\(\int \frac {-1+x^8}{\sqrt [4]{x^2+x^6} (1+x^8)} \, dx\) [2195]

Optimal result

Integrand size = 24, antiderivative size = 162 \[ \int \frac {-1+x^8}{\sqrt [4]{x^2+x^6} \left (1+x^8\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [8]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{2 \sqrt [8]{2}}+\frac {\arctan \left (\frac {2^{5/8} x \sqrt [4]{x^2+x^6}}{\sqrt [4]{2} x^2-\sqrt {x^2+x^6}}\right )}{2\ 2^{5/8}}-\frac {\text {arctanh}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{2 \sqrt [8]{2}}-\frac {\text {arctanh}\left (\frac {\frac {x^2}{2^{3/8}}+\frac {\sqrt {x^2+x^6}}{2^{5/8}}}{x \sqrt [4]{x^2+x^6}}\right )}{2\ 2^{5/8}} \] Output:

-1/4*arctan(2^(1/8)*x/(x^6+x^2)^(1/4))*2^(7/8)+1/4*arctan(2^(5/8)*x*(x^6+x 
^2)^(1/4)/(x^2*2^(1/4)-(x^6+x^2)^(1/2)))*2^(3/8)-1/4*arctanh(2^(1/8)*x/(x^ 
6+x^2)^(1/4))*2^(7/8)-1/4*arctanh((1/2*x^2*2^(5/8)+1/2*(x^6+x^2)^(1/2)*2^( 
3/8))/x/(x^6+x^2)^(1/4))*2^(3/8)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.92 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.09 \[ \int \frac {-1+x^8}{\sqrt [4]{x^2+x^6} \left (1+x^8\right )} \, dx=-\frac {\sqrt {x} \sqrt [4]{1+x^4} \left (\sqrt {2} \arctan \left (\frac {\sqrt [8]{2} \sqrt {x}}{\sqrt [4]{1+x^4}}\right )-\arctan \left (\frac {2^{5/8} \sqrt {x} \sqrt [4]{1+x^4}}{\sqrt [4]{2} x-\sqrt {1+x^4}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt [8]{2} \sqrt {x}}{\sqrt [4]{1+x^4}}\right )+\text {arctanh}\left (\frac {2\ 2^{3/8} \sqrt {x} \sqrt [4]{1+x^4}}{2 x+2^{3/4} \sqrt {1+x^4}}\right )\right )}{2\ 2^{5/8} \sqrt [4]{x^2+x^6}} \] Input:

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

Output:

-1/2*(Sqrt[x]*(1 + x^4)^(1/4)*(Sqrt[2]*ArcTan[(2^(1/8)*Sqrt[x])/(1 + x^4)^ 
(1/4)] - ArcTan[(2^(5/8)*Sqrt[x]*(1 + x^4)^(1/4))/(2^(1/4)*x - Sqrt[1 + x^ 
4])] + Sqrt[2]*ArcTanh[(2^(1/8)*Sqrt[x])/(1 + x^4)^(1/4)] + ArcTanh[(2*2^( 
3/8)*Sqrt[x]*(1 + x^4)^(1/4))/(2*x + 2^(3/4)*Sqrt[1 + x^4])]))/(2^(5/8)*(x 
^2 + x^6)^(1/4))
 

Rubi [C] (warning: unable to verify)

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

Time = 0.60 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.62, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2467, 25, 1388, 2035, 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

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

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

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

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

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

Time = 98.31 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.23

method	result	size
pseudoelliptic	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}}\right )}{4}\)	\(37\)
trager	\(\text {Expression too large to display}\)	\(2800\)

Input:

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

Output:

1/4*sum(ln((-_R*x+(x^2*(x^4+1))^(1/4))/x)/_R,_R=RootOf(_Z^8-2))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 68.39 (sec) , antiderivative size = 1624, normalized size of antiderivative = 10.02 \[ \int \frac {-1+x^8}{\sqrt [4]{x^2+x^6} \left (1+x^8\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/16*2^(7/8)*log(-(2^(7/8)*(x^9 - 2*x^7 + 4*x^5 - 2*x^3 + x) + 2*(x^6 + x 
^2)^(3/4)*(2*x^4 - 2*x^2 - sqrt(2)*(x^4 - 2*x^2 + 1) + 2) - 2*sqrt(x^6 + x 
^2)*(2^(5/8)*(x^5 - 2*x^3 + x) - 2*2^(1/8)*(x^5 - x^3 + x)) - 2^(3/8)*(x^9 
 - 4*x^7 + 4*x^5 - 4*x^3 + x) - 2*(x^6 + x^2)^(1/4)*(2^(3/4)*(x^6 - 2*x^4 
+ x^2) - 2*2^(1/4)*(x^6 - x^4 + x^2)))/(x^9 + x)) + 1/16*2^(7/8)*log((2^(7 
/8)*(x^9 - 2*x^7 + 4*x^5 - 2*x^3 + x) - 2*(x^6 + x^2)^(3/4)*(2*x^4 - 2*x^2 
 - sqrt(2)*(x^4 - 2*x^2 + 1) + 2) - 2*sqrt(x^6 + x^2)*(2^(5/8)*(x^5 - 2*x^ 
3 + x) - 2*2^(1/8)*(x^5 - x^3 + x)) - 2^(3/8)*(x^9 - 4*x^7 + 4*x^5 - 4*x^3 
 + x) + 2*(x^6 + x^2)^(1/4)*(2^(3/4)*(x^6 - 2*x^4 + x^2) - 2*2^(1/4)*(x^6 
- x^4 + x^2)))/(x^9 + x)) + 1/16*I*2^(7/8)*log((2^(7/8)*(I*x^9 - 2*I*x^7 + 
 4*I*x^5 - 2*I*x^3 + I*x) - 2*(x^6 + x^2)^(3/4)*(2*x^4 - 2*x^2 - sqrt(2)*( 
x^4 - 2*x^2 + 1) + 2) - 2*sqrt(x^6 + x^2)*(2^(5/8)*(-I*x^5 + 2*I*x^3 - I*x 
) + 2*2^(1/8)*(I*x^5 - I*x^3 + I*x)) + 2^(3/8)*(-I*x^9 + 4*I*x^7 - 4*I*x^5 
 + 4*I*x^3 - I*x) - 2*(x^6 + x^2)^(1/4)*(2^(3/4)*(x^6 - 2*x^4 + x^2) - 2*2 
^(1/4)*(x^6 - x^4 + x^2)))/(x^9 + x)) - 1/16*I*2^(7/8)*log((2^(7/8)*(-I*x^ 
9 + 2*I*x^7 - 4*I*x^5 + 2*I*x^3 - I*x) - 2*(x^6 + x^2)^(3/4)*(2*x^4 - 2*x^ 
2 - sqrt(2)*(x^4 - 2*x^2 + 1) + 2) - 2*sqrt(x^6 + x^2)*(2^(5/8)*(I*x^5 - 2 
*I*x^3 + I*x) + 2*2^(1/8)*(-I*x^5 + I*x^3 - I*x)) + 2^(3/8)*(I*x^9 - 4*I*x 
^7 + 4*I*x^5 - 4*I*x^3 + I*x) - 2*(x^6 + x^2)^(1/4)*(2^(3/4)*(x^6 - 2*x^4 
+ x^2) - 2*2^(1/4)*(x^6 - x^4 + x^2)))/(x^9 + x)) + (1/16*I + 1/16)*2^(...
 

Sympy [F]

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

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

Output:

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