Integrand size = 83, antiderivative size = 195 \[ \int \frac {-\sqrt {3}+x+4 \sqrt {2} x+2 \sqrt {3} x^2-2 \sqrt {6} x^2+2 \sqrt {2} x^3}{1-2 \sqrt {3} x+2 x^2+\sqrt {3} x^3+\sqrt {2} x^4} \, dx=-\sqrt {-1+4 \sqrt {2}} \arctan \left (\sqrt {\frac {113}{279}+\frac {80 \sqrt {2}}{279}}+\sqrt {\frac {8}{93}+\frac {32 \sqrt {2}}{93}} x\right )-\sqrt {-1+4 \sqrt {2}} \arctan \left (\sqrt {\frac {49}{31}+\frac {196 \sqrt {2}}{31}}-\sqrt {\frac {108}{31}+\frac {432 \sqrt {2}}{31}} x-\sqrt {\frac {236}{31}+\frac {200 \sqrt {2}}{31}} x^2-\sqrt {\frac {24}{31}+\frac {96 \sqrt {2}}{31}} x^3\right )+\frac {1}{2} \log \left (\sqrt {2}-2 \sqrt {6} x+2 \sqrt {2} x^2+\sqrt {6} x^3+2 x^4\right ) \] Output:
-(-1+4*2^(1/2))^(1/2)*arctan(1/93*(3503+2480*2^(1/2))^(1/2)+2/93*(186+744* 2^(1/2))^(1/2)*x)+(-1+4*2^(1/2))^(1/2)*arctan(-7/31*(31+124*2^(1/2))^(1/2) +6/31*(93+372*2^(1/2))^(1/2)*x+2/31*(1829+1550*2^(1/2))^(1/2)*x^2+2/31*(18 6+744*2^(1/2))^(1/2)*x^3)+1/2*ln(2^(1/2)-2*x*6^(1/2)+2*x^2*2^(1/2)+6^(1/2) *x^3+2*x^4)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.10 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.89 \[ \int \frac {-\sqrt {3}+x+4 \sqrt {2} x+2 \sqrt {3} x^2-2 \sqrt {6} x^2+2 \sqrt {2} x^3}{1-2 \sqrt {3} x+2 x^2+\sqrt {3} x^3+\sqrt {2} x^4} \, dx=-\text {RootSum}\left [1-2 \sqrt {3} \text {$\#$1}+2 \text {$\#$1}^2+\sqrt {3} \text {$\#$1}^3+\sqrt {2} \text {$\#$1}^4\&,\frac {\sqrt {3} \log (x-\text {$\#$1})-\log (x-\text {$\#$1}) \text {$\#$1}-4 \sqrt {2} \log (x-\text {$\#$1}) \text {$\#$1}-2 \sqrt {3} \log (x-\text {$\#$1}) \text {$\#$1}^2+2 \sqrt {6} \log (x-\text {$\#$1}) \text {$\#$1}^2-2 \sqrt {2} \log (x-\text {$\#$1}) \text {$\#$1}^3}{-2 \sqrt {3}+4 \text {$\#$1}+3 \sqrt {3} \text {$\#$1}^2+4 \sqrt {2} \text {$\#$1}^3}\&\right ] \] Input:
Integrate[(-Sqrt[3] + x + 4*Sqrt[2]*x + 2*Sqrt[3]*x^2 - 2*Sqrt[6]*x^2 + 2* Sqrt[2]*x^3)/(1 - 2*Sqrt[3]*x + 2*x^2 + Sqrt[3]*x^3 + Sqrt[2]*x^4),x]
Output:
-RootSum[1 - 2*Sqrt[3]*#1 + 2*#1^2 + Sqrt[3]*#1^3 + Sqrt[2]*#1^4 & , (Sqrt [3]*Log[x - #1] - Log[x - #1]*#1 - 4*Sqrt[2]*Log[x - #1]*#1 - 2*Sqrt[3]*Lo g[x - #1]*#1^2 + 2*Sqrt[6]*Log[x - #1]*#1^2 - 2*Sqrt[2]*Log[x - #1]*#1^3)/ (-2*Sqrt[3] + 4*#1 + 3*Sqrt[3]*#1^2 + 4*Sqrt[2]*#1^3) & ]
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.
| \(\displaystyle \int \frac {2 \sqrt {2} x^3-2 \sqrt {6} x^2+2 \sqrt {3} x^2+4 \sqrt {2} x+x-\sqrt {3}}{\sqrt {2} x^4+\sqrt {3} x^3+2 x^2-2 \sqrt {3} x+1} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {2 \sqrt {2} x^3-2 \sqrt {6} x^2+2 \sqrt {3} x^2+\left (1+4 \sqrt {2}\right ) x-\sqrt {3}}{\sqrt {2} x^4+\sqrt {3} x^3+2 x^2-2 \sqrt {3} x+1}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {2 \sqrt {2} x^3+\left (2 \sqrt {3}-2 \sqrt {6}\right ) x^2+\left (1+4 \sqrt {2}\right ) x-\sqrt {3}}{\sqrt {2} x^4+\sqrt {3} x^3+2 x^2-2 \sqrt {3} x+1}dx\) |
| \(\Big \downarrow \) 2525 |
| \(\displaystyle \frac {\int \frac {2 \left (\sqrt {6} \left (1-4 \sqrt {2}\right ) x^2+2 \left (8-\sqrt {2}\right ) x\right )}{\sqrt {2} x^4+\sqrt {3} x^3+2 x^2-2 \sqrt {3} x+1}dx}{4 \sqrt {2}}+\frac {1}{2} \log \left (\sqrt {2} x^4+\sqrt {3} x^3+2 x^2-2 \sqrt {3} x+1\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {6} \left (1-4 \sqrt {2}\right ) x^2+2 \left (8-\sqrt {2}\right ) x}{\sqrt {2} x^4+\sqrt {3} x^3+2 x^2-2 \sqrt {3} x+1}dx}{2 \sqrt {2}}+\frac {1}{2} \log \left (\sqrt {2} x^4+\sqrt {3} x^3+2 x^2-2 \sqrt {3} x+1\right )\) |
| \(\Big \downarrow \) 2027 |
| \(\displaystyle \frac {\int \frac {x \left (\sqrt {6} \left (1-4 \sqrt {2}\right ) x+2 \left (8-\sqrt {2}\right )\right )}{\sqrt {2} x^4+\sqrt {3} x^3+2 x^2-2 \sqrt {3} x+1}dx}{2 \sqrt {2}}+\frac {1}{2} \log \left (\sqrt {2} x^4+\sqrt {3} x^3+2 x^2-2 \sqrt {3} x+1\right )\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\int \left (-\frac {\sqrt {6} \left (-1+4 \sqrt {2}\right ) x^2}{\sqrt {2} x^4+\sqrt {3} x^3+2 x^2-2 \sqrt {3} x+1}-\frac {2 \left (-8+\sqrt {2}\right ) x}{\sqrt {2} x^4+\sqrt {3} x^3+2 x^2-2 \sqrt {3} x+1}\right )dx}{2 \sqrt {2}}+\frac {1}{2} \log \left (\sqrt {2} x^4+\sqrt {3} x^3+2 x^2-2 \sqrt {3} x+1\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \left (8-\sqrt {2}\right ) \int \frac {x}{\sqrt {2} x^4+\sqrt {3} x^3+2 x^2-2 \sqrt {3} x+1}dx+\sqrt {6} \left (1-4 \sqrt {2}\right ) \int \frac {x^2}{\sqrt {2} x^4+\sqrt {3} x^3+2 x^2-2 \sqrt {3} x+1}dx}{2 \sqrt {2}}+\frac {1}{2} \log \left (\sqrt {2} x^4+\sqrt {3} x^3+2 x^2-2 \sqrt {3} x+1\right )\) |
Input:
Int[(-Sqrt[3] + x + 4*Sqrt[2]*x + 2*Sqrt[3]*x^2 - 2*Sqrt[6]*x^2 + 2*Sqrt[2 ]*x^3)/(1 - 2*Sqrt[3]*x + 2*x^2 + Sqrt[3]*x^3 + Sqrt[2]*x^4),x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.56 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.63
| method | result | size |
| default | \(\frac {\sqrt {2}\, \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\sqrt {3}\, \sqrt {2}\, \textit {\_Z}^{3}+2 \textit {\_Z}^{4}-2 \sqrt {3}\, \sqrt {2}\, \textit {\_Z} +2 \sqrt {2}\, \textit {\_Z}^{2}+\sqrt {2}\right )}{\sum }\frac {\left (4 \sqrt {2}\, \textit {\_R}^{3}+2 \sqrt {3}\, \sqrt {2}\, \textit {\_R}^{2} \left (\sqrt {2}-2\right )+\sqrt {2}\, \left (8+\sqrt {2}\right ) \textit {\_R} -2 \sqrt {3}\right ) \ln \left (x -\textit {\_R} \right )}{8 \textit {\_R}^{3}+\sqrt {2}\, \left (3 \sqrt {3}\, \textit {\_R}^{2}-2 \sqrt {3}+4 \textit {\_R} \right )}\right )}{2}\) | \(122\) |
Input:
int((-3^(1/2)+x+4*2^(1/2)*x+2*3^(1/2)*x^2-2*6^(1/2)*x^2+2*2^(1/2)*x^3)/(1- 2*3^(1/2)*x+2*x^2+3^(1/2)*x^3+2^(1/2)*x^4),x,method=_RETURNVERBOSE)
Output:
1/2*2^(1/2)*sum((4*2^(1/2)*_R^3+2*3^(1/2)*2^(1/2)*_R^2*(2^(1/2)-2)+2^(1/2) *(8+2^(1/2))*_R-2*3^(1/2))/(8*_R^3+2^(1/2)*(3*3^(1/2)*_R^2-2*3^(1/2)+4*_R) )*ln(x-_R),_R=RootOf(3^(1/2)*2^(1/2)*_Z^3+2*_Z^4-2*3^(1/2)*2^(1/2)*_Z+2*2^ (1/2)*_Z^2+2^(1/2)))
Timed out. \[ \int \frac {-\sqrt {3}+x+4 \sqrt {2} x+2 \sqrt {3} x^2-2 \sqrt {6} x^2+2 \sqrt {2} x^3}{1-2 \sqrt {3} x+2 x^2+\sqrt {3} x^3+\sqrt {2} x^4} \, dx=\text {Timed out} \] Input:
integrate((-3^(1/2)+x+4*2^(1/2)*x+2*3^(1/2)*x^2-2*6^(1/2)*x^2+2*2^(1/2)*x^ 3)/(1-2*3^(1/2)*x+2*x^2+3^(1/2)*x^3+2^(1/2)*x^4),x, algorithm="fricas")
Output:
Timed out
Exception generated. \[ \int \frac {-\sqrt {3}+x+4 \sqrt {2} x+2 \sqrt {3} x^2-2 \sqrt {6} x^2+2 \sqrt {2} x^3}{1-2 \sqrt {3} x+2 x^2+\sqrt {3} x^3+\sqrt {2} x^4} \, dx=\text {Exception raised: PolynomialError} \] Input:
integrate((-3**(1/2)+x+4*2**(1/2)*x+2*3**(1/2)*x**2-2*6**(1/2)*x**2+2*2**( 1/2)*x**3)/(1-2*3**(1/2)*x+2*x**2+3**(1/2)*x**3+2**(1/2)*x**4),x)
Output:
Exception raised: PolynomialError >> 1/(-593*_t**4 + 288*sqrt(2)*_t**4 - 1 360*sqrt(2)*_t**3 + 1196*_t**3 - 1044*_t**2 + 244*sqrt(2)*_t**2 - 800*_t + 584*sqrt(2)*_t - 136 + 96*sqrt(2)) contains an element of the set of gene rators.
\[ \int \frac {-\sqrt {3}+x+4 \sqrt {2} x+2 \sqrt {3} x^2-2 \sqrt {6} x^2+2 \sqrt {2} x^3}{1-2 \sqrt {3} x+2 x^2+\sqrt {3} x^3+\sqrt {2} x^4} \, dx=\int { \frac {2 \, \sqrt {2} x^{3} - 2 \, \sqrt {6} x^{2} + 2 \, \sqrt {3} x^{2} + 4 \, \sqrt {2} x + x - \sqrt {3}}{\sqrt {2} x^{4} + \sqrt {3} x^{3} + 2 \, x^{2} - 2 \, \sqrt {3} x + 1} \,d x } \] Input:
integrate((-3^(1/2)+x+4*2^(1/2)*x+2*3^(1/2)*x^2-2*6^(1/2)*x^2+2*2^(1/2)*x^ 3)/(1-2*3^(1/2)*x+2*x^2+3^(1/2)*x^3+2^(1/2)*x^4),x, algorithm="maxima")
Output:
integrate((2*sqrt(2)*x^3 - 2*sqrt(6)*x^2 + 2*sqrt(3)*x^2 + 4*sqrt(2)*x + x - sqrt(3))/(sqrt(2)*x^4 + sqrt(3)*x^3 + 2*x^2 - 2*sqrt(3)*x + 1), x)
Exception generated. \[ \int \frac {-\sqrt {3}+x+4 \sqrt {2} x+2 \sqrt {3} x^2-2 \sqrt {6} x^2+2 \sqrt {2} x^3}{1-2 \sqrt {3} x+2 x^2+\sqrt {3} x^3+\sqrt {2} x^4} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-3^(1/2)+x+4*2^(1/2)*x+2*3^(1/2)*x^2-2*6^(1/2)*x^2+2*2^(1/2)*x^ 3)/(1-2*3^(1/2)*x+2*x^2+3^(1/2)*x^3+2^(1/2)*x^4),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Precision problem choosing root in common_EXT, current precision 14Precision problem choosing root in common_ EXT, curr
Time = 14.34 (sec) , antiderivative size = 1698, normalized size of antiderivative = 8.71 \[ \int \frac {-\sqrt {3}+x+4 \sqrt {2} x+2 \sqrt {3} x^2-2 \sqrt {6} x^2+2 \sqrt {2} x^3}{1-2 \sqrt {3} x+2 x^2+\sqrt {3} x^3+\sqrt {2} x^4} \, dx=\text {Too large to display} \] Input:
int((x + 4*2^(1/2)*x - 3^(1/2) + 2*2^(1/2)*x^3 + 2*3^(1/2)*x^2 - 2*6^(1/2) *x^2)/(2^(1/2)*x^4 - 2*3^(1/2)*x + 3^(1/2)*x^3 + 2*x^2 + 1),x)
Output:
symsum(log(163*x + (291*3^(1/2)*root(2880*2^(1/2)*z^4 + 3496*z^4 - 5760*2^ (1/2)*z^3 - 6992*z^3 - 2372*2^(1/2)*3^(1/2)*6^(1/2)*z^2 - 4832*3^(1/2)*6^( 1/2)*z^2 + 24368*2^(1/2)*z^2 + 29248*z^2 + 3180*2^(1/2)*3^(1/2)*6^(1/2)*z + 6304*3^(1/2)*6^(1/2)*z - 25904*2^(1/2)*z - 30600*z - 13872*2^(1/2)*3^(1/ 2)*6^(1/2) - 19608*3^(1/2)*6^(1/2) + 64584*2^(1/2) + 90224, z, k))/4 + (23 1*6^(1/2)*root(2880*2^(1/2)*z^4 + 3496*z^4 - 5760*2^(1/2)*z^3 - 6992*z^3 - 2372*2^(1/2)*3^(1/2)*6^(1/2)*z^2 - 4832*3^(1/2)*6^(1/2)*z^2 + 24368*2^(1/ 2)*z^2 + 29248*z^2 + 3180*2^(1/2)*3^(1/2)*6^(1/2)*z + 6304*3^(1/2)*6^(1/2) *z - 25904*2^(1/2)*z - 30600*z - 13872*2^(1/2)*3^(1/2)*6^(1/2) - 19608*3^( 1/2)*6^(1/2) + 64584*2^(1/2) + 90224, z, k))/4 - (1209*root(2880*2^(1/2)*z ^4 + 3496*z^4 - 5760*2^(1/2)*z^3 - 6992*z^3 - 2372*2^(1/2)*3^(1/2)*6^(1/2) *z^2 - 4832*3^(1/2)*6^(1/2)*z^2 + 24368*2^(1/2)*z^2 + 29248*z^2 + 3180*2^( 1/2)*3^(1/2)*6^(1/2)*z + 6304*3^(1/2)*6^(1/2)*z - 25904*2^(1/2)*z - 30600* z - 13872*2^(1/2)*3^(1/2)*6^(1/2) - 19608*3^(1/2)*6^(1/2) + 64584*2^(1/2) + 90224, z, k)*x)/4 + (337*2^(1/2)*x)/4 - 39*3^(1/2) - (27*6^(1/2))/4 - 45 *3^(1/2)*root(2880*2^(1/2)*z^4 + 3496*z^4 - 5760*2^(1/2)*z^3 - 6992*z^3 - 2372*2^(1/2)*3^(1/2)*6^(1/2)*z^2 - 4832*3^(1/2)*6^(1/2)*z^2 + 24368*2^(1/2 )*z^2 + 29248*z^2 + 3180*2^(1/2)*3^(1/2)*6^(1/2)*z + 6304*3^(1/2)*6^(1/2)* z - 25904*2^(1/2)*z - 30600*z - 13872*2^(1/2)*3^(1/2)*6^(1/2) - 19608*3^(1 /2)*6^(1/2) + 64584*2^(1/2) + 90224, z, k)^2 + (153*3^(1/2)*root(2880*2...
\[ \int \frac {-\sqrt {3}+x+4 \sqrt {2} x+2 \sqrt {3} x^2-2 \sqrt {6} x^2+2 \sqrt {2} x^3}{1-2 \sqrt {3} x+2 x^2+\sqrt {3} x^3+\sqrt {2} x^4} \, dx=\text {too large to display} \] Input:
int((-3^(1/2)+x+4*2^(1/2)*x+2*3^(1/2)*x^2-2*6^(1/2)*x^2+2*2^(1/2)*x^3)/(1- 2*3^(1/2)*x+2*x^2+3^(1/2)*x^3+2^(1/2)*x^4),x)
Output:
(48*sqrt(6)*int(x**12/(4*x**16 - 12*x**14 + 41*x**12 - 160*x**10 + 300*x** 8 - 262*x**6 + 96*x**4 - 16*x**2 + 1),x) + 184*sqrt(6)*int(x**10/(4*x**16 - 12*x**14 + 41*x**12 - 160*x**10 + 300*x**8 - 262*x**6 + 96*x**4 - 16*x** 2 + 1),x) - 896*sqrt(6)*int(x**8/(4*x**16 - 12*x**14 + 41*x**12 - 160*x**1 0 + 300*x**8 - 262*x**6 + 96*x**4 - 16*x**2 + 1),x) + 1088*sqrt(6)*int(x** 6/(4*x**16 - 12*x**14 + 41*x**12 - 160*x**10 + 300*x**8 - 262*x**6 + 96*x* *4 - 16*x**2 + 1),x) - 408*sqrt(6)*int(x**4/(4*x**16 - 12*x**14 + 41*x**12 - 160*x**10 + 300*x**8 - 262*x**6 + 96*x**4 - 16*x**2 + 1),x) + 48*sqrt(6 )*int(x**2/(4*x**16 - 12*x**14 + 41*x**12 - 160*x**10 + 300*x**8 - 262*x** 6 + 96*x**4 - 16*x**2 + 1),x) - 64*sqrt(3)*int(x**14/(4*x**16 - 12*x**14 + 41*x**12 - 160*x**10 + 300*x**8 - 262*x**6 + 96*x**4 - 16*x**2 + 1),x) + 144*sqrt(3)*int(x**12/(4*x**16 - 12*x**14 + 41*x**12 - 160*x**10 + 300*x** 8 - 262*x**6 + 96*x**4 - 16*x**2 + 1),x) - 232*sqrt(3)*int(x**10/(4*x**16 - 12*x**14 + 41*x**12 - 160*x**10 + 300*x**8 - 262*x**6 + 96*x**4 - 16*x** 2 + 1),x) + 352*sqrt(3)*int(x**8/(4*x**16 - 12*x**14 + 41*x**12 - 160*x**1 0 + 300*x**8 - 262*x**6 + 96*x**4 - 16*x**2 + 1),x) - 136*sqrt(3)*int(x**6 /(4*x**16 - 12*x**14 + 41*x**12 - 160*x**10 + 300*x**8 - 262*x**6 + 96*x** 4 - 16*x**2 + 1),x) - 232*sqrt(3)*int(x**4/(4*x**16 - 12*x**14 + 41*x**12 - 160*x**10 + 300*x**8 - 262*x**6 + 96*x**4 - 16*x**2 + 1),x) + 80*sqrt(3) *int(x**2/(4*x**16 - 12*x**14 + 41*x**12 - 160*x**10 + 300*x**8 - 262*x...