\(\int \frac {x^m}{1-2 x^4+x^8} \, dx\) [290]

Optimal result

Integrand size = 16, antiderivative size = 30 \[ \int \frac {x^m}{1-2 x^4+x^8} \, dx=\frac {x^{1+m} \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{4},\frac {5+m}{4},x^4\right )}{1+m} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {28, 371} \[ \int \frac {x^m}{1-2 x^4+x^8} \, dx=\frac {x^{m+1} \operatorname {Hypergeometric2F1}\left (2,\frac {m+1}{4},\frac {m+5}{4},x^4\right )}{m+1} \]

[In]

Int[x^m/(1 - 2*x^4 + x^8),x]

[Out]

(x^(1 + m)*Hypergeometric2F1[2, (1 + m)/4, (5 + m)/4, x^4])/(1 + m)

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^m}{\left (-1+x^4\right )^2} \, dx \\ & = \frac {x^{1+m} \, _2F_1\left (2,\frac {1+m}{4};\frac {5+m}{4};x^4\right )}{1+m} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {x^m}{1-2 x^4+x^8} \, dx=\frac {x^{1+m} \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{4},1+\frac {1+m}{4},x^4\right )}{1+m} \]

[In]

Integrate[x^m/(1 - 2*x^4 + x^8),x]

[Out]

(x^(1 + m)*Hypergeometric2F1[2, (1 + m)/4, 1 + (1 + m)/4, x^4])/(1 + m)

Maple [F]

\[\int \frac {x^{m}}{x^{8}-2 x^{4}+1}d x\]

[In]

int(x^m/(x^8-2*x^4+1),x)

[Out]

int(x^m/(x^8-2*x^4+1),x)

Fricas [F]

\[ \int \frac {x^m}{1-2 x^4+x^8} \, dx=\int { \frac {x^{m}}{x^{8} - 2 \, x^{4} + 1} \,d x } \]

[In]

integrate(x^m/(x^8-2*x^4+1),x, algorithm="fricas")

[Out]

integral(x^m/(x^8 - 2*x^4 + 1), x)

Sympy [F]

\[ \int \frac {x^m}{1-2 x^4+x^8} \, dx=\int \frac {x^{m}}{\left (x - 1\right )^{2} \left (x + 1\right )^{2} \left (x^{2} + 1\right )^{2}}\, dx \]

[In]

integrate(x**m/(x**8-2*x**4+1),x)

[Out]

Integral(x**m/((x - 1)**2*(x + 1)**2*(x**2 + 1)**2), x)

Maxima [F]

\[ \int \frac {x^m}{1-2 x^4+x^8} \, dx=\int { \frac {x^{m}}{x^{8} - 2 \, x^{4} + 1} \,d x } \]

[In]

integrate(x^m/(x^8-2*x^4+1),x, algorithm="maxima")

[Out]

integrate(x^m/(x^8 - 2*x^4 + 1), x)

Giac [F]

\[ \int \frac {x^m}{1-2 x^4+x^8} \, dx=\int { \frac {x^{m}}{x^{8} - 2 \, x^{4} + 1} \,d x } \]

[In]

integrate(x^m/(x^8-2*x^4+1),x, algorithm="giac")

[Out]

integrate(x^m/(x^8 - 2*x^4 + 1), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^m}{1-2 x^4+x^8} \, dx=\int \frac {x^m}{x^8-2\,x^4+1} \,d x \]

[In]

int(x^m/(x^8 - 2*x^4 + 1),x)

[Out]

int(x^m/(x^8 - 2*x^4 + 1), x)