Integrand size = 205, antiderivative size = 26 \[ \int \frac {-2 x^2 \log (3)+2 x \log (3) \log (4)+\left (2 x^3+4 x^5+\left (-2 x^2-6 x^4\right ) \log (4)+2 x^3 \log ^2(4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^4 \log (3)-6 x^3 \log (3) \log (4)+2 x^2 \log (3) \log ^2(4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )+\left (-2 \log (3)+\left (2 x+4 x^3-2 x^2 \log (4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^2 \log (3)-2 x \log (3) \log (4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )\right ) \log \left (\frac {x+\log (3) \log \left (\log \left (\frac {3}{x}\right )\right )}{\log (3)}\right )}{x^2 \log \left (\frac {3}{x}\right )+x \log (3) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )} \, dx=\left (x^2-x \log (4)+\log \left (\frac {x}{\log (3)}+\log \left (\log \left (\frac {3}{x}\right )\right )\right )\right )^2 \]
[Out]
Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.015, Rules used = {6820, 12, 6818} \[ \int \frac {-2 x^2 \log (3)+2 x \log (3) \log (4)+\left (2 x^3+4 x^5+\left (-2 x^2-6 x^4\right ) \log (4)+2 x^3 \log ^2(4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^4 \log (3)-6 x^3 \log (3) \log (4)+2 x^2 \log (3) \log ^2(4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )+\left (-2 \log (3)+\left (2 x+4 x^3-2 x^2 \log (4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^2 \log (3)-2 x \log (3) \log (4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )\right ) \log \left (\frac {x+\log (3) \log \left (\log \left (\frac {3}{x}\right )\right )}{\log (3)}\right )}{x^2 \log \left (\frac {3}{x}\right )+x \log (3) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )} \, dx=\left (x (x-\log (4))+\log \left (\frac {x}{\log (3)}+\log \left (\log \left (\frac {3}{x}\right )\right )\right )\right )^2 \]
[In]
[Out]
Rule 12
Rule 6818
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \frac {2 \left (\log (3)-x \log \left (\frac {3}{x}\right ) \left (1+2 x^2-x \log (4)+(-\log (3) \log (4)+x \log (9)) \log \left (\log \left (\frac {3}{x}\right )\right )\right )\right ) \left (-x (x-\log (4))-\log \left (\frac {x}{\log (3)}+\log \left (\log \left (\frac {3}{x}\right )\right )\right )\right )}{x \log \left (\frac {3}{x}\right ) \left (x+\log (3) \log \left (\log \left (\frac {3}{x}\right )\right )\right )} \, dx \\ & = 2 \int \frac {\left (\log (3)-x \log \left (\frac {3}{x}\right ) \left (1+2 x^2-x \log (4)+(-\log (3) \log (4)+x \log (9)) \log \left (\log \left (\frac {3}{x}\right )\right )\right )\right ) \left (-x (x-\log (4))-\log \left (\frac {x}{\log (3)}+\log \left (\log \left (\frac {3}{x}\right )\right )\right )\right )}{x \log \left (\frac {3}{x}\right ) \left (x+\log (3) \log \left (\log \left (\frac {3}{x}\right )\right )\right )} \, dx \\ & = \left (x (x-\log (4))+\log \left (\frac {x}{\log (3)}+\log \left (\log \left (\frac {3}{x}\right )\right )\right )\right )^2 \\ \end{align*}
\[ \int \frac {-2 x^2 \log (3)+2 x \log (3) \log (4)+\left (2 x^3+4 x^5+\left (-2 x^2-6 x^4\right ) \log (4)+2 x^3 \log ^2(4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^4 \log (3)-6 x^3 \log (3) \log (4)+2 x^2 \log (3) \log ^2(4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )+\left (-2 \log (3)+\left (2 x+4 x^3-2 x^2 \log (4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^2 \log (3)-2 x \log (3) \log (4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )\right ) \log \left (\frac {x+\log (3) \log \left (\log \left (\frac {3}{x}\right )\right )}{\log (3)}\right )}{x^2 \log \left (\frac {3}{x}\right )+x \log (3) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )} \, dx=\int \frac {-2 x^2 \log (3)+2 x \log (3) \log (4)+\left (2 x^3+4 x^5+\left (-2 x^2-6 x^4\right ) \log (4)+2 x^3 \log ^2(4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^4 \log (3)-6 x^3 \log (3) \log (4)+2 x^2 \log (3) \log ^2(4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )+\left (-2 \log (3)+\left (2 x+4 x^3-2 x^2 \log (4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^2 \log (3)-2 x \log (3) \log (4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )\right ) \log \left (\frac {x+\log (3) \log \left (\log \left (\frac {3}{x}\right )\right )}{\log (3)}\right )}{x^2 \log \left (\frac {3}{x}\right )+x \log (3) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )} \, dx \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(86\) vs. \(2(26)=52\).
Time = 22.31 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.35
method | result | size |
parallelrisch | \(4 x^{2} \ln \left (2\right )^{2}-4 x^{3} \ln \left (2\right )+x^{4}-4 \ln \left (2\right ) x \ln \left (\frac {\ln \left (3\right ) \ln \left (\ln \left (\frac {3}{x}\right )\right )+x}{\ln \left (3\right )}\right )+2 \ln \left (\frac {\ln \left (3\right ) \ln \left (\ln \left (\frac {3}{x}\right )\right )+x}{\ln \left (3\right )}\right ) x^{2}+\ln \left (\frac {\ln \left (3\right ) \ln \left (\ln \left (\frac {3}{x}\right )\right )+x}{\ln \left (3\right )}\right )^{2}\) | \(87\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (26) = 52\).
Time = 0.31 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.65 \[ \int \frac {-2 x^2 \log (3)+2 x \log (3) \log (4)+\left (2 x^3+4 x^5+\left (-2 x^2-6 x^4\right ) \log (4)+2 x^3 \log ^2(4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^4 \log (3)-6 x^3 \log (3) \log (4)+2 x^2 \log (3) \log ^2(4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )+\left (-2 \log (3)+\left (2 x+4 x^3-2 x^2 \log (4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^2 \log (3)-2 x \log (3) \log (4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )\right ) \log \left (\frac {x+\log (3) \log \left (\log \left (\frac {3}{x}\right )\right )}{\log (3)}\right )}{x^2 \log \left (\frac {3}{x}\right )+x \log (3) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )} \, dx=x^{4} - 4 \, x^{3} \log \left (2\right ) + 4 \, x^{2} \log \left (2\right )^{2} + 2 \, {\left (x^{2} - 2 \, x \log \left (2\right )\right )} \log \left (\frac {\log \left (3\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) + x}{\log \left (3\right )}\right ) + \log \left (\frac {\log \left (3\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) + x}{\log \left (3\right )}\right )^{2} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (24) = 48\).
Time = 0.41 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.62 \[ \int \frac {-2 x^2 \log (3)+2 x \log (3) \log (4)+\left (2 x^3+4 x^5+\left (-2 x^2-6 x^4\right ) \log (4)+2 x^3 \log ^2(4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^4 \log (3)-6 x^3 \log (3) \log (4)+2 x^2 \log (3) \log ^2(4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )+\left (-2 \log (3)+\left (2 x+4 x^3-2 x^2 \log (4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^2 \log (3)-2 x \log (3) \log (4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )\right ) \log \left (\frac {x+\log (3) \log \left (\log \left (\frac {3}{x}\right )\right )}{\log (3)}\right )}{x^2 \log \left (\frac {3}{x}\right )+x \log (3) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )} \, dx=x^{4} - 4 x^{3} \log {\left (2 \right )} + 4 x^{2} \log {\left (2 \right )}^{2} + \left (2 x^{2} - 4 x \log {\left (2 \right )}\right ) \log {\left (\frac {x + \log {\left (3 \right )} \log {\left (\log {\left (\frac {3}{x} \right )} \right )}}{\log {\left (3 \right )}} \right )} + \log {\left (\frac {x + \log {\left (3 \right )} \log {\left (\log {\left (\frac {3}{x} \right )} \right )}}{\log {\left (3 \right )}} \right )}^{2} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (26) = 52\).
Time = 0.34 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.15 \[ \int \frac {-2 x^2 \log (3)+2 x \log (3) \log (4)+\left (2 x^3+4 x^5+\left (-2 x^2-6 x^4\right ) \log (4)+2 x^3 \log ^2(4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^4 \log (3)-6 x^3 \log (3) \log (4)+2 x^2 \log (3) \log ^2(4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )+\left (-2 \log (3)+\left (2 x+4 x^3-2 x^2 \log (4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^2 \log (3)-2 x \log (3) \log (4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )\right ) \log \left (\frac {x+\log (3) \log \left (\log \left (\frac {3}{x}\right )\right )}{\log (3)}\right )}{x^2 \log \left (\frac {3}{x}\right )+x \log (3) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )} \, dx=x^{4} - 4 \, x^{3} \log \left (2\right ) + 2 \, {\left (2 \, \log \left (2\right )^{2} - \log \left (\log \left (3\right )\right )\right )} x^{2} + 4 \, x \log \left (2\right ) \log \left (\log \left (3\right )\right ) + 2 \, {\left (x^{2} - 2 \, x \log \left (2\right ) - \log \left (\log \left (3\right )\right )\right )} \log \left (\log \left (3\right ) \log \left (\log \left (3\right ) - \log \left (x\right )\right ) + x\right ) + \log \left (\log \left (3\right ) \log \left (\log \left (3\right ) - \log \left (x\right )\right ) + x\right )^{2} \]
[In]
[Out]
\[ \int \frac {-2 x^2 \log (3)+2 x \log (3) \log (4)+\left (2 x^3+4 x^5+\left (-2 x^2-6 x^4\right ) \log (4)+2 x^3 \log ^2(4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^4 \log (3)-6 x^3 \log (3) \log (4)+2 x^2 \log (3) \log ^2(4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )+\left (-2 \log (3)+\left (2 x+4 x^3-2 x^2 \log (4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^2 \log (3)-2 x \log (3) \log (4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )\right ) \log \left (\frac {x+\log (3) \log \left (\log \left (\frac {3}{x}\right )\right )}{\log (3)}\right )}{x^2 \log \left (\frac {3}{x}\right )+x \log (3) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )} \, dx=\int { -\frac {2 \, {\left (x^{2} \log \left (3\right ) - 2 \, x \log \left (3\right ) \log \left (2\right ) - 2 \, {\left (x^{4} \log \left (3\right ) - 3 \, x^{3} \log \left (3\right ) \log \left (2\right ) + 2 \, x^{2} \log \left (3\right ) \log \left (2\right )^{2}\right )} \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) - {\left (2 \, {\left (x^{2} \log \left (3\right ) - x \log \left (3\right ) \log \left (2\right )\right )} \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) + {\left (2 \, x^{3} - 2 \, x^{2} \log \left (2\right ) + x\right )} \log \left (\frac {3}{x}\right ) - \log \left (3\right )\right )} \log \left (\frac {\log \left (3\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) + x}{\log \left (3\right )}\right ) - {\left (2 \, x^{5} + 4 \, x^{3} \log \left (2\right )^{2} + x^{3} - 2 \, {\left (3 \, x^{4} + x^{2}\right )} \log \left (2\right )\right )} \log \left (\frac {3}{x}\right )\right )}}{x \log \left (3\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) + x^{2} \log \left (\frac {3}{x}\right )} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {-2 x^2 \log (3)+2 x \log (3) \log (4)+\left (2 x^3+4 x^5+\left (-2 x^2-6 x^4\right ) \log (4)+2 x^3 \log ^2(4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^4 \log (3)-6 x^3 \log (3) \log (4)+2 x^2 \log (3) \log ^2(4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )+\left (-2 \log (3)+\left (2 x+4 x^3-2 x^2 \log (4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^2 \log (3)-2 x \log (3) \log (4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )\right ) \log \left (\frac {x+\log (3) \log \left (\log \left (\frac {3}{x}\right )\right )}{\log (3)}\right )}{x^2 \log \left (\frac {3}{x}\right )+x \log (3) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )} \, dx=\int \frac {\ln \left (\frac {3}{x}\right )\,\left (8\,x^3\,{\ln \left (2\right )}^2-2\,\ln \left (2\right )\,\left (6\,x^4+2\,x^2\right )+2\,x^3+4\,x^5\right )-2\,x^2\,\ln \left (3\right )+\ln \left (\frac {x+\ln \left (\ln \left (\frac {3}{x}\right )\right )\,\ln \left (3\right )}{\ln \left (3\right )}\right )\,\left (\ln \left (\frac {3}{x}\right )\,\left (4\,x^3-4\,\ln \left (2\right )\,x^2+2\,x\right )-2\,\ln \left (3\right )+\ln \left (\ln \left (\frac {3}{x}\right )\right )\,\ln \left (\frac {3}{x}\right )\,\left (4\,x^2\,\ln \left (3\right )-4\,x\,\ln \left (2\right )\,\ln \left (3\right )\right )\right )+\ln \left (\ln \left (\frac {3}{x}\right )\right )\,\ln \left (\frac {3}{x}\right )\,\left (4\,\ln \left (3\right )\,x^4-12\,\ln \left (2\right )\,\ln \left (3\right )\,x^3+8\,{\ln \left (2\right )}^2\,\ln \left (3\right )\,x^2\right )+4\,x\,\ln \left (2\right )\,\ln \left (3\right )}{x^2\,\ln \left (\frac {3}{x}\right )+x\,\ln \left (\ln \left (\frac {3}{x}\right )\right )\,\ln \left (3\right )\,\ln \left (\frac {3}{x}\right )} \,d x \]
[In]
[Out]