Integrand size = 226, antiderivative size = 32 \[ \int \frac {14 x+36 x^2+28 x^3+6 x^4+\left (16+7 x+17 x^2+7 x^3+x^4\right ) \log \left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )}{64+28 x+68 x^2+28 x^3+4 x^4+\left (-64-28 x-68 x^2-28 x^3-4 x^4\right ) \log \left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )+\left (16+7 x+17 x^2+7 x^3+x^4\right ) \log ^2\left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )} \, dx=\frac {x}{-2+\log \left (\frac {e^4}{x^2 \left (1+x^2\right ) \left (x-(4+x)^2\right )^2}\right )} \]
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\[ \int \frac {14 x+36 x^2+28 x^3+6 x^4+\left (16+7 x+17 x^2+7 x^3+x^4\right ) \log \left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )}{64+28 x+68 x^2+28 x^3+4 x^4+\left (-64-28 x-68 x^2-28 x^3-4 x^4\right ) \log \left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )+\left (16+7 x+17 x^2+7 x^3+x^4\right ) \log ^2\left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )} \, dx=\int \frac {14 x+36 x^2+28 x^3+6 x^4+\left (16+7 x+17 x^2+7 x^3+x^4\right ) \log \left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )}{64+28 x+68 x^2+28 x^3+4 x^4+\left (-64-28 x-68 x^2-28 x^3-4 x^4\right ) \log \left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )+\left (16+7 x+17 x^2+7 x^3+x^4\right ) \log ^2\left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {2 \left (32+21 x+52 x^2+28 x^3+5 x^4\right )+\left (16+7 x+17 x^2+7 x^3+x^4\right ) \log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )}{\left (16+7 x+17 x^2+7 x^3+x^4\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx \\ & = \int \left (\frac {2 \left (16+14 x+35 x^2+21 x^3+4 x^4\right )}{\left (1+x^2\right ) \left (16+7 x+x^2\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2}+\frac {1}{2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )}\right ) \, dx \\ & = 2 \int \frac {16+14 x+35 x^2+21 x^3+4 x^4}{\left (1+x^2\right ) \left (16+7 x+x^2\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx+\int \frac {1}{2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )} \, dx \\ & = 2 \int \left (\frac {4}{\left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2}-\frac {1}{\left (1+x^2\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2}+\frac {-32-7 x}{\left (16+7 x+x^2\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2}\right ) \, dx+\int \frac {1}{2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )} \, dx \\ & = -\left (2 \int \frac {1}{\left (1+x^2\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx\right )+2 \int \frac {-32-7 x}{\left (16+7 x+x^2\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx+8 \int \frac {1}{\left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx+\int \frac {1}{2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )} \, dx \\ & = -\left (2 \int \left (\frac {i}{2 (i-x) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2}+\frac {i}{2 (i+x) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2}\right ) \, dx\right )+2 \int \left (-\frac {32}{\left (16+7 x+x^2\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2}-\frac {7 x}{\left (16+7 x+x^2\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2}\right ) \, dx+8 \int \frac {1}{\left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx+\int \frac {1}{2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )} \, dx \\ & = -\left (i \int \frac {1}{(i-x) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx\right )-i \int \frac {1}{(i+x) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx+8 \int \frac {1}{\left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx-14 \int \frac {x}{\left (16+7 x+x^2\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx-64 \int \frac {1}{\left (16+7 x+x^2\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx+\int \frac {1}{2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )} \, dx \\ & = -\left (i \int \frac {1}{(i-x) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx\right )-i \int \frac {1}{(i+x) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx+8 \int \frac {1}{\left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx-14 \int \left (\frac {1+\frac {7 i}{\sqrt {15}}}{\left (7-i \sqrt {15}+2 x\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2}+\frac {1-\frac {7 i}{\sqrt {15}}}{\left (7+i \sqrt {15}+2 x\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2}\right ) \, dx-64 \int \left (\frac {2 i}{\sqrt {15} \left (-7+i \sqrt {15}-2 x\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2}+\frac {2 i}{\sqrt {15} \left (7+i \sqrt {15}+2 x\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2}\right ) \, dx+\int \frac {1}{2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )} \, dx \\ & = -\left (i \int \frac {1}{(i-x) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx\right )-i \int \frac {1}{(i+x) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx+8 \int \frac {1}{\left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx-\frac {(128 i) \int \frac {1}{\left (-7+i \sqrt {15}-2 x\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx}{\sqrt {15}}-\frac {(128 i) \int \frac {1}{\left (7+i \sqrt {15}+2 x\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx}{\sqrt {15}}-\frac {1}{15} \left (14 \left (15-7 i \sqrt {15}\right )\right ) \int \frac {1}{\left (7+i \sqrt {15}+2 x\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx-\frac {1}{15} \left (14 \left (15+7 i \sqrt {15}\right )\right ) \int \frac {1}{\left (7-i \sqrt {15}+2 x\right ) \left (2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )\right )^2} \, dx+\int \frac {1}{2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )} \, dx \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \frac {14 x+36 x^2+28 x^3+6 x^4+\left (16+7 x+17 x^2+7 x^3+x^4\right ) \log \left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )}{64+28 x+68 x^2+28 x^3+4 x^4+\left (-64-28 x-68 x^2-28 x^3-4 x^4\right ) \log \left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )+\left (16+7 x+17 x^2+7 x^3+x^4\right ) \log ^2\left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )} \, dx=\frac {x}{2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )} \]
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Time = 3.83 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.38
method | result | size |
parallelrisch | \(\frac {x}{\ln \left (\frac {{\mathrm e}^{4}}{x^{2} \left (x^{6}+14 x^{5}+82 x^{4}+238 x^{3}+337 x^{2}+224 x +256\right )}\right )-2}\) | \(44\) |
norman | \(\frac {x}{\ln \left (\frac {{\mathrm e}^{4}}{x^{8}+14 x^{7}+82 x^{6}+238 x^{5}+337 x^{4}+224 x^{3}+256 x^{2}}\right )-2}\) | \(47\) |
risch | \(\frac {x}{\ln \left (\frac {{\mathrm e}^{4}}{x^{8}+14 x^{7}+82 x^{6}+238 x^{5}+337 x^{4}+224 x^{3}+256 x^{2}}\right )-2}\) | \(47\) |
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Time = 0.27 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.44 \[ \int \frac {14 x+36 x^2+28 x^3+6 x^4+\left (16+7 x+17 x^2+7 x^3+x^4\right ) \log \left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )}{64+28 x+68 x^2+28 x^3+4 x^4+\left (-64-28 x-68 x^2-28 x^3-4 x^4\right ) \log \left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )+\left (16+7 x+17 x^2+7 x^3+x^4\right ) \log ^2\left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )} \, dx=\frac {x}{\log \left (\frac {e^{4}}{x^{8} + 14 \, x^{7} + 82 \, x^{6} + 238 \, x^{5} + 337 \, x^{4} + 224 \, x^{3} + 256 \, x^{2}}\right ) - 2} \]
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Time = 0.14 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.28 \[ \int \frac {14 x+36 x^2+28 x^3+6 x^4+\left (16+7 x+17 x^2+7 x^3+x^4\right ) \log \left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )}{64+28 x+68 x^2+28 x^3+4 x^4+\left (-64-28 x-68 x^2-28 x^3-4 x^4\right ) \log \left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )+\left (16+7 x+17 x^2+7 x^3+x^4\right ) \log ^2\left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )} \, dx=\frac {x}{\log {\left (\frac {e^{4}}{x^{8} + 14 x^{7} + 82 x^{6} + 238 x^{5} + 337 x^{4} + 224 x^{3} + 256 x^{2}} \right )} - 2} \]
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Time = 0.34 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \frac {14 x+36 x^2+28 x^3+6 x^4+\left (16+7 x+17 x^2+7 x^3+x^4\right ) \log \left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )}{64+28 x+68 x^2+28 x^3+4 x^4+\left (-64-28 x-68 x^2-28 x^3-4 x^4\right ) \log \left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )+\left (16+7 x+17 x^2+7 x^3+x^4\right ) \log ^2\left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )} \, dx=-\frac {x}{2 \, \log \left (x^{2} + 7 \, x + 16\right ) + \log \left (x^{2} + 1\right ) + 2 \, \log \left (x\right ) - 2} \]
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Time = 0.41 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.31 \[ \int \frac {14 x+36 x^2+28 x^3+6 x^4+\left (16+7 x+17 x^2+7 x^3+x^4\right ) \log \left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )}{64+28 x+68 x^2+28 x^3+4 x^4+\left (-64-28 x-68 x^2-28 x^3-4 x^4\right ) \log \left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )+\left (16+7 x+17 x^2+7 x^3+x^4\right ) \log ^2\left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )} \, dx=-\frac {x}{\log \left (x^{8} + 14 \, x^{7} + 82 \, x^{6} + 238 \, x^{5} + 337 \, x^{4} + 224 \, x^{3} + 256 \, x^{2}\right ) - 2} \]
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Time = 16.21 (sec) , antiderivative size = 164, normalized size of antiderivative = 5.12 \[ \int \frac {14 x+36 x^2+28 x^3+6 x^4+\left (16+7 x+17 x^2+7 x^3+x^4\right ) \log \left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )}{64+28 x+68 x^2+28 x^3+4 x^4+\left (-64-28 x-68 x^2-28 x^3-4 x^4\right ) \log \left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )+\left (16+7 x+17 x^2+7 x^3+x^4\right ) \log ^2\left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )} \, dx=\frac {7\,\ln \left (x^8+14\,x^7+82\,x^6+238\,x^5+337\,x^4+224\,x^3+256\,x^2\right )}{32\,\left (\ln \left (x^8+14\,x^7+82\,x^6+238\,x^5+337\,x^4+224\,x^3+256\,x^2\right )-2\right )}-\frac {7}{16\,\left (\ln \left (x^8+14\,x^7+82\,x^6+238\,x^5+337\,x^4+224\,x^3+256\,x^2\right )-2\right )}-\frac {x}{\ln \left (x^8+14\,x^7+82\,x^6+238\,x^5+337\,x^4+224\,x^3+256\,x^2\right )-2} \]
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