Integrand size = 175, antiderivative size = 33 \[ \int \frac {-45+39 x-56 x^2+40 x^3-11 x^4+x^5+\left (-45+30 x-170 x^2+119 x^3-58 x^4+13 x^5-x^6\right ) \log (x)+\left (9 x^2-6 x^3+13 x^4-11 x^5-x^6-4 x^7-5 x^8+x^9\right ) \log ^2(x)}{\left (45 x-39 x^2+56 x^3-40 x^4+11 x^5-x^6\right ) \log (x)+\left (9 x^3-6 x^4+19 x^5-12 x^6+11 x^7-6 x^8+x^9\right ) \log ^2(x)} \, dx=-\frac {x^2}{3-x}+\log \left (x+\frac {5-x}{\left (x+x^3\right ) \log (x)}\right ) \]
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\[ \int \frac {-45+39 x-56 x^2+40 x^3-11 x^4+x^5+\left (-45+30 x-170 x^2+119 x^3-58 x^4+13 x^5-x^6\right ) \log (x)+\left (9 x^2-6 x^3+13 x^4-11 x^5-x^6-4 x^7-5 x^8+x^9\right ) \log ^2(x)}{\left (45 x-39 x^2+56 x^3-40 x^4+11 x^5-x^6\right ) \log (x)+\left (9 x^3-6 x^4+19 x^5-12 x^6+11 x^7-6 x^8+x^9\right ) \log ^2(x)} \, dx=\int \frac {-45+39 x-56 x^2+40 x^3-11 x^4+x^5+\left (-45+30 x-170 x^2+119 x^3-58 x^4+13 x^5-x^6\right ) \log (x)+\left (9 x^2-6 x^3+13 x^4-11 x^5-x^6-4 x^7-5 x^8+x^9\right ) \log ^2(x)}{\left (45 x-39 x^2+56 x^3-40 x^4+11 x^5-x^6\right ) \log (x)+\left (9 x^3-6 x^4+19 x^5-12 x^6+11 x^7-6 x^8+x^9\right ) \log ^2(x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {(-3+x)^2 \left (-5+x-5 x^2+x^3\right )-\left (45-30 x+170 x^2-119 x^3+58 x^4-13 x^5+x^6\right ) \log (x)+\left (x+x^3\right )^2 \left (9-6 x-5 x^2+x^3\right ) \log ^2(x)}{(3-x)^2 x \left (1+x^2\right ) \log (x) \left (5-x+\left (x^2+x^4\right ) \log (x)\right )} \, dx \\ & = \int \left (\frac {9-6 x-5 x^2+x^3}{(-3+x)^2 x}-\frac {1}{x \log (x)}+\frac {x \left (1+x^2\right )}{5-x+x^2 \log (x)+x^4 \log (x)}+\frac {-10+x-20 x^2+3 x^3}{x \left (1+x^2\right ) \left (5-x+x^2 \log (x)+x^4 \log (x)\right )}\right ) \, dx \\ & = \int \frac {9-6 x-5 x^2+x^3}{(-3+x)^2 x} \, dx-\int \frac {1}{x \log (x)} \, dx+\int \frac {x \left (1+x^2\right )}{5-x+x^2 \log (x)+x^4 \log (x)} \, dx+\int \frac {-10+x-20 x^2+3 x^3}{x \left (1+x^2\right ) \left (5-x+x^2 \log (x)+x^4 \log (x)\right )} \, dx \\ & = \int \left (1-\frac {9}{(-3+x)^2}+\frac {1}{x}\right ) \, dx+\int \left (\frac {x}{5-x+x^2 \log (x)+x^4 \log (x)}+\frac {x^3}{5-x+x^2 \log (x)+x^4 \log (x)}\right ) \, dx+\int \left (\frac {3}{5-x+x^2 \log (x)+x^4 \log (x)}-\frac {10}{x \left (5-x+x^2 \log (x)+x^4 \log (x)\right )}-\frac {2 (1+5 x)}{\left (1+x^2\right ) \left (5-x+x^2 \log (x)+x^4 \log (x)\right )}\right ) \, dx-\text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right ) \\ & = -\frac {9}{3-x}+x+\log (x)-\log (\log (x))-2 \int \frac {1+5 x}{\left (1+x^2\right ) \left (5-x+x^2 \log (x)+x^4 \log (x)\right )} \, dx+3 \int \frac {1}{5-x+x^2 \log (x)+x^4 \log (x)} \, dx-10 \int \frac {1}{x \left (5-x+x^2 \log (x)+x^4 \log (x)\right )} \, dx+\int \frac {x}{5-x+x^2 \log (x)+x^4 \log (x)} \, dx+\int \frac {x^3}{5-x+x^2 \log (x)+x^4 \log (x)} \, dx \\ & = -\frac {9}{3-x}+x+\log (x)-\log (\log (x))-2 \int \left (\frac {1}{\left (1+x^2\right ) \left (5-x+x^2 \log (x)+x^4 \log (x)\right )}+\frac {5 x}{\left (1+x^2\right ) \left (5-x+x^2 \log (x)+x^4 \log (x)\right )}\right ) \, dx+3 \int \frac {1}{5-x+x^2 \log (x)+x^4 \log (x)} \, dx-10 \int \frac {1}{x \left (5-x+x^2 \log (x)+x^4 \log (x)\right )} \, dx+\int \frac {x}{5-x+x^2 \log (x)+x^4 \log (x)} \, dx+\int \frac {x^3}{5-x+x^2 \log (x)+x^4 \log (x)} \, dx \\ & = -\frac {9}{3-x}+x+\log (x)-\log (\log (x))-2 \int \frac {1}{\left (1+x^2\right ) \left (5-x+x^2 \log (x)+x^4 \log (x)\right )} \, dx+3 \int \frac {1}{5-x+x^2 \log (x)+x^4 \log (x)} \, dx-10 \int \frac {1}{x \left (5-x+x^2 \log (x)+x^4 \log (x)\right )} \, dx-10 \int \frac {x}{\left (1+x^2\right ) \left (5-x+x^2 \log (x)+x^4 \log (x)\right )} \, dx+\int \frac {x}{5-x+x^2 \log (x)+x^4 \log (x)} \, dx+\int \frac {x^3}{5-x+x^2 \log (x)+x^4 \log (x)} \, dx \\ & = -\frac {9}{3-x}+x+\log (x)-\log (\log (x))-2 \int \left (\frac {i}{2 (i-x) \left (5-x+x^2 \log (x)+x^4 \log (x)\right )}+\frac {i}{2 (i+x) \left (5-x+x^2 \log (x)+x^4 \log (x)\right )}\right ) \, dx+3 \int \frac {1}{5-x+x^2 \log (x)+x^4 \log (x)} \, dx-10 \int \frac {1}{x \left (5-x+x^2 \log (x)+x^4 \log (x)\right )} \, dx-10 \int \left (-\frac {1}{2 (i-x) \left (5-x+x^2 \log (x)+x^4 \log (x)\right )}+\frac {1}{2 (i+x) \left (5-x+x^2 \log (x)+x^4 \log (x)\right )}\right ) \, dx+\int \frac {x}{5-x+x^2 \log (x)+x^4 \log (x)} \, dx+\int \frac {x^3}{5-x+x^2 \log (x)+x^4 \log (x)} \, dx \\ & = -\frac {9}{3-x}+x+\log (x)-\log (\log (x))-i \int \frac {1}{(i-x) \left (5-x+x^2 \log (x)+x^4 \log (x)\right )} \, dx-i \int \frac {1}{(i+x) \left (5-x+x^2 \log (x)+x^4 \log (x)\right )} \, dx+3 \int \frac {1}{5-x+x^2 \log (x)+x^4 \log (x)} \, dx+5 \int \frac {1}{(i-x) \left (5-x+x^2 \log (x)+x^4 \log (x)\right )} \, dx-5 \int \frac {1}{(i+x) \left (5-x+x^2 \log (x)+x^4 \log (x)\right )} \, dx-10 \int \frac {1}{x \left (5-x+x^2 \log (x)+x^4 \log (x)\right )} \, dx+\int \frac {x}{5-x+x^2 \log (x)+x^4 \log (x)} \, dx+\int \frac {x^3}{5-x+x^2 \log (x)+x^4 \log (x)} \, dx \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.33 \[ \int \frac {-45+39 x-56 x^2+40 x^3-11 x^4+x^5+\left (-45+30 x-170 x^2+119 x^3-58 x^4+13 x^5-x^6\right ) \log (x)+\left (9 x^2-6 x^3+13 x^4-11 x^5-x^6-4 x^7-5 x^8+x^9\right ) \log ^2(x)}{\left (45 x-39 x^2+56 x^3-40 x^4+11 x^5-x^6\right ) \log (x)+\left (9 x^3-6 x^4+19 x^5-12 x^6+11 x^7-6 x^8+x^9\right ) \log ^2(x)} \, dx=\frac {9}{-3+x}+x-\log (x)-\log \left (1+x^2\right )-\log (\log (x))+\log \left (5-x+x^2 \log (x)+x^4 \log (x)\right ) \]
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Time = 0.54 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.30
method | result | size |
default | \(-\ln \left (x \right )+x +\frac {9}{-3+x}+\ln \left (x^{4}+x^{2}-\frac {x}{\ln \left (x \right )}+\frac {5}{\ln \left (x \right )}\right )-\ln \left (x^{2}+1\right )\) | \(43\) |
risch | \(\frac {x \ln \left (x \right )+x^{2}-3 \ln \left (x \right )-3 x +9}{-3+x}-\ln \left (\ln \left (x \right )\right )+\ln \left (\ln \left (x \right )-\frac {-5+x}{x^{2} \left (x^{2}+1\right )}\right )\) | \(48\) |
parallelrisch | \(\frac {-x \ln \left (\ln \left (x \right )\right )-\ln \left (x^{2}+1\right ) x +\ln \left (x^{4} \ln \left (x \right )+x^{2} \ln \left (x \right )-x +5\right ) x +x^{2}-x \ln \left (x \right )+3 \ln \left (\ln \left (x \right )\right )+3 \ln \left (x^{2}+1\right )-3 \ln \left (x^{4} \ln \left (x \right )+x^{2} \ln \left (x \right )-x +5\right )+3 \ln \left (x \right )}{-3+x}\) | \(88\) |
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Time = 0.26 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.76 \[ \int \frac {-45+39 x-56 x^2+40 x^3-11 x^4+x^5+\left (-45+30 x-170 x^2+119 x^3-58 x^4+13 x^5-x^6\right ) \log (x)+\left (9 x^2-6 x^3+13 x^4-11 x^5-x^6-4 x^7-5 x^8+x^9\right ) \log ^2(x)}{\left (45 x-39 x^2+56 x^3-40 x^4+11 x^5-x^6\right ) \log (x)+\left (9 x^3-6 x^4+19 x^5-12 x^6+11 x^7-6 x^8+x^9\right ) \log ^2(x)} \, dx=\frac {x^{2} + {\left (x - 3\right )} \log \left (x\right ) + {\left (x - 3\right )} \log \left (\frac {{\left (x^{4} + x^{2}\right )} \log \left (x\right ) - x + 5}{x^{4} + x^{2}}\right ) - {\left (x - 3\right )} \log \left (\log \left (x\right )\right ) - 3 \, x + 9}{x - 3} \]
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Exception generated. \[ \int \frac {-45+39 x-56 x^2+40 x^3-11 x^4+x^5+\left (-45+30 x-170 x^2+119 x^3-58 x^4+13 x^5-x^6\right ) \log (x)+\left (9 x^2-6 x^3+13 x^4-11 x^5-x^6-4 x^7-5 x^8+x^9\right ) \log ^2(x)}{\left (45 x-39 x^2+56 x^3-40 x^4+11 x^5-x^6\right ) \log (x)+\left (9 x^3-6 x^4+19 x^5-12 x^6+11 x^7-6 x^8+x^9\right ) \log ^2(x)} \, dx=\text {Exception raised: PolynomialError} \]
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Time = 0.32 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.45 \[ \int \frac {-45+39 x-56 x^2+40 x^3-11 x^4+x^5+\left (-45+30 x-170 x^2+119 x^3-58 x^4+13 x^5-x^6\right ) \log (x)+\left (9 x^2-6 x^3+13 x^4-11 x^5-x^6-4 x^7-5 x^8+x^9\right ) \log ^2(x)}{\left (45 x-39 x^2+56 x^3-40 x^4+11 x^5-x^6\right ) \log (x)+\left (9 x^3-6 x^4+19 x^5-12 x^6+11 x^7-6 x^8+x^9\right ) \log ^2(x)} \, dx=\frac {x^{2} - 3 \, x + 9}{x - 3} + \log \left (x\right ) + \log \left (\frac {{\left (x^{4} + x^{2}\right )} \log \left (x\right ) - x + 5}{x^{4} + x^{2}}\right ) - \log \left (\log \left (x\right )\right ) \]
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Time = 0.29 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.33 \[ \int \frac {-45+39 x-56 x^2+40 x^3-11 x^4+x^5+\left (-45+30 x-170 x^2+119 x^3-58 x^4+13 x^5-x^6\right ) \log (x)+\left (9 x^2-6 x^3+13 x^4-11 x^5-x^6-4 x^7-5 x^8+x^9\right ) \log ^2(x)}{\left (45 x-39 x^2+56 x^3-40 x^4+11 x^5-x^6\right ) \log (x)+\left (9 x^3-6 x^4+19 x^5-12 x^6+11 x^7-6 x^8+x^9\right ) \log ^2(x)} \, dx=x + \frac {9}{x - 3} + \log \left (x^{4} \log \left (x\right ) + x^{2} \log \left (x\right ) - x + 5\right ) - \log \left (x^{2} + 1\right ) - \log \left (x\right ) - \log \left (\log \left (x\right )\right ) \]
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Timed out. \[ \int \frac {-45+39 x-56 x^2+40 x^3-11 x^4+x^5+\left (-45+30 x-170 x^2+119 x^3-58 x^4+13 x^5-x^6\right ) \log (x)+\left (9 x^2-6 x^3+13 x^4-11 x^5-x^6-4 x^7-5 x^8+x^9\right ) \log ^2(x)}{\left (45 x-39 x^2+56 x^3-40 x^4+11 x^5-x^6\right ) \log (x)+\left (9 x^3-6 x^4+19 x^5-12 x^6+11 x^7-6 x^8+x^9\right ) \log ^2(x)} \, dx=\int -\frac {\ln \left (x\right )\,\left (x^6-13\,x^5+58\,x^4-119\,x^3+170\,x^2-30\,x+45\right )-39\,x+{\ln \left (x\right )}^2\,\left (-x^9+5\,x^8+4\,x^7+x^6+11\,x^5-13\,x^4+6\,x^3-9\,x^2\right )+56\,x^2-40\,x^3+11\,x^4-x^5+45}{\left (x^9-6\,x^8+11\,x^7-12\,x^6+19\,x^5-6\,x^4+9\,x^3\right )\,{\ln \left (x\right )}^2+\left (-x^6+11\,x^5-40\,x^4+56\,x^3-39\,x^2+45\,x\right )\,\ln \left (x\right )} \,d x \]
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