\(\int \frac {1}{x^2 (a+b x^4+c x^8)^2} \, dx\) [62]

Optimal result

Integrand size = 18, antiderivative size = 553 \[ \int \frac {1}{x^2 \left (a+b x^4+c x^8\right )^2} \, dx=-\frac {5 b^2-18 a c}{4 a^2 \left (b^2-4 a c\right ) x}+\frac {b^2-2 a c+b c x^4}{4 a \left (b^2-4 a c\right ) x \left (a+b x^4+c x^8\right )}+\frac {\sqrt [4]{c} \left (5 b^3-28 a b c-\left (5 b^2-18 a c\right ) \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{8\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt [4]{-b-\sqrt {b^2-4 a c}}}-\frac {\sqrt [4]{c} \left (5 b^3-28 a b c+\left (5 b^2-18 a c\right ) \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{8\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt [4]{-b+\sqrt {b^2-4 a c}}}-\frac {\sqrt [4]{c} \left (5 b^3-28 a b c-\left (5 b^2-18 a c\right ) \sqrt {b^2-4 a c}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{8\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt [4]{-b-\sqrt {b^2-4 a c}}}+\frac {\sqrt [4]{c} \left (5 b^3-28 a b c+\left (5 b^2-18 a c\right ) \sqrt {b^2-4 a c}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{8\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt [4]{-b+\sqrt {b^2-4 a c}}} \] Output:

-1/4*(-18*a*c+5*b^2)/a^2/(-4*a*c+b^2)/x+1/4*(b*c*x^4-2*a*c+b^2)/a/(-4*a*c+ 
b^2)/x/(c*x^8+b*x^4+a)+1/16*c^(1/4)*(5*b^3-28*a*b*c-(-18*a*c+5*b^2)*(-4*a* 
c+b^2)^(1/2))*arctan(2^(1/4)*c^(1/4)*x/(-b-(-4*a*c+b^2)^(1/2))^(1/4))*2^(1 
/4)/a^2/(-4*a*c+b^2)^(3/2)/(-b-(-4*a*c+b^2)^(1/2))^(1/4)-1/16*c^(1/4)*(5*b 
^3-28*a*b*c+(-18*a*c+5*b^2)*(-4*a*c+b^2)^(1/2))*arctan(2^(1/4)*c^(1/4)*x/( 
-b+(-4*a*c+b^2)^(1/2))^(1/4))*2^(1/4)/a^2/(-4*a*c+b^2)^(3/2)/(-b+(-4*a*c+b 
^2)^(1/2))^(1/4)-1/16*c^(1/4)*(5*b^3-28*a*b*c-(-18*a*c+5*b^2)*(-4*a*c+b^2) 
^(1/2))*arctanh(2^(1/4)*c^(1/4)*x/(-b-(-4*a*c+b^2)^(1/2))^(1/4))*2^(1/4)/a 
^2/(-4*a*c+b^2)^(3/2)/(-b-(-4*a*c+b^2)^(1/2))^(1/4)+1/16*c^(1/4)*(5*b^3-28 
*a*b*c+(-18*a*c+5*b^2)*(-4*a*c+b^2)^(1/2))*arctanh(2^(1/4)*c^(1/4)*x/(-b+( 
-4*a*c+b^2)^(1/2))^(1/4))*2^(1/4)/a^2/(-4*a*c+b^2)^(3/2)/(-b+(-4*a*c+b^2)^ 
(1/2))^(1/4)
 

Mathematica [C] (verified)

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

Time = 0.22 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.31 \[ \int \frac {1}{x^2 \left (a+b x^4+c x^8\right )^2} \, dx=-\frac {\frac {16}{x}+\frac {4 x^3 \left (b^3-3 a b c+b^2 c x^4-2 a c^2 x^4\right )}{\left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}+\frac {\text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {5 b^3 \log (x-\text {$\#$1})-23 a b c \log (x-\text {$\#$1})+5 b^2 c \log (x-\text {$\#$1}) \text {$\#$1}^4-18 a c^2 \log (x-\text {$\#$1}) \text {$\#$1}^4}{b \text {$\#$1}+2 c \text {$\#$1}^5}\&\right ]}{b^2-4 a c}}{16 a^2} \] Input:

Integrate[1/(x^2*(a + b*x^4 + c*x^8)^2),x]
 

Output:

-1/16*(16/x + (4*x^3*(b^3 - 3*a*b*c + b^2*c*x^4 - 2*a*c^2*x^4))/((b^2 - 4* 
a*c)*(a + b*x^4 + c*x^8)) + RootSum[a + b*#1^4 + c*#1^8 & , (5*b^3*Log[x - 
 #1] - 23*a*b*c*Log[x - #1] + 5*b^2*c*Log[x - #1]*#1^4 - 18*a*c^2*Log[x - 
#1]*#1^4)/(b*#1 + 2*c*#1^5) & ]/(b^2 - 4*a*c))/a^2
 

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 465, normalized size of antiderivative = 0.84, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {1702, 25, 1828, 1834, 27, 827, 218, 221}

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.

\(\displaystyle \int \frac {1}{x^2 \left (a+b x^4+c x^8\right )^2} \, dx\)

\(\Big \downarrow \) 1702

\(\displaystyle \frac {-2 a c+b^2+b c x^4}{4 a x \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}-\frac {\int -\frac {5 b c x^4+5 b^2-18 a c}{x^2 \left (c x^8+b x^4+a\right )}dx}{4 a \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {5 b c x^4+5 b^2-18 a c}{x^2 \left (c x^8+b x^4+a\right )}dx}{4 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^4}{4 a x \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\)

\(\Big \downarrow \) 1828

\(\displaystyle \frac {-\frac {\int \frac {x^2 \left (c \left (5 b^2-18 a c\right ) x^4+b \left (5 b^2-23 a c\right )\right )}{c x^8+b x^4+a}dx}{a}-\frac {5 b^2-18 a c}{a x}}{4 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^4}{4 a x \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\)

\(\Big \downarrow \) 1834

\(\displaystyle \frac {-\frac {\frac {1}{2} c \left (-\frac {28 a b c}{\sqrt {b^2-4 a c}}+\frac {5 b^3}{\sqrt {b^2-4 a c}}-18 a c+5 b^2\right ) \int \frac {2 x^2}{2 c x^4+b-\sqrt {b^2-4 a c}}dx-\frac {c \left (-\left (5 b^2-18 a c\right ) \sqrt {b^2-4 a c}-28 a b c+5 b^3\right ) \int \frac {2 x^2}{2 c x^4+b+\sqrt {b^2-4 a c}}dx}{2 \sqrt {b^2-4 a c}}}{a}-\frac {5 b^2-18 a c}{a x}}{4 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^4}{4 a x \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {c \left (-\frac {28 a b c}{\sqrt {b^2-4 a c}}+\frac {5 b^3}{\sqrt {b^2-4 a c}}-18 a c+5 b^2\right ) \int \frac {x^2}{2 c x^4+b-\sqrt {b^2-4 a c}}dx-\frac {c \left (-\left (5 b^2-18 a c\right ) \sqrt {b^2-4 a c}-28 a b c+5 b^3\right ) \int \frac {x^2}{2 c x^4+b+\sqrt {b^2-4 a c}}dx}{\sqrt {b^2-4 a c}}}{a}-\frac {5 b^2-18 a c}{a x}}{4 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^4}{4 a x \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\)

\(\Big \downarrow \) 827

\(\displaystyle \frac {-\frac {c \left (-\frac {28 a b c}{\sqrt {b^2-4 a c}}+\frac {5 b^3}{\sqrt {b^2-4 a c}}-18 a c+5 b^2\right ) \left (\frac {\int \frac {1}{\sqrt {2} \sqrt {c} x^2+\sqrt {\sqrt {b^2-4 a c}-b}}dx}{2 \sqrt {2} \sqrt {c}}-\frac {\int \frac {1}{\sqrt {\sqrt {b^2-4 a c}-b}-\sqrt {2} \sqrt {c} x^2}dx}{2 \sqrt {2} \sqrt {c}}\right )-\frac {c \left (-\left (5 b^2-18 a c\right ) \sqrt {b^2-4 a c}-28 a b c+5 b^3\right ) \left (\frac {\int \frac {1}{\sqrt {2} \sqrt {c} x^2+\sqrt {-b-\sqrt {b^2-4 a c}}}dx}{2 \sqrt {2} \sqrt {c}}-\frac {\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x^2}dx}{2 \sqrt {2} \sqrt {c}}\right )}{\sqrt {b^2-4 a c}}}{a}-\frac {5 b^2-18 a c}{a x}}{4 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^4}{4 a x \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {-\frac {c \left (-\frac {28 a b c}{\sqrt {b^2-4 a c}}+\frac {5 b^3}{\sqrt {b^2-4 a c}}-18 a c+5 b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{\sqrt {b^2-4 a c}-b}}-\frac {\int \frac {1}{\sqrt {\sqrt {b^2-4 a c}-b}-\sqrt {2} \sqrt {c} x^2}dx}{2 \sqrt {2} \sqrt {c}}\right )-\frac {c \left (-\left (5 b^2-18 a c\right ) \sqrt {b^2-4 a c}-28 a b c+5 b^3\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}-\frac {\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x^2}dx}{2 \sqrt {2} \sqrt {c}}\right )}{\sqrt {b^2-4 a c}}}{a}-\frac {5 b^2-18 a c}{a x}}{4 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^4}{4 a x \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {-\frac {c \left (-\frac {28 a b c}{\sqrt {b^2-4 a c}}+\frac {5 b^3}{\sqrt {b^2-4 a c}}-18 a c+5 b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{\sqrt {b^2-4 a c}-b}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )-\frac {c \left (-\left (5 b^2-18 a c\right ) \sqrt {b^2-4 a c}-28 a b c+5 b^3\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{\sqrt {b^2-4 a c}}}{a}-\frac {5 b^2-18 a c}{a x}}{4 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^4}{4 a x \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\)

Input:

Int[1/(x^2*(a + b*x^4 + c*x^8)^2),x]
 

Output:

(b^2 - 2*a*c + b*c*x^4)/(4*a*(b^2 - 4*a*c)*x*(a + b*x^4 + c*x^8)) + (-((5* 
b^2 - 18*a*c)/(a*x)) - (-((c*(5*b^3 - 28*a*b*c - (5*b^2 - 18*a*c)*Sqrt[b^2 
 - 4*a*c])*(ArcTan[(2^(1/4)*c^(1/4)*x)/(-b - Sqrt[b^2 - 4*a*c])^(1/4)]/(2* 
2^(3/4)*c^(3/4)*(-b - Sqrt[b^2 - 4*a*c])^(1/4)) - ArcTanh[(2^(1/4)*c^(1/4) 
*x)/(-b - Sqrt[b^2 - 4*a*c])^(1/4)]/(2*2^(3/4)*c^(3/4)*(-b - Sqrt[b^2 - 4* 
a*c])^(1/4))))/Sqrt[b^2 - 4*a*c]) + c*(5*b^2 - 18*a*c + (5*b^3)/Sqrt[b^2 - 
 4*a*c] - (28*a*b*c)/Sqrt[b^2 - 4*a*c])*(ArcTan[(2^(1/4)*c^(1/4)*x)/(-b + 
Sqrt[b^2 - 4*a*c])^(1/4)]/(2*2^(3/4)*c^(3/4)*(-b + Sqrt[b^2 - 4*a*c])^(1/4 
)) - ArcTanh[(2^(1/4)*c^(1/4)*x)/(-b + Sqrt[b^2 - 4*a*c])^(1/4)]/(2*2^(3/4 
)*c^(3/4)*(-b + Sqrt[b^2 - 4*a*c])^(1/4))))/a)/(4*a*(b^2 - 4*a*c))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 827

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 
 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b)   Int[1/(r + s*x^2), x], 
x] - Simp[s/(2*b)   Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ 
[a/b, 0]
 

rule 1702

Int[((d_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x 
_Symbol] :> Simp[(-(d*x)^(m + 1))*(b^2 - 2*a*c + b*c*x^n)*((a + b*x^n + c*x 
^(2*n))^(p + 1)/(a*d*n*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(a*n*(p + 1)*(b 
^2 - 4*a*c))   Int[(d*x)^m*(a + b*x^n + c*x^(2*n))^(p + 1)*Simp[b^2*(m + n* 
(p + 1) + 1) - 2*a*c*(m + 2*n*(p + 1) + 1) + b*c*(m + n*(2*p + 3) + 1)*x^n, 
 x], x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c 
, 0] && IGtQ[n, 0] && ILtQ[p, -1]
 

rule 1828

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + ( 
c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Simp[d*(f*x)^(m + 1)*((a + b*x^n + c*x^ 
(2*n))^(p + 1)/(a*f*(m + 1))), x] + Simp[1/(a*f^n*(m + 1))   Int[(f*x)^(m + 
 n)*(a + b*x^n + c*x^(2*n))^p*Simp[a*e*(m + 1) - b*d*(m + n*(p + 1) + 1) - 
c*d*(m + 2*n*(p + 1) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x 
] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[m, -1] && Int 
egerQ[p]
 

rule 1834

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_)))/((a_) + (b_.)*(x_)^(n_) + 
 (c_.)*(x_)^(n2_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + 
 (2*c*d - b*e)/(2*q))   Int[(f*x)^m/(b/2 - q/2 + c*x^n), x], x] + Simp[(e/2 
 - (2*c*d - b*e)/(2*q))   Int[(f*x)^m/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ 
[{a, b, c, d, e, f, m}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n 
, 0]
 

Maple [C] (verified)

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

Time = 0.10 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.31

method	result	size
default	\(-\frac {\frac {\frac {c \left (2 a c -b^{2}\right ) x^{7}}{16 a c -4 b^{2}}+\frac {b \left (3 a c -b^{2}\right ) x^{3}}{16 a c -4 b^{2}}}{c \,x^{8}+b \,x^{4}+a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8} c +\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (c \left (18 a c -5 b^{2}\right ) \textit {\_R}^{6}+b \left (23 a c -5 b^{2}\right ) \textit {\_R}^{2}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}}{64 a c -16 b^{2}}}{a^{2}}-\frac {1}{a^{2} x}\)	\(170\)
risch	\(\frac {-\frac {c \left (18 a c -5 b^{2}\right ) x^{8}}{4 a^{2} \left (4 a c -b^{2}\right )}-\frac {b \left (19 a c -5 b^{2}\right ) x^{4}}{4 \left (4 a c -b^{2}\right ) a^{2}}-\frac {1}{a}}{x \left (c \,x^{8}+b \,x^{4}+a \right )}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (16777216 a^{21} c^{12}-50331648 a^{20} b^{2} c^{11}[phone] a^{19} b^{4} c^{10}-57671680 a^{18} b^{6} c^{9}[phone] a^{17} b^{8} c^{8}-12976128 a^{16} b^{10} c^{7}+3784704 a^{15} b^{12} c^{6}-811008 a^{14} b^{14} c^{5}+126720 a^{13} b^{16} c^{4}-14080 a^{12} b^{18} c^{3}+1056 a^{11} b^{20} c^{2}-48 a^{10} b^{22} c +a^{9} b^{24}\right ) \textit {\_Z}^{8}+\left (3105423360 a^{12} b \,c^{12}-12575047680 a^{11} b^{3} c^{11}[phone] a^{10} b^{5} c^{10}-21122310144 a^{9} b^{7} c^{9}[phone] a^{8} b^{9} c^{8}-5996689920 a^{7} b^{11} c^{7}[phone] a^{6} b^{13} c^{6}-434478624 a^{5} b^{15} c^{5}[phone] a^{4} b^{17} c^{4}-8264990 a^{3} b^{19} c^{3}+638475 a^{2} b^{21} c^{2}-29625 a \,b^{23} c +625 b^{25}\right ) \textit {\_Z}^{4}[phone] a^{4} c^{13}-8843178240 a^{3} b^{2} c^{12}[phone] a^{2} b^{4} c^{11}-355914000 a \,b^{6} c^{10}[phone] b^{8} c^{9}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (75497472 a^{21} c^{12}-228589568 a^{20} b^{2} c^{11}[phone] a^{19} b^{4} c^{10}-266731520 a^{18} b^{6} c^{9}[phone] a^{17} b^{8} c^{8}-61095936 a^{16} b^{10} c^{7}[phone] a^{15} b^{12} c^{6}-3886080 a^{14} b^{14} c^{5}+612480 a^{13} b^{16} c^{4}-68640 a^{12} b^{18} c^{3}+5192 a^{11} b^{20} c^{2}-238 a^{10} b^{22} c +5 a^{9} b^{24}\right ) \textit {\_R}^{8}+\left (12983058432 a^{12} b \,c^{12}-51854413824 a^{11} b^{3} c^{11}[phone] a^{10} b^{5} c^{10}-85793418496 a^{9} b^{7} c^{9}[phone] a^{8} b^{9} c^{8}-24164185008 a^{7} b^{11} c^{7}[phone] a^{6} b^{13} c^{6}-1743126511 a^{5} b^{15} c^{5}[phone] a^{4} b^{17} c^{4}-33085835 a^{3} b^{19} c^{3}+2554525 a^{2} b^{21} c^{2}-118500 a \,b^{23} c +2500 b^{25}\right ) \textit {\_R}^{4}[phone] a^{4} c^{13}-35372712960 a^{3} b^{2} c^{12}[phone] a^{2} b^{4} c^{11}-1423656000 a \,b^{6} c^{10}[phone] b^{8} c^{9}\right ) x +\left (18874368 a^{19} c^{12}-76546048 a^{18} b^{2} c^{11}[phone] a^{17} b^{4} c^{10}-131072000 a^{16} b^{6} c^{9}[phone] a^{15} b^{8} c^{8}-38559744 a^{14} b^{10} c^{7}[phone] a^{13} b^{12} c^{6}-2933760 a^{12} b^{14} c^{5}+496800 a^{11} b^{16} c^{4}-59280 a^{10} b^{18} c^{3}+4738 a^{9} b^{20} c^{2}-228 a^{8} b^{22} c +5 a^{7} b^{24}\right ) \textit {\_R}^{7}+\left (886837248 a^{10} b \,c^{12}-1702176768 a^{9} b^{3} c^{11}[phone] a^{8} b^{5} c^{10}-671208448 a^{7} b^{7} c^{9}[phone] a^{6} b^{9} c^{8}-36851720 a^{5} b^{11} c^{7}+4280425 a^{4} b^{13} c^{6}-282625 a^{3} b^{15} c^{5}+8125 a^{2} b^{17} c^{4}\right ) \textit {\_R}^{3}\right )\right )}{16}\)	\(989\)

Input:

int(1/x^2/(c*x^8+b*x^4+a)^2,x,method=_RETURNVERBOSE)
 

Output:

-1/a^2*((1/4*c*(2*a*c-b^2)/(4*a*c-b^2)*x^7+1/4*b*(3*a*c-b^2)/(4*a*c-b^2)*x 
^3)/(c*x^8+b*x^4+a)+1/16/(4*a*c-b^2)*sum((c*(18*a*c-5*b^2)*_R^6+b*(23*a*c- 
5*b^2)*_R^2)/(2*_R^7*c+_R^3*b)*ln(x-_R),_R=RootOf(_Z^8*c+_Z^4*b+a)))-1/a^2 
/x
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 15984 vs. \(2 (461) = 922\).

Time = 23.94 (sec) , antiderivative size = 15984, normalized size of antiderivative = 28.90 \[ \int \frac {1}{x^2 \left (a+b x^4+c x^8\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(1/x^2/(c*x^8+b*x^4+a)^2,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x^2 \left (a+b x^4+c x^8\right )^2} \, dx=\text {Timed out} \] Input:

integrate(1/x**2/(c*x**8+b*x**4+a)**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{x^2 \left (a+b x^4+c x^8\right )^2} \, dx=\int { \frac {1}{{\left (c x^{8} + b x^{4} + a\right )}^{2} x^{2}} \,d x } \] Input:

integrate(1/x^2/(c*x^8+b*x^4+a)^2,x, algorithm="maxima")
 

Output:

-1/4*((5*b^2*c - 18*a*c^2)*x^8 + (5*b^3 - 19*a*b*c)*x^4 + 4*a*b^2 - 16*a^2 
*c)/((a^2*b^2*c - 4*a^3*c^2)*x^9 + (a^2*b^3 - 4*a^3*b*c)*x^5 + (a^3*b^2 - 
4*a^4*c)*x) + 1/4*integrate(-((5*b^2*c - 18*a*c^2)*x^6 + (5*b^3 - 23*a*b*c 
)*x^2)/(c*x^8 + b*x^4 + a), x)/(a^2*b^2 - 4*a^3*c)
 

Giac [F]

\[ \int \frac {1}{x^2 \left (a+b x^4+c x^8\right )^2} \, dx=\int { \frac {1}{{\left (c x^{8} + b x^{4} + a\right )}^{2} x^{2}} \,d x } \] Input:

integrate(1/x^2/(c*x^8+b*x^4+a)^2,x, algorithm="giac")
 

Output:

integrate(1/((c*x^8 + b*x^4 + a)^2*x^2), x)
 

Mupad [B] (verification not implemented)

Time = 27.30 (sec) , antiderivative size = 31085, normalized size of antiderivative = 56.21 \[ \int \frac {1}{x^2 \left (a+b x^4+c x^8\right )^2} \, dx=\text {Too large to display} \] Input:

int(1/(x^2*(a + b*x^4 + c*x^8)^2),x)
 

Output:

atan((((-(625*b^25 - 625*b^10*(-(4*a*c - b^2)^15)^(1/2) + 3105423360*a^12* 
b*c^12 + 638475*a^2*b^21*c^2 - 8264990*a^3*b^19*c^3 + 71483001*a^4*b^17*c^ 
4 - 434478624*a^5*b^15*c^5 + 1898983360*a^6*b^13*c^6 - 5996689920*a^7*b^11 
*c^7 + 13524825600*a^8*b^9*c^8 - 21122310144*a^9*b^7*c^9 + 21483012096*a^1 
0*b^5*c^10 - 12575047680*a^11*b^3*c^11 + 26244*a^5*c^5*(-(4*a*c - b^2)^15) 
^(1/2) - 29625*a*b^23*c - 68475*a^2*b^6*c^2*(-(4*a*c - b^2)^15)^(1/2) + 18 
1990*a^3*b^4*c^3*(-(4*a*c - b^2)^15)^(1/2) - 171801*a^4*b^2*c^4*(-(4*a*c - 
 b^2)^15)^(1/2) + 10875*a*b^8*c*(-(4*a*c - b^2)^15)^(1/2))/(131072*(a^9*b^ 
24 + 16777216*a^21*c^12 - 48*a^10*b^22*c + 1056*a^11*b^20*c^2 - 14080*a^12 
*b^18*c^3 + 126720*a^13*b^16*c^4 - 811008*a^14*b^14*c^5 + 3784704*a^15*b^1 
2*c^6 - 12976128*a^16*b^10*c^7 + 32440320*a^17*b^8*c^8 - 57671680*a^18*b^6 
*c^9 + 69206016*a^19*b^4*c^10 - 50331648*a^20*b^2*c^11)))^(3/4)*(x*(-(625* 
b^25 - 625*b^10*(-(4*a*c - b^2)^15)^(1/2) + 3105423360*a^12*b*c^12 + 63847 
5*a^2*b^21*c^2 - 8264990*a^3*b^19*c^3 + 71483001*a^4*b^17*c^4 - 434478624* 
a^5*b^15*c^5 + 1898983360*a^6*b^13*c^6 - 5996689920*a^7*b^11*c^7 + 1352482 
5600*a^8*b^9*c^8 - 21122310144*a^9*b^7*c^9 + 21483012096*a^10*b^5*c^10 - 1 
2575047680*a^11*b^3*c^11 + 26244*a^5*c^5*(-(4*a*c - b^2)^15)^(1/2) - 29625 
*a*b^23*c - 68475*a^2*b^6*c^2*(-(4*a*c - b^2)^15)^(1/2) + 181990*a^3*b^4*c 
^3*(-(4*a*c - b^2)^15)^(1/2) - 171801*a^4*b^2*c^4*(-(4*a*c - b^2)^15)^(1/2 
) + 10875*a*b^8*c*(-(4*a*c - b^2)^15)^(1/2))/(131072*(a^9*b^24 + 167772...
 

Reduce [F]

\[ \int \frac {1}{x^2 \left (a+b x^4+c x^8\right )^2} \, dx=\int \frac {1}{x^{2} \left (c \,x^{8}+b \,x^{4}+a \right )^{2}}d x \] Input:

int(1/x^2/(c*x^8+b*x^4+a)^2,x)
 

Output:

int(1/x^2/(c*x^8+b*x^4+a)^2,x)