Integrand size = 182, antiderivative size = 28 \[ \int \frac {e^{10} (-4-3 x)-e^{10} x \log (\log (3))+\left (e^{10} (5+15 x)+5 e^{10} x \log (\log (3))\right ) \log \left (\frac {1+3 x+x \log (\log (3))}{x}\right )+\left (e^{10} (-4-12 x)-4 e^{10} x \log (\log (3))\right ) \log ^2\left (\frac {1+3 x+x \log (\log (3))}{x}\right )}{x^2+3 x^3+x^3 \log (\log (3))+\left (-8 x^2-24 x^3-8 x^3 \log (\log (3))\right ) \log \left (\frac {1+3 x+x \log (\log (3))}{x}\right )+\left (16 x^2+48 x^3+16 x^3 \log (\log (3))\right ) \log ^2\left (\frac {1+3 x+x \log (\log (3))}{x}\right )} \, dx=\frac {e^{10}}{x \left (4-\frac {3}{1-\log \left (3+\frac {1}{x}+\log (\log (3))\right )}\right )} \]
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\[ \int \frac {e^{10} (-4-3 x)-e^{10} x \log (\log (3))+\left (e^{10} (5+15 x)+5 e^{10} x \log (\log (3))\right ) \log \left (\frac {1+3 x+x \log (\log (3))}{x}\right )+\left (e^{10} (-4-12 x)-4 e^{10} x \log (\log (3))\right ) \log ^2\left (\frac {1+3 x+x \log (\log (3))}{x}\right )}{x^2+3 x^3+x^3 \log (\log (3))+\left (-8 x^2-24 x^3-8 x^3 \log (\log (3))\right ) \log \left (\frac {1+3 x+x \log (\log (3))}{x}\right )+\left (16 x^2+48 x^3+16 x^3 \log (\log (3))\right ) \log ^2\left (\frac {1+3 x+x \log (\log (3))}{x}\right )} \, dx=\int \frac {e^{10} (-4-3 x)-e^{10} x \log (\log (3))+\left (e^{10} (5+15 x)+5 e^{10} x \log (\log (3))\right ) \log \left (\frac {1+3 x+x \log (\log (3))}{x}\right )+\left (e^{10} (-4-12 x)-4 e^{10} x \log (\log (3))\right ) \log ^2\left (\frac {1+3 x+x \log (\log (3))}{x}\right )}{x^2+3 x^3+x^3 \log (\log (3))+\left (-8 x^2-24 x^3-8 x^3 \log (\log (3))\right ) \log \left (\frac {1+3 x+x \log (\log (3))}{x}\right )+\left (16 x^2+48 x^3+16 x^3 \log (\log (3))\right ) \log ^2\left (\frac {1+3 x+x \log (\log (3))}{x}\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{10} (-4-3 x)-e^{10} x \log (\log (3))+\left (e^{10} (5+15 x)+5 e^{10} x \log (\log (3))\right ) \log \left (\frac {1+3 x+x \log (\log (3))}{x}\right )+\left (e^{10} (-4-12 x)-4 e^{10} x \log (\log (3))\right ) \log ^2\left (\frac {1+3 x+x \log (\log (3))}{x}\right )}{x^2+x^3 (3+\log (\log (3)))+\left (-8 x^2-24 x^3-8 x^3 \log (\log (3))\right ) \log \left (\frac {1+3 x+x \log (\log (3))}{x}\right )+\left (16 x^2+48 x^3+16 x^3 \log (\log (3))\right ) \log ^2\left (\frac {1+3 x+x \log (\log (3))}{x}\right )} \, dx \\ & = \int \frac {e^{10} \left (-4-x (3+\log (\log (3)))+5 (1+x (3+\log (\log (3)))) \log \left (3+\frac {1}{x}+\log (\log (3))\right )-4 (1+x (3+\log (\log (3)))) \log ^2\left (3+\frac {1}{x}+\log (\log (3))\right )\right )}{(1+x (3+\log (\log (3)))) \left (x-4 x \log \left (3+\frac {1}{x}+\log (\log (3))\right )\right )^2} \, dx \\ & = e^{10} \int \frac {-4-x (3+\log (\log (3)))+5 (1+x (3+\log (\log (3)))) \log \left (3+\frac {1}{x}+\log (\log (3))\right )-4 (1+x (3+\log (\log (3)))) \log ^2\left (3+\frac {1}{x}+\log (\log (3))\right )}{(1+x (3+\log (\log (3)))) \left (x-4 x \log \left (3+\frac {1}{x}+\log (\log (3))\right )\right )^2} \, dx \\ & = e^{10} \int \left (-\frac {1}{4 x^2}+\frac {3}{x^2 (-1-x (3+\log (\log (3)))) \left (1-4 \log \left (3+\frac {1}{x}+\log (\log (3))\right )\right )^2}+\frac {3}{4 x^2 \left (-1+4 \log \left (3+\frac {1}{x}+\log (\log (3))\right )\right )}\right ) \, dx \\ & = \frac {e^{10}}{4 x}+\frac {1}{4} \left (3 e^{10}\right ) \int \frac {1}{x^2 \left (-1+4 \log \left (3+\frac {1}{x}+\log (\log (3))\right )\right )} \, dx+\left (3 e^{10}\right ) \int \frac {1}{x^2 (-1+x (-3-\log (\log (3)))) \left (1-4 \log \left (3+\frac {1}{x}+\log (\log (3))\right )\right )^2} \, dx \\ & = \frac {e^{10}}{4 x}-\frac {1}{4} \left (3 e^{10}\right ) \text {Subst}\left (\int \frac {1}{-1+4 \log (3+x+\log (\log (3)))} \, dx,x,\frac {1}{x}\right )+\left (3 e^{10}\right ) \int \left (-\frac {1}{x^2 \left (1-4 \log \left (3+\frac {1}{x}+\log (\log (3))\right )\right )^2}+\frac {3+\log (\log (3))}{x \left (1-4 \log \left (3+\frac {1}{x}+\log (\log (3))\right )\right )^2}+\frac {(3+\log (\log (3)))^2}{(-1-x (3+\log (\log (3)))) \left (1-4 \log \left (3+\frac {1}{x}+\log (\log (3))\right )\right )^2}\right ) \, dx \\ & = \frac {e^{10}}{4 x}-\frac {1}{4} \left (3 e^{10}\right ) \text {Subst}\left (\int \frac {1}{-1+4 \log (x)} \, dx,x,3+\frac {1}{x}+\log (\log (3))\right )-\left (3 e^{10}\right ) \int \frac {1}{x^2 \left (1-4 \log \left (3+\frac {1}{x}+\log (\log (3))\right )\right )^2} \, dx+\left (3 e^{10} (3+\log (\log (3)))\right ) \int \frac {1}{x \left (1-4 \log \left (3+\frac {1}{x}+\log (\log (3))\right )\right )^2} \, dx+\left (3 e^{10} (3+\log (\log (3)))^2\right ) \int \frac {1}{(-1+x (-3-\log (\log (3)))) \left (1-4 \log \left (3+\frac {1}{x}+\log (\log (3))\right )\right )^2} \, dx \\ & = \frac {e^{10}}{4 x}-\frac {1}{4} \left (3 e^{10}\right ) \text {Subst}\left (\int \frac {e^x}{-1+4 x} \, dx,x,\log \left (3+\frac {1}{x}+\log (\log (3))\right )\right )+\left (3 e^{10}\right ) \text {Subst}\left (\int \frac {1}{(1-4 \log (3+x+\log (\log (3))))^2} \, dx,x,\frac {1}{x}\right )+\left (3 e^{10} (3+\log (\log (3)))\right ) \int \frac {1}{x \left (1-4 \log \left (3+\frac {1}{x}+\log (\log (3))\right )\right )^2} \, dx+\left (3 e^{10} (3+\log (\log (3)))^2\right ) \int \frac {1}{(-1+x (-3-\log (\log (3)))) \left (1-4 \log \left (3+\frac {1}{x}+\log (\log (3))\right )\right )^2} \, dx \\ & = \frac {e^{10}}{4 x}-\frac {3}{16} e^{41/4} \text {Ei}\left (\frac {1}{4} \left (-1+4 \log \left (3+\frac {1}{x}+\log (\log (3))\right )\right )\right )+\left (3 e^{10}\right ) \text {Subst}\left (\int \frac {1}{(1-4 \log (x))^2} \, dx,x,3+\frac {1}{x}+\log (\log (3))\right )+\left (3 e^{10} (3+\log (\log (3)))\right ) \int \frac {1}{x \left (1-4 \log \left (3+\frac {1}{x}+\log (\log (3))\right )\right )^2} \, dx+\left (3 e^{10} (3+\log (\log (3)))^2\right ) \int \frac {1}{(-1+x (-3-\log (\log (3)))) \left (1-4 \log \left (3+\frac {1}{x}+\log (\log (3))\right )\right )^2} \, dx \\ & = \frac {e^{10}}{4 x}-\frac {3}{16} e^{41/4} \text {Ei}\left (\frac {1}{4} \left (-1+4 \log \left (3+\frac {1}{x}+\log (\log (3))\right )\right )\right )+\frac {3 e^{10} \left (3+\frac {1}{x}+\log (\log (3))\right )}{4 \left (1-4 \log \left (3+\frac {1}{x}+\log (\log (3))\right )\right )}-\frac {1}{4} \left (3 e^{10}\right ) \text {Subst}\left (\int \frac {1}{1-4 \log (x)} \, dx,x,3+\frac {1}{x}+\log (\log (3))\right )+\left (3 e^{10} (3+\log (\log (3)))\right ) \int \frac {1}{x \left (1-4 \log \left (3+\frac {1}{x}+\log (\log (3))\right )\right )^2} \, dx+\left (3 e^{10} (3+\log (\log (3)))^2\right ) \int \frac {1}{(-1+x (-3-\log (\log (3)))) \left (1-4 \log \left (3+\frac {1}{x}+\log (\log (3))\right )\right )^2} \, dx \\ & = \frac {e^{10}}{4 x}-\frac {3}{16} e^{41/4} \text {Ei}\left (\frac {1}{4} \left (-1+4 \log \left (3+\frac {1}{x}+\log (\log (3))\right )\right )\right )+\frac {3 e^{10} \left (3+\frac {1}{x}+\log (\log (3))\right )}{4 \left (1-4 \log \left (3+\frac {1}{x}+\log (\log (3))\right )\right )}-\frac {1}{4} \left (3 e^{10}\right ) \text {Subst}\left (\int \frac {e^x}{1-4 x} \, dx,x,\log \left (3+\frac {1}{x}+\log (\log (3))\right )\right )+\left (3 e^{10} (3+\log (\log (3)))\right ) \int \frac {1}{x \left (1-4 \log \left (3+\frac {1}{x}+\log (\log (3))\right )\right )^2} \, dx+\left (3 e^{10} (3+\log (\log (3)))^2\right ) \int \frac {1}{(-1+x (-3-\log (\log (3)))) \left (1-4 \log \left (3+\frac {1}{x}+\log (\log (3))\right )\right )^2} \, dx \\ & = \frac {e^{10}}{4 x}+\frac {3 e^{10} \left (3+\frac {1}{x}+\log (\log (3))\right )}{4 \left (1-4 \log \left (3+\frac {1}{x}+\log (\log (3))\right )\right )}+\left (3 e^{10} (3+\log (\log (3)))\right ) \int \frac {1}{x \left (1-4 \log \left (3+\frac {1}{x}+\log (\log (3))\right )\right )^2} \, dx+\left (3 e^{10} (3+\log (\log (3)))^2\right ) \int \frac {1}{(-1+x (-3-\log (\log (3)))) \left (1-4 \log \left (3+\frac {1}{x}+\log (\log (3))\right )\right )^2} \, dx \\ \end{align*}
Time = 0.57 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {e^{10} (-4-3 x)-e^{10} x \log (\log (3))+\left (e^{10} (5+15 x)+5 e^{10} x \log (\log (3))\right ) \log \left (\frac {1+3 x+x \log (\log (3))}{x}\right )+\left (e^{10} (-4-12 x)-4 e^{10} x \log (\log (3))\right ) \log ^2\left (\frac {1+3 x+x \log (\log (3))}{x}\right )}{x^2+3 x^3+x^3 \log (\log (3))+\left (-8 x^2-24 x^3-8 x^3 \log (\log (3))\right ) \log \left (\frac {1+3 x+x \log (\log (3))}{x}\right )+\left (16 x^2+48 x^3+16 x^3 \log (\log (3))\right ) \log ^2\left (\frac {1+3 x+x \log (\log (3))}{x}\right )} \, dx=-\frac {e^{10} \left (1-\log \left (3+\frac {1}{x}+\log (\log (3))\right )\right )}{x \left (-1+4 \log \left (3+\frac {1}{x}+\log (\log (3))\right )\right )} \]
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Time = 1.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32
method | result | size |
risch | \(\frac {{\mathrm e}^{10}}{4 x}-\frac {3 \,{\mathrm e}^{10}}{4 x \left (-1+4 \ln \left (\frac {\ln \left (\ln \left (3\right )\right ) x +3 x +1}{x}\right )\right )}\) | \(37\) |
norman | \(\frac {{\mathrm e}^{10} \ln \left (\frac {\ln \left (\ln \left (3\right )\right ) x +3 x +1}{x}\right )-{\mathrm e}^{10}}{x \left (-1+4 \ln \left (\frac {\ln \left (\ln \left (3\right )\right ) x +3 x +1}{x}\right )\right )}\) | \(53\) |
derivativedivides | \(\frac {3 \,{\mathrm e}^{10} \ln \left (\ln \left (3\right )\right )}{4 \left (4 \ln \left (3+\frac {1}{x}+\ln \left (\ln \left (3\right )\right )\right )-1\right )}+{\mathrm e}^{10} \left (\frac {9}{4 \left (4 \ln \left (3+\frac {1}{x}+\ln \left (\ln \left (3\right )\right )\right )-1\right )}-\frac {3 \left (3+\frac {1}{x}+\ln \left (\ln \left (3\right )\right )\right )}{16 \left (\ln \left (3+\frac {1}{x}+\ln \left (\ln \left (3\right )\right )\right )-\frac {1}{4}\right )}+\frac {\ln \left (\ln \left (3\right )\right )}{4}+\frac {3}{4}+\frac {1}{4 x}\right )\) | \(83\) |
default | \(\frac {3 \,{\mathrm e}^{10} \ln \left (\ln \left (3\right )\right )}{4 \left (4 \ln \left (3+\frac {1}{x}+\ln \left (\ln \left (3\right )\right )\right )-1\right )}+{\mathrm e}^{10} \left (\frac {9}{4 \left (4 \ln \left (3+\frac {1}{x}+\ln \left (\ln \left (3\right )\right )\right )-1\right )}-\frac {3 \left (3+\frac {1}{x}+\ln \left (\ln \left (3\right )\right )\right )}{16 \left (\ln \left (3+\frac {1}{x}+\ln \left (\ln \left (3\right )\right )\right )-\frac {1}{4}\right )}+\frac {\ln \left (\ln \left (3\right )\right )}{4}+\frac {3}{4}+\frac {1}{4 x}\right )\) | \(83\) |
parallelrisch | \(-\frac {8 \ln \left (\ln \left (3\right )\right ) {\mathrm e}^{10} x \ln \left (\frac {\ln \left (\ln \left (3\right )\right ) x +3 x +1}{x}\right )-2 x \,{\mathrm e}^{10} \ln \left (\ln \left (3\right )\right )+24 \,{\mathrm e}^{10} x \ln \left (\frac {\ln \left (\ln \left (3\right )\right ) x +3 x +1}{x}\right )-6 x \,{\mathrm e}^{10}-4 \,{\mathrm e}^{10} \ln \left (\frac {\ln \left (\ln \left (3\right )\right ) x +3 x +1}{x}\right )+4 \,{\mathrm e}^{10}}{4 x \left (-1+4 \ln \left (\frac {\ln \left (\ln \left (3\right )\right ) x +3 x +1}{x}\right )\right )}\) | \(119\) |
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Time = 0.24 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.71 \[ \int \frac {e^{10} (-4-3 x)-e^{10} x \log (\log (3))+\left (e^{10} (5+15 x)+5 e^{10} x \log (\log (3))\right ) \log \left (\frac {1+3 x+x \log (\log (3))}{x}\right )+\left (e^{10} (-4-12 x)-4 e^{10} x \log (\log (3))\right ) \log ^2\left (\frac {1+3 x+x \log (\log (3))}{x}\right )}{x^2+3 x^3+x^3 \log (\log (3))+\left (-8 x^2-24 x^3-8 x^3 \log (\log (3))\right ) \log \left (\frac {1+3 x+x \log (\log (3))}{x}\right )+\left (16 x^2+48 x^3+16 x^3 \log (\log (3))\right ) \log ^2\left (\frac {1+3 x+x \log (\log (3))}{x}\right )} \, dx=\frac {e^{10} \log \left (\frac {x \log \left (\log \left (3\right )\right ) + 3 \, x + 1}{x}\right ) - e^{10}}{4 \, x \log \left (\frac {x \log \left (\log \left (3\right )\right ) + 3 \, x + 1}{x}\right ) - x} \]
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Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {e^{10} (-4-3 x)-e^{10} x \log (\log (3))+\left (e^{10} (5+15 x)+5 e^{10} x \log (\log (3))\right ) \log \left (\frac {1+3 x+x \log (\log (3))}{x}\right )+\left (e^{10} (-4-12 x)-4 e^{10} x \log (\log (3))\right ) \log ^2\left (\frac {1+3 x+x \log (\log (3))}{x}\right )}{x^2+3 x^3+x^3 \log (\log (3))+\left (-8 x^2-24 x^3-8 x^3 \log (\log (3))\right ) \log \left (\frac {1+3 x+x \log (\log (3))}{x}\right )+\left (16 x^2+48 x^3+16 x^3 \log (\log (3))\right ) \log ^2\left (\frac {1+3 x+x \log (\log (3))}{x}\right )} \, dx=- \frac {3 e^{10}}{16 x \log {\left (\frac {x \log {\left (\log {\left (3 \right )} \right )} + 3 x + 1}{x} \right )} - 4 x} + \frac {e^{10}}{4 x} \]
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Time = 0.33 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75 \[ \int \frac {e^{10} (-4-3 x)-e^{10} x \log (\log (3))+\left (e^{10} (5+15 x)+5 e^{10} x \log (\log (3))\right ) \log \left (\frac {1+3 x+x \log (\log (3))}{x}\right )+\left (e^{10} (-4-12 x)-4 e^{10} x \log (\log (3))\right ) \log ^2\left (\frac {1+3 x+x \log (\log (3))}{x}\right )}{x^2+3 x^3+x^3 \log (\log (3))+\left (-8 x^2-24 x^3-8 x^3 \log (\log (3))\right ) \log \left (\frac {1+3 x+x \log (\log (3))}{x}\right )+\left (16 x^2+48 x^3+16 x^3 \log (\log (3))\right ) \log ^2\left (\frac {1+3 x+x \log (\log (3))}{x}\right )} \, dx=\frac {e^{10} \log \left (x {\left (\log \left (\log \left (3\right )\right ) + 3\right )} + 1\right ) - e^{10} \log \left (x\right ) - e^{10}}{4 \, x \log \left (x {\left (\log \left (\log \left (3\right )\right ) + 3\right )} + 1\right ) - 4 \, x \log \left (x\right ) - x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (25) = 50\).
Time = 0.35 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.82 \[ \int \frac {e^{10} (-4-3 x)-e^{10} x \log (\log (3))+\left (e^{10} (5+15 x)+5 e^{10} x \log (\log (3))\right ) \log \left (\frac {1+3 x+x \log (\log (3))}{x}\right )+\left (e^{10} (-4-12 x)-4 e^{10} x \log (\log (3))\right ) \log ^2\left (\frac {1+3 x+x \log (\log (3))}{x}\right )}{x^2+3 x^3+x^3 \log (\log (3))+\left (-8 x^2-24 x^3-8 x^3 \log (\log (3))\right ) \log \left (\frac {1+3 x+x \log (\log (3))}{x}\right )+\left (16 x^2+48 x^3+16 x^3 \log (\log (3))\right ) \log ^2\left (\frac {1+3 x+x \log (\log (3))}{x}\right )} \, dx=\frac {e^{10} \log \left (x \log \left (\log \left (3\right )\right ) + 3 \, x + 1\right ) - e^{10} \log \left (x\right ) - e^{10}}{4 \, x \log \left (x \log \left (\log \left (3\right )\right ) + 3 \, x + 1\right ) - 4 \, x \log \left (x\right ) - x} \]
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Time = 12.83 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.71 \[ \int \frac {e^{10} (-4-3 x)-e^{10} x \log (\log (3))+\left (e^{10} (5+15 x)+5 e^{10} x \log (\log (3))\right ) \log \left (\frac {1+3 x+x \log (\log (3))}{x}\right )+\left (e^{10} (-4-12 x)-4 e^{10} x \log (\log (3))\right ) \log ^2\left (\frac {1+3 x+x \log (\log (3))}{x}\right )}{x^2+3 x^3+x^3 \log (\log (3))+\left (-8 x^2-24 x^3-8 x^3 \log (\log (3))\right ) \log \left (\frac {1+3 x+x \log (\log (3))}{x}\right )+\left (16 x^2+48 x^3+16 x^3 \log (\log (3))\right ) \log ^2\left (\frac {1+3 x+x \log (\log (3))}{x}\right )} \, dx=\frac {{\mathrm {e}}^{10}\,\left (36\,x+12\,x\,\ln \left (\ln \left (3\right )\right )+16\right )}{64\,x}-\frac {\frac {{\mathrm {e}}^{10}\,\left (9\,x+3\,x\,\ln \left (\ln \left (3\right )\right )+16\right )}{16}-\frac {{\mathrm {e}}^{10}\,\left (36\,x+12\,x\,\ln \left (\ln \left (3\right )\right )+16\right )}{64}}{x\,\left (4\,\ln \left (\frac {3\,x+x\,\ln \left (\ln \left (3\right )\right )+1}{x}\right )-1\right )} \]
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