Integrand size = 162, antiderivative size = 26 \[ \int \frac {108 x+198 x^2+96 x^3+e^{2 x} \left (-54 x^2-441 x^3-696 x^4-192 x^5\right )}{288+1536 x+2048 x^2+e^{2 x} \left (720 x+3840 x^2+5120 x^3\right )+e^{4 x} \left (720 x^2+3840 x^3+5120 x^4\right )+e^{6 x} \left (360 x^3+1920 x^4+2560 x^5\right )+e^{8 x} \left (90 x^4+480 x^5+640 x^6\right )+e^{10 x} \left (9 x^5+48 x^6+64 x^7\right )} \, dx=\frac {3 x (3+x)}{\left (8+\frac {3}{x}\right ) \left (2+e^{2 x} x\right )^4} \]
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\[ \int \frac {108 x+198 x^2+96 x^3+e^{2 x} \left (-54 x^2-441 x^3-696 x^4-192 x^5\right )}{288+1536 x+2048 x^2+e^{2 x} \left (720 x+3840 x^2+5120 x^3\right )+e^{4 x} \left (720 x^2+3840 x^3+5120 x^4\right )+e^{6 x} \left (360 x^3+1920 x^4+2560 x^5\right )+e^{8 x} \left (90 x^4+480 x^5+640 x^6\right )+e^{10 x} \left (9 x^5+48 x^6+64 x^7\right )} \, dx=\int \frac {108 x+198 x^2+96 x^3+e^{2 x} \left (-54 x^2-441 x^3-696 x^4-192 x^5\right )}{288+1536 x+2048 x^2+e^{2 x} \left (720 x+3840 x^2+5120 x^3\right )+e^{4 x} \left (720 x^2+3840 x^3+5120 x^4\right )+e^{6 x} \left (360 x^3+1920 x^4+2560 x^5\right )+e^{8 x} \left (90 x^4+480 x^5+640 x^6\right )+e^{10 x} \left (9 x^5+48 x^6+64 x^7\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {3 x \left (36-6 \left (-11+3 e^{2 x}\right ) x-\left (-32+147 e^{2 x}\right ) x^2-232 e^{2 x} x^3-64 e^{2 x} x^4\right )}{(3+8 x)^2 \left (2+e^{2 x} x\right )^5} \, dx \\ & = 3 \int \frac {x \left (36-6 \left (-11+3 e^{2 x}\right ) x-\left (-32+147 e^{2 x}\right ) x^2-232 e^{2 x} x^3-64 e^{2 x} x^4\right )}{(3+8 x)^2 \left (2+e^{2 x} x\right )^5} \, dx \\ & = 3 \int \left (\frac {8 x \left (3+7 x+2 x^2\right )}{(3+8 x) \left (2+e^{2 x} x\right )^5}-\frac {x \left (18+147 x+232 x^2+64 x^3\right )}{(3+8 x)^2 \left (2+e^{2 x} x\right )^4}\right ) \, dx \\ & = -\left (3 \int \frac {x \left (18+147 x+232 x^2+64 x^3\right )}{(3+8 x)^2 \left (2+e^{2 x} x\right )^4} \, dx\right )+24 \int \frac {x \left (3+7 x+2 x^2\right )}{(3+8 x) \left (2+e^{2 x} x\right )^5} \, dx \\ & = -\left (3 \int \left (\frac {23 x}{8 \left (2+e^{2 x} x\right )^4}+\frac {x^2}{\left (2+e^{2 x} x\right )^4}+\frac {189}{64 (3+8 x)^2 \left (2+e^{2 x} x\right )^4}-\frac {63}{64 (3+8 x) \left (2+e^{2 x} x\right )^4}\right ) \, dx\right )+24 \int \left (\frac {21}{256 \left (2+e^{2 x} x\right )^5}+\frac {25 x}{32 \left (2+e^{2 x} x\right )^5}+\frac {x^2}{4 \left (2+e^{2 x} x\right )^5}-\frac {63}{256 (3+8 x) \left (2+e^{2 x} x\right )^5}\right ) \, dx \\ & = \frac {63}{32} \int \frac {1}{\left (2+e^{2 x} x\right )^5} \, dx+\frac {189}{64} \int \frac {1}{(3+8 x) \left (2+e^{2 x} x\right )^4} \, dx-3 \int \frac {x^2}{\left (2+e^{2 x} x\right )^4} \, dx-\frac {189}{32} \int \frac {1}{(3+8 x) \left (2+e^{2 x} x\right )^5} \, dx+6 \int \frac {x^2}{\left (2+e^{2 x} x\right )^5} \, dx-\frac {69}{8} \int \frac {x}{\left (2+e^{2 x} x\right )^4} \, dx-\frac {567}{64} \int \frac {1}{(3+8 x)^2 \left (2+e^{2 x} x\right )^4} \, dx+\frac {75}{4} \int \frac {x}{\left (2+e^{2 x} x\right )^5} \, dx \\ \end{align*}
Time = 2.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {108 x+198 x^2+96 x^3+e^{2 x} \left (-54 x^2-441 x^3-696 x^4-192 x^5\right )}{288+1536 x+2048 x^2+e^{2 x} \left (720 x+3840 x^2+5120 x^3\right )+e^{4 x} \left (720 x^2+3840 x^3+5120 x^4\right )+e^{6 x} \left (360 x^3+1920 x^4+2560 x^5\right )+e^{8 x} \left (90 x^4+480 x^5+640 x^6\right )+e^{10 x} \left (9 x^5+48 x^6+64 x^7\right )} \, dx=\frac {3 x^2 (3+x)}{(3+8 x) \left (2+e^{2 x} x\right )^4} \]
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Time = 0.65 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00
method | result | size |
risch | \(\frac {3 x^{2} \left (3+x \right )}{\left (8 x +3\right ) \left (x \,{\mathrm e}^{2 x}+2\right )^{4}}\) | \(26\) |
parallelrisch | \(\frac {72 x^{3}+216 x^{2}}{192 \,{\mathrm e}^{8 x} x^{5}+72 \,{\mathrm e}^{8 x} x^{4}+1536 \,{\mathrm e}^{6 x} x^{4}+576 \,{\mathrm e}^{6 x} x^{3}+4608 x^{3} {\mathrm e}^{4 x}+1728 x^{2} {\mathrm e}^{4 x}+6144 \,{\mathrm e}^{2 x} x^{2}+2304 x \,{\mathrm e}^{2 x}+3072 x +1152}\) | \(91\) |
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Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (25) = 50\).
Time = 0.29 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.19 \[ \int \frac {108 x+198 x^2+96 x^3+e^{2 x} \left (-54 x^2-441 x^3-696 x^4-192 x^5\right )}{288+1536 x+2048 x^2+e^{2 x} \left (720 x+3840 x^2+5120 x^3\right )+e^{4 x} \left (720 x^2+3840 x^3+5120 x^4\right )+e^{6 x} \left (360 x^3+1920 x^4+2560 x^5\right )+e^{8 x} \left (90 x^4+480 x^5+640 x^6\right )+e^{10 x} \left (9 x^5+48 x^6+64 x^7\right )} \, dx=\frac {3 \, {\left (x^{3} + 3 \, x^{2}\right )}}{{\left (8 \, x^{5} + 3 \, x^{4}\right )} e^{\left (8 \, x\right )} + 8 \, {\left (8 \, x^{4} + 3 \, x^{3}\right )} e^{\left (6 \, x\right )} + 24 \, {\left (8 \, x^{3} + 3 \, x^{2}\right )} e^{\left (4 \, x\right )} + 32 \, {\left (8 \, x^{2} + 3 \, x\right )} e^{\left (2 \, x\right )} + 128 \, x + 48} \]
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Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (20) = 40\).
Time = 0.14 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.81 \[ \int \frac {108 x+198 x^2+96 x^3+e^{2 x} \left (-54 x^2-441 x^3-696 x^4-192 x^5\right )}{288+1536 x+2048 x^2+e^{2 x} \left (720 x+3840 x^2+5120 x^3\right )+e^{4 x} \left (720 x^2+3840 x^3+5120 x^4\right )+e^{6 x} \left (360 x^3+1920 x^4+2560 x^5\right )+e^{8 x} \left (90 x^4+480 x^5+640 x^6\right )+e^{10 x} \left (9 x^5+48 x^6+64 x^7\right )} \, dx=\frac {3 x^{3} + 9 x^{2}}{128 x + \left (256 x^{2} + 96 x\right ) e^{2 x} + \left (192 x^{3} + 72 x^{2}\right ) e^{4 x} + \left (64 x^{4} + 24 x^{3}\right ) e^{6 x} + \left (8 x^{5} + 3 x^{4}\right ) e^{8 x} + 48} \]
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Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (25) = 50\).
Time = 0.31 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.19 \[ \int \frac {108 x+198 x^2+96 x^3+e^{2 x} \left (-54 x^2-441 x^3-696 x^4-192 x^5\right )}{288+1536 x+2048 x^2+e^{2 x} \left (720 x+3840 x^2+5120 x^3\right )+e^{4 x} \left (720 x^2+3840 x^3+5120 x^4\right )+e^{6 x} \left (360 x^3+1920 x^4+2560 x^5\right )+e^{8 x} \left (90 x^4+480 x^5+640 x^6\right )+e^{10 x} \left (9 x^5+48 x^6+64 x^7\right )} \, dx=\frac {3 \, {\left (x^{3} + 3 \, x^{2}\right )}}{{\left (8 \, x^{5} + 3 \, x^{4}\right )} e^{\left (8 \, x\right )} + 8 \, {\left (8 \, x^{4} + 3 \, x^{3}\right )} e^{\left (6 \, x\right )} + 24 \, {\left (8 \, x^{3} + 3 \, x^{2}\right )} e^{\left (4 \, x\right )} + 32 \, {\left (8 \, x^{2} + 3 \, x\right )} e^{\left (2 \, x\right )} + 128 \, x + 48} \]
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Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (25) = 50\).
Time = 0.31 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.38 \[ \int \frac {108 x+198 x^2+96 x^3+e^{2 x} \left (-54 x^2-441 x^3-696 x^4-192 x^5\right )}{288+1536 x+2048 x^2+e^{2 x} \left (720 x+3840 x^2+5120 x^3\right )+e^{4 x} \left (720 x^2+3840 x^3+5120 x^4\right )+e^{6 x} \left (360 x^3+1920 x^4+2560 x^5\right )+e^{8 x} \left (90 x^4+480 x^5+640 x^6\right )+e^{10 x} \left (9 x^5+48 x^6+64 x^7\right )} \, dx=\frac {3 \, {\left (x^{3} + 3 \, x^{2}\right )}}{8 \, x^{5} e^{\left (8 \, x\right )} + 3 \, x^{4} e^{\left (8 \, x\right )} + 64 \, x^{4} e^{\left (6 \, x\right )} + 24 \, x^{3} e^{\left (6 \, x\right )} + 192 \, x^{3} e^{\left (4 \, x\right )} + 72 \, x^{2} e^{\left (4 \, x\right )} + 256 \, x^{2} e^{\left (2 \, x\right )} + 96 \, x e^{\left (2 \, x\right )} + 128 \, x + 48} \]
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Time = 12.72 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.85 \[ \int \frac {108 x+198 x^2+96 x^3+e^{2 x} \left (-54 x^2-441 x^3-696 x^4-192 x^5\right )}{288+1536 x+2048 x^2+e^{2 x} \left (720 x+3840 x^2+5120 x^3\right )+e^{4 x} \left (720 x^2+3840 x^3+5120 x^4\right )+e^{6 x} \left (360 x^3+1920 x^4+2560 x^5\right )+e^{8 x} \left (90 x^4+480 x^5+640 x^6\right )+e^{10 x} \left (9 x^5+48 x^6+64 x^7\right )} \, dx=\frac {3\,\left (16\,x^5+62\,x^4+45\,x^3+9\,x^2\right )}{\left (2\,x+1\right )\,{\left (8\,x+3\right )}^2\,\left (32\,x\,{\mathrm {e}}^{2\,x}+24\,x^2\,{\mathrm {e}}^{4\,x}+8\,x^3\,{\mathrm {e}}^{6\,x}+x^4\,{\mathrm {e}}^{8\,x}+16\right )} \]
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