Integrand size = 96, antiderivative size = 26 \[ \int \frac {-4 x^5 \log (x)+\left (-3 x-x^5+\left (-6 x-2 x^5\right ) \log (x)\right ) \log \left (\frac {e^5}{3+x^4}\right )+\left (24+24 x+8 x^4+8 x^5\right ) \log ^2\left (\frac {e^5}{3+x^4}\right )}{\left (12+4 x^4\right ) \log ^2\left (\frac {e^5}{3+x^4}\right )} \, dx=x \left (2+x-\frac {x \log (x)}{4 \log \left (\frac {e^5}{3+x^4}\right )}\right ) \]
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\[ \int \frac {-4 x^5 \log (x)+\left (-3 x-x^5+\left (-6 x-2 x^5\right ) \log (x)\right ) \log \left (\frac {e^5}{3+x^4}\right )+\left (24+24 x+8 x^4+8 x^5\right ) \log ^2\left (\frac {e^5}{3+x^4}\right )}{\left (12+4 x^4\right ) \log ^2\left (\frac {e^5}{3+x^4}\right )} \, dx=\int \frac {-4 x^5 \log (x)+\left (-3 x-x^5+\left (-6 x-2 x^5\right ) \log (x)\right ) \log \left (\frac {e^5}{3+x^4}\right )+\left (24+24 x+8 x^4+8 x^5\right ) \log ^2\left (\frac {e^5}{3+x^4}\right )}{\left (12+4 x^4\right ) \log ^2\left (\frac {e^5}{3+x^4}\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (2 (1+x)-\frac {x^5 \log (x)}{\left (3+x^4\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2}-\frac {x (1+2 \log (x))}{4 \left (5+\log \left (\frac {1}{3+x^4}\right )\right )}\right ) \, dx \\ & = (1+x)^2-\frac {1}{4} \int \frac {x (1+2 \log (x))}{5+\log \left (\frac {1}{3+x^4}\right )} \, dx-\int \frac {x^5 \log (x)}{\left (3+x^4\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx \\ & = (1+x)^2-\frac {1}{4} \int \left (\frac {x}{5+\log \left (\frac {1}{3+x^4}\right )}+\frac {2 x \log (x)}{5+\log \left (\frac {1}{3+x^4}\right )}\right ) \, dx-\int \left (\frac {x \log (x)}{\left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2}-\frac {3 x \log (x)}{\left (3+x^4\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2}\right ) \, dx \\ & = (1+x)^2-\frac {1}{4} \int \frac {x}{5+\log \left (\frac {1}{3+x^4}\right )} \, dx-\frac {1}{2} \int \frac {x \log (x)}{5+\log \left (\frac {1}{3+x^4}\right )} \, dx+3 \int \frac {x \log (x)}{\left (3+x^4\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx-\int \frac {x \log (x)}{\left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx \\ & = (1+x)^2-\frac {1}{4} \int \frac {x}{5+\log \left (\frac {1}{3+x^4}\right )} \, dx-\frac {1}{2} \int \frac {x \log (x)}{5+\log \left (\frac {1}{3+x^4}\right )} \, dx+3 \int \left (-\frac {i x \log (x)}{2 \sqrt {3} \left (-i \sqrt {3}+x^2\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2}+\frac {i x \log (x)}{2 \sqrt {3} \left (i \sqrt {3}+x^2\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2}\right ) \, dx-\int \frac {x \log (x)}{\left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx \\ & = (1+x)^2-\frac {1}{4} \int \frac {x}{5+\log \left (\frac {1}{3+x^4}\right )} \, dx-\frac {1}{2} \int \frac {x \log (x)}{5+\log \left (\frac {1}{3+x^4}\right )} \, dx-\frac {1}{2} \left (i \sqrt {3}\right ) \int \frac {x \log (x)}{\left (-i \sqrt {3}+x^2\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx+\frac {1}{2} \left (i \sqrt {3}\right ) \int \frac {x \log (x)}{\left (i \sqrt {3}+x^2\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx-\int \frac {x \log (x)}{\left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx \\ & = (1+x)^2-\frac {1}{4} \int \frac {x}{5+\log \left (\frac {1}{3+x^4}\right )} \, dx-\frac {1}{2} \int \frac {x \log (x)}{5+\log \left (\frac {1}{3+x^4}\right )} \, dx-\frac {1}{2} \left (i \sqrt {3}\right ) \int \left (-\frac {\log (x)}{2 \left (\sqrt [4]{-3}-x\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2}+\frac {\log (x)}{2 \left (\sqrt [4]{-3}+x\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2}\right ) \, dx+\frac {1}{2} \left (i \sqrt {3}\right ) \int \left (-\frac {\log (x)}{2 \left (-(-1)^{3/4} \sqrt [4]{3}-x\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2}+\frac {\log (x)}{2 \left (-(-1)^{3/4} \sqrt [4]{3}+x\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2}\right ) \, dx-\int \frac {x \log (x)}{\left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx \\ & = (1+x)^2-\frac {1}{4} \int \frac {x}{5+\log \left (\frac {1}{3+x^4}\right )} \, dx-\frac {1}{2} \int \frac {x \log (x)}{5+\log \left (\frac {1}{3+x^4}\right )} \, dx+\frac {1}{4} \left (i \sqrt {3}\right ) \int \frac {\log (x)}{\left (\sqrt [4]{-3}-x\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx-\frac {1}{4} \left (i \sqrt {3}\right ) \int \frac {\log (x)}{\left (-(-1)^{3/4} \sqrt [4]{3}-x\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx-\frac {1}{4} \left (i \sqrt {3}\right ) \int \frac {\log (x)}{\left (\sqrt [4]{-3}+x\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx+\frac {1}{4} \left (i \sqrt {3}\right ) \int \frac {\log (x)}{\left (-(-1)^{3/4} \sqrt [4]{3}+x\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx-\int \frac {x \log (x)}{\left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx \\ \end{align*}
Time = 1.50 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {-4 x^5 \log (x)+\left (-3 x-x^5+\left (-6 x-2 x^5\right ) \log (x)\right ) \log \left (\frac {e^5}{3+x^4}\right )+\left (24+24 x+8 x^4+8 x^5\right ) \log ^2\left (\frac {e^5}{3+x^4}\right )}{\left (12+4 x^4\right ) \log ^2\left (\frac {e^5}{3+x^4}\right )} \, dx=\frac {1}{4} x \left (8+4 x-\frac {x \log (x)}{5+\log \left (\frac {1}{3+x^4}\right )}\right ) \]
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Result contains complex when optimal does not.
Time = 3.12 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15
method | result | size |
risch | \(x^{2}+2 x -\frac {i x^{2} \ln \left (x \right )}{2 \left (-2 i \ln \left (x^{4}+3\right )+10 i\right )}\) | \(30\) |
parallelrisch | \(-\frac {6 x^{2} \ln \left (x \right )-24 x^{2} \ln \left (\frac {{\mathrm e}^{5}}{x^{4}+3}\right )-48 \ln \left (\frac {{\mathrm e}^{5}}{x^{4}+3}\right ) x}{24 \ln \left (\frac {{\mathrm e}^{5}}{x^{4}+3}\right )}\) | \(54\) |
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Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62 \[ \int \frac {-4 x^5 \log (x)+\left (-3 x-x^5+\left (-6 x-2 x^5\right ) \log (x)\right ) \log \left (\frac {e^5}{3+x^4}\right )+\left (24+24 x+8 x^4+8 x^5\right ) \log ^2\left (\frac {e^5}{3+x^4}\right )}{\left (12+4 x^4\right ) \log ^2\left (\frac {e^5}{3+x^4}\right )} \, dx=-\frac {x^{2} \log \left (x\right ) - 4 \, {\left (x^{2} + 2 \, x\right )} \log \left (\frac {e^{5}}{x^{4} + 3}\right )}{4 \, \log \left (\frac {e^{5}}{x^{4} + 3}\right )} \]
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Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {-4 x^5 \log (x)+\left (-3 x-x^5+\left (-6 x-2 x^5\right ) \log (x)\right ) \log \left (\frac {e^5}{3+x^4}\right )+\left (24+24 x+8 x^4+8 x^5\right ) \log ^2\left (\frac {e^5}{3+x^4}\right )}{\left (12+4 x^4\right ) \log ^2\left (\frac {e^5}{3+x^4}\right )} \, dx=- \frac {x^{2} \log {\left (x \right )}}{4 \log {\left (\frac {e^{5}}{x^{4} + 3} \right )}} + x^{2} + 2 x \]
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Time = 0.32 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62 \[ \int \frac {-4 x^5 \log (x)+\left (-3 x-x^5+\left (-6 x-2 x^5\right ) \log (x)\right ) \log \left (\frac {e^5}{3+x^4}\right )+\left (24+24 x+8 x^4+8 x^5\right ) \log ^2\left (\frac {e^5}{3+x^4}\right )}{\left (12+4 x^4\right ) \log ^2\left (\frac {e^5}{3+x^4}\right )} \, dx=\frac {x^{2} \log \left (x\right ) - 20 \, x^{2} + 4 \, {\left (x^{2} + 2 \, x\right )} \log \left (x^{4} + 3\right ) - 40 \, x}{4 \, {\left (\log \left (x^{4} + 3\right ) - 5\right )}} \]
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Time = 0.43 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {-4 x^5 \log (x)+\left (-3 x-x^5+\left (-6 x-2 x^5\right ) \log (x)\right ) \log \left (\frac {e^5}{3+x^4}\right )+\left (24+24 x+8 x^4+8 x^5\right ) \log ^2\left (\frac {e^5}{3+x^4}\right )}{\left (12+4 x^4\right ) \log ^2\left (\frac {e^5}{3+x^4}\right )} \, dx=x^{2} + \frac {x^{2} \log \left (x\right )}{4 \, {\left (\log \left (x^{4} + 3\right ) - 5\right )}} + 2 \, x \]
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Time = 10.05 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.96 \[ \int \frac {-4 x^5 \log (x)+\left (-3 x-x^5+\left (-6 x-2 x^5\right ) \log (x)\right ) \log \left (\frac {e^5}{3+x^4}\right )+\left (24+24 x+8 x^4+8 x^5\right ) \log ^2\left (\frac {e^5}{3+x^4}\right )}{\left (12+4 x^4\right ) \log ^2\left (\frac {e^5}{3+x^4}\right )} \, dx=2\,x+\frac {3}{16\,x^2}+\frac {17\,x^2}{16}-\frac {\frac {x^2\,\ln \left (x\right )}{4}+\frac {\ln \left (\frac {{\mathrm {e}}^5}{x^4+3}\right )\,\left (x^4+3\right )\,\left (2\,\ln \left (x\right )+1\right )}{16\,x^2}}{\ln \left (\frac {{\mathrm {e}}^5}{x^4+3}\right )}+\frac {\ln \left (x\right )\,\left (\frac {x^4}{8}+\frac {3}{8}\right )}{x^2} \]
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