\(\int \frac {-1+x^8}{\sqrt [4]{-x^2+x^6} (1+x^8)} \, dx\) [3058]

Optimal result

Integrand size = 26, antiderivative size = 469 \[ \int \frac {-1+x^8}{\sqrt [4]{-x^2+x^6} \left (1+x^8\right )} \, dx=-\frac {1}{4} \sqrt [4]{-4+3 \sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}} x}{-\sqrt {2+\sqrt {2}} x+2^{7/8} \sqrt [4]{-x^2+x^6}}\right )-\frac {1}{4} \sqrt [4]{-4+3 \sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}} x}{\sqrt {2+\sqrt {2}} x+2^{7/8} \sqrt [4]{-x^2+x^6}}\right )-\frac {1}{4} \sqrt [4]{4+3 \sqrt {2}} \arctan \left (\frac {2^{7/8} \sqrt {2+\sqrt {2}} x \sqrt [4]{-x^2+x^6}}{-2 x^2+2^{3/4} \sqrt {-x^2+x^6}}\right )-\frac {1}{4} \sqrt [4]{-4+3 \sqrt {2}} \text {arctanh}\left (\frac {\frac {\sqrt [8]{2} x^2}{\sqrt {2-\sqrt {2}}}+\frac {\sqrt {-x^2+x^6}}{\sqrt [8]{2} \sqrt {2-\sqrt {2}}}}{x \sqrt [4]{-x^2+x^6}}\right )+\frac {1}{8} \sqrt [4]{4+3 \sqrt {2}} \log \left (-2 x^2+2^{7/8} \sqrt {2+\sqrt {2}} x \sqrt [4]{-x^2+x^6}-2^{3/4} \sqrt {-x^2+x^6}\right )-\frac {1}{8} \sqrt [4]{4+3 \sqrt {2}} \log \left (2 \sqrt {2-\sqrt {2}} x^2+2\ 2^{3/8} x \sqrt [4]{-x^2+x^6}+2^{3/4} \sqrt {2-\sqrt {2}} \sqrt {-x^2+x^6}\right ) \] Output:

-1/4*(-4+3*2^(1/2))^(1/4)*arctan((2-2^(1/2))^(1/2)*x/(-(2+2^(1/2))^(1/2)*x 
+2^(7/8)*(x^6-x^2)^(1/4)))-1/4*(-4+3*2^(1/2))^(1/4)*arctan((2-2^(1/2))^(1/ 
2)*x/((2+2^(1/2))^(1/2)*x+2^(7/8)*(x^6-x^2)^(1/4)))-1/4*(4+3*2^(1/2))^(1/4 
)*arctan(2^(7/8)*(2+2^(1/2))^(1/2)*x*(x^6-x^2)^(1/4)/(-2*x^2+2^(3/4)*(x^6- 
x^2)^(1/2)))-1/4*(-4+3*2^(1/2))^(1/4)*arctanh((2^(1/8)*x^2/(2-2^(1/2))^(1/ 
2)+1/2*(x^6-x^2)^(1/2)*2^(7/8)/(2-2^(1/2))^(1/2))/x/(x^6-x^2)^(1/4))+1/8*( 
4+3*2^(1/2))^(1/4)*ln(-2*x^2+2^(7/8)*(2+2^(1/2))^(1/2)*x*(x^6-x^2)^(1/4)-2 
^(3/4)*(x^6-x^2)^(1/2))-1/8*(4+3*2^(1/2))^(1/4)*ln(2*(2-2^(1/2))^(1/2)*x^2 
+2*2^(3/8)*x*(x^6-x^2)^(1/4)+2^(3/4)*(2-2^(1/2))^(1/2)*(x^6-x^2)^(1/2))
                                                                                    
                                                                                    
 

Mathematica [F]

\[ \int \frac {-1+x^8}{\sqrt [4]{-x^2+x^6} \left (1+x^8\right )} \, dx=\int \frac {-1+x^8}{\sqrt [4]{-x^2+x^6} \left (1+x^8\right )} \, dx \] Input:

Integrate[(-1 + x^8)/((-x^2 + x^6)^(1/4)*(1 + x^8)),x]
 

Output:

Integrate[(-1 + x^8)/((-x^2 + x^6)^(1/4)*(1 + x^8)), x]
 

Rubi [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 0.57 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.29, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {2467, 25, 1388, 2035, 25, 7276, 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.

\(\displaystyle \int \frac {x^8-1}{\sqrt [4]{x^6-x^2} \left (x^8+1\right )} \, dx\)

\(\Big \downarrow \) 2467

\(\displaystyle \frac {\sqrt {x} \sqrt [4]{x^4-1} \int -\frac {1-x^8}{\sqrt {x} \sqrt [4]{x^4-1} \left (x^8+1\right )}dx}{\sqrt [4]{x^6-x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\sqrt {x} \sqrt [4]{x^4-1} \int \frac {1-x^8}{\sqrt {x} \sqrt [4]{x^4-1} \left (x^8+1\right )}dx}{\sqrt [4]{x^6-x^2}}\)

\(\Big \downarrow \) 1388

\(\displaystyle -\frac {\sqrt {x} \sqrt [4]{x^4-1} \int \frac {\left (-x^4-1\right ) \left (x^4-1\right )^{3/4}}{\sqrt {x} \left (x^8+1\right )}dx}{\sqrt [4]{x^6-x^2}}\)

\(\Big \downarrow \) 2035

\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^4-1} \int -\frac {\left (x^4-1\right )^{3/4} \left (x^4+1\right )}{x^8+1}d\sqrt {x}}{\sqrt [4]{x^6-x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^4-1} \int \frac {\left (x^4-1\right )^{3/4} \left (x^4+1\right )}{x^8+1}d\sqrt {x}}{\sqrt [4]{x^6-x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^4-1} \int \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (x^4-1\right )^{3/4}}{x^4+i}-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \left (x^4-1\right )^{3/4}}{i-x^4}\right )d\sqrt {x}}{\sqrt [4]{x^6-x^2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^4-1} \left (-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {x} \left (x^4-1\right )^{3/4} \operatorname {AppellF1}\left (\frac {1}{8},-\frac {3}{4},1,\frac {9}{8},x^4,i x^4\right )}{\left (1-x^4\right )^{3/4}}-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {x} \left (x^4-1\right )^{3/4} \operatorname {AppellF1}\left (\frac {1}{8},1,-\frac {3}{4},\frac {9}{8},-i x^4,x^4\right )}{\left (1-x^4\right )^{3/4}}\right )}{\sqrt [4]{x^6-x^2}}\)

Input:

Int[(-1 + x^8)/((-x^2 + x^6)^(1/4)*(1 + x^8)),x]
 

Output:

(-2*Sqrt[x]*(-1 + x^4)^(1/4)*(((-1/2 + I/2)*Sqrt[x]*(-1 + x^4)^(3/4)*Appel 
lF1[1/8, -3/4, 1, 9/8, x^4, I*x^4])/(1 - x^4)^(3/4) - ((1/2 + I/2)*Sqrt[x] 
*(-1 + x^4)^(3/4)*AppellF1[1/8, 1, -3/4, 9/8, (-I)*x^4, x^4])/(1 - x^4)^(3 
/4)))/(-x^2 + x^6)^(1/4)
 

Defintions of rubi rules used

rule 25

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

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]))
 

rule 2009

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

rule 2035

Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k   Subst 
[Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti 
onQ[m] && AlgebraicFunctionQ[Fx, x]
 

rule 2467

Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F 
racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p])   Int[x^( 
p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P 
olyQ[Px, x] &&  !IntegerQ[p] &&  !MonomialQ[Px, x] &&  !PolyQ[Fx, x]
 

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

Maple [F(-1)]

Timed out.

\[\int \frac {x^{8}-1}{\left (x^{6}-x^{2}\right )^{\frac {1}{4}} \left (x^{8}+1\right )}d x\]

Input:

int((x^8-1)/(x^6-x^2)^(1/4)/(x^8+1),x)
 

Output:

int((x^8-1)/(x^6-x^2)^(1/4)/(x^8+1),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {-1+x^8}{\sqrt [4]{-x^2+x^6} \left (1+x^8\right )} \, dx=\text {Timed out} \] Input:

integrate((x^8-1)/(x^6-x^2)^(1/4)/(x^8+1),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {-1+x^8}{\sqrt [4]{-x^2+x^6} \left (1+x^8\right )} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )}{\sqrt [4]{x^{2} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x^{8} + 1\right )}\, dx \] Input:

integrate((x**8-1)/(x**6-x**2)**(1/4)/(x**8+1),x)
 

Output:

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

Maxima [F]

\[ \int \frac {-1+x^8}{\sqrt [4]{-x^2+x^6} \left (1+x^8\right )} \, dx=\int { \frac {x^{8} - 1}{{\left (x^{8} + 1\right )} {\left (x^{6} - x^{2}\right )}^{\frac {1}{4}}} \,d x } \] Input:

integrate((x^8-1)/(x^6-x^2)^(1/4)/(x^8+1),x, algorithm="maxima")
 

Output:

integrate((x^8 - 1)/((x^8 + 1)*(x^6 - x^2)^(1/4)), x)
 

Giac [F]

\[ \int \frac {-1+x^8}{\sqrt [4]{-x^2+x^6} \left (1+x^8\right )} \, dx=\int { \frac {x^{8} - 1}{{\left (x^{8} + 1\right )} {\left (x^{6} - x^{2}\right )}^{\frac {1}{4}}} \,d x } \] Input:

integrate((x^8-1)/(x^6-x^2)^(1/4)/(x^8+1),x, algorithm="giac")
 

Output:

integrate((x^8 - 1)/((x^8 + 1)*(x^6 - x^2)^(1/4)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-1+x^8}{\sqrt [4]{-x^2+x^6} \left (1+x^8\right )} \, dx=\int \frac {x^8-1}{\left (x^8+1\right )\,{\left (x^6-x^2\right )}^{1/4}} \,d x \] Input:

int((x^8 - 1)/((x^8 + 1)*(x^6 - x^2)^(1/4)),x)
 

Output:

int((x^8 - 1)/((x^8 + 1)*(x^6 - x^2)^(1/4)), x)
 

Reduce [F]

\[ \int \frac {-1+x^8}{\sqrt [4]{-x^2+x^6} \left (1+x^8\right )} \, dx=\frac {\frac {4 \sqrt {x}\, \left (x^{4}-1\right )^{\frac {5}{4}}}{3}+\frac {2 \sqrt {x}\, \left (x^{4}-1\right )^{\frac {1}{4}} x^{4}}{3}-\frac {2 \sqrt {x}\, \left (x^{4}-1\right )^{\frac {1}{4}}}{3}+2 \sqrt {x^{4}-1}\, \left (\int \frac {\left (x^{4}-1\right )^{\frac {3}{4}}}{\sqrt {x}\, x^{16}-2 \sqrt {x}\, x^{12}+2 \sqrt {x}\, x^{8}-2 \sqrt {x}\, x^{4}+\sqrt {x}}d x \right ) x^{4}-2 \sqrt {x^{4}-1}\, \left (\int \frac {\left (x^{4}-1\right )^{\frac {3}{4}}}{\sqrt {x}\, x^{16}-2 \sqrt {x}\, x^{12}+2 \sqrt {x}\, x^{8}-2 \sqrt {x}\, x^{4}+\sqrt {x}}d x \right )+\frac {2 \sqrt {x^{4}-1}\, \left (\int \frac {\sqrt {x}\, \left (x^{4}-1\right )^{\frac {3}{4}} x^{11}}{x^{16}-2 x^{12}+2 x^{8}-2 x^{4}+1}d x \right ) x^{4}}{3}-\frac {2 \sqrt {x^{4}-1}\, \left (\int \frac {\sqrt {x}\, \left (x^{4}-1\right )^{\frac {3}{4}} x^{11}}{x^{16}-2 x^{12}+2 x^{8}-2 x^{4}+1}d x \right )}{3}-\frac {4 \sqrt {x^{4}-1}\, \left (\int \frac {\sqrt {x}\, \left (x^{4}-1\right )^{\frac {3}{4}} x^{3}}{x^{16}-2 x^{12}+2 x^{8}-2 x^{4}+1}d x \right ) x^{4}}{3}+\frac {4 \sqrt {x^{4}-1}\, \left (\int \frac {\sqrt {x}\, \left (x^{4}-1\right )^{\frac {3}{4}} x^{3}}{x^{16}-2 x^{12}+2 x^{8}-2 x^{4}+1}d x \right )}{3}}{\sqrt {x^{4}-1}\, \left (x^{4}-1\right )} \] Input:

int((x^8-1)/(x^6-x^2)^(1/4)/(x^8+1),x)
 

Output:

(2*(2*sqrt(x)*(x**4 - 1)**(5/4) + sqrt(x)*(x**4 - 1)**(1/4)*x**4 - sqrt(x) 
*(x**4 - 1)**(1/4) + 3*sqrt(x**4 - 1)*int((x**4 - 1)**(3/4)/(sqrt(x)*x**16 
 - 2*sqrt(x)*x**12 + 2*sqrt(x)*x**8 - 2*sqrt(x)*x**4 + sqrt(x)),x)*x**4 - 
3*sqrt(x**4 - 1)*int((x**4 - 1)**(3/4)/(sqrt(x)*x**16 - 2*sqrt(x)*x**12 + 
2*sqrt(x)*x**8 - 2*sqrt(x)*x**4 + sqrt(x)),x) + sqrt(x**4 - 1)*int((sqrt(x 
)*(x**4 - 1)**(3/4)*x**11)/(x**16 - 2*x**12 + 2*x**8 - 2*x**4 + 1),x)*x**4 
 - sqrt(x**4 - 1)*int((sqrt(x)*(x**4 - 1)**(3/4)*x**11)/(x**16 - 2*x**12 + 
 2*x**8 - 2*x**4 + 1),x) - 2*sqrt(x**4 - 1)*int((sqrt(x)*(x**4 - 1)**(3/4) 
*x**3)/(x**16 - 2*x**12 + 2*x**8 - 2*x**4 + 1),x)*x**4 + 2*sqrt(x**4 - 1)* 
int((sqrt(x)*(x**4 - 1)**(3/4)*x**3)/(x**16 - 2*x**12 + 2*x**8 - 2*x**4 + 
1),x)))/(3*sqrt(x**4 - 1)*(x**4 - 1))