Integrand size = 49, antiderivative size = 21 \[ \int \frac {25-100 e^{-1+4 x}+e^{e^{-1+4 x}} \left (-6+2 x+x^2\right )}{25+e^{e^{-1+4 x}} \left (-6+x^2\right )} \, dx=x+\log \left (-6+25 e^{-e^{-1+4 x}}+x^2\right ) \]
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\[ \int \frac {25-100 e^{-1+4 x}+e^{e^{-1+4 x}} \left (-6+2 x+x^2\right )}{25+e^{e^{-1+4 x}} \left (-6+x^2\right )} \, dx=\int \frac {25-100 e^{-1+4 x}+e^{e^{-1+4 x}} \left (-6+2 x+x^2\right )}{25+e^{e^{-1+4 x}} \left (-6+x^2\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {100 e^{-1+4 x}}{25-6 e^{e^{-1+4 x}}+e^{e^{-1+4 x}} x^2}+\frac {25-6 e^{e^{-1+4 x}}+2 e^{e^{-1+4 x}} x+e^{e^{-1+4 x}} x^2}{25-6 e^{e^{-1+4 x}}+e^{e^{-1+4 x}} x^2}\right ) \, dx \\ & = -\left (100 \int \frac {e^{-1+4 x}}{25-6 e^{e^{-1+4 x}}+e^{e^{-1+4 x}} x^2} \, dx\right )+\int \frac {25-6 e^{e^{-1+4 x}}+2 e^{e^{-1+4 x}} x+e^{e^{-1+4 x}} x^2}{25-6 e^{e^{-1+4 x}}+e^{e^{-1+4 x}} x^2} \, dx \\ & = -\left (100 \int \frac {e^{-1+4 x}}{25+e^{e^{-1+4 x}} \left (-6+x^2\right )} \, dx\right )+\int \frac {25+e^{e^{-1+4 x}} \left (-6+2 x+x^2\right )}{25+e^{e^{-1+4 x}} \left (-6+x^2\right )} \, dx \\ & = -\left (100 \int \frac {e^{-1+4 x}}{25+e^{e^{-1+4 x}} \left (-6+x^2\right )} \, dx\right )+\int \left (\frac {-6+2 x+x^2}{-6+x^2}-\frac {50 x}{\left (-6+x^2\right ) \left (25-6 e^{e^{-1+4 x}}+e^{e^{-1+4 x}} x^2\right )}\right ) \, dx \\ & = -\left (50 \int \frac {x}{\left (-6+x^2\right ) \left (25-6 e^{e^{-1+4 x}}+e^{e^{-1+4 x}} x^2\right )} \, dx\right )-100 \int \frac {e^{-1+4 x}}{25+e^{e^{-1+4 x}} \left (-6+x^2\right )} \, dx+\int \frac {-6+2 x+x^2}{-6+x^2} \, dx \\ & = -\left (50 \int \frac {x}{\left (-6+x^2\right ) \left (25+e^{e^{-1+4 x}} \left (-6+x^2\right )\right )} \, dx\right )-100 \int \frac {e^{-1+4 x}}{25+e^{e^{-1+4 x}} \left (-6+x^2\right )} \, dx+\int \left (1+\frac {2 x}{-6+x^2}\right ) \, dx \\ & = x+2 \int \frac {x}{-6+x^2} \, dx-50 \int \left (-\frac {1}{2 \left (\sqrt {6}-x\right ) \left (25+e^{e^{-1+4 x}} \left (-6+x^2\right )\right )}+\frac {1}{2 \left (\sqrt {6}+x\right ) \left (25+e^{e^{-1+4 x}} \left (-6+x^2\right )\right )}\right ) \, dx-100 \int \frac {e^{-1+4 x}}{25+e^{e^{-1+4 x}} \left (-6+x^2\right )} \, dx \\ & = x+\log \left (6-x^2\right )+25 \int \frac {1}{\left (\sqrt {6}-x\right ) \left (25+e^{e^{-1+4 x}} \left (-6+x^2\right )\right )} \, dx-25 \int \frac {1}{\left (\sqrt {6}+x\right ) \left (25+e^{e^{-1+4 x}} \left (-6+x^2\right )\right )} \, dx-100 \int \frac {e^{-1+4 x}}{25+e^{e^{-1+4 x}} \left (-6+x^2\right )} \, dx \\ \end{align*}
Time = 1.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.38 \[ \int \frac {25-100 e^{-1+4 x}+e^{e^{-1+4 x}} \left (-6+2 x+x^2\right )}{25+e^{e^{-1+4 x}} \left (-6+x^2\right )} \, dx=-e^{-1+4 x}+x+\log \left (25+e^{e^{-1+4 x}} \left (-6+x^2\right )\right ) \]
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Time = 0.17 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.62
method | result | size |
norman | \(x -{\mathrm e}^{-1+4 x}+\ln \left ({\mathrm e}^{{\mathrm e}^{-1+4 x}} x^{2}-6 \,{\mathrm e}^{{\mathrm e}^{-1+4 x}}+25\right )\) | \(34\) |
parallelrisch | \(x -{\mathrm e}^{-1+4 x}+\ln \left ({\mathrm e}^{{\mathrm e}^{-1+4 x}} x^{2}-6 \,{\mathrm e}^{{\mathrm e}^{-1+4 x}}+25\right )\) | \(34\) |
risch | \(x +\ln \left (x^{2}-6\right )-{\mathrm e}^{-1+4 x}+\ln \left ({\mathrm e}^{{\mathrm e}^{-1+4 x}}+\frac {25}{x^{2}-6}\right )\) | \(35\) |
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Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (19) = 38\).
Time = 0.28 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.90 \[ \int \frac {25-100 e^{-1+4 x}+e^{e^{-1+4 x}} \left (-6+2 x+x^2\right )}{25+e^{e^{-1+4 x}} \left (-6+x^2\right )} \, dx=x - e^{\left (4 \, x - 1\right )} + \log \left (x^{2} - 6\right ) + \log \left (\frac {{\left (x^{2} - 6\right )} e^{\left (e^{\left (4 \, x - 1\right )}\right )} + 25}{x^{2} - 6}\right ) \]
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Time = 0.21 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.48 \[ \int \frac {25-100 e^{-1+4 x}+e^{e^{-1+4 x}} \left (-6+2 x+x^2\right )}{25+e^{e^{-1+4 x}} \left (-6+x^2\right )} \, dx=x - e^{4 x - 1} + \log {\left (x^{2} - 6 \right )} + \log {\left (e^{e^{4 x - 1}} + \frac {25}{x^{2} - 6} \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (19) = 38\).
Time = 0.22 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.14 \[ \int \frac {25-100 e^{-1+4 x}+e^{e^{-1+4 x}} \left (-6+2 x+x^2\right )}{25+e^{e^{-1+4 x}} \left (-6+x^2\right )} \, dx={\left (x e - e^{\left (4 \, x\right )}\right )} e^{\left (-1\right )} + \log \left (x^{2} - 6\right ) + \log \left (\frac {{\left (x^{2} - 6\right )} e^{\left (e^{\left (4 \, x - 1\right )}\right )} + 25}{x^{2} - 6}\right ) \]
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\[ \int \frac {25-100 e^{-1+4 x}+e^{e^{-1+4 x}} \left (-6+2 x+x^2\right )}{25+e^{e^{-1+4 x}} \left (-6+x^2\right )} \, dx=\int { \frac {{\left (x^{2} + 2 \, x - 6\right )} e^{\left (e^{\left (4 \, x - 1\right )}\right )} - 100 \, e^{\left (4 \, x - 1\right )} + 25}{{\left (x^{2} - 6\right )} e^{\left (e^{\left (4 \, x - 1\right )}\right )} + 25} \,d x } \]
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Time = 9.71 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.57 \[ \int \frac {25-100 e^{-1+4 x}+e^{e^{-1+4 x}} \left (-6+2 x+x^2\right )}{25+e^{e^{-1+4 x}} \left (-6+x^2\right )} \, dx=x+\ln \left (x^2\,{\mathrm {e}}^{{\mathrm {e}}^{4\,x-1}}-6\,{\mathrm {e}}^{{\mathrm {e}}^{4\,x-1}}+25\right )-{\mathrm {e}}^{4\,x-1} \]
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