Integrand size = 116, antiderivative size = 29 \[ \int \frac {2 x^2 \log (12)+\left (2 x^2+\left (-4 x+x^2\right ) \log (12)\right ) \log (x)+\left (-2 x^2+(2-2 x) \log (12)\right ) \log ^2(x)+\log (12) \log ^3(x)+\left (x^2 \log (12) \log (x)-2 x \log (12) \log ^2(x)+\log (12) \log ^3(x)\right ) \log \left (\frac {5}{x \log ^2(x)}\right )}{x^4 \log (x)-2 x^3 \log ^2(x)+x^2 \log ^3(x)} \, dx=\frac {2 x}{x-\log (x)}-\frac {\log (12) \log \left (\frac {5}{x \log ^2(x)}\right )}{x} \]
Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {2 x^2 \log (12)+\left (2 x^2+\left (-4 x+x^2\right ) \log (12)\right ) \log (x)+\left (-2 x^2+(2-2 x) \log (12)\right ) \log ^2(x)+\log (12) \log ^3(x)+\left (x^2 \log (12) \log (x)-2 x \log (12) \log ^2(x)+\log (12) \log ^3(x)\right ) \log \left (\frac {5}{x \log ^2(x)}\right )}{x^4 \log (x)-2 x^3 \log ^2(x)+x^2 \log ^3(x)} \, dx=\frac {2 x}{x-\log (x)}-\frac {\log (12) \log \left (\frac {5}{x \log ^2(x)}\right )}{x} \]
Integrate[(2*x^2*Log[12] + (2*x^2 + (-4*x + x^2)*Log[12])*Log[x] + (-2*x^2 + (2 - 2*x)*Log[12])*Log[x]^2 + Log[12]*Log[x]^3 + (x^2*Log[12]*Log[x] - 2*x*Log[12]*Log[x]^2 + Log[12]*Log[x]^3)*Log[5/(x*Log[x]^2)])/(x^4*Log[x] - 2*x^3*Log[x]^2 + x^2*Log[x]^3),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.
\(\displaystyle \int \frac {\left ((2-2 x) \log (12)-2 x^2\right ) \log ^2(x)+\left (x^2 \log (12) \log (x)+\log (12) \log ^3(x)-2 x \log (12) \log ^2(x)\right ) \log \left (\frac {5}{x \log ^2(x)}\right )+\left (2 x^2+\left (x^2-4 x\right ) \log (12)\right ) \log (x)+2 x^2 \log (12)+\log (12) \log ^3(x)}{x^4 \log (x)-2 x^3 \log ^2(x)+x^2 \log ^3(x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {\left ((2-2 x) \log (12)-2 x^2\right ) \log ^2(x)+\left (x^2 \log (12) \log (x)+\log (12) \log ^3(x)-2 x \log (12) \log ^2(x)\right ) \log \left (\frac {5}{x \log ^2(x)}\right )+\left (2 x^2+\left (x^2-4 x\right ) \log (12)\right ) \log (x)+2 x^2 \log (12)+\log (12) \log ^3(x)}{x^2 (x-\log (x))^2 \log (x)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {\log (12) \log ^2(x)}{x^2 (x-\log (x))^2}+\frac {\log (12) \log \left (\frac {5}{x \log ^2(x)}\right )}{x^2}-\frac {2 \left (x^2+x \log (12)-\log (12)\right ) \log (x)}{x^2 (x-\log (x))^2}+\frac {x (2+\log (12))-4 \log (12)}{x (x-\log (x))^2}+\frac {2 \log (12)}{(x-\log (x))^2 \log (x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle (2+\log (12)) \int \frac {1}{(x-\log (x))^2}dx-\log (12) \int \frac {1}{(x-\log (x))^2}dx-2 \int \frac {x}{(x-\log (x))^2}dx-2 \int \frac {1}{\log (x)-x}dx+\log (12) \log (x) \operatorname {ExpIntegralEi}(-\log (x))-\log (12) (\log (x)+2) \operatorname {ExpIntegralEi}(-\log (x))+2 \log (12) \operatorname {ExpIntegralEi}(-\log (x))-\frac {\log (12) \log \left (\frac {5}{x \log ^2(x)}\right )}{x}\) |
Int[(2*x^2*Log[12] + (2*x^2 + (-4*x + x^2)*Log[12])*Log[x] + (-2*x^2 + (2 - 2*x)*Log[12])*Log[x]^2 + Log[12]*Log[x]^3 + (x^2*Log[12]*Log[x] - 2*x*Lo g[12]*Log[x]^2 + Log[12]*Log[x]^3)*Log[5/(x*Log[x]^2)])/(x^4*Log[x] - 2*x^ 3*Log[x]^2 + x^2*Log[x]^3),x]
3.27.81.3.1 Defintions of rubi rules used
Time = 2.06 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.31
method | result | size |
default | \(-\frac {\ln \left (12\right ) \ln \left (\frac {1}{x \ln \left (x \right )^{2}}\right )}{x}-\frac {\ln \left (12\right ) \ln \left (5\right )}{x}-\frac {2 x}{\ln \left (x \right )-x}\) | \(38\) |
parallelrisch | \(\frac {-2 \ln \left (12\right ) x \ln \left (\frac {5}{x \ln \left (x \right )^{2}}\right )+2 \ln \left (12\right ) \ln \left (\frac {5}{x \ln \left (x \right )^{2}}\right ) \ln \left (x \right )+4 x^{2}}{2 x \left (x -\ln \left (x \right )\right )}\) | \(51\) |
risch | \(\text {Expression too large to display}\) | \(794\) |
int(((ln(12)*ln(x)^3-2*x*ln(12)*ln(x)^2+x^2*ln(12)*ln(x))*ln(5/x/ln(x)^2)+ ln(12)*ln(x)^3+((2-2*x)*ln(12)-2*x^2)*ln(x)^2+((x^2-4*x)*ln(12)+2*x^2)*ln( x)+2*x^2*ln(12))/(x^2*ln(x)^3-2*x^3*ln(x)^2+x^4*ln(x)),x,method=_RETURNVER BOSE)
Time = 0.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.41 \[ \int \frac {2 x^2 \log (12)+\left (2 x^2+\left (-4 x+x^2\right ) \log (12)\right ) \log (x)+\left (-2 x^2+(2-2 x) \log (12)\right ) \log ^2(x)+\log (12) \log ^3(x)+\left (x^2 \log (12) \log (x)-2 x \log (12) \log ^2(x)+\log (12) \log ^3(x)\right ) \log \left (\frac {5}{x \log ^2(x)}\right )}{x^4 \log (x)-2 x^3 \log ^2(x)+x^2 \log ^3(x)} \, dx=\frac {2 \, x^{2} - {\left (x \log \left (12\right ) - \log \left (12\right ) \log \left (x\right )\right )} \log \left (\frac {5}{x \log \left (x\right )^{2}}\right )}{x^{2} - x \log \left (x\right )} \]
integrate(((log(12)*log(x)^3-2*x*log(12)*log(x)^2+x^2*log(12)*log(x))*log( 5/x/log(x)^2)+log(12)*log(x)^3+((2-2*x)*log(12)-2*x^2)*log(x)^2+((x^2-4*x) *log(12)+2*x^2)*log(x)+2*x^2*log(12))/(x^2*log(x)^3-2*x^3*log(x)^2+x^4*log (x)),x, algorithm=\
Time = 0.18 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {2 x^2 \log (12)+\left (2 x^2+\left (-4 x+x^2\right ) \log (12)\right ) \log (x)+\left (-2 x^2+(2-2 x) \log (12)\right ) \log ^2(x)+\log (12) \log ^3(x)+\left (x^2 \log (12) \log (x)-2 x \log (12) \log ^2(x)+\log (12) \log ^3(x)\right ) \log \left (\frac {5}{x \log ^2(x)}\right )}{x^4 \log (x)-2 x^3 \log ^2(x)+x^2 \log ^3(x)} \, dx=- \frac {2 x}{- x + \log {\left (x \right )}} - \frac {\log {\left (12 \right )} \log {\left (\frac {5}{x \log {\left (x \right )}^{2}} \right )}}{x} \]
integrate(((ln(12)*ln(x)**3-2*x*ln(12)*ln(x)**2+x**2*ln(12)*ln(x))*ln(5/x/ ln(x)**2)+ln(12)*ln(x)**3+((2-2*x)*ln(12)-2*x**2)*ln(x)**2+((x**2-4*x)*ln( 12)+2*x**2)*ln(x)+2*x**2*ln(12))/(x**2*ln(x)**3-2*x**3*ln(x)**2+x**4*ln(x) ),x)
Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (29) = 58\).
Time = 0.32 (sec) , antiderivative size = 96, normalized size of antiderivative = 3.31 \[ \int \frac {2 x^2 \log (12)+\left (2 x^2+\left (-4 x+x^2\right ) \log (12)\right ) \log (x)+\left (-2 x^2+(2-2 x) \log (12)\right ) \log ^2(x)+\log (12) \log ^3(x)+\left (x^2 \log (12) \log (x)-2 x \log (12) \log ^2(x)+\log (12) \log ^3(x)\right ) \log \left (\frac {5}{x \log ^2(x)}\right )}{x^4 \log (x)-2 x^3 \log ^2(x)+x^2 \log ^3(x)} \, dx=-\frac {{\left (\log \left (3\right ) + 2 \, \log \left (2\right )\right )} \log \left (x\right )^{2} + {\left (\log \left (5\right ) \log \left (3\right ) + 2 \, \log \left (5\right ) \log \left (2\right )\right )} x - 2 \, x^{2} - {\left (x {\left (\log \left (3\right ) + 2 \, \log \left (2\right )\right )} + \log \left (5\right ) \log \left (3\right ) + 2 \, \log \left (5\right ) \log \left (2\right )\right )} \log \left (x\right ) - 2 \, {\left (x {\left (\log \left (3\right ) + 2 \, \log \left (2\right )\right )} - {\left (\log \left (3\right ) + 2 \, \log \left (2\right )\right )} \log \left (x\right )\right )} \log \left (\log \left (x\right )\right )}{x^{2} - x \log \left (x\right )} \]
integrate(((log(12)*log(x)^3-2*x*log(12)*log(x)^2+x^2*log(12)*log(x))*log( 5/x/log(x)^2)+log(12)*log(x)^3+((2-2*x)*log(12)-2*x^2)*log(x)^2+((x^2-4*x) *log(12)+2*x^2)*log(x)+2*x^2*log(12))/(x^2*log(x)^3-2*x^3*log(x)^2+x^4*log (x)),x, algorithm=\
-((log(3) + 2*log(2))*log(x)^2 + (log(5)*log(3) + 2*log(5)*log(2))*x - 2*x ^2 - (x*(log(3) + 2*log(2)) + log(5)*log(3) + 2*log(5)*log(2))*log(x) - 2* (x*(log(3) + 2*log(2)) - (log(3) + 2*log(2))*log(x))*log(log(x)))/(x^2 - x *log(x))
Time = 0.28 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.38 \[ \int \frac {2 x^2 \log (12)+\left (2 x^2+\left (-4 x+x^2\right ) \log (12)\right ) \log (x)+\left (-2 x^2+(2-2 x) \log (12)\right ) \log ^2(x)+\log (12) \log ^3(x)+\left (x^2 \log (12) \log (x)-2 x \log (12) \log ^2(x)+\log (12) \log ^3(x)\right ) \log \left (\frac {5}{x \log ^2(x)}\right )}{x^4 \log (x)-2 x^3 \log ^2(x)+x^2 \log ^3(x)} \, dx=-\frac {\log \left (12\right ) \log \left (5\right )}{x} + \frac {\log \left (12\right ) \log \left (\log \left (x\right )^{2}\right )}{x} + \frac {\log \left (12\right ) \log \left (x\right )}{x} + \frac {2 \, x}{x - \log \left (x\right )} \]
integrate(((log(12)*log(x)^3-2*x*log(12)*log(x)^2+x^2*log(12)*log(x))*log( 5/x/log(x)^2)+log(12)*log(x)^3+((2-2*x)*log(12)-2*x^2)*log(x)^2+((x^2-4*x) *log(12)+2*x^2)*log(x)+2*x^2*log(12))/(x^2*log(x)^3-2*x^3*log(x)^2+x^4*log (x)),x, algorithm=\
Time = 9.84 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.86 \[ \int \frac {2 x^2 \log (12)+\left (2 x^2+\left (-4 x+x^2\right ) \log (12)\right ) \log (x)+\left (-2 x^2+(2-2 x) \log (12)\right ) \log ^2(x)+\log (12) \log ^3(x)+\left (x^2 \log (12) \log (x)-2 x \log (12) \log ^2(x)+\log (12) \log ^3(x)\right ) \log \left (\frac {5}{x \log ^2(x)}\right )}{x^4 \log (x)-2 x^3 \log ^2(x)+x^2 \log ^3(x)} \, dx=\frac {2}{x-1}-\frac {\frac {2\,x}{x-1}-\frac {2\,x\,\ln \left (x\right )}{x-1}}{x-\ln \left (x\right )}-\frac {\ln \left (12\right )\,\ln \left (\frac {5}{x\,{\ln \left (x\right )}^2}\right )}{x} \]
int((log(12)*log(x)^3 - log(x)*(log(12)*(4*x - x^2) - 2*x^2) - log(x)^2*(l og(12)*(2*x - 2) + 2*x^2) + 2*x^2*log(12) + log(5/(x*log(x)^2))*(log(12)*l og(x)^3 - 2*x*log(12)*log(x)^2 + x^2*log(12)*log(x)))/(x^4*log(x) + x^2*lo g(x)^3 - 2*x^3*log(x)^2),x)