\(\int \frac {624 x^3+144 x^7+24 x^{11}}{(-460-936 x^4-376 x^8-36 x^{12}-x^{16})^2} \, dx\) [46]

Optimal result

Integrand size = 41, antiderivative size = 136 \[ \int \frac {624 x^3+144 x^7+24 x^{11}}{\left (-460-936 x^4-376 x^8-36 x^{12}-x^{16}\right )^2} \, dx=\frac {-566-720 x^4-190 x^8-9 x^{12}}{636 \left (460+936 x^4+376 x^8+36 x^{12}+x^{16}\right )}-\frac {2791 \text {arctanh}\left (\frac {1}{18} \left (8 \sqrt {6}+\sqrt {6} x^4\right )\right )}{202248 \sqrt {6}}+\frac {2827 \text {arctanh}\left (\frac {1}{18} \left (10 \sqrt {6}+\sqrt {6} x^4\right )\right )}{202248 \sqrt {6}}-\frac {153 \log \left (10+16 x^4+x^8\right )}{44944}+\frac {153 \log \left (46+20 x^4+x^8\right )}{44944} \] Output:

(-9*x^12-190*x^8-720*x^4-566)/(636*x^16+22896*x^12+239136*x^8+595296*x^4+2 
92560)-2791/1213488*arctanh(4/9*6^(1/2)+1/18*6^(1/2)*x^4)*6^(1/2)+2827/121 
3488*arctanh(5/9*6^(1/2)+1/18*6^(1/2)*x^4)*6^(1/2)-153/44944*ln(x^8+16*x^4 
+10)+153/44944*ln(x^8+20*x^4+46)
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.05 \[ \int \frac {624 x^3+144 x^7+24 x^{11}}{\left (-460-936 x^4-376 x^8-36 x^{12}-x^{16}\right )^2} \, dx=\frac {-\frac {3816 \left (566+720 x^4+190 x^8+9 x^{12}\right )}{460+936 x^4+376 x^8+36 x^{12}+x^{16}}+\left (8262-2827 \sqrt {6}\right ) \log \left (-10+3 \sqrt {6}-x^4\right )+\left (-8262+2791 \sqrt {6}\right ) \log \left (-8+3 \sqrt {6}-x^4\right )-\left (8262+2791 \sqrt {6}\right ) \log \left (8+3 \sqrt {6}+x^4\right )+\left (8262+2827 \sqrt {6}\right ) \log \left (10+3 \sqrt {6}+x^4\right )}{2426976} \] Input:

Integrate[(624*x^3 + 144*x^7 + 24*x^11)/(-460 - 936*x^4 - 376*x^8 - 36*x^1 
2 - x^16)^2,x]
 

Output:

((-3816*(566 + 720*x^4 + 190*x^8 + 9*x^12))/(460 + 936*x^4 + 376*x^8 + 36* 
x^12 + x^16) + (8262 - 2827*Sqrt[6])*Log[-10 + 3*Sqrt[6] - x^4] + (-8262 + 
 2791*Sqrt[6])*Log[-8 + 3*Sqrt[6] - x^4] - (8262 + 2791*Sqrt[6])*Log[8 + 3 
*Sqrt[6] + x^4] + (8262 + 2827*Sqrt[6])*Log[10 + 3*Sqrt[6] + x^4])/2426976
 

Rubi [A] (verified)

Time = 1.27 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.76, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.073, Rules used = {2028, 2460, 2009}

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.

\(\displaystyle \int \frac {24 x^{11}+144 x^7+624 x^3}{\left (-x^{16}-36 x^{12}-376 x^8-936 x^4-460\right )^2} \, dx\)

\(\Big \downarrow \) 2028

\(\displaystyle \int \frac {x^3 \left (24 x^8+144 x^4+624\right )}{\left (-x^{16}-36 x^{12}-376 x^8-936 x^4-460\right )^2}dx\)

\(\Big \downarrow \) 2460

\(\displaystyle \int \left (-\frac {9 \left (17 x^4+13\right ) x^3}{5618 \left (x^8+16 x^4+10\right )}+\frac {9 \left (17 x^4+81\right ) x^3}{5618 \left (x^8+20 x^4+46\right )}-\frac {3 \left (7 x^4-4\right ) x^3}{53 \left (x^8+16 x^4+10\right )^2}-\frac {3 \left (5 x^4+2\right ) x^3}{53 \left (x^8+20 x^4+46\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 \left (102-41 \sqrt {6}\right ) \log \left (x^4-3 \sqrt {6}+8\right )}{89888}-\frac {5 \log \left (x^4-3 \sqrt {6}+8\right )}{3816 \sqrt {6}}+\frac {\left (306-89 \sqrt {6}\right ) \log \left (x^4-3 \sqrt {6}+10\right )}{89888}-\frac {\log \left (x^4-3 \sqrt {6}+10\right )}{954 \sqrt {6}}-\frac {3 \left (102+41 \sqrt {6}\right ) \log \left (x^4+3 \sqrt {6}+8\right )}{89888}+\frac {5 \log \left (x^4+3 \sqrt {6}+8\right )}{3816 \sqrt {6}}+\frac {\left (306+89 \sqrt {6}\right ) \log \left (x^4+3 \sqrt {6}+10\right )}{89888}+\frac {\log \left (x^4+3 \sqrt {6}+10\right )}{954 \sqrt {6}}-\frac {8 x^4+35}{1272 \left (x^8+20 x^4+46\right )}-\frac {10 x^4+17}{1272 \left (x^8+16 x^4+10\right )}\)

Input:

Int[(624*x^3 + 144*x^7 + 24*x^11)/(-460 - 936*x^4 - 376*x^8 - 36*x^12 - x^ 
16)^2,x]
 

Output:

-1/1272*(17 + 10*x^4)/(10 + 16*x^4 + x^8) - (35 + 8*x^4)/(1272*(46 + 20*x^ 
4 + x^8)) - (5*Log[8 - 3*Sqrt[6] + x^4])/(3816*Sqrt[6]) - (3*(102 - 41*Sqr 
t[6])*Log[8 - 3*Sqrt[6] + x^4])/89888 - Log[10 - 3*Sqrt[6] + x^4]/(954*Sqr 
t[6]) + ((306 - 89*Sqrt[6])*Log[10 - 3*Sqrt[6] + x^4])/89888 + (5*Log[8 + 
3*Sqrt[6] + x^4])/(3816*Sqrt[6]) - (3*(102 + 41*Sqrt[6])*Log[8 + 3*Sqrt[6] 
 + x^4])/89888 + Log[10 + 3*Sqrt[6] + x^4]/(954*Sqrt[6]) + ((306 + 89*Sqrt 
[6])*Log[10 + 3*Sqrt[6] + x^4])/89888
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2028

Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.) + (c_.)*(x_)^(t_.))^(p_.), 
x_Symbol] :> Int[x^(p*r)*(a + b*x^(s - r) + c*x^(t - r))^p*Fx, x] /; FreeQ[ 
{a, b, c, r, s, t}, x] && IntegerQ[p] && PosQ[s - r] && PosQ[t - r] &&  !(E 
qQ[p, 1] && EqQ[u, 1])
 

rule 2460

Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px /. x -> Sqrt[x]]}, 
Int[ExpandIntegrand[u*(Qx /. x -> x^2)^p, x], x] /;  !SumQ[NonfreeFactors[Q 
x, x]]] /; PolyQ[Px, x^2] && GtQ[Expon[Px, x], 2] &&  !BinomialQ[Px, x] && 
 !TrinomialQ[Px, x] && ILtQ[p, 0] && RationalFunctionQ[u, x]
 

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.78

method	result	size
default	\(\frac {-\frac {x^{4}}{159}-\frac {35}{1272}}{x^{8}+20 x^{4}+46}+\frac {153 \ln \left (x^{8}+20 x^{4}+46\right )}{44944}+\frac {2827 \sqrt {6}\, \operatorname {arctanh}\left (\frac {\left (2 x^{4}+20\right ) \sqrt {6}}{36}\right )}{1213488}-\frac {3 \left (\frac {530 x^{4}}{9}+\frac {901}{9}\right )}{22472 \left (x^{8}+16 x^{4}+10\right )}-\frac {153 \ln \left (x^{8}+16 x^{4}+10\right )}{44944}-\frac {2791 \sqrt {6}\, \operatorname {arctanh}\left (\frac {\left (2 x^{4}+16\right ) \sqrt {6}}{36}\right )}{1213488}\)	\(106\)
risch	\(\frac {-\frac {283}{318}-\frac {60}{53} x^{4}-\frac {95}{318} x^{8}-\frac {3}{212} x^{12}}{x^{16}+36 x^{12}+376 x^{8}+936 x^{4}+460}-\frac {153 \ln \left (x^{4}+8-3 \sqrt {6}\right )}{44944}+\frac {2791 \ln \left (x^{4}+8-3 \sqrt {6}\right ) \sqrt {6}}{2426976}-\frac {153 \ln \left (x^{4}+8+3 \sqrt {6}\right )}{44944}-\frac {2791 \ln \left (x^{4}+8+3 \sqrt {6}\right ) \sqrt {6}}{2426976}+\frac {153 \ln \left (x^{4}+10+3 \sqrt {6}\right )}{44944}+\frac {2827 \ln \left (x^{4}+10+3 \sqrt {6}\right ) \sqrt {6}}{2426976}+\frac {153 \ln \left (x^{4}+10-3 \sqrt {6}\right )}{44944}-\frac {2827 \ln \left (x^{4}+10-3 \sqrt {6}\right ) \sqrt {6}}{2426976}\)	\(158\)

Input:

int((24*x^11+144*x^7+624*x^3)/(-x^16-36*x^12-376*x^8-936*x^4-460)^2,x,meth 
od=_RETURNVERBOSE)
 

Output:

3/22472*(-424/9*x^4-1855/9)/(x^8+20*x^4+46)+153/44944*ln(x^8+20*x^4+46)+28 
27/1213488*6^(1/2)*arctanh(1/36*(2*x^4+20)*6^(1/2))-3/22472*(530/9*x^4+901 
/9)/(x^8+16*x^4+10)-153/44944*ln(x^8+16*x^4+10)-2791/1213488*6^(1/2)*arcta 
nh(1/36*(2*x^4+16)*6^(1/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 225 vs. \(2 (108) = 216\).

Time = 0.09 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.65 \[ \int \frac {624 x^3+144 x^7+24 x^{11}}{\left (-460-936 x^4-376 x^8-36 x^{12}-x^{16}\right )^2} \, dx=-\frac {34344 \, x^{12} + 725040 \, x^{8} + 2747520 \, x^{4} - 2827 \, \sqrt {6} {\left (x^{16} + 36 \, x^{12} + 376 \, x^{8} + 936 \, x^{4} + 460\right )} \log \left (\frac {x^{8} + 20 \, x^{4} + 6 \, \sqrt {6} {\left (x^{4} + 10\right )} + 154}{x^{8} + 20 \, x^{4} + 46}\right ) - 2791 \, \sqrt {6} {\left (x^{16} + 36 \, x^{12} + 376 \, x^{8} + 936 \, x^{4} + 460\right )} \log \left (\frac {x^{8} + 16 \, x^{4} - 6 \, \sqrt {6} {\left (x^{4} + 8\right )} + 118}{x^{8} + 16 \, x^{4} + 10}\right ) - 8262 \, {\left (x^{16} + 36 \, x^{12} + 376 \, x^{8} + 936 \, x^{4} + 460\right )} \log \left (x^{8} + 20 \, x^{4} + 46\right ) + 8262 \, {\left (x^{16} + 36 \, x^{12} + 376 \, x^{8} + 936 \, x^{4} + 460\right )} \log \left (x^{8} + 16 \, x^{4} + 10\right ) + 2159856}{2426976 \, {\left (x^{16} + 36 \, x^{12} + 376 \, x^{8} + 936 \, x^{4} + 460\right )}} \] Input:

integrate((24*x^11+144*x^7+624*x^3)/(-x^16-36*x^12-376*x^8-936*x^4-460)^2, 
x, algorithm="fricas")
 

Output:

-1/2426976*(34344*x^12 + 725040*x^8 + 2747520*x^4 - 2827*sqrt(6)*(x^16 + 3 
6*x^12 + 376*x^8 + 936*x^4 + 460)*log((x^8 + 20*x^4 + 6*sqrt(6)*(x^4 + 10) 
 + 154)/(x^8 + 20*x^4 + 46)) - 2791*sqrt(6)*(x^16 + 36*x^12 + 376*x^8 + 93 
6*x^4 + 460)*log((x^8 + 16*x^4 - 6*sqrt(6)*(x^4 + 8) + 118)/(x^8 + 16*x^4 
+ 10)) - 8262*(x^16 + 36*x^12 + 376*x^8 + 936*x^4 + 460)*log(x^8 + 20*x^4 
+ 46) + 8262*(x^16 + 36*x^12 + 376*x^8 + 936*x^4 + 460)*log(x^8 + 16*x^4 + 
 10) + 2159856)/(x^16 + 36*x^12 + 376*x^8 + 936*x^4 + 460)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (128) = 256\).

Time = 0.29 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.18 \[ \int \frac {624 x^3+144 x^7+24 x^{11}}{\left (-460-936 x^4-376 x^8-36 x^{12}-x^{16}\right )^2} \, dx=\frac {- 9 x^{12} - 190 x^{8} - 720 x^{4} - 566}{636 x^{16} + 22896 x^{12} + 239136 x^{8} + 595296 x^{4} + 292560} + \left (- \frac {153}{44944} + \frac {2791 \sqrt {6}}{2426976}\right ) \log {\left (x^{4} - \frac {156994779317323815343}{34770727855109164050} - \frac {28935357430731508002816 \left (- \frac {153}{44944} + \frac {2791 \sqrt {6}}{2426976}\right )^{3}}{687684977950025} + \frac {262251655755486967296 \left (- \frac {153}{44944} + \frac {2791 \sqrt {6}}{2426976}\right )^{2}}{687684977950025} + \frac {1549415193319901 \sqrt {6}}{755040071552700} \right )} + \left (\frac {153}{44944} - \frac {2827 \sqrt {6}}{2426976}\right ) \log {\left (x^{4} - \frac {17043567126518911 \sqrt {6}}{8199676419080100} - \frac {28935357430731508002816 \left (\frac {153}{44944} - \frac {2827 \sqrt {6}}{2426976}\right )^{3}}{687684977950025} + \frac {262251655755486967296 \left (\frac {153}{44944} - \frac {2827 \sqrt {6}}{2426976}\right )^{2}}{687684977950025} + \frac {265447075480573912703}{34770727855109164050} \right )} + \left (- \frac {153}{44944} - \frac {2791 \sqrt {6}}{2426976}\right ) \log {\left (x^{4} - \frac {1549415193319901 \sqrt {6}}{755040071552700} - \frac {156994779317323815343}{34770727855109164050} - \frac {28935357430731508002816 \left (- \frac {153}{44944} - \frac {2791 \sqrt {6}}{2426976}\right )^{3}}{687684977950025} + \frac {262251655755486967296 \left (- \frac {153}{44944} - \frac {2791 \sqrt {6}}{2426976}\right )^{2}}{687684977950025} \right )} + \left (\frac {2827 \sqrt {6}}{2426976} + \frac {153}{44944}\right ) \log {\left (x^{4} - \frac {28935357430731508002816 \left (\frac {2827 \sqrt {6}}{2426976} + \frac {153}{44944}\right )^{3}}{687684977950025} + \frac {17043567126518911 \sqrt {6}}{8199676419080100} + \frac {265447075480573912703}{34770727855109164050} + \frac {262251655755486967296 \left (\frac {2827 \sqrt {6}}{2426976} + \frac {153}{44944}\right )^{2}}{687684977950025} \right )} \] Input:

integrate((24*x**11+144*x**7+624*x**3)/(-x**16-36*x**12-376*x**8-936*x**4- 
460)**2,x)
 

Output:

(-9*x**12 - 190*x**8 - 720*x**4 - 566)/(636*x**16 + 22896*x**12 + 239136*x 
**8 + 595296*x**4 + 292560) + (-153/44944 + 2791*sqrt(6)/2426976)*log(x**4 
 - 156994779317323815343/34770727855109164050 - 28935357430731508002816*(- 
153/44944 + 2791*sqrt(6)/2426976)**3/687684977950025 + 2622516557554869672 
96*(-153/44944 + 2791*sqrt(6)/2426976)**2/687684977950025 + 15494151933199 
01*sqrt(6)/755040071552700) + (153/44944 - 2827*sqrt(6)/2426976)*log(x**4 
- 17043567126518911*sqrt(6)/8199676419080100 - 28935357430731508002816*(15 
3/44944 - 2827*sqrt(6)/2426976)**3/687684977950025 + 262251655755486967296 
*(153/44944 - 2827*sqrt(6)/2426976)**2/687684977950025 + 26544707548057391 
2703/34770727855109164050) + (-153/44944 - 2791*sqrt(6)/2426976)*log(x**4 
- 1549415193319901*sqrt(6)/755040071552700 - 156994779317323815343/3477072 
7855109164050 - 28935357430731508002816*(-153/44944 - 2791*sqrt(6)/2426976 
)**3/687684977950025 + 262251655755486967296*(-153/44944 - 2791*sqrt(6)/24 
26976)**2/687684977950025) + (2827*sqrt(6)/2426976 + 153/44944)*log(x**4 - 
 28935357430731508002816*(2827*sqrt(6)/2426976 + 153/44944)**3/68768497795 
0025 + 17043567126518911*sqrt(6)/8199676419080100 + 265447075480573912703/ 
34770727855109164050 + 262251655755486967296*(2827*sqrt(6)/2426976 + 153/4 
4944)**2/687684977950025)
 

Maxima [F]

\[ \int \frac {624 x^3+144 x^7+24 x^{11}}{\left (-460-936 x^4-376 x^8-36 x^{12}-x^{16}\right )^2} \, dx=\int { \frac {24 \, {\left (x^{11} + 6 \, x^{7} + 26 \, x^{3}\right )}}{{\left (x^{16} + 36 \, x^{12} + 376 \, x^{8} + 936 \, x^{4} + 460\right )}^{2}} \,d x } \] Input:

integrate((24*x^11+144*x^7+624*x^3)/(-x^16-36*x^12-376*x^8-936*x^4-460)^2, 
x, algorithm="maxima")
 

Output:

-1/636*(9*x^12 + 190*x^8 + 720*x^4 + 566)/(x^16 + 36*x^12 + 376*x^8 + 936* 
x^4 + 460) + 1/16854*integrate((459*x^7 + 1763*x^3)/(x^8 + 20*x^4 + 46), x 
) - 1/16854*integrate((459*x^7 + 881*x^3)/(x^8 + 16*x^4 + 10), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {624 x^3+144 x^7+24 x^{11}}{\left (-460-936 x^4-376 x^8-36 x^{12}-x^{16}\right )^2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((24*x^11+144*x^7+624*x^3)/(-x^16-36*x^12-376*x^8-936*x^4-460)^2, 
x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to find common minimal polyn 
omial Error: Bad Argument ValueUnable to find common minimal polynomial Er 
ror: Bad
 

Mupad [B] (verification not implemented)

Time = 9.54 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.88 \[ \int \frac {624 x^3+144 x^7+24 x^{11}}{\left (-460-936 x^4-376 x^8-36 x^{12}-x^{16}\right )^2} \, dx=\ln \left (x^4-3\,\sqrt {6}+8\right )\,\left (\frac {2791\,\sqrt {6}}{2426976}-\frac {153}{44944}\right )-\frac {\frac {3\,x^{12}}{212}+\frac {95\,x^8}{318}+\frac {60\,x^4}{53}+\frac {283}{318}}{x^{16}+36\,x^{12}+376\,x^8+936\,x^4+460}-\ln \left (x^4+3\,\sqrt {6}+8\right )\,\left (\frac {2791\,\sqrt {6}}{2426976}+\frac {153}{44944}\right )-\ln \left (x^4-3\,\sqrt {6}+10\right )\,\left (\frac {2827\,\sqrt {6}}{2426976}-\frac {153}{44944}\right )+\ln \left (x^4+3\,\sqrt {6}+10\right )\,\left (\frac {2827\,\sqrt {6}}{2426976}+\frac {153}{44944}\right ) \] Input:

int((624*x^3 + 144*x^7 + 24*x^11)/(936*x^4 + 376*x^8 + 36*x^12 + x^16 + 46 
0)^2,x)
 

Output:

log(x^4 - 3*6^(1/2) + 8)*((2791*6^(1/2))/2426976 - 153/44944) - ((60*x^4)/ 
53 + (95*x^8)/318 + (3*x^12)/212 + 283/318)/(936*x^4 + 376*x^8 + 36*x^12 + 
 x^16 + 460) - log(3*6^(1/2) + x^4 + 8)*((2791*6^(1/2))/2426976 + 153/4494 
4) - log(x^4 - 3*6^(1/2) + 10)*((2827*6^(1/2))/2426976 - 153/44944) + log( 
3*6^(1/2) + x^4 + 10)*((2827*6^(1/2))/2426976 + 153/44944)
 

Reduce [F]

\[ \int \frac {624 x^3+144 x^7+24 x^{11}}{\left (-460-936 x^4-376 x^8-36 x^{12}-x^{16}\right )^2} \, dx=\int \frac {24 x^{11}+144 x^{7}+624 x^{3}}{\left (-x^{16}-36 x^{12}-376 x^{8}-936 x^{4}-460\right )^{2}}d x \] Input:

int((24*x^11+144*x^7+624*x^3)/(-x^16-36*x^12-376*x^8-936*x^4-460)^2,x)
 

Output:

int((24*x^11+144*x^7+624*x^3)/(-x^16-36*x^12-376*x^8-936*x^4-460)^2,x)