Integrand size = 96, antiderivative size = 22 \[ \int \frac {3-3 x+e^x \left (-9 x-28 x^2+23 x^3+14 x^4\right )+\left (e^x \left (9 x+14 x^2\right )-3 \log (x)\right ) \log \left (\frac {1}{3} \left (e^x \left (-9 x-14 x^2\right )+3 \log (x)\right )\right )}{e^x \left (-9 x^3-14 x^4\right )+3 x^2 \log (x)} \, dx=\left (-1+\frac {1}{x}\right ) \log \left (e^x \left (-3-\frac {14 x}{3}\right ) x+\log (x)\right ) \]
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\[ \int \frac {3-3 x+e^x \left (-9 x-28 x^2+23 x^3+14 x^4\right )+\left (e^x \left (9 x+14 x^2\right )-3 \log (x)\right ) \log \left (\frac {1}{3} \left (e^x \left (-9 x-14 x^2\right )+3 \log (x)\right )\right )}{e^x \left (-9 x^3-14 x^4\right )+3 x^2 \log (x)} \, dx=\int \frac {3-3 x+e^x \left (-9 x-28 x^2+23 x^3+14 x^4\right )+\left (e^x \left (9 x+14 x^2\right )-3 \log (x)\right ) \log \left (\frac {1}{3} \left (e^x \left (-9 x-14 x^2\right )+3 \log (x)\right )\right )}{e^x \left (-9 x^3-14 x^4\right )+3 x^2 \log (x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {3 (-1+x) \left (-9-14 x+9 \log (x)+37 x \log (x)+14 x^2 \log (x)\right )}{x^2 (9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}+\frac {9+28 x-23 x^2-14 x^3-9 \log \left (-\frac {1}{3} e^x x (9+14 x)+\log (x)\right )-14 x \log \left (-\frac {1}{3} e^x x (9+14 x)+\log (x)\right )}{x^2 (9+14 x)}\right ) \, dx \\ & = -\left (3 \int \frac {(-1+x) \left (-9-14 x+9 \log (x)+37 x \log (x)+14 x^2 \log (x)\right )}{x^2 (9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx\right )+\int \frac {9+28 x-23 x^2-14 x^3-9 \log \left (-\frac {1}{3} e^x x (9+14 x)+\log (x)\right )-14 x \log \left (-\frac {1}{3} e^x x (9+14 x)+\log (x)\right )}{x^2 (9+14 x)} \, dx \\ & = -\left (3 \int \left (-\frac {-9-14 x+9 \log (x)+37 x \log (x)+14 x^2 \log (x)}{9 x^2 \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}+\frac {23 \left (-9-14 x+9 \log (x)+37 x \log (x)+14 x^2 \log (x)\right )}{81 x \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}-\frac {322 \left (-9-14 x+9 \log (x)+37 x \log (x)+14 x^2 \log (x)\right )}{81 (9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}\right ) \, dx\right )+\int \frac {9+28 x-23 x^2-14 x^3-(9+14 x) \log \left (-\frac {1}{3} e^x x (9+14 x)+\log (x)\right )}{x^2 (9+14 x)} \, dx \\ & = \frac {1}{3} \int \frac {-9-14 x+9 \log (x)+37 x \log (x)+14 x^2 \log (x)}{x^2 \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx-\frac {23}{27} \int \frac {-9-14 x+9 \log (x)+37 x \log (x)+14 x^2 \log (x)}{x \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx+\frac {322}{27} \int \frac {-9-14 x+9 \log (x)+37 x \log (x)+14 x^2 \log (x)}{(9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx+\int \left (\frac {9+28 x-23 x^2-14 x^3}{x^2 (9+14 x)}-\frac {\log \left (-\frac {1}{3} e^x x (9+14 x)+\log (x)\right )}{x^2}\right ) \, dx \\ & = \frac {1}{3} \int \left (-\frac {9}{x^2 \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}-\frac {14}{x \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}+\frac {14 \log (x)}{9 e^x x+14 e^x x^2-3 \log (x)}+\frac {9 \log (x)}{x^2 \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}+\frac {37 \log (x)}{x \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}\right ) \, dx-\frac {23}{27} \int \left (-\frac {14}{9 e^x x+14 e^x x^2-3 \log (x)}-\frac {9}{x \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}+\frac {37 \log (x)}{9 e^x x+14 e^x x^2-3 \log (x)}+\frac {9 \log (x)}{x \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}+\frac {14 x \log (x)}{9 e^x x+14 e^x x^2-3 \log (x)}\right ) \, dx+\frac {322}{27} \int \left (-\frac {9}{(9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}-\frac {14 x}{(9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}+\frac {9 \log (x)}{(9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}+\frac {37 x \log (x)}{(9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}+\frac {14 x^2 \log (x)}{(9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}\right ) \, dx+\int \frac {9+28 x-23 x^2-14 x^3}{x^2 (9+14 x)} \, dx-\int \frac {\log \left (-\frac {1}{3} e^x x (9+14 x)+\log (x)\right )}{x^2} \, dx \\ & = -\left (3 \int \frac {1}{x^2 \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx\right )+3 \int \frac {\log (x)}{x^2 \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx-\frac {14}{3} \int \frac {1}{x \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx+\frac {14}{3} \int \frac {\log (x)}{9 e^x x+14 e^x x^2-3 \log (x)} \, dx+\frac {23}{3} \int \frac {1}{x \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx-\frac {23}{3} \int \frac {\log (x)}{x \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx+\frac {322}{27} \int \frac {1}{9 e^x x+14 e^x x^2-3 \log (x)} \, dx-\frac {322}{27} \int \frac {x \log (x)}{9 e^x x+14 e^x x^2-3 \log (x)} \, dx+\frac {37}{3} \int \frac {\log (x)}{x \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx-\frac {851}{27} \int \frac {\log (x)}{9 e^x x+14 e^x x^2-3 \log (x)} \, dx-\frac {322}{3} \int \frac {1}{(9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx+\frac {322}{3} \int \frac {\log (x)}{(9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx-\frac {4508}{27} \int \frac {x}{(9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx+\frac {4508}{27} \int \frac {x^2 \log (x)}{(9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx+\frac {11914}{27} \int \frac {x \log (x)}{(9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx+\int \left (-1+\frac {1}{x^2}+\frac {14}{9 x}-\frac {322}{9 (9+14 x)}\right ) \, dx-\int \frac {\log \left (-\frac {1}{3} e^x x (9+14 x)+\log (x)\right )}{x^2} \, dx \\ & = -\frac {1}{x}-x+\frac {14 \log (x)}{9}-\frac {23}{9} \log (9+14 x)-3 \int \frac {1}{x^2 \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx+3 \int \frac {\log (x)}{x^2 \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx-\frac {14}{3} \int \frac {1}{x \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx+\frac {14}{3} \int \frac {\log (x)}{9 e^x x+14 e^x x^2-3 \log (x)} \, dx+\frac {23}{3} \int \frac {1}{x \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx-\frac {23}{3} \int \frac {\log (x)}{x \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx+\frac {322}{27} \int \frac {1}{9 e^x x+14 e^x x^2-3 \log (x)} \, dx-\frac {322}{27} \int \frac {x \log (x)}{9 e^x x+14 e^x x^2-3 \log (x)} \, dx+\frac {37}{3} \int \frac {\log (x)}{x \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx-\frac {851}{27} \int \frac {\log (x)}{9 e^x x+14 e^x x^2-3 \log (x)} \, dx-\frac {322}{3} \int \frac {1}{(9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx+\frac {322}{3} \int \frac {\log (x)}{(9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx-\frac {4508}{27} \int \left (\frac {1}{14 \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}-\frac {9}{14 (9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}\right ) \, dx+\frac {4508}{27} \int \left (-\frac {9 \log (x)}{196 \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}+\frac {x \log (x)}{14 \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}+\frac {81 \log (x)}{196 (9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}\right ) \, dx+\frac {11914}{27} \int \left (\frac {\log (x)}{14 \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}-\frac {9 \log (x)}{14 (9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}\right ) \, dx-\int \frac {\log \left (-\frac {1}{3} e^x x (9+14 x)+\log (x)\right )}{x^2} \, dx \\ & = -\frac {1}{x}-x+\frac {14 \log (x)}{9}-\frac {23}{9} \log (9+14 x)-3 \int \frac {1}{x^2 \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx+3 \int \frac {\log (x)}{x^2 \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx-\frac {14}{3} \int \frac {1}{x \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx+\frac {14}{3} \int \frac {\log (x)}{9 e^x x+14 e^x x^2-3 \log (x)} \, dx+\frac {23}{3} \int \frac {1}{x \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx-\frac {23}{3} \int \frac {\log (x)}{9 e^x x+14 e^x x^2-3 \log (x)} \, dx-\frac {23}{3} \int \frac {\log (x)}{x \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx+\frac {37}{3} \int \frac {\log (x)}{x \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx+69 \int \frac {\log (x)}{(9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx+\frac {322}{3} \int \frac {\log (x)}{(9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx-\frac {851}{3} \int \frac {\log (x)}{(9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx-\int \frac {\log \left (-\frac {1}{3} e^x x (9+14 x)+\log (x)\right )}{x^2} \, dx \\ \end{align*}
Time = 0.97 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.00 \[ \int \frac {3-3 x+e^x \left (-9 x-28 x^2+23 x^3+14 x^4\right )+\left (e^x \left (9 x+14 x^2\right )-3 \log (x)\right ) \log \left (\frac {1}{3} \left (e^x \left (-9 x-14 x^2\right )+3 \log (x)\right )\right )}{e^x \left (-9 x^3-14 x^4\right )+3 x^2 \log (x)} \, dx=-\log \left (9 e^x x+14 e^x x^2-3 \log (x)\right )+\frac {\log \left (-\frac {1}{3} e^x x (9+14 x)+\log (x)\right )}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(40\) vs. \(2(19)=38\).
Time = 4.12 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.86
method | result | size |
risch | \(\frac {\ln \left (\ln \left (x \right )+\frac {\left (-14 x^{2}-9 x \right ) {\mathrm e}^{x}}{3}\right )}{x}-\ln \left (\ln \left (x \right )-\frac {14 \,{\mathrm e}^{x} x^{2}}{3}-3 \,{\mathrm e}^{x} x \right )\) | \(41\) |
parallelrisch | \(\frac {420 \ln \left ({\mathrm e}^{x} x^{2}+\frac {9 \,{\mathrm e}^{x} x}{14}-\frac {3 \ln \left (x \right )}{14}\right ) x -1176 \ln \left (\ln \left (x \right )+\frac {\left (-14 x^{2}-9 x \right ) {\mathrm e}^{x}}{3}\right ) x +756 \ln \left (\ln \left (x \right )+\frac {\left (-14 x^{2}-9 x \right ) {\mathrm e}^{x}}{3}\right )}{756 x}\) | \(66\) |
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Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {3-3 x+e^x \left (-9 x-28 x^2+23 x^3+14 x^4\right )+\left (e^x \left (9 x+14 x^2\right )-3 \log (x)\right ) \log \left (\frac {1}{3} \left (e^x \left (-9 x-14 x^2\right )+3 \log (x)\right )\right )}{e^x \left (-9 x^3-14 x^4\right )+3 x^2 \log (x)} \, dx=-\frac {{\left (x - 1\right )} \log \left (-\frac {1}{3} \, {\left (14 \, x^{2} + 9 \, x\right )} e^{x} + \log \left (x\right )\right )}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (22) = 44\).
Time = 0.72 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.23 \[ \int \frac {3-3 x+e^x \left (-9 x-28 x^2+23 x^3+14 x^4\right )+\left (e^x \left (9 x+14 x^2\right )-3 \log (x)\right ) \log \left (\frac {1}{3} \left (e^x \left (-9 x-14 x^2\right )+3 \log (x)\right )\right )}{e^x \left (-9 x^3-14 x^4\right )+3 x^2 \log (x)} \, dx=- \log {\left (14 x^{2} + 9 x \right )} - \log {\left (e^{x} - \frac {3 \log {\left (x \right )}}{14 x^{2} + 9 x} \right )} + \frac {\log {\left (\left (- \frac {14 x^{2}}{3} - 3 x\right ) e^{x} + \log {\left (x \right )} \right )}}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (20) = 40\).
Time = 0.31 (sec) , antiderivative size = 74, normalized size of antiderivative = 3.36 \[ \int \frac {3-3 x+e^x \left (-9 x-28 x^2+23 x^3+14 x^4\right )+\left (e^x \left (9 x+14 x^2\right )-3 \log (x)\right ) \log \left (\frac {1}{3} \left (e^x \left (-9 x-14 x^2\right )+3 \log (x)\right )\right )}{e^x \left (-9 x^3-14 x^4\right )+3 x^2 \log (x)} \, dx=-\frac {\log \left (3\right ) - \log \left (-{\left (14 \, x^{2} + 9 \, x\right )} e^{x} + 3 \, \log \left (x\right )\right )}{x} - \log \left (14 \, x + 9\right ) - \log \left (x\right ) - \log \left (\frac {{\left (14 \, x^{2} + 9 \, x\right )} e^{x} - 3 \, \log \left (x\right )}{14 \, x^{2} + 9 \, x}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (20) = 40\).
Time = 0.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.18 \[ \int \frac {3-3 x+e^x \left (-9 x-28 x^2+23 x^3+14 x^4\right )+\left (e^x \left (9 x+14 x^2\right )-3 \log (x)\right ) \log \left (\frac {1}{3} \left (e^x \left (-9 x-14 x^2\right )+3 \log (x)\right )\right )}{e^x \left (-9 x^3-14 x^4\right )+3 x^2 \log (x)} \, dx=-\frac {x \log \left (14 \, x^{2} e^{x} + 9 \, x e^{x} - 3 \, \log \left (x\right )\right ) + \log \left (3\right ) - \log \left (-14 \, x^{2} e^{x} - 9 \, x e^{x} + 3 \, \log \left (x\right )\right )}{x} \]
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Timed out. \[ \int \frac {3-3 x+e^x \left (-9 x-28 x^2+23 x^3+14 x^4\right )+\left (e^x \left (9 x+14 x^2\right )-3 \log (x)\right ) \log \left (\frac {1}{3} \left (e^x \left (-9 x-14 x^2\right )+3 \log (x)\right )\right )}{e^x \left (-9 x^3-14 x^4\right )+3 x^2 \log (x)} \, dx=\int \frac {3\,x+\ln \left (\ln \left (x\right )-\frac {{\mathrm {e}}^x\,\left (14\,x^2+9\,x\right )}{3}\right )\,\left (3\,\ln \left (x\right )-{\mathrm {e}}^x\,\left (14\,x^2+9\,x\right )\right )+{\mathrm {e}}^x\,\left (-14\,x^4-23\,x^3+28\,x^2+9\,x\right )-3}{{\mathrm {e}}^x\,\left (14\,x^4+9\,x^3\right )-3\,x^2\,\ln \left (x\right )} \,d x \]
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