Integrand size = 86, antiderivative size = 27 \[ \int \frac {e^x \left (-12+12 x+512 x^3-128 x^4\right )+e^{\frac {1}{4} (20+3 x)} \left (-12+9 x+512 x^3-96 x^4\right )}{8 e^{2 x}+8 e^{\frac {1}{2} (20+3 x)}+16 e^{x+\frac {1}{4} (20+3 x)}} \, dx=\frac {-\frac {3 x}{2}+16 x^4}{e^{5+\frac {3 x}{4}}+e^x} \]
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\[ \int \frac {e^x \left (-12+12 x+512 x^3-128 x^4\right )+e^{\frac {1}{4} (20+3 x)} \left (-12+9 x+512 x^3-96 x^4\right )}{8 e^{2 x}+8 e^{\frac {1}{2} (20+3 x)}+16 e^{x+\frac {1}{4} (20+3 x)}} \, dx=\int \frac {e^x \left (-12+12 x+512 x^3-128 x^4\right )+e^{\frac {1}{4} (20+3 x)} \left (-12+9 x+512 x^3-96 x^4\right )}{8 e^{2 x}+8 e^{\frac {1}{2} (20+3 x)}+16 e^{x+\frac {1}{4} (20+3 x)}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{-3 x/2} \left (e^x \left (-12+12 x+512 x^3-128 x^4\right )+e^{\frac {1}{4} (20+3 x)} \left (-12+9 x+512 x^3-96 x^4\right )\right )}{8 \left (e^5+e^{x/4}\right )^2} \, dx \\ & = \frac {1}{8} \int \frac {e^{-3 x/2} \left (e^x \left (-12+12 x+512 x^3-128 x^4\right )+e^{\frac {1}{4} (20+3 x)} \left (-12+9 x+512 x^3-96 x^4\right )\right )}{\left (e^5+e^{x/4}\right )^2} \, dx \\ & = \frac {1}{8} \int \left (-\frac {e^{20-\frac {3 x}{2}} x \left (-3+32 x^3\right )}{\left (e^5+e^{x/4}\right )^2}-4 e^{-x} \left (3-3 x-128 x^3+32 x^4\right )-2 e^{10-\frac {3 x}{2}} \left (6-9 x-256 x^3+96 x^4\right )+e^{5-\frac {5 x}{4}} \left (12-15 x-512 x^3+160 x^4\right )+\frac {e^{15-\frac {3 x}{2}} \left (12-21 x-512 x^3+224 x^4\right )}{e^5+e^{x/4}}\right ) \, dx \\ & = -\left (\frac {1}{8} \int \frac {e^{20-\frac {3 x}{2}} x \left (-3+32 x^3\right )}{\left (e^5+e^{x/4}\right )^2} \, dx\right )+\frac {1}{8} \int e^{5-\frac {5 x}{4}} \left (12-15 x-512 x^3+160 x^4\right ) \, dx+\frac {1}{8} \int \frac {e^{15-\frac {3 x}{2}} \left (12-21 x-512 x^3+224 x^4\right )}{e^5+e^{x/4}} \, dx-\frac {1}{4} \int e^{10-\frac {3 x}{2}} \left (6-9 x-256 x^3+96 x^4\right ) \, dx-\frac {1}{2} \int e^{-x} \left (3-3 x-128 x^3+32 x^4\right ) \, dx \\ & = \frac {1}{8} \int \left (12 e^{5-\frac {5 x}{4}}-15 e^{5-\frac {5 x}{4}} x-512 e^{5-\frac {5 x}{4}} x^3+160 e^{5-\frac {5 x}{4}} x^4\right ) \, dx-\frac {1}{8} \int \left (-\frac {3 e^{20-\frac {3 x}{2}} x}{\left (e^5+e^{x/4}\right )^2}+\frac {32 e^{20-\frac {3 x}{2}} x^4}{\left (e^5+e^{x/4}\right )^2}\right ) \, dx+\frac {1}{8} \int \left (\frac {12 e^{15-\frac {3 x}{2}}}{e^5+e^{x/4}}-\frac {21 e^{15-\frac {3 x}{2}} x}{e^5+e^{x/4}}-\frac {512 e^{15-\frac {3 x}{2}} x^3}{e^5+e^{x/4}}+\frac {224 e^{15-\frac {3 x}{2}} x^4}{e^5+e^{x/4}}\right ) \, dx-\frac {1}{4} \int \left (6 e^{10-\frac {3 x}{2}}-9 e^{10-\frac {3 x}{2}} x-256 e^{10-\frac {3 x}{2}} x^3+96 e^{10-\frac {3 x}{2}} x^4\right ) \, dx-\frac {1}{2} \int \left (3 e^{-x}-3 e^{-x} x-128 e^{-x} x^3+32 e^{-x} x^4\right ) \, dx \\ & = \frac {3}{8} \int \frac {e^{20-\frac {3 x}{2}} x}{\left (e^5+e^{x/4}\right )^2} \, dx-\frac {3}{2} \int e^{10-\frac {3 x}{2}} \, dx+\frac {3}{2} \int e^{5-\frac {5 x}{4}} \, dx-\frac {3}{2} \int e^{-x} \, dx+\frac {3}{2} \int \frac {e^{15-\frac {3 x}{2}}}{e^5+e^{x/4}} \, dx+\frac {3}{2} \int e^{-x} x \, dx-\frac {15}{8} \int e^{5-\frac {5 x}{4}} x \, dx+\frac {9}{4} \int e^{10-\frac {3 x}{2}} x \, dx-\frac {21}{8} \int \frac {e^{15-\frac {3 x}{2}} x}{e^5+e^{x/4}} \, dx-4 \int \frac {e^{20-\frac {3 x}{2}} x^4}{\left (e^5+e^{x/4}\right )^2} \, dx-16 \int e^{-x} x^4 \, dx+20 \int e^{5-\frac {5 x}{4}} x^4 \, dx-24 \int e^{10-\frac {3 x}{2}} x^4 \, dx+28 \int \frac {e^{15-\frac {3 x}{2}} x^4}{e^5+e^{x/4}} \, dx+64 \int e^{10-\frac {3 x}{2}} x^3 \, dx-64 \int e^{5-\frac {5 x}{4}} x^3 \, dx+64 \int e^{-x} x^3 \, dx-64 \int \frac {e^{15-\frac {3 x}{2}} x^3}{e^5+e^{x/4}} \, dx \\ & = e^{10-\frac {3 x}{2}}-\frac {6}{5} e^{5-\frac {5 x}{4}}+\frac {3 e^{-x}}{2}-\frac {3}{2} e^{10-\frac {3 x}{2}} x+\frac {3}{2} e^{5-\frac {5 x}{4}} x-\frac {3 e^{-x} x}{2}-\frac {128}{3} e^{10-\frac {3 x}{2}} x^3+\frac {256}{5} e^{5-\frac {5 x}{4}} x^3-64 e^{-x} x^3+16 e^{10-\frac {3 x}{2}} x^4-16 e^{5-\frac {5 x}{4}} x^4+16 e^{-x} x^4+\frac {3}{8} \int \frac {e^{20-\frac {3 x}{2}} x}{\left (e^5+e^{x/4}\right )^2} \, dx+\frac {3}{2} \int e^{10-\frac {3 x}{2}} \, dx-\frac {3}{2} \int e^{5-\frac {5 x}{4}} \, dx+\frac {3}{2} \int e^{-x} \, dx-\frac {21}{8} \int \frac {e^{15-\frac {3 x}{2}} x}{e^5+e^{x/4}} \, dx-4 \int \frac {e^{20-\frac {3 x}{2}} x^4}{\left (e^5+e^{x/4}\right )^2} \, dx+28 \int \frac {e^{15-\frac {3 x}{2}} x^4}{e^5+e^{x/4}} \, dx-64 \int e^{10-\frac {3 x}{2}} x^3 \, dx+64 \int e^{5-\frac {5 x}{4}} x^3 \, dx-64 \int e^{-x} x^3 \, dx-64 \int \frac {e^{15-\frac {3 x}{2}} x^3}{e^5+e^{x/4}} \, dx+128 \int e^{10-\frac {3 x}{2}} x^2 \, dx-\frac {768}{5} \int e^{5-\frac {5 x}{4}} x^2 \, dx+192 \int e^{-x} x^2 \, dx+\left (6 e^{15}\right ) \text {Subst}\left (\int \frac {1}{x^7 \left (e^5+x\right )} \, dx,x,e^{x/4}\right ) \\ & = -\frac {3}{2} e^{10-\frac {3 x}{2}} x+\frac {3}{2} e^{5-\frac {5 x}{4}} x-\frac {3 e^{-x} x}{2}-\frac {256}{3} e^{10-\frac {3 x}{2}} x^2+\frac {3072}{25} e^{5-\frac {5 x}{4}} x^2-192 e^{-x} x^2+16 e^{10-\frac {3 x}{2}} x^4-16 e^{5-\frac {5 x}{4}} x^4+16 e^{-x} x^4+\frac {3}{8} \int \frac {e^{20-\frac {3 x}{2}} x}{\left (e^5+e^{x/4}\right )^2} \, dx-\frac {21}{8} \int \frac {e^{15-\frac {3 x}{2}} x}{e^5+e^{x/4}} \, dx-4 \int \frac {e^{20-\frac {3 x}{2}} x^4}{\left (e^5+e^{x/4}\right )^2} \, dx+28 \int \frac {e^{15-\frac {3 x}{2}} x^4}{e^5+e^{x/4}} \, dx-64 \int \frac {e^{15-\frac {3 x}{2}} x^3}{e^5+e^{x/4}} \, dx-128 \int e^{10-\frac {3 x}{2}} x^2 \, dx+\frac {768}{5} \int e^{5-\frac {5 x}{4}} x^2 \, dx+\frac {512}{3} \int e^{10-\frac {3 x}{2}} x \, dx-192 \int e^{-x} x^2 \, dx-\frac {6144}{25} \int e^{5-\frac {5 x}{4}} x \, dx+384 \int e^{-x} x \, dx+\left (6 e^{15}\right ) \text {Subst}\left (\int \left (\frac {1}{e^5 x^7}-\frac {1}{e^{10} x^6}+\frac {1}{e^{15} x^5}-\frac {1}{e^{20} x^4}+\frac {1}{e^{25} x^3}-\frac {1}{e^{30} x^2}+\frac {1}{e^{35} x}-\frac {1}{e^{35} \left (e^5+x\right )}\right ) \, dx,x,e^{x/4}\right ) \\ & = -e^{10-\frac {3 x}{2}}+\frac {6}{5} e^{5-\frac {5 x}{4}}+2 e^{-5-\frac {3 x}{4}}-3 e^{-10-\frac {x}{2}}+6 e^{-15-\frac {x}{4}}-\frac {3 e^{-x}}{2}+\frac {3 x}{2 e^{20}}-\frac {2075}{18} e^{10-\frac {3 x}{2}} x+\frac {49527}{250} e^{5-\frac {5 x}{4}} x-\frac {771 e^{-x} x}{2}+16 e^{10-\frac {3 x}{2}} x^4-16 e^{5-\frac {5 x}{4}} x^4+16 e^{-x} x^4-\frac {6 \log \left (e^5+e^{x/4}\right )}{e^{20}}+\frac {3}{8} \int \frac {e^{20-\frac {3 x}{2}} x}{\left (e^5+e^{x/4}\right )^2} \, dx-\frac {21}{8} \int \frac {e^{15-\frac {3 x}{2}} x}{e^5+e^{x/4}} \, dx-4 \int \frac {e^{20-\frac {3 x}{2}} x^4}{\left (e^5+e^{x/4}\right )^2} \, dx+28 \int \frac {e^{15-\frac {3 x}{2}} x^4}{e^5+e^{x/4}} \, dx-64 \int \frac {e^{15-\frac {3 x}{2}} x^3}{e^5+e^{x/4}} \, dx+\frac {1024}{9} \int e^{10-\frac {3 x}{2}} \, dx-\frac {512}{3} \int e^{10-\frac {3 x}{2}} x \, dx-\frac {24576}{125} \int e^{5-\frac {5 x}{4}} \, dx+\frac {6144}{25} \int e^{5-\frac {5 x}{4}} x \, dx+384 \int e^{-x} \, dx-384 \int e^{-x} x \, dx \\ & = -\frac {2075}{27} e^{10-\frac {3 x}{2}}+\frac {99054}{625} e^{5-\frac {5 x}{4}}+2 e^{-5-\frac {3 x}{4}}-3 e^{-10-\frac {x}{2}}+6 e^{-15-\frac {x}{4}}-\frac {771 e^{-x}}{2}+\frac {3 x}{2 e^{20}}-\frac {3}{2} e^{10-\frac {3 x}{2}} x+\frac {3}{2} e^{5-\frac {5 x}{4}} x-\frac {3 e^{-x} x}{2}+16 e^{10-\frac {3 x}{2}} x^4-16 e^{5-\frac {5 x}{4}} x^4+16 e^{-x} x^4-\frac {6 \log \left (e^5+e^{x/4}\right )}{e^{20}}+\frac {3}{8} \int \frac {e^{20-\frac {3 x}{2}} x}{\left (e^5+e^{x/4}\right )^2} \, dx-\frac {21}{8} \int \frac {e^{15-\frac {3 x}{2}} x}{e^5+e^{x/4}} \, dx-4 \int \frac {e^{20-\frac {3 x}{2}} x^4}{\left (e^5+e^{x/4}\right )^2} \, dx+28 \int \frac {e^{15-\frac {3 x}{2}} x^4}{e^5+e^{x/4}} \, dx-64 \int \frac {e^{15-\frac {3 x}{2}} x^3}{e^5+e^{x/4}} \, dx-\frac {1024}{9} \int e^{10-\frac {3 x}{2}} \, dx+\frac {24576}{125} \int e^{5-\frac {5 x}{4}} \, dx-384 \int e^{-x} \, dx \\ & = -e^{10-\frac {3 x}{2}}+\frac {6}{5} e^{5-\frac {5 x}{4}}+2 e^{-5-\frac {3 x}{4}}-3 e^{-10-\frac {x}{2}}+6 e^{-15-\frac {x}{4}}-\frac {3 e^{-x}}{2}+\frac {3 x}{2 e^{20}}-\frac {3}{2} e^{10-\frac {3 x}{2}} x+\frac {3}{2} e^{5-\frac {5 x}{4}} x-\frac {3 e^{-x} x}{2}+16 e^{10-\frac {3 x}{2}} x^4-16 e^{5-\frac {5 x}{4}} x^4+16 e^{-x} x^4-\frac {6 \log \left (e^5+e^{x/4}\right )}{e^{20}}+\frac {3}{8} \int \frac {e^{20-\frac {3 x}{2}} x}{\left (e^5+e^{x/4}\right )^2} \, dx-\frac {21}{8} \int \frac {e^{15-\frac {3 x}{2}} x}{e^5+e^{x/4}} \, dx-4 \int \frac {e^{20-\frac {3 x}{2}} x^4}{\left (e^5+e^{x/4}\right )^2} \, dx+28 \int \frac {e^{15-\frac {3 x}{2}} x^4}{e^5+e^{x/4}} \, dx-64 \int \frac {e^{15-\frac {3 x}{2}} x^3}{e^5+e^{x/4}} \, dx \\ \end{align*}
Time = 6.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {e^x \left (-12+12 x+512 x^3-128 x^4\right )+e^{\frac {1}{4} (20+3 x)} \left (-12+9 x+512 x^3-96 x^4\right )}{8 e^{2 x}+8 e^{\frac {1}{2} (20+3 x)}+16 e^{x+\frac {1}{4} (20+3 x)}} \, dx=\frac {x \left (-3+32 x^3\right )}{2 \left (e^{5+\frac {3 x}{4}}+e^x\right )} \]
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Time = 0.20 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85
method | result | size |
parallelrisch | \(\frac {128 x^{4}-12 x}{8 \,{\mathrm e}^{x}+8 \,{\mathrm e}^{\frac {3 x}{4}+5}}\) | \(23\) |
risch | \(\frac {x \left (32 x^{3}-3\right ) {\mathrm e}^{-\frac {3 x}{4}}}{2 \,{\mathrm e}^{5}+2 \,{\mathrm e}^{\frac {x}{4}}}\) | \(24\) |
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Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {e^x \left (-12+12 x+512 x^3-128 x^4\right )+e^{\frac {1}{4} (20+3 x)} \left (-12+9 x+512 x^3-96 x^4\right )}{8 e^{2 x}+8 e^{\frac {1}{2} (20+3 x)}+16 e^{x+\frac {1}{4} (20+3 x)}} \, dx=\frac {{\left (32 \, x^{4} - 3 \, x\right )} e^{\frac {20}{3}}}{2 \, {\left (e^{\left (x + \frac {20}{3}\right )} + e^{\left (\frac {3}{4} \, x + \frac {35}{3}\right )}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (20) = 40\).
Time = 0.21 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.33 \[ \int \frac {e^x \left (-12+12 x+512 x^3-128 x^4\right )+e^{\frac {1}{4} (20+3 x)} \left (-12+9 x+512 x^3-96 x^4\right )}{8 e^{2 x}+8 e^{\frac {1}{2} (20+3 x)}+16 e^{x+\frac {1}{4} (20+3 x)}} \, dx=\frac {- 32 x^{4} + 3 x}{2 e^{15} e^{\frac {x}{4}} + 2 e^{20}} + \frac {\left (128 x^{4} e^{15} - 12 x e^{15}\right ) e^{- \frac {x}{4}} + \left (- 128 x^{4} e^{20} + 12 x e^{20}\right ) e^{- \frac {x}{2}} + \frac {128 x^{4} e^{25} - 12 x e^{25}}{\left (e^{x}\right )^{\frac {3}{4}}}}{8 e^{30}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (22) = 44\).
Time = 0.24 (sec) , antiderivative size = 106, normalized size of antiderivative = 3.93 \[ \int \frac {e^x \left (-12+12 x+512 x^3-128 x^4\right )+e^{\frac {1}{4} (20+3 x)} \left (-12+9 x+512 x^3-96 x^4\right )}{8 e^{2 x}+8 e^{\frac {1}{2} (20+3 x)}+16 e^{x+\frac {1}{4} (20+3 x)}} \, dx=2 \, {\left (3 \, x + 20\right )} e^{\left (-20\right )} - 24 \, e^{\left (-20\right )} \log \left (e^{\frac {20}{3}} + e^{\left (\frac {1}{4} \, x + \frac {5}{3}\right )}\right ) + 24 \, e^{\left (-20\right )} \log \left (e^{5} + e^{\left (\frac {1}{4} \, x\right )}\right ) - \frac {6 \, {\left (x e^{5} + x e^{\left (\frac {1}{4} \, x\right )} + 4 \, e^{5}\right )}}{e^{25} + e^{\left (\frac {1}{4} \, x + 20\right )}} + \frac {2 \, {\left (e^{20} + 12 \, e^{\left (\frac {3}{4} \, x + 5\right )} + 6 \, e^{\left (\frac {1}{2} \, x + 10\right )} - 2 \, e^{\left (\frac {1}{4} \, x + 15\right )}\right )}}{e^{\left (x + 20\right )} + e^{\left (\frac {3}{4} \, x + 25\right )}} \]
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Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {e^x \left (-12+12 x+512 x^3-128 x^4\right )+e^{\frac {1}{4} (20+3 x)} \left (-12+9 x+512 x^3-96 x^4\right )}{8 e^{2 x}+8 e^{\frac {1}{2} (20+3 x)}+16 e^{x+\frac {1}{4} (20+3 x)}} \, dx=\frac {32 \, x^{4} - 3 \, x}{2 \, {\left (e^{x} + e^{\left (\frac {3}{4} \, x + 5\right )}\right )}} \]
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Time = 0.50 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.07 \[ \int \frac {e^x \left (-12+12 x+512 x^3-128 x^4\right )+e^{\frac {1}{4} (20+3 x)} \left (-12+9 x+512 x^3-96 x^4\right )}{8 e^{2 x}+8 e^{\frac {1}{2} (20+3 x)}+16 e^{x+\frac {1}{4} (20+3 x)}} \, dx={\mathrm {e}}^{-\frac {x}{2}}\,\left (\frac {3\,x\,{\mathrm {e}}^{-10}}{2}-16\,x^4\,{\mathrm {e}}^{-10}\right )-{\mathrm {e}}^{-\frac {3\,x}{4}}\,\left (\frac {3\,x\,{\mathrm {e}}^{-5}}{2}-16\,x^4\,{\mathrm {e}}^{-5}\right )-{\mathrm {e}}^{-\frac {x}{4}}\,\left (\frac {3\,x\,{\mathrm {e}}^{-15}}{2}-16\,x^4\,{\mathrm {e}}^{-15}\right )+\frac {16\,{\mathrm {e}}^{-15}\,\left (3\,x-32\,x^4\right )}{32\,{\mathrm {e}}^{x/4}+32\,{\mathrm {e}}^5} \]
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