Integrand size = 77, antiderivative size = 26 \[ \int \frac {e^5 \left (54 e x+e^2 \left (36+6 x^3\right )\right )+\left (-18 e x+e^2 \left (-12-2 x^3\right )\right ) \log \left (\frac {e x^2+x^3}{-3+x^3}\right )}{-3 x^2+x^5+e \left (-3 x+x^4\right )} \, dx=e \left (3 e^5-\log \left (\frac {e+x}{-\frac {3}{x^2}+x}\right )\right )^2 \]
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Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 2.06 (sec) , antiderivative size = 1214, normalized size of antiderivative = 46.69, number of steps used = 86, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.221, Rules used = {6820, 12, 2608, 2604, 2404, 2338, 2354, 2438, 2375, 2465, 2439, 2437, 266, 2463, 2441, 2440, 2352} \[ \int \frac {e^5 \left (54 e x+e^2 \left (36+6 x^3\right )\right )+\left (-18 e x+e^2 \left (-12-2 x^3\right )\right ) \log \left (\frac {e x^2+x^3}{-3+x^3}\right )}{-3 x^2+x^5+e \left (-3 x+x^4\right )} \, dx =\text {Too large to display} \]
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Rule 12
Rule 266
Rule 2338
Rule 2352
Rule 2354
Rule 2375
Rule 2404
Rule 2437
Rule 2438
Rule 2439
Rule 2440
Rule 2441
Rule 2463
Rule 2465
Rule 2604
Rule 2608
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \frac {2 e \left (6 e+9 x+e x^3\right ) \left (-3 e^5+\log \left (\frac {x^2 (e+x)}{-3+x^3}\right )\right )}{x (e+x) \left (3-x^3\right )} \, dx \\ & = (2 e) \int \frac {\left (6 e+9 x+e x^3\right ) \left (-3 e^5+\log \left (\frac {x^2 (e+x)}{-3+x^3}\right )\right )}{x (e+x) \left (3-x^3\right )} \, dx \\ & = (2 e) \int \left (\frac {2 \left (-3 e^5+\log \left (\frac {x^2 (e+x)}{-3+x^3}\right )\right )}{x}+\frac {-3 e^5+\log \left (\frac {x^2 (e+x)}{-3+x^3}\right )}{e+x}-\frac {3 x^2 \left (-3 e^5+\log \left (\frac {x^2 (e+x)}{-3+x^3}\right )\right )}{-3+x^3}\right ) \, dx \\ & = (2 e) \int \frac {-3 e^5+\log \left (\frac {x^2 (e+x)}{-3+x^3}\right )}{e+x} \, dx+(4 e) \int \frac {-3 e^5+\log \left (\frac {x^2 (e+x)}{-3+x^3}\right )}{x} \, dx-(6 e) \int \frac {x^2 \left (-3 e^5+\log \left (\frac {x^2 (e+x)}{-3+x^3}\right )\right )}{-3+x^3} \, dx \\ & = -4 e \log (x) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )-2 e \log (e+x) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )-(2 e) \int \frac {\left (-3+x^3\right ) \left (-\frac {3 x^4 (e+x)}{\left (-3+x^3\right )^2}+\frac {x^2}{-3+x^3}+\frac {2 x (e+x)}{-3+x^3}\right ) \log (e+x)}{x^2 (e+x)} \, dx-(4 e) \int \frac {\left (-3+x^3\right ) \left (-\frac {3 x^4 (e+x)}{\left (-3+x^3\right )^2}+\frac {x^2}{-3+x^3}+\frac {2 x (e+x)}{-3+x^3}\right ) \log (x)}{x^2 (e+x)} \, dx-(6 e) \int \left (-\frac {-3 e^5+\log \left (\frac {x^2 (e+x)}{-3+x^3}\right )}{3 \left (-\sqrt [3]{-3}-x\right )}-\frac {-3 e^5+\log \left (\frac {x^2 (e+x)}{-3+x^3}\right )}{3 \left (\sqrt [3]{3}-x\right )}-\frac {-3 e^5+\log \left (\frac {x^2 (e+x)}{-3+x^3}\right )}{3 \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}\right ) \, dx \\ & = -4 e \log (x) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )-2 e \log (e+x) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )-(2 e) \int \left (\frac {2 \log (e+x)}{x}+\frac {\log (e+x)}{e+x}-\frac {3 x^2 \log (e+x)}{-3+x^3}\right ) \, dx+(2 e) \int \frac {-3 e^5+\log \left (\frac {x^2 (e+x)}{-3+x^3}\right )}{-\sqrt [3]{-3}-x} \, dx+(2 e) \int \frac {-3 e^5+\log \left (\frac {x^2 (e+x)}{-3+x^3}\right )}{\sqrt [3]{3}-x} \, dx+(2 e) \int \frac {-3 e^5+\log \left (\frac {x^2 (e+x)}{-3+x^3}\right )}{(-1)^{2/3} \sqrt [3]{3}-x} \, dx-(4 e) \int \left (\frac {2 \log (x)}{x}+\frac {\log (x)}{e+x}-\frac {3 x^2 \log (x)}{-3+x^3}\right ) \, dx \\ & = 2 e \log \left (-\sqrt [3]{-3}-x\right ) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )+2 e \log \left (\sqrt [3]{3}-x\right ) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )+2 e \log \left ((-1)^{2/3} \sqrt [3]{3}-x\right ) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )-4 e \log (x) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )-2 e \log (e+x) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )+(2 e) \int \frac {\left (-3+x^3\right ) \left (-\frac {3 x^4 (e+x)}{\left (-3+x^3\right )^2}+\frac {x^2}{-3+x^3}+\frac {2 x (e+x)}{-3+x^3}\right ) \log \left (-\sqrt [3]{-3}-x\right )}{x^2 (e+x)} \, dx+(2 e) \int \frac {\left (-3+x^3\right ) \left (-\frac {3 x^4 (e+x)}{\left (-3+x^3\right )^2}+\frac {x^2}{-3+x^3}+\frac {2 x (e+x)}{-3+x^3}\right ) \log \left (\sqrt [3]{3}-x\right )}{x^2 (e+x)} \, dx+(2 e) \int \frac {\left (-3+x^3\right ) \left (-\frac {3 x^4 (e+x)}{\left (-3+x^3\right )^2}+\frac {x^2}{-3+x^3}+\frac {2 x (e+x)}{-3+x^3}\right ) \log \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}{x^2 (e+x)} \, dx-(2 e) \int \frac {\log (e+x)}{e+x} \, dx-(4 e) \int \frac {\log (x)}{e+x} \, dx-(4 e) \int \frac {\log (e+x)}{x} \, dx+(6 e) \int \frac {x^2 \log (e+x)}{-3+x^3} \, dx-(8 e) \int \frac {\log (x)}{x} \, dx+(12 e) \int \frac {x^2 \log (x)}{-3+x^3} \, dx \\ & = -4 e \log (x)-4 e \log ^2(x)-4 e \log (x) \log \left (1+\frac {x}{e}\right )+2 e \log \left (-\sqrt [3]{-3}-x\right ) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )+2 e \log \left (\sqrt [3]{3}-x\right ) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )+2 e \log \left ((-1)^{2/3} \sqrt [3]{3}-x\right ) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )-4 e \log (x) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )-2 e \log (e+x) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )+4 e \log (x) \log \left (1-\frac {x^3}{3}\right )+(2 e) \int \left (\frac {2 \log \left (-\sqrt [3]{-3}-x\right )}{x}+\frac {\log \left (-\sqrt [3]{-3}-x\right )}{e+x}-\frac {3 x^2 \log \left (-\sqrt [3]{-3}-x\right )}{-3+x^3}\right ) \, dx+(2 e) \int \left (\frac {2 \log \left (\sqrt [3]{3}-x\right )}{x}+\frac {\log \left (\sqrt [3]{3}-x\right )}{e+x}-\frac {3 x^2 \log \left (\sqrt [3]{3}-x\right )}{-3+x^3}\right ) \, dx+(2 e) \int \left (\frac {2 \log \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}{x}+\frac {\log \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}{e+x}-\frac {3 x^2 \log \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}{-3+x^3}\right ) \, dx-(2 e) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,e+x\right )-(4 e) \int \frac {\log \left (1-\frac {x^3}{3}\right )}{x} \, dx+(6 e) \int \left (-\frac {\log (e+x)}{3 \left (-\sqrt [3]{-3}-x\right )}-\frac {\log (e+x)}{3 \left (\sqrt [3]{3}-x\right )}-\frac {\log (e+x)}{3 \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}\right ) \, dx \\ & = -4 e \log (x)-4 e \log ^2(x)-e \log ^2(e+x)-4 e \log (x) \log \left (1+\frac {x}{e}\right )+2 e \log \left (-\sqrt [3]{-3}-x\right ) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )+2 e \log \left (\sqrt [3]{3}-x\right ) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )+2 e \log \left ((-1)^{2/3} \sqrt [3]{3}-x\right ) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )-4 e \log (x) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )-2 e \log (e+x) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )+4 e \log (x) \log \left (1-\frac {x^3}{3}\right )+\frac {4}{3} e \operatorname {PolyLog}\left (2,\frac {x^3}{3}\right )+(2 e) \int \frac {\log \left (-\sqrt [3]{-3}-x\right )}{e+x} \, dx+(2 e) \int \frac {\log \left (\sqrt [3]{3}-x\right )}{e+x} \, dx+(2 e) \int \frac {\log \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}{e+x} \, dx-(2 e) \int \frac {\log (e+x)}{-\sqrt [3]{-3}-x} \, dx-(2 e) \int \frac {\log (e+x)}{\sqrt [3]{3}-x} \, dx-(2 e) \int \frac {\log (e+x)}{(-1)^{2/3} \sqrt [3]{3}-x} \, dx+(4 e) \int \frac {\log \left (-\sqrt [3]{-3}-x\right )}{x} \, dx+(4 e) \int \frac {\log \left (\sqrt [3]{3}-x\right )}{x} \, dx+(4 e) \int \frac {\log \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}{x} \, dx-(6 e) \int \frac {x^2 \log \left (-\sqrt [3]{-3}-x\right )}{-3+x^3} \, dx-(6 e) \int \frac {x^2 \log \left (\sqrt [3]{3}-x\right )}{-3+x^3} \, dx-(6 e) \int \frac {x^2 \log \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}{-3+x^3} \, dx \\ & = -4 e \log (x)+\frac {4}{3} e \log (3) \log (x)-4 e \log ^2(x)+4 e \log \left ((-1)^{2/3} \sqrt [3]{3}-x\right ) \log \left (-\sqrt [3]{-\frac {1}{3}} x\right )+4 e \log \left (-\sqrt [3]{-3}-x\right ) \log \left (\frac {(-1)^{2/3} x}{\sqrt [3]{3}}\right )+2 e \log \left (\frac {\sqrt [3]{3}-x}{\sqrt [3]{3}+e}\right ) \log (e+x)+2 e \log \left (\frac {\sqrt [3]{-3}+x}{\sqrt [3]{-3}-e}\right ) \log (e+x)-e \log ^2(e+x)+2 e \log \left (-\sqrt [3]{-3}-x\right ) \log \left (-\frac {e+x}{\sqrt [3]{-3}-e}\right )+2 e \log \left (\sqrt [3]{3}-x\right ) \log \left (\frac {e+x}{\sqrt [3]{3}+e}\right )+2 e \log \left ((-1)^{2/3} \sqrt [3]{3}-x\right ) \log \left (\frac {e+x}{(-1)^{2/3} \sqrt [3]{3}+e}\right )+2 e \log (e+x) \log \left (\frac {(-1)^{2/3} \left (\sqrt [3]{3}+\sqrt [3]{-1} x\right )}{(-1)^{2/3} \sqrt [3]{3}+e}\right )-4 e \log (x) \log \left (1+\frac {x}{e}\right )+2 e \log \left (-\sqrt [3]{-3}-x\right ) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )+2 e \log \left (\sqrt [3]{3}-x\right ) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )+2 e \log \left ((-1)^{2/3} \sqrt [3]{3}-x\right ) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )-4 e \log (x) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )-2 e \log (e+x) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )+4 e \log (x) \log \left (1-\frac {x^3}{3}\right )+\frac {4}{3} e \operatorname {PolyLog}\left (2,\frac {x^3}{3}\right )-(2 e) \int \frac {\log \left (\frac {-\sqrt [3]{-3}-x}{-\sqrt [3]{-3}+e}\right )}{e+x} \, dx-(2 e) \int \frac {\log \left (\frac {\sqrt [3]{3}-x}{\sqrt [3]{3}+e}\right )}{e+x} \, dx-(2 e) \int \frac {\log \left (\frac {(-1)^{2/3} \sqrt [3]{3}-x}{(-1)^{2/3} \sqrt [3]{3}+e}\right )}{e+x} \, dx+(2 e) \int \frac {\log \left (\frac {-e-x}{\sqrt [3]{-3}-e}\right )}{-\sqrt [3]{-3}-x} \, dx+(2 e) \int \frac {\log \left (\frac {-e-x}{-\sqrt [3]{3}-e}\right )}{\sqrt [3]{3}-x} \, dx+(2 e) \int \frac {\log \left (\frac {-e-x}{-(-1)^{2/3} \sqrt [3]{3}-e}\right )}{(-1)^{2/3} \sqrt [3]{3}-x} \, dx+(4 e) \int \frac {\log \left (-\sqrt [3]{-\frac {1}{3}} x\right )}{(-1)^{2/3} \sqrt [3]{3}-x} \, dx+(4 e) \int \frac {\log \left (\frac {(-1)^{2/3} x}{\sqrt [3]{3}}\right )}{-\sqrt [3]{-3}-x} \, dx+(4 e) \int \frac {\log \left (1-\frac {x}{\sqrt [3]{3}}\right )}{x} \, dx-(6 e) \int \left (-\frac {\log \left (-\sqrt [3]{-3}-x\right )}{3 \left (-\sqrt [3]{-3}-x\right )}-\frac {\log \left (-\sqrt [3]{-3}-x\right )}{3 \left (\sqrt [3]{3}-x\right )}-\frac {\log \left (-\sqrt [3]{-3}-x\right )}{3 \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}\right ) \, dx-(6 e) \int \left (-\frac {\log \left (\sqrt [3]{3}-x\right )}{3 \left (-\sqrt [3]{-3}-x\right )}-\frac {\log \left (\sqrt [3]{3}-x\right )}{3 \left (\sqrt [3]{3}-x\right )}-\frac {\log \left (\sqrt [3]{3}-x\right )}{3 \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}\right ) \, dx-(6 e) \int \left (-\frac {\log \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}{3 \left (-\sqrt [3]{-3}-x\right )}-\frac {\log \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}{3 \left (\sqrt [3]{3}-x\right )}-\frac {\log \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}{3 \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}\right ) \, dx \\ & = -4 e \log (x)+\frac {4}{3} e \log (3) \log (x)-4 e \log ^2(x)+4 e \log \left ((-1)^{2/3} \sqrt [3]{3}-x\right ) \log \left (-\sqrt [3]{-\frac {1}{3}} x\right )+4 e \log \left (-\sqrt [3]{-3}-x\right ) \log \left (\frac {(-1)^{2/3} x}{\sqrt [3]{3}}\right )+2 e \log \left (\frac {\sqrt [3]{3}-x}{\sqrt [3]{3}+e}\right ) \log (e+x)+2 e \log \left (\frac {\sqrt [3]{-3}+x}{\sqrt [3]{-3}-e}\right ) \log (e+x)-e \log ^2(e+x)+2 e \log \left (-\sqrt [3]{-3}-x\right ) \log \left (-\frac {e+x}{\sqrt [3]{-3}-e}\right )+2 e \log \left (\sqrt [3]{3}-x\right ) \log \left (\frac {e+x}{\sqrt [3]{3}+e}\right )+2 e \log \left ((-1)^{2/3} \sqrt [3]{3}-x\right ) \log \left (\frac {e+x}{(-1)^{2/3} \sqrt [3]{3}+e}\right )+2 e \log (e+x) \log \left (\frac {(-1)^{2/3} \left (\sqrt [3]{3}+\sqrt [3]{-1} x\right )}{(-1)^{2/3} \sqrt [3]{3}+e}\right )-4 e \log (x) \log \left (1+\frac {x}{e}\right )+2 e \log \left (-\sqrt [3]{-3}-x\right ) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )+2 e \log \left (\sqrt [3]{3}-x\right ) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )+2 e \log \left ((-1)^{2/3} \sqrt [3]{3}-x\right ) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )-4 e \log (x) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )-2 e \log (e+x) \left (3 e^5-\log \left (-\frac {x^2 (e+x)}{3-x^3}\right )\right )+4 e \log (x) \log \left (1-\frac {x^3}{3}\right )-4 e \operatorname {PolyLog}\left (2,\frac {x}{\sqrt [3]{3}}\right )+\frac {4}{3} e \operatorname {PolyLog}\left (2,\frac {x^3}{3}\right )+4 e \operatorname {PolyLog}\left (2,1+\sqrt [3]{-\frac {1}{3}} x\right )+4 e \operatorname {PolyLog}\left (2,1-\frac {(-1)^{2/3} x}{\sqrt [3]{3}}\right )+(2 e) \int \frac {\log \left (-\sqrt [3]{-3}-x\right )}{-\sqrt [3]{-3}-x} \, dx+(2 e) \int \frac {\log \left (-\sqrt [3]{-3}-x\right )}{\sqrt [3]{3}-x} \, dx+(2 e) \int \frac {\log \left (-\sqrt [3]{-3}-x\right )}{(-1)^{2/3} \sqrt [3]{3}-x} \, dx+(2 e) \int \frac {\log \left (\sqrt [3]{3}-x\right )}{-\sqrt [3]{-3}-x} \, dx+(2 e) \int \frac {\log \left (\sqrt [3]{3}-x\right )}{\sqrt [3]{3}-x} \, dx+(2 e) \int \frac {\log \left (\sqrt [3]{3}-x\right )}{(-1)^{2/3} \sqrt [3]{3}-x} \, dx+(2 e) \int \frac {\log \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}{-\sqrt [3]{-3}-x} \, dx+(2 e) \int \frac {\log \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}{\sqrt [3]{3}-x} \, dx+(2 e) \int \frac {\log \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}{(-1)^{2/3} \sqrt [3]{3}-x} \, dx-(2 e) \text {Subst}\left (\int \frac {\log \left (1+\frac {x}{\sqrt [3]{-3}-e}\right )}{x} \, dx,x,-\sqrt [3]{-3}-x\right )-(2 e) \text {Subst}\left (\int \frac {\log \left (1+\frac {x}{-\sqrt [3]{3}-e}\right )}{x} \, dx,x,\sqrt [3]{3}-x\right )-(2 e) \text {Subst}\left (\int \frac {\log \left (1+\frac {x}{-(-1)^{2/3} \sqrt [3]{3}-e}\right )}{x} \, dx,x,(-1)^{2/3} \sqrt [3]{3}-x\right )-(2 e) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{-\sqrt [3]{-3}+e}\right )}{x} \, dx,x,e+x\right )-(2 e) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{\sqrt [3]{3}+e}\right )}{x} \, dx,x,e+x\right )-(2 e) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{(-1)^{2/3} \sqrt [3]{3}+e}\right )}{x} \, dx,x,e+x\right ) \\ & = \text {Too large to display} \\ \end{align*}
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.12 (sec) , antiderivative size = 1186, normalized size of antiderivative = 45.62 \[ \int \frac {e^5 \left (54 e x+e^2 \left (36+6 x^3\right )\right )+\left (-18 e x+e^2 \left (-12-2 x^3\right )\right ) \log \left (\frac {e x^2+x^3}{-3+x^3}\right )}{-3 x^2+x^5+e \left (-3 x+x^4\right )} \, dx =\text {Too large to display} \]
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Time = 0.51 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.96
method | result | size |
norman | \(-6 \,{\mathrm e} \,{\mathrm e}^{5} \ln \left (\frac {x^{2} {\mathrm e}+x^{3}}{x^{3}-3}\right )+{\mathrm e} \ln \left (\frac {x^{2} {\mathrm e}+x^{3}}{x^{3}-3}\right )^{2}\) | \(51\) |
risch | \(\text {Expression too large to display}\) | \(1571\) |
default | \(\text {Expression too large to display}\) | \(1579\) |
parts | \(\text {Expression too large to display}\) | \(1579\) |
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Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.85 \[ \int \frac {e^5 \left (54 e x+e^2 \left (36+6 x^3\right )\right )+\left (-18 e x+e^2 \left (-12-2 x^3\right )\right ) \log \left (\frac {e x^2+x^3}{-3+x^3}\right )}{-3 x^2+x^5+e \left (-3 x+x^4\right )} \, dx=e \log \left (\frac {x^{3} + x^{2} e}{x^{3} - 3}\right )^{2} - 6 \, e^{6} \log \left (\frac {x^{3} + x^{2} e}{x^{3} - 3}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (22) = 44\).
Time = 1.23 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.04 \[ \int \frac {e^5 \left (54 e x+e^2 \left (36+6 x^3\right )\right )+\left (-18 e x+e^2 \left (-12-2 x^3\right )\right ) \log \left (\frac {e x^2+x^3}{-3+x^3}\right )}{-3 x^2+x^5+e \left (-3 x+x^4\right )} \, dx=- 12 e^{6} \log {\left (x \right )} + e \log {\left (\frac {x^{3} + e x^{2}}{x^{3} - 3} \right )}^{2} - 6 e^{6} \log {\left (x + e \right )} + 6 e^{6} \log {\left (x^{3} - 3 \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 513 vs. \(2 (27) = 54\).
Time = 2.93 (sec) , antiderivative size = 513, normalized size of antiderivative = 19.73 \[ \int \frac {e^5 \left (54 e x+e^2 \left (36+6 x^3\right )\right )+\left (-18 e x+e^2 \left (-12-2 x^3\right )\right ) \log \left (\frac {e x^2+x^3}{-3+x^3}\right )}{-3 x^2+x^5+e \left (-3 x+x^4\right )} \, dx=e \log \left (x^{3} - 3\right )^{2} + e \log \left (x + e\right )^{2} + 4 \, e \log \left (x + e\right ) \log \left (x\right ) + 4 \, e \log \left (x\right )^{2} - 2 \, {\left (6 \, e^{\left (-1\right )} \log \left (x\right ) - \frac {6 \cdot 3^{\frac {1}{6}} {\left (3^{\frac {1}{3}} e + 3^{\frac {2}{3}}\right )} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x + 3^{\frac {1}{3}}\right )}\right )}{3^{\frac {2}{3}} e^{3} + 3 \cdot 3^{\frac {2}{3}}} - \frac {{\left (2 \cdot 3^{\frac {2}{3}} e^{2} - 3 \cdot 3^{\frac {1}{3}} + 3 \, e\right )} \log \left (x^{2} + 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right )}{3^{\frac {2}{3}} e^{3} + 3 \cdot 3^{\frac {2}{3}}} - \frac {2 \, {\left (3^{\frac {2}{3}} e^{2} + 3 \cdot 3^{\frac {1}{3}} - 3 \, e\right )} \log \left (x - 3^{\frac {1}{3}}\right )}{3^{\frac {2}{3}} e^{3} + 3 \cdot 3^{\frac {2}{3}}} - \frac {18 \, \log \left (x + e\right )}{e^{4} + 3 \, e}\right )} e^{7} + {\left (\frac {6 \cdot 3^{\frac {1}{6}} {\left (3^{\frac {1}{3}} e + 3^{\frac {2}{3}}\right )} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x + 3^{\frac {1}{3}}\right )}\right )}{3^{\frac {2}{3}} e^{3} + 3 \cdot 3^{\frac {2}{3}}} + \frac {{\left (2 \cdot 3^{\frac {2}{3}} e^{2} - 3 \cdot 3^{\frac {1}{3}} + 3 \, e\right )} \log \left (x^{2} + 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right )}{3^{\frac {2}{3}} e^{3} + 3 \cdot 3^{\frac {2}{3}}} + \frac {2 \, {\left (3^{\frac {2}{3}} e^{2} + 3 \cdot 3^{\frac {1}{3}} - 3 \, e\right )} \log \left (x - 3^{\frac {1}{3}}\right )}{3^{\frac {2}{3}} e^{3} + 3 \cdot 3^{\frac {2}{3}}} - \frac {6 \, e^{2} \log \left (x + e\right )}{e^{3} + 3}\right )} e^{7} - 9 \, {\left (\frac {2 \cdot 3^{\frac {1}{6}} {\left (3^{\frac {2}{3}} e + 3^{\frac {1}{3}} e^{2}\right )} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x + 3^{\frac {1}{3}}\right )}\right )}{3^{\frac {2}{3}} e^{3} + 3 \cdot 3^{\frac {2}{3}}} - \frac {{\left (3^{\frac {1}{3}} e + 2 \cdot 3^{\frac {2}{3}} - e^{2}\right )} \log \left (x^{2} + 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right )}{3^{\frac {2}{3}} e^{3} + 3 \cdot 3^{\frac {2}{3}}} + \frac {2 \, {\left (3^{\frac {1}{3}} e - 3^{\frac {2}{3}} - e^{2}\right )} \log \left (x - 3^{\frac {1}{3}}\right )}{3^{\frac {2}{3}} e^{3} + 3 \cdot 3^{\frac {2}{3}}} + \frac {6 \, \log \left (x + e\right )}{e^{3} + 3}\right )} e^{6} - 2 \, {\left (e \log \left (x + e\right ) + 2 \, e \log \left (x\right )\right )} \log \left (x^{3} - 3\right ) \]
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\[ \int \frac {e^5 \left (54 e x+e^2 \left (36+6 x^3\right )\right )+\left (-18 e x+e^2 \left (-12-2 x^3\right )\right ) \log \left (\frac {e x^2+x^3}{-3+x^3}\right )}{-3 x^2+x^5+e \left (-3 x+x^4\right )} \, dx=\int { \frac {2 \, {\left (3 \, {\left ({\left (x^{3} + 6\right )} e^{2} + 9 \, x e\right )} e^{5} - {\left ({\left (x^{3} + 6\right )} e^{2} + 9 \, x e\right )} \log \left (\frac {x^{3} + x^{2} e}{x^{3} - 3}\right )\right )}}{x^{5} - 3 \, x^{2} + {\left (x^{4} - 3 \, x\right )} e} \,d x } \]
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Time = 9.06 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.92 \[ \int \frac {e^5 \left (54 e x+e^2 \left (36+6 x^3\right )\right )+\left (-18 e x+e^2 \left (-12-2 x^3\right )\right ) \log \left (\frac {e x^2+x^3}{-3+x^3}\right )}{-3 x^2+x^5+e \left (-3 x+x^4\right )} \, dx=\mathrm {e}\,{\ln \left (\frac {x^3+\mathrm {e}\,x^2}{x^3-3}\right )}^2+6\,{\mathrm {e}}^6\,\ln \left (x^3-3\right )-6\,{\mathrm {e}}^6\,\ln \left (x+\mathrm {e}\right )-12\,{\mathrm {e}}^6\,\ln \left (x\right ) \]
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