Integrand size = 188, antiderivative size = 31 \[ \int \frac {36-288 x-128 x^2+82 x^3+56 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (9-30 x+13 x^2+20 x^3+4 x^4\right )+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (18-51 x-4 x^2+53 x^3+28 x^4+4 x^5\right ) \log (x)}{18 x-60 x^2+26 x^3+40 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (9 x-30 x^2+13 x^3+20 x^4+4 x^5\right ) \log (x)} \, dx=x+\log \left (x^2 \left (2 e^{-\frac {3}{-x+\frac {3}{5+2 x}}}+\log (x)\right )\right ) \]
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\[ \int \frac {36-288 x-128 x^2+82 x^3+56 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (9-30 x+13 x^2+20 x^3+4 x^4\right )+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (18-51 x-4 x^2+53 x^3+28 x^4+4 x^5\right ) \log (x)}{18 x-60 x^2+26 x^3+40 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (9 x-30 x^2+13 x^3+20 x^4+4 x^5\right ) \log (x)} \, dx=\int \frac {36-288 x-128 x^2+82 x^3+56 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (9-30 x+13 x^2+20 x^3+4 x^4\right )+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (18-51 x-4 x^2+53 x^3+28 x^4+4 x^5\right ) \log (x)}{18 x-60 x^2+26 x^3+40 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (9 x-30 x^2+13 x^3+20 x^4+4 x^5\right ) \log (x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (-3+5 x+2 x^2\right )^2+2 e^{\frac {3 (5+2 x)}{-3+5 x+2 x^2}} \left (18-144 x-64 x^2+41 x^3+28 x^4+4 x^5\right )+(2+x) \left (-3+5 x+2 x^2\right )^2 \log (x)}{x \left (3-5 x-2 x^2\right )^2 \left (2 e^{\frac {3 (5+2 x)}{-3+5 x+2 x^2}}+\log (x)\right )} \, dx \\ & = \int \left (\frac {18-144 x-64 x^2+41 x^3+28 x^4+4 x^5}{x (3+x)^2 (-1+2 x)^2}+\frac {9-30 x+13 x^2+20 x^3+4 x^4+93 x \log (x)+60 x^2 \log (x)+12 x^3 \log (x)}{x (3+x)^2 (-1+2 x)^2 \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )}\right ) \, dx \\ & = \int \frac {18-144 x-64 x^2+41 x^3+28 x^4+4 x^5}{x (3+x)^2 (-1+2 x)^2} \, dx+\int \frac {9-30 x+13 x^2+20 x^3+4 x^4+93 x \log (x)+60 x^2 \log (x)+12 x^3 \log (x)}{x (3+x)^2 (-1+2 x)^2 \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )} \, dx \\ & = \int \left (1+\frac {2}{x}-\frac {3}{7 (3+x)^2}-\frac {72}{7 (-1+2 x)^2}\right ) \, dx+\int \frac {\left (-3+5 x+2 x^2\right )^2+3 x \left (31+20 x+4 x^2\right ) \log (x)}{(1-2 x)^2 x (3+x)^2 \left (2 e^{\frac {3 (5+2 x)}{-3+5 x+2 x^2}}+\log (x)\right )} \, dx \\ & = -\frac {36}{7 (1-2 x)}+x+\frac {3}{7 (3+x)}+2 \log (x)+\int \left (\frac {9-30 x+13 x^2+20 x^3+4 x^4+93 x \log (x)+60 x^2 \log (x)+12 x^3 \log (x)}{9 x \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )}-\frac {9-30 x+13 x^2+20 x^3+4 x^4+93 x \log (x)+60 x^2 \log (x)+12 x^3 \log (x)}{147 (3+x)^2 \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )}-\frac {19 \left (9-30 x+13 x^2+20 x^3+4 x^4+93 x \log (x)+60 x^2 \log (x)+12 x^3 \log (x)\right )}{3087 (3+x) \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )}+\frac {8 \left (9-30 x+13 x^2+20 x^3+4 x^4+93 x \log (x)+60 x^2 \log (x)+12 x^3 \log (x)\right )}{49 (-1+2 x)^2 \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )}-\frac {72 \left (9-30 x+13 x^2+20 x^3+4 x^4+93 x \log (x)+60 x^2 \log (x)+12 x^3 \log (x)\right )}{343 (-1+2 x) \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )}\right ) \, dx \\ & = -\frac {36}{7 (1-2 x)}+x+\frac {3}{7 (3+x)}+2 \log (x)-\frac {19 \int \frac {9-30 x+13 x^2+20 x^3+4 x^4+93 x \log (x)+60 x^2 \log (x)+12 x^3 \log (x)}{(3+x) \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )} \, dx}{3087}-\frac {1}{147} \int \frac {9-30 x+13 x^2+20 x^3+4 x^4+93 x \log (x)+60 x^2 \log (x)+12 x^3 \log (x)}{(3+x)^2 \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )} \, dx+\frac {1}{9} \int \frac {9-30 x+13 x^2+20 x^3+4 x^4+93 x \log (x)+60 x^2 \log (x)+12 x^3 \log (x)}{x \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )} \, dx+\frac {8}{49} \int \frac {9-30 x+13 x^2+20 x^3+4 x^4+93 x \log (x)+60 x^2 \log (x)+12 x^3 \log (x)}{(-1+2 x)^2 \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )} \, dx-\frac {72}{343} \int \frac {9-30 x+13 x^2+20 x^3+4 x^4+93 x \log (x)+60 x^2 \log (x)+12 x^3 \log (x)}{(-1+2 x) \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )} \, dx \\ & = -\frac {36}{7 (1-2 x)}+x+\frac {3}{7 (3+x)}+2 \log (x)-\frac {19 \int \frac {\left (-3+5 x+2 x^2\right )^2+3 x \left (31+20 x+4 x^2\right ) \log (x)}{(3+x) \left (2 e^{\frac {3 (5+2 x)}{-3+5 x+2 x^2}}+\log (x)\right )} \, dx}{3087}-\frac {1}{147} \int \frac {\left (-3+5 x+2 x^2\right )^2+3 x \left (31+20 x+4 x^2\right ) \log (x)}{(3+x)^2 \left (2 e^{\frac {3 (5+2 x)}{-3+5 x+2 x^2}}+\log (x)\right )} \, dx+\frac {1}{9} \int \frac {\left (-3+5 x+2 x^2\right )^2+3 x \left (31+20 x+4 x^2\right ) \log (x)}{x \left (2 e^{\frac {3 (5+2 x)}{-3+5 x+2 x^2}}+\log (x)\right )} \, dx+\frac {8}{49} \int \frac {\left (-3+5 x+2 x^2\right )^2+3 x \left (31+20 x+4 x^2\right ) \log (x)}{(1-2 x)^2 \left (2 e^{\frac {3 (5+2 x)}{-3+5 x+2 x^2}}+\log (x)\right )} \, dx-\frac {72}{343} \int \left (\frac {9}{(-1+2 x) \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )}-\frac {30 x}{(-1+2 x) \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )}+\frac {13 x^2}{(-1+2 x) \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )}+\frac {20 x^3}{(-1+2 x) \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )}+\frac {4 x^4}{(-1+2 x) \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )}+\frac {93 x \log (x)}{(-1+2 x) \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )}+\frac {60 x^2 \log (x)}{(-1+2 x) \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )}+\frac {12 x^3 \log (x)}{(-1+2 x) \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )}\right ) \, dx \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {36-288 x-128 x^2+82 x^3+56 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (9-30 x+13 x^2+20 x^3+4 x^4\right )+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (18-51 x-4 x^2+53 x^3+28 x^4+4 x^5\right ) \log (x)}{18 x-60 x^2+26 x^3+40 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (9 x-30 x^2+13 x^3+20 x^4+4 x^5\right ) \log (x)} \, dx=x+2 \log (x)+\log \left (2 e^{\frac {3 (5+2 x)}{-3+5 x+2 x^2}}+\log (x)\right ) \]
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Time = 5.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06
method | result | size |
risch | \(2 \ln \left (x \right )+x +\ln \left (\ln \left (x \right )+2 \,{\mathrm e}^{\frac {6 x +15}{\left (3+x \right ) \left (-1+2 x \right )}}\right )\) | \(33\) |
parallelrisch | \(-\frac {-2016 x^{2} \ln \left (x \right )-1008 \ln \left ({\mathrm e}^{-\frac {3 \left (5+2 x \right )}{2 x^{2}+5 x -3}} \ln \left (x \right )+2\right ) x^{2}-1008 x^{3}-9840-5040 x \ln \left (x \right )-2520 \ln \left ({\mathrm e}^{-\frac {3 \left (5+2 x \right )}{2 x^{2}+5 x -3}} \ln \left (x \right )+2\right ) x -1000 x^{2}+3024 \ln \left (x \right )+1512 \ln \left ({\mathrm e}^{-\frac {3 \left (5+2 x \right )}{2 x^{2}+5 x -3}} \ln \left (x \right )+2\right )+2288 x}{504 \left (2 x^{2}+5 x -3\right )}\) | \(134\) |
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Time = 0.29 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.71 \[ \int \frac {36-288 x-128 x^2+82 x^3+56 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (9-30 x+13 x^2+20 x^3+4 x^4\right )+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (18-51 x-4 x^2+53 x^3+28 x^4+4 x^5\right ) \log (x)}{18 x-60 x^2+26 x^3+40 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (9 x-30 x^2+13 x^3+20 x^4+4 x^5\right ) \log (x)} \, dx=x + \log \left ({\left (e^{\left (-\frac {3 \, {\left (2 \, x + 5\right )}}{2 \, x^{2} + 5 \, x - 3}\right )} \log \left (x\right ) + 2\right )} e^{\left (\frac {3 \, {\left (2 \, x + 5\right )}}{2 \, x^{2} + 5 \, x - 3}\right )}\right ) + 2 \, \log \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (22) = 44\).
Time = 0.60 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.65 \[ \int \frac {36-288 x-128 x^2+82 x^3+56 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (9-30 x+13 x^2+20 x^3+4 x^4\right )+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (18-51 x-4 x^2+53 x^3+28 x^4+4 x^5\right ) \log (x)}{18 x-60 x^2+26 x^3+40 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (9 x-30 x^2+13 x^3+20 x^4+4 x^5\right ) \log (x)} \, dx=x + \frac {6 x + 15}{2 x^{2} + 5 x - 3} + 2 \log {\left (x \right )} + \log {\left (e^{\frac {- 6 x - 15}{2 x^{2} + 5 x - 3}} + \frac {2}{\log {\left (x \right )}} \right )} + \log {\left (\log {\left (x \right )} \right )} \]
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Time = 0.30 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.81 \[ \int \frac {36-288 x-128 x^2+82 x^3+56 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (9-30 x+13 x^2+20 x^3+4 x^4\right )+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (18-51 x-4 x^2+53 x^3+28 x^4+4 x^5\right ) \log (x)}{18 x-60 x^2+26 x^3+40 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (9 x-30 x^2+13 x^3+20 x^4+4 x^5\right ) \log (x)} \, dx=\frac {7 \, x^{2} + 21 \, x + 3}{7 \, {\left (x + 3\right )}} + \log \left (\frac {1}{2} \, {\left (2 \, e^{\left (\frac {36}{7 \, {\left (2 \, x - 1\right )}} + \frac {3}{7 \, {\left (x + 3\right )}}\right )} + \log \left (x\right )\right )} e^{\left (-\frac {3}{7 \, {\left (x + 3\right )}}\right )}\right ) + 2 \, \log \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 150 vs. \(2 (28) = 56\).
Time = 0.53 (sec) , antiderivative size = 150, normalized size of antiderivative = 4.84 \[ \int \frac {36-288 x-128 x^2+82 x^3+56 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (9-30 x+13 x^2+20 x^3+4 x^4\right )+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (18-51 x-4 x^2+53 x^3+28 x^4+4 x^5\right ) \log (x)}{18 x-60 x^2+26 x^3+40 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (9 x-30 x^2+13 x^3+20 x^4+4 x^5\right ) \log (x)} \, dx=\frac {2 \, x^{3} + 2 \, x^{2} \log \left (e^{\left (-\frac {10 \, x^{2} + 31 \, x}{2 \, x^{2} + 5 \, x - 3} + 5\right )} \log \left (x\right ) + 2\right ) + 4 \, x^{2} \log \left (x\right ) + 5 \, x^{2} + 5 \, x \log \left (e^{\left (-\frac {10 \, x^{2} + 31 \, x}{2 \, x^{2} + 5 \, x - 3} + 5\right )} \log \left (x\right ) + 2\right ) + 10 \, x \log \left (x\right ) + 3 \, x - 3 \, \log \left (e^{\left (-\frac {10 \, x^{2} + 31 \, x}{2 \, x^{2} + 5 \, x - 3} + 5\right )} \log \left (x\right ) + 2\right ) - 6 \, \log \left (x\right ) + 15}{2 \, x^{2} + 5 \, x - 3} \]
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Time = 9.82 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.81 \[ \int \frac {36-288 x-128 x^2+82 x^3+56 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (9-30 x+13 x^2+20 x^3+4 x^4\right )+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (18-51 x-4 x^2+53 x^3+28 x^4+4 x^5\right ) \log (x)}{18 x-60 x^2+26 x^3+40 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (9 x-30 x^2+13 x^3+20 x^4+4 x^5\right ) \log (x)} \, dx=x+\ln \left (\ln \left (x\right )\right )+\ln \left (\frac {{\mathrm {e}}^{-\frac {6\,x+15}{2\,x^2+5\,x-3}}\,\ln \left (x\right )+2}{\ln \left (x\right )}\right )+2\,\ln \left (x\right )+\frac {3\,x+\frac {15}{2}}{x^2+\frac {5\,x}{2}-\frac {3}{2}} \]
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