Integrand size = 84, antiderivative size = 404 \[ \int \frac {x \left (-2+x+18 x^2-7 x^3-56 x^4+15 x^5+72 x^6-12 x^7-36 x^8+3 x^9+5 x^{10}\right )}{1-12 x^2+54 x^4-113 x^6+111 x^8-45 x^{10}+5 x^{12}} \, dx=-\frac {1}{15} \sqrt {5-2 \sqrt {5}} \text {arctanh}\left (\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} x\right )+\sqrt {\frac {1}{45}+\frac {2}{45 \sqrt {5}}} \text {arctanh}\left (\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} x\right )-\frac {\text {arctanh}\left (\sqrt {5}-2 \sqrt {5} x^2\right )}{6 \sqrt {5}}+\frac {1}{30} \sqrt {5-2 \sqrt {5}} \log \left (1-\sqrt {5}-2 \sqrt {5-2 \sqrt {5}} x+2 x^2\right )-\frac {1}{30} \sqrt {5-2 \sqrt {5}} \log \left (1-\sqrt {5}+2 \sqrt {5-2 \sqrt {5}} x+2 x^2\right )+\sqrt {\frac {1}{180}+\frac {1}{90 \sqrt {5}}} \log \left (1+\sqrt {5}-2 \sqrt {5+2 \sqrt {5}} x+2 x^2\right )-\sqrt {\frac {1}{180}+\frac {1}{90 \sqrt {5}}} \log \left (1+\sqrt {5}+2 \sqrt {5+2 \sqrt {5}} x+2 x^2\right )+\frac {\log \left (3+\sqrt {5}-8 x^2-2 \sqrt {5} x^2+2 x^4\right )}{12 \sqrt {5}}-\frac {\log \left (3-\sqrt {5}-8 x^2+2 \sqrt {5} x^2+2 x^4\right )}{12 \sqrt {5}}+\frac {1}{12} \log \left (1-12 x^2+54 x^4-113 x^6+111 x^8-45 x^{10}+5 x^{12}\right ) \] Output:
-1/15*(5-2*5^(1/2))^(1/2)*arctanh(1/2*(10-2*5^(1/2))^(1/2)*x)+1/15*(5+2*5^ (1/2))^(1/2)*arctanh(1/2*(10+2*5^(1/2))^(1/2)*x)+1/30*arctanh(-5^(1/2)+2*5 ^(1/2)*x^2)*5^(1/2)+1/30*(5-2*5^(1/2))^(1/2)*ln(1-5^(1/2)-2*(5-2*5^(1/2))^ (1/2)*x+2*x^2)-1/30*(5-2*5^(1/2))^(1/2)*ln(1-5^(1/2)+2*(5-2*5^(1/2))^(1/2) *x+2*x^2)+1/30*(5+2*5^(1/2))^(1/2)*ln(1+5^(1/2)-2*(5+2*5^(1/2))^(1/2)*x+2* x^2)-1/30*(5+2*5^(1/2))^(1/2)*ln(1+5^(1/2)+2*(5+2*5^(1/2))^(1/2)*x+2*x^2)+ 1/60*ln(3+5^(1/2)-8*x^2-2*5^(1/2)*x^2+2*x^4)*5^(1/2)-1/60*ln(3-5^(1/2)-8*x ^2+2*5^(1/2)*x^2+2*x^4)*5^(1/2)+1/12*ln(5*x^12-45*x^10+111*x^8-113*x^6+54* x^4-12*x^2+1)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.04 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.21 \[ \int \frac {x \left (-2+x+18 x^2-7 x^3-56 x^4+15 x^5+72 x^6-12 x^7-36 x^8+3 x^9+5 x^{10}\right )}{1-12 x^2+54 x^4-113 x^6+111 x^8-45 x^{10}+5 x^{12}} \, dx=\text {RootSum}\left [1-120 \text {$\#$1}+4500 \text {$\#$1}^2-54000 \text {$\#$1}^3+162000 \text {$\#$1}^4\&,\log \left (3-9 x^2-x^3-150 \text {$\#$1}+450 x^2 \text {$\#$1}+1800 \text {$\#$1}^2-5400 x^2 \text {$\#$1}^2-5400 \text {$\#$1}^3+16200 x^2 \text {$\#$1}^3\right ) \text {$\#$1}\&\right ] \] Input:
Integrate[(x*(-2 + x + 18*x^2 - 7*x^3 - 56*x^4 + 15*x^5 + 72*x^6 - 12*x^7 - 36*x^8 + 3*x^9 + 5*x^10))/(1 - 12*x^2 + 54*x^4 - 113*x^6 + 111*x^8 - 45* x^10 + 5*x^12),x]
Output:
RootSum[1 - 120*#1 + 4500*#1^2 - 54000*#1^3 + 162000*#1^4 & , Log[3 - 9*x^ 2 - x^3 - 150*#1 + 450*x^2*#1 + 1800*#1^2 - 5400*x^2*#1^2 - 5400*#1^3 + 16 200*x^2*#1^3]*#1 & ]
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.
| \(\displaystyle \int \frac {x \left (5 x^{10}+3 x^9-36 x^8-12 x^7+72 x^6+15 x^5-56 x^4-7 x^3+18 x^2+x-2\right )}{5 x^{12}-45 x^{10}+111 x^8-113 x^6+54 x^4-12 x^2+1} \, dx\) |
| \(\Big \downarrow \) 2460 |
| \(\displaystyle \int \left (\frac {5 x^3-x^2-3 x+1}{3 \left (5 x^4-5 x^2+1\right )}+\frac {2 x^7+2 x^6-11 x^5-7 x^4+13 x^3+6 x^2-3 x-1}{3 \left (x^8-8 x^6+14 x^4-7 x^2+1\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{12} \text {Subst}\left (\int \frac {1}{x^4-8 x^3+14 x^2-7 x+1}dx,x,x^2\right )-\frac {1}{6} \text {Subst}\left (\int \frac {x}{x^4-8 x^3+14 x^2-7 x+1}dx,x,x^2\right )+\frac {1}{6} \text {Subst}\left (\int \frac {x^2}{x^4-8 x^3+14 x^2-7 x+1}dx,x,x^2\right )-\frac {1}{3} \int \frac {1}{x^8-8 x^6+14 x^4-7 x^2+1}dx+2 \int \frac {x^2}{x^8-8 x^6+14 x^4-7 x^2+1}dx-\frac {7}{3} \int \frac {x^4}{x^8-8 x^6+14 x^4-7 x^2+1}dx+\frac {2}{3} \int \frac {x^6}{x^8-8 x^6+14 x^4-7 x^2+1}dx-\frac {1}{15} \sqrt {5-2 \sqrt {5}} \text {arctanh}\left (\sqrt {\frac {10}{5+\sqrt {5}}} x\right )+\frac {1}{15} \sqrt {5+2 \sqrt {5}} \text {arctanh}\left (\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} x\right )+\frac {1}{60} \left (5+\sqrt {5}\right ) \log \left (-10 x^2-\sqrt {5}+5\right )+\frac {1}{60} \left (5-\sqrt {5}\right ) \log \left (-10 x^2+\sqrt {5}+5\right )+\frac {1}{12} \log \left (x^8-8 x^6+14 x^4-7 x^2+1\right )\) |
Input:
Int[(x*(-2 + x + 18*x^2 - 7*x^3 - 56*x^4 + 15*x^5 + 72*x^6 - 12*x^7 - 36*x ^8 + 3*x^9 + 5*x^10))/(1 - 12*x^2 + 54*x^4 - 113*x^6 + 111*x^8 - 45*x^10 + 5*x^12),x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.11 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.17
| method | result | size |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (125 \textit {\_Z}^{4}-250 \textit {\_Z}^{3}+125 \textit {\_Z}^{2}-20 \textit {\_Z} +1\right )}{\sum }\textit {\_R} \ln \left (x^{3}+\left (-75 \textit {\_R}^{3}+150 \textit {\_R}^{2}-75 \textit {\_R} +9\right ) x^{2}+25 \textit {\_R}^{3}-50 \textit {\_R}^{2}+25 \textit {\_R} -3\right )\right )}{6}\) | \(67\) |
| default | \(\frac {\left (5+\sqrt {5}\right ) \ln \left (10 x^{2}+\sqrt {5}-5\right )}{60}-\frac {\left (-\sqrt {5}-1\right ) \operatorname {arctanh}\left (\frac {10 x}{\sqrt {50-10 \sqrt {5}}}\right )}{3 \sqrt {50-10 \sqrt {5}}}-\frac {\left (\sqrt {5}-5\right ) \ln \left (10 x^{2}-\sqrt {5}-5\right )}{60}+\frac {\left (-\sqrt {5}+1\right ) \operatorname {arctanh}\left (\frac {10 x}{\sqrt {50+10 \sqrt {5}}}\right )}{3 \sqrt {50+10 \sqrt {5}}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (125 \textit {\_Z}^{4}-250 \textit {\_Z}^{3}+125 \textit {\_Z}^{2}-20 \textit {\_Z} +1\right )}{\sum }\textit {\_R} \ln \left (x^{2}+\left (-250 \textit {\_R}^{3}+475 \textit {\_R}^{2}-205 \textit {\_R} +20\right ) x -250 \textit {\_R}^{3}+475 \textit {\_R}^{2}-200 \textit {\_R} +18\right )\right )}{6}\) | \(166\) |
Input:
int(x*(5*x^10+3*x^9-36*x^8-12*x^7+72*x^6+15*x^5-56*x^4-7*x^3+18*x^2+x-2)/( 5*x^12-45*x^10+111*x^8-113*x^6+54*x^4-12*x^2+1),x,method=_RETURNVERBOSE)
Output:
1/6*sum(_R*ln(x^3+(-75*_R^3+150*_R^2-75*_R+9)*x^2+25*_R^3-50*_R^2+25*_R-3) ,_R=RootOf(125*_Z^4-250*_Z^3+125*_Z^2-20*_Z+1))
Time = 0.10 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.54 \[ \int \frac {x \left (-2+x+18 x^2-7 x^3-56 x^4+15 x^5+72 x^6-12 x^7-36 x^8+3 x^9+5 x^{10}\right )}{1-12 x^2+54 x^4-113 x^6+111 x^8-45 x^{10}+5 x^{12}} \, dx=\frac {1}{60} \, {\left (\sqrt {5} - 2 \, \sqrt {2 \, \sqrt {5} + 5} + 5\right )} \log \left (10 \, x^{3} + {\left (15 \, x^{2} - \sqrt {5} {\left (3 \, x^{2} - 1\right )} - 5\right )} \sqrt {2 \, \sqrt {5} + 5}\right ) + \frac {1}{60} \, {\left (\sqrt {5} + 2 \, \sqrt {2 \, \sqrt {5} + 5} + 5\right )} \log \left (10 \, x^{3} - {\left (15 \, x^{2} - \sqrt {5} {\left (3 \, x^{2} - 1\right )} - 5\right )} \sqrt {2 \, \sqrt {5} + 5}\right ) - \frac {1}{60} \, {\left (\sqrt {5} + 2 \, \sqrt {-2 \, \sqrt {5} + 5} - 5\right )} \log \left (10 \, x^{3} + {\left (15 \, x^{2} + \sqrt {5} {\left (3 \, x^{2} - 1\right )} - 5\right )} \sqrt {-2 \, \sqrt {5} + 5}\right ) - \frac {1}{60} \, {\left (\sqrt {5} - 2 \, \sqrt {-2 \, \sqrt {5} + 5} - 5\right )} \log \left (10 \, x^{3} - {\left (15 \, x^{2} + \sqrt {5} {\left (3 \, x^{2} - 1\right )} - 5\right )} \sqrt {-2 \, \sqrt {5} + 5}\right ) \] Input:
integrate(x*(5*x^10+3*x^9-36*x^8-12*x^7+72*x^6+15*x^5-56*x^4-7*x^3+18*x^2+ x-2)/(5*x^12-45*x^10+111*x^8-113*x^6+54*x^4-12*x^2+1),x, algorithm="fricas ")
Output:
1/60*(sqrt(5) - 2*sqrt(2*sqrt(5) + 5) + 5)*log(10*x^3 + (15*x^2 - sqrt(5)* (3*x^2 - 1) - 5)*sqrt(2*sqrt(5) + 5)) + 1/60*(sqrt(5) + 2*sqrt(2*sqrt(5) + 5) + 5)*log(10*x^3 - (15*x^2 - sqrt(5)*(3*x^2 - 1) - 5)*sqrt(2*sqrt(5) + 5)) - 1/60*(sqrt(5) + 2*sqrt(-2*sqrt(5) + 5) - 5)*log(10*x^3 + (15*x^2 + s qrt(5)*(3*x^2 - 1) - 5)*sqrt(-2*sqrt(5) + 5)) - 1/60*(sqrt(5) - 2*sqrt(-2* sqrt(5) + 5) - 5)*log(10*x^3 - (15*x^2 + sqrt(5)*(3*x^2 - 1) - 5)*sqrt(-2* sqrt(5) + 5))
Time = 0.23 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.15 \[ \int \frac {x \left (-2+x+18 x^2-7 x^3-56 x^4+15 x^5+72 x^6-12 x^7-36 x^8+3 x^9+5 x^{10}\right )}{1-12 x^2+54 x^4-113 x^6+111 x^8-45 x^{10}+5 x^{12}} \, dx=\operatorname {RootSum} {\left (162000 t^{4} - 54000 t^{3} + 4500 t^{2} - 120 t + 1, \left ( t \mapsto t \log {\left (5400 t^{3} - 1800 t^{2} + 150 t + x^{3} + x^{2} \left (- 16200 t^{3} + 5400 t^{2} - 450 t + 9\right ) - 3 \right )} \right )\right )} \] Input:
integrate(x*(5*x**10+3*x**9-36*x**8-12*x**7+72*x**6+15*x**5-56*x**4-7*x**3 +18*x**2+x-2)/(5*x**12-45*x**10+111*x**8-113*x**6+54*x**4-12*x**2+1),x)
Output:
RootSum(162000*_t**4 - 54000*_t**3 + 4500*_t**2 - 120*_t + 1, Lambda(_t, _ t*log(5400*_t**3 - 1800*_t**2 + 150*_t + x**3 + x**2*(-16200*_t**3 + 5400* _t**2 - 450*_t + 9) - 3)))
\[ \int \frac {x \left (-2+x+18 x^2-7 x^3-56 x^4+15 x^5+72 x^6-12 x^7-36 x^8+3 x^9+5 x^{10}\right )}{1-12 x^2+54 x^4-113 x^6+111 x^8-45 x^{10}+5 x^{12}} \, dx=\int { \frac {{\left (5 \, x^{10} + 3 \, x^{9} - 36 \, x^{8} - 12 \, x^{7} + 72 \, x^{6} + 15 \, x^{5} - 56 \, x^{4} - 7 \, x^{3} + 18 \, x^{2} + x - 2\right )} x}{5 \, x^{12} - 45 \, x^{10} + 111 \, x^{8} - 113 \, x^{6} + 54 \, x^{4} - 12 \, x^{2} + 1} \,d x } \] Input:
integrate(x*(5*x^10+3*x^9-36*x^8-12*x^7+72*x^6+15*x^5-56*x^4-7*x^3+18*x^2+ x-2)/(5*x^12-45*x^10+111*x^8-113*x^6+54*x^4-12*x^2+1),x, algorithm="maxima ")
Output:
integrate((5*x^10 + 3*x^9 - 36*x^8 - 12*x^7 + 72*x^6 + 15*x^5 - 56*x^4 - 7 *x^3 + 18*x^2 + x - 2)*x/(5*x^12 - 45*x^10 + 111*x^8 - 113*x^6 + 54*x^4 - 12*x^2 + 1), x)
\[ \int \frac {x \left (-2+x+18 x^2-7 x^3-56 x^4+15 x^5+72 x^6-12 x^7-36 x^8+3 x^9+5 x^{10}\right )}{1-12 x^2+54 x^4-113 x^6+111 x^8-45 x^{10}+5 x^{12}} \, dx=\int { \frac {{\left (5 \, x^{10} + 3 \, x^{9} - 36 \, x^{8} - 12 \, x^{7} + 72 \, x^{6} + 15 \, x^{5} - 56 \, x^{4} - 7 \, x^{3} + 18 \, x^{2} + x - 2\right )} x}{5 \, x^{12} - 45 \, x^{10} + 111 \, x^{8} - 113 \, x^{6} + 54 \, x^{4} - 12 \, x^{2} + 1} \,d x } \] Input:
integrate(x*(5*x^10+3*x^9-36*x^8-12*x^7+72*x^6+15*x^5-56*x^4-7*x^3+18*x^2+ x-2)/(5*x^12-45*x^10+111*x^8-113*x^6+54*x^4-12*x^2+1),x, algorithm="giac")
Output:
integrate((5*x^10 + 3*x^9 - 36*x^8 - 12*x^7 + 72*x^6 + 15*x^5 - 56*x^4 - 7 *x^3 + 18*x^2 + x - 2)*x/(5*x^12 - 45*x^10 + 111*x^8 - 113*x^6 + 54*x^4 - 12*x^2 + 1), x)
Time = 11.27 (sec) , antiderivative size = 1145, normalized size of antiderivative = 2.83 \[ \int \frac {x \left (-2+x+18 x^2-7 x^3-56 x^4+15 x^5+72 x^6-12 x^7-36 x^8+3 x^9+5 x^{10}\right )}{1-12 x^2+54 x^4-113 x^6+111 x^8-45 x^{10}+5 x^{12}} \, dx=\text {Too large to display} \] Input:
int((x*(x + 18*x^2 - 7*x^3 - 56*x^4 + 15*x^5 + 72*x^6 - 12*x^7 - 36*x^8 + 3*x^9 + 5*x^10 - 2))/(54*x^4 - 12*x^2 - 113*x^6 + 111*x^8 - 45*x^10 + 5*x^ 12 + 1),x)
Output:
symsum(log((146752739407872000000*x + 3914915958685440000*root(z^4 - 120*z ^3 + 4500*z^2 - 54000*z + 162000, z, k)^2 - 877535262010152000*root(z^4 - 120*z^3 + 4500*z^2 - 54000*z + 162000, z, k)^3 + 98732772668808000*root(z^ 4 - 120*z^3 + 4500*z^2 - 54000*z + 162000, z, k)^4 - 6106360017121200*root (z^4 - 120*z^3 + 4500*z^2 - 54000*z + 162000, z, k)^5 + 213667660002240*ro ot(z^4 - 120*z^3 + 4500*z^2 - 54000*z + 162000, z, k)^6 - 4206592440936*ro ot(z^4 - 120*z^3 + 4500*z^2 - 54000*z + 162000, z, k)^7 - 183131208*root(z ^4 - 120*z^3 + 4500*z^2 - 54000*z + 162000, z, k)^9 - 14828663965694400000 *x^2*root(z^4 - 120*z^3 + 4500*z^2 - 54000*z + 162000, z, k)^2 + 333318757 2573816000*x^2*root(z^4 - 120*z^3 + 4500*z^2 - 54000*z + 162000, z, k)^3 - 111817024500716640000*x^3*root(z^4 - 120*z^3 + 4500*z^2 - 54000*z + 16200 0, z, k)^2 - 376069644198360000*x^2*root(z^4 - 120*z^3 + 4500*z^2 - 54000* z + 162000, z, k)^4 + 17011847867603976000*x^3*root(z^4 - 120*z^3 + 4500*z ^2 - 54000*z + 162000, z, k)^3 + 23321109901309200*x^2*root(z^4 - 120*z^3 + 4500*z^2 - 54000*z + 162000, z, k)^5 - 1489935486584556000*x^3*root(z^4 - 120*z^3 + 4500*z^2 - 54000*z + 162000, z, k)^4 - 818119549632960*x^2*roo t(z^4 - 120*z^3 + 4500*z^2 - 54000*z + 162000, z, k)^6 + 77798117717812800 *x^3*root(z^4 - 120*z^3 + 4500*z^2 - 54000*z + 162000, z, k)^5 + 443376954 04968*x^3*root(z^4 - 120*z^3 + 4500*z^2 - 54000*z + 162000, z, k)^7 + 7064 06168*x^2*root(z^4 - 120*z^3 + 4500*z^2 - 54000*z + 162000, z, k)^9 + 1...
\[ \int \frac {x \left (-2+x+18 x^2-7 x^3-56 x^4+15 x^5+72 x^6-12 x^7-36 x^8+3 x^9+5 x^{10}\right )}{1-12 x^2+54 x^4-113 x^6+111 x^8-45 x^{10}+5 x^{12}} \, dx=\frac {3 \sqrt {\sqrt {5}-5}\, \sqrt {10}\, \mathit {atan} \left (\frac {10 x}{\sqrt {\sqrt {5}-5}\, \sqrt {10}}\right )}{5}+\frac {33 \sqrt {\sqrt {5}-5}\, \sqrt {2}\, \mathit {atan} \left (\frac {10 x}{\sqrt {\sqrt {5}-5}\, \sqrt {10}}\right )}{10}-\frac {3 \sqrt {\sqrt {5}+5}\, \sqrt {10}\, \mathrm {log}\left (-\sqrt {\sqrt {5}+5}+\sqrt {10}\, x \right )}{10}+\frac {3 \sqrt {\sqrt {5}+5}\, \sqrt {10}\, \mathrm {log}\left (\sqrt {\sqrt {5}+5}+\sqrt {10}\, x \right )}{10}+\frac {33 \sqrt {\sqrt {5}+5}\, \sqrt {2}\, \mathrm {log}\left (-\sqrt {\sqrt {5}+5}+\sqrt {10}\, x \right )}{20}-\frac {33 \sqrt {\sqrt {5}+5}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {5}+5}+\sqrt {10}\, x \right )}{20}-\frac {7 \sqrt {5}\, \mathrm {log}\left (-\sqrt {\sqrt {5}+5}+\sqrt {10}\, x \right )}{120}-\frac {7 \sqrt {5}\, \mathrm {log}\left (\sqrt {\sqrt {5}+5}+\sqrt {10}\, x \right )}{120}+\frac {7 \sqrt {5}\, \mathrm {log}\left (\sqrt {5}+10 x^{2}-5\right )}{120}+69 \left (\int \frac {x^{6}}{5 x^{12}-45 x^{10}+111 x^{8}-113 x^{6}+54 x^{4}-12 x^{2}+1}d x \right )+\frac {73 \left (\int \frac {x^{5}}{5 x^{12}-45 x^{10}+111 x^{8}-113 x^{6}+54 x^{4}-12 x^{2}+1}d x \right )}{24}-154 \left (\int \frac {x^{4}}{5 x^{12}-45 x^{10}+111 x^{8}-113 x^{6}+54 x^{4}-12 x^{2}+1}d x \right )-\frac {53 \left (\int \frac {x^{3}}{5 x^{12}-45 x^{10}+111 x^{8}-113 x^{6}+54 x^{4}-12 x^{2}+1}d x \right )}{24}+82 \left (\int \frac {x^{2}}{5 x^{12}-45 x^{10}+111 x^{8}-113 x^{6}+54 x^{4}-12 x^{2}+1}d x \right )+\frac {3 \left (\int \frac {x}{5 x^{12}-45 x^{10}+111 x^{8}-113 x^{6}+54 x^{4}-12 x^{2}+1}d x \right )}{8}-12 \left (\int \frac {1}{5 x^{12}-45 x^{10}+111 x^{8}-113 x^{6}+54 x^{4}-12 x^{2}+1}d x \right )+\frac {17 \,\mathrm {log}\left (x^{8}-8 x^{6}+14 x^{4}-7 x^{2}+1\right )}{144}+\frac {\mathrm {log}\left (-\sqrt {\sqrt {5}+5}+\sqrt {10}\, x \right )}{72}+\frac {\mathrm {log}\left (\sqrt {\sqrt {5}+5}+\sqrt {10}\, x \right )}{72}+\frac {\mathrm {log}\left (\sqrt {5}+10 x^{2}-5\right )}{72} \] Input:
int(x*(5*x^10+3*x^9-36*x^8-12*x^7+72*x^6+15*x^5-56*x^4-7*x^3+18*x^2+x-2)/( 5*x^12-45*x^10+111*x^8-113*x^6+54*x^4-12*x^2+1),x)
Output:
(432*sqrt(sqrt(5) - 5)*sqrt(10)*atan((10*x)/(sqrt(sqrt(5) - 5)*sqrt(10))) + 2376*sqrt(sqrt(5) - 5)*sqrt(2)*atan((10*x)/(sqrt(sqrt(5) - 5)*sqrt(10))) - 216*sqrt(sqrt(5) + 5)*sqrt(10)*log( - sqrt(sqrt(5) + 5) + sqrt(10)*x) + 216*sqrt(sqrt(5) + 5)*sqrt(10)*log(sqrt(sqrt(5) + 5) + sqrt(10)*x) + 1188 *sqrt(sqrt(5) + 5)*sqrt(2)*log( - sqrt(sqrt(5) + 5) + sqrt(10)*x) - 1188*s qrt(sqrt(5) + 5)*sqrt(2)*log(sqrt(sqrt(5) + 5) + sqrt(10)*x) - 42*sqrt(5)* log( - sqrt(sqrt(5) + 5) + sqrt(10)*x) - 42*sqrt(5)*log(sqrt(sqrt(5) + 5) + sqrt(10)*x) + 42*sqrt(5)*log(sqrt(5) + 10*x**2 - 5) + 49680*int(x**6/(5* x**12 - 45*x**10 + 111*x**8 - 113*x**6 + 54*x**4 - 12*x**2 + 1),x) + 2190* int(x**5/(5*x**12 - 45*x**10 + 111*x**8 - 113*x**6 + 54*x**4 - 12*x**2 + 1 ),x) - 110880*int(x**4/(5*x**12 - 45*x**10 + 111*x**8 - 113*x**6 + 54*x**4 - 12*x**2 + 1),x) - 1590*int(x**3/(5*x**12 - 45*x**10 + 111*x**8 - 113*x* *6 + 54*x**4 - 12*x**2 + 1),x) + 59040*int(x**2/(5*x**12 - 45*x**10 + 111* x**8 - 113*x**6 + 54*x**4 - 12*x**2 + 1),x) + 270*int(x/(5*x**12 - 45*x**1 0 + 111*x**8 - 113*x**6 + 54*x**4 - 12*x**2 + 1),x) - 8640*int(1/(5*x**12 - 45*x**10 + 111*x**8 - 113*x**6 + 54*x**4 - 12*x**2 + 1),x) + 85*log(x**8 - 8*x**6 + 14*x**4 - 7*x**2 + 1) + 10*log( - sqrt(sqrt(5) + 5) + sqrt(10) *x) + 10*log(sqrt(sqrt(5) + 5) + sqrt(10)*x) + 10*log(sqrt(5) + 10*x**2 - 5))/720