3.24.24 \(\int \frac {\sqrt [4]{x^2+x^6} (1+x^8)}{x^4 (-1+x^4)} \, dx\) [2324]

3.24.24.1 Optimal result

Integrand size = 27, antiderivative size = 180 \[ \int \frac {\sqrt [4]{x^2+x^6} \left (1+x^8\right )}{x^4 \left (-1+x^4\right )} \, dx=\frac {2 \left (1+x^4\right ) \sqrt [4]{x^2+x^6}}{5 x^3}+\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{2^{3/4}}+\frac {\arctan \left (\frac {2^{3/4} x \sqrt [4]{x^2+x^6}}{\sqrt {2} x^2-\sqrt {x^2+x^6}}\right )}{2 \sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{2^{3/4}}+\frac {\text {arctanh}\left (\frac {\frac {x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^2+x^6}}{2^{3/4}}}{x \sqrt [4]{x^2+x^6}}\right )}{2 \sqrt [4]{2}} \]

output

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

3.24.24.2 Mathematica [A] (verified)

Time = 0.80 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.27 \[ \int \frac {\sqrt [4]{x^2+x^6} \left (1+x^8\right )}{x^4 \left (-1+x^4\right )} \, dx=\frac {\sqrt [4]{x^2+x^6} \left (8 \sqrt [4]{1+x^4}+8 x^4 \sqrt [4]{1+x^4}+10 \sqrt [4]{2} x^{5/2} \arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^4}}\right )+5\ 2^{3/4} x^{5/2} \arctan \left (\frac {2^{3/4} \sqrt {x} \sqrt [4]{1+x^4}}{\sqrt {2} x-\sqrt {1+x^4}}\right )-10 \sqrt [4]{2} x^{5/2} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^4}}\right )+5\ 2^{3/4} x^{5/2} \text {arctanh}\left (\frac {2 \sqrt [4]{2} \sqrt {x} \sqrt [4]{1+x^4}}{2 x+\sqrt {2} \sqrt {1+x^4}}\right )\right )}{20 x^3 \sqrt [4]{1+x^4}} \]

input

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

output

((x^2 + x^6)^(1/4)*(8*(1 + x^4)^(1/4) + 8*x^4*(1 + x^4)^(1/4) + 10*2^(1/4) 
*x^(5/2)*ArcTan[(2^(1/4)*Sqrt[x])/(1 + x^4)^(1/4)] + 5*2^(3/4)*x^(5/2)*Arc 
Tan[(2^(3/4)*Sqrt[x]*(1 + x^4)^(1/4))/(Sqrt[2]*x - Sqrt[1 + x^4])] - 10*2^ 
(1/4)*x^(5/2)*ArcTanh[(2^(1/4)*Sqrt[x])/(1 + x^4)^(1/4)] + 5*2^(3/4)*x^(5/ 
2)*ArcTanh[(2*2^(1/4)*Sqrt[x]*(1 + x^4)^(1/4))/(2*x + Sqrt[2]*Sqrt[1 + x^4 
])]))/(20*x^3*(1 + x^4)^(1/4))
 

3.24.24.3 Rubi [F]

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.

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

\(\Big \downarrow \) 2467

\(\displaystyle \frac {\sqrt [4]{x^6+x^2} \int -\frac {\sqrt [4]{x^4+1} \left (x^8+1\right )}{x^{7/2} \left (1-x^4\right )}dx}{\sqrt {x} \sqrt [4]{x^4+1}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\sqrt [4]{x^6+x^2} \int \frac {\sqrt [4]{x^4+1} \left (x^8+1\right )}{x^{7/2} \left (1-x^4\right )}dx}{\sqrt {x} \sqrt [4]{x^4+1}}\)

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

input

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

output

$Aborted
 

3.24.24.3.1 Defintions of rubi rules used

rule 25

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

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 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, 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]
 

3.24.24.4 Maple [A] (verified)

Time = 47.08 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.44

method	result	size
pseudoelliptic	\(\frac {5 \ln \left (\frac {2^{\frac {3}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} x +\sqrt {2}\, x^{2}+\sqrt {x^{2} \left (x^{4}+1\right )}}{-2^{\frac {3}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} x +\sqrt {2}\, x^{2}+\sqrt {x^{2} \left (x^{4}+1\right )}}\right ) 2^{\frac {3}{4}} x^{3}+10 \arctan \left (\frac {2^{\frac {1}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}+x}{x}\right ) 2^{\frac {3}{4}} x^{3}+10 \arctan \left (\frac {2^{\frac {1}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}-x}{x}\right ) 2^{\frac {3}{4}} x^{3}-10 \ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}\right ) 2^{\frac {1}{4}} x^{3}-20 \arctan \left (\frac {\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2 x}\right ) 2^{\frac {1}{4}} x^{3}+16 x^{4} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}+16 \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}{40 x^{3}}\)	\(260\)
trager	\(\text {Expression too large to display}\)	\(656\)
risch	\(\text {Expression too large to display}\)	\(1587\)

input

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

output

1/40/x^3*(5*ln((2^(3/4)*(x^2*(x^4+1))^(1/4)*x+2^(1/2)*x^2+(x^2*(x^4+1))^(1 
/2))/(-2^(3/4)*(x^2*(x^4+1))^(1/4)*x+2^(1/2)*x^2+(x^2*(x^4+1))^(1/2)))*2^( 
3/4)*x^3+10*arctan((2^(1/4)*(x^2*(x^4+1))^(1/4)+x)/x)*2^(3/4)*x^3+10*arcta 
n((2^(1/4)*(x^2*(x^4+1))^(1/4)-x)/x)*2^(3/4)*x^3-10*ln((-2^(1/4)*x-(x^2*(x 
^4+1))^(1/4))/(2^(1/4)*x-(x^2*(x^4+1))^(1/4)))*2^(1/4)*x^3-20*arctan(1/2*( 
x^2*(x^4+1))^(1/4)/x*2^(3/4))*2^(1/4)*x^3+16*x^4*(x^2*(x^4+1))^(1/4)+16*(x 
^2*(x^4+1))^(1/4))
 

3.24.24.5 Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 8.37 (sec) , antiderivative size = 724, normalized size of antiderivative = 4.02 \[ \int \frac {\sqrt [4]{x^2+x^6} \left (1+x^8\right )}{x^4 \left (-1+x^4\right )} \, dx=\text {Too large to display} \]

input

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

output

1/320*((5*I + 5)*8^(3/4)*sqrt(2)*x^3*log(((I + 1)*8^(3/4)*sqrt(2)*sqrt(x^6 
 + x^2)*x + 8*I*sqrt(2)*(x^6 + x^2)^(1/4)*x^2 + 8^(1/4)*sqrt(2)*(-(I - 1)* 
x^5 + (2*I - 2)*x^3 - (I - 1)*x) + 8*(x^6 + x^2)^(3/4))/(x^5 + 2*x^3 + x)) 
 - (5*I - 5)*8^(3/4)*sqrt(2)*x^3*log((-(I - 1)*8^(3/4)*sqrt(2)*sqrt(x^6 + 
x^2)*x - 8*I*sqrt(2)*(x^6 + x^2)^(1/4)*x^2 + 8^(1/4)*sqrt(2)*((I + 1)*x^5 
- (2*I + 2)*x^3 + (I + 1)*x) + 8*(x^6 + x^2)^(3/4))/(x^5 + 2*x^3 + x)) + ( 
5*I - 5)*8^(3/4)*sqrt(2)*x^3*log(((I - 1)*8^(3/4)*sqrt(2)*sqrt(x^6 + x^2)* 
x - 8*I*sqrt(2)*(x^6 + x^2)^(1/4)*x^2 + 8^(1/4)*sqrt(2)*(-(I + 1)*x^5 + (2 
*I + 2)*x^3 - (I + 1)*x) + 8*(x^6 + x^2)^(3/4))/(x^5 + 2*x^3 + x)) - (5*I 
+ 5)*8^(3/4)*sqrt(2)*x^3*log((-(I + 1)*8^(3/4)*sqrt(2)*sqrt(x^6 + x^2)*x + 
 8*I*sqrt(2)*(x^6 + x^2)^(1/4)*x^2 + 8^(1/4)*sqrt(2)*((I - 1)*x^5 - (2*I - 
 2)*x^3 + (I - 1)*x) + 8*(x^6 + x^2)^(3/4))/(x^5 + 2*x^3 + x)) - 10*8^(3/4 
)*x^3*log(-(4*sqrt(2)*(x^6 + x^2)^(1/4)*x^2 + 8^(3/4)*sqrt(x^6 + x^2)*x + 
8^(1/4)*(x^5 + 2*x^3 + x) + 4*(x^6 + x^2)^(3/4))/(x^5 - 2*x^3 + x)) + 10*I 
*8^(3/4)*x^3*log((4*sqrt(2)*(x^6 + x^2)^(1/4)*x^2 + I*8^(3/4)*sqrt(x^6 + x 
^2)*x + 8^(1/4)*(-I*x^5 - 2*I*x^3 - I*x) - 4*(x^6 + x^2)^(3/4))/(x^5 - 2*x 
^3 + x)) - 10*I*8^(3/4)*x^3*log((4*sqrt(2)*(x^6 + x^2)^(1/4)*x^2 - I*8^(3/ 
4)*sqrt(x^6 + x^2)*x + 8^(1/4)*(I*x^5 + 2*I*x^3 + I*x) - 4*(x^6 + x^2)^(3/ 
4))/(x^5 - 2*x^3 + x)) + 10*8^(3/4)*x^3*log(-(4*sqrt(2)*(x^6 + x^2)^(1/4)* 
x^2 - 8^(3/4)*sqrt(x^6 + x^2)*x - 8^(1/4)*(x^5 + 2*x^3 + x) + 4*(x^6 + ...
 

3.24.24.6 Sympy [F]

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

input

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

output

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

3.24.24.7 Maxima [F]

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

input

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

output

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

3.24.24.8 Giac [F]

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

input

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

output

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

3.24.24.9 Mupad [F(-1)]

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

input

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

output

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