Integrand size = 178, antiderivative size = 28 \[ \int \frac {e^{27} \left (-250 e^3-750 e^5-750 e^7-250 e^9\right )}{-216+e^9 \left (e (-1080+540 x)+e^3 (-1080+540 x)\right )+e^{18} \left (e^4 \left (-3600+3600 x-900 x^2\right )+e^2 \left (-1800+1800 x-450 x^2\right )+e^6 \left (-1800+1800 x-450 x^2\right )\right )+e^{27} \left (e^3 \left (-1000+1500 x-750 x^2+125 x^3\right )+e^9 \left (-1000+1500 x-750 x^2+125 x^3\right )+e^5 \left (-3000+4500 x-2250 x^2+375 x^3\right )+e^7 \left (-3000+4500 x-2250 x^2+375 x^3\right )\right )} \, dx=\frac {x^2}{\left (-2 x-\frac {6 x}{5 e^9 \left (e+e^3\right )}+x^2\right )^2} \]
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Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {12, 2083, 6, 32} \[ \int \frac {e^{27} \left (-250 e^3-750 e^5-750 e^7-250 e^9\right )}{-216+e^9 \left (e (-1080+540 x)+e^3 (-1080+540 x)\right )+e^{18} \left (e^4 \left (-3600+3600 x-900 x^2\right )+e^2 \left (-1800+1800 x-450 x^2\right )+e^6 \left (-1800+1800 x-450 x^2\right )\right )+e^{27} \left (e^3 \left (-1000+1500 x-750 x^2+125 x^3\right )+e^9 \left (-1000+1500 x-750 x^2+125 x^3\right )+e^5 \left (-3000+4500 x-2250 x^2+375 x^3\right )+e^7 \left (-3000+4500 x-2250 x^2+375 x^3\right )\right )} \, dx=\frac {25 e^{20} \left (1+e^2\right )^2}{\left (2 \left (3+5 e^{10}+5 e^{12}\right )-5 e^{10} \left (1+e^2\right ) x\right )^2} \]
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Rule 6
Rule 12
Rule 32
Rule 2083
Rubi steps \begin{align*} \text {integral}& = -\left (\left (250 e^{30} \left (1+e^2\right )^3\right ) \int \frac {1}{-216+e^9 \left (e (-1080+540 x)+e^3 (-1080+540 x)\right )+e^{18} \left (e^4 \left (-3600+3600 x-900 x^2\right )+e^2 \left (-1800+1800 x-450 x^2\right )+e^6 \left (-1800+1800 x-450 x^2\right )\right )+e^{27} \left (e^3 \left (-1000+1500 x-750 x^2+125 x^3\right )+e^9 \left (-1000+1500 x-750 x^2+125 x^3\right )+e^5 \left (-3000+4500 x-2250 x^2+375 x^3\right )+e^7 \left (-3000+4500 x-2250 x^2+375 x^3\right )\right )} \, dx\right ) \\ & = -\left (\left (250 e^{30} \left (1+e^2\right )^3\right ) \int \frac {1}{\left (-6-10 e^{10}-10 e^{12}+5 e^{10} x+5 e^{12} x\right )^3} \, dx\right ) \\ & = -\left (\left (250 e^{30} \left (1+e^2\right )^3\right ) \int \frac {1}{\left (-6-10 e^{10}-10 e^{12}+5 \left (e^{10}+e^{12}\right ) x\right )^3} \, dx\right ) \\ & = \frac {25 e^{20} \left (1+e^2\right )^2}{\left (2 \left (3+5 e^{10}+5 e^{12}\right )-5 e^{10} \left (1+e^2\right ) x\right )^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {e^{27} \left (-250 e^3-750 e^5-750 e^7-250 e^9\right )}{-216+e^9 \left (e (-1080+540 x)+e^3 (-1080+540 x)\right )+e^{18} \left (e^4 \left (-3600+3600 x-900 x^2\right )+e^2 \left (-1800+1800 x-450 x^2\right )+e^6 \left (-1800+1800 x-450 x^2\right )\right )+e^{27} \left (e^3 \left (-1000+1500 x-750 x^2+125 x^3\right )+e^9 \left (-1000+1500 x-750 x^2+125 x^3\right )+e^5 \left (-3000+4500 x-2250 x^2+375 x^3\right )+e^7 \left (-3000+4500 x-2250 x^2+375 x^3\right )\right )} \, dx=\frac {25 e^{20} \left (1+e^2\right )^2}{\left (-6+5 e^{10} (-2+x)+5 e^{12} (-2+x)\right )^2} \]
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Time = 0.40 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.86
method | result | size |
norman | \(\frac {25 \,{\mathrm e}^{18} \left ({\mathrm e}^{2}+2 \,{\mathrm e} \,{\mathrm e}^{3}+{\mathrm e}^{6}\right )}{\left (5 \,{\mathrm e} \,{\mathrm e}^{9} x +5 \,{\mathrm e}^{3} {\mathrm e}^{9} x -10 \,{\mathrm e} \,{\mathrm e}^{9}-10 \,{\mathrm e}^{3} {\mathrm e}^{9}-6\right )^{2}}\) | \(52\) |
gosper | \(\frac {25 \,{\mathrm e}^{18} \left ({\mathrm e}^{2}+2 \,{\mathrm e} \,{\mathrm e}^{3}+{\mathrm e}^{6}\right )}{25 \,{\mathrm e}^{2} {\mathrm e}^{18} x^{2}+50 \,{\mathrm e} \,{\mathrm e}^{3} {\mathrm e}^{18} x^{2}+25 \,{\mathrm e}^{6} {\mathrm e}^{18} x^{2}-100 \,{\mathrm e}^{2} {\mathrm e}^{18} x -200 \,{\mathrm e} \,{\mathrm e}^{3} {\mathrm e}^{18} x -100 \,{\mathrm e}^{6} {\mathrm e}^{18} x +100 \,{\mathrm e}^{2} {\mathrm e}^{18}+200 \,{\mathrm e} \,{\mathrm e}^{3} {\mathrm e}^{18}+100 \,{\mathrm e}^{6} {\mathrm e}^{18}-60 \,{\mathrm e} \,{\mathrm e}^{9} x -60 \,{\mathrm e}^{3} {\mathrm e}^{9} x +120 \,{\mathrm e} \,{\mathrm e}^{9}+120 \,{\mathrm e}^{3} {\mathrm e}^{9}+36}\) | \(154\) |
parallelrisch | \(\frac {\left (-250 \,{\mathrm e}^{9}-750 \,{\mathrm e} \,{\mathrm e}^{6}-750 \,{\mathrm e}^{2} {\mathrm e}^{3}-250 \,{\mathrm e}^{3}\right ) {\mathrm e}^{9} \left (-5 \,{\mathrm e} \,{\mathrm e}^{9}-5 \,{\mathrm e}^{3} {\mathrm e}^{9}\right )}{50 \left ({\mathrm e}+{\mathrm e}^{3}\right )^{2} \left (25 \,{\mathrm e}^{2} {\mathrm e}^{18} x^{2}+50 \,{\mathrm e} \,{\mathrm e}^{3} {\mathrm e}^{18} x^{2}+25 \,{\mathrm e}^{6} {\mathrm e}^{18} x^{2}-100 \,{\mathrm e}^{2} {\mathrm e}^{18} x -200 \,{\mathrm e} \,{\mathrm e}^{3} {\mathrm e}^{18} x -100 \,{\mathrm e}^{6} {\mathrm e}^{18} x +100 \,{\mathrm e}^{2} {\mathrm e}^{18}+200 \,{\mathrm e} \,{\mathrm e}^{3} {\mathrm e}^{18}+100 \,{\mathrm e}^{6} {\mathrm e}^{18}-60 \,{\mathrm e} \,{\mathrm e}^{9} x -60 \,{\mathrm e}^{3} {\mathrm e}^{9} x +120 \,{\mathrm e} \,{\mathrm e}^{9}+120 \,{\mathrm e}^{3} {\mathrm e}^{9}+36\right )}\) | \(186\) |
default | \(\frac {50 \left (-{\mathrm e}^{9}-3 \,{\mathrm e} \,{\mathrm e}^{6}-3 \,{\mathrm e}^{2} {\mathrm e}^{3}-{\mathrm e}^{3}\right ) {\mathrm e}^{27} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (125 \,{\mathrm e}^{36}+125 \,{\mathrm e}^{30}+375 \,{\mathrm e}^{32}+375 \,{\mathrm e}^{34}\right ) \textit {\_Z}^{3}+\left (-450 \,{\mathrm e}^{20}-750 \,{\mathrm e}^{30}-750 \,{\mathrm e}^{36}-450 \,{\mathrm e}^{24}-900 \,{\mathrm e}^{22}-2250 \,{\mathrm e}^{32}-2250 \,{\mathrm e}^{34}\right ) \textit {\_Z}^{2}+\left (540 \,{\mathrm e}^{10}+1800 \,{\mathrm e}^{20}+1500 \,{\mathrm e}^{30}+1500 \,{\mathrm e}^{36}+1800 \,{\mathrm e}^{24}+540 \,{\mathrm e}^{12}+4500 \,{\mathrm e}^{34}+3600 \,{\mathrm e}^{22}+4500 \,{\mathrm e}^{32}\right ) \textit {\_Z} -216-1000 \,{\mathrm e}^{36}-3000 \,{\mathrm e}^{32}-3600 \,{\mathrm e}^{22}-1080 \,{\mathrm e}^{12}-1800 \,{\mathrm e}^{24}-1800 \,{\mathrm e}^{20}-1080 \,{\mathrm e}^{10}-3000 \,{\mathrm e}^{34}-1000 \,{\mathrm e}^{30}\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{-60 \textit {\_R} \,{\mathrm e}^{20}+36 \,{\mathrm e}^{10}+120 \,{\mathrm e}^{20}+100 \,{\mathrm e}^{30}+100 \,{\mathrm e}^{36}+25 \textit {\_R}^{2} {\mathrm e}^{36}-100 \textit {\_R} \,{\mathrm e}^{30}-100 \textit {\_R} \,{\mathrm e}^{36}+120 \,{\mathrm e}^{24}+25 \textit {\_R}^{2} {\mathrm e}^{30}+36 \,{\mathrm e}^{12}-60 \textit {\_R} \,{\mathrm e}^{24}+300 \,{\mathrm e}^{34}-120 \textit {\_R} \,{\mathrm e}^{22}+240 \,{\mathrm e}^{22}+75 \textit {\_R}^{2} {\mathrm e}^{32}+75 \textit {\_R}^{2} {\mathrm e}^{34}-300 \textit {\_R} \,{\mathrm e}^{32}-300 \textit {\_R} \,{\mathrm e}^{34}+300 \,{\mathrm e}^{32}}\right )}{3}\) | \(280\) |
risch | \(\frac {{\mathrm e}^{17} {\mathrm e}^{9}}{\left ({\mathrm e}^{2}+1\right ) \left (x^{2} {\mathrm e}^{24}-4 x \,{\mathrm e}^{24}+4 \,{\mathrm e}^{24}+2 x^{2} {\mathrm e}^{22}-8 x \,{\mathrm e}^{22}+8 \,{\mathrm e}^{22}+x^{2} {\mathrm e}^{20}-4 x \,{\mathrm e}^{20}+4 \,{\mathrm e}^{20}-\frac {12 x \,{\mathrm e}^{12}}{5}+\frac {24 \,{\mathrm e}^{12}}{5}-\frac {12 x \,{\mathrm e}^{10}}{5}+\frac {24 \,{\mathrm e}^{10}}{5}+\frac {36}{25}\right )}+\frac {3 \,{\mathrm e}^{17} {\mathrm e}^{7}}{\left ({\mathrm e}^{2}+1\right ) \left (x^{2} {\mathrm e}^{24}-4 x \,{\mathrm e}^{24}+4 \,{\mathrm e}^{24}+2 x^{2} {\mathrm e}^{22}-8 x \,{\mathrm e}^{22}+8 \,{\mathrm e}^{22}+x^{2} {\mathrm e}^{20}-4 x \,{\mathrm e}^{20}+4 \,{\mathrm e}^{20}-\frac {12 x \,{\mathrm e}^{12}}{5}+\frac {24 \,{\mathrm e}^{12}}{5}-\frac {12 x \,{\mathrm e}^{10}}{5}+\frac {24 \,{\mathrm e}^{10}}{5}+\frac {36}{25}\right )}+\frac {3 \,{\mathrm e}^{17} {\mathrm e}^{5}}{\left ({\mathrm e}^{2}+1\right ) \left (x^{2} {\mathrm e}^{24}-4 x \,{\mathrm e}^{24}+4 \,{\mathrm e}^{24}+2 x^{2} {\mathrm e}^{22}-8 x \,{\mathrm e}^{22}+8 \,{\mathrm e}^{22}+x^{2} {\mathrm e}^{20}-4 x \,{\mathrm e}^{20}+4 \,{\mathrm e}^{20}-\frac {12 x \,{\mathrm e}^{12}}{5}+\frac {24 \,{\mathrm e}^{12}}{5}-\frac {12 x \,{\mathrm e}^{10}}{5}+\frac {24 \,{\mathrm e}^{10}}{5}+\frac {36}{25}\right )}+\frac {{\mathrm e}^{17} {\mathrm e}^{3}}{\left ({\mathrm e}^{2}+1\right ) \left (x^{2} {\mathrm e}^{24}-4 x \,{\mathrm e}^{24}+4 \,{\mathrm e}^{24}+2 x^{2} {\mathrm e}^{22}-8 x \,{\mathrm e}^{22}+8 \,{\mathrm e}^{22}+x^{2} {\mathrm e}^{20}-4 x \,{\mathrm e}^{20}+4 \,{\mathrm e}^{20}-\frac {12 x \,{\mathrm e}^{12}}{5}+\frac {24 \,{\mathrm e}^{12}}{5}-\frac {12 x \,{\mathrm e}^{10}}{5}+\frac {24 \,{\mathrm e}^{10}}{5}+\frac {36}{25}\right )}\) | \(320\) |
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Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (28) = 56\).
Time = 0.26 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.32 \[ \int \frac {e^{27} \left (-250 e^3-750 e^5-750 e^7-250 e^9\right )}{-216+e^9 \left (e (-1080+540 x)+e^3 (-1080+540 x)\right )+e^{18} \left (e^4 \left (-3600+3600 x-900 x^2\right )+e^2 \left (-1800+1800 x-450 x^2\right )+e^6 \left (-1800+1800 x-450 x^2\right )\right )+e^{27} \left (e^3 \left (-1000+1500 x-750 x^2+125 x^3\right )+e^9 \left (-1000+1500 x-750 x^2+125 x^3\right )+e^5 \left (-3000+4500 x-2250 x^2+375 x^3\right )+e^7 \left (-3000+4500 x-2250 x^2+375 x^3\right )\right )} \, dx=\frac {25 \, {\left (e^{24} + 2 \, e^{22} + e^{20}\right )}}{25 \, {\left (x^{2} - 4 \, x + 4\right )} e^{24} + 50 \, {\left (x^{2} - 4 \, x + 4\right )} e^{22} + 25 \, {\left (x^{2} - 4 \, x + 4\right )} e^{20} - 60 \, {\left (x - 2\right )} e^{12} - 60 \, {\left (x - 2\right )} e^{10} + 36} \]
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Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (26) = 52\).
Time = 0.64 (sec) , antiderivative size = 131, normalized size of antiderivative = 4.68 \[ \int \frac {e^{27} \left (-250 e^3-750 e^5-750 e^7-250 e^9\right )}{-216+e^9 \left (e (-1080+540 x)+e^3 (-1080+540 x)\right )+e^{18} \left (e^4 \left (-3600+3600 x-900 x^2\right )+e^2 \left (-1800+1800 x-450 x^2\right )+e^6 \left (-1800+1800 x-450 x^2\right )\right )+e^{27} \left (e^3 \left (-1000+1500 x-750 x^2+125 x^3\right )+e^9 \left (-1000+1500 x-750 x^2+125 x^3\right )+e^5 \left (-3000+4500 x-2250 x^2+375 x^3\right )+e^7 \left (-3000+4500 x-2250 x^2+375 x^3\right )\right )} \, dx=- \frac {- 250 e^{36} - 750 e^{34} - 750 e^{32} - 250 e^{30}}{x^{2} \cdot \left (250 e^{30} + 750 e^{32} + 750 e^{34} + 250 e^{36}\right ) + x \left (- 1000 e^{36} - 3000 e^{34} - 3000 e^{32} - 1000 e^{30} - 600 e^{24} - 1200 e^{22} - 600 e^{20}\right ) + 360 e^{10} + 360 e^{12} + 1200 e^{20} + 2400 e^{22} + 1200 e^{24} + 1000 e^{30} + 3000 e^{32} + 3000 e^{34} + 1000 e^{36}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (28) = 56\).
Time = 0.19 (sec) , antiderivative size = 106, normalized size of antiderivative = 3.79 \[ \int \frac {e^{27} \left (-250 e^3-750 e^5-750 e^7-250 e^9\right )}{-216+e^9 \left (e (-1080+540 x)+e^3 (-1080+540 x)\right )+e^{18} \left (e^4 \left (-3600+3600 x-900 x^2\right )+e^2 \left (-1800+1800 x-450 x^2\right )+e^6 \left (-1800+1800 x-450 x^2\right )\right )+e^{27} \left (e^3 \left (-1000+1500 x-750 x^2+125 x^3\right )+e^9 \left (-1000+1500 x-750 x^2+125 x^3\right )+e^5 \left (-3000+4500 x-2250 x^2+375 x^3\right )+e^7 \left (-3000+4500 x-2250 x^2+375 x^3\right )\right )} \, dx=\frac {25 \, {\left (e^{9} + 3 \, e^{7} + 3 \, e^{5} + e^{3}\right )} e^{27}}{25 \, x^{2} {\left (e^{36} + 3 \, e^{34} + 3 \, e^{32} + e^{30}\right )} - 20 \, x {\left (5 \, e^{36} + 15 \, e^{34} + 15 \, e^{32} + 5 \, e^{30} + 3 \, e^{24} + 6 \, e^{22} + 3 \, e^{20}\right )} + 100 \, e^{36} + 300 \, e^{34} + 300 \, e^{32} + 100 \, e^{30} + 120 \, e^{24} + 240 \, e^{22} + 120 \, e^{20} + 36 \, e^{12} + 36 \, e^{10}} \]
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Time = 0.30 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64 \[ \int \frac {e^{27} \left (-250 e^3-750 e^5-750 e^7-250 e^9\right )}{-216+e^9 \left (e (-1080+540 x)+e^3 (-1080+540 x)\right )+e^{18} \left (e^4 \left (-3600+3600 x-900 x^2\right )+e^2 \left (-1800+1800 x-450 x^2\right )+e^6 \left (-1800+1800 x-450 x^2\right )\right )+e^{27} \left (e^3 \left (-1000+1500 x-750 x^2+125 x^3\right )+e^9 \left (-1000+1500 x-750 x^2+125 x^3\right )+e^5 \left (-3000+4500 x-2250 x^2+375 x^3\right )+e^7 \left (-3000+4500 x-2250 x^2+375 x^3\right )\right )} \, dx=\frac {25 \, {\left (e^{9} + 3 \, e^{7} + 3 \, e^{5} + e^{3}\right )} e^{27}}{{\left (5 \, x e^{12} + 5 \, x e^{10} - 10 \, e^{12} - 10 \, e^{10} - 6\right )}^{2} {\left (e^{12} + e^{10}\right )}} \]
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Time = 12.71 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.68 \[ \int \frac {e^{27} \left (-250 e^3-750 e^5-750 e^7-250 e^9\right )}{-216+e^9 \left (e (-1080+540 x)+e^3 (-1080+540 x)\right )+e^{18} \left (e^4 \left (-3600+3600 x-900 x^2\right )+e^2 \left (-1800+1800 x-450 x^2\right )+e^6 \left (-1800+1800 x-450 x^2\right )\right )+e^{27} \left (e^3 \left (-1000+1500 x-750 x^2+125 x^3\right )+e^9 \left (-1000+1500 x-750 x^2+125 x^3\right )+e^5 \left (-3000+4500 x-2250 x^2+375 x^3\right )+e^7 \left (-3000+4500 x-2250 x^2+375 x^3\right )\right )} \, dx=\frac {25\,{\mathrm {e}}^{20}\,{\left ({\mathrm {e}}^2+1\right )}^2}{\left (25\,{\mathrm {e}}^{20}+50\,{\mathrm {e}}^{22}+25\,{\mathrm {e}}^{24}\right )\,x^2+\left (-60\,{\mathrm {e}}^{10}-60\,{\mathrm {e}}^{12}-100\,{\mathrm {e}}^{20}-200\,{\mathrm {e}}^{22}-100\,{\mathrm {e}}^{24}\right )\,x+120\,{\mathrm {e}}^{10}+120\,{\mathrm {e}}^{12}+100\,{\mathrm {e}}^{20}+200\,{\mathrm {e}}^{22}+100\,{\mathrm {e}}^{24}+36} \]
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