Integrand size = 140, antiderivative size = 33 \[ \int \frac {6 x+3 x^2+e^4 \left (6 x+3 x^2\right )+\left (6 x+3 x^2\right ) \log (3)+\left (-170-414 x-241 x^2-53 x^3-4 x^4+e^4 \left (-170-244 x-82 x^2-8 x^3\right )+\left (-170-244 x-82 x^2-8 x^3\right ) \log (3)\right ) \log \left (\frac {17+4 x}{5+x}\right )}{\left (170 x+159 x^2+45 x^3+4 x^4\right ) \log \left (\frac {17+4 x}{5+x}\right )} \, dx=-x+\left (1+e^4+\log (3)\right ) \log \left (\frac {5 \log \left (4-\frac {3}{5+x}\right )}{x (2+x)}\right ) \]
[Out]
Time = 0.71 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.73, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {6, 6820, 6874, 907, 2561, 2339, 29} \[ \int \frac {6 x+3 x^2+e^4 \left (6 x+3 x^2\right )+\left (6 x+3 x^2\right ) \log (3)+\left (-170-414 x-241 x^2-53 x^3-4 x^4+e^4 \left (-170-244 x-82 x^2-8 x^3\right )+\left (-170-244 x-82 x^2-8 x^3\right ) \log (3)\right ) \log \left (\frac {17+4 x}{5+x}\right )}{\left (170 x+159 x^2+45 x^3+4 x^4\right ) \log \left (\frac {17+4 x}{5+x}\right )} \, dx=-x-\frac {1}{2} \left (2+2 e^4+\log (9)\right ) \log (x)-\frac {1}{2} \left (2+2 e^4+\log (9)\right ) \log (x+2)+\left (1+e^4+\log (3)\right ) \log \left (\log \left (\frac {4 x+17}{x+5}\right )\right ) \]
[In]
[Out]
Rule 6
Rule 29
Rule 907
Rule 2339
Rule 2561
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {6 x+3 x^2+\left (6 x+3 x^2\right ) \left (e^4+\log (3)\right )+\left (-170-414 x-241 x^2-53 x^3-4 x^4+e^4 \left (-170-244 x-82 x^2-8 x^3\right )+\left (-170-244 x-82 x^2-8 x^3\right ) \log (3)\right ) \log \left (\frac {17+4 x}{5+x}\right )}{\left (170 x+159 x^2+45 x^3+4 x^4\right ) \log \left (\frac {17+4 x}{5+x}\right )} \, dx \\ & = \int \frac {6 x+3 x^2+3 x (2+x) \left (e^4+\log (3)\right )-\left (85+37 x+4 x^2\right ) \left (2+x^2+2 e^4 (1+x)+\log (9)+x (4+\log (9))\right ) \log \left (\frac {17+4 x}{5+x}\right )}{x \left (170+159 x+45 x^2+4 x^3\right ) \log \left (\frac {17+4 x}{5+x}\right )} \, dx \\ & = \int \left (\frac {-2-2 e^4-x^2-\log (9)-x \left (4+2 e^4+\log (9)\right )}{x (2+x)}+\frac {3 \left (1+e^4+\log (3)\right )}{(5+x) (17+4 x) \log \left (\frac {17+4 x}{5+x}\right )}\right ) \, dx \\ & = \left (3 \left (1+e^4+\log (3)\right )\right ) \int \frac {1}{(5+x) (17+4 x) \log \left (\frac {17+4 x}{5+x}\right )} \, dx+\int \frac {-2-2 e^4-x^2-\log (9)-x \left (4+2 e^4+\log (9)\right )}{x (2+x)} \, dx \\ & = \left (1+e^4+\log (3)\right ) \text {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,\frac {17+4 x}{5+x}\right )+\int \left (-1+\frac {-2-2 e^4-\log (9)}{2 x}+\frac {-2-2 e^4-\log (9)}{2 (2+x)}\right ) \, dx \\ & = -x-\frac {1}{2} \left (2+2 e^4+\log (9)\right ) \log (x)-\frac {1}{2} \left (2+2 e^4+\log (9)\right ) \log (2+x)+\left (1+e^4+\log (3)\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (\frac {17+4 x}{5+x}\right )\right ) \\ & = -x-\frac {1}{2} \left (2+2 e^4+\log (9)\right ) \log (x)-\frac {1}{2} \left (2+2 e^4+\log (9)\right ) \log (2+x)+\left (1+e^4+\log (3)\right ) \log \left (\log \left (\frac {17+4 x}{5+x}\right )\right ) \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.73 \[ \int \frac {6 x+3 x^2+e^4 \left (6 x+3 x^2\right )+\left (6 x+3 x^2\right ) \log (3)+\left (-170-414 x-241 x^2-53 x^3-4 x^4+e^4 \left (-170-244 x-82 x^2-8 x^3\right )+\left (-170-244 x-82 x^2-8 x^3\right ) \log (3)\right ) \log \left (\frac {17+4 x}{5+x}\right )}{\left (170 x+159 x^2+45 x^3+4 x^4\right ) \log \left (\frac {17+4 x}{5+x}\right )} \, dx=-x-\frac {1}{2} \left (2+2 e^4+\log (9)\right ) \log (x)-\frac {1}{2} \left (2+2 e^4+\log (9)\right ) \log (2+x)+\left (1+e^4+\log (3)\right ) \log \left (\log \left (\frac {17+4 x}{5+x}\right )\right ) \]
[In]
[Out]
Time = 0.76 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.52
method | result | size |
parts | \(\frac {\left (3+3 \,{\mathrm e}^{4}+3 \ln \left (3\right )\right ) \ln \left (\ln \left (4-\frac {3}{5+x}\right )\right )}{3}-x -\left (1+{\mathrm e}^{4}+\ln \left (3\right )\right ) \ln \left (x \right )-\left (1+{\mathrm e}^{4}+\ln \left (3\right )\right ) \ln \left (2+x \right )\) | \(50\) |
norman | \(-x +\left (1+{\mathrm e}^{4}+\ln \left (3\right )\right ) \ln \left (\ln \left (\frac {4 x +17}{5+x}\right )\right )+\left (-{\mathrm e}^{4}-\ln \left (3\right )-1\right ) \ln \left (x \right )+\left (-{\mathrm e}^{4}-\ln \left (3\right )-1\right ) \ln \left (2+x \right )\) | \(53\) |
risch | \(-\ln \left (x^{2}+2 x \right ) {\mathrm e}^{4}+\ln \left (\ln \left (\frac {4 x +17}{5+x}\right )\right ) {\mathrm e}^{4}-\ln \left (x^{2}+2 x \right ) \ln \left (3\right )+\ln \left (\ln \left (\frac {4 x +17}{5+x}\right )\right ) \ln \left (3\right )-\ln \left (x^{2}+2 x \right )+\ln \left (\ln \left (\frac {4 x +17}{5+x}\right )\right )-x\) | \(84\) |
parallelrisch | \(-{\mathrm e}^{4} \ln \left (x \right )+\ln \left (\ln \left (\frac {4 x +17}{5+x}\right )\right ) {\mathrm e}^{4}-{\mathrm e}^{4} \ln \left (2+x \right )-\ln \left (3\right ) \ln \left (x \right )+\ln \left (\ln \left (\frac {4 x +17}{5+x}\right )\right ) \ln \left (3\right )-\ln \left (3\right ) \ln \left (2+x \right )+\frac {37}{2}-\ln \left (x \right )+\ln \left (\ln \left (\frac {4 x +17}{5+x}\right )\right )-\ln \left (2+x \right )-x\) | \(89\) |
derivativedivides | \(\ln \left (3\right ) \left (\ln \left (\ln \left (4-\frac {3}{5+x}\right )\right )-\ln \left (1-\frac {3}{5+x}\right )+2 \ln \left (-\frac {3}{5+x}\right )-\ln \left (3-\frac {15}{5+x}\right )\right )+3 \left (\frac {{\mathrm e}^{4}}{3}+\frac {1}{3}\right ) \ln \left (\ln \left (4-\frac {3}{5+x}\right )\right )+\left (-{\mathrm e}^{4}-1\right ) \ln \left (1-\frac {3}{5+x}\right )+\left (2 \,{\mathrm e}^{4}+2\right ) \ln \left (-\frac {3}{5+x}\right )-5-x +\left (-{\mathrm e}^{4}-1\right ) \ln \left (3-\frac {15}{5+x}\right )\) | \(123\) |
default | \(\ln \left (3\right ) \left (\ln \left (\ln \left (4-\frac {3}{5+x}\right )\right )-\ln \left (1-\frac {3}{5+x}\right )+2 \ln \left (-\frac {3}{5+x}\right )-\ln \left (3-\frac {15}{5+x}\right )\right )+3 \left (\frac {{\mathrm e}^{4}}{3}+\frac {1}{3}\right ) \ln \left (\ln \left (4-\frac {3}{5+x}\right )\right )+\left (-{\mathrm e}^{4}-1\right ) \ln \left (1-\frac {3}{5+x}\right )+\left (2 \,{\mathrm e}^{4}+2\right ) \ln \left (-\frac {3}{5+x}\right )-5-x +\left (-{\mathrm e}^{4}-1\right ) \ln \left (3-\frac {15}{5+x}\right )\) | \(123\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.21 \[ \int \frac {6 x+3 x^2+e^4 \left (6 x+3 x^2\right )+\left (6 x+3 x^2\right ) \log (3)+\left (-170-414 x-241 x^2-53 x^3-4 x^4+e^4 \left (-170-244 x-82 x^2-8 x^3\right )+\left (-170-244 x-82 x^2-8 x^3\right ) \log (3)\right ) \log \left (\frac {17+4 x}{5+x}\right )}{\left (170 x+159 x^2+45 x^3+4 x^4\right ) \log \left (\frac {17+4 x}{5+x}\right )} \, dx=-{\left (e^{4} + \log \left (3\right ) + 1\right )} \log \left (x^{2} + 2 \, x\right ) + {\left (e^{4} + \log \left (3\right ) + 1\right )} \log \left (\log \left (\frac {4 \, x + 17}{x + 5}\right )\right ) - x \]
[In]
[Out]
Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {6 x+3 x^2+e^4 \left (6 x+3 x^2\right )+\left (6 x+3 x^2\right ) \log (3)+\left (-170-414 x-241 x^2-53 x^3-4 x^4+e^4 \left (-170-244 x-82 x^2-8 x^3\right )+\left (-170-244 x-82 x^2-8 x^3\right ) \log (3)\right ) \log \left (\frac {17+4 x}{5+x}\right )}{\left (170 x+159 x^2+45 x^3+4 x^4\right ) \log \left (\frac {17+4 x}{5+x}\right )} \, dx=- x - \left (1 + \log {\left (3 \right )} + e^{4}\right ) \log {\left (x^{2} + 2 x \right )} + \left (1 + \log {\left (3 \right )} + e^{4}\right ) \log {\left (\log {\left (\frac {4 x + 17}{x + 5} \right )} \right )} \]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.42 \[ \int \frac {6 x+3 x^2+e^4 \left (6 x+3 x^2\right )+\left (6 x+3 x^2\right ) \log (3)+\left (-170-414 x-241 x^2-53 x^3-4 x^4+e^4 \left (-170-244 x-82 x^2-8 x^3\right )+\left (-170-244 x-82 x^2-8 x^3\right ) \log (3)\right ) \log \left (\frac {17+4 x}{5+x}\right )}{\left (170 x+159 x^2+45 x^3+4 x^4\right ) \log \left (\frac {17+4 x}{5+x}\right )} \, dx=-{\left (e^{4} + \log \left (3\right ) + 1\right )} \log \left (x + 2\right ) - {\left (e^{4} + \log \left (3\right ) + 1\right )} \log \left (x\right ) + {\left (e^{4} + \log \left (3\right ) + 1\right )} \log \left (\log \left (4 \, x + 17\right ) - \log \left (x + 5\right )\right ) - x \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 502 vs. \(2 (32) = 64\).
Time = 0.35 (sec) , antiderivative size = 502, normalized size of antiderivative = 15.21 \[ \int \frac {6 x+3 x^2+e^4 \left (6 x+3 x^2\right )+\left (6 x+3 x^2\right ) \log (3)+\left (-170-414 x-241 x^2-53 x^3-4 x^4+e^4 \left (-170-244 x-82 x^2-8 x^3\right )+\left (-170-244 x-82 x^2-8 x^3\right ) \log (3)\right ) \log \left (\frac {17+4 x}{5+x}\right )}{\left (170 x+159 x^2+45 x^3+4 x^4\right ) \log \left (\frac {17+4 x}{5+x}\right )} \, dx=-\frac {\frac {{\left (4 \, x + 17\right )} e^{4} \log \left (\frac {5 \, {\left (4 \, x + 17\right )}^{2}}{{\left (x + 5\right )}^{2}} - \frac {32 \, {\left (4 \, x + 17\right )}}{x + 5} + 51\right )}{x + 5} - 4 \, e^{4} \log \left (\frac {5 \, {\left (4 \, x + 17\right )}^{2}}{{\left (x + 5\right )}^{2}} - \frac {32 \, {\left (4 \, x + 17\right )}}{x + 5} + 51\right ) + \frac {{\left (4 \, x + 17\right )} \log \left (3\right ) \log \left (\frac {5 \, {\left (4 \, x + 17\right )}^{2}}{{\left (x + 5\right )}^{2}} - \frac {32 \, {\left (4 \, x + 17\right )}}{x + 5} + 51\right )}{x + 5} - 4 \, \log \left (3\right ) \log \left (\frac {5 \, {\left (4 \, x + 17\right )}^{2}}{{\left (x + 5\right )}^{2}} - \frac {32 \, {\left (4 \, x + 17\right )}}{x + 5} + 51\right ) - \frac {2 \, {\left (4 \, x + 17\right )} e^{4} \log \left (\frac {4 \, x + 17}{x + 5} - 4\right )}{x + 5} + 8 \, e^{4} \log \left (\frac {4 \, x + 17}{x + 5} - 4\right ) - \frac {2 \, {\left (4 \, x + 17\right )} \log \left (3\right ) \log \left (\frac {4 \, x + 17}{x + 5} - 4\right )}{x + 5} + 8 \, \log \left (3\right ) \log \left (\frac {4 \, x + 17}{x + 5} - 4\right ) - \frac {{\left (4 \, x + 17\right )} e^{4} \log \left (\log \left (\frac {4 \, x + 17}{x + 5}\right )\right )}{x + 5} + 4 \, e^{4} \log \left (\log \left (\frac {4 \, x + 17}{x + 5}\right )\right ) - \frac {{\left (4 \, x + 17\right )} \log \left (3\right ) \log \left (\log \left (\frac {4 \, x + 17}{x + 5}\right )\right )}{x + 5} + 4 \, \log \left (3\right ) \log \left (\log \left (\frac {4 \, x + 17}{x + 5}\right )\right ) + \frac {{\left (4 \, x + 17\right )} \log \left (\frac {5 \, {\left (4 \, x + 17\right )}^{2}}{{\left (x + 5\right )}^{2}} - \frac {32 \, {\left (4 \, x + 17\right )}}{x + 5} + 51\right )}{x + 5} - \frac {2 \, {\left (4 \, x + 17\right )} \log \left (\frac {4 \, x + 17}{x + 5} - 4\right )}{x + 5} - \frac {{\left (4 \, x + 17\right )} \log \left (\log \left (\frac {4 \, x + 17}{x + 5}\right )\right )}{x + 5} - 4 \, \log \left (\frac {5 \, {\left (4 \, x + 17\right )}^{2}}{{\left (x + 5\right )}^{2}} - \frac {32 \, {\left (4 \, x + 17\right )}}{x + 5} + 51\right ) + 8 \, \log \left (\frac {4 \, x + 17}{x + 5} - 4\right ) + 4 \, \log \left (\log \left (\frac {4 \, x + 17}{x + 5}\right )\right ) - 3}{\frac {4 \, x + 17}{x + 5} - 4} \]
[In]
[Out]
Time = 0.73 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.15 \[ \int \frac {6 x+3 x^2+e^4 \left (6 x+3 x^2\right )+\left (6 x+3 x^2\right ) \log (3)+\left (-170-414 x-241 x^2-53 x^3-4 x^4+e^4 \left (-170-244 x-82 x^2-8 x^3\right )+\left (-170-244 x-82 x^2-8 x^3\right ) \log (3)\right ) \log \left (\frac {17+4 x}{5+x}\right )}{\left (170 x+159 x^2+45 x^3+4 x^4\right ) \log \left (\frac {17+4 x}{5+x}\right )} \, dx=\ln \left (\ln \left (\frac {4\,x+17}{x+5}\right )\right )\,\left ({\mathrm {e}}^4+\ln \left (3\right )+1\right )-x-\ln \left (x\,\left (x+2\right )\right )\,\left ({\mathrm {e}}^4+\ln \left (3\right )+1\right ) \]
[In]
[Out]