Integrand size = 94, antiderivative size = 28 \[ \int \frac {10 x}{125-75 x+15 x^2-x^3+\left (600 x-240 x^2+24 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^2+\left (960 x^2-192 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^4+512 x^3 \left (i \pi +\log \left (\frac {e}{4}\right )\right )^6} \, dx=\frac {x^2}{\left (5-x+8 x \left (i \pi +\log \left (\frac {e}{4}\right )\right )^2\right )^2} \]
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Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(133\) vs. \(2(28)=56\).
Time = 0.35 (sec) , antiderivative size = 133, normalized size of antiderivative = 4.75, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {6, 12, 2099} \[ \int \frac {10 x}{125-75 x+15 x^2-x^3+\left (600 x-240 x^2+24 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^2+\left (960 x^2-192 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^4+512 x^3 \left (i \pi +\log \left (\frac {e}{4}\right )\right )^6} \, dx=\frac {25}{\left (7-8 \pi ^2+8 \log ^2(4)+16 i \pi (1-\log (4))-16 \log (4)\right )^2 \left (5+x \left (7-8 \pi ^2+8 \log ^2(4)+16 i \pi (1-\log (4))-16 \log (4)\right )\right )^2}-\frac {10}{\left (7-8 \pi ^2+8 \log ^2(4)+16 i \pi (1-\log (4))-16 \log (4)\right )^2 \left (5+x \left (7-8 \pi ^2+8 \log ^2(4)+16 i \pi (1-\log (4))-16 \log (4)\right )\right )} \]
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Rule 6
Rule 12
Rule 2099
Rubi steps \begin{align*} \text {integral}& = \int \frac {10 x}{125-75 x+15 x^2+\left (600 x-240 x^2+24 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^2+\left (960 x^2-192 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^4+x^3 \left (-1+512 \left (i \pi +\log \left (\frac {e}{4}\right )\right )^6\right )} \, dx \\ & = 10 \int \frac {x}{125-75 x+15 x^2+\left (600 x-240 x^2+24 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^2+\left (960 x^2-192 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^4+x^3 \left (-1+512 \left (i \pi +\log \left (\frac {e}{4}\right )\right )^6\right )} \, dx \\ & = 10 \int \left (\frac {5}{\left (-7+8 \pi ^2-16 i \pi (1-\log (4))+16 \log (4)-8 \log ^2(4)\right ) \left (5+x \left (7-8 \pi ^2+16 i \pi (1-\log (4))-16 \log (4)+8 \log ^2(4)\right )\right )^3}+\frac {1}{\left (7-8 \pi ^2+16 i \pi (1-\log (4))-16 \log (4)+8 \log ^2(4)\right ) \left (5+x \left (7-8 \pi ^2+16 i \pi (1-\log (4))-16 \log (4)+8 \log ^2(4)\right )\right )^2}\right ) \, dx \\ & = \frac {25}{\left (7-8 \pi ^2+16 i \pi (1-\log (4))-16 \log (4)+8 \log ^2(4)\right )^2 \left (5+x \left (7-8 \pi ^2+16 i \pi (1-\log (4))-16 \log (4)+8 \log ^2(4)\right )\right )^2}-\frac {10}{\left (7-8 \pi ^2+16 i \pi (1-\log (4))-16 \log (4)+8 \log ^2(4)\right )^2 \left (5+x \left (7-8 \pi ^2+16 i \pi (1-\log (4))-16 \log (4)+8 \log ^2(4)\right )\right )} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(93\) vs. \(2(28)=56\).
Time = 0.07 (sec) , antiderivative size = 93, normalized size of antiderivative = 3.32 \[ \int \frac {10 x}{125-75 x+15 x^2-x^3+\left (600 x-240 x^2+24 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^2+\left (960 x^2-192 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^4+512 x^3 \left (i \pi +\log \left (\frac {e}{4}\right )\right )^6} \, dx=-\frac {5 \left (5-2 x \left (-7+8 \pi ^2+16 i \pi (-1+\log (4))+16 \log (4)-8 \log ^2(4)\right )\right )}{\left (-7+8 \pi ^2+16 i \pi (-1+\log (4))+16 \log (4)-8 \log ^2(4)\right )^2 \left (-5+x \left (-7+8 \pi ^2+16 i \pi (-1+\log (4))+16 \log (4)-8 \log ^2(4)\right )\right )^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(48\) vs. \(2(21)=42\).
Time = 2.84 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75
method | result | size |
parallelrisch | \(\frac {x^{2}}{64 \ln \left (-\frac {{\mathrm e}}{4}\right )^{4} x^{2}-16 \ln \left (-\frac {{\mathrm e}}{4}\right )^{2} x^{2}+80 x \ln \left (-\frac {{\mathrm e}}{4}\right )^{2}+x^{2}-10 x +25}\) | \(49\) |
default | \(-\frac {10}{\left (8 \ln \left (-\frac {{\mathrm e}}{4}\right )^{2}-1\right )^{2} \left (8 x \ln \left (-\frac {{\mathrm e}}{4}\right )^{2}+5-x \right )}+\frac {25}{\left (8 \ln \left (-\frac {{\mathrm e}}{4}\right )^{2}-1\right )^{2} \left (8 x \ln \left (-\frac {{\mathrm e}}{4}\right )^{2}+5-x \right )^{2}}\) | \(66\) |
gosper | \(-\frac {5 \left (16 x \ln \left (-\frac {{\mathrm e}}{4}\right )^{2}-2 x +5\right )}{\left (64 \ln \left (-\frac {{\mathrm e}}{4}\right )^{4} x^{2}-16 \ln \left (-\frac {{\mathrm e}}{4}\right )^{2} x^{2}+80 x \ln \left (-\frac {{\mathrm e}}{4}\right )^{2}+x^{2}-10 x +25\right ) \left (8 \ln \left (-\frac {{\mathrm e}}{4}\right )^{2}-1\right )^{2}}\) | \(75\) |
risch | \(\frac {\frac {10 x}{-448+2048 \ln \left (2\right )-1024 i \pi -2048 \ln \left (2\right )^{2}+2048 i \ln \left (2\right ) \pi +512 \pi ^{2}}-\frac {25}{64 \left (-7+32 \ln \left (2\right )-16 i \pi -32 \ln \left (2\right )^{2}+32 i \ln \left (2\right ) \pi +8 \pi ^{2}\right )^{2}}}{-23 i \pi \ln \left (2\right ) x^{2}+\frac {7 i \pi \,x^{2}}{2}+\pi ^{4} x^{2}+\frac {5 i \pi x}{2}-24 \pi ^{2} \ln \left (2\right )^{2} x^{2}+8 i \pi ^{3} \ln \left (2\right ) x^{2}+16 x^{2} \ln \left (2\right )^{4}+24 \pi ^{2} \ln \left (2\right ) x^{2}-5 i \pi \ln \left (2\right ) x -32 x^{2} \ln \left (2\right )^{3}-\frac {23 \pi ^{2} x^{2}}{4}-32 i \pi \ln \left (2\right )^{3} x^{2}-4 i \pi ^{3} x^{2}+23 x^{2} \ln \left (2\right )^{2}-\frac {5 x \,\pi ^{2}}{4}+48 i \pi \ln \left (2\right )^{2} x^{2}+5 x \ln \left (2\right )^{2}-7 x^{2} \ln \left (2\right )-5 x \ln \left (2\right )+\frac {49 x^{2}}{64}+\frac {35 x}{32}+\frac {25}{64}}\) | \(238\) |
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 199, normalized size of antiderivative = 7.11 \[ \int \frac {10 x}{125-75 x+15 x^2-x^3+\left (600 x-240 x^2+24 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^2+\left (960 x^2-192 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^4+512 x^3 \left (i \pi +\log \left (\frac {e}{4}\right )\right )^6} \, dx=-\frac {5 \, {\left (16 \, {\left (i \, \pi + 2 \, \log \left (2\right )\right )}^{2} x - 32 \, {\left (i \, \pi + 2 \, \log \left (2\right )\right )} x + 14 \, x + 5\right )}}{4096 \, {\left (i \, \pi + 2 \, \log \left (2\right )\right )}^{8} x^{2} - 32768 \, {\left (i \, \pi + 2 \, \log \left (2\right )\right )}^{7} x^{2} + 5120 \, {\left (i \, \pi + 2 \, \log \left (2\right )\right )}^{6} {\left (22 \, x^{2} + x\right )} - 2048 \, {\left (i \, \pi + 2 \, \log \left (2\right )\right )}^{5} {\left (106 \, x^{2} + 15 \, x\right )} + 64 \, {\left (i \, \pi + 2 \, \log \left (2\right )\right )}^{4} {\left (4006 \, x^{2} + 1170 \, x + 25\right )} - 256 \, {\left (i \, \pi + 2 \, \log \left (2\right )\right )}^{3} {\left (742 \, x^{2} + 370 \, x + 25\right )} + 80 \, {\left (i \, \pi + 2 \, \log \left (2\right )\right )}^{2} {\left (1078 \, x^{2} + 819 \, x + 115\right )} - 224 \, {\left (i \, \pi + 2 \, \log \left (2\right )\right )} {\left (98 \, x^{2} + 105 \, x + 25\right )} + 2401 \, x^{2} + 3430 \, x + 1225} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 765 vs. \(2 (22) = 44\).
Time = 4.15 (sec) , antiderivative size = 765, normalized size of antiderivative = 27.32 \[ \int \frac {10 x}{125-75 x+15 x^2-x^3+\left (600 x-240 x^2+24 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^2+\left (960 x^2-192 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^4+512 x^3 \left (i \pi +\log \left (\frac {e}{4}\right )\right )^6} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (21) = 42\).
Time = 0.20 (sec) , antiderivative size = 114, normalized size of antiderivative = 4.07 \[ \int \frac {10 x}{125-75 x+15 x^2-x^3+\left (600 x-240 x^2+24 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^2+\left (960 x^2-192 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^4+512 x^3 \left (i \pi +\log \left (\frac {e}{4}\right )\right )^6} \, dx=-\frac {5 \, {\left (2 \, {\left (8 \, \log \left (-\frac {1}{4} \, e\right )^{2} - 1\right )} x + 5\right )}}{1600 \, \log \left (-\frac {1}{4} \, e\right )^{4} + {\left (4096 \, \log \left (-\frac {1}{4} \, e\right )^{8} - 2048 \, \log \left (-\frac {1}{4} \, e\right )^{6} + 384 \, \log \left (-\frac {1}{4} \, e\right )^{4} - 32 \, \log \left (-\frac {1}{4} \, e\right )^{2} + 1\right )} x^{2} + 10 \, {\left (512 \, \log \left (-\frac {1}{4} \, e\right )^{6} - 192 \, \log \left (-\frac {1}{4} \, e\right )^{4} + 24 \, \log \left (-\frac {1}{4} \, e\right )^{2} - 1\right )} x - 400 \, \log \left (-\frac {1}{4} \, e\right )^{2} + 25} \]
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Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (21) = 42\).
Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.00 \[ \int \frac {10 x}{125-75 x+15 x^2-x^3+\left (600 x-240 x^2+24 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^2+\left (960 x^2-192 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^4+512 x^3 \left (i \pi +\log \left (\frac {e}{4}\right )\right )^6} \, dx=-\frac {5 \, {\left (16 \, x \log \left (-\frac {1}{4} \, e\right )^{2} - 2 \, x + 5\right )}}{{\left (64 \, \log \left (-\frac {1}{4} \, e\right )^{4} - 16 \, \log \left (-\frac {1}{4} \, e\right )^{2} + 1\right )} {\left (8 \, x \log \left (-\frac {1}{4} \, e\right )^{2} - x + 5\right )}^{2}} \]
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Time = 0.22 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.68 \[ \int \frac {10 x}{125-75 x+15 x^2-x^3+\left (600 x-240 x^2+24 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^2+\left (960 x^2-192 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^4+512 x^3 \left (i \pi +\log \left (\frac {e}{4}\right )\right )^6} \, dx=-\frac {5\,\left (16\,x\,{\ln \left (-\frac {\mathrm {e}}{4}\right )}^2-2\,x+5\right )}{{\left (8\,{\ln \left (-\frac {\mathrm {e}}{4}\right )}^2-1\right )}^2\,{\left (8\,x\,{\ln \left (-\frac {\mathrm {e}}{4}\right )}^2-x+5\right )}^2} \]
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