Integrand size = 116, antiderivative size = 24 \[ \int \frac {96+e^3 (-48-8 x)+16 x+4 e^{4+2 x^2} x^2+e^6 (6+x)+e^{2+x^2} \left (8+48 x^2+e^3 \left (-2-12 x^2\right )\right )+\left (32-16 e^3+2 e^6+e^{2+x^2} \left (16 x^2-4 e^3 x^2\right )\right ) \log (x)}{16 x-8 e^3 x+e^6 x} \, dx=x+\left (3-\frac {e^{2+x^2}}{-4+e^3}+\log (x)\right )^2 \]
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Leaf count is larger than twice the leaf count of optimal. \(62\) vs. \(2(24)=48\).
Time = 0.09 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.58, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.060, Rules used = {6, 12, 14, 2240, 45, 2338, 2326} \[ \int \frac {96+e^3 (-48-8 x)+16 x+4 e^{4+2 x^2} x^2+e^6 (6+x)+e^{2+x^2} \left (8+48 x^2+e^3 \left (-2-12 x^2\right )\right )+\left (32-16 e^3+2 e^6+e^{2+x^2} \left (16 x^2-4 e^3 x^2\right )\right ) \log (x)}{16 x-8 e^3 x+e^6 x} \, dx=\frac {e^{2 x^2+4}}{\left (4-e^3\right )^2}+\frac {2 e^{x^2+2} \left (3 x^2+x^2 \log (x)\right )}{\left (4-e^3\right ) x^2}+x+\log ^2(x)+6 \log (x) \]
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Rule 6
Rule 12
Rule 14
Rule 45
Rule 2240
Rule 2326
Rule 2338
Rubi steps \begin{align*} \text {integral}& = \int \frac {96+e^3 (-48-8 x)+16 x+4 e^{4+2 x^2} x^2+e^6 (6+x)+e^{2+x^2} \left (8+48 x^2+e^3 \left (-2-12 x^2\right )\right )+\left (32-16 e^3+2 e^6+e^{2+x^2} \left (16 x^2-4 e^3 x^2\right )\right ) \log (x)}{e^6 x+\left (16-8 e^3\right ) x} \, dx \\ & = \int \frac {96+e^3 (-48-8 x)+16 x+4 e^{4+2 x^2} x^2+e^6 (6+x)+e^{2+x^2} \left (8+48 x^2+e^3 \left (-2-12 x^2\right )\right )+\left (32-16 e^3+2 e^6+e^{2+x^2} \left (16 x^2-4 e^3 x^2\right )\right ) \log (x)}{\left (16-8 e^3+e^6\right ) x} \, dx \\ & = \frac {\int \frac {96+e^3 (-48-8 x)+16 x+4 e^{4+2 x^2} x^2+e^6 (6+x)+e^{2+x^2} \left (8+48 x^2+e^3 \left (-2-12 x^2\right )\right )+\left (32-16 e^3+2 e^6+e^{2+x^2} \left (16 x^2-4 e^3 x^2\right )\right ) \log (x)}{x} \, dx}{16-8 e^3+e^6} \\ & = \frac {\int \left (4 e^{4+2 x^2} x+\frac {\left (-4+e^3\right )^2 (6+x+2 \log (x))}{x}-\frac {2 e^{2+x^2} \left (-4+e^3\right ) \left (1+6 x^2+2 x^2 \log (x)\right )}{x}\right ) \, dx}{16-8 e^3+e^6} \\ & = \frac {4 \int e^{4+2 x^2} x \, dx}{\left (4-e^3\right )^2}+\frac {2 \int \frac {e^{2+x^2} \left (1+6 x^2+2 x^2 \log (x)\right )}{x} \, dx}{4-e^3}+\int \frac {6+x+2 \log (x)}{x} \, dx \\ & = \frac {e^{4+2 x^2}}{\left (4-e^3\right )^2}+\frac {2 e^{2+x^2} \left (3 x^2+x^2 \log (x)\right )}{\left (4-e^3\right ) x^2}+\int \left (\frac {6+x}{x}+\frac {2 \log (x)}{x}\right ) \, dx \\ & = \frac {e^{4+2 x^2}}{\left (4-e^3\right )^2}+\frac {2 e^{2+x^2} \left (3 x^2+x^2 \log (x)\right )}{\left (4-e^3\right ) x^2}+2 \int \frac {\log (x)}{x} \, dx+\int \frac {6+x}{x} \, dx \\ & = \frac {e^{4+2 x^2}}{\left (4-e^3\right )^2}+\log ^2(x)+\frac {2 e^{2+x^2} \left (3 x^2+x^2 \log (x)\right )}{\left (4-e^3\right ) x^2}+\int \left (1+\frac {6}{x}\right ) \, dx \\ & = \frac {e^{4+2 x^2}}{\left (4-e^3\right )^2}+x+6 \log (x)+\log ^2(x)+\frac {2 e^{2+x^2} \left (3 x^2+x^2 \log (x)\right )}{\left (4-e^3\right ) x^2} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(80\) vs. \(2(24)=48\).
Time = 0.08 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.33 \[ \int \frac {96+e^3 (-48-8 x)+16 x+4 e^{4+2 x^2} x^2+e^6 (6+x)+e^{2+x^2} \left (8+48 x^2+e^3 \left (-2-12 x^2\right )\right )+\left (32-16 e^3+2 e^6+e^{2+x^2} \left (16 x^2-4 e^3 x^2\right )\right ) \log (x)}{16 x-8 e^3 x+e^6 x} \, dx=\frac {e^{4+2 x^2}-6 e^{2+x^2} \left (-4+e^3\right )+\left (-4+e^3\right )^2 x-2 e^{2+x^2} \left (-4+e^3\right ) \log (x)+6 \left (-4+e^3\right )^2 \log (x)+\left (-4+e^3\right )^2 \log ^2(x)}{\left (-4+e^3\right )^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(61\) vs. \(2(22)=44\).
Time = 0.34 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.58
method | result | size |
default | \(x +6 \ln \left (x \right )-\frac {6 \,{\mathrm e}^{x^{2}+2}}{{\mathrm e}^{3}-4}-\frac {2 \,{\mathrm e}^{x^{2}+2} \ln \left (x \right )}{{\mathrm e}^{3}-4}+\ln \left (x \right )^{2}+\frac {{\mathrm e}^{2 x^{2}+4}}{16+{\mathrm e}^{6}-8 \,{\mathrm e}^{3}}\) | \(62\) |
parts | \(x +6 \ln \left (x \right )-\frac {6 \,{\mathrm e}^{x^{2}+2}}{{\mathrm e}^{3}-4}-\frac {2 \,{\mathrm e}^{x^{2}+2} \ln \left (x \right )}{{\mathrm e}^{3}-4}+\ln \left (x \right )^{2}+\frac {{\mathrm e}^{2 x^{2}+4}}{16+{\mathrm e}^{6}-8 \,{\mathrm e}^{3}}\) | \(62\) |
norman | \(\frac {\frac {{\mathrm e}^{2 x^{2}+4}}{{\mathrm e}^{3}-4}+\left ({\mathrm e}^{3}-4\right ) x +\left ({\mathrm e}^{3}-4\right ) \ln \left (x \right )^{2}+\left (6 \,{\mathrm e}^{3}-24\right ) \ln \left (x \right )-2 \,{\mathrm e}^{x^{2}+2} \ln \left (x \right )-6 \,{\mathrm e}^{x^{2}+2}}{{\mathrm e}^{3}-4}\) | \(66\) |
parallelrisch | \(\frac {\ln \left (x \right )^{2} {\mathrm e}^{6}+x \,{\mathrm e}^{6}+6 \,{\mathrm e}^{6} \ln \left (x \right )-8 \,{\mathrm e}^{3} \ln \left (x \right )^{2}-2 \,{\mathrm e}^{x^{2}+2} \ln \left (x \right ) {\mathrm e}^{3}-8 x \,{\mathrm e}^{3}-48 \ln \left (x \right ) {\mathrm e}^{3}-6 \,{\mathrm e}^{3} {\mathrm e}^{x^{2}+2}+16 \ln \left (x \right )^{2}+8 \,{\mathrm e}^{x^{2}+2} \ln \left (x \right )+{\mathrm e}^{2 x^{2}+4}+16 x +96 \ln \left (x \right )+24 \,{\mathrm e}^{x^{2}+2}}{16+{\mathrm e}^{6}-8 \,{\mathrm e}^{3}}\) | \(118\) |
risch | \(\ln \left (x \right )^{2}-\frac {2 \,{\mathrm e}^{x^{2}+2} \ln \left (x \right )}{{\mathrm e}^{3}-4}-\frac {48 \ln \left (x \right ) {\mathrm e}^{3}}{\left ({\mathrm e}^{3}-4\right )^{2}}+\frac {6 \,{\mathrm e}^{6} \ln \left (x \right )}{\left ({\mathrm e}^{3}-4\right )^{2}}-\frac {8 x \,{\mathrm e}^{3}}{\left ({\mathrm e}^{3}-4\right )^{2}}+\frac {x \,{\mathrm e}^{6}}{\left ({\mathrm e}^{3}-4\right )^{2}}+\frac {96 \ln \left (x \right )}{\left ({\mathrm e}^{3}-4\right )^{2}}+\frac {24 \,{\mathrm e}^{x^{2}+2}}{\left ({\mathrm e}^{3}-4\right )^{2}}-\frac {6 \,{\mathrm e}^{x^{2}+5}}{\left ({\mathrm e}^{3}-4\right )^{2}}+\frac {{\mathrm e}^{2 x^{2}+4}}{\left ({\mathrm e}^{3}-4\right )^{2}}+\frac {16 x}{\left ({\mathrm e}^{3}-4\right )^{2}}\) | \(129\) |
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Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (23) = 46\).
Time = 0.25 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.42 \[ \int \frac {96+e^3 (-48-8 x)+16 x+4 e^{4+2 x^2} x^2+e^6 (6+x)+e^{2+x^2} \left (8+48 x^2+e^3 \left (-2-12 x^2\right )\right )+\left (32-16 e^3+2 e^6+e^{2+x^2} \left (16 x^2-4 e^3 x^2\right )\right ) \log (x)}{16 x-8 e^3 x+e^6 x} \, dx=\frac {{\left (e^{6} - 8 \, e^{3} + 16\right )} \log \left (x\right )^{2} + x e^{6} - 8 \, x e^{3} - 6 \, {\left (e^{3} - 4\right )} e^{\left (x^{2} + 2\right )} - 2 \, {\left ({\left (e^{3} - 4\right )} e^{\left (x^{2} + 2\right )} - 3 \, e^{6} + 24 \, e^{3} - 48\right )} \log \left (x\right ) + 16 \, x + e^{\left (2 \, x^{2} + 4\right )}}{e^{6} - 8 \, e^{3} + 16} \]
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Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (19) = 38\).
Time = 0.27 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.33 \[ \int \frac {96+e^3 (-48-8 x)+16 x+4 e^{4+2 x^2} x^2+e^6 (6+x)+e^{2+x^2} \left (8+48 x^2+e^3 \left (-2-12 x^2\right )\right )+\left (32-16 e^3+2 e^6+e^{2+x^2} \left (16 x^2-4 e^3 x^2\right )\right ) \log (x)}{16 x-8 e^3 x+e^6 x} \, dx=x + \frac {\left (- 2 e^{6} \log {\left (x \right )} - 32 \log {\left (x \right )} + 16 e^{3} \log {\left (x \right )} - 6 e^{6} - 96 + 48 e^{3}\right ) e^{x^{2} + 2} + \left (-4 + e^{3}\right ) e^{2 x^{2} + 4}}{- 12 e^{6} - 64 + 48 e^{3} + e^{9}} + \log {\left (x \right )}^{2} + 6 \log {\left (x \right )} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.32 (sec) , antiderivative size = 457, normalized size of antiderivative = 19.04 \[ \int \frac {96+e^3 (-48-8 x)+16 x+4 e^{4+2 x^2} x^2+e^6 (6+x)+e^{2+x^2} \left (8+48 x^2+e^3 \left (-2-12 x^2\right )\right )+\left (32-16 e^3+2 e^6+e^{2+x^2} \left (16 x^2-4 e^3 x^2\right )\right ) \log (x)}{16 x-8 e^3 x+e^6 x} \, dx=-{\left (\frac {2 \, \log \left (x e^{6} - 8 \, x e^{3} + 16 \, x\right ) \log \left (x\right )}{e^{6} - 8 \, e^{3} + 16} - \frac {\log \left (x\right )^{2}}{e^{6} - 8 \, e^{3} + 16}\right )} e^{6} + 8 \, {\left (\frac {2 \, \log \left (x e^{6} - 8 \, x e^{3} + 16 \, x\right ) \log \left (x\right )}{e^{6} - 8 \, e^{3} + 16} - \frac {\log \left (x\right )^{2}}{e^{6} - 8 \, e^{3} + 16}\right )} e^{3} + \frac {2 \, e^{6} \log \left (x e^{6} - 8 \, x e^{3} + 16 \, x\right ) \log \left (x\right )}{e^{6} - 8 \, e^{3} + 16} - \frac {16 \, e^{3} \log \left (x e^{6} - 8 \, x e^{3} + 16 \, x\right ) \log \left (x\right )}{e^{6} - 8 \, e^{3} + 16} + \frac {x e^{6}}{e^{6} - 8 \, e^{3} + 16} + \frac {{\rm Ei}\left (x^{2}\right ) e^{5}}{e^{6} - 8 \, e^{3} + 16} - \frac {8 \, x e^{3}}{e^{6} - 8 \, e^{3} + 16} - \frac {4 \, {\rm Ei}\left (x^{2}\right ) e^{2}}{e^{6} - 8 \, e^{3} + 16} + \frac {6 \, e^{6} \log \left (x e^{6} - 8 \, x e^{3} + 16 \, x\right )}{e^{6} - 8 \, e^{3} + 16} - \frac {48 \, e^{3} \log \left (x e^{6} - 8 \, x e^{3} + 16 \, x\right )}{e^{6} - 8 \, e^{3} + 16} - \frac {2 \, e^{\left (x^{2} + 5\right )} \log \left (x\right )}{e^{6} - 8 \, e^{3} + 16} + \frac {8 \, e^{\left (x^{2} + 2\right )} \log \left (x\right )}{e^{6} - 8 \, e^{3} + 16} + \frac {16 \, \log \left (x\right )^{2}}{e^{6} - 8 \, e^{3} + 16} + \frac {16 \, x}{e^{6} - 8 \, e^{3} + 16} - \frac {{\rm Ei}\left (x^{2}\right ) e^{5}}{{\left (e^{3} - 4\right )}^{2}} + \frac {4 \, {\rm Ei}\left (x^{2}\right ) e^{2}}{{\left (e^{3} - 4\right )}^{2}} + \frac {e^{\left (2 \, x^{2} + 4\right )}}{e^{6} - 8 \, e^{3} + 16} - \frac {6 \, e^{\left (x^{2} + 5\right )}}{e^{6} - 8 \, e^{3} + 16} + \frac {24 \, e^{\left (x^{2} + 2\right )}}{e^{6} - 8 \, e^{3} + 16} + \frac {96 \, \log \left (x e^{6} - 8 \, x e^{3} + 16 \, x\right )}{e^{6} - 8 \, e^{3} + 16} \]
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Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (23) = 46\).
Time = 0.27 (sec) , antiderivative size = 105, normalized size of antiderivative = 4.38 \[ \int \frac {96+e^3 (-48-8 x)+16 x+4 e^{4+2 x^2} x^2+e^6 (6+x)+e^{2+x^2} \left (8+48 x^2+e^3 \left (-2-12 x^2\right )\right )+\left (32-16 e^3+2 e^6+e^{2+x^2} \left (16 x^2-4 e^3 x^2\right )\right ) \log (x)}{16 x-8 e^3 x+e^6 x} \, dx=\frac {e^{6} \log \left (x\right )^{2} - 8 \, e^{3} \log \left (x\right )^{2} + x e^{6} - 8 \, x e^{3} + 6 \, e^{6} \log \left (x\right ) - 48 \, e^{3} \log \left (x\right ) - 2 \, e^{\left (x^{2} + 5\right )} \log \left (x\right ) + 8 \, e^{\left (x^{2} + 2\right )} \log \left (x\right ) + 16 \, \log \left (x\right )^{2} + 16 \, x + e^{\left (2 \, x^{2} + 4\right )} - 6 \, e^{\left (x^{2} + 5\right )} + 24 \, e^{\left (x^{2} + 2\right )} + 96 \, \log \left (x\right )}{e^{6} - 8 \, e^{3} + 16} \]
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Time = 11.62 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.17 \[ \int \frac {96+e^3 (-48-8 x)+16 x+4 e^{4+2 x^2} x^2+e^6 (6+x)+e^{2+x^2} \left (8+48 x^2+e^3 \left (-2-12 x^2\right )\right )+\left (32-16 e^3+2 e^6+e^{2+x^2} \left (16 x^2-4 e^3 x^2\right )\right ) \log (x)}{16 x-8 e^3 x+e^6 x} \, dx=x+6\,\ln \left (x\right )+{\ln \left (x\right )}^2-{\mathrm {e}}^{x^2+2}\,\left (\frac {6}{{\mathrm {e}}^3-4}+\frac {2\,\ln \left (x\right )}{{\mathrm {e}}^3-4}\right )+\frac {{\mathrm {e}}^{2\,x^2+4}}{{\left ({\mathrm {e}}^3-4\right )}^2} \]
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