Integrand size = 171, antiderivative size = 177 \[ \int \frac {-16 \left (3-2 \sqrt {3}\right ) x-16 \left (15-10 \sqrt {3}\right ) x^3-16 \left (90+30 \sqrt {3}\right ) x^5+960 \sqrt {3} x^7+240 \sqrt {3} x^9-1440 \sqrt {3} x^{11}}{14-8 \sqrt {3}+\left (140-100 \sqrt {3}\right ) x^2+\left (1578+158 \sqrt {3}\right ) x^4+\left (440-820 \sqrt {3}\right ) x^6+\left (1682+1343 \sqrt {3}\right ) x^8+\left (-2580-540 \sqrt {3}\right ) x^{10}+\left (3132-468 \sqrt {3}\right ) x^{12}-2160 x^{14}+648 x^{16}} \, dx=\sqrt [8]{3} \arctan \left (\frac {1}{\sqrt [8]{3}}-3^{3/8} x\right )+\sqrt [8]{3} \arctan \left (\frac {1}{\sqrt [8]{3}}+3^{3/8} x\right )+\sqrt [8]{3} \arctan \left (\frac {1}{2} \sqrt {\frac {1}{3} \left (-6+\sqrt {6 \left (3+2 \sqrt {3}\right )}\right )}+\frac {x^2}{\sqrt [8]{3}}\right )+\sqrt [8]{3} \text {arctanh}\left (\frac {1}{\sqrt [8]{3}}-3^{3/8} x\right )+\sqrt [8]{3} \text {arctanh}\left (\frac {1}{\sqrt [8]{3}}+3^{3/8} x\right )-\sqrt [8]{3} \text {arctanh}\left (\frac {1}{2} \sqrt {\frac {1}{3} \left (6+\sqrt {6 \left (3+2 \sqrt {3}\right )}\right )}-\frac {x^2}{\sqrt [8]{3}}\right ) \] Output:
-3^(1/8)*arctan(-1/3*3^(7/8)+3^(3/8)*x)+3^(1/8)*arctan(1/3*3^(7/8)+3^(3/8) *x)+3^(1/8)*arctan(1/6*(-18+3*(18+12*3^(1/2))^(1/2))^(1/2)+1/3*x^2*3^(7/8) )-3^(1/8)*arctanh(-1/3*3^(7/8)+3^(3/8)*x)+3^(1/8)*arctanh(1/3*3^(7/8)+3^(3 /8)*x)+3^(1/8)*arctanh(-1/6*(18+3*(18+12*3^(1/2))^(1/2))^(1/2)+1/3*x^2*3^( 7/8))
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.86 (sec) , antiderivative size = 377, normalized size of antiderivative = 2.13 \[ \int \frac {-16 \left (3-2 \sqrt {3}\right ) x-16 \left (15-10 \sqrt {3}\right ) x^3-16 \left (90+30 \sqrt {3}\right ) x^5+960 \sqrt {3} x^7+240 \sqrt {3} x^9-1440 \sqrt {3} x^{11}}{14-8 \sqrt {3}+\left (140-100 \sqrt {3}\right ) x^2+\left (1578+158 \sqrt {3}\right ) x^4+\left (440-820 \sqrt {3}\right ) x^6+\left (1682+1343 \sqrt {3}\right ) x^8+\left (-2580-540 \sqrt {3}\right ) x^{10}+\left (3132-468 \sqrt {3}\right ) x^{12}-2160 x^{14}+648 x^{16}} \, dx=2 \text {RootSum}\left [-14+8 \sqrt {3}-140 \text {$\#$1}^2+100 \sqrt {3} \text {$\#$1}^2-1578 \text {$\#$1}^4-158 \sqrt {3} \text {$\#$1}^4-440 \text {$\#$1}^6+820 \sqrt {3} \text {$\#$1}^6-1682 \text {$\#$1}^8-1343 \sqrt {3} \text {$\#$1}^8+2580 \text {$\#$1}^{10}+540 \sqrt {3} \text {$\#$1}^{10}-3132 \text {$\#$1}^{12}+468 \sqrt {3} \text {$\#$1}^{12}+2160 \text {$\#$1}^{14}-648 \text {$\#$1}^{16}\&,\frac {3 \log (x-\text {$\#$1})-2 \sqrt {3} \log (x-\text {$\#$1})+15 \log (x-\text {$\#$1}) \text {$\#$1}^2-10 \sqrt {3} \log (x-\text {$\#$1}) \text {$\#$1}^2+90 \log (x-\text {$\#$1}) \text {$\#$1}^4+30 \sqrt {3} \log (x-\text {$\#$1}) \text {$\#$1}^4-60 \sqrt {3} \log (x-\text {$\#$1}) \text {$\#$1}^6-15 \sqrt {3} \log (x-\text {$\#$1}) \text {$\#$1}^8+90 \sqrt {3} \log (x-\text {$\#$1}) \text {$\#$1}^{10}}{-35+25 \sqrt {3}-789 \text {$\#$1}^2-79 \sqrt {3} \text {$\#$1}^2-330 \text {$\#$1}^4+615 \sqrt {3} \text {$\#$1}^4-1682 \text {$\#$1}^6-1343 \sqrt {3} \text {$\#$1}^6+3225 \text {$\#$1}^8+675 \sqrt {3} \text {$\#$1}^8-4698 \text {$\#$1}^{10}+702 \sqrt {3} \text {$\#$1}^{10}+3780 \text {$\#$1}^{12}-1296 \text {$\#$1}^{14}}\&\right ] \] Input:
Integrate[(-16*(3 - 2*Sqrt[3])*x - 16*(15 - 10*Sqrt[3])*x^3 - 16*(90 + 30* Sqrt[3])*x^5 + 960*Sqrt[3]*x^7 + 240*Sqrt[3]*x^9 - 1440*Sqrt[3]*x^11)/(14 - 8*Sqrt[3] + (140 - 100*Sqrt[3])*x^2 + (1578 + 158*Sqrt[3])*x^4 + (440 - 820*Sqrt[3])*x^6 + (1682 + 1343*Sqrt[3])*x^8 + (-2580 - 540*Sqrt[3])*x^10 + (3132 - 468*Sqrt[3])*x^12 - 2160*x^14 + 648*x^16),x]
Output:
2*RootSum[-14 + 8*Sqrt[3] - 140*#1^2 + 100*Sqrt[3]*#1^2 - 1578*#1^4 - 158* Sqrt[3]*#1^4 - 440*#1^6 + 820*Sqrt[3]*#1^6 - 1682*#1^8 - 1343*Sqrt[3]*#1^8 + 2580*#1^10 + 540*Sqrt[3]*#1^10 - 3132*#1^12 + 468*Sqrt[3]*#1^12 + 2160* #1^14 - 648*#1^16 & , (3*Log[x - #1] - 2*Sqrt[3]*Log[x - #1] + 15*Log[x - #1]*#1^2 - 10*Sqrt[3]*Log[x - #1]*#1^2 + 90*Log[x - #1]*#1^4 + 30*Sqrt[3]* Log[x - #1]*#1^4 - 60*Sqrt[3]*Log[x - #1]*#1^6 - 15*Sqrt[3]*Log[x - #1]*#1 ^8 + 90*Sqrt[3]*Log[x - #1]*#1^10)/(-35 + 25*Sqrt[3] - 789*#1^2 - 79*Sqrt[ 3]*#1^2 - 330*#1^4 + 615*Sqrt[3]*#1^4 - 1682*#1^6 - 1343*Sqrt[3]*#1^6 + 32 25*#1^8 + 675*Sqrt[3]*#1^8 - 4698*#1^10 + 702*Sqrt[3]*#1^10 + 3780*#1^12 - 1296*#1^14) & ]
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.
| \(\displaystyle \int \frac {-1440 \sqrt {3} x^{11}+240 \sqrt {3} x^9+960 \sqrt {3} x^7-16 \left (90+30 \sqrt {3}\right ) x^5-16 \left (15-10 \sqrt {3}\right ) x^3-16 \left (3-2 \sqrt {3}\right ) x}{648 x^{16}-2160 x^{14}+\left (3132-468 \sqrt {3}\right ) x^{12}+\left (-2580-540 \sqrt {3}\right ) x^{10}+\left (1682+1343 \sqrt {3}\right ) x^8+\left (440-820 \sqrt {3}\right ) x^6+\left (1578+158 \sqrt {3}\right ) x^4+\left (140-100 \sqrt {3}\right ) x^2-8 \sqrt {3}+14} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {16 x \left (-90 \sqrt {3} x^{10}+15 \sqrt {3} x^8+60 \sqrt {3} x^6-90 \left (1+\frac {1}{\sqrt {3}}\right ) x^4-15 \left (1-\frac {2}{\sqrt {3}}\right ) x^2-3 \left (1-\frac {2}{\sqrt {3}}\right )\right )}{648 x^{16}-2160 x^{14}+\left (3132-468 \sqrt {3}\right ) x^{12}+\left (-2580-540 \sqrt {3}\right ) x^{10}+\left (1682+1343 \sqrt {3}\right ) x^8+\left (440-820 \sqrt {3}\right ) x^6+\left (1578+158 \sqrt {3}\right ) x^4+\left (140-100 \sqrt {3}\right ) x^2+14 \left (1-\frac {4 \sqrt {3}}{7}\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 16 \int -\frac {x \left (90 \sqrt {3} x^{10}-15 \sqrt {3} x^8-60 \sqrt {3} x^6+30 \left (3+\sqrt {3}\right ) x^4+5 \left (3-2 \sqrt {3}\right ) x^2-2 \sqrt {3}+3\right )}{648 x^{16}-2160 x^{14}+36 \left (87-13 \sqrt {3}\right ) x^{12}-60 \left (43+9 \sqrt {3}\right ) x^{10}+\left (1682+1343 \sqrt {3}\right ) x^8+20 \left (22-41 \sqrt {3}\right ) x^6+2 \left (789+79 \sqrt {3}\right ) x^4+20 \left (7-5 \sqrt {3}\right ) x^2+2 \left (7-4 \sqrt {3}\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -16 \int \frac {x \left (90 \sqrt {3} x^{10}-15 \sqrt {3} x^8-60 \sqrt {3} x^6+30 \left (3+\sqrt {3}\right ) x^4+5 \left (3-2 \sqrt {3}\right ) x^2-2 \sqrt {3}+3\right )}{648 x^{16}-2160 x^{14}+36 \left (87-13 \sqrt {3}\right ) x^{12}-60 \left (43+9 \sqrt {3}\right ) x^{10}+\left (1682+1343 \sqrt {3}\right ) x^8+20 \left (22-41 \sqrt {3}\right ) x^6+2 \left (789+79 \sqrt {3}\right ) x^4+20 \left (7-5 \sqrt {3}\right ) x^2+2 \left (7-4 \sqrt {3}\right )}dx\) |
| \(\Big \downarrow \) 7266 |
| \(\displaystyle -8 \int \frac {90 \sqrt {3} x^{10}-15 \sqrt {3} x^8-60 \sqrt {3} x^6+30 \left (3+\sqrt {3}\right ) x^4+5 \left (3-2 \sqrt {3}\right ) x^2-2 \sqrt {3}+3}{648 x^{16}-2160 x^{14}+36 \left (87-13 \sqrt {3}\right ) x^{12}-60 \left (43+9 \sqrt {3}\right ) x^{10}+\left (1682+1343 \sqrt {3}\right ) x^8+20 \left (22-41 \sqrt {3}\right ) x^6+2 \left (789+79 \sqrt {3}\right ) x^4+20 \left (7-5 \sqrt {3}\right ) x^2+2 \left (7-4 \sqrt {3}\right )}dx^2\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -8 \int \frac {90 \sqrt {3} x^{10}-15 \sqrt {3} x^8-60 \sqrt {3} x^6+30 \left (3+\sqrt {3}\right ) x^4+5 \left (3-2 \sqrt {3}\right ) x^2+3 \left (1-\frac {2}{\sqrt {3}}\right )}{648 x^{16}-2160 x^{14}+36 \left (87-13 \sqrt {3}\right ) x^{12}-60 \left (43+9 \sqrt {3}\right ) x^{10}+\left (1682+1343 \sqrt {3}\right ) x^8+20 \left (22-41 \sqrt {3}\right ) x^6+2 \left (789+79 \sqrt {3}\right ) x^4+20 \left (7-5 \sqrt {3}\right ) x^2+2 \left (7-4 \sqrt {3}\right )}dx^2\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -8 \int \left (\frac {90 \sqrt {3} x^{10}}{648 x^{16}-2160 x^{14}+3132 \left (1-\frac {13}{29 \sqrt {3}}\right ) x^{12}-2580 \left (1+\frac {9 \sqrt {3}}{43}\right ) x^{10}+1682 \left (1+\frac {1343 \sqrt {3}}{1682}\right ) x^8+440 \left (1-\frac {41 \sqrt {3}}{22}\right ) x^6+1578 \left (1+\frac {79}{263 \sqrt {3}}\right ) x^4+140 \left (1-\frac {5 \sqrt {3}}{7}\right ) x^2+14 \left (1-\frac {4 \sqrt {3}}{7}\right )}+\frac {15 \sqrt {3} x^8}{-648 x^{16}+2160 x^{14}-3132 \left (1-\frac {13}{29 \sqrt {3}}\right ) x^{12}+2580 \left (1+\frac {9 \sqrt {3}}{43}\right ) x^{10}-1682 \left (1+\frac {1343 \sqrt {3}}{1682}\right ) x^8-440 \left (1-\frac {41 \sqrt {3}}{22}\right ) x^6-1578 \left (1+\frac {79}{263 \sqrt {3}}\right ) x^4-140 \left (1-\frac {5 \sqrt {3}}{7}\right ) x^2-14 \left (1-\frac {4 \sqrt {3}}{7}\right )}+\frac {60 \sqrt {3} x^6}{-648 x^{16}+2160 x^{14}-3132 \left (1-\frac {13}{29 \sqrt {3}}\right ) x^{12}+2580 \left (1+\frac {9 \sqrt {3}}{43}\right ) x^{10}-1682 \left (1+\frac {1343 \sqrt {3}}{1682}\right ) x^8-440 \left (1-\frac {41 \sqrt {3}}{22}\right ) x^6-1578 \left (1+\frac {79}{263 \sqrt {3}}\right ) x^4-140 \left (1-\frac {5 \sqrt {3}}{7}\right ) x^2-14 \left (1-\frac {4 \sqrt {3}}{7}\right )}+\frac {30 \left (3+\sqrt {3}\right ) x^4}{648 x^{16}-2160 x^{14}+3132 \left (1-\frac {13}{29 \sqrt {3}}\right ) x^{12}-2580 \left (1+\frac {9 \sqrt {3}}{43}\right ) x^{10}+1682 \left (1+\frac {1343 \sqrt {3}}{1682}\right ) x^8+440 \left (1-\frac {41 \sqrt {3}}{22}\right ) x^6+1578 \left (1+\frac {79}{263 \sqrt {3}}\right ) x^4+140 \left (1-\frac {5 \sqrt {3}}{7}\right ) x^2+14 \left (1-\frac {4 \sqrt {3}}{7}\right )}+\frac {5 \left (3-2 \sqrt {3}\right ) x^2}{648 x^{16}-2160 x^{14}+3132 \left (1-\frac {13}{29 \sqrt {3}}\right ) x^{12}-2580 \left (1+\frac {9 \sqrt {3}}{43}\right ) x^{10}+1682 \left (1+\frac {1343 \sqrt {3}}{1682}\right ) x^8+440 \left (1-\frac {41 \sqrt {3}}{22}\right ) x^6+1578 \left (1+\frac {79}{263 \sqrt {3}}\right ) x^4+140 \left (1-\frac {5 \sqrt {3}}{7}\right ) x^2+14 \left (1-\frac {4 \sqrt {3}}{7}\right )}+\frac {3-2 \sqrt {3}}{648 x^{16}-2160 x^{14}+3132 \left (1-\frac {13}{29 \sqrt {3}}\right ) x^{12}-2580 \left (1+\frac {9 \sqrt {3}}{43}\right ) x^{10}+1682 \left (1+\frac {1343 \sqrt {3}}{1682}\right ) x^8+440 \left (1-\frac {41 \sqrt {3}}{22}\right ) x^6+1578 \left (1+\frac {79}{263 \sqrt {3}}\right ) x^4+140 \left (1-\frac {5 \sqrt {3}}{7}\right ) x^2+14 \left (1-\frac {4 \sqrt {3}}{7}\right )}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -8 \left (60 \sqrt {3} \int \frac {x^6}{-648 x^{16}+2160 x^{14}-3132 \left (1-\frac {13}{29 \sqrt {3}}\right ) x^{12}+2580 \left (1+\frac {9 \sqrt {3}}{43}\right ) x^{10}-1682 \left (1+\frac {1343 \sqrt {3}}{1682}\right ) x^8-440 \left (1-\frac {41 \sqrt {3}}{22}\right ) x^6-1578 \left (1+\frac {79}{263 \sqrt {3}}\right ) x^4-140 \left (1-\frac {5 \sqrt {3}}{7}\right ) x^2-14 \left (1-\frac {4 \sqrt {3}}{7}\right )}dx^2+15 \sqrt {3} \int \frac {x^8}{-648 x^{16}+2160 x^{14}-3132 \left (1-\frac {13}{29 \sqrt {3}}\right ) x^{12}+2580 \left (1+\frac {9 \sqrt {3}}{43}\right ) x^{10}-1682 \left (1+\frac {1343 \sqrt {3}}{1682}\right ) x^8-440 \left (1-\frac {41 \sqrt {3}}{22}\right ) x^6-1578 \left (1+\frac {79}{263 \sqrt {3}}\right ) x^4-140 \left (1-\frac {5 \sqrt {3}}{7}\right ) x^2-14 \left (1-\frac {4 \sqrt {3}}{7}\right )}dx^2+\left (3-2 \sqrt {3}\right ) \int \frac {1}{648 x^{16}-2160 x^{14}+3132 \left (1-\frac {13}{29 \sqrt {3}}\right ) x^{12}-2580 \left (1+\frac {9 \sqrt {3}}{43}\right ) x^{10}+1682 \left (1+\frac {1343 \sqrt {3}}{1682}\right ) x^8+440 \left (1-\frac {41 \sqrt {3}}{22}\right ) x^6+1578 \left (1+\frac {79}{263 \sqrt {3}}\right ) x^4+140 \left (1-\frac {5 \sqrt {3}}{7}\right ) x^2+14 \left (1-\frac {4 \sqrt {3}}{7}\right )}dx^2+5 \left (3-2 \sqrt {3}\right ) \int \frac {x^2}{648 x^{16}-2160 x^{14}+3132 \left (1-\frac {13}{29 \sqrt {3}}\right ) x^{12}-2580 \left (1+\frac {9 \sqrt {3}}{43}\right ) x^{10}+1682 \left (1+\frac {1343 \sqrt {3}}{1682}\right ) x^8+440 \left (1-\frac {41 \sqrt {3}}{22}\right ) x^6+1578 \left (1+\frac {79}{263 \sqrt {3}}\right ) x^4+140 \left (1-\frac {5 \sqrt {3}}{7}\right ) x^2+14 \left (1-\frac {4 \sqrt {3}}{7}\right )}dx^2+30 \left (3+\sqrt {3}\right ) \int \frac {x^4}{648 x^{16}-2160 x^{14}+3132 \left (1-\frac {13}{29 \sqrt {3}}\right ) x^{12}-2580 \left (1+\frac {9 \sqrt {3}}{43}\right ) x^{10}+1682 \left (1+\frac {1343 \sqrt {3}}{1682}\right ) x^8+440 \left (1-\frac {41 \sqrt {3}}{22}\right ) x^6+1578 \left (1+\frac {79}{263 \sqrt {3}}\right ) x^4+140 \left (1-\frac {5 \sqrt {3}}{7}\right ) x^2+14 \left (1-\frac {4 \sqrt {3}}{7}\right )}dx^2+90 \sqrt {3} \int \frac {x^{10}}{648 x^{16}-2160 x^{14}+3132 \left (1-\frac {13}{29 \sqrt {3}}\right ) x^{12}-2580 \left (1+\frac {9 \sqrt {3}}{43}\right ) x^{10}+1682 \left (1+\frac {1343 \sqrt {3}}{1682}\right ) x^8+440 \left (1-\frac {41 \sqrt {3}}{22}\right ) x^6+1578 \left (1+\frac {79}{263 \sqrt {3}}\right ) x^4+140 \left (1-\frac {5 \sqrt {3}}{7}\right ) x^2+14 \left (1-\frac {4 \sqrt {3}}{7}\right )}dx^2\right )\) |
Input:
Int[(-16*(3 - 2*Sqrt[3])*x - 16*(15 - 10*Sqrt[3])*x^3 - 16*(90 + 30*Sqrt[3 ])*x^5 + 960*Sqrt[3]*x^7 + 240*Sqrt[3]*x^9 - 1440*Sqrt[3]*x^11)/(14 - 8*Sq rt[3] + (140 - 100*Sqrt[3])*x^2 + (1578 + 158*Sqrt[3])*x^4 + (440 - 820*Sq rt[3])*x^6 + (1682 + 1343*Sqrt[3])*x^8 + (-2580 - 540*Sqrt[3])*x^10 + (313 2 - 468*Sqrt[3])*x^12 - 2160*x^14 + 648*x^16),x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.17 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.98
| method | result | size |
| default | \(2 \sqrt {3}\, \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (8 \textit {\_Z}^{4}-16 \textit {\_Z}^{3}+\left (-4 \sqrt {3}+12\right ) \textit {\_Z}^{2}+\left (-12 \sqrt {3}-4\right ) \textit {\_Z} +2-\sqrt {3}\right )}{\sum }\frac {\left (1+2 \textit {\_R} \right ) \ln \left (x^{2}-\textit {\_R} \right )}{-8 \textit {\_R}^{3}+12 \textit {\_R}^{2}-6 \textit {\_R} +1+\sqrt {3}\, \left (3+2 \textit {\_R} \right )}\right )-2 \sqrt {3}\, \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (81 \textit {\_Z}^{4}-108 \textit {\_Z}^{3}+\left (-18 \sqrt {3}+54\right ) \textit {\_Z}^{2}+\left (-36 \sqrt {3}-12\right ) \textit {\_Z} -2 \sqrt {3}+4\right )}{\sum }\frac {\left (3 \textit {\_R} +1\right ) \ln \left (x^{2}-\textit {\_R} \right )}{-27 \textit {\_R}^{3}+27 \textit {\_R}^{2}-9 \textit {\_R} +1+3 \sqrt {3}\, \left (1+\textit {\_R} \right )}\right )\) | \(173\) |
Input:
int((-16*(3-2*3^(1/2))*x-16*(15-10*3^(1/2))*x^3-16*(90+30*3^(1/2))*x^5+960 *3^(1/2)*x^7+240*3^(1/2)*x^9-1440*3^(1/2)*x^11)/(14-8*3^(1/2)+(140-100*3^( 1/2))*x^2+(1578+158*3^(1/2))*x^4+(440-820*3^(1/2))*x^6+(1682+1343*3^(1/2)) *x^8+(-2580-540*3^(1/2))*x^10+(3132-468*3^(1/2))*x^12-2160*x^14+648*x^16), x,method=_RETURNVERBOSE)
Output:
2*3^(1/2)*sum((1+2*_R)/(-8*_R^3+12*_R^2-6*_R+1+3^(1/2)*(3+2*_R))*ln(x^2-_R ),_R=RootOf(8*_Z^4-16*_Z^3+(-4*3^(1/2)+12)*_Z^2+(-12*3^(1/2)-4)*_Z+2-3^(1/ 2)))-2*3^(1/2)*sum((3*_R+1)/(-27*_R^3+27*_R^2-9*_R+1+3*3^(1/2)*(1+_R))*ln( x^2-_R),_R=RootOf(81*_Z^4-108*_Z^3+(-18*3^(1/2)+54)*_Z^2+(-36*3^(1/2)-12)* _Z-2*3^(1/2)+4))
Time = 0.39 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.12 \[ \int \frac {-16 \left (3-2 \sqrt {3}\right ) x-16 \left (15-10 \sqrt {3}\right ) x^3-16 \left (90+30 \sqrt {3}\right ) x^5+960 \sqrt {3} x^7+240 \sqrt {3} x^9-1440 \sqrt {3} x^{11}}{14-8 \sqrt {3}+\left (140-100 \sqrt {3}\right ) x^2+\left (1578+158 \sqrt {3}\right ) x^4+\left (440-820 \sqrt {3}\right ) x^6+\left (1682+1343 \sqrt {3}\right ) x^8+\left (-2580-540 \sqrt {3}\right ) x^{10}+\left (3132-468 \sqrt {3}\right ) x^{12}-2160 x^{14}+648 x^{16}} \, dx=3^{\frac {1}{8}} \arctan \left (\frac {1}{6} \cdot 3^{\frac {7}{8}} {\left (6 \, x^{2} - 5\right )} + \frac {5}{2} \cdot 3^{\frac {1}{8}}\right ) - 3^{\frac {1}{8}} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{8}} {\left (162 \, x^{6} - 630 \, x^{4} + 576 \, x^{2} + \sqrt {3} {\left (90 \, x^{6} - 330 \, x^{4} + 319 \, x^{2} - 5\right )}\right )} - \frac {1}{2} \cdot 3^{\frac {1}{8}} {\left (72 \, x^{6} - 270 \, x^{4} + 240 \, x^{2} + 6 \, \sqrt {3} {\left (6 \, x^{6} - 25 \, x^{4} + 24 \, x^{2}\right )} - 5\right )}\right ) + \frac {1}{2} \cdot 3^{\frac {1}{8}} \log \left (6 \, x^{4} + 2 \cdot 3^{\frac {1}{8}} x^{2} - 5 \, x^{2} - 3^{\frac {1}{4}} {\left (5 \, x^{2} + 2\right )} + \sqrt {3} + 1\right ) - \frac {1}{2} \cdot 3^{\frac {1}{8}} \log \left (6 \, x^{4} - 2 \cdot 3^{\frac {1}{8}} x^{2} - 5 \, x^{2} - 3^{\frac {1}{4}} {\left (5 \, x^{2} + 2\right )} + \sqrt {3} + 1\right ) \] Input:
integrate((-16*(3-2*3^(1/2))*x-16*(15-10*3^(1/2))*x^3-16*(90+30*3^(1/2))*x ^5+960*3^(1/2)*x^7+240*3^(1/2)*x^9-1440*3^(1/2)*x^11)/(14-8*3^(1/2)+(140-1 00*3^(1/2))*x^2+(1578+158*3^(1/2))*x^4+(440-820*3^(1/2))*x^6+(1682+1343*3^ (1/2))*x^8+(-2580-540*3^(1/2))*x^10+(3132-468*3^(1/2))*x^12-2160*x^14+648* x^16),x, algorithm="fricas")
Output:
3^(1/8)*arctan(1/6*3^(7/8)*(6*x^2 - 5) + 5/2*3^(1/8)) - 3^(1/8)*arctan(1/6 *3^(3/8)*(162*x^6 - 630*x^4 + 576*x^2 + sqrt(3)*(90*x^6 - 330*x^4 + 319*x^ 2 - 5)) - 1/2*3^(1/8)*(72*x^6 - 270*x^4 + 240*x^2 + 6*sqrt(3)*(6*x^6 - 25* x^4 + 24*x^2) - 5)) + 1/2*3^(1/8)*log(6*x^4 + 2*3^(1/8)*x^2 - 5*x^2 - 3^(1 /4)*(5*x^2 + 2) + sqrt(3) + 1) - 1/2*3^(1/8)*log(6*x^4 - 2*3^(1/8)*x^2 - 5 *x^2 - 3^(1/4)*(5*x^2 + 2) + sqrt(3) + 1)
Timed out. \[ \int \frac {-16 \left (3-2 \sqrt {3}\right ) x-16 \left (15-10 \sqrt {3}\right ) x^3-16 \left (90+30 \sqrt {3}\right ) x^5+960 \sqrt {3} x^7+240 \sqrt {3} x^9-1440 \sqrt {3} x^{11}}{14-8 \sqrt {3}+\left (140-100 \sqrt {3}\right ) x^2+\left (1578+158 \sqrt {3}\right ) x^4+\left (440-820 \sqrt {3}\right ) x^6+\left (1682+1343 \sqrt {3}\right ) x^8+\left (-2580-540 \sqrt {3}\right ) x^{10}+\left (3132-468 \sqrt {3}\right ) x^{12}-2160 x^{14}+648 x^{16}} \, dx=\text {Timed out} \] Input:
integrate((-16*(3-2*3**(1/2))*x-16*(15-10*3**(1/2))*x**3-16*(90+30*3**(1/2 ))*x**5+960*3**(1/2)*x**7+240*3**(1/2)*x**9-1440*3**(1/2)*x**11)/(14-8*3** (1/2)+(140-100*3**(1/2))*x**2+(1578+158*3**(1/2))*x**4+(440-820*3**(1/2))* x**6+(1682+1343*3**(1/2))*x**8+(-2580-540*3**(1/2))*x**10+(3132-468*3**(1/ 2))*x**12-2160*x**14+648*x**16),x)
Output:
Timed out
\[ \int \frac {-16 \left (3-2 \sqrt {3}\right ) x-16 \left (15-10 \sqrt {3}\right ) x^3-16 \left (90+30 \sqrt {3}\right ) x^5+960 \sqrt {3} x^7+240 \sqrt {3} x^9-1440 \sqrt {3} x^{11}}{14-8 \sqrt {3}+\left (140-100 \sqrt {3}\right ) x^2+\left (1578+158 \sqrt {3}\right ) x^4+\left (440-820 \sqrt {3}\right ) x^6+\left (1682+1343 \sqrt {3}\right ) x^8+\left (-2580-540 \sqrt {3}\right ) x^{10}+\left (3132-468 \sqrt {3}\right ) x^{12}-2160 x^{14}+648 x^{16}} \, dx=\int { -\frac {16 \, {\left (90 \, \sqrt {3} x^{11} - 15 \, \sqrt {3} x^{9} - 60 \, \sqrt {3} x^{7} + 30 \, x^{5} {\left (\sqrt {3} + 3\right )} - 5 \, x^{3} {\left (2 \, \sqrt {3} - 3\right )} - x {\left (2 \, \sqrt {3} - 3\right )}\right )}}{648 \, x^{16} - 2160 \, x^{14} - 36 \, x^{12} {\left (13 \, \sqrt {3} - 87\right )} - 60 \, x^{10} {\left (9 \, \sqrt {3} + 43\right )} + x^{8} {\left (1343 \, \sqrt {3} + 1682\right )} - 20 \, x^{6} {\left (41 \, \sqrt {3} - 22\right )} + 2 \, x^{4} {\left (79 \, \sqrt {3} + 789\right )} - 20 \, x^{2} {\left (5 \, \sqrt {3} - 7\right )} - 8 \, \sqrt {3} + 14} \,d x } \] Input:
integrate((-16*(3-2*3^(1/2))*x-16*(15-10*3^(1/2))*x^3-16*(90+30*3^(1/2))*x ^5+960*3^(1/2)*x^7+240*3^(1/2)*x^9-1440*3^(1/2)*x^11)/(14-8*3^(1/2)+(140-1 00*3^(1/2))*x^2+(1578+158*3^(1/2))*x^4+(440-820*3^(1/2))*x^6+(1682+1343*3^ (1/2))*x^8+(-2580-540*3^(1/2))*x^10+(3132-468*3^(1/2))*x^12-2160*x^14+648* x^16),x, algorithm="maxima")
Output:
-16*integrate((90*sqrt(3)*x^11 - 15*sqrt(3)*x^9 - 60*sqrt(3)*x^7 + 30*x^5* (sqrt(3) + 3) - 5*x^3*(2*sqrt(3) - 3) - x*(2*sqrt(3) - 3))/(648*x^16 - 216 0*x^14 - 36*x^12*(13*sqrt(3) - 87) - 60*x^10*(9*sqrt(3) + 43) + x^8*(1343* sqrt(3) + 1682) - 20*x^6*(41*sqrt(3) - 22) + 2*x^4*(79*sqrt(3) + 789) - 20 *x^2*(5*sqrt(3) - 7) - 8*sqrt(3) + 14), x)
Exception generated. \[ \int \frac {-16 \left (3-2 \sqrt {3}\right ) x-16 \left (15-10 \sqrt {3}\right ) x^3-16 \left (90+30 \sqrt {3}\right ) x^5+960 \sqrt {3} x^7+240 \sqrt {3} x^9-1440 \sqrt {3} x^{11}}{14-8 \sqrt {3}+\left (140-100 \sqrt {3}\right ) x^2+\left (1578+158 \sqrt {3}\right ) x^4+\left (440-820 \sqrt {3}\right ) x^6+\left (1682+1343 \sqrt {3}\right ) x^8+\left (-2580-540 \sqrt {3}\right ) x^{10}+\left (3132-468 \sqrt {3}\right ) x^{12}-2160 x^{14}+648 x^{16}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-16*(3-2*3^(1/2))*x-16*(15-10*3^(1/2))*x^3-16*(90+30*3^(1/2))*x ^5+960*3^(1/2)*x^7+240*3^(1/2)*x^9-1440*3^(1/2)*x^11)/(14-8*3^(1/2)+(140-1 00*3^(1/2))*x^2+(1578+158*3^(1/2))*x^4+(440-820*3^(1/2))*x^6+(1682+1343*3^ (1/2))*x^8+(-2580-540*3^(1/2))*x^10+(3132-468*3^(1/2))*x^12-2160*x^14+648* x^16),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Francis algorithm failure for[infin ity,0.0,infinity,infinity,infinity]proot error [undef,0.0,undef,undef,unde f]proot e
Time = 20.52 (sec) , antiderivative size = 837, normalized size of antiderivative = 4.73 \[ \int \frac {-16 \left (3-2 \sqrt {3}\right ) x-16 \left (15-10 \sqrt {3}\right ) x^3-16 \left (90+30 \sqrt {3}\right ) x^5+960 \sqrt {3} x^7+240 \sqrt {3} x^9-1440 \sqrt {3} x^{11}}{14-8 \sqrt {3}+\left (140-100 \sqrt {3}\right ) x^2+\left (1578+158 \sqrt {3}\right ) x^4+\left (440-820 \sqrt {3}\right ) x^6+\left (1682+1343 \sqrt {3}\right ) x^8+\left (-2580-540 \sqrt {3}\right ) x^{10}+\left (3132-468 \sqrt {3}\right ) x^{12}-2160 x^{14}+648 x^{16}} \, dx=\text {Too large to display} \] Input:
int(-(16*x^3*(10*3^(1/2) - 15) - 16*x^5*(30*3^(1/2) + 90) + 960*3^(1/2)*x^ 7 + 240*3^(1/2)*x^9 - 1440*3^(1/2)*x^11 + 16*x*(2*3^(1/2) - 3))/(x^2*(100* 3^(1/2) - 140) + x^6*(820*3^(1/2) - 440) - x^4*(158*3^(1/2) + 1578) - x^8* (1343*3^(1/2) + 1682) + x^10*(540*3^(1/2) + 2580) + x^12*(468*3^(1/2) - 31 32) + 8*3^(1/2) + 2160*x^14 - 648*x^16 - 14),x)
Output:
symsum(log(root(185290476904455051041625341952*3^(1/2)*z^8 + 4542967087543 22101063029817344*z^8 - 56787088594290262632878727168*3^(1/2)*z^4 - 694839 28839170644140609503232*z^4 + 2171372776224082629394046976*3^(1/2) + 53237 89555714712121832380672, z, k)*(x^2*((116452365523047784926105240954688839 1307053779667*3^(1/2))/320403904639134924968393214218988672 + 134082915512 7388744734478240447187948901339593923/213602603092756616645595476145992448 ) + root(185290476904455051041625341952*3^(1/2)*z^8 + 45429670875432210106 3029817344*z^8 - 56787088594290262632878727168*3^(1/2)*z^4 - 6948392883917 0644140609503232*z^4 + 2171372776224082629394046976*3^(1/2) + 532378955571 4712121832380672, z, k)*((14814514368105567951390204291701229791620991467* 3^(1/2))/53400650773189154161398869036498112 - x^2*((137301582705476336508 19936627817610377868308256655*3^(1/2))/42720520618551323329119095229198489 6 + 71331809161667252793264507457982328197470692629905/1281615618556539699 873572856875954688) + root(185290476904455051041625341952*3^(1/2)*z^8 + 45 4296708754322101063029817344*z^8 - 56787088594290262632878727168*3^(1/2)*z ^4 - 69483928839170644140609503232*z^4 + 2171372776224082629394046976*3^(1 /2) + 5323789555714712121832380672, z, k)*((490612725533053311932446343021 52037598418395*3^(1/2))/20025244039945932810524575888686792 - root(1852904 76904455051041625341952*3^(1/2)*z^8 + 454296708754322101063029817344*z^8 - 56787088594290262632878727168*3^(1/2)*z^4 - 69483928839170644140609503...
\[ \int \frac {-16 \left (3-2 \sqrt {3}\right ) x-16 \left (15-10 \sqrt {3}\right ) x^3-16 \left (90+30 \sqrt {3}\right ) x^5+960 \sqrt {3} x^7+240 \sqrt {3} x^9-1440 \sqrt {3} x^{11}}{14-8 \sqrt {3}+\left (140-100 \sqrt {3}\right ) x^2+\left (1578+158 \sqrt {3}\right ) x^4+\left (440-820 \sqrt {3}\right ) x^6+\left (1682+1343 \sqrt {3}\right ) x^8+\left (-2580-540 \sqrt {3}\right ) x^{10}+\left (3132-468 \sqrt {3}\right ) x^{12}-2160 x^{14}+648 x^{16}} \, dx=\text {too large to display} \] Input:
int((-16*(3-2*3^(1/2))*x-16*(15-10*3^(1/2))*x^3-16*(90+30*3^(1/2))*x^5+960 *3^(1/2)*x^7+240*3^(1/2)*x^9-1440*3^(1/2)*x^11)/(14-8*3^(1/2)+(140-100*3^( 1/2))*x^2+(1578+158*3^(1/2))*x^4+(440-820*3^(1/2))*x^6+(1682+1343*3^(1/2)) *x^8+(-2580-540*3^(1/2))*x^10+(3132-468*3^(1/2))*x^12-2160*x^14+648*x^16), x)
Output:
16*( - 58320*sqrt(3)*int(x**27/(419904*x**32 - 2799360*x**30 + 8724672*x** 28 - 16873920*x**26 + 22477824*x**24 - 24373440*x**22 + 20233080*x**20 - 1 0509720*x**18 + 2232577*x**16 + 992840*x**14 + 1230460*x**12 + 3344600*x** 10 + 2157952*x**8 + 509600*x**6 + 41368*x**4 - 880*x**2 + 4),x) + 204120*s qrt(3)*int(x**25/(419904*x**32 - 2799360*x**30 + 8724672*x**28 - 16873920* x**26 + 22477824*x**24 - 24373440*x**22 + 20233080*x**20 - 10509720*x**18 + 2232577*x**16 + 992840*x**14 + 1230460*x**12 + 3344600*x**10 + 2157952*x **8 + 509600*x**6 + 41368*x**4 - 880*x**2 + 4),x) - 275400*sqrt(3)*int(x** 23/(419904*x**32 - 2799360*x**30 + 8724672*x**28 - 16873920*x**26 + 224778 24*x**24 - 24373440*x**22 + 20233080*x**20 - 10509720*x**18 + 2232577*x**1 6 + 992840*x**14 + 1230460*x**12 + 3344600*x**10 + 2157952*x**8 + 509600*x **6 + 41368*x**4 - 880*x**2 + 4),x) + 130140*sqrt(3)*int(x**21/(419904*x** 32 - 2799360*x**30 + 8724672*x**28 - 16873920*x**26 + 22477824*x**24 - 243 73440*x**22 + 20233080*x**20 - 10509720*x**18 + 2232577*x**16 + 992840*x** 14 + 1230460*x**12 + 3344600*x**10 + 2157952*x**8 + 509600*x**6 + 41368*x* *4 - 880*x**2 + 4),x) + 69120*sqrt(3)*int(x**19/(419904*x**32 - 2799360*x* *30 + 8724672*x**28 - 16873920*x**26 + 22477824*x**24 - 24373440*x**22 + 2 0233080*x**20 - 10509720*x**18 + 2232577*x**16 + 992840*x**14 + 1230460*x* *12 + 3344600*x**10 + 2157952*x**8 + 509600*x**6 + 41368*x**4 - 880*x**2 + 4),x) - 325554*sqrt(3)*int(x**17/(419904*x**32 - 2799360*x**30 + 87246...