\(\int (1-x^4)^p (1-x^8)^p \, dx\) [23]

Optimal result

Integrand size = 19, antiderivative size = 23 \[ \int \left (1-x^4\right )^p \left (1-x^8\right )^p \, dx=x \operatorname {AppellF1}\left (\frac {1}{4},-2 p,-p,\frac {5}{4},x^4,-x^4\right ) \] Output:

x*AppellF1(1/4,-2*p,-p,5/4,x^4,-x^4)
 

Mathematica [F]

\[ \int \left (1-x^4\right )^p \left (1-x^8\right )^p \, dx=\int \left (1-x^4\right )^p \left (1-x^8\right )^p \, dx \] Input:

Integrate[(1 - x^4)^p*(1 - x^8)^p,x]
 

Output:

Integrate[(1 - x^4)^p*(1 - x^8)^p, x]
 

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {1388, 936}

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.

\(\displaystyle \int \left (1-x^4\right )^p \left (1-x^8\right )^p \, dx\)

\(\Big \downarrow \) 1388

\(\displaystyle \int \left (1-x^4\right )^{2 p} \left (x^4+1\right )^pdx\)

\(\Big \downarrow \) 936

\(\displaystyle x \operatorname {AppellF1}\left (\frac {1}{4},-2 p,-p,\frac {5}{4},x^4,-x^4\right )\)

Input:

Int[(1 - x^4)^p*(1 - x^8)^p,x]
 

Output:

x*AppellF1[1/4, -2*p, -p, 5/4, x^4, -x^4]
 

Defintions of rubi rules used

rule 936

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[a^p*c^q*x*AppellF1[1/n, -p, -q, 1 + 1/n, (-b)*(x^n/a), (-d)*(x^n/c) 
], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] 
 && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 1388

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), 
x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, 
 c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer 
Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
 

Maple [F]

\[\int \left (-x^{4}+1\right )^{p} \left (-x^{8}+1\right )^{p}d x\]

Input:

int((-x^4+1)^p*(-x^8+1)^p,x)
 

Output:

int((-x^4+1)^p*(-x^8+1)^p,x)
 

Fricas [F]

\[ \int \left (1-x^4\right )^p \left (1-x^8\right )^p \, dx=\int { {\left (-x^{8} + 1\right )}^{p} {\left (-x^{4} + 1\right )}^{p} \,d x } \] Input:

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

Output:

integral((-x^8 + 1)^p*(-x^4 + 1)^p, x)
 

Sympy [F]

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

integrate((-x**4+1)**p*(-x**8+1)**p,x)
 

Output:

Integral((-(x - 1)*(x + 1)*(x**2 + 1))**p*(-(x - 1)*(x + 1)*(x**2 + 1)*(x* 
*4 + 1))**p, x)
 

Maxima [F]

\[ \int \left (1-x^4\right )^p \left (1-x^8\right )^p \, dx=\int { {\left (-x^{8} + 1\right )}^{p} {\left (-x^{4} + 1\right )}^{p} \,d x } \] Input:

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

Output:

integrate((-x^8 + 1)^p*(-x^4 + 1)^p, x)
 

Giac [F]

\[ \int \left (1-x^4\right )^p \left (1-x^8\right )^p \, dx=\int { {\left (-x^{8} + 1\right )}^{p} {\left (-x^{4} + 1\right )}^{p} \,d x } \] Input:

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

Output:

integrate((-x^8 + 1)^p*(-x^4 + 1)^p, x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (1-x^4\right )^p \left (1-x^8\right )^p \, dx=\int {\left (1-x^4\right )}^p\,{\left (1-x^8\right )}^p \,d x \] Input:

int((1 - x^4)^p*(1 - x^8)^p,x)
 

Output:

int((1 - x^4)^p*(1 - x^8)^p, x)
 

Reduce [F]

\[ \int \left (1-x^4\right )^p \left (1-x^8\right )^p \, dx=\frac {\left (-x^{4}+1\right )^{p} \left (-x^{8}+1\right )^{p} x -48 \left (\int \frac {\left (-x^{4}+1\right )^{p} \left (-x^{8}+1\right )^{p} x^{4}}{12 p \,x^{8}+x^{8}-12 p -1}d x \right ) p^{2}-4 \left (\int \frac {\left (-x^{4}+1\right )^{p} \left (-x^{8}+1\right )^{p} x^{4}}{12 p \,x^{8}+x^{8}-12 p -1}d x \right ) p -144 \left (\int \frac {\left (-x^{4}+1\right )^{p} \left (-x^{8}+1\right )^{p}}{12 p \,x^{8}+x^{8}-12 p -1}d x \right ) p^{2}-12 \left (\int \frac {\left (-x^{4}+1\right )^{p} \left (-x^{8}+1\right )^{p}}{12 p \,x^{8}+x^{8}-12 p -1}d x \right ) p}{12 p +1} \] Input:

int((-x^4+1)^p*(-x^8+1)^p,x)
 

Output:

(( - x**4 + 1)**p*( - x**8 + 1)**p*x - 48*int((( - x**4 + 1)**p*( - x**8 + 
 1)**p*x**4)/(12*p*x**8 - 12*p + x**8 - 1),x)*p**2 - 4*int((( - x**4 + 1)* 
*p*( - x**8 + 1)**p*x**4)/(12*p*x**8 - 12*p + x**8 - 1),x)*p - 144*int((( 
- x**4 + 1)**p*( - x**8 + 1)**p)/(12*p*x**8 - 12*p + x**8 - 1),x)*p**2 - 1 
2*int((( - x**4 + 1)**p*( - x**8 + 1)**p)/(12*p*x**8 - 12*p + x**8 - 1),x) 
*p)/(12*p + 1)