Integrand size = 68, antiderivative size = 212 \[ \int \frac {x^3 \left (2-15 x^2+36 x^4-27 x^6-4 x^8+6 x^{10}\right )}{1-12 x^2+54 x^4-108 x^6+80 x^8+6 x^{10}-9 x^{12}+x^{16}} \, dx=\frac {1}{4} \arctan \left (\frac {1}{9} \left (2-9 \sqrt {3}+6 x^2\right )\right )+\frac {1}{4} \arctan \left (\frac {1}{9} \left (2+9 \sqrt {3}+6 x^2\right )\right )+\frac {1}{4} \arctan \left (18+\sqrt {3}-54 x^2-2 x^4-18 \sqrt {3} x^4-6 x^6\right )+\frac {1}{4} \arctan \left (18-\sqrt {3}-54 x^2-2 x^4+18 \sqrt {3} x^4-6 x^6\right )+\frac {1}{8} \sqrt {3} \log \left (1-6 x^2+9 x^4+\sqrt {3} x^4-3 \sqrt {3} x^6+x^8\right )-\frac {1}{8} \sqrt {3} \log \left (1-6 x^2+9 x^4-\sqrt {3} x^4+3 \sqrt {3} x^6+x^8\right ) \] Output:
1/4*arctan(2/9-3^(1/2)+2/3*x^2)+1/4*arctan(2/9+3^(1/2)+2/3*x^2)-1/4*arctan (-18-3^(1/2)+54*x^2+2*x^4+18*3^(1/2)*x^4+6*x^6)-1/4*arctan(-18+3^(1/2)+54* x^2+2*x^4-18*3^(1/2)*x^4+6*x^6)+1/8*3^(1/2)*ln(1-6*x^2+9*x^4+3^(1/2)*x^4-3 *x^6*3^(1/2)+x^8)-1/8*3^(1/2)*ln(1-6*x^2+9*x^4-3^(1/2)*x^4+3*x^6*3^(1/2)+x ^8)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.80 \[ \int \frac {x^3 \left (2-15 x^2+36 x^4-27 x^6-4 x^8+6 x^{10}\right )}{1-12 x^2+54 x^4-108 x^6+80 x^8+6 x^{10}-9 x^{12}+x^{16}} \, dx=\frac {1}{4} \text {RootSum}\left [1-12 \text {$\#$1}^2+54 \text {$\#$1}^4-108 \text {$\#$1}^6+80 \text {$\#$1}^8+6 \text {$\#$1}^{10}-9 \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {2 \log (x-\text {$\#$1}) \text {$\#$1}^2-15 \log (x-\text {$\#$1}) \text {$\#$1}^4+36 \log (x-\text {$\#$1}) \text {$\#$1}^6-27 \log (x-\text {$\#$1}) \text {$\#$1}^8-4 \log (x-\text {$\#$1}) \text {$\#$1}^{10}+6 \log (x-\text {$\#$1}) \text {$\#$1}^{12}}{-6+54 \text {$\#$1}^2-162 \text {$\#$1}^4+160 \text {$\#$1}^6+15 \text {$\#$1}^8-27 \text {$\#$1}^{10}+4 \text {$\#$1}^{14}}\&\right ] \] Input:
Integrate[(x^3*(2 - 15*x^2 + 36*x^4 - 27*x^6 - 4*x^8 + 6*x^10))/(1 - 12*x^ 2 + 54*x^4 - 108*x^6 + 80*x^8 + 6*x^10 - 9*x^12 + x^16),x]
Output:
RootSum[1 - 12*#1^2 + 54*#1^4 - 108*#1^6 + 80*#1^8 + 6*#1^10 - 9*#1^12 + # 1^16 & , (2*Log[x - #1]*#1^2 - 15*Log[x - #1]*#1^4 + 36*Log[x - #1]*#1^6 - 27*Log[x - #1]*#1^8 - 4*Log[x - #1]*#1^10 + 6*Log[x - #1]*#1^12)/(-6 + 54 *#1^2 - 162*#1^4 + 160*#1^6 + 15*#1^8 - 27*#1^10 + 4*#1^14) & ]/4
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.
| \(\displaystyle \int \frac {x^3 \left (6 x^{10}-4 x^8-27 x^6+36 x^4-15 x^2+2\right )}{x^{16}-9 x^{12}+6 x^{10}+80 x^8-108 x^6+54 x^4-12 x^2+1} \, dx\) |
| \(\Big \downarrow \) 7283 |
| \(\displaystyle \frac {1}{2} \int \frac {x^2 \left (6 x^{10}-4 x^8-27 x^6+36 x^4-15 x^2+2\right )}{x^{16}-9 x^{12}+6 x^{10}+80 x^8-108 x^6+54 x^4-12 x^2+1}dx^2\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {6 x^{12}}{x^{16}-9 x^{12}+6 x^{10}+80 x^8-108 x^6+54 x^4-12 x^2+1}-\frac {4 x^{10}}{x^{16}-9 x^{12}+6 x^{10}+80 x^8-108 x^6+54 x^4-12 x^2+1}-\frac {27 x^8}{x^{16}-9 x^{12}+6 x^{10}+80 x^8-108 x^6+54 x^4-12 x^2+1}+\frac {36 x^6}{x^{16}-9 x^{12}+6 x^{10}+80 x^8-108 x^6+54 x^4-12 x^2+1}-\frac {15 x^4}{x^{16}-9 x^{12}+6 x^{10}+80 x^8-108 x^6+54 x^4-12 x^2+1}+\frac {2 x^2}{x^{16}-9 x^{12}+6 x^{10}+80 x^8-108 x^6+54 x^4-12 x^2+1}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (2 \int \frac {x^2}{x^{16}-9 x^{12}+6 x^{10}+80 x^8-108 x^6+54 x^4-12 x^2+1}dx^2-15 \int \frac {x^4}{x^{16}-9 x^{12}+6 x^{10}+80 x^8-108 x^6+54 x^4-12 x^2+1}dx^2+36 \int \frac {x^6}{x^{16}-9 x^{12}+6 x^{10}+80 x^8-108 x^6+54 x^4-12 x^2+1}dx^2-27 \int \frac {x^8}{x^{16}-9 x^{12}+6 x^{10}+80 x^8-108 x^6+54 x^4-12 x^2+1}dx^2-4 \int \frac {x^{10}}{x^{16}-9 x^{12}+6 x^{10}+80 x^8-108 x^6+54 x^4-12 x^2+1}dx^2+6 \int \frac {x^{12}}{x^{16}-9 x^{12}+6 x^{10}+80 x^8-108 x^6+54 x^4-12 x^2+1}dx^2\right )\) |
Input:
Int[(x^3*(2 - 15*x^2 + 36*x^4 - 27*x^6 - 4*x^8 + 6*x^10))/(1 - 12*x^2 + 54 *x^4 - 108*x^6 + 80*x^8 + 6*x^10 - 9*x^12 + x^16),x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.11 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.20
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-\textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (x^{4}+\left (3 \textit {\_R}^{3}-3 \textit {\_R} \right ) x^{2}-\textit {\_R}^{3}+\textit {\_R} \right )\right )}{4}\) | \(43\) |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-\textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (x^{4}+\left (3 \textit {\_R}^{3}-3 \textit {\_R} \right ) x^{2}-\textit {\_R}^{3}+\textit {\_R} \right )\right )}{4}\) | \(43\) |
Input:
int(x^3*(6*x^10-4*x^8-27*x^6+36*x^4-15*x^2+2)/(x^16-9*x^12+6*x^10+80*x^8-1 08*x^6+54*x^4-12*x^2+1),x,method=_RETURNVERBOSE)
Output:
1/4*sum(_R*ln(x^4+(3*_R^3-3*_R)*x^2-_R^3+_R),_R=RootOf(_Z^4-_Z^2+1))
Time = 0.08 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.76 \[ \int \frac {x^3 \left (2-15 x^2+36 x^4-27 x^6-4 x^8+6 x^{10}\right )}{1-12 x^2+54 x^4-108 x^6+80 x^8+6 x^{10}-9 x^{12}+x^{16}} \, dx=-\frac {1}{8} \, \sqrt {3} \log \left (x^{8} + 9 \, x^{4} - 6 \, x^{2} + \sqrt {3} {\left (3 \, x^{6} - x^{4}\right )} + 1\right ) + \frac {1}{8} \, \sqrt {3} \log \left (x^{8} + 9 \, x^{4} - 6 \, x^{2} - \sqrt {3} {\left (3 \, x^{6} - x^{4}\right )} + 1\right ) - \frac {1}{4} \, \arctan \left (6 \, x^{6} + 2 \, x^{4} + 54 \, x^{2} + \sqrt {3} {\left (18 \, x^{4} - 1\right )} - 18\right ) + \frac {1}{4} \, \arctan \left (-6 \, x^{6} - 2 \, x^{4} - 54 \, x^{2} + \sqrt {3} {\left (18 \, x^{4} - 1\right )} + 18\right ) + \frac {1}{4} \, \arctan \left (\frac {2}{3} \, x^{2} + \sqrt {3} + \frac {2}{9}\right ) - \frac {1}{4} \, \arctan \left (-\frac {2}{3} \, x^{2} + \sqrt {3} - \frac {2}{9}\right ) \] Input:
integrate(x^3*(6*x^10-4*x^8-27*x^6+36*x^4-15*x^2+2)/(x^16-9*x^12+6*x^10+80 *x^8-108*x^6+54*x^4-12*x^2+1),x, algorithm="fricas")
Output:
-1/8*sqrt(3)*log(x^8 + 9*x^4 - 6*x^2 + sqrt(3)*(3*x^6 - x^4) + 1) + 1/8*sq rt(3)*log(x^8 + 9*x^4 - 6*x^2 - sqrt(3)*(3*x^6 - x^4) + 1) - 1/4*arctan(6* x^6 + 2*x^4 + 54*x^2 + sqrt(3)*(18*x^4 - 1) - 18) + 1/4*arctan(-6*x^6 - 2* x^4 - 54*x^2 + sqrt(3)*(18*x^4 - 1) + 18) + 1/4*arctan(2/3*x^2 + sqrt(3) + 2/9) - 1/4*arctan(-2/3*x^2 + sqrt(3) - 2/9)
Time = 0.17 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.85 \[ \int \frac {x^3 \left (2-15 x^2+36 x^4-27 x^6-4 x^8+6 x^{10}\right )}{1-12 x^2+54 x^4-108 x^6+80 x^8+6 x^{10}-9 x^{12}+x^{16}} \, dx=\frac {\sqrt {3} \log {\left (x^{8} - 3 \sqrt {3} x^{6} + x^{4} \left (\sqrt {3} + 9\right ) - 6 x^{2} + 1 \right )}}{8} - \frac {\sqrt {3} \log {\left (x^{8} + 3 \sqrt {3} x^{6} + x^{4} \cdot \left (9 - \sqrt {3}\right ) - 6 x^{2} + 1 \right )}}{8} + \frac {\operatorname {atan}{\left (\frac {2 x^{2}}{3} + \frac {2}{9} + \sqrt {3} \right )}}{4} + \frac {\operatorname {atan}{\left (\frac {2 x^{2}}{3} - \sqrt {3} + \frac {2}{9} \right )}}{4} + \frac {\operatorname {atan}{\left (- 6 x^{6} + x^{4} \left (- 18 \sqrt {3} - 2\right ) - 54 x^{2} + \sqrt {3} + 18 \right )}}{4} - \frac {\operatorname {atan}{\left (6 x^{6} - x^{4} \left (-2 + 18 \sqrt {3}\right ) + 54 x^{2} - 18 + \sqrt {3} \right )}}{4} \] Input:
integrate(x**3*(6*x**10-4*x**8-27*x**6+36*x**4-15*x**2+2)/(x**16-9*x**12+6 *x**10+80*x**8-108*x**6+54*x**4-12*x**2+1),x)
Output:
sqrt(3)*log(x**8 - 3*sqrt(3)*x**6 + x**4*(sqrt(3) + 9) - 6*x**2 + 1)/8 - s qrt(3)*log(x**8 + 3*sqrt(3)*x**6 + x**4*(9 - sqrt(3)) - 6*x**2 + 1)/8 + at an(2*x**2/3 + 2/9 + sqrt(3))/4 + atan(2*x**2/3 - sqrt(3) + 2/9)/4 + atan(- 6*x**6 + x**4*(-18*sqrt(3) - 2) - 54*x**2 + sqrt(3) + 18)/4 - atan(6*x**6 - x**4*(-2 + 18*sqrt(3)) + 54*x**2 - 18 + sqrt(3))/4
\[ \int \frac {x^3 \left (2-15 x^2+36 x^4-27 x^6-4 x^8+6 x^{10}\right )}{1-12 x^2+54 x^4-108 x^6+80 x^8+6 x^{10}-9 x^{12}+x^{16}} \, dx=\int { \frac {{\left (6 \, x^{10} - 4 \, x^{8} - 27 \, x^{6} + 36 \, x^{4} - 15 \, x^{2} + 2\right )} x^{3}}{x^{16} - 9 \, x^{12} + 6 \, x^{10} + 80 \, x^{8} - 108 \, x^{6} + 54 \, x^{4} - 12 \, x^{2} + 1} \,d x } \] Input:
integrate(x^3*(6*x^10-4*x^8-27*x^6+36*x^4-15*x^2+2)/(x^16-9*x^12+6*x^10+80 *x^8-108*x^6+54*x^4-12*x^2+1),x, algorithm="maxima")
Output:
integrate((6*x^10 - 4*x^8 - 27*x^6 + 36*x^4 - 15*x^2 + 2)*x^3/(x^16 - 9*x^ 12 + 6*x^10 + 80*x^8 - 108*x^6 + 54*x^4 - 12*x^2 + 1), x)
Time = 0.17 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.77 \[ \int \frac {x^3 \left (2-15 x^2+36 x^4-27 x^6-4 x^8+6 x^{10}\right )}{1-12 x^2+54 x^4-108 x^6+80 x^8+6 x^{10}-9 x^{12}+x^{16}} \, dx=-\frac {1}{8} \, \sqrt {3} \log \left ({\left (2 \, x^{4} + 3 \, \sqrt {3} x^{2} - \sqrt {3}\right )}^{2} + {\left (3 \, x^{2} - 1\right )}^{2}\right ) + \frac {1}{8} \, \sqrt {3} \log \left ({\left (2 \, x^{4} - 3 \, \sqrt {3} x^{2} + \sqrt {3}\right )}^{2} + {\left (3 \, x^{2} - 1\right )}^{2}\right ) - \frac {1}{4} \, \arctan \left (6 \, x^{6} + 2 \, x^{4} {\left (9 \, \sqrt {3} + 1\right )} + 54 \, x^{2} - \sqrt {3} - 18\right ) - \frac {1}{4} \, \arctan \left (6 \, x^{6} - 2 \, x^{4} {\left (9 \, \sqrt {3} - 1\right )} + 54 \, x^{2} + \sqrt {3} - 18\right ) + \frac {1}{4} \, \arctan \left (\frac {2}{3} \, x^{2} + \sqrt {3} + \frac {2}{9}\right ) + \frac {1}{4} \, \arctan \left (\frac {2}{3} \, x^{2} - \sqrt {3} + \frac {2}{9}\right ) \] Input:
integrate(x^3*(6*x^10-4*x^8-27*x^6+36*x^4-15*x^2+2)/(x^16-9*x^12+6*x^10+80 *x^8-108*x^6+54*x^4-12*x^2+1),x, algorithm="giac")
Output:
-1/8*sqrt(3)*log((2*x^4 + 3*sqrt(3)*x^2 - sqrt(3))^2 + (3*x^2 - 1)^2) + 1/ 8*sqrt(3)*log((2*x^4 - 3*sqrt(3)*x^2 + sqrt(3))^2 + (3*x^2 - 1)^2) - 1/4*a rctan(6*x^6 + 2*x^4*(9*sqrt(3) + 1) + 54*x^2 - sqrt(3) - 18) - 1/4*arctan( 6*x^6 - 2*x^4*(9*sqrt(3) - 1) + 54*x^2 + sqrt(3) - 18) + 1/4*arctan(2/3*x^ 2 + sqrt(3) + 2/9) + 1/4*arctan(2/3*x^2 - sqrt(3) + 2/9)
Time = 11.81 (sec) , antiderivative size = 1081, normalized size of antiderivative = 5.10 \[ \int \frac {x^3 \left (2-15 x^2+36 x^4-27 x^6-4 x^8+6 x^{10}\right )}{1-12 x^2+54 x^4-108 x^6+80 x^8+6 x^{10}-9 x^{12}+x^{16}} \, dx=\text {Too large to display} \] Input:
int(-(x^3*(15*x^2 - 36*x^4 + 27*x^6 + 4*x^8 - 6*x^10 - 2))/(54*x^4 - 12*x^ 2 - 108*x^6 + 80*x^8 + 6*x^10 - 9*x^12 + x^16 + 1),x)
Output:
atan(85712904/(3^(1/2)*x^2*621355131i - 3^(1/2)*207118377i + 3^(1/2)*x^4*1 22611968i - 1890235035*x^2 + 539261712*x^4 + 630078345) - (257138712*x^2)/ (3^(1/2)*x^2*621355131i - 3^(1/2)*207118377i + 3^(1/2)*x^4*122611968i - 18 90235035*x^2 + 539261712*x^4 + 630078345) - (4361607*x^4)/(3^(1/2)*x^2*621 355131i - 3^(1/2)*207118377i + 3^(1/2)*x^4*122611968i - 1890235035*x^2 + 5 39261712*x^4 + 630078345) + (3^(1/2)*330936840i)/(3^(1/2)*x^2*621355131i - 3^(1/2)*207118377i + 3^(1/2)*x^4*122611968i - 1890235035*x^2 + 539261712* x^4 + 630078345) - (3^(1/2)*x^2*992810520i)/(3^(1/2)*x^2*621355131i - 3^(1 /2)*207118377i + 3^(1/2)*x^4*122611968i - 1890235035*x^2 + 539261712*x^4 + 630078345) + (3^(1/2)*x^4*418598361i)/(3^(1/2)*x^2*621355131i - 3^(1/2)*2 07118377i + 3^(1/2)*x^4*122611968i - 1890235035*x^2 + 539261712*x^4 + 6300 78345))/4 - atan((257138712*x^2)/(3^(1/2)*207118377i - 3^(1/2)*x^2*6213551 31i - 3^(1/2)*x^4*122611968i - 1890235035*x^2 + 539261712*x^4 + 630078345) - 85712904/(3^(1/2)*207118377i - 3^(1/2)*x^2*621355131i - 3^(1/2)*x^4*122 611968i - 1890235035*x^2 + 539261712*x^4 + 630078345) + (4361607*x^4)/(3^( 1/2)*207118377i - 3^(1/2)*x^2*621355131i - 3^(1/2)*x^4*122611968i - 189023 5035*x^2 + 539261712*x^4 + 630078345) + (3^(1/2)*330936840i)/(3^(1/2)*2071 18377i - 3^(1/2)*x^2*621355131i - 3^(1/2)*x^4*122611968i - 1890235035*x^2 + 539261712*x^4 + 630078345) - (3^(1/2)*x^2*992810520i)/(3^(1/2)*207118377 i - 3^(1/2)*x^2*621355131i - 3^(1/2)*x^4*122611968i - 1890235035*x^2 + ...
\[ \int \frac {x^3 \left (2-15 x^2+36 x^4-27 x^6-4 x^8+6 x^{10}\right )}{1-12 x^2+54 x^4-108 x^6+80 x^8+6 x^{10}-9 x^{12}+x^{16}} \, dx=6 \left (\int \frac {x^{13}}{x^{16}-9 x^{12}+6 x^{10}+80 x^{8}-108 x^{6}+54 x^{4}-12 x^{2}+1}d x \right )-4 \left (\int \frac {x^{11}}{x^{16}-9 x^{12}+6 x^{10}+80 x^{8}-108 x^{6}+54 x^{4}-12 x^{2}+1}d x \right )-27 \left (\int \frac {x^{9}}{x^{16}-9 x^{12}+6 x^{10}+80 x^{8}-108 x^{6}+54 x^{4}-12 x^{2}+1}d x \right )+36 \left (\int \frac {x^{7}}{x^{16}-9 x^{12}+6 x^{10}+80 x^{8}-108 x^{6}+54 x^{4}-12 x^{2}+1}d x \right )-15 \left (\int \frac {x^{5}}{x^{16}-9 x^{12}+6 x^{10}+80 x^{8}-108 x^{6}+54 x^{4}-12 x^{2}+1}d x \right )+2 \left (\int \frac {x^{3}}{x^{16}-9 x^{12}+6 x^{10}+80 x^{8}-108 x^{6}+54 x^{4}-12 x^{2}+1}d x \right ) \] Input:
int(x^3*(6*x^10-4*x^8-27*x^6+36*x^4-15*x^2+2)/(x^16-9*x^12+6*x^10+80*x^8-1 08*x^6+54*x^4-12*x^2+1),x)
Output:
6*int(x**13/(x**16 - 9*x**12 + 6*x**10 + 80*x**8 - 108*x**6 + 54*x**4 - 12 *x**2 + 1),x) - 4*int(x**11/(x**16 - 9*x**12 + 6*x**10 + 80*x**8 - 108*x** 6 + 54*x**4 - 12*x**2 + 1),x) - 27*int(x**9/(x**16 - 9*x**12 + 6*x**10 + 8 0*x**8 - 108*x**6 + 54*x**4 - 12*x**2 + 1),x) + 36*int(x**7/(x**16 - 9*x** 12 + 6*x**10 + 80*x**8 - 108*x**6 + 54*x**4 - 12*x**2 + 1),x) - 15*int(x** 5/(x**16 - 9*x**12 + 6*x**10 + 80*x**8 - 108*x**6 + 54*x**4 - 12*x**2 + 1) ,x) + 2*int(x**3/(x**16 - 9*x**12 + 6*x**10 + 80*x**8 - 108*x**6 + 54*x**4 - 12*x**2 + 1),x)