Integrand size = 171, antiderivative size = 194 \[ \int \frac {-80 \left (-3+2 \sqrt {3}\right ) x-80 \left (3-2 \sqrt {3}\right ) x^3-80 \left (90+30 \sqrt {3}\right ) x^5-960 \sqrt {3} x^7-7440 \sqrt {3} x^9-1440 \sqrt {3} x^{11}}{14-8 \sqrt {3}+\left (-28+20 \sqrt {3}\right ) x^2+\left (-1110-418 \sqrt {3}\right ) x^4+\left (296-268 \sqrt {3}\right ) x^6+\left (434-2401 \sqrt {3}\right ) x^8+\left (204-828 \sqrt {3}\right ) x^{10}+\left (-324-468 \sqrt {3}\right ) x^{12}-432 x^{14}+648 x^{16}} \, dx=\sqrt [8]{3} \arctan \left (\frac {1}{2} \sqrt {\frac {1}{3} \left (-6+\sqrt {6 \left (3+2 \sqrt {3}\right )}\right )}+\frac {x^2}{\sqrt [8]{3}}\right )-\sqrt [8]{3} \arctan \left (\frac {1}{2} \sqrt {\frac {1}{3} \left (-6+\sqrt {6 \left (3+2 \sqrt {3}\right )}\right )}-\frac {1}{2} 3^{7/8} x^2\right )-\sqrt [8]{3} \text {arctanh}\left (\frac {1}{2} \sqrt {\frac {1}{3} \left (6+\sqrt {6 \left (3+2 \sqrt {3}\right )}\right )}-\frac {x^2}{\sqrt [8]{3}}\right )+\sqrt [8]{3} \text {arctanh}\left (\frac {1}{2} \sqrt {\frac {1}{3} \left (6+\sqrt {6 \left (3+2 \sqrt {3}\right )}\right )}+\frac {1}{2} 3^{7/8} x^2\right ) \] Output:
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)*arctan(-1/6*(-18+3*(18+12*3^(1/2))^(1/2))^(1/2)+1/2*x^2*3^(7/8))+3^( 1/8)*arctanh(-1/6*(18+3*(18+12*3^(1/2))^(1/2))^(1/2)+1/3*x^2*3^(7/8))+3^(1 /8)*arctanh(1/6*(18+3*(18+12*3^(1/2))^(1/2))^(1/2)+1/2*x^2*3^(7/8))
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.85 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.94 \[ \int \frac {-80 \left (-3+2 \sqrt {3}\right ) x-80 \left (3-2 \sqrt {3}\right ) x^3-80 \left (90+30 \sqrt {3}\right ) x^5-960 \sqrt {3} x^7-7440 \sqrt {3} x^9-1440 \sqrt {3} x^{11}}{14-8 \sqrt {3}+\left (-28+20 \sqrt {3}\right ) x^2+\left (-1110-418 \sqrt {3}\right ) x^4+\left (296-268 \sqrt {3}\right ) x^6+\left (434-2401 \sqrt {3}\right ) x^8+\left (204-828 \sqrt {3}\right ) x^{10}+\left (-324-468 \sqrt {3}\right ) x^{12}-432 x^{14}+648 x^{16}} \, dx=10 \text {RootSum}\left [-14+8 \sqrt {3}+28 \text {$\#$1}^2-20 \sqrt {3} \text {$\#$1}^2+1110 \text {$\#$1}^4+418 \sqrt {3} \text {$\#$1}^4-296 \text {$\#$1}^6+268 \sqrt {3} \text {$\#$1}^6-434 \text {$\#$1}^8+2401 \sqrt {3} \text {$\#$1}^8-204 \text {$\#$1}^{10}+828 \sqrt {3} \text {$\#$1}^{10}+324 \text {$\#$1}^{12}+468 \sqrt {3} \text {$\#$1}^{12}+432 \text {$\#$1}^{14}-648 \text {$\#$1}^{16}\&,\frac {-3 \log (x-\text {$\#$1})+2 \sqrt {3} \log (x-\text {$\#$1})+3 \log (x-\text {$\#$1}) \text {$\#$1}^2-2 \sqrt {3} \log (x-\text {$\#$1}) \text {$\#$1}^2+90 \log (x-\text {$\#$1}) \text {$\#$1}^4+30 \sqrt {3} \log (x-\text {$\#$1}) \text {$\#$1}^4+12 \sqrt {3} \log (x-\text {$\#$1}) \text {$\#$1}^6+93 \sqrt {3} \log (x-\text {$\#$1}) \text {$\#$1}^8+18 \sqrt {3} \log (x-\text {$\#$1}) \text {$\#$1}^{10}}{7-5 \sqrt {3}+555 \text {$\#$1}^2+209 \sqrt {3} \text {$\#$1}^2-222 \text {$\#$1}^4+201 \sqrt {3} \text {$\#$1}^4-434 \text {$\#$1}^6+2401 \sqrt {3} \text {$\#$1}^6-255 \text {$\#$1}^8+1035 \sqrt {3} \text {$\#$1}^8+486 \text {$\#$1}^{10}+702 \sqrt {3} \text {$\#$1}^{10}+756 \text {$\#$1}^{12}-1296 \text {$\#$1}^{14}}\&\right ] \] Input:
Integrate[(-80*(-3 + 2*Sqrt[3])*x - 80*(3 - 2*Sqrt[3])*x^3 - 80*(90 + 30*S qrt[3])*x^5 - 960*Sqrt[3]*x^7 - 7440*Sqrt[3]*x^9 - 1440*Sqrt[3]*x^11)/(14 - 8*Sqrt[3] + (-28 + 20*Sqrt[3])*x^2 + (-1110 - 418*Sqrt[3])*x^4 + (296 - 268*Sqrt[3])*x^6 + (434 - 2401*Sqrt[3])*x^8 + (204 - 828*Sqrt[3])*x^10 + ( -324 - 468*Sqrt[3])*x^12 - 432*x^14 + 648*x^16),x]
Output:
10*RootSum[-14 + 8*Sqrt[3] + 28*#1^2 - 20*Sqrt[3]*#1^2 + 1110*#1^4 + 418*S qrt[3]*#1^4 - 296*#1^6 + 268*Sqrt[3]*#1^6 - 434*#1^8 + 2401*Sqrt[3]*#1^8 - 204*#1^10 + 828*Sqrt[3]*#1^10 + 324*#1^12 + 468*Sqrt[3]*#1^12 + 432*#1^14 - 648*#1^16 & , (-3*Log[x - #1] + 2*Sqrt[3]*Log[x - #1] + 3*Log[x - #1]*# 1^2 - 2*Sqrt[3]*Log[x - #1]*#1^2 + 90*Log[x - #1]*#1^4 + 30*Sqrt[3]*Log[x - #1]*#1^4 + 12*Sqrt[3]*Log[x - #1]*#1^6 + 93*Sqrt[3]*Log[x - #1]*#1^8 + 1 8*Sqrt[3]*Log[x - #1]*#1^10)/(7 - 5*Sqrt[3] + 555*#1^2 + 209*Sqrt[3]*#1^2 - 222*#1^4 + 201*Sqrt[3]*#1^4 - 434*#1^6 + 2401*Sqrt[3]*#1^6 - 255*#1^8 + 1035*Sqrt[3]*#1^8 + 486*#1^10 + 702*Sqrt[3]*#1^10 + 756*#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.
| \(\displaystyle \int \frac {-1440 \sqrt {3} x^{11}-7440 \sqrt {3} x^9-960 \sqrt {3} x^7-80 \left (90+30 \sqrt {3}\right ) x^5-80 \left (3-2 \sqrt {3}\right ) x^3-80 \left (2 \sqrt {3}-3\right ) x}{648 x^{16}-432 x^{14}+\left (-324-468 \sqrt {3}\right ) x^{12}+\left (204-828 \sqrt {3}\right ) x^{10}+\left (434-2401 \sqrt {3}\right ) x^8+\left (296-268 \sqrt {3}\right ) x^6+\left (-1110-418 \sqrt {3}\right ) x^4+\left (20 \sqrt {3}-28\right ) x^2-8 \sqrt {3}+14} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {80 x \left (-18 \sqrt {3} x^{10}-93 \sqrt {3} x^8-12 \sqrt {3} x^6-90 \left (1+\frac {1}{\sqrt {3}}\right ) x^4-3 \left (1-\frac {2}{\sqrt {3}}\right ) x^2+3 \left (1-\frac {2}{\sqrt {3}}\right )\right )}{648 x^{16}-432 x^{14}+\left (-324-468 \sqrt {3}\right ) x^{12}+\left (204-828 \sqrt {3}\right ) x^{10}+\left (434-2401 \sqrt {3}\right ) x^8+\left (296-268 \sqrt {3}\right ) x^6+\left (-1110-418 \sqrt {3}\right ) x^4+\left (20 \sqrt {3}-28\right ) x^2+14 \left (1-\frac {4 \sqrt {3}}{7}\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 80 \int \frac {x \left (-18 \sqrt {3} x^{10}-93 \sqrt {3} x^8-12 \sqrt {3} x^6-30 \left (3+\sqrt {3}\right ) x^4-\left (3-2 \sqrt {3}\right ) x^2-2 \sqrt {3}+3\right )}{648 x^{16}-432 x^{14}-36 \left (9+13 \sqrt {3}\right ) x^{12}+12 \left (17-69 \sqrt {3}\right ) x^{10}+7 \left (62-343 \sqrt {3}\right ) x^8+4 \left (74-67 \sqrt {3}\right ) x^6-2 \left (555+209 \sqrt {3}\right ) x^4-4 \left (7-5 \sqrt {3}\right ) x^2+2 \left (7-4 \sqrt {3}\right )}dx\) |
| \(\Big \downarrow \) 7266 |
| \(\displaystyle 40 \int \frac {-18 \sqrt {3} x^{10}-93 \sqrt {3} x^8-12 \sqrt {3} x^6-30 \left (3+\sqrt {3}\right ) x^4-\left (3-2 \sqrt {3}\right ) x^2-2 \sqrt {3}+3}{648 x^{16}-432 x^{14}-36 \left (9+13 \sqrt {3}\right ) x^{12}+12 \left (17-69 \sqrt {3}\right ) x^{10}+7 \left (62-343 \sqrt {3}\right ) x^8+4 \left (74-67 \sqrt {3}\right ) x^6-2 \left (555+209 \sqrt {3}\right ) x^4-4 \left (7-5 \sqrt {3}\right ) x^2+2 \left (7-4 \sqrt {3}\right )}dx^2\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle 40 \int \frac {-18 \sqrt {3} x^{10}-93 \sqrt {3} x^8-12 \sqrt {3} x^6-30 \left (3+\sqrt {3}\right ) x^4-\left (3-2 \sqrt {3}\right ) x^2+3 \left (1-\frac {2}{\sqrt {3}}\right )}{648 x^{16}-432 x^{14}-36 \left (9+13 \sqrt {3}\right ) x^{12}+12 \left (17-69 \sqrt {3}\right ) x^{10}+7 \left (62-343 \sqrt {3}\right ) x^8+4 \left (74-67 \sqrt {3}\right ) x^6-2 \left (555+209 \sqrt {3}\right ) x^4-4 \left (7-5 \sqrt {3}\right ) x^2+2 \left (7-4 \sqrt {3}\right )}dx^2\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 40 \int \left (\frac {18 \sqrt {3} x^{10}}{-648 x^{16}+432 x^{14}+324 \left (1+\frac {13}{3 \sqrt {3}}\right ) x^{12}-204 \left (1-\frac {69 \sqrt {3}}{17}\right ) x^{10}-434 \left (1-\frac {343 \sqrt {3}}{62}\right ) x^8-296 \left (1-\frac {67 \sqrt {3}}{74}\right ) x^6+1110 \left (1+\frac {209}{185 \sqrt {3}}\right ) x^4+28 \left (1-\frac {5 \sqrt {3}}{7}\right ) x^2-14 \left (1-\frac {4 \sqrt {3}}{7}\right )}+\frac {93 \sqrt {3} x^8}{-648 x^{16}+432 x^{14}+324 \left (1+\frac {13}{3 \sqrt {3}}\right ) x^{12}-204 \left (1-\frac {69 \sqrt {3}}{17}\right ) x^{10}-434 \left (1-\frac {343 \sqrt {3}}{62}\right ) x^8-296 \left (1-\frac {67 \sqrt {3}}{74}\right ) x^6+1110 \left (1+\frac {209}{185 \sqrt {3}}\right ) x^4+28 \left (1-\frac {5 \sqrt {3}}{7}\right ) x^2-14 \left (1-\frac {4 \sqrt {3}}{7}\right )}+\frac {12 \sqrt {3} x^6}{-648 x^{16}+432 x^{14}+324 \left (1+\frac {13}{3 \sqrt {3}}\right ) x^{12}-204 \left (1-\frac {69 \sqrt {3}}{17}\right ) x^{10}-434 \left (1-\frac {343 \sqrt {3}}{62}\right ) x^8-296 \left (1-\frac {67 \sqrt {3}}{74}\right ) x^6+1110 \left (1+\frac {209}{185 \sqrt {3}}\right ) x^4+28 \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}-432 x^{14}-324 \left (1+\frac {13}{3 \sqrt {3}}\right ) x^{12}+204 \left (1-\frac {69 \sqrt {3}}{17}\right ) x^{10}+434 \left (1-\frac {343 \sqrt {3}}{62}\right ) x^8+296 \left (1-\frac {67 \sqrt {3}}{74}\right ) x^6-1110 \left (1+\frac {209}{185 \sqrt {3}}\right ) x^4-28 \left (1-\frac {5 \sqrt {3}}{7}\right ) x^2+14 \left (1-\frac {4 \sqrt {3}}{7}\right )}+\frac {\left (-3+2 \sqrt {3}\right ) x^2}{648 x^{16}-432 x^{14}-324 \left (1+\frac {13}{3 \sqrt {3}}\right ) x^{12}+204 \left (1-\frac {69 \sqrt {3}}{17}\right ) x^{10}+434 \left (1-\frac {343 \sqrt {3}}{62}\right ) x^8+296 \left (1-\frac {67 \sqrt {3}}{74}\right ) x^6-1110 \left (1+\frac {209}{185 \sqrt {3}}\right ) x^4-28 \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}-432 x^{14}-324 \left (1+\frac {13}{3 \sqrt {3}}\right ) x^{12}+204 \left (1-\frac {69 \sqrt {3}}{17}\right ) x^{10}+434 \left (1-\frac {343 \sqrt {3}}{62}\right ) x^8+296 \left (1-\frac {67 \sqrt {3}}{74}\right ) x^6-1110 \left (1+\frac {209}{185 \sqrt {3}}\right ) x^4-28 \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 40 \left (12 \sqrt {3} \int \frac {x^6}{-648 x^{16}+432 x^{14}+324 \left (1+\frac {13}{3 \sqrt {3}}\right ) x^{12}-204 \left (1-\frac {69 \sqrt {3}}{17}\right ) x^{10}-434 \left (1-\frac {343 \sqrt {3}}{62}\right ) x^8-296 \left (1-\frac {67 \sqrt {3}}{74}\right ) x^6+1110 \left (1+\frac {209}{185 \sqrt {3}}\right ) x^4+28 \left (1-\frac {5 \sqrt {3}}{7}\right ) x^2-14 \left (1-\frac {4 \sqrt {3}}{7}\right )}dx^2+93 \sqrt {3} \int \frac {x^8}{-648 x^{16}+432 x^{14}+324 \left (1+\frac {13}{3 \sqrt {3}}\right ) x^{12}-204 \left (1-\frac {69 \sqrt {3}}{17}\right ) x^{10}-434 \left (1-\frac {343 \sqrt {3}}{62}\right ) x^8-296 \left (1-\frac {67 \sqrt {3}}{74}\right ) x^6+1110 \left (1+\frac {209}{185 \sqrt {3}}\right ) x^4+28 \left (1-\frac {5 \sqrt {3}}{7}\right ) x^2-14 \left (1-\frac {4 \sqrt {3}}{7}\right )}dx^2+18 \sqrt {3} \int \frac {x^{10}}{-648 x^{16}+432 x^{14}+324 \left (1+\frac {13}{3 \sqrt {3}}\right ) x^{12}-204 \left (1-\frac {69 \sqrt {3}}{17}\right ) x^{10}-434 \left (1-\frac {343 \sqrt {3}}{62}\right ) x^8-296 \left (1-\frac {67 \sqrt {3}}{74}\right ) x^6+1110 \left (1+\frac {209}{185 \sqrt {3}}\right ) x^4+28 \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}-432 x^{14}-324 \left (1+\frac {13}{3 \sqrt {3}}\right ) x^{12}+204 \left (1-\frac {69 \sqrt {3}}{17}\right ) x^{10}+434 \left (1-\frac {343 \sqrt {3}}{62}\right ) x^8+296 \left (1-\frac {67 \sqrt {3}}{74}\right ) x^6-1110 \left (1+\frac {209}{185 \sqrt {3}}\right ) x^4-28 \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 {x^2}{648 x^{16}-432 x^{14}-324 \left (1+\frac {13}{3 \sqrt {3}}\right ) x^{12}+204 \left (1-\frac {69 \sqrt {3}}{17}\right ) x^{10}+434 \left (1-\frac {343 \sqrt {3}}{62}\right ) x^8+296 \left (1-\frac {67 \sqrt {3}}{74}\right ) x^6-1110 \left (1+\frac {209}{185 \sqrt {3}}\right ) x^4-28 \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}-432 x^{14}-324 \left (1+\frac {13}{3 \sqrt {3}}\right ) x^{12}+204 \left (1-\frac {69 \sqrt {3}}{17}\right ) x^{10}+434 \left (1-\frac {343 \sqrt {3}}{62}\right ) x^8+296 \left (1-\frac {67 \sqrt {3}}{74}\right ) x^6-1110 \left (1+\frac {209}{185 \sqrt {3}}\right ) x^4-28 \left (1-\frac {5 \sqrt {3}}{7}\right ) x^2+14 \left (1-\frac {4 \sqrt {3}}{7}\right )}dx^2\right )\) |
Input:
Int[(-80*(-3 + 2*Sqrt[3])*x - 80*(3 - 2*Sqrt[3])*x^3 - 80*(90 + 30*Sqrt[3] )*x^5 - 960*Sqrt[3]*x^7 - 7440*Sqrt[3]*x^9 - 1440*Sqrt[3]*x^11)/(14 - 8*Sq rt[3] + (-28 + 20*Sqrt[3])*x^2 + (-1110 - 418*Sqrt[3])*x^4 + (296 - 268*Sq rt[3])*x^6 + (434 - 2401*Sqrt[3])*x^8 + (204 - 828*Sqrt[3])*x^10 + (-324 - 468*Sqrt[3])*x^12 - 432*x^14 + 648*x^16),x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.20 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.89
| 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((-80*(-3+2*3^(1/2))*x-80*(3-2*3^(1/2))*x^3-80*(90+30*3^(1/2))*x^5-960* 3^(1/2)*x^7-7440*3^(1/2)*x^9-1440*3^(1/2)*x^11)/(14-8*3^(1/2)+(-28+20*3^(1 /2))*x^2+(-1110-418*3^(1/2))*x^4+(296-268*3^(1/2))*x^6+(434-2401*3^(1/2))* x^8+(204-828*3^(1/2))*x^10+(-324-468*3^(1/2))*x^12-432*x^14+648*x^16),x,me thod=_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 = 0.41 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.02 \[ \int \frac {-80 \left (-3+2 \sqrt {3}\right ) x-80 \left (3-2 \sqrt {3}\right ) x^3-80 \left (90+30 \sqrt {3}\right ) x^5-960 \sqrt {3} x^7-7440 \sqrt {3} x^9-1440 \sqrt {3} x^{11}}{14-8 \sqrt {3}+\left (-28+20 \sqrt {3}\right ) x^2+\left (-1110-418 \sqrt {3}\right ) x^4+\left (296-268 \sqrt {3}\right ) x^6+\left (434-2401 \sqrt {3}\right ) x^8+\left (204-828 \sqrt {3}\right ) x^{10}+\left (-324-468 \sqrt {3}\right ) x^{12}-432 x^{14}+648 x^{16}} \, dx=3^{\frac {1}{8}} \arctan \left (\frac {1}{30} \cdot 3^{\frac {7}{8}} {\left (6 \, x^{2} - 1\right )} + \frac {1}{10} \cdot 3^{\frac {1}{8}}\right ) + 3^{\frac {1}{8}} \arctan \left (\frac {1}{30} \cdot 3^{\frac {3}{8}} {\left (162 \, x^{6} - 126 \, x^{4} - 576 \, x^{2} + \sqrt {3} {\left (90 \, x^{6} - 66 \, x^{4} - 293 \, x^{2} + 1\right )}\right )} - \frac {1}{10} \cdot 3^{\frac {1}{8}} {\left (72 \, x^{6} - 54 \, x^{4} - 240 \, x^{2} + 6 \, \sqrt {3} {\left (6 \, x^{6} - 5 \, x^{4} - 24 \, x^{2}\right )} + 1\right )}\right ) + \frac {1}{2} \cdot 3^{\frac {1}{8}} \log \left (6 \, x^{4} + 10 \cdot 3^{\frac {1}{8}} x^{2} - x^{2} - 3^{\frac {1}{4}} {\left (x^{2} - 2\right )} - \sqrt {3} - 1\right ) - \frac {1}{2} \cdot 3^{\frac {1}{8}} \log \left (6 \, x^{4} - 10 \cdot 3^{\frac {1}{8}} x^{2} - x^{2} - 3^{\frac {1}{4}} {\left (x^{2} - 2\right )} - \sqrt {3} - 1\right ) \] Input:
integrate((-80*(-3+2*3^(1/2))*x-80*(3-2*3^(1/2))*x^3-80*(90+30*3^(1/2))*x^ 5-960*3^(1/2)*x^7-7440*3^(1/2)*x^9-1440*3^(1/2)*x^11)/(14-8*3^(1/2)+(-28+2 0*3^(1/2))*x^2+(-1110-418*3^(1/2))*x^4+(296-268*3^(1/2))*x^6+(434-2401*3^( 1/2))*x^8+(204-828*3^(1/2))*x^10+(-324-468*3^(1/2))*x^12-432*x^14+648*x^16 ),x, algorithm="fricas")
Output:
3^(1/8)*arctan(1/30*3^(7/8)*(6*x^2 - 1) + 1/10*3^(1/8)) + 3^(1/8)*arctan(1 /30*3^(3/8)*(162*x^6 - 126*x^4 - 576*x^2 + sqrt(3)*(90*x^6 - 66*x^4 - 293* x^2 + 1)) - 1/10*3^(1/8)*(72*x^6 - 54*x^4 - 240*x^2 + 6*sqrt(3)*(6*x^6 - 5 *x^4 - 24*x^2) + 1)) + 1/2*3^(1/8)*log(6*x^4 + 10*3^(1/8)*x^2 - x^2 - 3^(1 /4)*(x^2 - 2) - sqrt(3) - 1) - 1/2*3^(1/8)*log(6*x^4 - 10*3^(1/8)*x^2 - x^ 2 - 3^(1/4)*(x^2 - 2) - sqrt(3) - 1)
Timed out. \[ \int \frac {-80 \left (-3+2 \sqrt {3}\right ) x-80 \left (3-2 \sqrt {3}\right ) x^3-80 \left (90+30 \sqrt {3}\right ) x^5-960 \sqrt {3} x^7-7440 \sqrt {3} x^9-1440 \sqrt {3} x^{11}}{14-8 \sqrt {3}+\left (-28+20 \sqrt {3}\right ) x^2+\left (-1110-418 \sqrt {3}\right ) x^4+\left (296-268 \sqrt {3}\right ) x^6+\left (434-2401 \sqrt {3}\right ) x^8+\left (204-828 \sqrt {3}\right ) x^{10}+\left (-324-468 \sqrt {3}\right ) x^{12}-432 x^{14}+648 x^{16}} \, dx=\text {Timed out} \] Input:
integrate((-80*(-3+2*3**(1/2))*x-80*(3-2*3**(1/2))*x**3-80*(90+30*3**(1/2) )*x**5-960*3**(1/2)*x**7-7440*3**(1/2)*x**9-1440*3**(1/2)*x**11)/(14-8*3** (1/2)+(-28+20*3**(1/2))*x**2+(-1110-418*3**(1/2))*x**4+(296-268*3**(1/2))* x**6+(434-2401*3**(1/2))*x**8+(204-828*3**(1/2))*x**10+(-324-468*3**(1/2)) *x**12-432*x**14+648*x**16),x)
Output:
Timed out
\[ \int \frac {-80 \left (-3+2 \sqrt {3}\right ) x-80 \left (3-2 \sqrt {3}\right ) x^3-80 \left (90+30 \sqrt {3}\right ) x^5-960 \sqrt {3} x^7-7440 \sqrt {3} x^9-1440 \sqrt {3} x^{11}}{14-8 \sqrt {3}+\left (-28+20 \sqrt {3}\right ) x^2+\left (-1110-418 \sqrt {3}\right ) x^4+\left (296-268 \sqrt {3}\right ) x^6+\left (434-2401 \sqrt {3}\right ) x^8+\left (204-828 \sqrt {3}\right ) x^{10}+\left (-324-468 \sqrt {3}\right ) x^{12}-432 x^{14}+648 x^{16}} \, dx=\int { -\frac {80 \, {\left (18 \, \sqrt {3} x^{11} + 93 \, \sqrt {3} x^{9} + 12 \, \sqrt {3} x^{7} + 30 \, x^{5} {\left (\sqrt {3} + 3\right )} - x^{3} {\left (2 \, \sqrt {3} - 3\right )} + x {\left (2 \, \sqrt {3} - 3\right )}\right )}}{648 \, x^{16} - 432 \, x^{14} - 36 \, x^{12} {\left (13 \, \sqrt {3} + 9\right )} - 12 \, x^{10} {\left (69 \, \sqrt {3} - 17\right )} - 7 \, x^{8} {\left (343 \, \sqrt {3} - 62\right )} - 4 \, x^{6} {\left (67 \, \sqrt {3} - 74\right )} - 2 \, x^{4} {\left (209 \, \sqrt {3} + 555\right )} + 4 \, x^{2} {\left (5 \, \sqrt {3} - 7\right )} - 8 \, \sqrt {3} + 14} \,d x } \] Input:
integrate((-80*(-3+2*3^(1/2))*x-80*(3-2*3^(1/2))*x^3-80*(90+30*3^(1/2))*x^ 5-960*3^(1/2)*x^7-7440*3^(1/2)*x^9-1440*3^(1/2)*x^11)/(14-8*3^(1/2)+(-28+2 0*3^(1/2))*x^2+(-1110-418*3^(1/2))*x^4+(296-268*3^(1/2))*x^6+(434-2401*3^( 1/2))*x^8+(204-828*3^(1/2))*x^10+(-324-468*3^(1/2))*x^12-432*x^14+648*x^16 ),x, algorithm="maxima")
Output:
-80*integrate((18*sqrt(3)*x^11 + 93*sqrt(3)*x^9 + 12*sqrt(3)*x^7 + 30*x^5* (sqrt(3) + 3) - x^3*(2*sqrt(3) - 3) + x*(2*sqrt(3) - 3))/(648*x^16 - 432*x ^14 - 36*x^12*(13*sqrt(3) + 9) - 12*x^10*(69*sqrt(3) - 17) - 7*x^8*(343*sq rt(3) - 62) - 4*x^6*(67*sqrt(3) - 74) - 2*x^4*(209*sqrt(3) + 555) + 4*x^2* (5*sqrt(3) - 7) - 8*sqrt(3) + 14), x)
Exception generated. \[ \int \frac {-80 \left (-3+2 \sqrt {3}\right ) x-80 \left (3-2 \sqrt {3}\right ) x^3-80 \left (90+30 \sqrt {3}\right ) x^5-960 \sqrt {3} x^7-7440 \sqrt {3} x^9-1440 \sqrt {3} x^{11}}{14-8 \sqrt {3}+\left (-28+20 \sqrt {3}\right ) x^2+\left (-1110-418 \sqrt {3}\right ) x^4+\left (296-268 \sqrt {3}\right ) x^6+\left (434-2401 \sqrt {3}\right ) x^8+\left (204-828 \sqrt {3}\right ) x^{10}+\left (-324-468 \sqrt {3}\right ) x^{12}-432 x^{14}+648 x^{16}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-80*(-3+2*3^(1/2))*x-80*(3-2*3^(1/2))*x^3-80*(90+30*3^(1/2))*x^ 5-960*3^(1/2)*x^7-7440*3^(1/2)*x^9-1440*3^(1/2)*x^11)/(14-8*3^(1/2)+(-28+2 0*3^(1/2))*x^2+(-1110-418*3^(1/2))*x^4+(296-268*3^(1/2))*x^6+(434-2401*3^( 1/2))*x^8+(204-828*3^(1/2))*x^10+(-324-468*3^(1/2))*x^12-432*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,infinity,infinity,infinity,infinity]proot error [undef,undef,undef,und ef,undef]
Time = 20.62 (sec) , antiderivative size = 841, normalized size of antiderivative = 4.34 \[ \int \frac {-80 \left (-3+2 \sqrt {3}\right ) x-80 \left (3-2 \sqrt {3}\right ) x^3-80 \left (90+30 \sqrt {3}\right ) x^5-960 \sqrt {3} x^7-7440 \sqrt {3} x^9-1440 \sqrt {3} x^{11}}{14-8 \sqrt {3}+\left (-28+20 \sqrt {3}\right ) x^2+\left (-1110-418 \sqrt {3}\right ) x^4+\left (296-268 \sqrt {3}\right ) x^6+\left (434-2401 \sqrt {3}\right ) x^8+\left (204-828 \sqrt {3}\right ) x^{10}+\left (-324-468 \sqrt {3}\right ) x^{12}-432 x^{14}+648 x^{16}} \, dx=\text {Too large to display} \] Input:
int((80*x^5*(30*3^(1/2) + 90) - 80*x^3*(2*3^(1/2) - 3) + 960*3^(1/2)*x^7 + 7440*3^(1/2)*x^9 + 1440*3^(1/2)*x^11 + 80*x*(2*3^(1/2) - 3))/(x^6*(268*3^ (1/2) - 296) - x^2*(20*3^(1/2) - 28) + x^12*(468*3^(1/2) + 324) + x^10*(82 8*3^(1/2) - 204) + x^4*(418*3^(1/2) + 1110) + x^8*(2401*3^(1/2) - 434) + 8 *3^(1/2) + 432*x^14 - 648*x^16 - 14),x)
Output:
symsum(log(2441392483387725377009512982635682733148193359375/5340065077318 9154161398869036498112 - root(3217350792006125968306176000000000*3^(1/2)*z ^8 - 6027263525562784202036505600000000*z^8 + 7534079406953480252545632000 00000*3^(1/2)*z^4 - 1206506547002297238114816000000000*z^4 + 3770332959382 1788691088000000000*3^(1/2) - 70631994440188877367615300000000, z, k)*(roo t(3217350792006125968306176000000000*3^(1/2)*z^8 - 60272635255627842020365 05600000000*z^8 + 753407940695348025254563200000000*3^(1/2)*z^4 - 12065065 47002297238114816000000000*z^4 + 37703329593821788691088000000000*3^(1/2) - 70631994440188877367615300000000, z, k)*((255221152149344477955891020733 652787446811767578125*3^(1/2))/53400650773189154161398869036498112 - x^2*( (6362576388018852111306272669458741480994398193359375*3^(1/2))/42720520618 5513233291190952291984896 - 1718436818983361349003909696867022298781071069 3359375/1281615618556539699873572856875954688) + root(32173507920061259683 06176000000000*3^(1/2)*z^8 - 6027263525562784202036505600000000*z^8 + 7534 07940695348025254563200000000*3^(1/2)*z^4 - 120650654700229723811481600000 0000*z^4 + 37703329593821788691088000000000*3^(1/2) - 70631994440188877367 615300000000, z, k)*(root(3217350792006125968306176000000000*3^(1/2)*z^8 - 6027263525562784202036505600000000*z^8 + 75340794069534802525456320000000 0*3^(1/2)*z^4 - 1206506547002297238114816000000000*z^4 + 37703329593821788 691088000000000*3^(1/2) - 70631994440188877367615300000000, z, k)*((137...
\[ \int \frac {-80 \left (-3+2 \sqrt {3}\right ) x-80 \left (3-2 \sqrt {3}\right ) x^3-80 \left (90+30 \sqrt {3}\right ) x^5-960 \sqrt {3} x^7-7440 \sqrt {3} x^9-1440 \sqrt {3} x^{11}}{14-8 \sqrt {3}+\left (-28+20 \sqrt {3}\right ) x^2+\left (-1110-418 \sqrt {3}\right ) x^4+\left (296-268 \sqrt {3}\right ) x^6+\left (434-2401 \sqrt {3}\right ) x^8+\left (204-828 \sqrt {3}\right ) x^{10}+\left (-324-468 \sqrt {3}\right ) x^{12}-432 x^{14}+648 x^{16}} \, dx=\text {too large to display} \] Input:
int((-80*(-3+2*3^(1/2))*x-80*(3-2*3^(1/2))*x^3-80*(90+30*3^(1/2))*x^5-960* 3^(1/2)*x^7-7440*3^(1/2)*x^9-1440*3^(1/2)*x^11)/(14-8*3^(1/2)+(-28+20*3^(1 /2))*x^2+(-1110-418*3^(1/2))*x^4+(296-268*3^(1/2))*x^6+(434-2401*3^(1/2))* x^8+(204-828*3^(1/2))*x^10+(-324-468*3^(1/2))*x^12-432*x^14+648*x^16),x)
Output:
80*( - 11664*sqrt(3)*int(x**27/(419904*x**32 - 559872*x**30 - 233280*x**28 + 544320*x**26 - 165888*x**24 - 2448576*x**22 - 10732680*x**20 - 11772696 *x**18 - 18728831*x**16 - 6071176*x**14 - 7056644*x**12 - 1099480*x**10 + 620416*x**8 + 107744*x**6 - 51560*x**4 + 176*x**2 + 4),x) - 52488*sqrt(3)* int(x**25/(419904*x**32 - 559872*x**30 - 233280*x**28 + 544320*x**26 - 165 888*x**24 - 2448576*x**22 - 10732680*x**20 - 11772696*x**18 - 18728831*x** 16 - 6071176*x**14 - 7056644*x**12 - 1099480*x**10 + 620416*x**8 + 107744* x**6 - 51560*x**4 + 176*x**2 + 4),x) + 38232*sqrt(3)*int(x**23/(419904*x** 32 - 559872*x**30 - 233280*x**28 + 544320*x**26 - 165888*x**24 - 2448576*x **22 - 10732680*x**20 - 11772696*x**18 - 18728831*x**16 - 6071176*x**14 - 7056644*x**12 - 1099480*x**10 + 620416*x**8 + 107744*x**6 - 51560*x**4 + 1 76*x**2 + 4),x) + 12204*sqrt(3)*int(x**21/(419904*x**32 - 559872*x**30 - 2 33280*x**28 + 544320*x**26 - 165888*x**24 - 2448576*x**22 - 10732680*x**20 - 11772696*x**18 - 18728831*x**16 - 6071176*x**14 - 7056644*x**12 - 10994 80*x**10 + 620416*x**8 + 107744*x**6 - 51560*x**4 + 176*x**2 + 4),x) - 864 0*sqrt(3)*int(x**19/(419904*x**32 - 559872*x**30 - 233280*x**28 + 544320*x **26 - 165888*x**24 - 2448576*x**22 - 10732680*x**20 - 11772696*x**18 - 18 728831*x**16 - 6071176*x**14 - 7056644*x**12 - 1099480*x**10 + 620416*x**8 + 107744*x**6 - 51560*x**4 + 176*x**2 + 4),x) - 82698*sqrt(3)*int(x**17/( 419904*x**32 - 559872*x**30 - 233280*x**28 + 544320*x**26 - 165888*x**2...