Integrand size = 22, antiderivative size = 366 \[ \int \frac {1}{\left (8+24 x+8 x^2-15 x^3+8 x^4\right )^2} \, dx=-\frac {3 \left (3359-107 \left (3+\frac {4}{x}\right )^2\right )}{208 \left (517-38 \left (3+\frac {4}{x}\right )^2+\left (3+\frac {4}{x}\right )^4\right )}+\frac {\left (3327931-129631 \left (3+\frac {4}{x}\right )^2\right ) \left (3+\frac {4}{x}\right )}{322608 \left (517-38 \left (3+\frac {4}{x}\right )^2+\left (3+\frac {4}{x}\right )^4\right )}-\frac {\sqrt {\frac {19+\sqrt {517}}{40326}} \left (1678181+74897 \sqrt {517}\right ) \arctan \left (\frac {6-\sqrt {2 \left (19+\sqrt {517}\right )}+\frac {8}{x}}{\sqrt {2 \left (-19+\sqrt {517}\right )}}\right )}{645216}-\frac {\sqrt {\frac {19+\sqrt {517}}{40326}} \left (1678181+74897 \sqrt {517}\right ) \arctan \left (\frac {6+\sqrt {2 \left (19+\sqrt {517}\right )}+\frac {8}{x}}{\sqrt {2 \left (-19+\sqrt {517}\right )}}\right )}{645216}+\frac {73}{208} \sqrt {\frac {3}{13}} \arctan \left (\frac {8+12 x-5 x^2}{\sqrt {39} x^2}\right )-\frac {\sqrt {\frac {-59644114671451+2623170438295 \sqrt {517}}{40326}} \log \left (\sqrt {517}-\sqrt {2 \left (19+\sqrt {517}\right )} \left (3+\frac {4}{x}\right )+\left (3+\frac {4}{x}\right )^2\right )}{645216}+\frac {\sqrt {\frac {-59644114671451+2623170438295 \sqrt {517}}{40326}} \log \left (\sqrt {517}+\sqrt {2 \left (19+\sqrt {517}\right )} \left (3+\frac {4}{x}\right )+\left (3+\frac {4}{x}\right )^2\right )}{645216} \]
-3/208*(3359-107*(3+4/x)^2)/(517-38*(3+4/x)^2+(3+4/x)^4)+1/322608*(3327931 -129631*(3+4/x)^2)*(3+4/x)/(517-38*(3+4/x)^2+(3+4/x)^4)+73/2704*arctan(1/3 9*(-5*x^2+12*x+8)/x^2*39^(1/2))*39^(1/2)-1/26018980416*arctan((6+8/x-(38+2 *517^(1/2))^(1/2))/(-38+2*517^(1/2))^(1/2))*(1678181+74897*517^(1/2))*(766 194+40326*517^(1/2))^(1/2)-1/26018980416*arctan((6+8/x+(38+2*517^(1/2))^(1 /2))/(-38+2*517^(1/2))^(1/2))*(1678181+74897*517^(1/2))*(766194+40326*517^ (1/2))^(1/2)-1/26018980416*ln((3+4/x)^2+517^(1/2)-(3+4/x)*(38+2*517^(1/2)) ^(1/2))*(-2405208568240933026+105781971094684170*517^(1/2))^(1/2)+1/260189 80416*ln((3+4/x)^2+517^(1/2)+(3+4/x)*(38+2*517^(1/2))^(1/2))*(-24052085682 40933026+105781971094684170*517^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.35 \[ \int \frac {1}{\left (8+24 x+8 x^2-15 x^3+8 x^4\right )^2} \, dx=\frac {72888+89033 x-94314 x^2+39280 x^3}{161304 \left (8+24 x+8 x^2-15 x^3+8 x^4\right )}+\frac {\text {RootSum}\left [8+24 \text {$\#$1}+8 \text {$\#$1}^2-15 \text {$\#$1}^3+8 \text {$\#$1}^4\&,\frac {74897 \log (x-\text {$\#$1})-57489 \log (x-\text {$\#$1}) \text {$\#$1}+19640 \log (x-\text {$\#$1}) \text {$\#$1}^2}{24+16 \text {$\#$1}-45 \text {$\#$1}^2+32 \text {$\#$1}^3}\&\right ]}{80652} \]
(72888 + 89033*x - 94314*x^2 + 39280*x^3)/(161304*(8 + 24*x + 8*x^2 - 15*x ^3 + 8*x^4)) + RootSum[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , (74897*Lo g[x - #1] - 57489*Log[x - #1]*#1 + 19640*Log[x - #1]*#1^2)/(24 + 16*#1 - 4 5*#1^2 + 32*#1^3) & ]/80652
Time = 0.89 (sec) , antiderivative size = 428, normalized size of antiderivative = 1.17, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.818, Rules used = {2504, 27, 2202, 2194, 27, 2191, 27, 1083, 217, 2206, 27, 1483, 27, 1142, 27, 1083, 217, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{\left (8 x^4-15 x^3+8 x^2+24 x+8\right )^2} \, dx\) |
| \(\Big \downarrow \) 2504 |
| \(\displaystyle -1024 \int \frac {\left (3-4 \left (\frac {3}{4}+\frac {1}{x}\right )\right )^6}{4096 \left (256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517\right )^2}d\left (\frac {3}{4}+\frac {1}{x}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{4} \int \frac {\left (3-4 \left (\frac {3}{4}+\frac {1}{x}\right )\right )^6}{\left (256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517\right )^2}d\left (\frac {3}{4}+\frac {1}{x}\right )\) |
| \(\Big \downarrow \) 2202 |
| \(\displaystyle \frac {1}{4} \left (-\int \frac {4096 \left (\frac {3}{4}+\frac {1}{x}\right )^6+34560 \left (\frac {3}{4}+\frac {1}{x}\right )^4+19440 \left (\frac {3}{4}+\frac {1}{x}\right )^2+729}{\left (256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517\right )^2}d\left (\frac {3}{4}+\frac {1}{x}\right )-\int \frac {\left (-18432 \left (\frac {3}{4}+\frac {1}{x}\right )^4-34560 \left (\frac {3}{4}+\frac {1}{x}\right )^2-5832\right ) \left (\frac {3}{4}+\frac {1}{x}\right )}{\left (256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517\right )^2}d\left (\frac {3}{4}+\frac {1}{x}\right )\right )\) |
| \(\Big \downarrow \) 2194 |
| \(\displaystyle \frac {1}{4} \left (-\frac {1}{2} \int -\frac {72 \left (256 \left (\frac {3}{4}+\frac {1}{x}\right )^4+480 \left (\frac {3}{4}+\frac {1}{x}\right )^2+81\right )}{\left (256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517\right )^2}d\left (\frac {3}{4}+\frac {1}{x}\right )^2-\int \frac {4096 \left (\frac {3}{4}+\frac {1}{x}\right )^6+34560 \left (\frac {3}{4}+\frac {1}{x}\right )^4+19440 \left (\frac {3}{4}+\frac {1}{x}\right )^2+729}{\left (256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517\right )^2}d\left (\frac {3}{4}+\frac {1}{x}\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (36 \int \frac {256 \left (\frac {3}{4}+\frac {1}{x}\right )^4+480 \left (\frac {3}{4}+\frac {1}{x}\right )^2+81}{\left (256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517\right )^2}d\left (\frac {3}{4}+\frac {1}{x}\right )^2-\int \frac {4096 \left (\frac {3}{4}+\frac {1}{x}\right )^6+34560 \left (\frac {3}{4}+\frac {1}{x}\right )^4+19440 \left (\frac {3}{4}+\frac {1}{x}\right )^2+729}{\left (256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517\right )^2}d\left (\frac {3}{4}+\frac {1}{x}\right )\right )\) |
| \(\Big \downarrow \) 2191 |
| \(\displaystyle \frac {1}{4} \left (36 \left (\frac {\int \frac {598016}{256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517}d\left (\frac {3}{4}+\frac {1}{x}\right )^2}{159744}-\frac {3359-1712 \left (\frac {1}{x}+\frac {3}{4}\right )^2}{624 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )}\right )-\int \frac {4096 \left (\frac {3}{4}+\frac {1}{x}\right )^6+34560 \left (\frac {3}{4}+\frac {1}{x}\right )^4+19440 \left (\frac {3}{4}+\frac {1}{x}\right )^2+729}{\left (256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517\right )^2}d\left (\frac {3}{4}+\frac {1}{x}\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (36 \left (\frac {146}{39} \int \frac {1}{256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517}d\left (\frac {3}{4}+\frac {1}{x}\right )^2-\frac {3359-1712 \left (\frac {1}{x}+\frac {3}{4}\right )^2}{624 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )}\right )-\int \frac {4096 \left (\frac {3}{4}+\frac {1}{x}\right )^6+34560 \left (\frac {3}{4}+\frac {1}{x}\right )^4+19440 \left (\frac {3}{4}+\frac {1}{x}\right )^2+729}{\left (256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517\right )^2}d\left (\frac {3}{4}+\frac {1}{x}\right )\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{4} \left (36 \left (-\frac {292}{39} \int \frac {1}{-\left (\frac {3}{4}+\frac {1}{x}\right )^4-159744}d\left (512 \left (\frac {3}{4}+\frac {1}{x}\right )^2-608\right )-\frac {3359-1712 \left (\frac {1}{x}+\frac {3}{4}\right )^2}{624 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )}\right )-\int \frac {4096 \left (\frac {3}{4}+\frac {1}{x}\right )^6+34560 \left (\frac {3}{4}+\frac {1}{x}\right )^4+19440 \left (\frac {3}{4}+\frac {1}{x}\right )^2+729}{\left (256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517\right )^2}d\left (\frac {3}{4}+\frac {1}{x}\right )\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{4} \left (36 \left (\frac {73 \arctan \left (\frac {512 \left (\frac {1}{x}+\frac {3}{4}\right )^2-608}{64 \sqrt {39}}\right )}{624 \sqrt {39}}-\frac {3359-1712 \left (\frac {1}{x}+\frac {3}{4}\right )^2}{624 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )}\right )-\int \frac {4096 \left (\frac {3}{4}+\frac {1}{x}\right )^6+34560 \left (\frac {3}{4}+\frac {1}{x}\right )^4+19440 \left (\frac {3}{4}+\frac {1}{x}\right )^2+729}{\left (256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517\right )^2}d\left (\frac {3}{4}+\frac {1}{x}\right )\right )\) |
| \(\Big \downarrow \) 2206 |
| \(\displaystyle \frac {1}{4} \left (-\frac {\int \frac {16384 \left (1198352 \left (\frac {3}{4}+\frac {1}{x}\right )^2+1678181\right )}{256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517}d\left (\frac {3}{4}+\frac {1}{x}\right )}{165175296}+36 \left (\frac {73 \arctan \left (\frac {512 \left (\frac {1}{x}+\frac {3}{4}\right )^2-608}{64 \sqrt {39}}\right )}{624 \sqrt {39}}-\frac {3359-1712 \left (\frac {1}{x}+\frac {3}{4}\right )^2}{624 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )}\right )+\frac {\left (3327931-2074096 \left (\frac {1}{x}+\frac {3}{4}\right )^2\right ) \left (\frac {1}{x}+\frac {3}{4}\right )}{20163 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (-\frac {2 \int \frac {1198352 \left (\frac {3}{4}+\frac {1}{x}\right )^2+1678181}{256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517}d\left (\frac {3}{4}+\frac {1}{x}\right )}{20163}+36 \left (\frac {73 \arctan \left (\frac {512 \left (\frac {1}{x}+\frac {3}{4}\right )^2-608}{64 \sqrt {39}}\right )}{624 \sqrt {39}}-\frac {3359-1712 \left (\frac {1}{x}+\frac {3}{4}\right )^2}{624 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )}\right )+\frac {\left (3327931-2074096 \left (\frac {1}{x}+\frac {3}{4}\right )^2\right ) \left (\frac {1}{x}+\frac {3}{4}\right )}{20163 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )}\right )\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {1}{4} \left (-\frac {2 \left (\frac {\int \frac {8 \left (1678181 \sqrt {\frac {1}{2} \left (19+\sqrt {517}\right )}-2 \left (1678181-74897 \sqrt {517}\right ) \left (\frac {3}{4}+\frac {1}{x}\right )\right )}{16 \left (\frac {3}{4}+\frac {1}{x}\right )^2-4 \sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {517}}d\left (\frac {3}{4}+\frac {1}{x}\right )}{8 \sqrt {1034 \left (19+\sqrt {517}\right )}}+\frac {\int \frac {8 \left (2 \left (1678181-74897 \sqrt {517}\right ) \left (\frac {3}{4}+\frac {1}{x}\right )+1678181 \sqrt {\frac {1}{2} \left (19+\sqrt {517}\right )}\right )}{16 \left (\frac {3}{4}+\frac {1}{x}\right )^2+4 \sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {517}}d\left (\frac {3}{4}+\frac {1}{x}\right )}{8 \sqrt {1034 \left (19+\sqrt {517}\right )}}\right )}{20163}+36 \left (\frac {73 \arctan \left (\frac {512 \left (\frac {1}{x}+\frac {3}{4}\right )^2-608}{64 \sqrt {39}}\right )}{624 \sqrt {39}}-\frac {3359-1712 \left (\frac {1}{x}+\frac {3}{4}\right )^2}{624 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )}\right )+\frac {\left (3327931-2074096 \left (\frac {1}{x}+\frac {3}{4}\right )^2\right ) \left (\frac {1}{x}+\frac {3}{4}\right )}{20163 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (-\frac {2 \left (\frac {\int \frac {1678181 \sqrt {\frac {1}{2} \left (19+\sqrt {517}\right )}-2 \left (1678181-74897 \sqrt {517}\right ) \left (\frac {3}{4}+\frac {1}{x}\right )}{16 \left (\frac {3}{4}+\frac {1}{x}\right )^2-4 \sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {517}}d\left (\frac {3}{4}+\frac {1}{x}\right )}{\sqrt {1034 \left (19+\sqrt {517}\right )}}+\frac {\int \frac {2 \left (1678181-74897 \sqrt {517}\right ) \left (\frac {3}{4}+\frac {1}{x}\right )+1678181 \sqrt {\frac {1}{2} \left (19+\sqrt {517}\right )}}{16 \left (\frac {3}{4}+\frac {1}{x}\right )^2+4 \sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {517}}d\left (\frac {3}{4}+\frac {1}{x}\right )}{\sqrt {1034 \left (19+\sqrt {517}\right )}}\right )}{20163}+36 \left (\frac {73 \arctan \left (\frac {512 \left (\frac {1}{x}+\frac {3}{4}\right )^2-608}{64 \sqrt {39}}\right )}{624 \sqrt {39}}-\frac {3359-1712 \left (\frac {1}{x}+\frac {3}{4}\right )^2}{624 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )}\right )+\frac {\left (3327931-2074096 \left (\frac {1}{x}+\frac {3}{4}\right )^2\right ) \left (\frac {1}{x}+\frac {3}{4}\right )}{20163 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{4} \left (-\frac {2 \left (\frac {\frac {1}{2} \sqrt {\frac {1}{2} \left (19+\sqrt {517}\right )} \left (1678181+74897 \sqrt {517}\right ) \int \frac {1}{16 \left (\frac {3}{4}+\frac {1}{x}\right )^2-4 \sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {517}}d\left (\frac {3}{4}+\frac {1}{x}\right )-\frac {1}{16} \left (1678181-74897 \sqrt {517}\right ) \int -\frac {4 \left (\sqrt {2 \left (19+\sqrt {517}\right )}-8 \left (\frac {3}{4}+\frac {1}{x}\right )\right )}{16 \left (\frac {3}{4}+\frac {1}{x}\right )^2-4 \sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {517}}d\left (\frac {3}{4}+\frac {1}{x}\right )}{\sqrt {1034 \left (19+\sqrt {517}\right )}}+\frac {\frac {1}{2} \sqrt {\frac {1}{2} \left (19+\sqrt {517}\right )} \left (1678181+74897 \sqrt {517}\right ) \int \frac {1}{16 \left (\frac {3}{4}+\frac {1}{x}\right )^2+4 \sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {517}}d\left (\frac {3}{4}+\frac {1}{x}\right )+\frac {1}{16} \left (1678181-74897 \sqrt {517}\right ) \int \frac {4 \left (8 \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {2 \left (19+\sqrt {517}\right )}\right )}{16 \left (\frac {3}{4}+\frac {1}{x}\right )^2+4 \sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {517}}d\left (\frac {3}{4}+\frac {1}{x}\right )}{\sqrt {1034 \left (19+\sqrt {517}\right )}}\right )}{20163}+36 \left (\frac {73 \arctan \left (\frac {512 \left (\frac {1}{x}+\frac {3}{4}\right )^2-608}{64 \sqrt {39}}\right )}{624 \sqrt {39}}-\frac {3359-1712 \left (\frac {1}{x}+\frac {3}{4}\right )^2}{624 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )}\right )+\frac {\left (3327931-2074096 \left (\frac {1}{x}+\frac {3}{4}\right )^2\right ) \left (\frac {1}{x}+\frac {3}{4}\right )}{20163 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (-\frac {2 \left (\frac {\frac {1}{2} \sqrt {\frac {1}{2} \left (19+\sqrt {517}\right )} \left (1678181+74897 \sqrt {517}\right ) \int \frac {1}{16 \left (\frac {3}{4}+\frac {1}{x}\right )^2-4 \sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {517}}d\left (\frac {3}{4}+\frac {1}{x}\right )+\frac {1}{4} \left (1678181-74897 \sqrt {517}\right ) \int \frac {\sqrt {2 \left (19+\sqrt {517}\right )}-8 \left (\frac {3}{4}+\frac {1}{x}\right )}{16 \left (\frac {3}{4}+\frac {1}{x}\right )^2-4 \sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {517}}d\left (\frac {3}{4}+\frac {1}{x}\right )}{\sqrt {1034 \left (19+\sqrt {517}\right )}}+\frac {\frac {1}{2} \sqrt {\frac {1}{2} \left (19+\sqrt {517}\right )} \left (1678181+74897 \sqrt {517}\right ) \int \frac {1}{16 \left (\frac {3}{4}+\frac {1}{x}\right )^2+4 \sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {517}}d\left (\frac {3}{4}+\frac {1}{x}\right )+\frac {1}{4} \left (1678181-74897 \sqrt {517}\right ) \int \frac {8 \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {2 \left (19+\sqrt {517}\right )}}{16 \left (\frac {3}{4}+\frac {1}{x}\right )^2+4 \sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {517}}d\left (\frac {3}{4}+\frac {1}{x}\right )}{\sqrt {1034 \left (19+\sqrt {517}\right )}}\right )}{20163}+36 \left (\frac {73 \arctan \left (\frac {512 \left (\frac {1}{x}+\frac {3}{4}\right )^2-608}{64 \sqrt {39}}\right )}{624 \sqrt {39}}-\frac {3359-1712 \left (\frac {1}{x}+\frac {3}{4}\right )^2}{624 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )}\right )+\frac {\left (3327931-2074096 \left (\frac {1}{x}+\frac {3}{4}\right )^2\right ) \left (\frac {1}{x}+\frac {3}{4}\right )}{20163 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{4} \left (-\frac {2 \left (\frac {\frac {1}{4} \left (1678181-74897 \sqrt {517}\right ) \int \frac {\sqrt {2 \left (19+\sqrt {517}\right )}-8 \left (\frac {3}{4}+\frac {1}{x}\right )}{16 \left (\frac {3}{4}+\frac {1}{x}\right )^2-4 \sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {517}}d\left (\frac {3}{4}+\frac {1}{x}\right )-\sqrt {\frac {1}{2} \left (19+\sqrt {517}\right )} \left (1678181+74897 \sqrt {517}\right ) \int \frac {1}{32 \left (19-\sqrt {517}\right )-\left (32 \left (\frac {3}{4}+\frac {1}{x}\right )-4 \sqrt {2 \left (19+\sqrt {517}\right )}\right )^2}d\left (32 \left (\frac {3}{4}+\frac {1}{x}\right )-4 \sqrt {2 \left (19+\sqrt {517}\right )}\right )}{\sqrt {1034 \left (19+\sqrt {517}\right )}}+\frac {\frac {1}{4} \left (1678181-74897 \sqrt {517}\right ) \int \frac {8 \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {2 \left (19+\sqrt {517}\right )}}{16 \left (\frac {3}{4}+\frac {1}{x}\right )^2+4 \sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {517}}d\left (\frac {3}{4}+\frac {1}{x}\right )-\sqrt {\frac {1}{2} \left (19+\sqrt {517}\right )} \left (1678181+74897 \sqrt {517}\right ) \int \frac {1}{32 \left (19-\sqrt {517}\right )-\left (32 \left (\frac {3}{4}+\frac {1}{x}\right )+4 \sqrt {2 \left (19+\sqrt {517}\right )}\right )^2}d\left (32 \left (\frac {3}{4}+\frac {1}{x}\right )+4 \sqrt {2 \left (19+\sqrt {517}\right )}\right )}{\sqrt {1034 \left (19+\sqrt {517}\right )}}\right )}{20163}+36 \left (\frac {73 \arctan \left (\frac {512 \left (\frac {1}{x}+\frac {3}{4}\right )^2-608}{64 \sqrt {39}}\right )}{624 \sqrt {39}}-\frac {3359-1712 \left (\frac {1}{x}+\frac {3}{4}\right )^2}{624 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )}\right )+\frac {\left (3327931-2074096 \left (\frac {1}{x}+\frac {3}{4}\right )^2\right ) \left (\frac {1}{x}+\frac {3}{4}\right )}{20163 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{4} \left (-\frac {2 \left (\frac {\frac {1}{4} \left (1678181-74897 \sqrt {517}\right ) \int \frac {\sqrt {2 \left (19+\sqrt {517}\right )}-8 \left (\frac {3}{4}+\frac {1}{x}\right )}{16 \left (\frac {3}{4}+\frac {1}{x}\right )^2-4 \sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {517}}d\left (\frac {3}{4}+\frac {1}{x}\right )+\frac {1}{8} \sqrt {\frac {19+\sqrt {517}}{\sqrt {517}-19}} \left (1678181+74897 \sqrt {517}\right ) \arctan \left (\frac {32 \left (\frac {1}{x}+\frac {3}{4}\right )-4 \sqrt {2 \left (19+\sqrt {517}\right )}}{4 \sqrt {2 \left (\sqrt {517}-19\right )}}\right )}{\sqrt {1034 \left (19+\sqrt {517}\right )}}+\frac {\frac {1}{4} \left (1678181-74897 \sqrt {517}\right ) \int \frac {8 \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {2 \left (19+\sqrt {517}\right )}}{16 \left (\frac {3}{4}+\frac {1}{x}\right )^2+4 \sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {517}}d\left (\frac {3}{4}+\frac {1}{x}\right )+\frac {1}{8} \sqrt {\frac {19+\sqrt {517}}{\sqrt {517}-19}} \left (1678181+74897 \sqrt {517}\right ) \arctan \left (\frac {32 \left (\frac {1}{x}+\frac {3}{4}\right )+4 \sqrt {2 \left (19+\sqrt {517}\right )}}{4 \sqrt {2 \left (\sqrt {517}-19\right )}}\right )}{\sqrt {1034 \left (19+\sqrt {517}\right )}}\right )}{20163}+36 \left (\frac {73 \arctan \left (\frac {512 \left (\frac {1}{x}+\frac {3}{4}\right )^2-608}{64 \sqrt {39}}\right )}{624 \sqrt {39}}-\frac {3359-1712 \left (\frac {1}{x}+\frac {3}{4}\right )^2}{624 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )}\right )+\frac {\left (3327931-2074096 \left (\frac {1}{x}+\frac {3}{4}\right )^2\right ) \left (\frac {1}{x}+\frac {3}{4}\right )}{20163 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{4} \left (36 \left (\frac {73 \arctan \left (\frac {512 \left (\frac {1}{x}+\frac {3}{4}\right )^2-608}{64 \sqrt {39}}\right )}{624 \sqrt {39}}-\frac {3359-1712 \left (\frac {1}{x}+\frac {3}{4}\right )^2}{624 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )}\right )-\frac {2 \left (\frac {\frac {1}{8} \sqrt {\frac {19+\sqrt {517}}{\sqrt {517}-19}} \left (1678181+74897 \sqrt {517}\right ) \arctan \left (\frac {32 \left (\frac {1}{x}+\frac {3}{4}\right )-4 \sqrt {2 \left (19+\sqrt {517}\right )}}{4 \sqrt {2 \left (\sqrt {517}-19\right )}}\right )-\frac {1}{16} \left (1678181-74897 \sqrt {517}\right ) \log \left (16 \left (\frac {1}{x}+\frac {3}{4}\right )^2-4 \sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {1}{x}+\frac {3}{4}\right )+\sqrt {517}\right )}{\sqrt {1034 \left (19+\sqrt {517}\right )}}+\frac {\frac {1}{8} \sqrt {\frac {19+\sqrt {517}}{\sqrt {517}-19}} \left (1678181+74897 \sqrt {517}\right ) \arctan \left (\frac {32 \left (\frac {1}{x}+\frac {3}{4}\right )+4 \sqrt {2 \left (19+\sqrt {517}\right )}}{4 \sqrt {2 \left (\sqrt {517}-19\right )}}\right )+\frac {1}{16} \left (1678181-74897 \sqrt {517}\right ) \log \left (16 \left (\frac {1}{x}+\frac {3}{4}\right )^2+4 \sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {1}{x}+\frac {3}{4}\right )+\sqrt {517}\right )}{\sqrt {1034 \left (19+\sqrt {517}\right )}}\right )}{20163}+\frac {\left (3327931-2074096 \left (\frac {1}{x}+\frac {3}{4}\right )^2\right ) \left (\frac {1}{x}+\frac {3}{4}\right )}{20163 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )}\right )\) |
(((3327931 - 2074096*(3/4 + x^(-1))^2)*(3/4 + x^(-1)))/(20163*(517 - 608*( 3/4 + x^(-1))^2 + 256*(3/4 + x^(-1))^4)) + 36*(-1/624*(3359 - 1712*(3/4 + x^(-1))^2)/(517 - 608*(3/4 + x^(-1))^2 + 256*(3/4 + x^(-1))^4) + (73*ArcTa n[(-608 + 512*(3/4 + x^(-1))^2)/(64*Sqrt[39])])/(624*Sqrt[39])) - (2*(((Sq rt[(19 + Sqrt[517])/(-19 + Sqrt[517])]*(1678181 + 74897*Sqrt[517])*ArcTan[ (-4*Sqrt[2*(19 + Sqrt[517])] + 32*(3/4 + x^(-1)))/(4*Sqrt[2*(-19 + Sqrt[51 7])])])/8 - ((1678181 - 74897*Sqrt[517])*Log[Sqrt[517] - 4*Sqrt[2*(19 + Sq rt[517])]*(3/4 + x^(-1)) + 16*(3/4 + x^(-1))^2])/16)/Sqrt[1034*(19 + Sqrt[ 517])] + ((Sqrt[(19 + Sqrt[517])/(-19 + Sqrt[517])]*(1678181 + 74897*Sqrt[ 517])*ArcTan[(4*Sqrt[2*(19 + Sqrt[517])] + 32*(3/4 + x^(-1)))/(4*Sqrt[2*(- 19 + Sqrt[517])])])/8 + ((1678181 - 74897*Sqrt[517])*Log[Sqrt[517] + 4*Sqr t[2*(19 + Sqrt[517])]*(3/4 + x^(-1)) + 16*(3/4 + x^(-1))^2])/16)/Sqrt[1034 *(19 + Sqrt[517])]))/20163)/4
3.1.62.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) In t[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && N eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[P q, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(b*f - 2*a*g + (2*c*f - b*g)*x)*((a + b*x + c*x^2)^ (p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)) Int [(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (2*p + 3)* (2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^ 2 - 4*a*c, 0] && LtQ[p, -1]
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] : > Simp[1/2 Subst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x + c*x^2) ^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x^2] && IntegerQ [(m - 1)/2]
Int[(Pn_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{n = Expon[Pn, x], k}, Int[Sum[Coeff[Pn, x, 2*k]*x^(2*k), {k, 0, n/2}]*(a + b *x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pn, x, 2*k + 1]*x^(2*k), {k, 0, (n - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pn, x] && !PolyQ[Pn, x^2]
Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Poly nomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^ 4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b ^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(a + b*x^2 + c *x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[Px, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4* p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x ^2] && Expon[Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
Int[(P4_)^(p_), x_Symbol] :> With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1] , c = Coeff[P4, x, 2], d = Coeff[P4, x, 3], e = Coeff[P4, x, 4]}, Simp[-16* a^2 Subst[Int[(1/(b - 4*a*x)^2)*(a*((-3*b^4 + 16*a*b^2*c - 64*a^2*b*d + 2 56*a^3*e - 32*a^2*(3*b^2 - 8*a*c)*x^2 + 256*a^4*x^4)/(b - 4*a*x)^4))^p, x], x, b/(4*a) + 1/x], x] /; NeQ[a, 0] && NeQ[b, 0] && EqQ[b^3 - 4*a*b*c + 8*a ^2*d, 0]] /; FreeQ[p, x] && PolyQ[P4, x, 4] && IntegerQ[2*p] && !IGtQ[p, 0 ]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.06 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.26
| method | result | size |
| default | \(\frac {\frac {2455}{80652} x^{3}-\frac {1429}{19552} x^{2}+\frac {89033}{1290432} x +\frac {3037}{53768}}{x^{4}-\frac {15}{8} x^{3}+x^{2}+3 x +1}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (8 \textit {\_Z}^{4}-15 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}+24 \textit {\_Z} +8\right )}{\sum }\frac {\left (19640 \textit {\_R}^{2}-57489 \textit {\_R} +74897\right ) \ln \left (x -\textit {\_R} \right )}{32 \textit {\_R}^{3}-45 \textit {\_R}^{2}+16 \textit {\_R} +24}\right )}{80652}\) | \(96\) |
| risch | \(\frac {\frac {2455}{80652} x^{3}-\frac {1429}{19552} x^{2}+\frac {89033}{1290432} x +\frac {3037}{53768}}{x^{4}-\frac {15}{8} x^{3}+x^{2}+3 x +1}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (8 \textit {\_Z}^{4}-15 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}+24 \textit {\_Z} +8\right )}{\sum }\frac {\left (19640 \textit {\_R}^{2}-57489 \textit {\_R} +74897\right ) \ln \left (x -\textit {\_R} \right )}{32 \textit {\_R}^{3}-45 \textit {\_R}^{2}+16 \textit {\_R} +24}\right )}{80652}\) | \(96\) |
(2455/80652*x^3-1429/19552*x^2+89033/1290432*x+3037/53768)/(x^4-15/8*x^3+x ^2+3*x+1)+1/80652*sum((19640*_R^2-57489*_R+74897)/(32*_R^3-45*_R^2+16*_R+2 4)*ln(x-_R),_R=RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8))
Result contains complex when optimal does not.
Time = 1.10 (sec) , antiderivative size = 1540, normalized size of antiderivative = 4.21 \[ \int \frac {1}{\left (8+24 x+8 x^2-15 x^3+8 x^4\right )^2} \, dx=\text {Too large to display} \]
1/26018980416*(6336021120*x^3 - 4811202*(8*x^4 - 15*x^3 + 8*x^2 + 24*x + 8 )*(-73*I*sqrt(13)*sqrt(3) + 2704*sqrt(537508757/26903625750144*I*sqrt(13)* sqrt(3) - 59644114671451/2098482808511232))*log(-131155175531952*(21770028 8287626155772963*I*sqrt(13)*sqrt(3) + 8063857253832070208357424*sqrt(-5375 08757/26903625750144*I*sqrt(13)*sqrt(3) - 59644114671451/2098482808511232) - 2904532176689925771712)*(73/5408*I*sqrt(13)*sqrt(3) - 1/2*sqrt(53750875 7/26903625750144*I*sqrt(13)*sqrt(3) - 59644114671451/2098482808511232))^2 + 2115233227181899165359763637490696823296*(-73/5408*I*sqrt(13)*sqrt(3) - 1/2*sqrt(-537508757/26903625750144*I*sqrt(13)*sqrt(3) - 59644114671451/209 8482808511232))^3 - 801867*(487774661427048094833332220336*(-73/5408*I*sqr t(13)*sqrt(3) - 1/2*sqrt(-537508757/26903625750144*I*sqrt(13)*sqrt(3) - 59 644114671451/2098482808511232))^2 + 3281707530577268651899)*(-73*I*sqrt(13 )*sqrt(3) + 2704*sqrt(537508757/26903625750144*I*sqrt(13)*sqrt(3) - 596441 14671451/2098482808511232)) - 3246265196161156614776051552784488493/4*I*sq rt(13)*sqrt(3) + 150930531402994079881533903215265*x - 3006130510417728591 2172751365511153716*sqrt(-537508757/26903625750144*I*sqrt(13)*sqrt(3) - 59 644114671451/2098482808511232) + 10905071149176173110139073138101752) - 48 11202*(8*x^4 - 15*x^3 + 8*x^2 + 24*x + 8)*(73*I*sqrt(13)*sqrt(3) + 2704*sq rt(-537508757/26903625750144*I*sqrt(13)*sqrt(3) - 59644114671451/209848280 8511232))*log(-16921865817455193322878109099925574586368*(-73/5408*I*sq...
Leaf count of result is larger than twice the leaf count of optimal. 3839 vs. \(2 (292) = 584\).
Time = 2.23 (sec) , antiderivative size = 3839, normalized size of antiderivative = 10.49 \[ \int \frac {1}{\left (8+24 x+8 x^2-15 x^3+8 x^4\right )^2} \, dx=\text {Too large to display} \]
(39280*x**3 - 94314*x**2 + 89033*x + 72888)/(1290432*x**4 - 2419560*x**3 + 1290432*x**2 + 3871296*x + 1290432) + sqrt(-59644114671451/16787862468089 856 + 5073830635*sqrt(517)/32471687559168)*log(x**2 + x*(-1123969950204685 03306932567484755463/603722125611976319526135612861060 - 29643869829812833 230907750777733957*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(5 17))/1936419398792394461637855141912238396080 - 181533261043120360732*sqrt (-7120427417275887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(5 17)) + 6263621568587150042935*sqrt(517) + 3557579971691991294769382675)/15 0930531402994079881533903215265 - 46926347979646613249222*sqrt(517)/297468 60362632912338339 + 994065243322493861977*sqrt(78)*sqrt(-59644114671451 + 2623170438295*sqrt(517))/1427849297406379792240272 + 994065243322493861977 *sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517))*sqrt(-7120427 417275887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517)) + 62 63621568587150042935*sqrt(517) + 3557579971691991294769382675)/12909462658 61596307758570094608158930720) - 45971497067730669689218547912235602388091 89313591735176029*sqrt(517)*sqrt(-7120427417275887*sqrt(40326)*sqrt(-59644 114671451 + 2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 3557579971691991294769382675)/18432767186998698626450604048374763890148748 05380627572841955840 - 102213276372026717588278042506361313108860193595830 3878081158710949715459967411486447/302201812380681690634631534385892067...
\[ \int \frac {1}{\left (8+24 x+8 x^2-15 x^3+8 x^4\right )^2} \, dx=\int { \frac {1}{{\left (8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8\right )}^{2}} \,d x } \]
1/161304*(39280*x^3 - 94314*x^2 + 89033*x + 72888)/(8*x^4 - 15*x^3 + 8*x^2 + 24*x + 8) + 1/80652*integrate((19640*x^2 - 57489*x + 74897)/(8*x^4 - 15 *x^3 + 8*x^2 + 24*x + 8), x)
\[ \int \frac {1}{\left (8+24 x+8 x^2-15 x^3+8 x^4\right )^2} \, dx=\int { \frac {1}{{\left (8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8\right )}^{2}} \,d x } \]
Time = 0.15 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.49 \[ \int \frac {1}{\left (8+24 x+8 x^2-15 x^3+8 x^4\right )^2} \, dx=\frac {\frac {2455\,x^3}{80652}-\frac {1429\,x^2}{19552}+\frac {89033\,x}{1290432}+\frac {3037}{53768}}{x^4-\frac {15\,x^3}{8}+x^2+3\,x+1}+\left (\sum _{k=1}^4\ln \left (\frac {2146659825\,\mathrm {root}\left (z^4+\frac {14911625619311\,z^2}{524620702127808}+\frac {39238139261\,z}{3730636104019968}+\frac {43023440}{44204510553294663},z,k\right )}{2960381771776}+\frac {2222183\,x}{338246745408}+\frac {\mathrm {root}\left (z^4+\frac {14911625619311\,z^2}{524620702127808}+\frac {39238139261\,z}{3730636104019968}+\frac {43023440}{44204510553294663},z,k\right )\,x\,924124364159}{26643435945984}-\frac {{\mathrm {root}\left (z^4+\frac {14911625619311\,z^2}{524620702127808}+\frac {39238139261\,z}{3730636104019968}+\frac {43023440}{44204510553294663},z,k\right )}^2\,x\,72451101}{8470528}-\frac {{\mathrm {root}\left (z^4+\frac {14911625619311\,z^2}{524620702127808}+\frac {39238139261\,z}{3730636104019968}+\frac {43023440}{44204510553294663},z,k\right )}^3\,x\,95745}{256}+\frac {389551\,{\mathrm {root}\left (z^4+\frac {14911625619311\,z^2}{524620702127808}+\frac {39238139261\,z}{3730636104019968}+\frac {43023440}{44204510553294663},z,k\right )}^2}{264704}-\frac {100737\,{\mathrm {root}\left (z^4+\frac {14911625619311\,z^2}{524620702127808}+\frac {39238139261\,z}{3730636104019968}+\frac {43023440}{44204510553294663},z,k\right )}^3}{512}+\frac {271033}{624455529984}\right )\,\mathrm {root}\left (z^4+\frac {14911625619311\,z^2}{524620702127808}+\frac {39238139261\,z}{3730636104019968}+\frac {43023440}{44204510553294663},z,k\right )\right ) \]
((89033*x)/1290432 - (1429*x^2)/19552 + (2455*x^3)/80652 + 3037/53768)/(3* x + x^2 - (15*x^3)/8 + x^4 + 1) + symsum(log((2146659825*root(z^4 + (14911 625619311*z^2)/524620702127808 + (39238139261*z)/3730636104019968 + 430234 40/44204510553294663, z, k))/2960381771776 + (2222183*x)/338246745408 + (9 24124364159*root(z^4 + (14911625619311*z^2)/524620702127808 + (39238139261 *z)/3730636104019968 + 43023440/44204510553294663, z, k)*x)/26643435945984 - (72451101*root(z^4 + (14911625619311*z^2)/524620702127808 + (3923813926 1*z)/3730636104019968 + 43023440/44204510553294663, z, k)^2*x)/8470528 - ( 95745*root(z^4 + (14911625619311*z^2)/524620702127808 + (39238139261*z)/37 30636104019968 + 43023440/44204510553294663, z, k)^3*x)/256 + (389551*root (z^4 + (14911625619311*z^2)/524620702127808 + (39238139261*z)/373063610401 9968 + 43023440/44204510553294663, z, k)^2)/264704 - (100737*root(z^4 + (1 4911625619311*z^2)/524620702127808 + (39238139261*z)/3730636104019968 + 43 023440/44204510553294663, z, k)^3)/512 + 271033/624455529984)*root(z^4 + ( 14911625619311*z^2)/524620702127808 + (39238139261*z)/3730636104019968 + 4 3023440/44204510553294663, z, k), k, 1, 4)