Integrand size = 184, antiderivative size = 30 \[ \int \frac {e^4 (-207360-810 x)+e^9 \left (17694720+138240 x+270 x^2\right )}{81+e^4 \left (-14155776-110592 x-216 x^2\right )+e^{15} \left (-201326592-2359296 x-9216 x^2-12 x^3\right )+e^{20} \left (4294967296+67108864 x+393216 x^2+1024 x^3+x^4\right )+e^8 \left (618475290624+9663676416 x+56623104 x^2+147456 x^3+144 x^4\right )+e^3 \left (e^2 (-27648-108 x)+e^6 \left (2415919104+28311552 x+110592 x^2+144 x^3\right )\right )+e^6 \left (e^4 \left (3538944+27648 x+54 x^2\right )+e^8 \left (-103079215104-1610612736 x-9437184 x^2-24576 x^3-24 x^4\right )\right )} \, dx=\frac {5}{\frac {4}{3}-\left (\frac {e^3}{3}-\frac {1}{e^2 (256+x)}\right )^2} \]
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Leaf count is larger than twice the leaf count of optimal. \(120\) vs. \(2(30)=60\).
Time = 0.12 (sec) , antiderivative size = 120, normalized size of antiderivative = 4.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {1694, 12, 1828, 8} \[ \int \frac {e^4 (-207360-810 x)+e^9 \left (17694720+138240 x+270 x^2\right )}{81+e^4 \left (-14155776-110592 x-216 x^2\right )+e^{15} \left (-201326592-2359296 x-9216 x^2-12 x^3\right )+e^{20} \left (4294967296+67108864 x+393216 x^2+1024 x^3+x^4\right )+e^8 \left (618475290624+9663676416 x+56623104 x^2+147456 x^3+144 x^4\right )+e^3 \left (e^2 (-27648-108 x)+e^6 \left (2415919104+28311552 x+110592 x^2+144 x^3\right )\right )+e^6 \left (e^4 \left (3538944+27648 x+54 x^2\right )+e^8 \left (-103079215104-1610612736 x-9437184 x^2-24576 x^3-24 x^4\right )\right )} \, dx=-\frac {135 \left (3 \left (144-e^{12}\right )-2 e^5 \left (12-e^6\right ) \left (\left (12-e^6\right ) x-256 e^6+3 e+3072\right )\right )}{\left (12-e^6\right )^2 \left (108-e^4 \left (12-e^6\right )^2 \left (x+\frac {147456 e^8+144 e^9-24576 e^{14}-12 e^{15}+1024 e^{20}}{4 \left (144 e^8-24 e^{14}+e^{20}\right )}\right )^2\right )} \]
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Rule 8
Rule 12
Rule 1694
Rule 1828
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {270 e^4 \left (108 e-3 \left (144-e^{12}\right ) x+e^5 \left (12-e^6\right )^2 x^2\right )}{\left (108-e^4 \left (12-e^6\right )^2 x^2\right )^2} \, dx,x,\frac {147456 e^8+144 e^9-24576 e^{14}-12 e^{15}+1024 e^{20}}{4 \left (144 e^8-24 e^{14}+e^{20}\right )}+x\right ) \\ & = \left (270 e^4\right ) \text {Subst}\left (\int \frac {108 e-3 \left (144-e^{12}\right ) x+e^5 \left (12-e^6\right )^2 x^2}{\left (108-e^4 \left (12-e^6\right )^2 x^2\right )^2} \, dx,x,\frac {147456 e^8+144 e^9-24576 e^{14}-12 e^{15}+1024 e^{20}}{4 \left (144 e^8-24 e^{14}+e^{20}\right )}+x\right ) \\ & = -\frac {135 \left (3 \left (144-e^{12}\right )-2 e^5 \left (12-e^6\right ) \left (3072+3 e-256 e^6+\left (12-e^6\right ) x\right )\right )}{\left (12-e^6\right )^2 \left (108-e^4 \left (12-e^6\right )^2 \left (\frac {3072+3 e-256 e^6}{12-e^6}+x\right )^2\right )}-\frac {1}{4} \left (5 e^4\right ) \text {Subst}\left (\int 0 \, dx,x,\frac {147456 e^8+144 e^9-24576 e^{14}-12 e^{15}+1024 e^{20}}{4 \left (144 e^8-24 e^{14}+e^{20}\right )}+x\right ) \\ & = -\frac {135 \left (3 \left (144-e^{12}\right )-2 e^5 \left (12-e^6\right ) \left (3072+3 e-256 e^6+\left (12-e^6\right ) x\right )\right )}{\left (12-e^6\right )^2 \left (108-e^4 \left (12-e^6\right )^2 \left (\frac {3072+3 e-256 e^6}{12-e^6}+x\right )^2\right )} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.67 \[ \int \frac {e^4 (-207360-810 x)+e^9 \left (17694720+138240 x+270 x^2\right )}{81+e^4 \left (-14155776-110592 x-216 x^2\right )+e^{15} \left (-201326592-2359296 x-9216 x^2-12 x^3\right )+e^{20} \left (4294967296+67108864 x+393216 x^2+1024 x^3+x^4\right )+e^8 \left (618475290624+9663676416 x+56623104 x^2+147456 x^3+144 x^4\right )+e^3 \left (e^2 (-27648-108 x)+e^6 \left (2415919104+28311552 x+110592 x^2+144 x^3\right )\right )+e^6 \left (e^4 \left (3538944+27648 x+54 x^2\right )+e^8 \left (-103079215104-1610612736 x-9437184 x^2-24576 x^3-24 x^4\right )\right )} \, dx=\frac {135 \left (3-2 e^5 (256+x)\right )}{\left (-12+e^6\right ) \left (9-6 e^5 (256+x)-12 e^4 (256+x)^2+e^{10} (256+x)^2\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(71\) vs. \(2(26)=52\).
Time = 3.08 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.40
method | result | size |
risch | \(\frac {-\frac {270 \,{\mathrm e}^{5} x}{{\mathrm e}^{6}-12}-\frac {135 \left (512 \,{\mathrm e}^{5}-3\right )}{{\mathrm e}^{6}-12}}{x^{2} {\mathrm e}^{10}+512 x \,{\mathrm e}^{10}+65536 \,{\mathrm e}^{10}-6 x \,{\mathrm e}^{5}-12 x^{2} {\mathrm e}^{4}-1536 \,{\mathrm e}^{5}-6144 x \,{\mathrm e}^{4}-786432 \,{\mathrm e}^{4}+9}\) | \(72\) |
gosper | \(-\frac {135 \left (2 x \,{\mathrm e}^{2} {\mathrm e}^{3}+512 \,{\mathrm e}^{2} {\mathrm e}^{3}-3\right )}{\left ({\mathrm e}^{4} {\mathrm e}^{6} x^{2}+512 \,{\mathrm e}^{4} {\mathrm e}^{6} x +65536 \,{\mathrm e}^{4} {\mathrm e}^{6}-12 x^{2} {\mathrm e}^{4}-6144 x \,{\mathrm e}^{4}-786432 \,{\mathrm e}^{4}-6 x \,{\mathrm e}^{2} {\mathrm e}^{3}-1536 \,{\mathrm e}^{2} {\mathrm e}^{3}+9\right ) \left ({\mathrm e}^{6}-12\right )}\) | \(106\) |
parallelrisch | \(-\frac {\left (270 \,{\mathrm e}^{6} {\mathrm e}^{3} x +69120 \,{\mathrm e}^{6} {\mathrm e}^{3}-405 \,{\mathrm e}^{4}\right ) {\mathrm e}^{-4}}{\left ({\mathrm e}^{6}-12\right ) \left ({\mathrm e}^{4} {\mathrm e}^{6} x^{2}+512 \,{\mathrm e}^{4} {\mathrm e}^{6} x +65536 \,{\mathrm e}^{4} {\mathrm e}^{6}-12 x^{2} {\mathrm e}^{4}-6144 x \,{\mathrm e}^{4}-786432 \,{\mathrm e}^{4}-6 x \,{\mathrm e}^{2} {\mathrm e}^{3}-1536 \,{\mathrm e}^{2} {\mathrm e}^{3}+9\right )}\) | \(115\) |
norman | \(\frac {-\frac {270 \,{\mathrm e}^{2} {\mathrm e}^{3} x}{{\mathrm e}^{6}-12}-\frac {135 \left (512 \,{\mathrm e}^{2} {\mathrm e}^{3}-3\right )}{{\mathrm e}^{6}-12}}{{\mathrm e}^{4} {\mathrm e}^{6} x^{2}+512 \,{\mathrm e}^{4} {\mathrm e}^{6} x +65536 \,{\mathrm e}^{4} {\mathrm e}^{6}-12 x^{2} {\mathrm e}^{4}-6144 x \,{\mathrm e}^{4}-786432 \,{\mathrm e}^{4}-6 x \,{\mathrm e}^{2} {\mathrm e}^{3}-1536 \,{\mathrm e}^{2} {\mathrm e}^{3}+9}\) | \(116\) |
default | \(-\frac {135 \,{\mathrm e}^{4} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (-144 \,{\mathrm e}^{8}-{\mathrm e}^{20}+24 \,{\mathrm e}^{14}\right ) \textit {\_Z}^{4}+\left (-147456 \,{\mathrm e}^{8}+12 \,{\mathrm e}^{15}-1024 \,{\mathrm e}^{20}-144 \,{\mathrm e}^{9}+24576 \,{\mathrm e}^{14}\right ) \textit {\_Z}^{3}+\left (216 \,{\mathrm e}^{4}-54 \,{\mathrm e}^{10}-56623104 \,{\mathrm e}^{8}+9216 \,{\mathrm e}^{15}-393216 \,{\mathrm e}^{20}-110592 \,{\mathrm e}^{9}+9437184 \,{\mathrm e}^{14}\right ) \textit {\_Z}^{2}+\left (108 \,{\mathrm e}^{5}+110592 \,{\mathrm e}^{4}-27648 \,{\mathrm e}^{10}-9663676416 \,{\mathrm e}^{8}+2359296 \,{\mathrm e}^{15}-67108864 \,{\mathrm e}^{20}-28311552 \,{\mathrm e}^{9}+1610612736 \,{\mathrm e}^{14}\right ) \textit {\_Z} -81+103079215104 \,{\mathrm e}^{14}+201326592 \,{\mathrm e}^{15}-4294967296 \,{\mathrm e}^{20}-618475290624 \,{\mathrm e}^{8}-2415919104 \,{\mathrm e}^{9}-3538944 \,{\mathrm e}^{10}+14155776 \,{\mathrm e}^{4}+27648 \,{\mathrm e}^{5}\right )}{\sum }\frac {\left (\textit {\_R}^{2} {\mathrm e}^{5}+\left (512 \,{\mathrm e}^{5}-3\right ) \textit {\_R} -768+65536 \,{\mathrm e}^{5}\right ) \ln \left (x -\textit {\_R} \right )}{-110592 \textit {\_R}^{2} {\mathrm e}^{8}-2415919104 \,{\mathrm e}^{8}+589824 \,{\mathrm e}^{15}-196608 \textit {\_R} \,{\mathrm e}^{20}-6912 \,{\mathrm e}^{10}-7077888 \,{\mathrm e}^{9}+108 \textit {\_R} \,{\mathrm e}^{4}+4608 \textit {\_R} \,{\mathrm e}^{15}-55296 \textit {\_R} \,{\mathrm e}^{9}-16777216 \,{\mathrm e}^{20}+27648 \,{\mathrm e}^{4}+27 \,{\mathrm e}^{5}-28311552 \textit {\_R} \,{\mathrm e}^{8}-144 \textit {\_R}^{3} {\mathrm e}^{8}+4718592 \textit {\_R} \,{\mathrm e}^{14}+402653184 \,{\mathrm e}^{14}-108 \textit {\_R}^{2} {\mathrm e}^{9}-\textit {\_R}^{3} {\mathrm e}^{20}+18432 \,{\mathrm e}^{14} \textit {\_R}^{2}+9 \textit {\_R}^{2} {\mathrm e}^{15}-768 \textit {\_R}^{2} {\mathrm e}^{20}+24 \textit {\_R}^{3} {\mathrm e}^{14}-27 \textit {\_R} \,{\mathrm e}^{10}}\right )}{2}\) | \(308\) |
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Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (22) = 44\).
Time = 0.29 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.27 \[ \int \frac {e^4 (-207360-810 x)+e^9 \left (17694720+138240 x+270 x^2\right )}{81+e^4 \left (-14155776-110592 x-216 x^2\right )+e^{15} \left (-201326592-2359296 x-9216 x^2-12 x^3\right )+e^{20} \left (4294967296+67108864 x+393216 x^2+1024 x^3+x^4\right )+e^8 \left (618475290624+9663676416 x+56623104 x^2+147456 x^3+144 x^4\right )+e^3 \left (e^2 (-27648-108 x)+e^6 \left (2415919104+28311552 x+110592 x^2+144 x^3\right )\right )+e^6 \left (e^4 \left (3538944+27648 x+54 x^2\right )+e^8 \left (-103079215104-1610612736 x-9437184 x^2-24576 x^3-24 x^4\right )\right )} \, dx=-\frac {135 \, {\left (2 \, {\left (x + 256\right )} e^{5} - 3\right )}}{{\left (x^{2} + 512 \, x + 65536\right )} e^{16} - 6 \, {\left (x + 256\right )} e^{11} - 24 \, {\left (x^{2} + 512 \, x + 65536\right )} e^{10} + 72 \, {\left (x + 256\right )} e^{5} + 144 \, {\left (x^{2} + 512 \, x + 65536\right )} e^{4} + 9 \, e^{6} - 108} \]
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Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (19) = 38\).
Time = 1.29 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.93 \[ \int \frac {e^4 (-207360-810 x)+e^9 \left (17694720+138240 x+270 x^2\right )}{81+e^4 \left (-14155776-110592 x-216 x^2\right )+e^{15} \left (-201326592-2359296 x-9216 x^2-12 x^3\right )+e^{20} \left (4294967296+67108864 x+393216 x^2+1024 x^3+x^4\right )+e^8 \left (618475290624+9663676416 x+56623104 x^2+147456 x^3+144 x^4\right )+e^3 \left (e^2 (-27648-108 x)+e^6 \left (2415919104+28311552 x+110592 x^2+144 x^3\right )\right )+e^6 \left (e^4 \left (3538944+27648 x+54 x^2\right )+e^8 \left (-103079215104-1610612736 x-9437184 x^2-24576 x^3-24 x^4\right )\right )} \, dx=\frac {- 270 x e^{5} - 69120 e^{5} + 405}{x^{2} \left (- 24 e^{10} + 144 e^{4} + e^{16}\right ) + x \left (- 12288 e^{10} - 6 e^{11} + 72 e^{5} + 73728 e^{4} + 512 e^{16}\right ) - 1572864 e^{10} - 1536 e^{11} - 108 + 9 e^{6} + 18432 e^{5} + 9437184 e^{4} + 65536 e^{16}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (22) = 44\).
Time = 0.18 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.67 \[ \int \frac {e^4 (-207360-810 x)+e^9 \left (17694720+138240 x+270 x^2\right )}{81+e^4 \left (-14155776-110592 x-216 x^2\right )+e^{15} \left (-201326592-2359296 x-9216 x^2-12 x^3\right )+e^{20} \left (4294967296+67108864 x+393216 x^2+1024 x^3+x^4\right )+e^8 \left (618475290624+9663676416 x+56623104 x^2+147456 x^3+144 x^4\right )+e^3 \left (e^2 (-27648-108 x)+e^6 \left (2415919104+28311552 x+110592 x^2+144 x^3\right )\right )+e^6 \left (e^4 \left (3538944+27648 x+54 x^2\right )+e^8 \left (-103079215104-1610612736 x-9437184 x^2-24576 x^3-24 x^4\right )\right )} \, dx=-\frac {135 \, {\left (2 \, x e^{5} + 512 \, e^{5} - 3\right )}}{x^{2} {\left (e^{16} - 24 \, e^{10} + 144 \, e^{4}\right )} + 2 \, x {\left (256 \, e^{16} - 3 \, e^{11} - 6144 \, e^{10} + 36 \, e^{5} + 36864 \, e^{4}\right )} + 65536 \, e^{16} - 1536 \, e^{11} - 1572864 \, e^{10} + 9 \, e^{6} + 18432 \, e^{5} + 9437184 \, e^{4} - 108} \]
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Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (22) = 44\).
Time = 0.30 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.10 \[ \int \frac {e^4 (-207360-810 x)+e^9 \left (17694720+138240 x+270 x^2\right )}{81+e^4 \left (-14155776-110592 x-216 x^2\right )+e^{15} \left (-201326592-2359296 x-9216 x^2-12 x^3\right )+e^{20} \left (4294967296+67108864 x+393216 x^2+1024 x^3+x^4\right )+e^8 \left (618475290624+9663676416 x+56623104 x^2+147456 x^3+144 x^4\right )+e^3 \left (e^2 (-27648-108 x)+e^6 \left (2415919104+28311552 x+110592 x^2+144 x^3\right )\right )+e^6 \left (e^4 \left (3538944+27648 x+54 x^2\right )+e^8 \left (-103079215104-1610612736 x-9437184 x^2-24576 x^3-24 x^4\right )\right )} \, dx=-\frac {135 \, {\left (2 \, x e^{5} + 512 \, e^{5} - 3\right )}}{{\left (x^{2} e^{10} - 12 \, x^{2} e^{4} + 512 \, x e^{10} - 6 \, x e^{5} - 6144 \, x e^{4} + 65536 \, e^{10} - 1536 \, e^{5} - 786432 \, e^{4} + 9\right )} {\left (e^{6} - 12\right )}} \]
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Time = 10.23 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.10 \[ \int \frac {e^4 (-207360-810 x)+e^9 \left (17694720+138240 x+270 x^2\right )}{81+e^4 \left (-14155776-110592 x-216 x^2\right )+e^{15} \left (-201326592-2359296 x-9216 x^2-12 x^3\right )+e^{20} \left (4294967296+67108864 x+393216 x^2+1024 x^3+x^4\right )+e^8 \left (618475290624+9663676416 x+56623104 x^2+147456 x^3+144 x^4\right )+e^3 \left (e^2 (-27648-108 x)+e^6 \left (2415919104+28311552 x+110592 x^2+144 x^3\right )\right )+e^6 \left (e^4 \left (3538944+27648 x+54 x^2\right )+e^8 \left (-103079215104-1610612736 x-9437184 x^2-24576 x^3-24 x^4\right )\right )} \, dx=\frac {135\,\left (512\,{\mathrm {e}}^5+2\,x\,{\mathrm {e}}^5-3\right )}{\left ({\mathrm {e}}^6-12\right )\,\left (\left (12\,{\mathrm {e}}^4-{\mathrm {e}}^{10}\right )\,x^2+\left (6144\,{\mathrm {e}}^4+6\,{\mathrm {e}}^5-512\,{\mathrm {e}}^{10}\right )\,x+786432\,{\mathrm {e}}^4+1536\,{\mathrm {e}}^5-65536\,{\mathrm {e}}^{10}-9\right )} \]
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