Integrand size = 135, antiderivative size = 27 \[ \int \frac {1+x^2+e^{16-40 x+33 x^2-10 x^3+x^4+\left (-32+56 x-28 x^2+4 x^3\right ) \log (2)+\left (24-26 x+6 x^2\right ) \log ^2(2)+(-8+4 x) \log ^3(2)+\log ^4(2)} \left (-1-40 x+66 x^2-30 x^3+4 x^4+\left (56 x-56 x^2+12 x^3\right ) \log (2)+\left (-26 x+12 x^2\right ) \log ^2(2)+4 x \log ^3(2)\right )}{x^2} \, dx=e-\frac {1-e^{\left (-x+(-2+x+\log (2))^2\right )^2}}{x}+x \]
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\[ \int \frac {1+x^2+e^{16-40 x+33 x^2-10 x^3+x^4+\left (-32+56 x-28 x^2+4 x^3\right ) \log (2)+\left (24-26 x+6 x^2\right ) \log ^2(2)+(-8+4 x) \log ^3(2)+\log ^4(2)} \left (-1-40 x+66 x^2-30 x^3+4 x^4+\left (56 x-56 x^2+12 x^3\right ) \log (2)+\left (-26 x+12 x^2\right ) \log ^2(2)+4 x \log ^3(2)\right )}{x^2} \, dx=\int \frac {1+x^2+\exp \left (16-40 x+33 x^2-10 x^3+x^4+\left (-32+56 x-28 x^2+4 x^3\right ) \log (2)+\left (24-26 x+6 x^2\right ) \log ^2(2)+(-8+4 x) \log ^3(2)+\log ^4(2)\right ) \left (-1-40 x+66 x^2-30 x^3+4 x^4+\left (56 x-56 x^2+12 x^3\right ) \log (2)+\left (-26 x+12 x^2\right ) \log ^2(2)+4 x \log ^3(2)\right )}{x^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1+x^2}{x^2}+\frac {16^{-8+14 x-7 x^2+x^3} \exp \left (16-10 x^3+x^4+24 \log ^2(2)-8 \log ^3(2)+\log ^4(2)+3 x^2 \left (11+2 \log ^2(2)\right )-2 x \left (20+13 \log ^2(2)-2 \log ^3(2)\right )\right ) \left (-1+4 x^4+2 x^2 \left (33-28 \log (2)+6 \log ^2(2)\right )-6 x^3 (5-\log (4))-2 x (2-\log (2))^2 (5-\log (4))\right )}{x^2}\right ) \, dx \\ & = \int \frac {1+x^2}{x^2} \, dx+\int \frac {16^{-8+14 x-7 x^2+x^3} \exp \left (16-10 x^3+x^4+24 \log ^2(2)-8 \log ^3(2)+\log ^4(2)+3 x^2 \left (11+2 \log ^2(2)\right )-2 x \left (20+13 \log ^2(2)-2 \log ^3(2)\right )\right ) \left (-1+4 x^4+2 x^2 \left (33-28 \log (2)+6 \log ^2(2)\right )-6 x^3 (5-\log (4))-2 x (2-\log (2))^2 (5-\log (4))\right )}{x^2} \, dx \\ & = \int \left (1+\frac {1}{x^2}\right ) \, dx+\int \left (-\frac {16^{-8+14 x-7 x^2+x^3} \exp \left (16-10 x^3+x^4+24 \log ^2(2)-8 \log ^3(2)+\log ^4(2)+3 x^2 \left (11+2 \log ^2(2)\right )-2 x \left (20+13 \log ^2(2)-2 \log ^3(2)\right )\right )}{x^2}+4^{-15+28 x-14 x^2+2 x^3} \exp \left (16-10 x^3+x^4+24 \log ^2(2)-8 \log ^3(2)+\log ^4(2)+3 x^2 \left (11+2 \log ^2(2)\right )-2 x \left (20+13 \log ^2(2)-2 \log ^3(2)\right )\right ) x^2+2^{-31+56 x-28 x^2+4 x^3} \exp \left (16-10 x^3+x^4+24 \log ^2(2)-8 \log ^3(2)+\log ^4(2)+3 x^2 \left (11+2 \log ^2(2)\right )-2 x \left (20+13 \log ^2(2)-2 \log ^3(2)\right )\right ) \left (33-28 \log (2)+6 \log ^2(2)\right )+3\ 2^{-31+56 x-28 x^2+4 x^3} \exp \left (16-10 x^3+x^4+24 \log ^2(2)-8 \log ^3(2)+\log ^4(2)+3 x^2 \left (11+2 \log ^2(2)\right )-2 x \left (20+13 \log ^2(2)-2 \log ^3(2)\right )\right ) x (-5+\log (4))+\frac {2^{-31+56 x-28 x^2+4 x^3} \exp \left (16-10 x^3+x^4+24 \log ^2(2)-8 \log ^3(2)+\log ^4(2)+3 x^2 \left (11+2 \log ^2(2)\right )-2 x \left (20+13 \log ^2(2)-2 \log ^3(2)\right )\right ) (-2+\log (2))^2 (-5+\log (4))}{x}\right ) \, dx \\ & = -\frac {1}{x}+x+\left (33-28 \log (2)+6 \log ^2(2)\right ) \int 2^{-31+56 x-28 x^2+4 x^3} \exp \left (16-10 x^3+x^4+24 \log ^2(2)-8 \log ^3(2)+\log ^4(2)+3 x^2 \left (11+2 \log ^2(2)\right )-2 x \left (20+13 \log ^2(2)-2 \log ^3(2)\right )\right ) \, dx-\left ((2-\log (2))^2 (5-\log (4))\right ) \int \frac {2^{-31+56 x-28 x^2+4 x^3} \exp \left (16-10 x^3+x^4+24 \log ^2(2)-8 \log ^3(2)+\log ^4(2)+3 x^2 \left (11+2 \log ^2(2)\right )-2 x \left (20+13 \log ^2(2)-2 \log ^3(2)\right )\right )}{x} \, dx+(3 (-5+\log (4))) \int 2^{-31+56 x-28 x^2+4 x^3} \exp \left (16-10 x^3+x^4+24 \log ^2(2)-8 \log ^3(2)+\log ^4(2)+3 x^2 \left (11+2 \log ^2(2)\right )-2 x \left (20+13 \log ^2(2)-2 \log ^3(2)\right )\right ) x \, dx-\int \frac {16^{-8+14 x-7 x^2+x^3} \exp \left (16-10 x^3+x^4+24 \log ^2(2)-8 \log ^3(2)+\log ^4(2)+3 x^2 \left (11+2 \log ^2(2)\right )-2 x \left (20+13 \log ^2(2)-2 \log ^3(2)\right )\right )}{x^2} \, dx+\int 4^{-15+28 x-14 x^2+2 x^3} \exp \left (16-10 x^3+x^4+24 \log ^2(2)-8 \log ^3(2)+\log ^4(2)+3 x^2 \left (11+2 \log ^2(2)\right )-2 x \left (20+13 \log ^2(2)-2 \log ^3(2)\right )\right ) x^2 \, dx \\ \end{align*}
\[ \int \frac {1+x^2+e^{16-40 x+33 x^2-10 x^3+x^4+\left (-32+56 x-28 x^2+4 x^3\right ) \log (2)+\left (24-26 x+6 x^2\right ) \log ^2(2)+(-8+4 x) \log ^3(2)+\log ^4(2)} \left (-1-40 x+66 x^2-30 x^3+4 x^4+\left (56 x-56 x^2+12 x^3\right ) \log (2)+\left (-26 x+12 x^2\right ) \log ^2(2)+4 x \log ^3(2)\right )}{x^2} \, dx=\int \frac {1+x^2+e^{16-40 x+33 x^2-10 x^3+x^4+\left (-32+56 x-28 x^2+4 x^3\right ) \log (2)+\left (24-26 x+6 x^2\right ) \log ^2(2)+(-8+4 x) \log ^3(2)+\log ^4(2)} \left (-1-40 x+66 x^2-30 x^3+4 x^4+\left (56 x-56 x^2+12 x^3\right ) \log (2)+\left (-26 x+12 x^2\right ) \log ^2(2)+4 x \log ^3(2)\right )}{x^2} \, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. \(75\) vs. \(2(27)=54\).
Time = 7.51 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.81
method | result | size |
norman | \(\frac {-1+x^{2}+{\mathrm e}^{\ln \left (2\right )^{4}+\left (4 x -8\right ) \ln \left (2\right )^{3}+\left (6 x^{2}-26 x +24\right ) \ln \left (2\right )^{2}+\left (4 x^{3}-28 x^{2}+56 x -32\right ) \ln \left (2\right )+x^{4}-10 x^{3}+33 x^{2}-40 x +16}}{x}\) | \(76\) |
parallelrisch | \(\frac {-1+x^{2}+{\mathrm e}^{\ln \left (2\right )^{4}+\left (4 x -8\right ) \ln \left (2\right )^{3}+\left (6 x^{2}-26 x +24\right ) \ln \left (2\right )^{2}+\left (4 x^{3}-28 x^{2}+56 x -32\right ) \ln \left (2\right )+x^{4}-10 x^{3}+33 x^{2}-40 x +16}}{x}\) | \(76\) |
parts | \(x -\frac {1}{x}+\frac {{\mathrm e}^{\ln \left (2\right )^{4}+\left (4 x -8\right ) \ln \left (2\right )^{3}+\left (6 x^{2}-26 x +24\right ) \ln \left (2\right )^{2}+\left (4 x^{3}-28 x^{2}+56 x -32\right ) \ln \left (2\right )+x^{4}-10 x^{3}+33 x^{2}-40 x +16}}{x}\) | \(78\) |
risch | \(x -\frac {1}{x}+\frac {16^{\left (-1+x \right ) \left (-2+x \right ) \left (x -4\right )} {\mathrm e}^{\ln \left (2\right )^{4}+4 x \ln \left (2\right )^{3}+6 x^{2} \ln \left (2\right )^{2}+x^{4}-8 \ln \left (2\right )^{3}-26 x \ln \left (2\right )^{2}-10 x^{3}+24 \ln \left (2\right )^{2}+33 x^{2}-40 x +16}}{x}\) | \(82\) |
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Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (24) = 48\).
Time = 0.26 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.74 \[ \int \frac {1+x^2+e^{16-40 x+33 x^2-10 x^3+x^4+\left (-32+56 x-28 x^2+4 x^3\right ) \log (2)+\left (24-26 x+6 x^2\right ) \log ^2(2)+(-8+4 x) \log ^3(2)+\log ^4(2)} \left (-1-40 x+66 x^2-30 x^3+4 x^4+\left (56 x-56 x^2+12 x^3\right ) \log (2)+\left (-26 x+12 x^2\right ) \log ^2(2)+4 x \log ^3(2)\right )}{x^2} \, dx=\frac {x^{2} + e^{\left (x^{4} + 4 \, {\left (x - 2\right )} \log \left (2\right )^{3} + \log \left (2\right )^{4} - 10 \, x^{3} + 2 \, {\left (3 \, x^{2} - 13 \, x + 12\right )} \log \left (2\right )^{2} + 33 \, x^{2} + 4 \, {\left (x^{3} - 7 \, x^{2} + 14 \, x - 8\right )} \log \left (2\right ) - 40 \, x + 16\right )} - 1}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (20) = 40\).
Time = 0.15 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.78 \[ \int \frac {1+x^2+e^{16-40 x+33 x^2-10 x^3+x^4+\left (-32+56 x-28 x^2+4 x^3\right ) \log (2)+\left (24-26 x+6 x^2\right ) \log ^2(2)+(-8+4 x) \log ^3(2)+\log ^4(2)} \left (-1-40 x+66 x^2-30 x^3+4 x^4+\left (56 x-56 x^2+12 x^3\right ) \log (2)+\left (-26 x+12 x^2\right ) \log ^2(2)+4 x \log ^3(2)\right )}{x^2} \, dx=x + \frac {e^{x^{4} - 10 x^{3} + 33 x^{2} - 40 x + \left (4 x - 8\right ) \log {\left (2 \right )}^{3} + \left (6 x^{2} - 26 x + 24\right ) \log {\left (2 \right )}^{2} + \left (4 x^{3} - 28 x^{2} + 56 x - 32\right ) \log {\left (2 \right )} + \log {\left (2 \right )}^{4} + 16}}{x} - \frac {1}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (24) = 48\).
Time = 0.48 (sec) , antiderivative size = 89, normalized size of antiderivative = 3.30 \[ \int \frac {1+x^2+e^{16-40 x+33 x^2-10 x^3+x^4+\left (-32+56 x-28 x^2+4 x^3\right ) \log (2)+\left (24-26 x+6 x^2\right ) \log ^2(2)+(-8+4 x) \log ^3(2)+\log ^4(2)} \left (-1-40 x+66 x^2-30 x^3+4 x^4+\left (56 x-56 x^2+12 x^3\right ) \log (2)+\left (-26 x+12 x^2\right ) \log ^2(2)+4 x \log ^3(2)\right )}{x^2} \, dx=x + \frac {e^{\left (x^{4} + 4 \, x^{3} \log \left (2\right ) + 6 \, x^{2} \log \left (2\right )^{2} + 4 \, x \log \left (2\right )^{3} + \log \left (2\right )^{4} - 10 \, x^{3} - 28 \, x^{2} \log \left (2\right ) - 26 \, x \log \left (2\right )^{2} - 8 \, \log \left (2\right )^{3} + 33 \, x^{2} + 56 \, x \log \left (2\right ) + 24 \, \log \left (2\right )^{2} - 40 \, x + 16\right )}}{4294967296 \, x} - \frac {1}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (24) = 48\).
Time = 0.56 (sec) , antiderivative size = 89, normalized size of antiderivative = 3.30 \[ \int \frac {1+x^2+e^{16-40 x+33 x^2-10 x^3+x^4+\left (-32+56 x-28 x^2+4 x^3\right ) \log (2)+\left (24-26 x+6 x^2\right ) \log ^2(2)+(-8+4 x) \log ^3(2)+\log ^4(2)} \left (-1-40 x+66 x^2-30 x^3+4 x^4+\left (56 x-56 x^2+12 x^3\right ) \log (2)+\left (-26 x+12 x^2\right ) \log ^2(2)+4 x \log ^3(2)\right )}{x^2} \, dx=\frac {4294967296 \, x^{2} + e^{\left (x^{4} + 4 \, x^{3} \log \left (2\right ) + 6 \, x^{2} \log \left (2\right )^{2} + 4 \, x \log \left (2\right )^{3} + \log \left (2\right )^{4} - 10 \, x^{3} - 28 \, x^{2} \log \left (2\right ) - 26 \, x \log \left (2\right )^{2} - 8 \, \log \left (2\right )^{3} + 33 \, x^{2} + 56 \, x \log \left (2\right ) + 24 \, \log \left (2\right )^{2} - 40 \, x + 16\right )} - 4294967296}{4294967296 \, x} \]
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Time = 0.45 (sec) , antiderivative size = 100, normalized size of antiderivative = 3.70 \[ \int \frac {1+x^2+e^{16-40 x+33 x^2-10 x^3+x^4+\left (-32+56 x-28 x^2+4 x^3\right ) \log (2)+\left (24-26 x+6 x^2\right ) \log ^2(2)+(-8+4 x) \log ^3(2)+\log ^4(2)} \left (-1-40 x+66 x^2-30 x^3+4 x^4+\left (56 x-56 x^2+12 x^3\right ) \log (2)+\left (-26 x+12 x^2\right ) \log ^2(2)+4 x \log ^3(2)\right )}{x^2} \, dx=x-\frac {1}{x}+\frac {2^{56\,x}\,2^{4\,x^3}\,{\mathrm {e}}^{4\,x\,{\ln \left (2\right )}^3}\,{\mathrm {e}}^{-26\,x\,{\ln \left (2\right )}^2}\,{\mathrm {e}}^{{\ln \left (2\right )}^4}\,{\mathrm {e}}^{-40\,x}\,{\mathrm {e}}^{x^4}\,{\mathrm {e}}^{16}\,{\mathrm {e}}^{6\,x^2\,{\ln \left (2\right )}^2}\,{\mathrm {e}}^{-8\,{\ln \left (2\right )}^3}\,{\mathrm {e}}^{24\,{\ln \left (2\right )}^2}\,{\mathrm {e}}^{-10\,x^3}\,{\mathrm {e}}^{33\,x^2}}{4294967296\,2^{28\,x^2}\,x} \]
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