\(\int \frac {136 x^3+1092 x^7+3136 x^{11}+508 x^{15}-192 x^{19}+20 x^{23}}{-25-211 x^4-424 x^8-3 x^{12}+48 x^{16}-11 x^{20}+x^{24}} \, dx\) [71]

Optimal result

Integrand size = 64, antiderivative size = 153 \[ \int \frac {136 x^3+1092 x^7+3136 x^{11}+508 x^{15}-192 x^{19}+20 x^{23}}{-25-211 x^4-424 x^8-3 x^{12}+48 x^{16}-11 x^{20}+x^{24}} \, dx=\sqrt {2 \left (5+\sqrt {29}\right )} \arctan \left (\frac {1}{2} \sqrt {-2+\sqrt {29}}-\frac {1}{2} \sqrt {\frac {1}{2} \left (-5+\sqrt {29}\right )} x^4\right )-\sqrt {29} \text {arctanh}\left (\frac {1}{29} \left (-5 \sqrt {29}+2 \sqrt {29} x^4\right )\right )-\sqrt {2 \left (-5+\sqrt {29}\right )} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {29}}}{2}+\frac {1}{2} \sqrt {\frac {1}{2} \left (5+\sqrt {29}\right )} x^4\right )+\frac {5}{2} \log \left (-1-5 x^4+x^8\right ) \] Output:

-(10+2*29^(1/2))^(1/2)*arctan(-1/2*(-2+29^(1/2))^(1/2)+1/4*(-10+2*29^(1/2) 
)^(1/2)*x^4)-29^(1/2)*arctanh(-5/29*29^(1/2)+2/29*29^(1/2)*x^4)-(-10+2*29^ 
(1/2))^(1/2)*arctanh(1/2*(2+29^(1/2))^(1/2)+1/4*(10+2*29^(1/2))^(1/2)*x^4) 
+5/2*ln(x^8-5*x^4-1)
 

Mathematica [C] (verified)

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

Time = 0.04 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.89 \[ \int \frac {136 x^3+1092 x^7+3136 x^{11}+508 x^{15}-192 x^{19}+20 x^{23}}{-25-211 x^4-424 x^8-3 x^{12}+48 x^{16}-11 x^{20}+x^{24}} \, dx=\frac {1}{2} \left (\left (5+\sqrt {29}\right ) \log \left (5+\sqrt {29}-2 x^4\right )-\left (-5+\sqrt {29}\right ) \log \left (-5+\sqrt {29}+2 x^4\right )-4 \text {RootSum}\left [25+86 \text {$\#$1}^4+19 \text {$\#$1}^8-6 \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {-4 \log (x-\text {$\#$1})+14 \log (x-\text {$\#$1}) \text {$\#$1}^4+5 \log (x-\text {$\#$1}) \text {$\#$1}^8}{43+19 \text {$\#$1}^4-9 \text {$\#$1}^8+2 \text {$\#$1}^{12}}\&\right ]\right ) \] Input:

Integrate[(136*x^3 + 1092*x^7 + 3136*x^11 + 508*x^15 - 192*x^19 + 20*x^23) 
/(-25 - 211*x^4 - 424*x^8 - 3*x^12 + 48*x^16 - 11*x^20 + x^24),x]
 

Output:

((5 + Sqrt[29])*Log[5 + Sqrt[29] - 2*x^4] - (-5 + Sqrt[29])*Log[-5 + Sqrt[ 
29] + 2*x^4] - 4*RootSum[25 + 86*#1^4 + 19*#1^8 - 6*#1^12 + #1^16 & , (-4* 
Log[x - #1] + 14*Log[x - #1]*#1^4 + 5*Log[x - #1]*#1^8)/(43 + 19*#1^4 - 9* 
#1^8 + 2*#1^12) & ])/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.

\(\displaystyle \int \frac {20 x^{23}-192 x^{19}+508 x^{15}+3136 x^{11}+1092 x^7+136 x^3}{x^{24}-11 x^{20}+48 x^{16}-3 x^{12}-424 x^8-211 x^4-25} \, dx\)

\(\Big \downarrow \) 2460

\(\displaystyle \int \left (\frac {4 x^3 \left (5 x^4+2\right )}{x^8-5 x^4-1}-\frac {16 x^3 \left (5 x^8+14 x^4-4\right )}{x^{16}-6 x^{12}+19 x^8+86 x^4+25}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 16 \text {Subst}\left (\int \frac {1}{x^4-6 x^3+19 x^2+86 x+25}dx,x,x^4\right )-56 \text {Subst}\left (\int \frac {x}{x^4-6 x^3+19 x^2+86 x+25}dx,x,x^4\right )-20 \text {Subst}\left (\int \frac {x^2}{x^4-6 x^3+19 x^2+86 x+25}dx,x,x^4\right )+\frac {1}{2} \left (5-\sqrt {29}\right ) \log \left (-2 x^4-\sqrt {29}+5\right )+\frac {1}{2} \left (5+\sqrt {29}\right ) \log \left (-2 x^4+\sqrt {29}+5\right )\)

Input:

Int[(136*x^3 + 1092*x^7 + 3136*x^11 + 508*x^15 - 192*x^19 + 20*x^23)/(-25 
- 211*x^4 - 424*x^8 - 3*x^12 + 48*x^16 - 11*x^20 + x^24),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 = 92, normalized size of antiderivative = 0.60

method	result	size
risch	\(\frac {5 \ln \left (2 x^{4}-\sqrt {29}-5\right )}{2}+\frac {\ln \left (2 x^{4}-\sqrt {29}-5\right ) \sqrt {29}}{2}+\frac {5 \ln \left (2 x^{4}+\sqrt {29}-5\right )}{2}-\frac {\ln \left (2 x^{4}+\sqrt {29}-5\right ) \sqrt {29}}{2}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (x^{4}+\textit {\_R}^{2}-2 \textit {\_R} +1\right )\right )\)	\(92\)
default	\(\frac {5 \ln \left (x^{8}-5 x^{4}-1\right )}{2}-\sqrt {29}\, \operatorname {arctanh}\left (\frac {\left (2 x^{4}-5\right ) \sqrt {29}}{29}\right )+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-6 \textit {\_Z}^{3}+19 \textit {\_Z}^{2}+86 \textit {\_Z} +25\right )}{\sum }\frac {\left (-5 \textit {\_R}^{2}-14 \textit {\_R} +4\right ) \ln \left (x^{4}-\textit {\_R} \right )}{2 \textit {\_R}^{3}-9 \textit {\_R}^{2}+19 \textit {\_R} +43}\right )\)	\(93\)

Input:

int((20*x^23-192*x^19+508*x^15+3136*x^11+1092*x^7+136*x^3)/(x^24-11*x^20+4 
8*x^16-3*x^12-424*x^8-211*x^4-25),x,method=_RETURNVERBOSE)
 

Output:

5/2*ln(2*x^4-29^(1/2)-5)+1/2*ln(2*x^4-29^(1/2)-5)*29^(1/2)+5/2*ln(2*x^4+29 
^(1/2)-5)-1/2*ln(2*x^4+29^(1/2)-5)*29^(1/2)+sum(_R*ln(x^4+_R^2-2*_R+1),_R= 
RootOf(_Z^4+5*_Z^2-1))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.06 \[ \int \frac {136 x^3+1092 x^7+3136 x^{11}+508 x^{15}-192 x^{19}+20 x^{23}}{-25-211 x^4-424 x^8-3 x^{12}+48 x^{16}-11 x^{20}+x^{24}} \, dx=-2 \, \sqrt {\frac {1}{2} \, \sqrt {29} + \frac {5}{2}} \arctan \left (-\frac {1}{4} \, {\left (5 \, x^{4} - \sqrt {29} {\left (x^{4} + 1\right )} + 7\right )} \sqrt {\frac {1}{2} \, \sqrt {29} + \frac {5}{2}}\right ) - \sqrt {\frac {1}{2} \, \sqrt {29} - \frac {5}{2}} \log \left (2 \, x^{4} + \sqrt {29} + 4 \, \sqrt {\frac {1}{2} \, \sqrt {29} - \frac {5}{2}} - 3\right ) + \sqrt {\frac {1}{2} \, \sqrt {29} - \frac {5}{2}} \log \left (2 \, x^{4} + \sqrt {29} - 4 \, \sqrt {\frac {1}{2} \, \sqrt {29} - \frac {5}{2}} - 3\right ) + \frac {1}{2} \, \sqrt {29} \log \left (\frac {2 \, x^{8} - 10 \, x^{4} - \sqrt {29} {\left (2 \, x^{4} - 5\right )} + 27}{x^{8} - 5 \, x^{4} - 1}\right ) + \frac {5}{2} \, \log \left (x^{8} - 5 \, x^{4} - 1\right ) \] Input:

integrate((20*x^23-192*x^19+508*x^15+3136*x^11+1092*x^7+136*x^3)/(x^24-11* 
x^20+48*x^16-3*x^12-424*x^8-211*x^4-25),x, algorithm="fricas")
 

Output:

-2*sqrt(1/2*sqrt(29) + 5/2)*arctan(-1/4*(5*x^4 - sqrt(29)*(x^4 + 1) + 7)*s 
qrt(1/2*sqrt(29) + 5/2)) - sqrt(1/2*sqrt(29) - 5/2)*log(2*x^4 + sqrt(29) + 
 4*sqrt(1/2*sqrt(29) - 5/2) - 3) + sqrt(1/2*sqrt(29) - 5/2)*log(2*x^4 + sq 
rt(29) - 4*sqrt(1/2*sqrt(29) - 5/2) - 3) + 1/2*sqrt(29)*log((2*x^8 - 10*x^ 
4 - sqrt(29)*(2*x^4 - 5) + 27)/(x^8 - 5*x^4 - 1)) + 5/2*log(x^8 - 5*x^4 - 
1)
 

Sympy [A] (verification not implemented)

Time = 0.51 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.49 \[ \int \frac {136 x^3+1092 x^7+3136 x^{11}+508 x^{15}-192 x^{19}+20 x^{23}}{-25-211 x^4-424 x^8-3 x^{12}+48 x^{16}-11 x^{20}+x^{24}} \, dx=\left (\frac {5}{2} - \frac {\sqrt {29}}{2}\right ) \log {\left (x^{4} - \frac {166}{35} - \frac {2 \left (\frac {5}{2} - \frac {\sqrt {29}}{2}\right )^{5}}{7} + \frac {51 \left (\frac {5}{2} - \frac {\sqrt {29}}{2}\right )^{4}}{35} - \frac {10 \left (\frac {5}{2} - \frac {\sqrt {29}}{2}\right )^{3}}{7} + \frac {58 \left (\frac {5}{2} - \frac {\sqrt {29}}{2}\right )^{2}}{7} + \frac {6 \sqrt {29}}{7} \right )} + \left (\frac {5}{2} + \frac {\sqrt {29}}{2}\right ) \log {\left (x^{4} - \frac {2 \left (\frac {5}{2} + \frac {\sqrt {29}}{2}\right )^{5}}{7} - \frac {10 \left (\frac {5}{2} + \frac {\sqrt {29}}{2}\right )^{3}}{7} - \frac {166}{35} - \frac {6 \sqrt {29}}{7} + \frac {58 \left (\frac {5}{2} + \frac {\sqrt {29}}{2}\right )^{2}}{7} + \frac {51 \left (\frac {5}{2} + \frac {\sqrt {29}}{2}\right )^{4}}{35} \right )} + \operatorname {RootSum} {\left (t^{4} + 5 t^{2} - 1, \left ( t \mapsto t \log {\left (- \frac {2 t^{5}}{7} + \frac {51 t^{4}}{35} - \frac {10 t^{3}}{7} + \frac {58 t^{2}}{7} - \frac {12 t}{7} + x^{4} - \frac {16}{35} \right )} \right )\right )} \] Input:

integrate((20*x**23-192*x**19+508*x**15+3136*x**11+1092*x**7+136*x**3)/(x* 
*24-11*x**20+48*x**16-3*x**12-424*x**8-211*x**4-25),x)
 

Output:

(5/2 - sqrt(29)/2)*log(x**4 - 166/35 - 2*(5/2 - sqrt(29)/2)**5/7 + 51*(5/2 
 - sqrt(29)/2)**4/35 - 10*(5/2 - sqrt(29)/2)**3/7 + 58*(5/2 - sqrt(29)/2)* 
*2/7 + 6*sqrt(29)/7) + (5/2 + sqrt(29)/2)*log(x**4 - 2*(5/2 + sqrt(29)/2)* 
*5/7 - 10*(5/2 + sqrt(29)/2)**3/7 - 166/35 - 6*sqrt(29)/7 + 58*(5/2 + sqrt 
(29)/2)**2/7 + 51*(5/2 + sqrt(29)/2)**4/35) + RootSum(_t**4 + 5*_t**2 - 1, 
 Lambda(_t, _t*log(-2*_t**5/7 + 51*_t**4/35 - 10*_t**3/7 + 58*_t**2/7 - 12 
*_t/7 + x**4 - 16/35)))
 

Maxima [F]

\[ \int \frac {136 x^3+1092 x^7+3136 x^{11}+508 x^{15}-192 x^{19}+20 x^{23}}{-25-211 x^4-424 x^8-3 x^{12}+48 x^{16}-11 x^{20}+x^{24}} \, dx=\int { \frac {4 \, {\left (5 \, x^{23} - 48 \, x^{19} + 127 \, x^{15} + 784 \, x^{11} + 273 \, x^{7} + 34 \, x^{3}\right )}}{x^{24} - 11 \, x^{20} + 48 \, x^{16} - 3 \, x^{12} - 424 \, x^{8} - 211 \, x^{4} - 25} \,d x } \] Input:

integrate((20*x^23-192*x^19+508*x^15+3136*x^11+1092*x^7+136*x^3)/(x^24-11* 
x^20+48*x^16-3*x^12-424*x^8-211*x^4-25),x, algorithm="maxima")
 

Output:

4*integrate((5*x^23 - 48*x^19 + 127*x^15 + 784*x^11 + 273*x^7 + 34*x^3)/(x 
^24 - 11*x^20 + 48*x^16 - 3*x^12 - 424*x^8 - 211*x^4 - 25), x)
 

Giac [A] (verification not implemented)

Time = 0.54 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.31 \[ \int \frac {136 x^3+1092 x^7+3136 x^{11}+508 x^{15}-192 x^{19}+20 x^{23}}{-25-211 x^4-424 x^8-3 x^{12}+48 x^{16}-11 x^{20}+x^{24}} \, dx=\frac {1}{2} \, \sqrt {29} \log \left (\frac {{\left | 2 \, x^{4} - \sqrt {29} - 5 \right |}}{{\left | 2 \, x^{4} + \sqrt {29} - 5 \right |}}\right ) + \frac {5}{2} \, \log \left ({\left | x^{8} - 5 \, x^{4} - 1 \right |}\right ) \] Input:

integrate((20*x^23-192*x^19+508*x^15+3136*x^11+1092*x^7+136*x^3)/(x^24-11* 
x^20+48*x^16-3*x^12-424*x^8-211*x^4-25),x, algorithm="giac")
 

Output:

1/2*sqrt(29)*log(abs(2*x^4 - sqrt(29) - 5)/abs(2*x^4 + sqrt(29) - 5)) + 5/ 
2*log(abs(x^8 - 5*x^4 - 1))
 

Mupad [B] (verification not implemented)

Time = 0.69 (sec) , antiderivative size = 384, normalized size of antiderivative = 2.51 \[ \int \frac {136 x^3+1092 x^7+3136 x^{11}+508 x^{15}-192 x^{19}+20 x^{23}}{-25-211 x^4-424 x^8-3 x^{12}+48 x^{16}-11 x^{20}+x^{24}} \, dx=\text {Too large to display} \] Input:

int(-(136*x^3 + 1092*x^7 + 3136*x^11 + 508*x^15 - 192*x^19 + 20*x^23)/(211 
*x^4 + 424*x^8 + 3*x^12 - 48*x^16 + 11*x^20 - x^24 + 25),x)
 

Output:

log(x^4 - 29^(1/2)/2 - 5/2)*(29^(1/2)/2 + 5/2) - log(29^(1/2)/2 + x^4 - 5/ 
2)*(29^(1/2)/2 - 5/2) + 2^(1/2)*atan((2^(1/2)*(- 29^(1/2) - 5)^(1/2)*89447 
86416240580834974380694522926529073777921316462966865920000000000000000000 
000i)/(1457838022316823349565294317840265749376152204574607005450240000000 
000000000000000*29^(1/2) + 46295538123172172900694516550692966436466410249 
18408984002560000000000000000000000*29^(1/2)*x^4 - 24930910262826055278263 
661804971438378100028050743841657978880000000000000000000000*x^4 - 7850698 
01228312255625499542865461897922948999129957822103552000000000000000000000 
0) + (2^(1/2)*x^4*(- 29^(1/2) - 5)^(1/2)*284053299610472748051489022838961 
08710549615273584907234836480000000000000000000000i)/(14578380223168233495 
65294317840265749376152204574607005450240000000000000000000000*29^(1/2) + 
46295538123172172900694516550692966436466410249184089840025600000000000000 
00000000*29^(1/2)*x^4 - 24930910262826055278263661804971438378100028050743 
841657978880000000000000000000000*x^4 - 7850698012283122556254995428654618 
979229489991299578221035520000000000000000000000) - (2^(1/2)*29^(1/2)*(- 2 
9^(1/2) - 5)^(1/2)*1661005138485294424129682667016151369593867072311527515 
095040000000000000000000000i)/(1457838022316823349565294317840265749376152 
204574607005450240000000000000000000000*29^(1/2) + 46295538123172172900694 
51655069296643646641024918408984002560000000000000000000000*29^(1/2)*x^4 - 
 2493091026282605527826366180497143837810002805074384165797888000000000...
 

Reduce [F]

\[ \int \frac {136 x^3+1092 x^7+3136 x^{11}+508 x^{15}-192 x^{19}+20 x^{23}}{-25-211 x^4-424 x^8-3 x^{12}+48 x^{16}-11 x^{20}+x^{24}} \, dx=-\frac {53 \sqrt {29}\, \mathrm {log}\left (-\left (\sqrt {29}-5\right )^{\frac {1}{4}} \sqrt {2}\, 2^{\frac {1}{4}} x +\sqrt {\sqrt {29}-5}+\sqrt {2}\, x^{2}\right )}{522}+\frac {53 \sqrt {29}\, \mathrm {log}\left (-\left (\sqrt {29}+5\right )^{\frac {1}{4}} 2^{\frac {1}{4}}+\sqrt {2}\, x \right )}{522}+\frac {53 \sqrt {29}\, \mathrm {log}\left (\sqrt {\sqrt {29}+5}+\sqrt {2}\, x^{2}\right )}{522}-\frac {53 \sqrt {29}\, \mathrm {log}\left (\left (\sqrt {29}-5\right )^{\frac {1}{4}} \sqrt {2}\, 2^{\frac {1}{4}} x +\sqrt {\sqrt {29}-5}+\sqrt {2}\, x^{2}\right )}{522}+\frac {53 \sqrt {29}\, \mathrm {log}\left (\left (\sqrt {29}+5\right )^{\frac {1}{4}} 2^{\frac {1}{4}}+\sqrt {2}\, x \right )}{522}+\frac {37520 \left (\int \frac {x^{11}}{x^{24}-11 x^{20}+48 x^{16}-3 x^{12}-424 x^{8}-211 x^{4}-25}d x \right )}{9}+\frac {4240 \left (\int \frac {x^{7}}{x^{24}-11 x^{20}+48 x^{16}-3 x^{12}-424 x^{8}-211 x^{4}-25}d x \right )}{3}-\frac {880 \left (\int \frac {x^{3}}{x^{24}-11 x^{20}+48 x^{16}-3 x^{12}-424 x^{8}-211 x^{4}-25}d x \right )}{3}+\frac {19 \,\mathrm {log}\left (x^{16}-6 x^{12}+19 x^{8}+86 x^{4}+25\right )}{9}-\frac {31 \,\mathrm {log}\left (-\left (\sqrt {29}-5\right )^{\frac {1}{4}} \sqrt {2}\, 2^{\frac {1}{4}} x +\sqrt {\sqrt {29}-5}+\sqrt {2}\, x^{2}\right )}{18}-\frac {31 \,\mathrm {log}\left (-\left (\sqrt {29}+5\right )^{\frac {1}{4}} 2^{\frac {1}{4}}+\sqrt {2}\, x \right )}{18}-\frac {31 \,\mathrm {log}\left (\sqrt {\sqrt {29}+5}+\sqrt {2}\, x^{2}\right )}{18}-\frac {31 \,\mathrm {log}\left (\left (\sqrt {29}-5\right )^{\frac {1}{4}} \sqrt {2}\, 2^{\frac {1}{4}} x +\sqrt {\sqrt {29}-5}+\sqrt {2}\, x^{2}\right )}{18}-\frac {31 \,\mathrm {log}\left (\left (\sqrt {29}+5\right )^{\frac {1}{4}} 2^{\frac {1}{4}}+\sqrt {2}\, x \right )}{18} \] Input:

int((20*x^23-192*x^19+508*x^15+3136*x^11+1092*x^7+136*x^3)/(x^24-11*x^20+4 
8*x^16-3*x^12-424*x^8-211*x^4-25),x)
 

Output:

( - 53*sqrt(29)*log( - (sqrt(29) - 5)**(1/4)*sqrt(2)*2**(1/4)*x + sqrt(sqr 
t(29) - 5) + sqrt(2)*x**2) + 53*sqrt(29)*log( - (sqrt(29) + 5)**(1/4)*2**( 
1/4) + sqrt(2)*x) + 53*sqrt(29)*log(sqrt(sqrt(29) + 5) + sqrt(2)*x**2) - 5 
3*sqrt(29)*log((sqrt(29) - 5)**(1/4)*sqrt(2)*2**(1/4)*x + sqrt(sqrt(29) - 
5) + sqrt(2)*x**2) + 53*sqrt(29)*log((sqrt(29) + 5)**(1/4)*2**(1/4) + sqrt 
(2)*x) + 2176160*int(x**11/(x**24 - 11*x**20 + 48*x**16 - 3*x**12 - 424*x* 
*8 - 211*x**4 - 25),x) + 737760*int(x**7/(x**24 - 11*x**20 + 48*x**16 - 3* 
x**12 - 424*x**8 - 211*x**4 - 25),x) - 153120*int(x**3/(x**24 - 11*x**20 + 
 48*x**16 - 3*x**12 - 424*x**8 - 211*x**4 - 25),x) + 1102*log(x**16 - 6*x* 
*12 + 19*x**8 + 86*x**4 + 25) - 899*log( - (sqrt(29) - 5)**(1/4)*sqrt(2)*2 
**(1/4)*x + sqrt(sqrt(29) - 5) + sqrt(2)*x**2) - 899*log( - (sqrt(29) + 5) 
**(1/4)*2**(1/4) + sqrt(2)*x) - 899*log(sqrt(sqrt(29) + 5) + sqrt(2)*x**2) 
 - 899*log((sqrt(29) - 5)**(1/4)*sqrt(2)*2**(1/4)*x + sqrt(sqrt(29) - 5) + 
 sqrt(2)*x**2) - 899*log((sqrt(29) + 5)**(1/4)*2**(1/4) + sqrt(2)*x))/522