\(\int \frac {-48 x^3-640 x^7-896 x^{11}-928 x^{15}-3960 x^{19}-1312 x^{23}+224 x^{27}}{16+76 x^4+217 x^8+576 x^{12}+771 x^{16}+460 x^{20}+238 x^{24}+88 x^{28}+8 x^{32}} \, dx\) [66]

Optimal result

Integrand size = 81, antiderivative size = 181 \[ \int \frac {-48 x^3-640 x^7-896 x^{11}-928 x^{15}-3960 x^{19}-1312 x^{23}+224 x^{27}}{16+76 x^4+217 x^8+576 x^{12}+771 x^{16}+460 x^{20}+238 x^{24}+88 x^{28}+8 x^{32}} \, dx=-\sqrt {7+\sqrt {41}} \arctan \left (\sqrt {\frac {1}{2} \left (7-\sqrt {41}\right )} x^4\right )-\sqrt {7-\sqrt {41}} \arctan \left (\sqrt {\frac {1}{2} \left (7+\sqrt {41}\right )} x^4\right )-\sqrt {7+\sqrt {41}} \text {arctanh}\left (\frac {\sqrt {29-\sqrt {41}}}{4}+\frac {1}{2} \sqrt {\frac {1}{2} \left (7-\sqrt {41}\right )} x^4\right )-\sqrt {7-\sqrt {41}} \text {arctanh}\left (\frac {\sqrt {29+\sqrt {41}}}{4}+\frac {1}{2} \sqrt {\frac {1}{2} \left (7+\sqrt {41}\right )} x^4\right ) \] Output:

-(7+41^(1/2))^(1/2)*arctan(1/2*(14-2*41^(1/2))^(1/2)*x^4)-(7-41^(1/2))^(1/ 
2)*arctan(1/2*(14+2*41^(1/2))^(1/2)*x^4)-(7+41^(1/2))^(1/2)*arctanh(1/4*(2 
9-41^(1/2))^(1/2)+1/4*(14-2*41^(1/2))^(1/2)*x^4)-(7-41^(1/2))^(1/2)*arctan 
h(1/4*(29+41^(1/2))^(1/2)+1/4*(14+2*41^(1/2))^(1/2)*x^4)
 

Mathematica [C] (verified)

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

Time = 0.08 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.86 \[ \int \frac {-48 x^3-640 x^7-896 x^{11}-928 x^{15}-3960 x^{19}-1312 x^{23}+224 x^{27}}{16+76 x^4+217 x^8+576 x^{12}+771 x^{16}+460 x^{20}+238 x^{24}+88 x^{28}+8 x^{32}} \, dx=8 \left (-\frac {1}{8} \text {RootSum}\left [1+7 \text {$\#$1}^8+2 \text {$\#$1}^{16}\&,\frac {2 \log (x-\text {$\#$1})+7 \log (x-\text {$\#$1}) \text {$\#$1}^8}{7 \text {$\#$1}^4+4 \text {$\#$1}^{12}}\&\right ]+\frac {1}{4} \text {RootSum}\left [16+76 \text {$\#$1}^4+105 \text {$\#$1}^8+44 \text {$\#$1}^{12}+4 \text {$\#$1}^{16}\&,\frac {13 \log (x-\text {$\#$1})+36 \log (x-\text {$\#$1}) \text {$\#$1}^4+14 \log (x-\text {$\#$1}) \text {$\#$1}^8}{38+105 \text {$\#$1}^4+66 \text {$\#$1}^8+8 \text {$\#$1}^{12}}\&\right ]\right ) \] Input:

Integrate[(-48*x^3 - 640*x^7 - 896*x^11 - 928*x^15 - 3960*x^19 - 1312*x^23 
 + 224*x^27)/(16 + 76*x^4 + 217*x^8 + 576*x^12 + 771*x^16 + 460*x^20 + 238 
*x^24 + 88*x^28 + 8*x^32),x]
 

Output:

8*(-1/8*RootSum[1 + 7*#1^8 + 2*#1^16 & , (2*Log[x - #1] + 7*Log[x - #1]*#1 
^8)/(7*#1^4 + 4*#1^12) & ] + RootSum[16 + 76*#1^4 + 105*#1^8 + 44*#1^12 + 
4*#1^16 & , (13*Log[x - #1] + 36*Log[x - #1]*#1^4 + 14*Log[x - #1]*#1^8)/( 
38 + 105*#1^4 + 66*#1^8 + 8*#1^12) & ]/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.

\(\displaystyle \int \frac {224 x^{27}-1312 x^{23}-3960 x^{19}-928 x^{15}-896 x^{11}-640 x^7-48 x^3}{8 x^{32}+88 x^{28}+238 x^{24}+460 x^{20}+771 x^{16}+576 x^{12}+217 x^8+76 x^4+16} \, dx\)

\(\Big \downarrow \) 2460

\(\displaystyle \int \left (\frac {16 x^3 \left (14 x^8+36 x^4+13\right )}{4 x^{16}+44 x^{12}+105 x^8+76 x^4+16}-\frac {8 x^3 \left (7 x^8+2\right )}{2 x^{16}+7 x^8+1}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 52 \text {Subst}\left (\int \frac {1}{4 x^4+44 x^3+105 x^2+76 x+16}dx,x,x^4\right )+144 \text {Subst}\left (\int \frac {x}{4 x^4+44 x^3+105 x^2+76 x+16}dx,x,x^4\right )+56 \text {Subst}\left (\int \frac {x^2}{4 x^4+44 x^3+105 x^2+76 x+16}dx,x,x^4\right )-\sqrt {7-\sqrt {41}} \arctan \left (\frac {2 x^4}{\sqrt {7-\sqrt {41}}}\right )-\sqrt {7+\sqrt {41}} \arctan \left (\frac {2 x^4}{\sqrt {7+\sqrt {41}}}\right )\)

Input:

Int[(-48*x^3 - 640*x^7 - 896*x^11 - 928*x^15 - 3960*x^19 - 1312*x^23 + 224 
*x^27)/(16 + 76*x^4 + 217*x^8 + 576*x^12 + 771*x^16 + 460*x^20 + 238*x^24 
+ 88*x^28 + 8*x^32),x]
 

Output:

$Aborted
 

Maple [C] (verified)

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

Time = 0.25 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.32

method	result	size
risch	\(\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{4}-7 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (x^{4}+\textit {\_R}^{2}-2 \textit {\_R} +1\right )\right )+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{4}+7 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (x^{4}-\textit {\_R} \right )\right )\)	\(58\)
default	\(-\frac {\left (41+7 \sqrt {41}\right ) \sqrt {41}\, \arctan \left (\frac {2 x^{4}}{\sqrt {7+\sqrt {41}}}\right )}{41 \sqrt {7+\sqrt {41}}}-\frac {\left (-41+7 \sqrt {41}\right ) \sqrt {41}\, \arctan \left (\frac {2 x^{4}}{\sqrt {7-\sqrt {41}}}\right )}{41 \sqrt {7-\sqrt {41}}}+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (4 \textit {\_Z}^{4}+44 \textit {\_Z}^{3}+105 \textit {\_Z}^{2}+76 \textit {\_Z} +16\right )}{\sum }\frac {\left (14 \textit {\_R}^{2}+36 \textit {\_R} +13\right ) \ln \left (x^{4}-\textit {\_R} \right )}{8 \textit {\_R}^{3}+66 \textit {\_R}^{2}+105 \textit {\_R} +38}\right )\)	\(132\)

Input:

int((224*x^27-1312*x^23-3960*x^19-928*x^15-896*x^11-640*x^7-48*x^3)/(8*x^3 
2+88*x^28+238*x^24+460*x^20+771*x^16+576*x^12+217*x^8+76*x^4+16),x,method= 
_RETURNVERBOSE)
 

Output:

sum(_R*ln(x^4+_R^2-2*_R+1),_R=RootOf(2*_Z^4-7*_Z^2+1))+sum(_R*ln(x^4-_R),_ 
R=RootOf(2*_Z^4+7*_Z^2+1))
 

Fricas [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.08 \[ \int \frac {-48 x^3-640 x^7-896 x^{11}-928 x^{15}-3960 x^{19}-1312 x^{23}+224 x^{27}}{16+76 x^4+217 x^8+576 x^{12}+771 x^{16}+460 x^{20}+238 x^{24}+88 x^{28}+8 x^{32}} \, dx=\sqrt {\sqrt {41} + 7} \arctan \left (\frac {1}{4} \, {\left (\sqrt {41} x^{4} - 7 \, x^{4}\right )} \sqrt {\sqrt {41} + 7}\right ) - \sqrt {-\sqrt {41} + 7} \arctan \left (\frac {1}{4} \, {\left (\sqrt {41} x^{4} + 7 \, x^{4}\right )} \sqrt {-\sqrt {41} + 7}\right ) - \frac {1}{2} \, \sqrt {\sqrt {41} + 7} \log \left (4 \, x^{4} + \sqrt {41} + 4 \, \sqrt {\sqrt {41} + 7} + 11\right ) + \frac {1}{2} \, \sqrt {\sqrt {41} + 7} \log \left (4 \, x^{4} + \sqrt {41} - 4 \, \sqrt {\sqrt {41} + 7} + 11\right ) - \frac {1}{2} \, \sqrt {-\sqrt {41} + 7} \log \left (4 \, x^{4} - \sqrt {41} + 4 \, \sqrt {-\sqrt {41} + 7} + 11\right ) + \frac {1}{2} \, \sqrt {-\sqrt {41} + 7} \log \left (4 \, x^{4} - \sqrt {41} - 4 \, \sqrt {-\sqrt {41} + 7} + 11\right ) \] Input:

integrate((224*x^27-1312*x^23-3960*x^19-928*x^15-896*x^11-640*x^7-48*x^3)/ 
(8*x^32+88*x^28+238*x^24+460*x^20+771*x^16+576*x^12+217*x^8+76*x^4+16),x, 
algorithm="fricas")
 

Output:

sqrt(sqrt(41) + 7)*arctan(1/4*(sqrt(41)*x^4 - 7*x^4)*sqrt(sqrt(41) + 7)) - 
 sqrt(-sqrt(41) + 7)*arctan(1/4*(sqrt(41)*x^4 + 7*x^4)*sqrt(-sqrt(41) + 7) 
) - 1/2*sqrt(sqrt(41) + 7)*log(4*x^4 + sqrt(41) + 4*sqrt(sqrt(41) + 7) + 1 
1) + 1/2*sqrt(sqrt(41) + 7)*log(4*x^4 + sqrt(41) - 4*sqrt(sqrt(41) + 7) + 
11) - 1/2*sqrt(-sqrt(41) + 7)*log(4*x^4 - sqrt(41) + 4*sqrt(-sqrt(41) + 7) 
 + 11) + 1/2*sqrt(-sqrt(41) + 7)*log(4*x^4 - sqrt(41) - 4*sqrt(-sqrt(41) + 
 7) + 11)
 

Sympy [A] (verification not implemented)

Time = 0.98 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.65 \[ \int \frac {-48 x^3-640 x^7-896 x^{11}-928 x^{15}-3960 x^{19}-1312 x^{23}+224 x^{27}}{16+76 x^4+217 x^8+576 x^{12}+771 x^{16}+460 x^{20}+238 x^{24}+88 x^{28}+8 x^{32}} \, dx=\operatorname {RootSum} {\left (2 t^{4} - 7 t^{2} + 1, \left ( t \mapsto t \log {\left (\frac {2 t^{7}}{7} - \frac {2 t^{6}}{7} + \frac {t^{4}}{7} - \frac {47 t^{3}}{14} + \frac {27 t^{2}}{7} - \frac {3 t}{2} + x^{4} + \frac {4}{7} \right )} \right )\right )} + \operatorname {RootSum} {\left (2 t^{4} + 7 t^{2} + 1, \left ( t \mapsto t \log {\left (\frac {2 t^{7}}{7} - \frac {2 t^{6}}{7} + \frac {t^{4}}{7} - \frac {47 t^{3}}{14} + \frac {27 t^{2}}{7} - \frac {3 t}{2} + x^{4} + \frac {4}{7} \right )} \right )\right )} \] Input:

integrate((224*x**27-1312*x**23-3960*x**19-928*x**15-896*x**11-640*x**7-48 
*x**3)/(8*x**32+88*x**28+238*x**24+460*x**20+771*x**16+576*x**12+217*x**8+ 
76*x**4+16),x)
 

Output:

RootSum(2*_t**4 - 7*_t**2 + 1, Lambda(_t, _t*log(2*_t**7/7 - 2*_t**6/7 + _ 
t**4/7 - 47*_t**3/14 + 27*_t**2/7 - 3*_t/2 + x**4 + 4/7))) + RootSum(2*_t* 
*4 + 7*_t**2 + 1, Lambda(_t, _t*log(2*_t**7/7 - 2*_t**6/7 + _t**4/7 - 47*_ 
t**3/14 + 27*_t**2/7 - 3*_t/2 + x**4 + 4/7)))
 

Maxima [F]

\[ \int \frac {-48 x^3-640 x^7-896 x^{11}-928 x^{15}-3960 x^{19}-1312 x^{23}+224 x^{27}}{16+76 x^4+217 x^8+576 x^{12}+771 x^{16}+460 x^{20}+238 x^{24}+88 x^{28}+8 x^{32}} \, dx=\int { \frac {8 \, {\left (28 \, x^{27} - 164 \, x^{23} - 495 \, x^{19} - 116 \, x^{15} - 112 \, x^{11} - 80 \, x^{7} - 6 \, x^{3}\right )}}{8 \, x^{32} + 88 \, x^{28} + 238 \, x^{24} + 460 \, x^{20} + 771 \, x^{16} + 576 \, x^{12} + 217 \, x^{8} + 76 \, x^{4} + 16} \,d x } \] Input:

integrate((224*x^27-1312*x^23-3960*x^19-928*x^15-896*x^11-640*x^7-48*x^3)/ 
(8*x^32+88*x^28+238*x^24+460*x^20+771*x^16+576*x^12+217*x^8+76*x^4+16),x, 
algorithm="maxima")
 

Output:

8*integrate((28*x^27 - 164*x^23 - 495*x^19 - 116*x^15 - 112*x^11 - 80*x^7 
- 6*x^3)/(8*x^32 + 88*x^28 + 238*x^24 + 460*x^20 + 771*x^16 + 576*x^12 + 2 
17*x^8 + 76*x^4 + 16), x)
 

Giac [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.46 \[ \int \frac {-48 x^3-640 x^7-896 x^{11}-928 x^{15}-3960 x^{19}-1312 x^{23}+224 x^{27}}{16+76 x^4+217 x^8+576 x^{12}+771 x^{16}+460 x^{20}+238 x^{24}+88 x^{28}+8 x^{32}} \, dx=\frac {1}{41} \, {\left (7 \, x^{8} \sqrt {-82 \, \sqrt {41} + 574} + 2 \, \sqrt {-82 \, \sqrt {41} + 574}\right )} \arctan \left (\frac {2 \, x^{4}}{\sqrt {\sqrt {41} + 7}}\right ) - \frac {1}{41} \, {\left (7 \, x^{8} \sqrt {82 \, \sqrt {41} + 574} + 2 \, \sqrt {82 \, \sqrt {41} + 574}\right )} \arctan \left (\frac {x^{4}}{\sqrt {-\frac {1}{4} \, \sqrt {41} + \frac {7}{4}}}\right ) \] Input:

integrate((224*x^27-1312*x^23-3960*x^19-928*x^15-896*x^11-640*x^7-48*x^3)/ 
(8*x^32+88*x^28+238*x^24+460*x^20+771*x^16+576*x^12+217*x^8+76*x^4+16),x, 
algorithm="giac")
 

Output:

1/41*(7*x^8*sqrt(-82*sqrt(41) + 574) + 2*sqrt(-82*sqrt(41) + 574))*arctan( 
2*x^4/sqrt(sqrt(41) + 7)) - 1/41*(7*x^8*sqrt(82*sqrt(41) + 574) + 2*sqrt(8 
2*sqrt(41) + 574))*arctan(x^4/sqrt(-1/4*sqrt(41) + 7/4))
 

Mupad [B] (verification not implemented)

Time = 10.37 (sec) , antiderivative size = 629, normalized size of antiderivative = 3.48 \[ \int \frac {-48 x^3-640 x^7-896 x^{11}-928 x^{15}-3960 x^{19}-1312 x^{23}+224 x^{27}}{16+76 x^4+217 x^8+576 x^{12}+771 x^{16}+460 x^{20}+238 x^{24}+88 x^{28}+8 x^{32}} \, dx=\text {Too large to display} \] Input:

int(-(48*x^3 + 640*x^7 + 896*x^11 + 928*x^15 + 3960*x^19 + 1312*x^23 - 224 
*x^27)/(76*x^4 + 217*x^8 + 576*x^12 + 771*x^16 + 460*x^20 + 238*x^24 + 88* 
x^28 + 8*x^32 + 16),x)
 

Output:

atan(((- 41^(1/2) - 7)^(1/2)*400088822273805431939150701133361978714566596 
82194924814904907552130859880405890234375i)/(128*((24471814588661885266627 
76646848037686590845300635962861999156225153644617329736752734375*41^(1/2) 
)/8192 - (6248337367708426852640351508190625035908461961705194913276183738 
721025221516073828125*41^(1/2)*x^4)/64 + (40008882227380543193915070113336 
197871456659682194924814904907552130859880405890234375*x^4)/64 - 156696051 
87166514361043137847068345981278008800225210012892253244081958050600228389 
453125/8192)) + (x^4*(- 41^(1/2) - 7)^(1/2)*105010838061839665060237903725 
123983509585359463825473716105588969936567832360402794140625i)/(16384*((24 
47181458866188526662776646848037686590845300635962861999156225153644617329 
736752734375*41^(1/2))/8192 - (6248337367708426852640351508190625035908461 
961705194913276183738721025221516073828125*41^(1/2)*x^4)/64 + (40008882227 
38054319391507011333619787145665968219492481490490755213085988040589023437 
5*x^4)/64 - 15669605187166514361043137847068345981278008800225210012892253 
244081958050600228389453125/8192)) - (41^(1/2)*(- 41^(1/2) - 7)^(1/2)*6248 
33736770842685264035150819062503590846196170519491327618373872102522151607 
3828125i)/(128*((244718145886618852666277664684803768659084530063596286199 
9156225153644617329736752734375*41^(1/2))/8192 - (624833736770842685264035 
1508190625035908461961705194913276183738721025221516073828125*41^(1/2)*x^4 
)/64 + (400088822273805431939150701133361978714566596821949248149049075...
 

Reduce [F]

\[ \int \frac {-48 x^3-640 x^7-896 x^{11}-928 x^{15}-3960 x^{19}-1312 x^{23}+224 x^{27}}{16+76 x^4+217 x^8+576 x^{12}+771 x^{16}+460 x^{20}+238 x^{24}+88 x^{28}+8 x^{32}} \, dx=224 \left (\int \frac {x^{27}}{8 x^{32}+88 x^{28}+238 x^{24}+460 x^{20}+771 x^{16}+576 x^{12}+217 x^{8}+76 x^{4}+16}d x \right )-1312 \left (\int \frac {x^{23}}{8 x^{32}+88 x^{28}+238 x^{24}+460 x^{20}+771 x^{16}+576 x^{12}+217 x^{8}+76 x^{4}+16}d x \right )-3960 \left (\int \frac {x^{19}}{8 x^{32}+88 x^{28}+238 x^{24}+460 x^{20}+771 x^{16}+576 x^{12}+217 x^{8}+76 x^{4}+16}d x \right )-928 \left (\int \frac {x^{15}}{8 x^{32}+88 x^{28}+238 x^{24}+460 x^{20}+771 x^{16}+576 x^{12}+217 x^{8}+76 x^{4}+16}d x \right )-896 \left (\int \frac {x^{11}}{8 x^{32}+88 x^{28}+238 x^{24}+460 x^{20}+771 x^{16}+576 x^{12}+217 x^{8}+76 x^{4}+16}d x \right )-640 \left (\int \frac {x^{7}}{8 x^{32}+88 x^{28}+238 x^{24}+460 x^{20}+771 x^{16}+576 x^{12}+217 x^{8}+76 x^{4}+16}d x \right )-48 \left (\int \frac {x^{3}}{8 x^{32}+88 x^{28}+238 x^{24}+460 x^{20}+771 x^{16}+576 x^{12}+217 x^{8}+76 x^{4}+16}d x \right ) \] Input:

int((224*x^27-1312*x^23-3960*x^19-928*x^15-896*x^11-640*x^7-48*x^3)/(8*x^3 
2+88*x^28+238*x^24+460*x^20+771*x^16+576*x^12+217*x^8+76*x^4+16),x)
 

Output:

8*(28*int(x**27/(8*x**32 + 88*x**28 + 238*x**24 + 460*x**20 + 771*x**16 + 
576*x**12 + 217*x**8 + 76*x**4 + 16),x) - 164*int(x**23/(8*x**32 + 88*x**2 
8 + 238*x**24 + 460*x**20 + 771*x**16 + 576*x**12 + 217*x**8 + 76*x**4 + 1 
6),x) - 495*int(x**19/(8*x**32 + 88*x**28 + 238*x**24 + 460*x**20 + 771*x* 
*16 + 576*x**12 + 217*x**8 + 76*x**4 + 16),x) - 116*int(x**15/(8*x**32 + 8 
8*x**28 + 238*x**24 + 460*x**20 + 771*x**16 + 576*x**12 + 217*x**8 + 76*x* 
*4 + 16),x) - 112*int(x**11/(8*x**32 + 88*x**28 + 238*x**24 + 460*x**20 + 
771*x**16 + 576*x**12 + 217*x**8 + 76*x**4 + 16),x) - 80*int(x**7/(8*x**32 
 + 88*x**28 + 238*x**24 + 460*x**20 + 771*x**16 + 576*x**12 + 217*x**8 + 7 
6*x**4 + 16),x) - 6*int(x**3/(8*x**32 + 88*x**28 + 238*x**24 + 460*x**20 + 
 771*x**16 + 576*x**12 + 217*x**8 + 76*x**4 + 16),x))