Integrand size = 63, antiderivative size = 250 \[ \int \frac {-2 x-5 x^2+5 x^4+4 x^5+x^6}{1+4 x+12 x^2+22 x^3+26 x^4+20 x^5+11 x^6+4 x^7+x^8} \, dx=\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \arctan \left (\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} x\right )-\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \arctan \left (\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )}+\sqrt {2+\sqrt {5}} x+\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} x^2+\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} x^3\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \log \left (\frac {1}{2} \left (2-\sqrt {2 \left (-1+\sqrt {5}\right )}\right )+\frac {1}{2} \left (2-\sqrt {2 \left (-1+\sqrt {5}\right )}\right ) x+x^2\right )+\frac {1}{2} \sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \log \left (\frac {1}{2} \left (2+\sqrt {2 \left (-1+\sqrt {5}\right )}\right )+\frac {1}{2} \left (2+\sqrt {2 \left (-1+\sqrt {5}\right )}\right ) x+x^2\right ) \] Output:
1/2*(2+2*5^(1/2))^(1/2)*arctan(1/2*(-2+2*5^(1/2))^(1/2)*x)-1/2*(2+2*5^(1/2 ))^(1/2)*arctan(1/2*(2+2*5^(1/2))^(1/2)+(2+5^(1/2))^(1/2)*x+1/2*(-2+2*5^(1 /2))^(1/2)*x^2+1/2*(-2+2*5^(1/2))^(1/2)*x^3)-1/4*(-2+2*5^(1/2))^(1/2)*ln(1 -1/2*(-2+2*5^(1/2))^(1/2)+1/2*(2-(-2+2*5^(1/2))^(1/2))*x+x^2)+1/4*(-2+2*5^ (1/2))^(1/2)*ln(1+1/2*(-2+2*5^(1/2))^(1/2)+1/2*(2+(-2+2*5^(1/2))^(1/2))*x+ x^2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.64 \[ \int \frac {-2 x-5 x^2+5 x^4+4 x^5+x^6}{1+4 x+12 x^2+22 x^3+26 x^4+20 x^5+11 x^6+4 x^7+x^8} \, dx=\frac {1}{2} \text {RootSum}\left [1+4 \text {$\#$1}+12 \text {$\#$1}^2+22 \text {$\#$1}^3+26 \text {$\#$1}^4+20 \text {$\#$1}^5+11 \text {$\#$1}^6+4 \text {$\#$1}^7+\text {$\#$1}^8\&,\frac {-2 \log (x-\text {$\#$1}) \text {$\#$1}-5 \log (x-\text {$\#$1}) \text {$\#$1}^2+5 \log (x-\text {$\#$1}) \text {$\#$1}^4+4 \log (x-\text {$\#$1}) \text {$\#$1}^5+\log (x-\text {$\#$1}) \text {$\#$1}^6}{2+12 \text {$\#$1}+33 \text {$\#$1}^2+52 \text {$\#$1}^3+50 \text {$\#$1}^4+33 \text {$\#$1}^5+14 \text {$\#$1}^6+4 \text {$\#$1}^7}\&\right ] \] Input:
Integrate[(-2*x - 5*x^2 + 5*x^4 + 4*x^5 + x^6)/(1 + 4*x + 12*x^2 + 22*x^3 + 26*x^4 + 20*x^5 + 11*x^6 + 4*x^7 + x^8),x]
Output:
RootSum[1 + 4*#1 + 12*#1^2 + 22*#1^3 + 26*#1^4 + 20*#1^5 + 11*#1^6 + 4*#1^ 7 + #1^8 & , (-2*Log[x - #1]*#1 - 5*Log[x - #1]*#1^2 + 5*Log[x - #1]*#1^4 + 4*Log[x - #1]*#1^5 + Log[x - #1]*#1^6)/(2 + 12*#1 + 33*#1^2 + 52*#1^3 + 50*#1^4 + 33*#1^5 + 14*#1^6 + 4*#1^7) & ]/2
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.
| \(\displaystyle \int \frac {x^6+4 x^5+5 x^4-5 x^2-2 x}{x^8+4 x^7+11 x^6+20 x^5+26 x^4+22 x^3+12 x^2+4 x+1} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {x \left (x^5+4 x^4+5 x^3-5 x-2\right )}{x^8+4 x^7+11 x^6+20 x^5+26 x^4+22 x^3+12 x^2+4 x+1}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x^6}{x^8+4 x^7+11 x^6+20 x^5+26 x^4+22 x^3+12 x^2+4 x+1}+\frac {4 x^5}{x^8+4 x^7+11 x^6+20 x^5+26 x^4+22 x^3+12 x^2+4 x+1}+\frac {5 x^4}{x^8+4 x^7+11 x^6+20 x^5+26 x^4+22 x^3+12 x^2+4 x+1}-\frac {5 x^2}{x^8+4 x^7+11 x^6+20 x^5+26 x^4+22 x^3+12 x^2+4 x+1}-\frac {2 x}{x^8+4 x^7+11 x^6+20 x^5+26 x^4+22 x^3+12 x^2+4 x+1}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \int \frac {x}{x^8+4 x^7+11 x^6+20 x^5+26 x^4+22 x^3+12 x^2+4 x+1}dx-5 \int \frac {x^2}{x^8+4 x^7+11 x^6+20 x^5+26 x^4+22 x^3+12 x^2+4 x+1}dx+5 \int \frac {x^4}{x^8+4 x^7+11 x^6+20 x^5+26 x^4+22 x^3+12 x^2+4 x+1}dx+4 \int \frac {x^5}{x^8+4 x^7+11 x^6+20 x^5+26 x^4+22 x^3+12 x^2+4 x+1}dx+\int \frac {x^6}{x^8+4 x^7+11 x^6+20 x^5+26 x^4+22 x^3+12 x^2+4 x+1}dx\) |
Input:
Int[(-2*x - 5*x^2 + 5*x^4 + 4*x^5 + x^6)/(1 + 4*x + 12*x^2 + 22*x^3 + 26*x ^4 + 20*x^5 + 11*x^6 + 4*x^7 + x^8),x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.05 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.12
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+\textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (x^{2}+\left (1+\textit {\_R} \right ) x +1+\textit {\_R} \right )\right )}{2}\) | \(29\) |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+\textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (x^{2}+\left (1+\textit {\_R} \right ) x +1+\textit {\_R} \right )\right )}{2}\) | \(29\) |
Input:
int((x^6+4*x^5+5*x^4-5*x^2-2*x)/(x^8+4*x^7+11*x^6+20*x^5+26*x^4+22*x^3+12* x^2+4*x+1),x,method=_RETURNVERBOSE)
Output:
1/2*sum(_R*ln(x^2+(1+_R)*x+1+_R),_R=RootOf(_Z^4+_Z^2-1))
Time = 0.07 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.57 \[ \int \frac {-2 x-5 x^2+5 x^4+4 x^5+x^6}{1+4 x+12 x^2+22 x^3+26 x^4+20 x^5+11 x^6+4 x^7+x^8} \, dx=-\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \arctan \left (-\frac {1}{2} \, {\left (x^{3} + x^{2} - \sqrt {5} {\left (x^{3} + x^{2} + x\right )} - x - 2\right )} \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}\right ) + \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \arctan \left (\frac {1}{2} \, {\left (\sqrt {5} x - x\right )} \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}\right ) + \frac {1}{2} \, \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \log \left (x^{2} + {\left (x + 1\right )} \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} + x + 1\right ) - \frac {1}{2} \, \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \log \left (x^{2} - {\left (x + 1\right )} \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} + x + 1\right ) \] Input:
integrate((x^6+4*x^5+5*x^4-5*x^2-2*x)/(x^8+4*x^7+11*x^6+20*x^5+26*x^4+22*x ^3+12*x^2+4*x+1),x, algorithm="fricas")
Output:
-sqrt(1/2*sqrt(5) + 1/2)*arctan(-1/2*(x^3 + x^2 - sqrt(5)*(x^3 + x^2 + x) - x - 2)*sqrt(1/2*sqrt(5) + 1/2)) + sqrt(1/2*sqrt(5) + 1/2)*arctan(1/2*(sq rt(5)*x - x)*sqrt(1/2*sqrt(5) + 1/2)) + 1/2*sqrt(1/2*sqrt(5) - 1/2)*log(x^ 2 + (x + 1)*sqrt(1/2*sqrt(5) - 1/2) + x + 1) - 1/2*sqrt(1/2*sqrt(5) - 1/2) *log(x^2 - (x + 1)*sqrt(1/2*sqrt(5) - 1/2) + x + 1)
Time = 0.19 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.12 \[ \int \frac {-2 x-5 x^2+5 x^4+4 x^5+x^6}{1+4 x+12 x^2+22 x^3+26 x^4+20 x^5+11 x^6+4 x^7+x^8} \, dx=\operatorname {RootSum} {\left (16 t^{4} + 4 t^{2} - 1, \left ( t \mapsto t \log {\left (2 t + x^{2} + x \left (2 t + 1\right ) + 1 \right )} \right )\right )} \] Input:
integrate((x**6+4*x**5+5*x**4-5*x**2-2*x)/(x**8+4*x**7+11*x**6+20*x**5+26* x**4+22*x**3+12*x**2+4*x+1),x)
Output:
RootSum(16*_t**4 + 4*_t**2 - 1, Lambda(_t, _t*log(2*_t + x**2 + x*(2*_t + 1) + 1)))
\[ \int \frac {-2 x-5 x^2+5 x^4+4 x^5+x^6}{1+4 x+12 x^2+22 x^3+26 x^4+20 x^5+11 x^6+4 x^7+x^8} \, dx=\int { \frac {x^{6} + 4 \, x^{5} + 5 \, x^{4} - 5 \, x^{2} - 2 \, x}{x^{8} + 4 \, x^{7} + 11 \, x^{6} + 20 \, x^{5} + 26 \, x^{4} + 22 \, x^{3} + 12 \, x^{2} + 4 \, x + 1} \,d x } \] Input:
integrate((x^6+4*x^5+5*x^4-5*x^2-2*x)/(x^8+4*x^7+11*x^6+20*x^5+26*x^4+22*x ^3+12*x^2+4*x+1),x, algorithm="maxima")
Output:
integrate((x^6 + 4*x^5 + 5*x^4 - 5*x^2 - 2*x)/(x^8 + 4*x^7 + 11*x^6 + 20*x ^5 + 26*x^4 + 22*x^3 + 12*x^2 + 4*x + 1), x)
Time = 0.19 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.68 \[ \int \frac {-2 x-5 x^2+5 x^4+4 x^5+x^6}{1+4 x+12 x^2+22 x^3+26 x^4+20 x^5+11 x^6+4 x^7+x^8} \, dx=\sqrt {\frac {1}{2}} \sqrt {\sqrt {5} + 1} {\left (\arctan \left (-\frac {1}{2} \, x^{3} \sqrt {2 \, \sqrt {5} - 2} - \frac {1}{2} \, x^{2} \sqrt {2 \, \sqrt {5} - 2} - x \sqrt {\sqrt {5} + 2} - \frac {1}{2} \, \sqrt {2 \, \sqrt {5} + 2}\right ) + \arctan \left (\frac {1}{2} \, x \sqrt {2 \, \sqrt {5} - 2}\right )\right )} + \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 1} \log \left (232 \, x^{2} + 116 \, x \sqrt {2 \, \sqrt {5} - 2} + 232 \, x + 116 \, \sqrt {2 \, \sqrt {5} - 2} + 232\right ) - \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 1} \log \left (232 \, x^{2} - 116 \, x \sqrt {2 \, \sqrt {5} - 2} + 232 \, x - 116 \, \sqrt {2 \, \sqrt {5} - 2} + 232\right ) \] Input:
integrate((x^6+4*x^5+5*x^4-5*x^2-2*x)/(x^8+4*x^7+11*x^6+20*x^5+26*x^4+22*x ^3+12*x^2+4*x+1),x, algorithm="giac")
Output:
sqrt(1/2)*sqrt(sqrt(5) + 1)*(arctan(-1/2*x^3*sqrt(2*sqrt(5) - 2) - 1/2*x^2 *sqrt(2*sqrt(5) - 2) - x*sqrt(sqrt(5) + 2) - 1/2*sqrt(2*sqrt(5) + 2)) + ar ctan(1/2*x*sqrt(2*sqrt(5) - 2))) + 1/2*sqrt(1/2)*sqrt(sqrt(5) - 1)*log(232 *x^2 + 116*x*sqrt(2*sqrt(5) - 2) + 232*x + 116*sqrt(2*sqrt(5) - 2) + 232) - 1/2*sqrt(1/2)*sqrt(sqrt(5) - 1)*log(232*x^2 - 116*x*sqrt(2*sqrt(5) - 2) + 232*x - 116*sqrt(2*sqrt(5) - 2) + 232)
Time = 10.02 (sec) , antiderivative size = 593, normalized size of antiderivative = 2.37 \[ \int \frac {-2 x-5 x^2+5 x^4+4 x^5+x^6}{1+4 x+12 x^2+22 x^3+26 x^4+20 x^5+11 x^6+4 x^7+x^8} \, dx=\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {750\,\sqrt {2}\,\sqrt {-\sqrt {5}-1}}{4500\,x-3750\,\sqrt {5}\,x-3750\,\sqrt {5}-6500\,\sqrt {5}\,x^2+12250\,x^2+4500}+\frac {750\,\sqrt {2}\,x\,\sqrt {-\sqrt {5}-1}}{4500\,x-3750\,\sqrt {5}\,x-3750\,\sqrt {5}-6500\,\sqrt {5}\,x^2+12250\,x^2+4500}-\frac {5375\,\sqrt {2}\,x^2\,\sqrt {-\sqrt {5}-1}}{4500\,x-3750\,\sqrt {5}\,x-3750\,\sqrt {5}-6500\,\sqrt {5}\,x^2+12250\,x^2+4500}-\frac {625\,\sqrt {2}\,\sqrt {5}\,\sqrt {-\sqrt {5}-1}}{4500\,x-3750\,\sqrt {5}\,x-3750\,\sqrt {5}-6500\,\sqrt {5}\,x^2+12250\,x^2+4500}-\frac {625\,\sqrt {2}\,\sqrt {5}\,x\,\sqrt {-\sqrt {5}-1}}{4500\,x-3750\,\sqrt {5}\,x-3750\,\sqrt {5}-6500\,\sqrt {5}\,x^2+12250\,x^2+4500}+\frac {2625\,\sqrt {2}\,\sqrt {5}\,x^2\,\sqrt {-\sqrt {5}-1}}{4500\,x-3750\,\sqrt {5}\,x-3750\,\sqrt {5}-6500\,\sqrt {5}\,x^2+12250\,x^2+4500}\right )\,\sqrt {-\sqrt {5}-1}}{2}+\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {750\,\sqrt {2}\,\sqrt {\sqrt {5}-1}}{4500\,x+3750\,\sqrt {5}\,x+3750\,\sqrt {5}+6500\,\sqrt {5}\,x^2+12250\,x^2+4500}-\frac {5375\,\sqrt {2}\,x^2\,\sqrt {\sqrt {5}-1}}{4500\,x+3750\,\sqrt {5}\,x+3750\,\sqrt {5}+6500\,\sqrt {5}\,x^2+12250\,x^2+4500}+\frac {625\,\sqrt {2}\,\sqrt {5}\,\sqrt {\sqrt {5}-1}}{4500\,x+3750\,\sqrt {5}\,x+3750\,\sqrt {5}+6500\,\sqrt {5}\,x^2+12250\,x^2+4500}+\frac {750\,\sqrt {2}\,x\,\sqrt {\sqrt {5}-1}}{4500\,x+3750\,\sqrt {5}\,x+3750\,\sqrt {5}+6500\,\sqrt {5}\,x^2+12250\,x^2+4500}+\frac {625\,\sqrt {2}\,\sqrt {5}\,x\,\sqrt {\sqrt {5}-1}}{4500\,x+3750\,\sqrt {5}\,x+3750\,\sqrt {5}+6500\,\sqrt {5}\,x^2+12250\,x^2+4500}-\frac {2625\,\sqrt {2}\,\sqrt {5}\,x^2\,\sqrt {\sqrt {5}-1}}{4500\,x+3750\,\sqrt {5}\,x+3750\,\sqrt {5}+6500\,\sqrt {5}\,x^2+12250\,x^2+4500}\right )\,\sqrt {\sqrt {5}-1}}{2} \] Input:
int((5*x^4 - 5*x^2 - 2*x + 4*x^5 + x^6)/(4*x + 12*x^2 + 22*x^3 + 26*x^4 + 20*x^5 + 11*x^6 + 4*x^7 + x^8 + 1),x)
Output:
(2^(1/2)*atanh((750*2^(1/2)*(- 5^(1/2) - 1)^(1/2))/(4500*x - 3750*5^(1/2)* x - 3750*5^(1/2) - 6500*5^(1/2)*x^2 + 12250*x^2 + 4500) + (750*2^(1/2)*x*( - 5^(1/2) - 1)^(1/2))/(4500*x - 3750*5^(1/2)*x - 3750*5^(1/2) - 6500*5^(1/ 2)*x^2 + 12250*x^2 + 4500) - (5375*2^(1/2)*x^2*(- 5^(1/2) - 1)^(1/2))/(450 0*x - 3750*5^(1/2)*x - 3750*5^(1/2) - 6500*5^(1/2)*x^2 + 12250*x^2 + 4500) - (625*2^(1/2)*5^(1/2)*(- 5^(1/2) - 1)^(1/2))/(4500*x - 3750*5^(1/2)*x - 3750*5^(1/2) - 6500*5^(1/2)*x^2 + 12250*x^2 + 4500) - (625*2^(1/2)*5^(1/2) *x*(- 5^(1/2) - 1)^(1/2))/(4500*x - 3750*5^(1/2)*x - 3750*5^(1/2) - 6500*5 ^(1/2)*x^2 + 12250*x^2 + 4500) + (2625*2^(1/2)*5^(1/2)*x^2*(- 5^(1/2) - 1) ^(1/2))/(4500*x - 3750*5^(1/2)*x - 3750*5^(1/2) - 6500*5^(1/2)*x^2 + 12250 *x^2 + 4500))*(- 5^(1/2) - 1)^(1/2))/2 + (2^(1/2)*atanh((750*2^(1/2)*(5^(1 /2) - 1)^(1/2))/(4500*x + 3750*5^(1/2)*x + 3750*5^(1/2) + 6500*5^(1/2)*x^2 + 12250*x^2 + 4500) - (5375*2^(1/2)*x^2*(5^(1/2) - 1)^(1/2))/(4500*x + 37 50*5^(1/2)*x + 3750*5^(1/2) + 6500*5^(1/2)*x^2 + 12250*x^2 + 4500) + (625* 2^(1/2)*5^(1/2)*(5^(1/2) - 1)^(1/2))/(4500*x + 3750*5^(1/2)*x + 3750*5^(1/ 2) + 6500*5^(1/2)*x^2 + 12250*x^2 + 4500) + (750*2^(1/2)*x*(5^(1/2) - 1)^( 1/2))/(4500*x + 3750*5^(1/2)*x + 3750*5^(1/2) + 6500*5^(1/2)*x^2 + 12250*x ^2 + 4500) + (625*2^(1/2)*5^(1/2)*x*(5^(1/2) - 1)^(1/2))/(4500*x + 3750*5^ (1/2)*x + 3750*5^(1/2) + 6500*5^(1/2)*x^2 + 12250*x^2 + 4500) - (2625*2^(1 /2)*5^(1/2)*x^2*(5^(1/2) - 1)^(1/2))/(4500*x + 3750*5^(1/2)*x + 3750*5^...
\[ \int \frac {-2 x-5 x^2+5 x^4+4 x^5+x^6}{1+4 x+12 x^2+22 x^3+26 x^4+20 x^5+11 x^6+4 x^7+x^8} \, dx=\int \frac {x^{6}}{x^{8}+4 x^{7}+11 x^{6}+20 x^{5}+26 x^{4}+22 x^{3}+12 x^{2}+4 x +1}d x +4 \left (\int \frac {x^{5}}{x^{8}+4 x^{7}+11 x^{6}+20 x^{5}+26 x^{4}+22 x^{3}+12 x^{2}+4 x +1}d x \right )+5 \left (\int \frac {x^{4}}{x^{8}+4 x^{7}+11 x^{6}+20 x^{5}+26 x^{4}+22 x^{3}+12 x^{2}+4 x +1}d x \right )-5 \left (\int \frac {x^{2}}{x^{8}+4 x^{7}+11 x^{6}+20 x^{5}+26 x^{4}+22 x^{3}+12 x^{2}+4 x +1}d x \right )-2 \left (\int \frac {x}{x^{8}+4 x^{7}+11 x^{6}+20 x^{5}+26 x^{4}+22 x^{3}+12 x^{2}+4 x +1}d x \right ) \] Input:
int((x^6+4*x^5+5*x^4-5*x^2-2*x)/(x^8+4*x^7+11*x^6+20*x^5+26*x^4+22*x^3+12* x^2+4*x+1),x)
Output:
int(x**6/(x**8 + 4*x**7 + 11*x**6 + 20*x**5 + 26*x**4 + 22*x**3 + 12*x**2 + 4*x + 1),x) + 4*int(x**5/(x**8 + 4*x**7 + 11*x**6 + 20*x**5 + 26*x**4 + 22*x**3 + 12*x**2 + 4*x + 1),x) + 5*int(x**4/(x**8 + 4*x**7 + 11*x**6 + 20 *x**5 + 26*x**4 + 22*x**3 + 12*x**2 + 4*x + 1),x) - 5*int(x**2/(x**8 + 4*x **7 + 11*x**6 + 20*x**5 + 26*x**4 + 22*x**3 + 12*x**2 + 4*x + 1),x) - 2*in t(x/(x**8 + 4*x**7 + 11*x**6 + 20*x**5 + 26*x**4 + 22*x**3 + 12*x**2 + 4*x + 1),x)