Integrand size = 84, antiderivative size = 20 \[ \int \frac {2-x-x^2+e^4 \left (-15 x^2+3 x^3\right )-3 e^4 x^2 \log (x)+\left (e^4 \left (-36 x-6 x^2+6 x^3\right )+e^4 \left (-6 x-3 x^2\right ) \log (x)\right ) \log (2+x)}{2 x+x^2} \, dx=(5-x+\log (x)) \left (1-3 e^4 x \log (2+x)\right ) \]
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Leaf count is larger than twice the leaf count of optimal. \(90\) vs. \(2(20)=40\).
Time = 0.49 (sec) , antiderivative size = 90, normalized size of antiderivative = 4.50, number of steps used = 27, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1607, 6874, 1634, 45, 2393, 2332, 2354, 2438, 2436, 2442, 2417, 2458, 2353, 2352} \[ \int \frac {2-x-x^2+e^4 \left (-15 x^2+3 x^3\right )-3 e^4 x^2 \log (x)+\left (e^4 \left (-36 x-6 x^2+6 x^3\right )+e^4 \left (-6 x-3 x^2\right ) \log (x)\right ) \log (2+x)}{2 x+x^2} \, dx=3 e^4 x^2 \log (x+2)-\left (1+21 e^4\right ) x+21 e^4 x+6 e^4 \log \left (\frac {x}{2}+1\right ) \log (x)+6 e^4 \log (2) \log (x)+\log (x)-15 e^4 (x+2) \log (x+2)-3 e^4 (x+2) \log (x) \log (x+2)+30 e^4 \log (x+2) \]
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Rule 45
Rule 1607
Rule 1634
Rule 2332
Rule 2352
Rule 2353
Rule 2354
Rule 2393
Rule 2417
Rule 2436
Rule 2438
Rule 2442
Rule 2458
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {2-x-x^2+e^4 \left (-15 x^2+3 x^3\right )-3 e^4 x^2 \log (x)+\left (e^4 \left (-36 x-6 x^2+6 x^3\right )+e^4 \left (-6 x-3 x^2\right ) \log (x)\right ) \log (2+x)}{x (2+x)} \, dx \\ & = \int \left (\frac {2-x-\left (1+15 e^4\right ) x^2+3 e^4 x^3-3 e^4 x^2 \log (x)}{x (2+x)}+3 e^4 (-6+2 x-\log (x)) \log (2+x)\right ) \, dx \\ & = \left (3 e^4\right ) \int (-6+2 x-\log (x)) \log (2+x) \, dx+\int \frac {2-x-\left (1+15 e^4\right ) x^2+3 e^4 x^3-3 e^4 x^2 \log (x)}{x (2+x)} \, dx \\ & = \left (3 e^4\right ) \int (-6 \log (2+x)+2 x \log (2+x)-\log (x) \log (2+x)) \, dx+\int \left (\frac {2-x-\left (1+15 e^4\right ) x^2+3 e^4 x^3}{x (2+x)}-\frac {3 e^4 x \log (x)}{2+x}\right ) \, dx \\ & = -\left (\left (3 e^4\right ) \int \frac {x \log (x)}{2+x} \, dx\right )-\left (3 e^4\right ) \int \log (x) \log (2+x) \, dx+\left (6 e^4\right ) \int x \log (2+x) \, dx-\left (18 e^4\right ) \int \log (2+x) \, dx+\int \frac {2-x-\left (1+15 e^4\right ) x^2+3 e^4 x^3}{x (2+x)} \, dx \\ & = 3 e^4 x \log (x)+3 e^4 x^2 \log (2+x)-3 e^4 (2+x) \log (x) \log (2+x)-\left (3 e^4\right ) \int \frac {x^2}{2+x} \, dx-\left (3 e^4\right ) \int \left (\log (x)-\frac {2 \log (x)}{2+x}\right ) \, dx+\left (3 e^4\right ) \int \left (-1+\frac {(2+x) \log (2+x)}{x}\right ) \, dx-\left (18 e^4\right ) \text {Subst}(\int \log (x) \, dx,x,2+x)+\int \left (-1-21 e^4+\frac {1}{x}+3 e^4 x+\frac {42 e^4}{2+x}\right ) \, dx \\ & = 15 e^4 x-\left (1+21 e^4\right ) x+\frac {3 e^4 x^2}{2}+\log (x)+3 e^4 x \log (x)+42 e^4 \log (2+x)+3 e^4 x^2 \log (2+x)-18 e^4 (2+x) \log (2+x)-3 e^4 (2+x) \log (x) \log (2+x)-\left (3 e^4\right ) \int \left (-2+x+\frac {4}{2+x}\right ) \, dx-\left (3 e^4\right ) \int \log (x) \, dx+\left (3 e^4\right ) \int \frac {(2+x) \log (2+x)}{x} \, dx+\left (6 e^4\right ) \int \frac {\log (x)}{2+x} \, dx \\ & = 24 e^4 x-\left (1+21 e^4\right ) x+\log (x)+6 e^4 \log \left (1+\frac {x}{2}\right ) \log (x)+30 e^4 \log (2+x)+3 e^4 x^2 \log (2+x)-18 e^4 (2+x) \log (2+x)-3 e^4 (2+x) \log (x) \log (2+x)+\left (3 e^4\right ) \text {Subst}\left (\int \frac {x \log (x)}{-2+x} \, dx,x,2+x\right )-\left (6 e^4\right ) \int \frac {\log \left (1+\frac {x}{2}\right )}{x} \, dx \\ & = 24 e^4 x-\left (1+21 e^4\right ) x+\log (x)+6 e^4 \log \left (1+\frac {x}{2}\right ) \log (x)+30 e^4 \log (2+x)+3 e^4 x^2 \log (2+x)-18 e^4 (2+x) \log (2+x)-3 e^4 (2+x) \log (x) \log (2+x)+6 e^4 \operatorname {PolyLog}\left (2,-\frac {x}{2}\right )+\left (3 e^4\right ) \text {Subst}\left (\int \left (\log (x)+\frac {2 \log (x)}{-2+x}\right ) \, dx,x,2+x\right ) \\ & = 24 e^4 x-\left (1+21 e^4\right ) x+\log (x)+6 e^4 \log \left (1+\frac {x}{2}\right ) \log (x)+30 e^4 \log (2+x)+3 e^4 x^2 \log (2+x)-18 e^4 (2+x) \log (2+x)-3 e^4 (2+x) \log (x) \log (2+x)+6 e^4 \operatorname {PolyLog}\left (2,-\frac {x}{2}\right )+\left (3 e^4\right ) \text {Subst}(\int \log (x) \, dx,x,2+x)+\left (6 e^4\right ) \text {Subst}\left (\int \frac {\log (x)}{-2+x} \, dx,x,2+x\right ) \\ & = 21 e^4 x-\left (1+21 e^4\right ) x+\log (x)+6 e^4 \log (2) \log (x)+6 e^4 \log \left (1+\frac {x}{2}\right ) \log (x)+30 e^4 \log (2+x)+3 e^4 x^2 \log (2+x)-15 e^4 (2+x) \log (2+x)-3 e^4 (2+x) \log (x) \log (2+x)+6 e^4 \operatorname {PolyLog}\left (2,-\frac {x}{2}\right )+\left (6 e^4\right ) \text {Subst}\left (\int \frac {\log \left (\frac {x}{2}\right )}{-2+x} \, dx,x,2+x\right ) \\ & = 21 e^4 x-\left (1+21 e^4\right ) x+\log (x)+6 e^4 \log (2) \log (x)+6 e^4 \log \left (1+\frac {x}{2}\right ) \log (x)+30 e^4 \log (2+x)+3 e^4 x^2 \log (2+x)-15 e^4 (2+x) \log (2+x)-3 e^4 (2+x) \log (x) \log (2+x) \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {2-x-x^2+e^4 \left (-15 x^2+3 x^3\right )-3 e^4 x^2 \log (x)+\left (e^4 \left (-36 x-6 x^2+6 x^3\right )+e^4 \left (-6 x-3 x^2\right ) \log (x)\right ) \log (2+x)}{2 x+x^2} \, dx=-x+\log (x)+3 e^4 x (-5+x-\log (x)) \log (2+x) \]
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Time = 1.11 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.60
method | result | size |
risch | \(\left (3 x^{2} {\mathrm e}^{4}-3 x \,{\mathrm e}^{4} \ln \left (x \right )-15 x \,{\mathrm e}^{4}\right ) \ln \left (2+x \right )+\ln \left (x \right )-x\) | \(32\) |
parallelrisch | \(3 \ln \left (2+x \right ) {\mathrm e}^{4} x^{2}-3 \ln \left (2+x \right ) \ln \left (x \right ) {\mathrm e}^{4} x -15 x \ln \left (2+x \right ) {\mathrm e}^{4}-x +\ln \left (x \right )+4\) | \(39\) |
default | \(48 \,{\mathrm e}^{4} \ln \left (2+x \right )+3 x \ln \left (2+x \right ) {\mathrm e}^{4}-24 x \,{\mathrm e}^{4}-3 \ln \left (2+x \right ) \ln \left (x \right ) {\mathrm e}^{4} x +\frac {3 x^{2} {\mathrm e}^{4}}{2}-x +\ln \left (x \right )+6 \,{\mathrm e}^{4} \left (-5 \left (2+x \right ) \ln \left (2+x \right )+5 x +10+\frac {\ln \left (2+x \right ) \left (2+x \right )^{2}}{2}-\frac {\left (2+x \right )^{2}}{4}\right )\) | \(83\) |
parts | \(48 \,{\mathrm e}^{4} \ln \left (2+x \right )+3 x \ln \left (2+x \right ) {\mathrm e}^{4}-24 x \,{\mathrm e}^{4}-3 \ln \left (2+x \right ) \ln \left (x \right ) {\mathrm e}^{4} x +\frac {3 x^{2} {\mathrm e}^{4}}{2}-x +\ln \left (x \right )+6 \,{\mathrm e}^{4} \left (-5 \left (2+x \right ) \ln \left (2+x \right )+5 x +10+\frac {\ln \left (2+x \right ) \left (2+x \right )^{2}}{2}-\frac {\left (2+x \right )^{2}}{4}\right )\) | \(83\) |
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Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.50 \[ \int \frac {2-x-x^2+e^4 \left (-15 x^2+3 x^3\right )-3 e^4 x^2 \log (x)+\left (e^4 \left (-36 x-6 x^2+6 x^3\right )+e^4 \left (-6 x-3 x^2\right ) \log (x)\right ) \log (2+x)}{2 x+x^2} \, dx=-3 \, {\left (x e^{4} \log \left (x\right ) - {\left (x^{2} - 5 \, x\right )} e^{4}\right )} \log \left (x + 2\right ) - x + \log \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (19) = 38\).
Time = 0.85 (sec) , antiderivative size = 65, normalized size of antiderivative = 3.25 \[ \int \frac {2-x-x^2+e^4 \left (-15 x^2+3 x^3\right )-3 e^4 x^2 \log (x)+\left (e^4 \left (-36 x-6 x^2+6 x^3\right )+e^4 \left (-6 x-3 x^2\right ) \log (x)\right ) \log (2+x)}{2 x+x^2} \, dx=- x + \left (3 x^{2} e^{4} - 3 x e^{4} \log {\left (x \right )} - 15 x e^{4} - \frac {35 e^{4}}{2}\right ) \log {\left (x + 2 \right )} + \log {\left (x \right )} + \frac {35 e^{4} \log {\left (x + \frac {-4 + 70 e^{4}}{-2 + 35 e^{4}} \right )}}{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (19) = 38\).
Time = 0.24 (sec) , antiderivative size = 76, normalized size of antiderivative = 3.80 \[ \int \frac {2-x-x^2+e^4 \left (-15 x^2+3 x^3\right )-3 e^4 x^2 \log (x)+\left (e^4 \left (-36 x-6 x^2+6 x^3\right )+e^4 \left (-6 x-3 x^2\right ) \log (x)\right ) \log (2+x)}{2 x+x^2} \, dx=-\frac {3}{2} \, x^{2} e^{4} + \frac {3}{2} \, {\left (x^{2} - 4 \, x + 8 \, \log \left (x + 2\right )\right )} e^{4} - 15 \, {\left (x - 2 \, \log \left (x + 2\right )\right )} e^{4} + 21 \, x e^{4} + 3 \, {\left (x^{2} e^{4} - x e^{4} \log \left (x\right ) - 5 \, x e^{4} - 14 \, e^{4}\right )} \log \left (x + 2\right ) - x + \log \left (x\right ) \]
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Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.85 \[ \int \frac {2-x-x^2+e^4 \left (-15 x^2+3 x^3\right )-3 e^4 x^2 \log (x)+\left (e^4 \left (-36 x-6 x^2+6 x^3\right )+e^4 \left (-6 x-3 x^2\right ) \log (x)\right ) \log (2+x)}{2 x+x^2} \, dx=3 \, x^{2} e^{4} \log \left (x + 2\right ) - 3 \, x e^{4} \log \left (x + 2\right ) \log \left (x\right ) - 15 \, x e^{4} \log \left (x + 2\right ) - x + \log \left (x\right ) \]
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Time = 12.86 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.60 \[ \int \frac {2-x-x^2+e^4 \left (-15 x^2+3 x^3\right )-3 e^4 x^2 \log (x)+\left (e^4 \left (-36 x-6 x^2+6 x^3\right )+e^4 \left (-6 x-3 x^2\right ) \log (x)\right ) \log (2+x)}{2 x+x^2} \, dx=\ln \left (x\right )-x-\ln \left (x+2\right )\,\left (15\,x\,{\mathrm {e}}^4-3\,x^2\,{\mathrm {e}}^4+3\,x\,{\mathrm {e}}^4\,\ln \left (x\right )\right ) \]
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