\(\int \frac {320 x^3+16 x^5+384 x^7+104 x^9-64 x^{11}-8 x^{13}}{-4-32 x^2-12 x^4-64 x^6-5 x^8+32 x^{10}+2 x^{12}+x^{16}} \, dx\) [72]

Optimal result

Integrand size = 69, antiderivative size = 151 \[ \int \frac {320 x^3+16 x^5+384 x^7+104 x^9-64 x^{11}-8 x^{13}}{-4-32 x^2-12 x^4-64 x^6-5 x^8+32 x^{10}+2 x^{12}+x^{16}} \, dx=-2 \sqrt {-1+\sqrt {2}} \arctan \left (\sqrt {1+\sqrt {2}} x^2\right )+2 \sqrt {1+\sqrt {2}} \arctan \left (\sqrt {-\frac {1}{2}+\frac {1}{\sqrt {2}}}-\frac {1}{2} \sqrt {-1+\sqrt {2}} x^2\right )-2 \sqrt {1+\sqrt {2}} \text {arctanh}\left (\sqrt {-1+\sqrt {2}} x^2\right )-2 \sqrt {-1+\sqrt {2}} \text {arctanh}\left (\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}}+\frac {1}{2} \sqrt {1+\sqrt {2}} x^2\right ) \] Output:

-2*(2^(1/2)-1)^(1/2)*arctan((1+2^(1/2))^(1/2)*x^2)-2*(1+2^(1/2))^(1/2)*arc 
tan(-1/2*(-2+2*2^(1/2))^(1/2)+1/2*(2^(1/2)-1)^(1/2)*x^2)-2*(1+2^(1/2))^(1/ 
2)*arctanh((2^(1/2)-1)^(1/2)*x^2)-2*(2^(1/2)-1)^(1/2)*arctanh(1/2*(2+2*2^( 
1/2))^(1/2)+1/2*(1+2^(1/2))^(1/2)*x^2)
 

Mathematica [C] (verified)

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

Time = 0.04 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.87 \[ \int \frac {320 x^3+16 x^5+384 x^7+104 x^9-64 x^{11}-8 x^{13}}{-4-32 x^2-12 x^4-64 x^6-5 x^8+32 x^{10}+2 x^{12}+x^{16}} \, dx=-8 \left (-\frac {1}{8} \text {RootSum}\left [-1-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x-\text {$\#$1})+\log (x-\text {$\#$1}) \text {$\#$1}^4}{-\text {$\#$1}^2+\text {$\#$1}^6}\&\right ]+\frac {1}{4} \text {RootSum}\left [4+32 \text {$\#$1}^2+4 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 \log (x-\text {$\#$1})+4 \log (x-\text {$\#$1}) \text {$\#$1}^2+\log (x-\text {$\#$1}) \text {$\#$1}^4}{8+2 \text {$\#$1}^2+\text {$\#$1}^6}\&\right ]\right ) \] Input:

Integrate[(320*x^3 + 16*x^5 + 384*x^7 + 104*x^9 - 64*x^11 - 8*x^13)/(-4 - 
32*x^2 - 12*x^4 - 64*x^6 - 5*x^8 + 32*x^10 + 2*x^12 + x^16),x]
 

Output:

-8*(-1/8*RootSum[-1 - 2*#1^4 + #1^8 & , (Log[x - #1] + Log[x - #1]*#1^4)/( 
-#1^2 + #1^6) & ] + RootSum[4 + 32*#1^2 + 4*#1^4 + #1^8 & , (-2*Log[x - #1 
] + 4*Log[x - #1]*#1^2 + Log[x - #1]*#1^4)/(8 + 2*#1^2 + #1^6) & ]/4)
 

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.

\(\displaystyle \int \frac {-8 x^{13}-64 x^{11}+104 x^9+384 x^7+16 x^5+320 x^3}{x^{16}+2 x^{12}+32 x^{10}-5 x^8-64 x^6-12 x^4-32 x^2-4} \, dx\)

\(\Big \downarrow \) 2460

\(\displaystyle \int \left (\frac {8 x \left (-x^4-1\right )}{-x^8+2 x^4+1}-\frac {16 x \left (x^4+4 x^2-2\right )}{x^8+4 x^4+32 x^2+4}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 16 \text {Subst}\left (\int \frac {1}{x^4+4 x^2+32 x+4}dx,x,x^2\right )-32 \text {Subst}\left (\int \frac {x}{x^4+4 x^2+32 x+4}dx,x,x^2\right )-8 \text {Subst}\left (\int \frac {x^2}{x^4+4 x^2+32 x+4}dx,x,x^2\right )-2 \sqrt {\sqrt {2}-1} \arctan \left (\frac {x^2}{\sqrt {\sqrt {2}-1}}\right )-2 \sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {x^2}{\sqrt {1+\sqrt {2}}}\right )\)

Input:

Int[(320*x^3 + 16*x^5 + 384*x^7 + 104*x^9 - 64*x^11 - 8*x^13)/(-4 - 32*x^2 
 - 12*x^4 - 64*x^6 - 5*x^8 + 32*x^10 + 2*x^12 + x^16),x]
 

Output:

$Aborted
 

Maple [C] (verified)

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

Time = 0.15 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.36

method	result	size
risch	\(\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (x^{2}-\textit {\_R} \right )\right )+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+2 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (\textit {\_R}^{2}+x^{2}-2 \textit {\_R} +1\right )\right )\)	\(54\)
default	\(2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+4 \textit {\_Z}^{2}+32 \textit {\_Z} +4\right )}{\sum }\frac {\left (-\textit {\_R}^{2}-4 \textit {\_R} +2\right ) \ln \left (x^{2}-\textit {\_R} \right )}{\textit {\_R}^{3}+2 \textit {\_R} +8}\right )-\frac {\left (2+\sqrt {2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {x^{2}}{\sqrt {1+\sqrt {2}}}\right )}{\sqrt {1+\sqrt {2}}}+\frac {\left (\sqrt {2}-2\right ) \sqrt {2}\, \arctan \left (\frac {x^{2}}{\sqrt {\sqrt {2}-1}}\right )}{\sqrt {\sqrt {2}-1}}\)	\(107\)

Input:

int((-8*x^13-64*x^11+104*x^9+384*x^7+16*x^5+320*x^3)/(x^16+2*x^12+32*x^10- 
5*x^8-64*x^6-12*x^4-32*x^2-4),x,method=_RETURNVERBOSE)
 

Output:

sum(_R*ln(x^2-_R),_R=RootOf(_Z^4-2*_Z^2-1))+sum(_R*ln(_R^2+x^2-2*_R+1),_R= 
RootOf(_Z^4+2*_Z^2-1))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.05 \[ \int \frac {320 x^3+16 x^5+384 x^7+104 x^9-64 x^{11}-8 x^{13}}{-4-32 x^2-12 x^4-64 x^6-5 x^8+32 x^{10}+2 x^{12}+x^{16}} \, dx=-2 \, \sqrt {\sqrt {2} + 1} \arctan \left (-\frac {1}{2} \, {\left (x^{2} - \sqrt {2} {\left (x^{2} + 1\right )} + 2\right )} \sqrt {\sqrt {2} + 1}\right ) - 2 \, \sqrt {\sqrt {2} - 1} \arctan \left ({\left (\sqrt {2} x^{2} + x^{2}\right )} \sqrt {\sqrt {2} - 1}\right ) - \sqrt {\sqrt {2} - 1} \log \left (x^{2} + \sqrt {2} + 2 \, \sqrt {\sqrt {2} - 1}\right ) + \sqrt {\sqrt {2} - 1} \log \left (x^{2} + \sqrt {2} - 2 \, \sqrt {\sqrt {2} - 1}\right ) - \sqrt {\sqrt {2} + 1} \log \left (x^{2} + \sqrt {\sqrt {2} + 1}\right ) + \sqrt {\sqrt {2} + 1} \log \left (x^{2} - \sqrt {\sqrt {2} + 1}\right ) \] Input:

integrate((-8*x^13-64*x^11+104*x^9+384*x^7+16*x^5+320*x^3)/(x^16+2*x^12+32 
*x^10-5*x^8-64*x^6-12*x^4-32*x^2-4),x, algorithm="fricas")
 

Output:

-2*sqrt(sqrt(2) + 1)*arctan(-1/2*(x^2 - sqrt(2)*(x^2 + 1) + 2)*sqrt(sqrt(2 
) + 1)) - 2*sqrt(sqrt(2) - 1)*arctan((sqrt(2)*x^2 + x^2)*sqrt(sqrt(2) - 1) 
) - sqrt(sqrt(2) - 1)*log(x^2 + sqrt(2) + 2*sqrt(sqrt(2) - 1)) + sqrt(sqrt 
(2) - 1)*log(x^2 + sqrt(2) - 2*sqrt(sqrt(2) - 1)) - sqrt(sqrt(2) + 1)*log( 
x^2 + sqrt(sqrt(2) + 1)) + sqrt(sqrt(2) + 1)*log(x^2 - sqrt(sqrt(2) + 1))
 

Sympy [A] (verification not implemented)

Time = 0.64 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.72 \[ \int \frac {320 x^3+16 x^5+384 x^7+104 x^9-64 x^{11}-8 x^{13}}{-4-32 x^2-12 x^4-64 x^6-5 x^8+32 x^{10}+2 x^{12}+x^{16}} \, dx=- \operatorname {RootSum} {\left (t^{4} - 2 t^{2} - 1, \left ( t \mapsto t \log {\left (- \frac {t^{7}}{4} - \frac {t^{6}}{4} - \frac {t^{4}}{4} + \frac {5 t^{3}}{4} + \frac {7 t^{2}}{4} + \frac {3 t}{2} + x^{2} + \frac {3}{4} \right )} \right )\right )} - \operatorname {RootSum} {\left (t^{4} + 2 t^{2} - 1, \left ( t \mapsto t \log {\left (- \frac {t^{7}}{4} - \frac {t^{6}}{4} - \frac {t^{4}}{4} + \frac {5 t^{3}}{4} + \frac {7 t^{2}}{4} + \frac {3 t}{2} + x^{2} + \frac {3}{4} \right )} \right )\right )} \] Input:

integrate((-8*x**13-64*x**11+104*x**9+384*x**7+16*x**5+320*x**3)/(x**16+2* 
x**12+32*x**10-5*x**8-64*x**6-12*x**4-32*x**2-4),x)
 

Output:

-RootSum(_t**4 - 2*_t**2 - 1, Lambda(_t, _t*log(-_t**7/4 - _t**6/4 - _t**4 
/4 + 5*_t**3/4 + 7*_t**2/4 + 3*_t/2 + x**2 + 3/4))) - RootSum(_t**4 + 2*_t 
**2 - 1, Lambda(_t, _t*log(-_t**7/4 - _t**6/4 - _t**4/4 + 5*_t**3/4 + 7*_t 
**2/4 + 3*_t/2 + x**2 + 3/4)))
 

Maxima [F]

\[ \int \frac {320 x^3+16 x^5+384 x^7+104 x^9-64 x^{11}-8 x^{13}}{-4-32 x^2-12 x^4-64 x^6-5 x^8+32 x^{10}+2 x^{12}+x^{16}} \, dx=\int { -\frac {8 \, {\left (x^{13} + 8 \, x^{11} - 13 \, x^{9} - 48 \, x^{7} - 2 \, x^{5} - 40 \, x^{3}\right )}}{x^{16} + 2 \, x^{12} + 32 \, x^{10} - 5 \, x^{8} - 64 \, x^{6} - 12 \, x^{4} - 32 \, x^{2} - 4} \,d x } \] Input:

integrate((-8*x^13-64*x^11+104*x^9+384*x^7+16*x^5+320*x^3)/(x^16+2*x^12+32 
*x^10-5*x^8-64*x^6-12*x^4-32*x^2-4),x, algorithm="maxima")
 

Output:

-8*integrate((x^13 + 8*x^11 - 13*x^9 - 48*x^7 - 2*x^5 - 40*x^3)/(x^16 + 2* 
x^12 + 32*x^10 - 5*x^8 - 64*x^6 - 12*x^4 - 32*x^2 - 4), x)
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.76 \[ \int \frac {320 x^3+16 x^5+384 x^7+104 x^9-64 x^{11}-8 x^{13}}{-4-32 x^2-12 x^4-64 x^6-5 x^8+32 x^{10}+2 x^{12}+x^{16}} \, dx=-{\left (x^{4} \sqrt {2 \, \sqrt {2} + 2} + \sqrt {2 \, \sqrt {2} + 2}\right )} \arctan \left (\frac {x^{2}}{\sqrt {\sqrt {2} - 1}}\right ) - \frac {1}{2} \, {\left (x^{4} \sqrt {2 \, \sqrt {2} - 2} + \sqrt {2 \, \sqrt {2} - 2}\right )} \log \left (x^{2} + \sqrt {\sqrt {2} + 1}\right ) + \frac {1}{2} \, {\left (x^{4} \sqrt {2 \, \sqrt {2} - 2} + \sqrt {2 \, \sqrt {2} - 2}\right )} \log \left ({\left | x^{2} - \sqrt {\sqrt {2} + 1} \right |}\right ) \] Input:

integrate((-8*x^13-64*x^11+104*x^9+384*x^7+16*x^5+320*x^3)/(x^16+2*x^12+32 
*x^10-5*x^8-64*x^6-12*x^4-32*x^2-4),x, algorithm="giac")
 

Output:

-(x^4*sqrt(2*sqrt(2) + 2) + sqrt(2*sqrt(2) + 2))*arctan(x^2/sqrt(sqrt(2) - 
 1)) - 1/2*(x^4*sqrt(2*sqrt(2) - 2) + sqrt(2*sqrt(2) - 2))*log(x^2 + sqrt( 
sqrt(2) + 1)) + 1/2*(x^4*sqrt(2*sqrt(2) - 2) + sqrt(2*sqrt(2) - 2))*log(ab 
s(x^2 - sqrt(sqrt(2) + 1)))
 

Mupad [B] (verification not implemented)

Time = 10.30 (sec) , antiderivative size = 629, normalized size of antiderivative = 4.17 \[ \int \frac {320 x^3+16 x^5+384 x^7+104 x^9-64 x^{11}-8 x^{13}}{-4-32 x^2-12 x^4-64 x^6-5 x^8+32 x^{10}+2 x^{12}+x^{16}} \, dx=\text {Too large to display} \] Input:

int(-(320*x^3 + 16*x^5 + 384*x^7 + 104*x^9 - 64*x^11 - 8*x^13)/(32*x^2 + 1 
2*x^4 + 64*x^6 + 5*x^8 - 32*x^10 - 2*x^12 - x^16 + 4),x)
 

Output:

atan(((- 2^(1/2) - 1)^(1/2)*841579333379105903449187258897975096522047488i 
)/(382994240663054868450414598110619467534827520*2^(1/2) + 301510773435543 
4857250577291096881798766919680*2^(1/2)*x^2 - 4264006249616297493976884726 
739467309770866688*x^2 - 541635645052411448624667998445701723706097664) - 
(2^(1/2)*(- 2^(1/2) - 1)^(1/2)*5950864520716020439635588383091151816160706 
56i)/(382994240663054868450414598110619467534827520*2^(1/2) + 301510773435 
5434857250577291096881798766919680*2^(1/2)*x^2 - 4264006249616297493976884 
726739467309770866688*x^2 - 541635645052411448624667998445701723706097664) 
 + (x^2*(- 2^(1/2) - 1)^(1/2)*66253019207824717584647134205028787791499100 
16i)/(382994240663054868450414598110619467534827520*2^(1/2) + 301510773435 
5434857250577291096881798766919680*2^(1/2)*x^2 - 4264006249616297493976884 
726739467309770866688*x^2 - 541635645052411448624667998445701723706097664) 
 - (2^(1/2)*x^2*(- 2^(1/2) - 1)^(1/2)*468479591630585044570147835618845485 
8031890432i)/(382994240663054868450414598110619467534827520*2^(1/2) + 3015 
107734355434857250577291096881798766919680*2^(1/2)*x^2 - 42640062496162974 
93976884726739467309770866688*x^2 - 54163564505241144862466799844570172370 
6097664))*(- 2^(1/2) - 1)^(1/2)*2i - atan(((1 - 2^(1/2))^(1/2)*39140865051 
705846084312643665536808386560000i)/(3817841140772752466962182843819437850 
6240000*2^(1/2) - 7985822117582362245183711943111653457920000*2^(1/2)*x^2 
- 39140865051705846084312643665536808386560000*x^2 + 745219368492329640...
 

Reduce [F]

\[ \int \frac {320 x^3+16 x^5+384 x^7+104 x^9-64 x^{11}-8 x^{13}}{-4-32 x^2-12 x^4-64 x^6-5 x^8+32 x^{10}+2 x^{12}+x^{16}} \, dx=-8 \left (\int \frac {x^{13}}{x^{16}+2 x^{12}+32 x^{10}-5 x^{8}-64 x^{6}-12 x^{4}-32 x^{2}-4}d x \right )-64 \left (\int \frac {x^{11}}{x^{16}+2 x^{12}+32 x^{10}-5 x^{8}-64 x^{6}-12 x^{4}-32 x^{2}-4}d x \right )+104 \left (\int \frac {x^{9}}{x^{16}+2 x^{12}+32 x^{10}-5 x^{8}-64 x^{6}-12 x^{4}-32 x^{2}-4}d x \right )+384 \left (\int \frac {x^{7}}{x^{16}+2 x^{12}+32 x^{10}-5 x^{8}-64 x^{6}-12 x^{4}-32 x^{2}-4}d x \right )+16 \left (\int \frac {x^{5}}{x^{16}+2 x^{12}+32 x^{10}-5 x^{8}-64 x^{6}-12 x^{4}-32 x^{2}-4}d x \right )+320 \left (\int \frac {x^{3}}{x^{16}+2 x^{12}+32 x^{10}-5 x^{8}-64 x^{6}-12 x^{4}-32 x^{2}-4}d x \right ) \] Input:

int((-8*x^13-64*x^11+104*x^9+384*x^7+16*x^5+320*x^3)/(x^16+2*x^12+32*x^10- 
5*x^8-64*x^6-12*x^4-32*x^2-4),x)
 

Output:

8*( - int(x**13/(x**16 + 2*x**12 + 32*x**10 - 5*x**8 - 64*x**6 - 12*x**4 - 
 32*x**2 - 4),x) - 8*int(x**11/(x**16 + 2*x**12 + 32*x**10 - 5*x**8 - 64*x 
**6 - 12*x**4 - 32*x**2 - 4),x) + 13*int(x**9/(x**16 + 2*x**12 + 32*x**10 
- 5*x**8 - 64*x**6 - 12*x**4 - 32*x**2 - 4),x) + 48*int(x**7/(x**16 + 2*x* 
*12 + 32*x**10 - 5*x**8 - 64*x**6 - 12*x**4 - 32*x**2 - 4),x) + 2*int(x**5 
/(x**16 + 2*x**12 + 32*x**10 - 5*x**8 - 64*x**6 - 12*x**4 - 32*x**2 - 4),x 
) + 40*int(x**3/(x**16 + 2*x**12 + 32*x**10 - 5*x**8 - 64*x**6 - 12*x**4 - 
 32*x**2 - 4),x))