Integrand size = 90, antiderivative size = 21 \[ \int \frac {\left (-8 x+12 x^2+72 x^3+32 x^4\right ) \log (x)+\left (64-216 x-148 x^2-44 x^3-16 x^4+\left (224 x+284 x^2+60 x^3+32 x^4\right ) \log (x)\right ) \log \left (8+x+x^2\right )}{\left (8 x+x^2+x^3\right ) \log ^2(x)} \, dx=\frac {4 (2+x) (-1+4 x) \log \left (8+x+x^2\right )}{\log (x)} \]
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\[ \int \frac {\left (-8 x+12 x^2+72 x^3+32 x^4\right ) \log (x)+\left (64-216 x-148 x^2-44 x^3-16 x^4+\left (224 x+284 x^2+60 x^3+32 x^4\right ) \log (x)\right ) \log \left (8+x+x^2\right )}{\left (8 x+x^2+x^3\right ) \log ^2(x)} \, dx=\int \frac {\left (-8 x+12 x^2+72 x^3+32 x^4\right ) \log (x)+\left (64-216 x-148 x^2-44 x^3-16 x^4+\left (224 x+284 x^2+60 x^3+32 x^4\right ) \log (x)\right ) \log \left (8+x+x^2\right )}{\left (8 x+x^2+x^3\right ) \log ^2(x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (-8 x+12 x^2+72 x^3+32 x^4\right ) \log (x)+\left (64-216 x-148 x^2-44 x^3-16 x^4+\left (224 x+284 x^2+60 x^3+32 x^4\right ) \log (x)\right ) \log \left (8+x+x^2\right )}{x \left (8+x+x^2\right ) \log ^2(x)} \, dx \\ & = \int \left (\frac {4 (2+x) (1+2 x) (-1+4 x)}{\left (8+x+x^2\right ) \log (x)}+\frac {4 \left (2-7 x-4 x^2+7 x \log (x)+8 x^2 \log (x)\right ) \log \left (8+x+x^2\right )}{x \log ^2(x)}\right ) \, dx \\ & = 4 \int \frac {(2+x) (1+2 x) (-1+4 x)}{\left (8+x+x^2\right ) \log (x)} \, dx+4 \int \frac {\left (2-7 x-4 x^2+7 x \log (x)+8 x^2 \log (x)\right ) \log \left (8+x+x^2\right )}{x \log ^2(x)} \, dx \\ & = 4 \int \frac {(2+x) (1+2 x) (-1+4 x)}{\left (8+x+x^2\right ) \log (x)} \, dx+4 \int \left (-\frac {7 \log \left (8+x+x^2\right )}{\log ^2(x)}+\frac {2 \log \left (8+x+x^2\right )}{x \log ^2(x)}-\frac {4 x \log \left (8+x+x^2\right )}{\log ^2(x)}+\frac {7 \log \left (8+x+x^2\right )}{\log (x)}+\frac {8 x \log \left (8+x+x^2\right )}{\log (x)}\right ) \, dx \\ & = 4 \int \frac {(2+x) (1+2 x) (-1+4 x)}{\left (8+x+x^2\right ) \log (x)} \, dx+8 \int \frac {\log \left (8+x+x^2\right )}{x \log ^2(x)} \, dx-16 \int \frac {x \log \left (8+x+x^2\right )}{\log ^2(x)} \, dx-28 \int \frac {\log \left (8+x+x^2\right )}{\log ^2(x)} \, dx+28 \int \frac {\log \left (8+x+x^2\right )}{\log (x)} \, dx+32 \int \frac {x \log \left (8+x+x^2\right )}{\log (x)} \, dx \\ & = 28 \log \left (8+x+x^2\right ) \operatorname {LogIntegral}(x)+4 \int \frac {(2+x) (1+2 x) (-1+4 x)}{\left (8+x+x^2\right ) \log (x)} \, dx+8 \int \frac {\log \left (8+x+x^2\right )}{x \log ^2(x)} \, dx-16 \int \frac {x \log \left (8+x+x^2\right )}{\log ^2(x)} \, dx-28 \int \frac {\log \left (8+x+x^2\right )}{\log ^2(x)} \, dx-28 \int \frac {(1+2 x) \operatorname {LogIntegral}(x)}{8+x+x^2} \, dx+32 \int \frac {x \log \left (8+x+x^2\right )}{\log (x)} \, dx \\ & = 28 \log \left (8+x+x^2\right ) \operatorname {LogIntegral}(x)+4 \int \frac {(2+x) (1+2 x) (-1+4 x)}{\left (8+x+x^2\right ) \log (x)} \, dx+8 \int \frac {\log \left (8+x+x^2\right )}{x \log ^2(x)} \, dx-16 \int \frac {x \log \left (8+x+x^2\right )}{\log ^2(x)} \, dx-28 \int \frac {\log \left (8+x+x^2\right )}{\log ^2(x)} \, dx-28 \int \left (\frac {\operatorname {LogIntegral}(x)}{8+x+x^2}+\frac {2 x \operatorname {LogIntegral}(x)}{8+x+x^2}\right ) \, dx+32 \int \frac {x \log \left (8+x+x^2\right )}{\log (x)} \, dx \\ & = 28 \log \left (8+x+x^2\right ) \operatorname {LogIntegral}(x)+4 \int \frac {(2+x) (1+2 x) (-1+4 x)}{\left (8+x+x^2\right ) \log (x)} \, dx+8 \int \frac {\log \left (8+x+x^2\right )}{x \log ^2(x)} \, dx-16 \int \frac {x \log \left (8+x+x^2\right )}{\log ^2(x)} \, dx-28 \int \frac {\log \left (8+x+x^2\right )}{\log ^2(x)} \, dx-28 \int \frac {\operatorname {LogIntegral}(x)}{8+x+x^2} \, dx+32 \int \frac {x \log \left (8+x+x^2\right )}{\log (x)} \, dx-56 \int \frac {x \operatorname {LogIntegral}(x)}{8+x+x^2} \, dx \\ & = 28 \log \left (8+x+x^2\right ) \operatorname {LogIntegral}(x)+4 \int \frac {(2+x) (1+2 x) (-1+4 x)}{\left (8+x+x^2\right ) \log (x)} \, dx+8 \int \frac {\log \left (8+x+x^2\right )}{x \log ^2(x)} \, dx-16 \int \frac {x \log \left (8+x+x^2\right )}{\log ^2(x)} \, dx-28 \int \frac {\log \left (8+x+x^2\right )}{\log ^2(x)} \, dx-28 \int \left (\frac {2 i \operatorname {LogIntegral}(x)}{\sqrt {31} \left (-1+i \sqrt {31}-2 x\right )}+\frac {2 i \operatorname {LogIntegral}(x)}{\sqrt {31} \left (1+i \sqrt {31}+2 x\right )}\right ) \, dx+32 \int \frac {x \log \left (8+x+x^2\right )}{\log (x)} \, dx-56 \int \left (\frac {\left (1+\frac {i}{\sqrt {31}}\right ) \operatorname {LogIntegral}(x)}{1-i \sqrt {31}+2 x}+\frac {\left (1-\frac {i}{\sqrt {31}}\right ) \operatorname {LogIntegral}(x)}{1+i \sqrt {31}+2 x}\right ) \, dx \\ & = 28 \log \left (8+x+x^2\right ) \operatorname {LogIntegral}(x)+4 \int \frac {(2+x) (1+2 x) (-1+4 x)}{\left (8+x+x^2\right ) \log (x)} \, dx+8 \int \frac {\log \left (8+x+x^2\right )}{x \log ^2(x)} \, dx-16 \int \frac {x \log \left (8+x+x^2\right )}{\log ^2(x)} \, dx-28 \int \frac {\log \left (8+x+x^2\right )}{\log ^2(x)} \, dx+32 \int \frac {x \log \left (8+x+x^2\right )}{\log (x)} \, dx-\frac {(56 i) \int \frac {\operatorname {LogIntegral}(x)}{-1+i \sqrt {31}-2 x} \, dx}{\sqrt {31}}-\frac {(56 i) \int \frac {\operatorname {LogIntegral}(x)}{1+i \sqrt {31}+2 x} \, dx}{\sqrt {31}}-\frac {1}{31} \left (56 \left (31-i \sqrt {31}\right )\right ) \int \frac {\operatorname {LogIntegral}(x)}{1+i \sqrt {31}+2 x} \, dx-\frac {1}{31} \left (56 \left (31+i \sqrt {31}\right )\right ) \int \frac {\operatorname {LogIntegral}(x)}{1-i \sqrt {31}+2 x} \, dx \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {\left (-8 x+12 x^2+72 x^3+32 x^4\right ) \log (x)+\left (64-216 x-148 x^2-44 x^3-16 x^4+\left (224 x+284 x^2+60 x^3+32 x^4\right ) \log (x)\right ) \log \left (8+x+x^2\right )}{\left (8 x+x^2+x^3\right ) \log ^2(x)} \, dx=\frac {4 \left (-2+7 x+4 x^2\right ) \log \left (8+x+x^2\right )}{\log (x)} \]
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Time = 1.60 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14
method | result | size |
risch | \(\frac {4 \left (4 x^{2}+7 x -2\right ) \ln \left (x^{2}+x +8\right )}{\ln \left (x \right )}\) | \(24\) |
parallelrisch | \(\frac {32 \ln \left (x^{2}+x +8\right ) x^{2}+56 \ln \left (x^{2}+x +8\right ) x -16 \ln \left (x^{2}+x +8\right )}{2 \ln \left (x \right )}\) | \(39\) |
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Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {\left (-8 x+12 x^2+72 x^3+32 x^4\right ) \log (x)+\left (64-216 x-148 x^2-44 x^3-16 x^4+\left (224 x+284 x^2+60 x^3+32 x^4\right ) \log (x)\right ) \log \left (8+x+x^2\right )}{\left (8 x+x^2+x^3\right ) \log ^2(x)} \, dx=\frac {4 \, {\left (4 \, x^{2} + 7 \, x - 2\right )} \log \left (x^{2} + x + 8\right )}{\log \left (x\right )} \]
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Exception generated. \[ \int \frac {\left (-8 x+12 x^2+72 x^3+32 x^4\right ) \log (x)+\left (64-216 x-148 x^2-44 x^3-16 x^4+\left (224 x+284 x^2+60 x^3+32 x^4\right ) \log (x)\right ) \log \left (8+x+x^2\right )}{\left (8 x+x^2+x^3\right ) \log ^2(x)} \, dx=\text {Exception raised: TypeError} \]
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Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {\left (-8 x+12 x^2+72 x^3+32 x^4\right ) \log (x)+\left (64-216 x-148 x^2-44 x^3-16 x^4+\left (224 x+284 x^2+60 x^3+32 x^4\right ) \log (x)\right ) \log \left (8+x+x^2\right )}{\left (8 x+x^2+x^3\right ) \log ^2(x)} \, dx=\frac {4 \, {\left (4 \, x^{2} + 7 \, x - 2\right )} \log \left (x^{2} + x + 8\right )}{\log \left (x\right )} \]
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Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {\left (-8 x+12 x^2+72 x^3+32 x^4\right ) \log (x)+\left (64-216 x-148 x^2-44 x^3-16 x^4+\left (224 x+284 x^2+60 x^3+32 x^4\right ) \log (x)\right ) \log \left (8+x+x^2\right )}{\left (8 x+x^2+x^3\right ) \log ^2(x)} \, dx=\frac {4 \, {\left (4 \, x^{2} + 7 \, x - 2\right )} \log \left (x^{2} + x + 8\right )}{\log \left (x\right )} \]
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Time = 13.34 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {\left (-8 x+12 x^2+72 x^3+32 x^4\right ) \log (x)+\left (64-216 x-148 x^2-44 x^3-16 x^4+\left (224 x+284 x^2+60 x^3+32 x^4\right ) \log (x)\right ) \log \left (8+x+x^2\right )}{\left (8 x+x^2+x^3\right ) \log ^2(x)} \, dx=\frac {4\,\ln \left (x^2+x+8\right )\,\left (4\,x^2+7\,x-2\right )}{\ln \left (x\right )} \]
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