\(\int \frac {-8+6 x-x^2}{-7-2 x+4 x^2+8 x^3-2 x^4} \, dx\) [17]

Optimal result

Integrand size = 33, antiderivative size = 182 \[ \int \frac {-8+6 x-x^2}{-7-2 x+4 x^2+8 x^3-2 x^4} \, dx=\sqrt {\frac {529}{836}+\frac {83}{38 \sqrt {11}}} \arctan \left (\sqrt {-\frac {13}{19}+\frac {10 \sqrt {11}}{19}}+\sqrt {\frac {20}{19}+\frac {8 \sqrt {11}}{19}} x\right )+\sqrt {-\frac {529}{836}+\frac {83}{38 \sqrt {11}}} \text {arctanh}\left (\sqrt {\frac {13}{19}+\frac {10 \sqrt {11}}{19}}-\sqrt {-\frac {20}{19}+\frac {8 \sqrt {11}}{19}} x\right )-\frac {\log \left (5+\sqrt {11}-4 x-2 \sqrt {11} x+2 x^2\right )}{4 \sqrt {11}}+\frac {\log \left (5-\sqrt {11}-4 x+2 \sqrt {11} x+2 x^2\right )}{4 \sqrt {11}} \] Output:

1/418*(110561+34694*11^(1/2))^(1/2)*arctan(1/19*(-247+190*11^(1/2))^(1/2)+ 
2/19*(95+38*11^(1/2))^(1/2)*x)-1/418*(-110561+34694*11^(1/2))^(1/2)*arctan 
h(-1/19*(247+190*11^(1/2))^(1/2)+2/19*(-95+38*11^(1/2))^(1/2)*x)-1/44*ln(5 
+11^(1/2)-4*x-2*11^(1/2)*x+2*x^2)*11^(1/2)+1/44*ln(5-11^(1/2)-4*x+2*11^(1/ 
2)*x+2*x^2)*11^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.01 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.47 \[ \int \frac {-8+6 x-x^2}{-7-2 x+4 x^2+8 x^3-2 x^4} \, dx=\frac {1}{2} \text {RootSum}\left [7+2 \text {$\#$1}-4 \text {$\#$1}^2-8 \text {$\#$1}^3+2 \text {$\#$1}^4\&,\frac {8 \log (x-\text {$\#$1})-6 \log (x-\text {$\#$1}) \text {$\#$1}+\log (x-\text {$\#$1}) \text {$\#$1}^2}{1-4 \text {$\#$1}-12 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ] \] Input:

Integrate[(-8 + 6*x - x^2)/(-7 - 2*x + 4*x^2 + 8*x^3 - 2*x^4),x]
 

Output:

RootSum[7 + 2*#1 - 4*#1^2 - 8*#1^3 + 2*#1^4 & , (8*Log[x - #1] - 6*Log[x - 
 #1]*#1 + Log[x - #1]*#1^2)/(1 - 4*#1 - 12*#1^2 + 4*#1^3) & ]/2
 

Rubi [F]

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.

\(\displaystyle \int \frac {-x^2+6 x-8}{-2 x^4+8 x^3+4 x^2-2 x-7} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {x^2}{2 x^4-8 x^3-4 x^2+2 x+7}-\frac {6 x}{2 x^4-8 x^3-4 x^2+2 x+7}+\frac {8}{2 x^4-8 x^3-4 x^2+2 x+7}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 8 \int \frac {1}{2 x^4-8 x^3-4 x^2+2 x+7}dx-6 \int \frac {x}{2 x^4-8 x^3-4 x^2+2 x+7}dx+\int \frac {x^2}{2 x^4-8 x^3-4 x^2+2 x+7}dx\)

Input:

Int[(-8 + 6*x - x^2)/(-7 - 2*x + 4*x^2 + 8*x^3 - 2*x^4),x]
 

Output:

$Aborted
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.32

method	result	size
default	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{4}-8 \textit {\_Z}^{3}-4 \textit {\_Z}^{2}+2 \textit {\_Z} +7\right )}{\sum }\frac {\left (\textit {\_R}^{2}-6 \textit {\_R} +8\right ) \ln \left (x -\textit {\_R} \right )}{4 \textit {\_R}^{3}-12 \textit {\_R}^{2}-4 \textit {\_R} +1}\right )}{2}\)	\(59\)
risch	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{4}-8 \textit {\_Z}^{3}-4 \textit {\_Z}^{2}+2 \textit {\_Z} +7\right )}{\sum }\frac {\left (\textit {\_R}^{2}-6 \textit {\_R} +8\right ) \ln \left (x -\textit {\_R} \right )}{4 \textit {\_R}^{3}-12 \textit {\_R}^{2}-4 \textit {\_R} +1}\right )}{2}\)	\(59\)

Input:

int((-x^2+6*x-8)/(-2*x^4+8*x^3+4*x^2-2*x-7),x,method=_RETURNVERBOSE)
 

Output:

1/2*sum((_R^2-6*_R+8)/(4*_R^3-12*_R^2-4*_R+1)*ln(x-_R),_R=RootOf(2*_Z^4-8* 
_Z^3-4*_Z^2+2*_Z+7))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.85 \[ \int \frac {-8+6 x-x^2}{-7-2 x+4 x^2+8 x^3-2 x^4} \, dx=-\frac {1}{44} \, {\left (\sqrt {11} + 11 \, \sqrt {\frac {166}{209} \, \sqrt {11} - \frac {529}{209}}\right )} \log \left ({\left (59 \, \sqrt {11} + 176\right )} \sqrt {\frac {166}{209} \, \sqrt {11} - \frac {529}{209}} + 70 \, x - 35 \, \sqrt {11} - 70\right ) - \frac {1}{44} \, {\left (\sqrt {11} - 11 \, \sqrt {\frac {166}{209} \, \sqrt {11} - \frac {529}{209}}\right )} \log \left (-{\left (59 \, \sqrt {11} + 176\right )} \sqrt {\frac {166}{209} \, \sqrt {11} - \frac {529}{209}} + 70 \, x - 35 \, \sqrt {11} - 70\right ) - \frac {1}{2} \, \sqrt {\frac {166}{209} \, \sqrt {11} + \frac {529}{209}} \arctan \left (\frac {1}{35} \, {\left (2 \, \sqrt {11} {\left (3 \, x - 14\right )} - 44 \, x + 77\right )} \sqrt {\frac {166}{209} \, \sqrt {11} + \frac {529}{209}}\right ) + \frac {1}{44} \, \sqrt {11} \log \left (2 \, x^{2} + \sqrt {11} {\left (2 \, x - 1\right )} - 4 \, x + 5\right ) \] Input:

integrate((-x^2+6*x-8)/(-2*x^4+8*x^3+4*x^2-2*x-7),x, algorithm="fricas")
 

Output:

-1/44*(sqrt(11) + 11*sqrt(166/209*sqrt(11) - 529/209))*log((59*sqrt(11) + 
176)*sqrt(166/209*sqrt(11) - 529/209) + 70*x - 35*sqrt(11) - 70) - 1/44*(s 
qrt(11) - 11*sqrt(166/209*sqrt(11) - 529/209))*log(-(59*sqrt(11) + 176)*sq 
rt(166/209*sqrt(11) - 529/209) + 70*x - 35*sqrt(11) - 70) - 1/2*sqrt(166/2 
09*sqrt(11) + 529/209)*arctan(1/35*(2*sqrt(11)*(3*x - 14) - 44*x + 77)*sqr 
t(166/209*sqrt(11) + 529/209)) + 1/44*sqrt(11)*log(2*x^2 + sqrt(11)*(2*x - 
 1) - 4*x + 5)
 

Sympy [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.23 \[ \int \frac {-8+6 x-x^2}{-7-2 x+4 x^2+8 x^3-2 x^4} \, dx=\operatorname {RootSum} {\left (147136 t^{4} + 44880 t^{2} + 7304 t - 37, \left ( t \mapsto t \log {\left (- \frac {30824992 t^{3}}{430955} - \frac {1699588 t^{2}}{430955} - \frac {8685072 t}{430955} + x - \frac {3675607}{861910} \right )} \right )\right )} \] Input:

integrate((-x**2+6*x-8)/(-2*x**4+8*x**3+4*x**2-2*x-7),x)
 

Output:

RootSum(147136*_t**4 + 44880*_t**2 + 7304*_t - 37, Lambda(_t, _t*log(-3082 
4992*_t**3/430955 - 1699588*_t**2/430955 - 8685072*_t/430955 + x - 3675607 
/861910)))
 

Maxima [F]

\[ \int \frac {-8+6 x-x^2}{-7-2 x+4 x^2+8 x^3-2 x^4} \, dx=\int { \frac {x^{2} - 6 \, x + 8}{2 \, x^{4} - 8 \, x^{3} - 4 \, x^{2} + 2 \, x + 7} \,d x } \] Input:

integrate((-x^2+6*x-8)/(-2*x^4+8*x^3+4*x^2-2*x-7),x, algorithm="maxima")
 

Output:

integrate((x^2 - 6*x + 8)/(2*x^4 - 8*x^3 - 4*x^2 + 2*x + 7), x)
 

Giac [F]

\[ \int \frac {-8+6 x-x^2}{-7-2 x+4 x^2+8 x^3-2 x^4} \, dx=\int { \frac {x^{2} - 6 \, x + 8}{2 \, x^{4} - 8 \, x^{3} - 4 \, x^{2} + 2 \, x + 7} \,d x } \] Input:

integrate((-x^2+6*x-8)/(-2*x^4+8*x^3+4*x^2-2*x-7),x, algorithm="giac")
 

Output:

integrate((x^2 - 6*x + 8)/(2*x^4 - 8*x^3 - 4*x^2 + 2*x + 7), x)
 

Mupad [B] (verification not implemented)

Time = 9.92 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.80 \[ \int \frac {-8+6 x-x^2}{-7-2 x+4 x^2+8 x^3-2 x^4} \, dx=\sum _{k=1}^4\ln \left (370\,\mathrm {root}\left (z^4+\frac {255\,z^2}{836}+\frac {83\,z}{1672}-\frac {37}{147136},z,k\right )+\frac {3\,x}{8}-\frac {\mathrm {root}\left (z^4+\frac {255\,z^2}{836}+\frac {83\,z}{1672}-\frac {37}{147136},z,k\right )\,x\,479}{2}+{\mathrm {root}\left (z^4+\frac {255\,z^2}{836}+\frac {83\,z}{1672}-\frac {37}{147136},z,k\right )}^2\,x\,1100-{\mathrm {root}\left (z^4+\frac {255\,z^2}{836}+\frac {83\,z}{1672}-\frac {37}{147136},z,k\right )}^3\,x\,132+\frac {363\,{\mathrm {root}\left (z^4+\frac {255\,z^2}{836}+\frac {83\,z}{1672}-\frac {37}{147136},z,k\right )}^2}{2}+2684\,{\mathrm {root}\left (z^4+\frac {255\,z^2}{836}+\frac {83\,z}{1672}-\frac {37}{147136},z,k\right )}^3+\frac {25}{16}\right )\,\mathrm {root}\left (z^4+\frac {255\,z^2}{836}+\frac {83\,z}{1672}-\frac {37}{147136},z,k\right ) \] Input:

int((x^2 - 6*x + 8)/(2*x - 4*x^2 - 8*x^3 + 2*x^4 + 7),x)
 

Output:

symsum(log(370*root(z^4 + (255*z^2)/836 + (83*z)/1672 - 37/147136, z, k) + 
 (3*x)/8 - (479*root(z^4 + (255*z^2)/836 + (83*z)/1672 - 37/147136, z, k)* 
x)/2 + 1100*root(z^4 + (255*z^2)/836 + (83*z)/1672 - 37/147136, z, k)^2*x 
- 132*root(z^4 + (255*z^2)/836 + (83*z)/1672 - 37/147136, z, k)^3*x + (363 
*root(z^4 + (255*z^2)/836 + (83*z)/1672 - 37/147136, z, k)^2)/2 + 2684*roo 
t(z^4 + (255*z^2)/836 + (83*z)/1672 - 37/147136, z, k)^3 + 25/16)*root(z^4 
 + (255*z^2)/836 + (83*z)/1672 - 37/147136, z, k), k, 1, 4)
 

Reduce [F]

\[ \int \frac {-8+6 x-x^2}{-7-2 x+4 x^2+8 x^3-2 x^4} \, dx=\int \frac {x^{2}}{2 x^{4}-8 x^{3}-4 x^{2}+2 x +7}d x -6 \left (\int \frac {x}{2 x^{4}-8 x^{3}-4 x^{2}+2 x +7}d x \right )+8 \left (\int \frac {1}{2 x^{4}-8 x^{3}-4 x^{2}+2 x +7}d x \right ) \] Input:

int((-x^2+6*x-8)/(-2*x^4+8*x^3+4*x^2-2*x-7),x)
 

Output:

int(x**2/(2*x**4 - 8*x**3 - 4*x**2 + 2*x + 7),x) - 6*int(x/(2*x**4 - 8*x** 
3 - 4*x**2 + 2*x + 7),x) + 8*int(1/(2*x**4 - 8*x**3 - 4*x**2 + 2*x + 7),x)