\(\int \frac {x^m}{1-x^4+x^8} \, dx\) [141]

Optimal result

Integrand size = 16, antiderivative size = 127 \[ \int \frac {x^m}{1-x^4+x^8} \, dx=\frac {2 x^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{4},\frac {5+m}{4},\frac {2 x^4}{1-i \sqrt {3}}\right )}{\sqrt {3} \left (i+\sqrt {3}\right ) (1+m)}-\frac {2 x^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{4},\frac {5+m}{4},\frac {2 x^4}{1+i \sqrt {3}}\right )}{\sqrt {3} \left (i-\sqrt {3}\right ) (1+m)} \] Output:

2/3*x^(1+m)*hypergeom([1, 1/4+1/4*m],[5/4+1/4*m],2*x^4/(1-I*3^(1/2)))*3^(1 
/2)/(3^(1/2)+I)/(1+m)-2/3*x^(1+m)*hypergeom([1, 1/4+1/4*m],[5/4+1/4*m],2*x 
^4/(1+I*3^(1/2)))*3^(1/2)/(I-3^(1/2))/(1+m)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.

Time = 0.12 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.62 \[ \int \frac {x^m}{1-x^4+x^8} \, dx=\frac {x^m \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\operatorname {Hypergeometric2F1}\left (-m,-m,1-m,-\frac {\text {$\#$1}}{x-\text {$\#$1}}\right ) \left (\frac {x}{x-\text {$\#$1}}\right )^{-m}}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ]}{4 m} \] Input:

Integrate[x^m/(1 - x^4 + x^8),x]
 

Output:

(x^m*RootSum[1 - #1^4 + #1^8 & , Hypergeometric2F1[-m, -m, 1 - m, -(#1/(x 
- #1))]/((x/(x - #1))^m*(-#1^3 + 2*#1^7)) & ])/(4*m)
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1711, 27, 888}

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.

\(\displaystyle \int \frac {x^m}{x^8-x^4+1} \, dx\)

\(\Big \downarrow \) 1711

\(\displaystyle \frac {i \int -\frac {2 x^m}{-2 x^4-i \sqrt {3}+1}dx}{\sqrt {3}}-\frac {i \int -\frac {2 x^m}{-2 x^4+i \sqrt {3}+1}dx}{\sqrt {3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 i \int \frac {x^m}{-2 x^4+i \sqrt {3}+1}dx}{\sqrt {3}}-\frac {2 i \int \frac {x^m}{-2 x^4-i \sqrt {3}+1}dx}{\sqrt {3}}\)

\(\Big \downarrow \) 888

\(\displaystyle \frac {2 i x^{m+1} \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{4},\frac {m+5}{4},\frac {2 x^4}{1+i \sqrt {3}}\right )}{\sqrt {3} \left (1+i \sqrt {3}\right ) (m+1)}-\frac {2 i x^{m+1} \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{4},\frac {m+5}{4},\frac {2 x^4}{1-i \sqrt {3}}\right )}{\sqrt {3} \left (1-i \sqrt {3}\right ) (m+1)}\)

Input:

Int[x^m/(1 - x^4 + x^8),x]
 

Output:

((-2*I)*x^(1 + m)*Hypergeometric2F1[1, (1 + m)/4, (5 + m)/4, (2*x^4)/(1 - 
I*Sqrt[3])])/(Sqrt[3]*(1 - I*Sqrt[3])*(1 + m)) + ((2*I)*x^(1 + m)*Hypergeo 
metric2F1[1, (1 + m)/4, (5 + m)/4, (2*x^4)/(1 + I*Sqrt[3])])/(Sqrt[3]*(1 + 
 I*Sqrt[3])*(1 + m))
 

Defintions of rubi rules used

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 888

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p 
*((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 
, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] && (ILt 
Q[p, 0] || GtQ[a, 0])
 

rule 1711

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

Maple [F]

\[\int \frac {x^{m}}{x^{8}-x^{4}+1}d x\]

Input:

int(x^m/(x^8-x^4+1),x)
 

Output:

int(x^m/(x^8-x^4+1),x)
 

Fricas [F]

\[ \int \frac {x^m}{1-x^4+x^8} \, dx=\int { \frac {x^{m}}{x^{8} - x^{4} + 1} \,d x } \] Input:

integrate(x^m/(x^8-x^4+1),x, algorithm="fricas")
 

Output:

integral(x^m/(x^8 - x^4 + 1), x)
 

Sympy [F]

\[ \int \frac {x^m}{1-x^4+x^8} \, dx=\int \frac {x^{m}}{x^{8} - x^{4} + 1}\, dx \] Input:

integrate(x**m/(x**8-x**4+1),x)
 

Output:

Integral(x**m/(x**8 - x**4 + 1), x)
 

Maxima [F]

\[ \int \frac {x^m}{1-x^4+x^8} \, dx=\int { \frac {x^{m}}{x^{8} - x^{4} + 1} \,d x } \] Input:

integrate(x^m/(x^8-x^4+1),x, algorithm="maxima")
 

Output:

integrate(x^m/(x^8 - x^4 + 1), x)
 

Giac [F]

\[ \int \frac {x^m}{1-x^4+x^8} \, dx=\int { \frac {x^{m}}{x^{8} - x^{4} + 1} \,d x } \] Input:

integrate(x^m/(x^8-x^4+1),x, algorithm="giac")
 

Output:

integrate(x^m/(x^8 - x^4 + 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^m}{1-x^4+x^8} \, dx=\int \frac {x^m}{x^8-x^4+1} \,d x \] Input:

int(x^m/(x^8 - x^4 + 1),x)
 

Output:

int(x^m/(x^8 - x^4 + 1), x)
 

Reduce [F]

\[ \int \frac {x^m}{1-x^4+x^8} \, dx=\int \frac {x^{m}}{x^{8}-x^{4}+1}d x \] Input:

int(x^m/(x^8-x^4+1),x)
 

Output:

int(x**m/(x**8 - x**4 + 1),x)