Integrand size = 206, antiderivative size = 22 \[ \int e^{100 x^2-40 x \log (4)+4 \log ^2(4)+4 \left (-10 x^3-20 x^4-10 x^5+\left (2 x^2+4 x^3+2 x^4\right ) \log (4)\right ) \log (x)+4 \left (x^4+4 x^5+6 x^6+4 x^7+x^8\right ) \log ^2(x)} \left (200 x-40 x^2-80 x^3-40 x^4+\left (-40+8 x+16 x^2+8 x^3\right ) \log (4)+\left (-120 x^2-312 x^3-168 x^4+48 x^5+32 x^6+8 x^7+\left (16 x+48 x^2+32 x^3\right ) \log (4)\right ) \log (x)+\left (16 x^3+80 x^4+144 x^5+112 x^6+32 x^7\right ) \log ^2(x)\right ) \, dx=e^{4 \left (-5 x+\log (4)+\left (x+x^2\right )^2 \log (x)\right )^2} \]
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\[ \int e^{100 x^2-40 x \log (4)+4 \log ^2(4)+4 \left (-10 x^3-20 x^4-10 x^5+\left (2 x^2+4 x^3+2 x^4\right ) \log (4)\right ) \log (x)+4 \left (x^4+4 x^5+6 x^6+4 x^7+x^8\right ) \log ^2(x)} \left (200 x-40 x^2-80 x^3-40 x^4+\left (-40+8 x+16 x^2+8 x^3\right ) \log (4)+\left (-120 x^2-312 x^3-168 x^4+48 x^5+32 x^6+8 x^7+\left (16 x+48 x^2+32 x^3\right ) \log (4)\right ) \log (x)+\left (16 x^3+80 x^4+144 x^5+112 x^6+32 x^7\right ) \log ^2(x)\right ) \, dx=\int \exp \left (100 x^2-40 x \log (4)+4 \log ^2(4)+4 \left (-10 x^3-20 x^4-10 x^5+\left (2 x^2+4 x^3+2 x^4\right ) \log (4)\right ) \log (x)+4 \left (x^4+4 x^5+6 x^6+4 x^7+x^8\right ) \log ^2(x)\right ) \left (200 x-40 x^2-80 x^3-40 x^4+\left (-40+8 x+16 x^2+8 x^3\right ) \log (4)+\left (-120 x^2-312 x^3-168 x^4+48 x^5+32 x^6+8 x^7+\left (16 x+48 x^2+32 x^3\right ) \log (4)\right ) \log (x)+\left (16 x^3+80 x^4+144 x^5+112 x^6+32 x^7\right ) \log ^2(x)\right ) \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int 2^{3-80 x} \exp \left (4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )\right ) x^{-8 x^2 (1+x)^2 (5 x-\log (4))} \left (5 x-\log (4)-x^2 (1+x)^2 \log (x)\right ) \left (5-x-2 x^2-x^3-2 x \left (1+3 x+2 x^2\right ) \log (x)\right ) \, dx \\ & = \int \left (-2^{3-80 x} \exp \left (4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )\right ) x^{-8 x^2 (1+x)^2 (5 x-\log (4))} \left (-5+x+2 x^2+x^3\right ) (5 x-\log (4))+2^{3-80 x} \exp \left (4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )\right ) x^{1-8 x^2 (1+x)^2 (5 x-\log (4))} (1+x) \left (-24 x^2+3 x^3+3 x^4+x^5-15 x \left (1-\frac {8 \log (2)}{15}\right )+\log (16)\right ) \log (x)+2^{4-80 x} \exp \left (4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )\right ) x^{3-8 x^2 (1+x)^2 (5 x-\log (4))} (1+x)^3 (1+2 x) \log ^2(x)\right ) \, dx \\ & = -\int 2^{3-80 x} \exp \left (4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )\right ) x^{-8 x^2 (1+x)^2 (5 x-\log (4))} \left (-5+x+2 x^2+x^3\right ) (5 x-\log (4)) \, dx+\int 2^{3-80 x} \exp \left (4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )\right ) x^{1-8 x^2 (1+x)^2 (5 x-\log (4))} (1+x) \left (-24 x^2+3 x^3+3 x^4+x^5-15 x \left (1-\frac {8 \log (2)}{15}\right )+\log (16)\right ) \log (x) \, dx+\int 2^{4-80 x} \exp \left (4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )\right ) x^{3-8 x^2 (1+x)^2 (5 x-\log (4))} (1+x)^3 (1+2 x) \log ^2(x) \, dx \\ & = -\int \left (5\ 2^{3-80 x} e^{4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )} x^{4-8 x^2 (1+x)^2 (5 x-\log (4))}+2^{3-80 x} e^{4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )} x^{2-8 x^2 (1+x)^2 (5 x-\log (4))} (5-2 \log (4))+2^{3-80 x} e^{4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )} x^{3-8 x^2 (1+x)^2 (5 x-\log (4))} (10-\log (4))+5\ 2^{3-80 x} e^{4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )} x^{-8 x^2 (1+x)^2 (5 x-\log (4))} \log (4)-2^{3-80 x} e^{4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )} x^{1-8 x^2 (1+x)^2 (5 x-\log (4))} (25+\log (4))\right ) \, dx+\int \left (-21 2^{3-80 x} e^{4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )} x^{4-8 x^2 (1+x)^2 (5 x-\log (4))} \log (x)+3\ 2^{4-80 x} e^{4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )} x^{5-8 x^2 (1+x)^2 (5 x-\log (4))} \log (x)+2^{5-80 x} e^{4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )} x^{6-8 x^2 (1+x)^2 (5 x-\log (4))} \log (x)+2^{3-80 x} e^{4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )} x^{7-8 x^2 (1+x)^2 (5 x-\log (4))} \log (x)+2^{3-80 x} e^{4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )} x^{1-8 x^2 (1+x)^2 (5 x-\log (4))} \log (16) \log (x)+2^{3-80 x} e^{4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )} x^{3-8 x^2 (1+x)^2 (5 x-\log (4))} (-39+\log (256)) \log (x)+2^{3-80 x} e^{4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )} x^{2-8 x^2 (1+x)^2 (5 x-\log (4))} (-15+\log (4096)) \log (x)\right ) \, dx+\int \left (2^{4-80 x} e^{4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )} x^{3-8 x^2 (1+x)^2 (5 x-\log (4))} \log ^2(x)+5\ 2^{4-80 x} e^{4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )} x^{4-8 x^2 (1+x)^2 (5 x-\log (4))} \log ^2(x)+9\ 2^{4-80 x} e^{4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )} x^{5-8 x^2 (1+x)^2 (5 x-\log (4))} \log ^2(x)+7\ 2^{4-80 x} e^{4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )} x^{6-8 x^2 (1+x)^2 (5 x-\log (4))} \log ^2(x)+2^{5-80 x} e^{4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )} x^{7-8 x^2 (1+x)^2 (5 x-\log (4))} \log ^2(x)\right ) \, dx \\ & = 3 \int 2^{4-80 x} e^{4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )} x^{5-8 x^2 (1+x)^2 (5 x-\log (4))} \log (x) \, dx-5 \int 2^{3-80 x} e^{4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )} x^{4-8 x^2 (1+x)^2 (5 x-\log (4))} \, dx+5 \int 2^{4-80 x} e^{4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )} x^{4-8 x^2 (1+x)^2 (5 x-\log (4))} \log ^2(x) \, dx+7 \int 2^{4-80 x} e^{4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )} x^{6-8 x^2 (1+x)^2 (5 x-\log (4))} \log ^2(x) \, dx+9 \int 2^{4-80 x} e^{4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )} x^{5-8 x^2 (1+x)^2 (5 x-\log (4))} \log ^2(x) \, dx-21 \int 2^{3-80 x} e^{4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )} x^{4-8 x^2 (1+x)^2 (5 x-\log (4))} \log (x) \, dx-(5-2 \log (4)) \int 2^{3-80 x} e^{4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )} x^{2-8 x^2 (1+x)^2 (5 x-\log (4))} \, dx-(-25-\log (4)) \int 2^{3-80 x} e^{4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )} x^{1-8 x^2 (1+x)^2 (5 x-\log (4))} \, dx-(10-\log (4)) \int 2^{3-80 x} e^{4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )} x^{3-8 x^2 (1+x)^2 (5 x-\log (4))} \, dx-(5 \log (4)) \int 2^{3-80 x} e^{4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )} x^{-8 x^2 (1+x)^2 (5 x-\log (4))} \, dx+\log (16) \int 2^{3-80 x} e^{4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )} x^{1-8 x^2 (1+x)^2 (5 x-\log (4))} \log (x) \, dx+(-39+\log (256)) \int 2^{3-80 x} e^{4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )} x^{3-8 x^2 (1+x)^2 (5 x-\log (4))} \log (x) \, dx+(-15+\log (4096)) \int 2^{3-80 x} e^{4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )} x^{2-8 x^2 (1+x)^2 (5 x-\log (4))} \log (x) \, dx+\int 2^{5-80 x} e^{4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )} x^{6-8 x^2 (1+x)^2 (5 x-\log (4))} \log (x) \, dx+\int 2^{3-80 x} e^{4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )} x^{7-8 x^2 (1+x)^2 (5 x-\log (4))} \log (x) \, dx+\int 2^{4-80 x} e^{4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )} x^{3-8 x^2 (1+x)^2 (5 x-\log (4))} \log ^2(x) \, dx+\int 2^{5-80 x} e^{4 \left (25 x^2+\log ^2(4)+x^4 (1+x)^4 \log ^2(x)\right )} x^{7-8 x^2 (1+x)^2 (5 x-\log (4))} \log ^2(x) \, dx \\ \end{align*}
\[ \int e^{100 x^2-40 x \log (4)+4 \log ^2(4)+4 \left (-10 x^3-20 x^4-10 x^5+\left (2 x^2+4 x^3+2 x^4\right ) \log (4)\right ) \log (x)+4 \left (x^4+4 x^5+6 x^6+4 x^7+x^8\right ) \log ^2(x)} \left (200 x-40 x^2-80 x^3-40 x^4+\left (-40+8 x+16 x^2+8 x^3\right ) \log (4)+\left (-120 x^2-312 x^3-168 x^4+48 x^5+32 x^6+8 x^7+\left (16 x+48 x^2+32 x^3\right ) \log (4)\right ) \log (x)+\left (16 x^3+80 x^4+144 x^5+112 x^6+32 x^7\right ) \log ^2(x)\right ) \, dx=\int e^{100 x^2-40 x \log (4)+4 \log ^2(4)+4 \left (-10 x^3-20 x^4-10 x^5+\left (2 x^2+4 x^3+2 x^4\right ) \log (4)\right ) \log (x)+4 \left (x^4+4 x^5+6 x^6+4 x^7+x^8\right ) \log ^2(x)} \left (200 x-40 x^2-80 x^3-40 x^4+\left (-40+8 x+16 x^2+8 x^3\right ) \log (4)+\left (-120 x^2-312 x^3-168 x^4+48 x^5+32 x^6+8 x^7+\left (16 x+48 x^2+32 x^3\right ) \log (4)\right ) \log (x)+\left (16 x^3+80 x^4+144 x^5+112 x^6+32 x^7\right ) \log ^2(x)\right ) \, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. \(86\) vs. \(2(23)=46\).
Time = 42.80 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.95
method | result | size |
risch | \(1048576^{-4 x} x^{8 x^{2} \left (1+x \right )^{2} \left (2 \ln \left (2\right )-5 x \right )} {\mathrm e}^{4 x^{8} \ln \left (x \right )^{2}+16 x^{7} \ln \left (x \right )^{2}+24 x^{6} \ln \left (x \right )^{2}+16 x^{5} \ln \left (x \right )^{2}+4 x^{4} \ln \left (x \right )^{2}+16 \ln \left (2\right )^{2}+100 x^{2}}\) | \(87\) |
parallelrisch | \({\mathrm e}^{4 \left (x^{8}+4 x^{7}+6 x^{6}+4 x^{5}+x^{4}\right ) \ln \left (x \right )^{2}+\left (8 \left (2 x^{4}+4 x^{3}+2 x^{2}\right ) \ln \left (2\right )-40 x^{5}-80 x^{4}-40 x^{3}\right ) \ln \left (x \right )+16 \ln \left (2\right )^{2}-80 x \ln \left (2\right )+100 x^{2}}\) | \(87\) |
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Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (23) = 46\).
Time = 0.25 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.73 \[ \int e^{100 x^2-40 x \log (4)+4 \log ^2(4)+4 \left (-10 x^3-20 x^4-10 x^5+\left (2 x^2+4 x^3+2 x^4\right ) \log (4)\right ) \log (x)+4 \left (x^4+4 x^5+6 x^6+4 x^7+x^8\right ) \log ^2(x)} \left (200 x-40 x^2-80 x^3-40 x^4+\left (-40+8 x+16 x^2+8 x^3\right ) \log (4)+\left (-120 x^2-312 x^3-168 x^4+48 x^5+32 x^6+8 x^7+\left (16 x+48 x^2+32 x^3\right ) \log (4)\right ) \log (x)+\left (16 x^3+80 x^4+144 x^5+112 x^6+32 x^7\right ) \log ^2(x)\right ) \, dx=e^{\left (4 \, {\left (x^{8} + 4 \, x^{7} + 6 \, x^{6} + 4 \, x^{5} + x^{4}\right )} \log \left (x\right )^{2} + 100 \, x^{2} - 80 \, x \log \left (2\right ) + 16 \, \log \left (2\right )^{2} - 8 \, {\left (5 \, x^{5} + 10 \, x^{4} + 5 \, x^{3} - 2 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} \log \left (2\right )\right )} \log \left (x\right )\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (22) = 44\).
Time = 0.39 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.95 \[ \int e^{100 x^2-40 x \log (4)+4 \log ^2(4)+4 \left (-10 x^3-20 x^4-10 x^5+\left (2 x^2+4 x^3+2 x^4\right ) \log (4)\right ) \log (x)+4 \left (x^4+4 x^5+6 x^6+4 x^7+x^8\right ) \log ^2(x)} \left (200 x-40 x^2-80 x^3-40 x^4+\left (-40+8 x+16 x^2+8 x^3\right ) \log (4)+\left (-120 x^2-312 x^3-168 x^4+48 x^5+32 x^6+8 x^7+\left (16 x+48 x^2+32 x^3\right ) \log (4)\right ) \log (x)+\left (16 x^3+80 x^4+144 x^5+112 x^6+32 x^7\right ) \log ^2(x)\right ) \, dx=e^{100 x^{2} - 80 x \log {\left (2 \right )} + 4 \left (- 10 x^{5} - 20 x^{4} - 10 x^{3} + \left (4 x^{4} + 8 x^{3} + 4 x^{2}\right ) \log {\left (2 \right )}\right ) \log {\left (x \right )} + 4 \left (x^{8} + 4 x^{7} + 6 x^{6} + 4 x^{5} + x^{4}\right ) \log {\left (x \right )}^{2} + 16 \log {\left (2 \right )}^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (23) = 46\).
Time = 0.60 (sec) , antiderivative size = 111, normalized size of antiderivative = 5.05 \[ \int e^{100 x^2-40 x \log (4)+4 \log ^2(4)+4 \left (-10 x^3-20 x^4-10 x^5+\left (2 x^2+4 x^3+2 x^4\right ) \log (4)\right ) \log (x)+4 \left (x^4+4 x^5+6 x^6+4 x^7+x^8\right ) \log ^2(x)} \left (200 x-40 x^2-80 x^3-40 x^4+\left (-40+8 x+16 x^2+8 x^3\right ) \log (4)+\left (-120 x^2-312 x^3-168 x^4+48 x^5+32 x^6+8 x^7+\left (16 x+48 x^2+32 x^3\right ) \log (4)\right ) \log (x)+\left (16 x^3+80 x^4+144 x^5+112 x^6+32 x^7\right ) \log ^2(x)\right ) \, dx=e^{\left (4 \, x^{8} \log \left (x\right )^{2} + 16 \, x^{7} \log \left (x\right )^{2} + 24 \, x^{6} \log \left (x\right )^{2} + 16 \, x^{5} \log \left (x\right )^{2} - 40 \, x^{5} \log \left (x\right ) + 16 \, x^{4} \log \left (2\right ) \log \left (x\right ) + 4 \, x^{4} \log \left (x\right )^{2} - 80 \, x^{4} \log \left (x\right ) + 32 \, x^{3} \log \left (2\right ) \log \left (x\right ) - 40 \, x^{3} \log \left (x\right ) + 16 \, x^{2} \log \left (2\right ) \log \left (x\right ) + 100 \, x^{2} - 80 \, x \log \left (2\right ) + 16 \, \log \left (2\right )^{2}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (23) = 46\).
Time = 0.38 (sec) , antiderivative size = 111, normalized size of antiderivative = 5.05 \[ \int e^{100 x^2-40 x \log (4)+4 \log ^2(4)+4 \left (-10 x^3-20 x^4-10 x^5+\left (2 x^2+4 x^3+2 x^4\right ) \log (4)\right ) \log (x)+4 \left (x^4+4 x^5+6 x^6+4 x^7+x^8\right ) \log ^2(x)} \left (200 x-40 x^2-80 x^3-40 x^4+\left (-40+8 x+16 x^2+8 x^3\right ) \log (4)+\left (-120 x^2-312 x^3-168 x^4+48 x^5+32 x^6+8 x^7+\left (16 x+48 x^2+32 x^3\right ) \log (4)\right ) \log (x)+\left (16 x^3+80 x^4+144 x^5+112 x^6+32 x^7\right ) \log ^2(x)\right ) \, dx=e^{\left (4 \, x^{8} \log \left (x\right )^{2} + 16 \, x^{7} \log \left (x\right )^{2} + 24 \, x^{6} \log \left (x\right )^{2} + 16 \, x^{5} \log \left (x\right )^{2} - 40 \, x^{5} \log \left (x\right ) + 16 \, x^{4} \log \left (2\right ) \log \left (x\right ) + 4 \, x^{4} \log \left (x\right )^{2} - 80 \, x^{4} \log \left (x\right ) + 32 \, x^{3} \log \left (2\right ) \log \left (x\right ) - 40 \, x^{3} \log \left (x\right ) + 16 \, x^{2} \log \left (2\right ) \log \left (x\right ) + 100 \, x^{2} - 80 \, x \log \left (2\right ) + 16 \, \log \left (2\right )^{2}\right )} \]
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Timed out. \[ \int e^{100 x^2-40 x \log (4)+4 \log ^2(4)+4 \left (-10 x^3-20 x^4-10 x^5+\left (2 x^2+4 x^3+2 x^4\right ) \log (4)\right ) \log (x)+4 \left (x^4+4 x^5+6 x^6+4 x^7+x^8\right ) \log ^2(x)} \left (200 x-40 x^2-80 x^3-40 x^4+\left (-40+8 x+16 x^2+8 x^3\right ) \log (4)+\left (-120 x^2-312 x^3-168 x^4+48 x^5+32 x^6+8 x^7+\left (16 x+48 x^2+32 x^3\right ) \log (4)\right ) \log (x)+\left (16 x^3+80 x^4+144 x^5+112 x^6+32 x^7\right ) \log ^2(x)\right ) \, dx=\int {\mathrm {e}}^{4\,{\ln \left (x\right )}^2\,\left (x^8+4\,x^7+6\,x^6+4\,x^5+x^4\right )-80\,x\,\ln \left (2\right )-4\,\ln \left (x\right )\,\left (10\,x^3-2\,\ln \left (2\right )\,\left (2\,x^4+4\,x^3+2\,x^2\right )+20\,x^4+10\,x^5\right )+16\,{\ln \left (2\right )}^2+100\,x^2}\,\left (200\,x+{\ln \left (x\right )}^2\,\left (32\,x^7+112\,x^6+144\,x^5+80\,x^4+16\,x^3\right )+\ln \left (x\right )\,\left (2\,\ln \left (2\right )\,\left (32\,x^3+48\,x^2+16\,x\right )-120\,x^2-312\,x^3-168\,x^4+48\,x^5+32\,x^6+8\,x^7\right )+2\,\ln \left (2\right )\,\left (8\,x^3+16\,x^2+8\,x-40\right )-40\,x^2-80\,x^3-40\,x^4\right ) \,d x \]
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