Integrand size = 182, antiderivative size = 28 \[ \int \frac {e \left (-30 x^4-5 x^5\right )+\left (x^3+30 x^5+5 x^6\right ) \log ^2(x)+\left (e \left (60 x^2+10 x^3\right )+\left (-x-60 x^3-10 x^4\right ) \log ^2(x)\right ) \log (2 x)+\left (e (-30-5 x)+\left (30 x+5 x^2\right ) \log ^2(x)\right ) \log ^2(2 x)+\left (6+x-12 x^2-2 x^3\right ) \log ^2(x) \log (6+x)}{\left (30 x^5+5 x^6\right ) \log ^2(x)+\left (-60 x^3-10 x^4\right ) \log ^2(x) \log (2 x)+\left (30 x+5 x^2\right ) \log ^2(x) \log ^2(2 x)} \, dx=x+\frac {e}{\log (x)}+\frac {\log (6+x)}{5 \left (x^2-\log (2 x)\right )} \]
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\[ \int \frac {e \left (-30 x^4-5 x^5\right )+\left (x^3+30 x^5+5 x^6\right ) \log ^2(x)+\left (e \left (60 x^2+10 x^3\right )+\left (-x-60 x^3-10 x^4\right ) \log ^2(x)\right ) \log (2 x)+\left (e (-30-5 x)+\left (30 x+5 x^2\right ) \log ^2(x)\right ) \log ^2(2 x)+\left (6+x-12 x^2-2 x^3\right ) \log ^2(x) \log (6+x)}{\left (30 x^5+5 x^6\right ) \log ^2(x)+\left (-60 x^3-10 x^4\right ) \log ^2(x) \log (2 x)+\left (30 x+5 x^2\right ) \log ^2(x) \log ^2(2 x)} \, dx=\int \frac {e \left (-30 x^4-5 x^5\right )+\left (x^3+30 x^5+5 x^6\right ) \log ^2(x)+\left (e \left (60 x^2+10 x^3\right )+\left (-x-60 x^3-10 x^4\right ) \log ^2(x)\right ) \log (2 x)+\left (e (-30-5 x)+\left (30 x+5 x^2\right ) \log ^2(x)\right ) \log ^2(2 x)+\left (6+x-12 x^2-2 x^3\right ) \log ^2(x) \log (6+x)}{\left (30 x^5+5 x^6\right ) \log ^2(x)+\left (-60 x^3-10 x^4\right ) \log ^2(x) \log (2 x)+\left (30 x+5 x^2\right ) \log ^2(x) \log ^2(2 x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {-\frac {5 e}{\log ^2(x)}+\frac {x^3+30 x^5+5 x^6-\left (x+60 x^3+10 x^4\right ) \log (2 x)+5 x (6+x) \log ^2(2 x)+\left (6+x-12 x^2-2 x^3\right ) \log (6+x)}{(6+x) \left (x^2-\log (2 x)\right )^2}}{5 x} \, dx \\ & = \frac {1}{5} \int \frac {-\frac {5 e}{\log ^2(x)}+\frac {x^3+30 x^5+5 x^6-\left (x+60 x^3+10 x^4\right ) \log (2 x)+5 x (6+x) \log ^2(2 x)+\left (6+x-12 x^2-2 x^3\right ) \log (6+x)}{(6+x) \left (x^2-\log (2 x)\right )^2}}{x} \, dx \\ & = \frac {1}{5} \int \left (-\frac {5 e}{x \log ^2(x)}+\frac {x^2}{(6+x) \left (x^2-\log (2 x)\right )^2}+\frac {30 x^4}{(6+x) \left (x^2-\log (2 x)\right )^2}+\frac {5 x^5}{(6+x) \left (x^2-\log (2 x)\right )^2}-\frac {\left (1+60 x^2+10 x^3\right ) \log (2 x)}{(6+x) \left (x^2-\log (2 x)\right )^2}+\frac {5 \log ^2(2 x)}{\left (x^2-\log (2 x)\right )^2}-\frac {\left (-1+2 x^2\right ) \log (6+x)}{x \left (x^2-\log (2 x)\right )^2}\right ) \, dx \\ & = \frac {1}{5} \int \frac {x^2}{(6+x) \left (x^2-\log (2 x)\right )^2} \, dx-\frac {1}{5} \int \frac {\left (1+60 x^2+10 x^3\right ) \log (2 x)}{(6+x) \left (x^2-\log (2 x)\right )^2} \, dx-\frac {1}{5} \int \frac {\left (-1+2 x^2\right ) \log (6+x)}{x \left (x^2-\log (2 x)\right )^2} \, dx+6 \int \frac {x^4}{(6+x) \left (x^2-\log (2 x)\right )^2} \, dx-e \int \frac {1}{x \log ^2(x)} \, dx+\int \frac {x^5}{(6+x) \left (x^2-\log (2 x)\right )^2} \, dx+\int \frac {\log ^2(2 x)}{\left (x^2-\log (2 x)\right )^2} \, dx \\ & = \frac {1}{5} \int \left (-\frac {6}{\left (x^2-\log (2 x)\right )^2}+\frac {x}{\left (x^2-\log (2 x)\right )^2}+\frac {36}{(6+x) \left (x^2-\log (2 x)\right )^2}\right ) \, dx-\frac {1}{5} \int \left (\frac {x^2 \left (1+60 x^2+10 x^3\right )}{(6+x) \left (x^2-\log (2 x)\right )^2}+\frac {-1-60 x^2-10 x^3}{(6+x) \left (x^2-\log (2 x)\right )}\right ) \, dx-\frac {1}{5} \int \left (-\frac {\log (6+x)}{x \left (x^2-\log (2 x)\right )^2}+\frac {2 x \log (6+x)}{\left (x^2-\log (2 x)\right )^2}\right ) \, dx+6 \int \left (-\frac {216}{\left (x^2-\log (2 x)\right )^2}+\frac {36 x}{\left (x^2-\log (2 x)\right )^2}-\frac {6 x^2}{\left (x^2-\log (2 x)\right )^2}+\frac {x^3}{\left (x^2-\log (2 x)\right )^2}+\frac {1296}{(6+x) \left (x^2-\log (2 x)\right )^2}\right ) \, dx-e \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (x)\right )+\int \left (\frac {1296}{\left (x^2-\log (2 x)\right )^2}-\frac {216 x}{\left (x^2-\log (2 x)\right )^2}+\frac {36 x^2}{\left (x^2-\log (2 x)\right )^2}-\frac {6 x^3}{\left (x^2-\log (2 x)\right )^2}+\frac {x^4}{\left (x^2-\log (2 x)\right )^2}-\frac {7776}{(6+x) \left (x^2-\log (2 x)\right )^2}\right ) \, dx+\int \left (1+\frac {x^4}{\left (x^2-\log (2 x)\right )^2}-\frac {2 x^2}{x^2-\log (2 x)}\right ) \, dx \\ & = x+\frac {e}{\log (x)}+\frac {1}{5} \int \frac {x}{\left (x^2-\log (2 x)\right )^2} \, dx-\frac {1}{5} \int \frac {x^2 \left (1+60 x^2+10 x^3\right )}{(6+x) \left (x^2-\log (2 x)\right )^2} \, dx-\frac {1}{5} \int \frac {-1-60 x^2-10 x^3}{(6+x) \left (x^2-\log (2 x)\right )} \, dx+\frac {1}{5} \int \frac {\log (6+x)}{x \left (x^2-\log (2 x)\right )^2} \, dx-\frac {2}{5} \int \frac {x \log (6+x)}{\left (x^2-\log (2 x)\right )^2} \, dx-\frac {6}{5} \int \frac {1}{\left (x^2-\log (2 x)\right )^2} \, dx-2 \int \frac {x^2}{x^2-\log (2 x)} \, dx+\frac {36}{5} \int \frac {1}{(6+x) \left (x^2-\log (2 x)\right )^2} \, dx+2 \int \frac {x^4}{\left (x^2-\log (2 x)\right )^2} \, dx \\ & = x+\frac {e}{\log (x)}-\frac {1}{5} \int \left (-\frac {6}{\left (x^2-\log (2 x)\right )^2}+\frac {x}{\left (x^2-\log (2 x)\right )^2}+\frac {10 x^4}{\left (x^2-\log (2 x)\right )^2}+\frac {36}{(6+x) \left (x^2-\log (2 x)\right )^2}\right ) \, dx-\frac {1}{5} \int \left (-\frac {10 x^2}{x^2-\log (2 x)}-\frac {1}{(6+x) \left (x^2-\log (2 x)\right )}\right ) \, dx+\frac {1}{5} \int \frac {x}{\left (x^2-\log (2 x)\right )^2} \, dx+\frac {1}{5} \int \frac {\log (6+x)}{x \left (x^2-\log (2 x)\right )^2} \, dx-\frac {2}{5} \int \frac {x \log (6+x)}{\left (x^2-\log (2 x)\right )^2} \, dx-\frac {6}{5} \int \frac {1}{\left (x^2-\log (2 x)\right )^2} \, dx-2 \int \frac {x^2}{x^2-\log (2 x)} \, dx+\frac {36}{5} \int \frac {1}{(6+x) \left (x^2-\log (2 x)\right )^2} \, dx+2 \int \frac {x^4}{\left (x^2-\log (2 x)\right )^2} \, dx \\ & = x+\frac {e}{\log (x)}+\frac {1}{5} \int \frac {1}{(6+x) \left (x^2-\log (2 x)\right )} \, dx+\frac {1}{5} \int \frac {\log (6+x)}{x \left (x^2-\log (2 x)\right )^2} \, dx-\frac {2}{5} \int \frac {x \log (6+x)}{\left (x^2-\log (2 x)\right )^2} \, dx-2 \int \frac {x^4}{\left (x^2-\log (2 x)\right )^2} \, dx+2 \int \frac {x^4}{\left (x^2-\log (2 x)\right )^2} \, dx \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18 \[ \int \frac {e \left (-30 x^4-5 x^5\right )+\left (x^3+30 x^5+5 x^6\right ) \log ^2(x)+\left (e \left (60 x^2+10 x^3\right )+\left (-x-60 x^3-10 x^4\right ) \log ^2(x)\right ) \log (2 x)+\left (e (-30-5 x)+\left (30 x+5 x^2\right ) \log ^2(x)\right ) \log ^2(2 x)+\left (6+x-12 x^2-2 x^3\right ) \log ^2(x) \log (6+x)}{\left (30 x^5+5 x^6\right ) \log ^2(x)+\left (-60 x^3-10 x^4\right ) \log ^2(x) \log (2 x)+\left (30 x+5 x^2\right ) \log ^2(x) \log ^2(2 x)} \, dx=\frac {1}{5} \left (5 x+\frac {5 e}{\log (x)}-\frac {\log (6+x)}{-x^2+\log (2 x)}\right ) \]
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Result contains complex when optimal does not.
Time = 64.44 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43
method | result | size |
risch | \(-\frac {2 i \ln \left (6+x \right )}{5 \left (-2 i x^{2}+2 i \ln \left (2\right )+2 i \ln \left (x \right )\right )}+\frac {x \ln \left (x \right )+{\mathrm e}}{\ln \left (x \right )}\) | \(40\) |
parallelrisch | \(\frac {5 x^{2} {\mathrm e}-5 x \ln \left (x \right ) \ln \left (2 x \right )+60 \ln \left (x \right ) \ln \left (2 x \right )+5 x^{3} \ln \left (x \right )-60 x^{2} \ln \left (x \right )+\ln \left (x \right ) \ln \left (6+x \right )-5 \,{\mathrm e} \ln \left (2 x \right )}{5 \left (x^{2}-\ln \left (2 x \right )\right ) \ln \left (x \right )}\) | \(73\) |
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Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (27) = 54\).
Time = 0.25 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.39 \[ \int \frac {e \left (-30 x^4-5 x^5\right )+\left (x^3+30 x^5+5 x^6\right ) \log ^2(x)+\left (e \left (60 x^2+10 x^3\right )+\left (-x-60 x^3-10 x^4\right ) \log ^2(x)\right ) \log (2 x)+\left (e (-30-5 x)+\left (30 x+5 x^2\right ) \log ^2(x)\right ) \log ^2(2 x)+\left (6+x-12 x^2-2 x^3\right ) \log ^2(x) \log (6+x)}{\left (30 x^5+5 x^6\right ) \log ^2(x)+\left (-60 x^3-10 x^4\right ) \log ^2(x) \log (2 x)+\left (30 x+5 x^2\right ) \log ^2(x) \log ^2(2 x)} \, dx=\frac {5 \, x^{2} e - 5 \, x \log \left (x\right )^{2} - 5 \, e \log \left (2\right ) + 5 \, {\left (x^{3} - x \log \left (2\right ) - e\right )} \log \left (x\right ) + \log \left (x + 6\right ) \log \left (x\right )}{5 \, {\left ({\left (x^{2} - \log \left (2\right )\right )} \log \left (x\right ) - \log \left (x\right )^{2}\right )}} \]
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Exception generated. \[ \int \frac {e \left (-30 x^4-5 x^5\right )+\left (x^3+30 x^5+5 x^6\right ) \log ^2(x)+\left (e \left (60 x^2+10 x^3\right )+\left (-x-60 x^3-10 x^4\right ) \log ^2(x)\right ) \log (2 x)+\left (e (-30-5 x)+\left (30 x+5 x^2\right ) \log ^2(x)\right ) \log ^2(2 x)+\left (6+x-12 x^2-2 x^3\right ) \log ^2(x) \log (6+x)}{\left (30 x^5+5 x^6\right ) \log ^2(x)+\left (-60 x^3-10 x^4\right ) \log ^2(x) \log (2 x)+\left (30 x+5 x^2\right ) \log ^2(x) \log ^2(2 x)} \, dx=\text {Exception raised: TypeError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (27) = 54\).
Time = 0.34 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.39 \[ \int \frac {e \left (-30 x^4-5 x^5\right )+\left (x^3+30 x^5+5 x^6\right ) \log ^2(x)+\left (e \left (60 x^2+10 x^3\right )+\left (-x-60 x^3-10 x^4\right ) \log ^2(x)\right ) \log (2 x)+\left (e (-30-5 x)+\left (30 x+5 x^2\right ) \log ^2(x)\right ) \log ^2(2 x)+\left (6+x-12 x^2-2 x^3\right ) \log ^2(x) \log (6+x)}{\left (30 x^5+5 x^6\right ) \log ^2(x)+\left (-60 x^3-10 x^4\right ) \log ^2(x) \log (2 x)+\left (30 x+5 x^2\right ) \log ^2(x) \log ^2(2 x)} \, dx=\frac {5 \, x^{2} e - 5 \, x \log \left (x\right )^{2} - 5 \, e \log \left (2\right ) + 5 \, {\left (x^{3} - x \log \left (2\right ) - e\right )} \log \left (x\right ) + \log \left (x + 6\right ) \log \left (x\right )}{5 \, {\left ({\left (x^{2} - \log \left (2\right )\right )} \log \left (x\right ) - \log \left (x\right )^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (27) = 54\).
Time = 0.34 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.54 \[ \int \frac {e \left (-30 x^4-5 x^5\right )+\left (x^3+30 x^5+5 x^6\right ) \log ^2(x)+\left (e \left (60 x^2+10 x^3\right )+\left (-x-60 x^3-10 x^4\right ) \log ^2(x)\right ) \log (2 x)+\left (e (-30-5 x)+\left (30 x+5 x^2\right ) \log ^2(x)\right ) \log ^2(2 x)+\left (6+x-12 x^2-2 x^3\right ) \log ^2(x) \log (6+x)}{\left (30 x^5+5 x^6\right ) \log ^2(x)+\left (-60 x^3-10 x^4\right ) \log ^2(x) \log (2 x)+\left (30 x+5 x^2\right ) \log ^2(x) \log ^2(2 x)} \, dx=\frac {5 \, x^{3} \log \left (x\right ) + 5 \, x^{2} e - 5 \, x \log \left (2\right ) \log \left (x\right ) - 5 \, x \log \left (x\right )^{2} - 5 \, e \log \left (2\right ) - 5 \, e \log \left (x\right ) + \log \left (x + 6\right ) \log \left (x\right )}{5 \, {\left (x^{2} \log \left (x\right ) - \log \left (2\right ) \log \left (x\right ) - \log \left (x\right )^{2}\right )}} \]
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Time = 12.23 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {e \left (-30 x^4-5 x^5\right )+\left (x^3+30 x^5+5 x^6\right ) \log ^2(x)+\left (e \left (60 x^2+10 x^3\right )+\left (-x-60 x^3-10 x^4\right ) \log ^2(x)\right ) \log (2 x)+\left (e (-30-5 x)+\left (30 x+5 x^2\right ) \log ^2(x)\right ) \log ^2(2 x)+\left (6+x-12 x^2-2 x^3\right ) \log ^2(x) \log (6+x)}{\left (30 x^5+5 x^6\right ) \log ^2(x)+\left (-60 x^3-10 x^4\right ) \log ^2(x) \log (2 x)+\left (30 x+5 x^2\right ) \log ^2(x) \log ^2(2 x)} \, dx=x+\frac {\mathrm {e}}{\ln \left (x\right )}-\frac {\ln \left (x+6\right )}{5\,\left (\ln \left (2\,x\right )-x^2\right )} \]
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