Integrand size = 79, antiderivative size = 145 \[ \int \frac {24 x-2304 x^3+4992 x^5+2304 x^7+1728 x^9+3072 x^{11}+1536 x^{13}-3072 x^{15}+384 x^{17}}{-3-624 x^4-212 x^8+640 x^{12}+240 x^{16}+256 x^{20}+64 x^{24}} \, dx=-\sqrt {2} \sqrt [4]{3} \arctan \left (\frac {\sqrt {2} x^2}{\sqrt [4]{3}}\right )-\sqrt {2} \sqrt [4]{3} \arctan \left (\sqrt {-\frac {1}{2}+\frac {7}{8 \sqrt {3}}}+\frac {x^4}{\sqrt {2} \sqrt [4]{3}}\right )-\sqrt {2} \sqrt [4]{3} \text {arctanh}\left (\frac {\sqrt {2} x^2}{\sqrt [4]{3}}\right )-\sqrt {2} \sqrt [4]{3} \text {arctanh}\left (\sqrt {\frac {1}{2}+\frac {7}{8 \sqrt {3}}}+\frac {x^4}{\sqrt {2} \sqrt [4]{3}}\right ) \] Output:
-2^(1/2)*3^(1/4)*arctan(1/3*2^(1/2)*x^2*3^(3/4))-2^(1/2)*3^(1/4)*arctan(1/ 12*(-72+42*3^(1/2))^(1/2)+1/6*x^4*2^(1/2)*3^(3/4))-2^(1/2)*3^(1/4)*arctanh (1/3*2^(1/2)*x^2*3^(3/4))-2^(1/2)*3^(1/4)*arctanh(1/12*(72+42*3^(1/2))^(1/ 2)+1/6*x^4*2^(1/2)*3^(3/4))
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.10 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.23 \[ \int \frac {24 x-2304 x^3+4992 x^5+2304 x^7+1728 x^9+3072 x^{11}+1536 x^{13}-3072 x^{15}+384 x^{17}}{-3-624 x^4-212 x^8+640 x^{12}+240 x^{16}+256 x^{20}+64 x^{24}} \, dx=\frac {1}{2} \left (\sqrt {2} \sqrt [4]{3} \left (2 \arctan \left (1-\frac {2^{3/4} x}{\sqrt [8]{3}}\right )+2 \arctan \left (1+\frac {2^{3/4} x}{\sqrt [8]{3}}\right )+\log \left (3-\sqrt [4]{2} 3^{7/8} x\right )+\log \left (3+\sqrt [4]{2} 3^{7/8} x\right )-\log \left (3+\sqrt {2} 3^{3/4} x^2\right )\right )-24 \text {RootSum}\left [1+208 \text {$\#$1}^4+72 \text {$\#$1}^8+64 \text {$\#$1}^{12}+16 \text {$\#$1}^{16}\&,\frac {-\log (x-\text {$\#$1})+\log (x-\text {$\#$1}) \text {$\#$1}^4}{13+9 \text {$\#$1}^4+12 \text {$\#$1}^8+4 \text {$\#$1}^{12}}\&\right ]\right ) \] Input:
Integrate[(24*x - 2304*x^3 + 4992*x^5 + 2304*x^7 + 1728*x^9 + 3072*x^11 + 1536*x^13 - 3072*x^15 + 384*x^17)/(-3 - 624*x^4 - 212*x^8 + 640*x^12 + 240 *x^16 + 256*x^20 + 64*x^24),x]
Output:
(Sqrt[2]*3^(1/4)*(2*ArcTan[1 - (2^(3/4)*x)/3^(1/8)] + 2*ArcTan[1 + (2^(3/4 )*x)/3^(1/8)] + Log[3 - 2^(1/4)*3^(7/8)*x] + Log[3 + 2^(1/4)*3^(7/8)*x] - Log[3 + Sqrt[2]*3^(3/4)*x^2]) - 24*RootSum[1 + 208*#1^4 + 72*#1^8 + 64*#1^ 12 + 16*#1^16 & , (-Log[x - #1] + Log[x - #1]*#1^4)/(13 + 9*#1^4 + 12*#1^8 + 4*#1^12) & ])/2
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.
| \(\displaystyle \int \frac {384 x^{17}-3072 x^{15}+1536 x^{13}+3072 x^{11}+1728 x^9+2304 x^7+4992 x^5-2304 x^3+24 x}{64 x^{24}+256 x^{20}+240 x^{16}+640 x^{12}-212 x^8-624 x^4-3} \, dx\) |
| \(\Big \downarrow \) 2460 |
| \(\displaystyle \int \left (\frac {24 x}{4 x^8-3}-\frac {768 x^3 \left (x^4-1\right )}{16 x^{16}+64 x^{12}+72 x^8+208 x^4+1}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -192 \text {Subst}\left (\int \frac {1}{-16 x^4-64 x^3-72 x^2-208 x-1}dx,x,x^4\right )-192 \text {Subst}\left (\int \frac {x}{16 x^4+64 x^3+72 x^2+208 x+1}dx,x,x^4\right )-\sqrt {2} \sqrt [4]{3} \arctan \left (\frac {\sqrt {2} x^2}{\sqrt [4]{3}}\right )-\sqrt {2} \sqrt [4]{3} \text {arctanh}\left (\frac {\sqrt {2} x^2}{\sqrt [4]{3}}\right )\) |
Input:
Int[(24*x - 2304*x^3 + 4992*x^5 + 2304*x^7 + 1728*x^9 + 3072*x^11 + 1536*x ^13 - 3072*x^15 + 384*x^17)/(-3 - 624*x^4 - 212*x^8 + 640*x^12 + 240*x^16 + 256*x^20 + 64*x^24),x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.18 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.37
| method | result | size |
| risch | \(3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (108 \textit {\_Z}^{4}-1\right )}{\sum }\textit {\_R} \ln \left (x^{6}-3 \textit {\_R} \,x^{4}+\left (9 \textit {\_R}^{2}-6 \textit {\_R} +1\right ) x^{2}-27 \textit {\_R}^{3}+18 \textit {\_R}^{2}-3 \textit {\_R} \right )\right )\) | \(54\) |
| default | \(-12 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (16 \textit {\_Z}^{4}+64 \textit {\_Z}^{3}+72 \textit {\_Z}^{2}+208 \textit {\_Z} +1\right )}{\sum }\frac {\left (-1+\textit {\_R} \right ) \ln \left (x^{4}-\textit {\_R} \right )}{4 \textit {\_R}^{3}+12 \textit {\_R}^{2}+9 \textit {\_R} +13}\right )-\frac {\sqrt {2}\, 3^{\frac {1}{4}} \left (\ln \left (\frac {x^{2}+\frac {\sqrt {2}\, 3^{\frac {1}{4}}}{2}}{x^{2}-\frac {\sqrt {2}\, 3^{\frac {1}{4}}}{2}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x^{2} 3^{\frac {3}{4}}}{3}\right )\right )}{2}\) | \(108\) |
Input:
int((384*x^17-3072*x^15+1536*x^13+3072*x^11+1728*x^9+2304*x^7+4992*x^5-230 4*x^3+24*x)/(64*x^24+256*x^20+240*x^16+640*x^12-212*x^8-624*x^4-3),x,metho d=_RETURNVERBOSE)
Output:
3*sum(_R*ln(x^6-3*_R*x^4+(9*_R^2-6*_R+1)*x^2-27*_R^3+18*_R^2-3*_R),_R=Root Of(108*_Z^4-1))
Leaf count of result is larger than twice the leaf count of optimal. 211 vs. \(2 (105) = 210\).
Time = 0.11 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.46 \[ \int \frac {24 x-2304 x^3+4992 x^5+2304 x^7+1728 x^9+3072 x^{11}+1536 x^{13}-3072 x^{15}+384 x^{17}}{-3-624 x^4-212 x^8+640 x^{12}+240 x^{16}+256 x^{20}+64 x^{24}} \, dx=-2 \, \left (\frac {3}{4}\right )^{\frac {1}{4}} \arctan \left (\frac {1}{24} \, \left (\frac {3}{4}\right )^{\frac {3}{4}} {\left (8 \, x^{10} - 4 \, x^{8} + 16 \, x^{6} - 4 \, x^{4} + 14 \, x^{2} + 21\right )} + \frac {1}{16} \, \left (\frac {3}{4}\right )^{\frac {1}{4}} {\left (4 \, x^{8} - 8 \, x^{6} + 12 \, x^{4} + 8 \, x^{2} - 5\right )}\right ) + 2 \, \left (\frac {3}{4}\right )^{\frac {1}{4}} \arctan \left (\frac {1}{12} \, \left (\frac {3}{4}\right )^{\frac {3}{4}} {\left (4 \, x^{6} - 10 \, x^{4} + 8 \, x^{2} - 5\right )} + \frac {1}{8} \, \left (\frac {3}{4}\right )^{\frac {1}{4}} {\left (2 \, x^{4} - 6 \, x^{2} + 11\right )}\right ) - 2 \, \left (\frac {3}{4}\right )^{\frac {1}{4}} \arctan \left (\frac {4}{3} \, \left (\frac {3}{4}\right )^{\frac {3}{4}} {\left (x^{2} - 2\right )}\right ) - \left (\frac {3}{4}\right )^{\frac {1}{4}} \log \left (x^{6} + x^{2} + \frac {1}{2} \, \sqrt {3} {\left (x^{2} + 2\right )} + \left (\frac {3}{4}\right )^{\frac {1}{4}} {\left (x^{4} + 2 \, x^{2} + 1\right )} + \left (\frac {3}{4}\right )^{\frac {3}{4}}\right ) + \left (\frac {3}{4}\right )^{\frac {1}{4}} \log \left (x^{6} + x^{2} + \frac {1}{2} \, \sqrt {3} {\left (x^{2} + 2\right )} - \left (\frac {3}{4}\right )^{\frac {1}{4}} {\left (x^{4} + 2 \, x^{2} + 1\right )} - \left (\frac {3}{4}\right )^{\frac {3}{4}}\right ) \] Input:
integrate((384*x^17-3072*x^15+1536*x^13+3072*x^11+1728*x^9+2304*x^7+4992*x ^5-2304*x^3+24*x)/(64*x^24+256*x^20+240*x^16+640*x^12-212*x^8-624*x^4-3),x , algorithm="fricas")
Output:
-2*(3/4)^(1/4)*arctan(1/24*(3/4)^(3/4)*(8*x^10 - 4*x^8 + 16*x^6 - 4*x^4 + 14*x^2 + 21) + 1/16*(3/4)^(1/4)*(4*x^8 - 8*x^6 + 12*x^4 + 8*x^2 - 5)) + 2* (3/4)^(1/4)*arctan(1/12*(3/4)^(3/4)*(4*x^6 - 10*x^4 + 8*x^2 - 5) + 1/8*(3/ 4)^(1/4)*(2*x^4 - 6*x^2 + 11)) - 2*(3/4)^(1/4)*arctan(4/3*(3/4)^(3/4)*(x^2 - 2)) - (3/4)^(1/4)*log(x^6 + x^2 + 1/2*sqrt(3)*(x^2 + 2) + (3/4)^(1/4)*( x^4 + 2*x^2 + 1) + (3/4)^(3/4)) + (3/4)^(1/4)*log(x^6 + x^2 + 1/2*sqrt(3)* (x^2 + 2) - (3/4)^(1/4)*(x^4 + 2*x^2 + 1) - (3/4)^(3/4))
Leaf count of result is larger than twice the leaf count of optimal. 473 vs. \(2 (136) = 272\).
Time = 0.38 (sec) , antiderivative size = 473, normalized size of antiderivative = 3.26 \[ \int \frac {24 x-2304 x^3+4992 x^5+2304 x^7+1728 x^9+3072 x^{11}+1536 x^{13}-3072 x^{15}+384 x^{17}}{-3-624 x^4-212 x^8+640 x^{12}+240 x^{16}+256 x^{20}+64 x^{24}} \, dx=\frac {\sqrt {2} \cdot \sqrt [4]{3} \left (- 2 \operatorname {atan}{\left (\frac {\sqrt {2} \cdot 3^{\frac {3}{4}} x^{2}}{3} - \frac {2 \sqrt {2} \cdot 3^{\frac {3}{4}}}{3} \right )} + 2 \operatorname {atan}{\left (\frac {\sqrt {2} \cdot 3^{\frac {3}{4}} x^{6}}{12} + x^{4} \left (- \frac {5 \sqrt {2} \cdot 3^{\frac {3}{4}}}{24} + \frac {\sqrt {2} \cdot \sqrt [4]{3}}{8}\right ) + x^{2} \left (- \frac {3 \sqrt {2} \cdot \sqrt [4]{3}}{8} + \frac {\sqrt {2} \cdot 3^{\frac {3}{4}}}{6}\right ) - \frac {5 \sqrt {2} \cdot 3^{\frac {3}{4}}}{48} + \frac {11 \sqrt {2} \cdot \sqrt [4]{3}}{16} \right )} + 2 \operatorname {atan}{\left (- \frac {\sqrt {2} \cdot 3^{\frac {3}{4}} x^{10}}{12} + x^{8} \left (- \frac {\sqrt {2} \cdot \sqrt [4]{3}}{8} + \frac {\sqrt {2} \cdot 3^{\frac {3}{4}}}{24}\right ) + x^{6} \left (- \frac {\sqrt {2} \cdot 3^{\frac {3}{4}}}{6} + \frac {\sqrt {2} \cdot \sqrt [4]{3}}{4}\right ) + x^{4} \left (- \frac {3 \sqrt {2} \cdot \sqrt [4]{3}}{8} + \frac {\sqrt {2} \cdot 3^{\frac {3}{4}}}{24}\right ) + x^{2} \left (- \frac {7 \sqrt {2} \cdot 3^{\frac {3}{4}}}{48} - \frac {\sqrt {2} \cdot \sqrt [4]{3}}{4}\right ) - \frac {7 \sqrt {2} \cdot 3^{\frac {3}{4}}}{32} + \frac {5 \sqrt {2} \cdot \sqrt [4]{3}}{32} \right )}\right )}{2} + \frac {\sqrt {2} \cdot \sqrt [4]{3} \log {\left (x^{6} - \frac {\sqrt {2} \cdot \sqrt [4]{3} x^{4}}{2} + x^{2} \left (- \sqrt {2} \cdot \sqrt [4]{3} + \frac {\sqrt {3}}{2} + 1\right ) - \frac {\sqrt {2} \cdot \sqrt [4]{3}}{2} - \frac {\sqrt {2} \cdot 3^{\frac {3}{4}}}{4} + \sqrt {3} \right )}}{2} - \frac {\sqrt {2} \cdot \sqrt [4]{3} \log {\left (x^{6} + \frac {\sqrt {2} \cdot \sqrt [4]{3} x^{4}}{2} + x^{2} \left (\frac {\sqrt {3}}{2} + 1 + \sqrt {2} \cdot \sqrt [4]{3}\right ) + \frac {\sqrt {2} \cdot 3^{\frac {3}{4}}}{4} + \frac {\sqrt {2} \cdot \sqrt [4]{3}}{2} + \sqrt {3} \right )}}{2} \] Input:
integrate((384*x**17-3072*x**15+1536*x**13+3072*x**11+1728*x**9+2304*x**7+ 4992*x**5-2304*x**3+24*x)/(64*x**24+256*x**20+240*x**16+640*x**12-212*x**8 -624*x**4-3),x)
Output:
sqrt(2)*3**(1/4)*(-2*atan(sqrt(2)*3**(3/4)*x**2/3 - 2*sqrt(2)*3**(3/4)/3) + 2*atan(sqrt(2)*3**(3/4)*x**6/12 + x**4*(-5*sqrt(2)*3**(3/4)/24 + sqrt(2) *3**(1/4)/8) + x**2*(-3*sqrt(2)*3**(1/4)/8 + sqrt(2)*3**(3/4)/6) - 5*sqrt( 2)*3**(3/4)/48 + 11*sqrt(2)*3**(1/4)/16) + 2*atan(-sqrt(2)*3**(3/4)*x**10/ 12 + x**8*(-sqrt(2)*3**(1/4)/8 + sqrt(2)*3**(3/4)/24) + x**6*(-sqrt(2)*3** (3/4)/6 + sqrt(2)*3**(1/4)/4) + x**4*(-3*sqrt(2)*3**(1/4)/8 + sqrt(2)*3**( 3/4)/24) + x**2*(-7*sqrt(2)*3**(3/4)/48 - sqrt(2)*3**(1/4)/4) - 7*sqrt(2)* 3**(3/4)/32 + 5*sqrt(2)*3**(1/4)/32))/2 + sqrt(2)*3**(1/4)*log(x**6 - sqrt (2)*3**(1/4)*x**4/2 + x**2*(-sqrt(2)*3**(1/4) + sqrt(3)/2 + 1) - sqrt(2)*3 **(1/4)/2 - sqrt(2)*3**(3/4)/4 + sqrt(3))/2 - sqrt(2)*3**(1/4)*log(x**6 + sqrt(2)*3**(1/4)*x**4/2 + x**2*(sqrt(3)/2 + 1 + sqrt(2)*3**(1/4)) + sqrt(2 )*3**(3/4)/4 + sqrt(2)*3**(1/4)/2 + sqrt(3))/2
\[ \int \frac {24 x-2304 x^3+4992 x^5+2304 x^7+1728 x^9+3072 x^{11}+1536 x^{13}-3072 x^{15}+384 x^{17}}{-3-624 x^4-212 x^8+640 x^{12}+240 x^{16}+256 x^{20}+64 x^{24}} \, dx=\int { \frac {24 \, {\left (16 \, x^{17} - 128 \, x^{15} + 64 \, x^{13} + 128 \, x^{11} + 72 \, x^{9} + 96 \, x^{7} + 208 \, x^{5} - 96 \, x^{3} + x\right )}}{64 \, x^{24} + 256 \, x^{20} + 240 \, x^{16} + 640 \, x^{12} - 212 \, x^{8} - 624 \, x^{4} - 3} \,d x } \] Input:
integrate((384*x^17-3072*x^15+1536*x^13+3072*x^11+1728*x^9+2304*x^7+4992*x ^5-2304*x^3+24*x)/(64*x^24+256*x^20+240*x^16+640*x^12-212*x^8-624*x^4-3),x , algorithm="maxima")
Output:
24*integrate((16*x^17 - 128*x^15 + 64*x^13 + 128*x^11 + 72*x^9 + 96*x^7 + 208*x^5 - 96*x^3 + x)/(64*x^24 + 256*x^20 + 240*x^16 + 640*x^12 - 212*x^8 - 624*x^4 - 3), x)
\[ \int \frac {24 x-2304 x^3+4992 x^5+2304 x^7+1728 x^9+3072 x^{11}+1536 x^{13}-3072 x^{15}+384 x^{17}}{-3-624 x^4-212 x^8+640 x^{12}+240 x^{16}+256 x^{20}+64 x^{24}} \, dx=\int { \frac {24 \, {\left (16 \, x^{17} - 128 \, x^{15} + 64 \, x^{13} + 128 \, x^{11} + 72 \, x^{9} + 96 \, x^{7} + 208 \, x^{5} - 96 \, x^{3} + x\right )}}{64 \, x^{24} + 256 \, x^{20} + 240 \, x^{16} + 640 \, x^{12} - 212 \, x^{8} - 624 \, x^{4} - 3} \,d x } \] Input:
integrate((384*x^17-3072*x^15+1536*x^13+3072*x^11+1728*x^9+2304*x^7+4992*x ^5-2304*x^3+24*x)/(64*x^24+256*x^20+240*x^16+640*x^12-212*x^8-624*x^4-3),x , algorithm="giac")
Output:
undef
Time = 10.53 (sec) , antiderivative size = 951, normalized size of antiderivative = 6.56 \[ \int \frac {24 x-2304 x^3+4992 x^5+2304 x^7+1728 x^9+3072 x^{11}+1536 x^{13}-3072 x^{15}+384 x^{17}}{-3-624 x^4-212 x^8+640 x^{12}+240 x^{16}+256 x^{20}+64 x^{24}} \, dx=\text {Too large to display} \] Input:
int((24*x - 2304*x^3 + 4992*x^5 + 2304*x^7 + 1728*x^9 + 3072*x^11 + 1536*x ^13 - 3072*x^15 + 384*x^17)/(640*x^12 - 212*x^8 - 624*x^4 + 240*x^16 + 256 *x^20 + 64*x^24 - 3),x)
Output:
2^(1/2)*3^(1/4)*atan((2738433225476105601148099171872214949560320*2^(1/2)* 3^(1/4))/(2942638004399783889991102112285232954605568*3^(1/2) - 3162070319 804927630841030047989339143536640*3^(1/2)*x^2 + 61105276816509490239526120 2093074533702434816*3^(1/2)*x^4 - 6566189314277520175612528538433890221597 65504*3^(1/2)*x^6 + 5476866450952211202296198343744429899120640*x^2 - 1058 374440567580622429902121601418936058904576*x^4 + 1137297350444485429720221 160781498768777281536*x^6 - 5096798531966923339035363210863670133260288) - (1581035159902463815420515023994669571768320*2^(1/2)*3^(3/4))/(2942638004 399783889991102112285232954605568*3^(1/2) - 316207031980492763084103004798 9339143536640*3^(1/2)*x^2 + 611052768165094902395261202093074533702434816* 3^(1/2)*x^4 - 656618931427752017561252853843389022159765504*3^(1/2)*x^6 + 5476866450952211202296198343744429899120640*x^2 - 105837444056758062242990 2121601418936058904576*x^4 + 113729735044448542972022116078149876877728153 6*x^6 - 5096798531966923339035363210863670133260288) - (294263800439978388 9991102112285232954605568*2^(1/2)*3^(1/4)*x^2)/(29426380043997838899911021 12285232954605568*3^(1/2) - 3162070319804927630841030047989339143536640*3^ (1/2)*x^2 + 611052768165094902395261202093074533702434816*3^(1/2)*x^4 - 65 6618931427752017561252853843389022159765504*3^(1/2)*x^6 + 5476866450952211 202296198343744429899120640*x^2 - 1058374440567580622429902121601418936058 904576*x^4 + 1137297350444485429720221160781498768777281536*x^6 - 50967...
\[ \int \frac {24 x-2304 x^3+4992 x^5+2304 x^7+1728 x^9+3072 x^{11}+1536 x^{13}-3072 x^{15}+384 x^{17}}{-3-624 x^4-212 x^8+640 x^{12}+240 x^{16}+256 x^{20}+64 x^{24}} \, dx=\sqrt {2}\, 3^{\frac {1}{4}} \mathit {atan} \left (\frac {\left (2^{\frac {1}{4}} 3^{\frac {1}{8}}-2 x \right ) 2^{\frac {3}{4}} 3^{\frac {7}{8}}}{6}\right )+\sqrt {2}\, 3^{\frac {1}{4}} \mathit {atan} \left (\frac {\left (2^{\frac {1}{4}} 3^{\frac {1}{8}}+2 x \right ) 2^{\frac {3}{4}} 3^{\frac {7}{8}}}{6}\right )-\frac {\sqrt {2}\, 3^{\frac {1}{4}} \mathrm {log}\left (3^{\frac {1}{4}}+\sqrt {2}\, x^{2}\right )}{2}+\frac {\sqrt {2}\, 3^{\frac {1}{4}} \mathrm {log}\left (-2^{\frac {1}{4}} 3^{\frac {1}{8}}+\sqrt {2}\, x \right )}{2}+\frac {\sqrt {2}\, 3^{\frac {1}{4}} \mathrm {log}\left (2^{\frac {1}{4}} 3^{\frac {1}{8}}+\sqrt {2}\, x \right )}{2}-4 \sqrt {3}\, \mathrm {log}\left (3^{\frac {1}{4}}+\sqrt {2}\, x^{2}\right )-4 \sqrt {3}\, \mathrm {log}\left (-2^{\frac {1}{4}} 3^{\frac {1}{8}}+\sqrt {2}\, x \right )+4 \sqrt {3}\, \mathrm {log}\left (-\sqrt {2}\, 2^{\frac {1}{4}} 3^{\frac {1}{8}} x +3^{\frac {1}{4}}+\sqrt {2}\, x^{2}\right )-4 \sqrt {3}\, \mathrm {log}\left (2^{\frac {1}{4}} 3^{\frac {1}{8}}+\sqrt {2}\, x \right )+4 \sqrt {3}\, \mathrm {log}\left (\sqrt {2}\, 2^{\frac {1}{4}} 3^{\frac {1}{8}} x +3^{\frac {1}{4}}+\sqrt {2}\, x^{2}\right )-19968 \left (\int \frac {x^{11}}{64 x^{24}+256 x^{20}+240 x^{16}+640 x^{12}-212 x^{8}-624 x^{4}-3}d x \right )+36864 \left (\int \frac {x^{7}}{64 x^{24}+256 x^{20}+240 x^{16}+640 x^{12}-212 x^{8}-624 x^{4}-3}d x \right )-9600 \left (\int \frac {x^{3}}{64 x^{24}+256 x^{20}+240 x^{16}+640 x^{12}-212 x^{8}-624 x^{4}-3}d x \right )+6 \,\mathrm {log}\left (3^{\frac {1}{4}}+\sqrt {2}\, x^{2}\right )+6 \,\mathrm {log}\left (-2^{\frac {1}{4}} 3^{\frac {1}{8}}+\sqrt {2}\, x \right )+6 \,\mathrm {log}\left (-\sqrt {2}\, 2^{\frac {1}{4}} 3^{\frac {1}{8}} x +3^{\frac {1}{4}}+\sqrt {2}\, x^{2}\right )+6 \,\mathrm {log}\left (2^{\frac {1}{4}} 3^{\frac {1}{8}}+\sqrt {2}\, x \right )+6 \,\mathrm {log}\left (\sqrt {2}\, 2^{\frac {1}{4}} 3^{\frac {1}{8}} x +3^{\frac {1}{4}}+\sqrt {2}\, x^{2}\right )-3 \,\mathrm {log}\left (16 x^{16}+64 x^{12}+72 x^{8}+208 x^{4}+1\right ) \] Input:
int((384*x^17-3072*x^15+1536*x^13+3072*x^11+1728*x^9+2304*x^7+4992*x^5-230 4*x^3+24*x)/(64*x^24+256*x^20+240*x^16+640*x^12-212*x^8-624*x^4-3),x)
Output:
(2*sqrt(2)*3**(1/4)*atan((2**(1/4)*3**(1/8) - 2*x)/(2**(1/4)*3**(1/8))) + 2*sqrt(2)*3**(1/4)*atan((2**(1/4)*3**(1/8) + 2*x)/(2**(1/4)*3**(1/8))) - s qrt(2)*3**(1/4)*log(3**(1/4) + sqrt(2)*x**2) + sqrt(2)*3**(1/4)*log( - 2** (1/4)*3**(1/8) + sqrt(2)*x) + sqrt(2)*3**(1/4)*log(2**(1/4)*3**(1/8) + sqr t(2)*x) - 8*sqrt(3)*log(3**(1/4) + sqrt(2)*x**2) - 8*sqrt(3)*log( - 2**(1/ 4)*3**(1/8) + sqrt(2)*x) + 8*sqrt(3)*log( - sqrt(2)*2**(1/4)*3**(1/8)*x + 3**(1/4) + sqrt(2)*x**2) - 8*sqrt(3)*log(2**(1/4)*3**(1/8) + sqrt(2)*x) + 8*sqrt(3)*log(sqrt(2)*2**(1/4)*3**(1/8)*x + 3**(1/4) + sqrt(2)*x**2) - 399 36*int(x**11/(64*x**24 + 256*x**20 + 240*x**16 + 640*x**12 - 212*x**8 - 62 4*x**4 - 3),x) + 73728*int(x**7/(64*x**24 + 256*x**20 + 240*x**16 + 640*x* *12 - 212*x**8 - 624*x**4 - 3),x) - 19200*int(x**3/(64*x**24 + 256*x**20 + 240*x**16 + 640*x**12 - 212*x**8 - 624*x**4 - 3),x) + 12*log(3**(1/4) + s qrt(2)*x**2) + 12*log( - 2**(1/4)*3**(1/8) + sqrt(2)*x) + 12*log( - sqrt(2 )*2**(1/4)*3**(1/8)*x + 3**(1/4) + sqrt(2)*x**2) + 12*log(2**(1/4)*3**(1/8 ) + sqrt(2)*x) + 12*log(sqrt(2)*2**(1/4)*3**(1/8)*x + 3**(1/4) + sqrt(2)*x **2) - 6*log(16*x**16 + 64*x**12 + 72*x**8 + 208*x**4 + 1))/2