Integrand size = 132, antiderivative size = 23 \[ \int \frac {\left (-30+5 x^3\right ) \log (3)+\left (\left (45+150 x^2+15 x^3\right ) \log (3)+\left (15+50 x^2+5 x^3\right ) \log (3) \log \left (\frac {-3-10 x^2-x^3}{2 x^2}\right )\right ) \log \left (3+\log \left (\frac {-3-10 x^2-x^3}{2 x^2}\right )\right )}{9+30 x^2+3 x^3+\left (3+10 x^2+x^3\right ) \log \left (\frac {-3-10 x^2-x^3}{2 x^2}\right )} \, dx=5 x \log (3) \log \left (3+\log \left (\frac {1}{2} \left (-10-\frac {3}{x^2}-x\right )\right )\right ) \]
Time = 0.06 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-30+5 x^3\right ) \log (3)+\left (\left (45+150 x^2+15 x^3\right ) \log (3)+\left (15+50 x^2+5 x^3\right ) \log (3) \log \left (\frac {-3-10 x^2-x^3}{2 x^2}\right )\right ) \log \left (3+\log \left (\frac {-3-10 x^2-x^3}{2 x^2}\right )\right )}{9+30 x^2+3 x^3+\left (3+10 x^2+x^3\right ) \log \left (\frac {-3-10 x^2-x^3}{2 x^2}\right )} \, dx=5 x \log (3) \log \left (3+\log \left (-5-\frac {3}{2 x^2}-\frac {x}{2}\right )\right ) \]
Integrate[((-30 + 5*x^3)*Log[3] + ((45 + 150*x^2 + 15*x^3)*Log[3] + (15 + 50*x^2 + 5*x^3)*Log[3]*Log[(-3 - 10*x^2 - x^3)/(2*x^2)])*Log[3 + Log[(-3 - 10*x^2 - x^3)/(2*x^2)]])/(9 + 30*x^2 + 3*x^3 + (3 + 10*x^2 + x^3)*Log[(-3 - 10*x^2 - x^3)/(2*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.
\(\displaystyle \int \frac {\left (5 x^3-30\right ) \log (3)+\left (\left (15 x^3+150 x^2+45\right ) \log (3)+\left (5 x^3+50 x^2+15\right ) \log (3) \log \left (\frac {-x^3-10 x^2-3}{2 x^2}\right )\right ) \log \left (\log \left (\frac {-x^3-10 x^2-3}{2 x^2}\right )+3\right )}{3 x^3+30 x^2+\left (x^3+10 x^2+3\right ) \log \left (\frac {-x^3-10 x^2-3}{2 x^2}\right )+9} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {\left (5 x^3-30\right ) \log (3)+\left (\left (15 x^3+150 x^2+45\right ) \log (3)+\left (5 x^3+50 x^2+15\right ) \log (3) \log \left (\frac {-x^3-10 x^2-3}{2 x^2}\right )\right ) \log \left (\log \left (\frac {-x^3-10 x^2-3}{2 x^2}\right )+3\right )}{\left (x^3+10 x^2+3\right ) \left (\log \left (-\frac {3}{2 x^2}-\frac {x}{2}-5\right )+3\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (5 \log (3) \log \left (\log \left (-\frac {3}{2 x^2}-\frac {x}{2}-5\right )+3\right )+\frac {5 \left (x^3-6\right ) \log (3)}{\left (x^3+10 x^2+3\right ) \left (\log \left (-\frac {3}{2 x^2}-\frac {x}{2}-5\right )+3\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 5 \log (3) \int \frac {1}{\log \left (-\frac {x}{2}-5-\frac {3}{2 x^2}\right )+3}dx+5 \log (3) \int \log \left (\log \left (-\frac {x}{2}-5-\frac {3}{2 x^2}\right )+3\right )dx-45 \log (3) \int \frac {1}{\left (x^3+10 x^2+3\right ) \left (\log \left (-\frac {x}{2}-5-\frac {3}{2 x^2}\right )+3\right )}dx-50 \log (3) \int \frac {x^2}{\left (x^3+10 x^2+3\right ) \left (\log \left (-\frac {x}{2}-5-\frac {3}{2 x^2}\right )+3\right )}dx\) |
Int[((-30 + 5*x^3)*Log[3] + ((45 + 150*x^2 + 15*x^3)*Log[3] + (15 + 50*x^2 + 5*x^3)*Log[3]*Log[(-3 - 10*x^2 - x^3)/(2*x^2)])*Log[3 + Log[(-3 - 10*x^ 2 - x^3)/(2*x^2)]])/(9 + 30*x^2 + 3*x^3 + (3 + 10*x^2 + x^3)*Log[(-3 - 10* x^2 - x^3)/(2*x^2)]),x]
3.16.65.3.1 Defintions of rubi rules used
Time = 3.11 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09
method | result | size |
parallelrisch | \(5 \ln \left (3\right ) x \ln \left (\ln \left (-\frac {x^{3}+10 x^{2}+3}{2 x^{2}}\right )+3\right )\) | \(25\) |
int((((5*x^3+50*x^2+15)*ln(3)*ln(1/2*(-x^3-10*x^2-3)/x^2)+(15*x^3+150*x^2+ 45)*ln(3))*ln(ln(1/2*(-x^3-10*x^2-3)/x^2)+3)+(5*x^3-30)*ln(3))/((x^3+10*x^ 2+3)*ln(1/2*(-x^3-10*x^2-3)/x^2)+3*x^3+30*x^2+9),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {\left (-30+5 x^3\right ) \log (3)+\left (\left (45+150 x^2+15 x^3\right ) \log (3)+\left (15+50 x^2+5 x^3\right ) \log (3) \log \left (\frac {-3-10 x^2-x^3}{2 x^2}\right )\right ) \log \left (3+\log \left (\frac {-3-10 x^2-x^3}{2 x^2}\right )\right )}{9+30 x^2+3 x^3+\left (3+10 x^2+x^3\right ) \log \left (\frac {-3-10 x^2-x^3}{2 x^2}\right )} \, dx=5 \, x \log \left (3\right ) \log \left (\log \left (-\frac {x^{3} + 10 \, x^{2} + 3}{2 \, x^{2}}\right ) + 3\right ) \]
integrate((((5*x^3+50*x^2+15)*log(3)*log(1/2*(-x^3-10*x^2-3)/x^2)+(15*x^3+ 150*x^2+45)*log(3))*log(log(1/2*(-x^3-10*x^2-3)/x^2)+3)+(5*x^3-30)*log(3)) /((x^3+10*x^2+3)*log(1/2*(-x^3-10*x^2-3)/x^2)+3*x^3+30*x^2+9),x, algorithm =\
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (24) = 48\).
Time = 0.60 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.87 \[ \int \frac {\left (-30+5 x^3\right ) \log (3)+\left (\left (45+150 x^2+15 x^3\right ) \log (3)+\left (15+50 x^2+5 x^3\right ) \log (3) \log \left (\frac {-3-10 x^2-x^3}{2 x^2}\right )\right ) \log \left (3+\log \left (\frac {-3-10 x^2-x^3}{2 x^2}\right )\right )}{9+30 x^2+3 x^3+\left (3+10 x^2+x^3\right ) \log \left (\frac {-3-10 x^2-x^3}{2 x^2}\right )} \, dx=\left (5 x \log {\left (3 \right )} + \frac {5 \log {\left (3 \right )}}{2}\right ) \log {\left (\log {\left (\frac {- \frac {x^{3}}{2} - 5 x^{2} - \frac {3}{2}}{x^{2}} \right )} + 3 \right )} - \frac {5 \log {\left (3 \right )} \log {\left (\log {\left (\frac {- \frac {x^{3}}{2} - 5 x^{2} - \frac {3}{2}}{x^{2}} \right )} + 3 \right )}}{2} \]
integrate((((5*x**3+50*x**2+15)*ln(3)*ln(1/2*(-x**3-10*x**2-3)/x**2)+(15*x **3+150*x**2+45)*ln(3))*ln(ln(1/2*(-x**3-10*x**2-3)/x**2)+3)+(5*x**3-30)*l n(3))/((x**3+10*x**2+3)*ln(1/2*(-x**3-10*x**2-3)/x**2)+3*x**3+30*x**2+9),x )
(5*x*log(3) + 5*log(3)/2)*log(log((-x**3/2 - 5*x**2 - 3/2)/x**2) + 3) - 5* log(3)*log(log((-x**3/2 - 5*x**2 - 3/2)/x**2) + 3)/2
Time = 0.33 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \[ \int \frac {\left (-30+5 x^3\right ) \log (3)+\left (\left (45+150 x^2+15 x^3\right ) \log (3)+\left (15+50 x^2+5 x^3\right ) \log (3) \log \left (\frac {-3-10 x^2-x^3}{2 x^2}\right )\right ) \log \left (3+\log \left (\frac {-3-10 x^2-x^3}{2 x^2}\right )\right )}{9+30 x^2+3 x^3+\left (3+10 x^2+x^3\right ) \log \left (\frac {-3-10 x^2-x^3}{2 x^2}\right )} \, dx=5 \, x \log \left (3\right ) \log \left (-\log \left (2\right ) + \log \left (-x^{3} - 10 \, x^{2} - 3\right ) - 2 \, \log \left (x\right ) + 3\right ) \]
integrate((((5*x^3+50*x^2+15)*log(3)*log(1/2*(-x^3-10*x^2-3)/x^2)+(15*x^3+ 150*x^2+45)*log(3))*log(log(1/2*(-x^3-10*x^2-3)/x^2)+3)+(5*x^3-30)*log(3)) /((x^3+10*x^2+3)*log(1/2*(-x^3-10*x^2-3)/x^2)+3*x^3+30*x^2+9),x, algorithm =\
Time = 0.60 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {\left (-30+5 x^3\right ) \log (3)+\left (\left (45+150 x^2+15 x^3\right ) \log (3)+\left (15+50 x^2+5 x^3\right ) \log (3) \log \left (\frac {-3-10 x^2-x^3}{2 x^2}\right )\right ) \log \left (3+\log \left (\frac {-3-10 x^2-x^3}{2 x^2}\right )\right )}{9+30 x^2+3 x^3+\left (3+10 x^2+x^3\right ) \log \left (\frac {-3-10 x^2-x^3}{2 x^2}\right )} \, dx=5 \, x \log \left (3\right ) \log \left (\log \left (-\frac {x^{3} + 10 \, x^{2} + 3}{2 \, x^{2}}\right ) + 3\right ) \]
integrate((((5*x^3+50*x^2+15)*log(3)*log(1/2*(-x^3-10*x^2-3)/x^2)+(15*x^3+ 150*x^2+45)*log(3))*log(log(1/2*(-x^3-10*x^2-3)/x^2)+3)+(5*x^3-30)*log(3)) /((x^3+10*x^2+3)*log(1/2*(-x^3-10*x^2-3)/x^2)+3*x^3+30*x^2+9),x, algorithm =\
Time = 12.94 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {\left (-30+5 x^3\right ) \log (3)+\left (\left (45+150 x^2+15 x^3\right ) \log (3)+\left (15+50 x^2+5 x^3\right ) \log (3) \log \left (\frac {-3-10 x^2-x^3}{2 x^2}\right )\right ) \log \left (3+\log \left (\frac {-3-10 x^2-x^3}{2 x^2}\right )\right )}{9+30 x^2+3 x^3+\left (3+10 x^2+x^3\right ) \log \left (\frac {-3-10 x^2-x^3}{2 x^2}\right )} \, dx=5\,x\,\ln \left (\ln \left (-\frac {\frac {x^3}{2}+5\,x^2+\frac {3}{2}}{x^2}\right )+3\right )\,\ln \left (3\right ) \]