Integrand size = 253, antiderivative size = 34 \[ \int \frac {24 e^{4+2 x}+8 e^x x^2+\left (4 e^x x^2+e^{2 x} \left (12 e^4 x+21 x^2-3 e x^2\right )\right ) \log \left (\frac {e^{-2 x} \left (16 x^2+e^x \left (96 e^4 x+168 x^2-24 e x^2\right )+e^{2 x} \left (144 e^8+441 x^2-126 e x^2+9 e^2 x^2+e^4 (504 x-72 e x)\right )\right )}{9 x^2}\right )}{\left (4 x^2+e^x \left (12 e^4 x+21 x^2-3 e x^2\right )\right ) \log ^2\left (\frac {e^{-2 x} \left (16 x^2+e^x \left (96 e^4 x+168 x^2-24 e x^2\right )+e^{2 x} \left (144 e^8+441 x^2-126 e x^2+9 e^2 x^2+e^4 (504 x-72 e x)\right )\right )}{9 x^2}\right )} \, dx=\frac {e^x}{\log \left (\left (3-e+\frac {4 \left (e^4+x+\frac {e^{-x} x}{3}\right )}{x}\right )^2\right )} \]
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Leaf count is larger than twice the leaf count of optimal. \(132\) vs. \(2(34)=68\).
Time = 0.67 (sec) , antiderivative size = 132, normalized size of antiderivative = 3.88, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.008, Rules used = {6820, 2326} \[ \int \frac {24 e^{4+2 x}+8 e^x x^2+\left (4 e^x x^2+e^{2 x} \left (12 e^4 x+21 x^2-3 e x^2\right )\right ) \log \left (\frac {e^{-2 x} \left (16 x^2+e^x \left (96 e^4 x+168 x^2-24 e x^2\right )+e^{2 x} \left (144 e^8+441 x^2-126 e x^2+9 e^2 x^2+e^4 (504 x-72 e x)\right )\right )}{9 x^2}\right )}{\left (4 x^2+e^x \left (12 e^4 x+21 x^2-3 e x^2\right )\right ) \log ^2\left (\frac {e^{-2 x} \left (16 x^2+e^x \left (96 e^4 x+168 x^2-24 e x^2\right )+e^{2 x} \left (144 e^8+441 x^2-126 e x^2+9 e^2 x^2+e^4 (504 x-72 e x)\right )\right )}{9 x^2}\right )} \, dx=\frac {e^x \left (21 e^x x-3 e^{x+1} x+4 x+12 e^{x+4}\right ) \log \left (\frac {e^{-2 x} \left (21 e^x x-3 e^{x+1} x+4 x+12 e^{x+4}\right )^2}{9 x^2}\right )}{\left (3 (7-e) e^x x+4 x+12 e^{x+4}\right ) \log ^2\left (\frac {e^{-2 x} \left (3 (7-e) e^x x+4 x+12 e^{x+4}\right )^2}{9 x^2}\right )} \]
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Rule 2326
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \frac {e^x \left (24 e^{4+x}+8 x^2+x \left (12 e^{4+x}+4 x+21 e^x x-3 e^{1+x} x\right ) \log \left (\frac {e^{-2 x} \left (12 e^{4+x}+4 x+21 e^x x-3 e^{1+x} x\right )^2}{9 x^2}\right )\right )}{x \left (12 e^{4+x}+4 x+21 \left (1-\frac {e}{7}\right ) e^x x\right ) \log ^2\left (\frac {e^{-2 x} \left (12 e^{4+x}+4 x+21 \left (1-\frac {e}{7}\right ) e^x x\right )^2}{9 x^2}\right )} \, dx \\ & = \frac {e^x \left (12 e^{4+x}+4 x+21 e^x x-3 e^{1+x} x\right ) \log \left (\frac {e^{-2 x} \left (12 e^{4+x}+4 x+21 e^x x-3 e^{1+x} x\right )^2}{9 x^2}\right )}{\left (12 e^{4+x}+4 x+3 (7-e) e^x x\right ) \log ^2\left (\frac {e^{-2 x} \left (12 e^{4+x}+4 x+3 (7-e) e^x x\right )^2}{9 x^2}\right )} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.35 \[ \int \frac {24 e^{4+2 x}+8 e^x x^2+\left (4 e^x x^2+e^{2 x} \left (12 e^4 x+21 x^2-3 e x^2\right )\right ) \log \left (\frac {e^{-2 x} \left (16 x^2+e^x \left (96 e^4 x+168 x^2-24 e x^2\right )+e^{2 x} \left (144 e^8+441 x^2-126 e x^2+9 e^2 x^2+e^4 (504 x-72 e x)\right )\right )}{9 x^2}\right )}{\left (4 x^2+e^x \left (12 e^4 x+21 x^2-3 e x^2\right )\right ) \log ^2\left (\frac {e^{-2 x} \left (16 x^2+e^x \left (96 e^4 x+168 x^2-24 e x^2\right )+e^{2 x} \left (144 e^8+441 x^2-126 e x^2+9 e^2 x^2+e^4 (504 x-72 e x)\right )\right )}{9 x^2}\right )} \, dx=\frac {e^x}{\log \left (\frac {e^{-2 x} \left (12 e^{4+x}+4 x+21 e^x x-3 e^{1+x} x\right )^2}{9 x^2}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(87\) vs. \(2(30)=60\).
Time = 1.66 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.59
method | result | size |
parallelrisch | \(\frac {{\mathrm e}^{x}}{\ln \left (\frac {\left (\left (144 \,{\mathrm e}^{8}+\left (-72 x \,{\mathrm e}+504 x \right ) {\mathrm e}^{4}+9 x^{2} {\mathrm e}^{2}-126 x^{2} {\mathrm e}+441 x^{2}\right ) {\mathrm e}^{2 x}+\left (96 x \,{\mathrm e}^{4}-24 x^{2} {\mathrm e}+168 x^{2}\right ) {\mathrm e}^{x}+16 x^{2}\right ) {\mathrm e}^{-2 x}}{9 x^{2}}\right )}\) | \(88\) |
risch | \(\text {Expression too large to display}\) | \(709\) |
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Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (35) = 70\).
Time = 0.26 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.38 \[ \int \frac {24 e^{4+2 x}+8 e^x x^2+\left (4 e^x x^2+e^{2 x} \left (12 e^4 x+21 x^2-3 e x^2\right )\right ) \log \left (\frac {e^{-2 x} \left (16 x^2+e^x \left (96 e^4 x+168 x^2-24 e x^2\right )+e^{2 x} \left (144 e^8+441 x^2-126 e x^2+9 e^2 x^2+e^4 (504 x-72 e x)\right )\right )}{9 x^2}\right )}{\left (4 x^2+e^x \left (12 e^4 x+21 x^2-3 e x^2\right )\right ) \log ^2\left (\frac {e^{-2 x} \left (16 x^2+e^x \left (96 e^4 x+168 x^2-24 e x^2\right )+e^{2 x} \left (144 e^8+441 x^2-126 e x^2+9 e^2 x^2+e^4 (504 x-72 e x)\right )\right )}{9 x^2}\right )} \, dx=\frac {e^{x}}{\log \left (\frac {{\left (16 \, x^{2} + 9 \, {\left (x^{2} e^{2} - 14 \, x^{2} e + 49 \, x^{2} - 8 \, x e^{5} + 56 \, x e^{4} + 16 \, e^{8}\right )} e^{\left (2 \, x\right )} - 24 \, {\left (x^{2} e - 7 \, x^{2} - 4 \, x e^{4}\right )} e^{x}\right )} e^{\left (-2 \, x\right )}}{9 \, x^{2}}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (29) = 58\).
Time = 0.21 (sec) , antiderivative size = 92, normalized size of antiderivative = 2.71 \[ \int \frac {24 e^{4+2 x}+8 e^x x^2+\left (4 e^x x^2+e^{2 x} \left (12 e^4 x+21 x^2-3 e x^2\right )\right ) \log \left (\frac {e^{-2 x} \left (16 x^2+e^x \left (96 e^4 x+168 x^2-24 e x^2\right )+e^{2 x} \left (144 e^8+441 x^2-126 e x^2+9 e^2 x^2+e^4 (504 x-72 e x)\right )\right )}{9 x^2}\right )}{\left (4 x^2+e^x \left (12 e^4 x+21 x^2-3 e x^2\right )\right ) \log ^2\left (\frac {e^{-2 x} \left (16 x^2+e^x \left (96 e^4 x+168 x^2-24 e x^2\right )+e^{2 x} \left (144 e^8+441 x^2-126 e x^2+9 e^2 x^2+e^4 (504 x-72 e x)\right )\right )}{9 x^2}\right )} \, dx=\frac {e^{x}}{\log {\left (\frac {\left (\frac {16 x^{2}}{9} + \frac {\left (- 24 e x^{2} + 168 x^{2} + 96 x e^{4}\right ) e^{x}}{9} + \frac {\left (- 126 e x^{2} + 9 x^{2} e^{2} + 441 x^{2} + \left (- 72 e x + 504 x\right ) e^{4} + 144 e^{8}\right ) e^{2 x}}{9}\right ) e^{- 2 x}}{x^{2}} \right )}} \]
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Time = 0.36 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {24 e^{4+2 x}+8 e^x x^2+\left (4 e^x x^2+e^{2 x} \left (12 e^4 x+21 x^2-3 e x^2\right )\right ) \log \left (\frac {e^{-2 x} \left (16 x^2+e^x \left (96 e^4 x+168 x^2-24 e x^2\right )+e^{2 x} \left (144 e^8+441 x^2-126 e x^2+9 e^2 x^2+e^4 (504 x-72 e x)\right )\right )}{9 x^2}\right )}{\left (4 x^2+e^x \left (12 e^4 x+21 x^2-3 e x^2\right )\right ) \log ^2\left (\frac {e^{-2 x} \left (16 x^2+e^x \left (96 e^4 x+168 x^2-24 e x^2\right )+e^{2 x} \left (144 e^8+441 x^2-126 e x^2+9 e^2 x^2+e^4 (504 x-72 e x)\right )\right )}{9 x^2}\right )} \, dx=-\frac {e^{x}}{2 \, {\left (x + \log \left (3\right ) - \log \left (3 \, {\left (x {\left (e - 7\right )} - 4 \, e^{4}\right )} e^{x} - 4 \, x\right ) + \log \left (x\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (35) = 70\).
Time = 4.25 (sec) , antiderivative size = 107, normalized size of antiderivative = 3.15 \[ \int \frac {24 e^{4+2 x}+8 e^x x^2+\left (4 e^x x^2+e^{2 x} \left (12 e^4 x+21 x^2-3 e x^2\right )\right ) \log \left (\frac {e^{-2 x} \left (16 x^2+e^x \left (96 e^4 x+168 x^2-24 e x^2\right )+e^{2 x} \left (144 e^8+441 x^2-126 e x^2+9 e^2 x^2+e^4 (504 x-72 e x)\right )\right )}{9 x^2}\right )}{\left (4 x^2+e^x \left (12 e^4 x+21 x^2-3 e x^2\right )\right ) \log ^2\left (\frac {e^{-2 x} \left (16 x^2+e^x \left (96 e^4 x+168 x^2-24 e x^2\right )+e^{2 x} \left (144 e^8+441 x^2-126 e x^2+9 e^2 x^2+e^4 (504 x-72 e x)\right )\right )}{9 x^2}\right )} \, dx=-\frac {e^{x}}{\log \left (9 \, x^{2}\right ) - \log \left ({\left (441 \, x^{2} e^{\left (2 \, x\right )} + 9 \, x^{2} e^{\left (2 \, x + 2\right )} - 126 \, x^{2} e^{\left (2 \, x + 1\right )} - 24 \, x^{2} e^{\left (x + 1\right )} + 168 \, x^{2} e^{x} + 16 \, x^{2} - 72 \, x e^{\left (2 \, x + 5\right )} + 504 \, x e^{\left (2 \, x + 4\right )} + 96 \, x e^{\left (x + 4\right )} + 144 \, e^{\left (2 \, x + 8\right )}\right )} e^{\left (-2 \, x\right )}\right )} \]
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Time = 8.87 (sec) , antiderivative size = 326, normalized size of antiderivative = 9.59 \[ \int \frac {24 e^{4+2 x}+8 e^x x^2+\left (4 e^x x^2+e^{2 x} \left (12 e^4 x+21 x^2-3 e x^2\right )\right ) \log \left (\frac {e^{-2 x} \left (16 x^2+e^x \left (96 e^4 x+168 x^2-24 e x^2\right )+e^{2 x} \left (144 e^8+441 x^2-126 e x^2+9 e^2 x^2+e^4 (504 x-72 e x)\right )\right )}{9 x^2}\right )}{\left (4 x^2+e^x \left (12 e^4 x+21 x^2-3 e x^2\right )\right ) \log ^2\left (\frac {e^{-2 x} \left (16 x^2+e^x \left (96 e^4 x+168 x^2-24 e x^2\right )+e^{2 x} \left (144 e^8+441 x^2-126 e x^2+9 e^2 x^2+e^4 (504 x-72 e x)\right )\right )}{9 x^2}\right )} \, dx=\frac {{\mathrm {e}}^x+\frac {x\,{\mathrm {e}}^x\,\ln \left (\frac {{\mathrm {e}}^{-2\,x}\,\left (\frac {{\mathrm {e}}^{2\,x}\,\left (144\,{\mathrm {e}}^8+{\mathrm {e}}^4\,\left (504\,x-72\,x\,\mathrm {e}\right )-126\,x^2\,\mathrm {e}+9\,x^2\,{\mathrm {e}}^2+441\,x^2\right )}{9}+\frac {{\mathrm {e}}^x\,\left (96\,x\,{\mathrm {e}}^4-24\,x^2\,\mathrm {e}+168\,x^2\right )}{9}+\frac {16\,x^2}{9}\right )}{x^2}\right )\,\left (4\,x+12\,{\mathrm {e}}^{x+4}-3\,x\,{\mathrm {e}}^{x+1}+21\,x\,{\mathrm {e}}^x\right )}{8\,\left (3\,{\mathrm {e}}^{x+4}+x^2\right )}}{\ln \left (\frac {{\mathrm {e}}^{-2\,x}\,\left (\frac {{\mathrm {e}}^{2\,x}\,\left (144\,{\mathrm {e}}^8+{\mathrm {e}}^4\,\left (504\,x-72\,x\,\mathrm {e}\right )-126\,x^2\,\mathrm {e}+9\,x^2\,{\mathrm {e}}^2+441\,x^2\right )}{9}+\frac {{\mathrm {e}}^x\,\left (96\,x\,{\mathrm {e}}^4-24\,x^2\,\mathrm {e}+168\,x^2\right )}{9}+\frac {16\,x^2}{9}\right )}{x^2}\right )}-\frac {x^2\,{\mathrm {e}}^{-4}}{6}+\frac {x^3\,{\mathrm {e}}^{-4}}{6}-{\mathrm {e}}^x\,\left (\frac {x}{2}-\frac {x^2\,{\mathrm {e}}^{-4}\,\left (\mathrm {e}-7\right )}{8}\right )-\frac {x^4\,{\mathrm {e}}^{-8}\,\left (\mathrm {e}-7\right )}{24}+\frac {{\mathrm {e}}^{-8}\,\left (2\,x^7\,\mathrm {e}+8\,x^5\,{\mathrm {e}}^4-x^8\,\mathrm {e}-12\,x^6\,{\mathrm {e}}^4+4\,x^7\,{\mathrm {e}}^4-14\,x^7+7\,x^8\right )}{72\,\left (2\,x\,{\mathrm {e}}^4-x^2\,{\mathrm {e}}^4\right )\,\left ({\mathrm {e}}^x+\frac {x^2\,{\mathrm {e}}^{-4}}{3}\right )} \]
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