}\left (\frac {x}{3}\right ) x^{3}-5 \,\mathrm {log}\left (\frac {x}{3}\right ) x^{2}+120 \,\mathrm {log}\left (\frac {x}{3}\right ) x -80 \,\mathrm {log}\left (\frac {x}{3}\right )+x^{3}}{\mathrm {log}\left (\frac {x}{3}\right ) x^{2}}\right ) x^{2}-\frac {13214 \,\mathrm {log}\left (\frac {-5 \,\mathrm {log}\left (\frac {x}{3}\right ) x^{4}-29 \,\mathrm {log}\left (\frac {x}{3}\right ) x^{3}-5 \,\mathrm {log}\left (\frac {x}{3}\right ) x^{2}+120 \,\mathrm {log}\left (\frac {x}{3}\right ) x -80 \,\mathrm {log}\left (\frac {x}{3}\right )+x^{3}}{\mathrm {log}\left (\frac {x}{3}\right ) x^{2}}\right )}{725}-\frac {27017 \,\mathrm {log}\left (\frac {x}{3}\right )}{725}+\frac {3489 \,\mathrm {log}\left (x \right )}{725} \] Input:
int((((10*x^6+58*x^5+10*x^4-240*x^3+160*x^2)*log(1/3*x)^2-2*x^5*log(1/3*x) )*log(((-5*x^4-29*x^3-5*x^2+120*x-80)*log(1/3*x)+x^3)/x^2/log(1/3*x))+(10* x^6+29*x^5-20*x^4+62*x^3-160*x^2-240*x+320)*log(1/3*x)^2+(-x^5+2*x^3)*log( 1/3*x)+x^5-2*x^3)/((5*x^5+29*x^4+5*x^3-120*x^2+80*x)*log(1/3*x)^2-x^4*log( 1/3*x)),x)
Output:
( - 11764*log(log(x/3)) + 11764*log(5*log(x/3)*x**4 + 29*log(x/3)*x**3 + 5 *log(x/3)*x**2 - 120*log(x/3)*x + 80*log(x/3) - x**3) + 725*log(( - 5*log( x/3)*x**4 - 29*log(x/3)*x**3 - 5*log(x/3)*x**2 + 120*log(x/3)*x - 80*log(x /3) + x**3)/(log(x/3)*x**2))*x**2 - 13214*log(( - 5*log(x/3)*x**4 - 29*log (x/3)*x**3 - 5*log(x/3)*x**2 + 120*log(x/3)*x - 80*log(x/3) + x**3)/(log(x /3)*x**2)) - 27017*log(x/3) + 3489*log(x))/725