Integrand size = 116, antiderivative size = 32 \[ \int \frac {2 x+\left (-6+2 e^2-2 x\right ) \log \left (3-e^2+x\right )+\left (-21+e^{4+2 x} \left (-3+e^2-x\right )-10 x-x^2+e^2 (7+x)+e^{2+x} \left (18+e^2 (-6-x)+9 x+x^2\right )\right ) \log ^2\left (3-e^2+x\right )}{\left (-6+2 e^2-2 x\right ) \log ^2\left (3-e^2+x\right )} \, dx=-1+x+\frac {1}{4} \left (5-e^{2+x}+x\right )^2+\frac {x}{\log \left (3-e^2+x\right )} \]
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Leaf count is larger than twice the leaf count of optimal. \(85\) vs. \(2(32)=64\).
Time = 0.37 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.66, number of steps used = 15, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {6820, 2225, 2207, 2458, 2395, 2334, 2335, 2339, 30, 2436} \[ \int \frac {2 x+\left (-6+2 e^2-2 x\right ) \log \left (3-e^2+x\right )+\left (-21+e^{4+2 x} \left (-3+e^2-x\right )-10 x-x^2+e^2 (7+x)+e^{2+x} \left (18+e^2 (-6-x)+9 x+x^2\right )\right ) \log ^2\left (3-e^2+x\right )}{\left (-6+2 e^2-2 x\right ) \log ^2\left (3-e^2+x\right )} \, dx=\frac {x^2}{4}+\frac {7 x}{2}+\frac {e^{x+2}}{2}+\frac {1}{4} e^{2 x+4}-\frac {1}{2} e^{x+2} (x+6)+\frac {x-e^2+3}{\log \left (x-e^2+3\right )}-\frac {3-e^2}{\log \left (x-e^2+3\right )} \]
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Rule 30
Rule 2207
Rule 2225
Rule 2334
Rule 2335
Rule 2339
Rule 2395
Rule 2436
Rule 2458
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{2} \left (7+e^{4+2 x}+x-e^{2+x} (6+x)\right )+\frac {x}{\left (-3+e^2-x\right ) \log ^2\left (3-e^2+x\right )}+\frac {1}{\log \left (3-e^2+x\right )}\right ) \, dx \\ & = \frac {1}{2} \int \left (7+e^{4+2 x}+x-e^{2+x} (6+x)\right ) \, dx+\int \frac {x}{\left (-3+e^2-x\right ) \log ^2\left (3-e^2+x\right )} \, dx+\int \frac {1}{\log \left (3-e^2+x\right )} \, dx \\ & = \frac {7 x}{2}+\frac {x^2}{4}+\frac {1}{2} \int e^{4+2 x} \, dx-\frac {1}{2} \int e^{2+x} (6+x) \, dx-\text {Subst}\left (\int \frac {-3+e^2+x}{x \log ^2(x)} \, dx,x,3-e^2+x\right )+\text {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,3-e^2+x\right ) \\ & = \frac {1}{4} e^{4+2 x}+\frac {7 x}{2}+\frac {x^2}{4}-\frac {1}{2} e^{2+x} (6+x)+\text {li}\left (3-e^2+x\right )+\frac {1}{2} \int e^{2+x} \, dx-\text {Subst}\left (\int \left (\frac {1}{\log ^2(x)}+\frac {-3+e^2}{x \log ^2(x)}\right ) \, dx,x,3-e^2+x\right ) \\ & = \frac {e^{2+x}}{2}+\frac {1}{4} e^{4+2 x}+\frac {7 x}{2}+\frac {x^2}{4}-\frac {1}{2} e^{2+x} (6+x)+\text {li}\left (3-e^2+x\right )-\left (-3+e^2\right ) \text {Subst}\left (\int \frac {1}{x \log ^2(x)} \, dx,x,3-e^2+x\right )-\text {Subst}\left (\int \frac {1}{\log ^2(x)} \, dx,x,3-e^2+x\right ) \\ & = \frac {e^{2+x}}{2}+\frac {1}{4} e^{4+2 x}+\frac {7 x}{2}+\frac {x^2}{4}-\frac {1}{2} e^{2+x} (6+x)+\frac {3-e^2+x}{\log \left (3-e^2+x\right )}+\text {li}\left (3-e^2+x\right )-\left (-3+e^2\right ) \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log \left (3-e^2+x\right )\right )-\text {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,3-e^2+x\right ) \\ & = \frac {e^{2+x}}{2}+\frac {1}{4} e^{4+2 x}+\frac {7 x}{2}+\frac {x^2}{4}-\frac {1}{2} e^{2+x} (6+x)-\frac {3-e^2}{\log \left (3-e^2+x\right )}+\frac {3-e^2+x}{\log \left (3-e^2+x\right )} \\ \end{align*}
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.10 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.41 \[ \int \frac {2 x+\left (-6+2 e^2-2 x\right ) \log \left (3-e^2+x\right )+\left (-21+e^{4+2 x} \left (-3+e^2-x\right )-10 x-x^2+e^2 (7+x)+e^{2+x} \left (18+e^2 (-6-x)+9 x+x^2\right )\right ) \log ^2\left (3-e^2+x\right )}{\left (-6+2 e^2-2 x\right ) \log ^2\left (3-e^2+x\right )} \, dx=\frac {1}{4} e^{4+2 x}+\frac {7 x}{2}+\frac {x^2}{4}+\frac {1}{2} e^x \left (-5 e^2-e^2 x\right )-\operatorname {ExpIntegralEi}\left (\log \left (3-e^2+x\right )\right )+\frac {x}{\log \left (3-e^2+x\right )}+\operatorname {LogIntegral}\left (3-e^2+x\right ) \]
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Time = 0.30 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.34
method | result | size |
risch | \(\frac {7 x}{2}+\frac {x^{2}}{4}+\frac {{\mathrm e}^{4+2 x}}{4}-\frac {5 \,{\mathrm e}^{2+x}}{2}-\frac {x \,{\mathrm e}^{2+x}}{2}+\frac {x}{\ln \left (-{\mathrm e}^{2}+3+x \right )}\) | \(43\) |
default | \(\frac {7 x}{2}+\frac {x^{2}}{4}+\frac {{\mathrm e}^{4+2 x}}{4}-\frac {{\mathrm e}^{2+x} \left (2+x \right )}{2}-\frac {3 \,{\mathrm e}^{2+x}}{2}+\frac {-{\mathrm e}^{2}+3+x}{\ln \left (-{\mathrm e}^{2}+3+x \right )}+\frac {{\mathrm e}^{2}}{\ln \left (-{\mathrm e}^{2}+3+x \right )}-\frac {3}{\ln \left (-{\mathrm e}^{2}+3+x \right )}\) | \(76\) |
parts | \(\frac {7 x}{2}+\frac {x^{2}}{4}+\frac {{\mathrm e}^{4+2 x}}{4}-\frac {{\mathrm e}^{2+x} \left (2+x \right )}{2}-\frac {3 \,{\mathrm e}^{2+x}}{2}+\frac {-{\mathrm e}^{2}+3+x}{\ln \left (-{\mathrm e}^{2}+3+x \right )}+\frac {{\mathrm e}^{2}}{\ln \left (-{\mathrm e}^{2}+3+x \right )}-\frac {3}{\ln \left (-{\mathrm e}^{2}+3+x \right )}\) | \(76\) |
parallelrisch | \(\frac {-{\mathrm e}^{4} \ln \left (-{\mathrm e}^{2}+3+x \right )+x^{2} \ln \left (-{\mathrm e}^{2}+3+x \right )-2 \,{\mathrm e}^{2+x} x \ln \left (-{\mathrm e}^{2}+3+x \right )+{\mathrm e}^{4+2 x} \ln \left (-{\mathrm e}^{2}+3+x \right )+34 \ln \left (-{\mathrm e}^{2}+3+x \right ) {\mathrm e}^{2}+14 \ln \left (-{\mathrm e}^{2}+3+x \right ) x -10 \,{\mathrm e}^{2+x} \ln \left (-{\mathrm e}^{2}+3+x \right )+4 x -93 \ln \left (-{\mathrm e}^{2}+3+x \right )}{4 \ln \left (-{\mathrm e}^{2}+3+x \right )}\) | \(120\) |
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Time = 0.23 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.47 \[ \int \frac {2 x+\left (-6+2 e^2-2 x\right ) \log \left (3-e^2+x\right )+\left (-21+e^{4+2 x} \left (-3+e^2-x\right )-10 x-x^2+e^2 (7+x)+e^{2+x} \left (18+e^2 (-6-x)+9 x+x^2\right )\right ) \log ^2\left (3-e^2+x\right )}{\left (-6+2 e^2-2 x\right ) \log ^2\left (3-e^2+x\right )} \, dx=\frac {{\left (x^{2} - 2 \, {\left (x + 5\right )} e^{\left (x + 2\right )} + 14 \, x + e^{\left (2 \, x + 4\right )}\right )} \log \left (x - e^{2} + 3\right ) + 4 \, x}{4 \, \log \left (x - e^{2} + 3\right )} \]
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Time = 0.20 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.28 \[ \int \frac {2 x+\left (-6+2 e^2-2 x\right ) \log \left (3-e^2+x\right )+\left (-21+e^{4+2 x} \left (-3+e^2-x\right )-10 x-x^2+e^2 (7+x)+e^{2+x} \left (18+e^2 (-6-x)+9 x+x^2\right )\right ) \log ^2\left (3-e^2+x\right )}{\left (-6+2 e^2-2 x\right ) \log ^2\left (3-e^2+x\right )} \, dx=\frac {x^{2}}{4} + \frac {7 x}{2} + \frac {x}{\log {\left (x - e^{2} + 3 \right )}} + \frac {\left (- 4 x - 20\right ) e^{x + 2}}{8} + \frac {e^{2 x + 4}}{4} \]
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Time = 0.22 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.59 \[ \int \frac {2 x+\left (-6+2 e^2-2 x\right ) \log \left (3-e^2+x\right )+\left (-21+e^{4+2 x} \left (-3+e^2-x\right )-10 x-x^2+e^2 (7+x)+e^{2+x} \left (18+e^2 (-6-x)+9 x+x^2\right )\right ) \log ^2\left (3-e^2+x\right )}{\left (-6+2 e^2-2 x\right ) \log ^2\left (3-e^2+x\right )} \, dx=\frac {{\left (x^{2} - 2 \, {\left (x e^{2} + 5 \, e^{2}\right )} e^{x} + 14 \, x + e^{\left (2 \, x + 4\right )}\right )} \log \left (x - e^{2} + 3\right ) + 4 \, x}{4 \, \log \left (x - e^{2} + 3\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (28) = 56\).
Time = 0.29 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.78 \[ \int \frac {2 x+\left (-6+2 e^2-2 x\right ) \log \left (3-e^2+x\right )+\left (-21+e^{4+2 x} \left (-3+e^2-x\right )-10 x-x^2+e^2 (7+x)+e^{2+x} \left (18+e^2 (-6-x)+9 x+x^2\right )\right ) \log ^2\left (3-e^2+x\right )}{\left (-6+2 e^2-2 x\right ) \log ^2\left (3-e^2+x\right )} \, dx=\frac {{\left (x + 2\right )}^{2} \log \left (x - e^{2} + 3\right ) - 2 \, {\left (x + 2\right )} e^{\left (x + 2\right )} \log \left (x - e^{2} + 3\right ) + 10 \, {\left (x + 2\right )} \log \left (x - e^{2} + 3\right ) + e^{\left (2 \, x + 4\right )} \log \left (x - e^{2} + 3\right ) - 6 \, e^{\left (x + 2\right )} \log \left (x - e^{2} + 3\right ) + 4 \, x}{4 \, \log \left (x - e^{2} + 3\right )} \]
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Time = 10.57 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.31 \[ \int \frac {2 x+\left (-6+2 e^2-2 x\right ) \log \left (3-e^2+x\right )+\left (-21+e^{4+2 x} \left (-3+e^2-x\right )-10 x-x^2+e^2 (7+x)+e^{2+x} \left (18+e^2 (-6-x)+9 x+x^2\right )\right ) \log ^2\left (3-e^2+x\right )}{\left (-6+2 e^2-2 x\right ) \log ^2\left (3-e^2+x\right )} \, dx=\frac {7\,x}{2}-\frac {5\,{\mathrm {e}}^{x+2}}{2}+\frac {{\mathrm {e}}^{2\,x+4}}{4}-\frac {x\,{\mathrm {e}}^{x+2}}{2}+\frac {x}{\ln \left (x-{\mathrm {e}}^2+3\right )}+\frac {x^2}{4} \]
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