\(\int \frac {(-1+x^4) \sqrt [4]{-x^2+x^4}}{-1-x^2+x^4} \, dx\) [2750]

Optimal result

Integrand size = 31, antiderivative size = 257 \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{-x^2+x^4}}{-1-x^2+x^4} \, dx=\frac {1}{2} x \sqrt [4]{-x^2+x^4}-\frac {3}{4} \arctan \left (\frac {x}{\sqrt [4]{-x^2+x^4}}\right )+\sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \arctan \left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-x^2+x^4}}\right )+\sqrt {\frac {1}{10} \left (-1+\sqrt {5}\right )} \arctan \left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-x^2+x^4}}\right )+\frac {3}{4} \text {arctanh}\left (\frac {x}{\sqrt [4]{-x^2+x^4}}\right )-\sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-x^2+x^4}}\right )-\sqrt {\frac {1}{10} \left (-1+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-x^2+x^4}}\right ) \] Output:

1/2*x*(x^4-x^2)^(1/4)-3/4*arctan(x/(x^4-x^2)^(1/4))+1/10*(10+10*5^(1/2))^( 
1/2)*arctan(1/2*(-2+2*5^(1/2))^(1/2)*x/(x^4-x^2)^(1/4))+1/10*(-10+10*5^(1/ 
2))^(1/2)*arctan(1/2*(2+2*5^(1/2))^(1/2)*x/(x^4-x^2)^(1/4))+3/4*arctanh(x/ 
(x^4-x^2)^(1/4))-1/10*(10+10*5^(1/2))^(1/2)*arctanh(1/2*(-2+2*5^(1/2))^(1/ 
2)*x/(x^4-x^2)^(1/4))-1/10*(-10+10*5^(1/2))^(1/2)*arctanh(1/2*(2+2*5^(1/2) 
)^(1/2)*x/(x^4-x^2)^(1/4))
 

Mathematica [A] (verified)

Time = 0.80 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.04 \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{-x^2+x^4}}{-1-x^2+x^4} \, dx=\frac {x^{3/2} \left (-1+x^2\right )^{3/4} \left (10 x^{3/2} \sqrt [4]{-1+x^2}-15 \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )+2 \sqrt {10 \left (1+\sqrt {5}\right )} \arctan \left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )+2 \sqrt {10 \left (-1+\sqrt {5}\right )} \arctan \left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )+15 \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )-2 \sqrt {10 \left (1+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )-2 \sqrt {10 \left (-1+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )\right )}{20 \left (x^2 \left (-1+x^2\right )\right )^{3/4}} \] Input:

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

Output:

(x^(3/2)*(-1 + x^2)^(3/4)*(10*x^(3/2)*(-1 + x^2)^(1/4) - 15*ArcTan[Sqrt[x] 
/(-1 + x^2)^(1/4)] + 2*Sqrt[10*(1 + Sqrt[5])]*ArcTan[(Sqrt[(-1 + Sqrt[5])/ 
2]*Sqrt[x])/(-1 + x^2)^(1/4)] + 2*Sqrt[10*(-1 + Sqrt[5])]*ArcTan[(Sqrt[(1 
+ Sqrt[5])/2]*Sqrt[x])/(-1 + x^2)^(1/4)] + 15*ArcTanh[Sqrt[x]/(-1 + x^2)^( 
1/4)] - 2*Sqrt[10*(1 + Sqrt[5])]*ArcTanh[(Sqrt[(-1 + Sqrt[5])/2]*Sqrt[x])/ 
(-1 + x^2)^(1/4)] - 2*Sqrt[10*(-1 + Sqrt[5])]*ArcTanh[(Sqrt[(1 + Sqrt[5])/ 
2]*Sqrt[x])/(-1 + x^2)^(1/4)]))/(20*(x^2*(-1 + x^2))^(3/4))
 

Rubi [C] (warning: unable to verify)

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

Time = 1.21 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.48, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {2467, 1388, 2035, 25, 7279, 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.

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 1388

\(\displaystyle \frac {\sqrt [4]{x^4-x^2} \int \frac {\sqrt {x} \left (-x^2-1\right ) \left (x^2-1\right )^{5/4}}{-x^4+x^2+1}dx}{\sqrt {x} \sqrt [4]{x^2-1}}\)

\(\Big \downarrow \) 2035

\(\displaystyle \frac {2 \sqrt [4]{x^4-x^2} \int -\frac {x \left (x^2-1\right )^{5/4} \left (x^2+1\right )}{-x^4+x^2+1}d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^2-1}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {2 \sqrt [4]{x^4-x^2} \int \frac {x \left (x^2-1\right )^{5/4} \left (x^2+1\right )}{-x^4+x^2+1}d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^2-1}}\)

\(\Big \downarrow \) 7279

\(\displaystyle -\frac {2 \sqrt [4]{x^4-x^2} \int \left (-\frac {\left (x^2-1\right )^{5/4} x^3}{x^4-x^2-1}-\frac {\left (x^2-1\right )^{5/4} x}{x^4-x^2-1}\right )d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^2-1}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \sqrt [4]{x^4-x^2} \left (-\frac {2 \sqrt [4]{x^2-1} x^{3/2} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {5}{4},1,\frac {7}{4},x^2,\frac {2 x^2}{1-\sqrt {5}}\right )}{3 \sqrt {5} \left (1-\sqrt {5}\right ) \sqrt [4]{1-x^2}}+\frac {2 \sqrt [4]{x^2-1} x^{3/2} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x^2,\frac {2 x^2}{1-\sqrt {5}}\right )}{3 \sqrt {5} \left (1-\sqrt {5}\right ) \sqrt [4]{1-x^2}}+\frac {2 \sqrt [4]{x^2-1} x^{3/2} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {5}{4},\frac {7}{4},\frac {2 x^2}{1+\sqrt {5}},x^2\right )}{3 \sqrt {5} \left (1+\sqrt {5}\right ) \sqrt [4]{1-x^2}}-\frac {2 \sqrt [4]{x^2-1} x^{3/2} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {2 x^2}{1+\sqrt {5}},x^2\right )}{3 \sqrt {5} \left (1+\sqrt {5}\right ) \sqrt [4]{1-x^2}}+\frac {1}{8} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{x^2-1}}\right )-\frac {1}{8} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2-1}}\right )+\frac {1}{4} \sqrt [4]{x^2-1} x^{3/2}\right )}{\sqrt {x} \sqrt [4]{x^2-1}}\)

Input:

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

Output:

(2*(-x^2 + x^4)^(1/4)*((x^(3/2)*(-1 + x^2)^(1/4))/4 - (2*x^(3/2)*(-1 + x^2 
)^(1/4)*AppellF1[3/4, -5/4, 1, 7/4, x^2, (2*x^2)/(1 - Sqrt[5])])/(3*Sqrt[5 
]*(1 - Sqrt[5])*(1 - x^2)^(1/4)) + (2*x^(3/2)*(-1 + x^2)^(1/4)*AppellF1[3/ 
4, -1/4, 1, 7/4, x^2, (2*x^2)/(1 - Sqrt[5])])/(3*Sqrt[5]*(1 - Sqrt[5])*(1 
- x^2)^(1/4)) + (2*x^(3/2)*(-1 + x^2)^(1/4)*AppellF1[3/4, 1, -5/4, 7/4, (2 
*x^2)/(1 + Sqrt[5]), x^2])/(3*Sqrt[5]*(1 + Sqrt[5])*(1 - x^2)^(1/4)) - (2* 
x^(3/2)*(-1 + x^2)^(1/4)*AppellF1[3/4, 1, -1/4, 7/4, (2*x^2)/(1 + Sqrt[5]) 
, x^2])/(3*Sqrt[5]*(1 + Sqrt[5])*(1 - x^2)^(1/4)) + ArcTan[Sqrt[x]/(-1 + x 
^2)^(1/4)]/8 - ArcTanh[Sqrt[x]/(-1 + x^2)^(1/4)]/8))/(Sqrt[x]*(-1 + x^2)^( 
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 7279

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ 
{v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su 
mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
 

Maple [A] (verified)

Time = 4.84 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.82

method	result	size
pseudoelliptic	\(\frac {x \left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{2}-\frac {\sqrt {5}\, \left (\operatorname {arctanh}\left (\frac {2 \left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{x \sqrt {2+2 \sqrt {5}}}\right )+\arctan \left (\frac {2 \left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{x \sqrt {2+2 \sqrt {5}}}\right )\right ) \sqrt {-2+2 \sqrt {5}}}{10}-\frac {\left (\arctan \left (\frac {2 \left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{x \sqrt {-2+2 \sqrt {5}}}\right )+\operatorname {arctanh}\left (\frac {2 \left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{x \sqrt {-2+2 \sqrt {5}}}\right )\right ) \sqrt {5}\, \sqrt {2+2 \sqrt {5}}}{10}-\frac {3 \ln \left (\frac {\left (x^{4}-x^{2}\right )^{\frac {1}{4}}-x}{x}\right )}{8}+\frac {3 \arctan \left (\frac {\left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{x}\right )}{4}+\frac {3 \ln \left (\frac {\left (x^{4}-x^{2}\right )^{\frac {1}{4}}+x}{x}\right )}{8}\)	\(210\)

Input:

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

Output:

1/2*x*(x^4-x^2)^(1/4)-1/10*5^(1/2)*(arctanh(2*(x^4-x^2)^(1/4)/x/(2+2*5^(1/ 
2))^(1/2))+arctan(2*(x^4-x^2)^(1/4)/x/(2+2*5^(1/2))^(1/2)))*(-2+2*5^(1/2)) 
^(1/2)-1/10*(arctan(2*(x^4-x^2)^(1/4)/x/(-2+2*5^(1/2))^(1/2))+arctanh(2*(x 
^4-x^2)^(1/4)/x/(-2+2*5^(1/2))^(1/2)))*5^(1/2)*(2+2*5^(1/2))^(1/2)-3/8*ln( 
((x^4-x^2)^(1/4)-x)/x)+3/4*arctan((x^4-x^2)^(1/4)/x)+3/8*ln(((x^4-x^2)^(1/ 
4)+x)/x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1026 vs. \(2 (187) = 374\).

Time = 22.31 (sec) , antiderivative size = 1026, normalized size of antiderivative = 3.99 \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{-x^2+x^4}}{-1-x^2+x^4} \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*sqrt(1/10*sqrt(5) + 1/10)*arctan(((x^4 - x^2)^(3/4)*(sqrt(5)*(2*x^2 - 
1) + 5) + (5*x^4 - 5*x^2 - sqrt(5)*(x^4 - 3*x^2))*(x^4 - x^2)^(1/4))*sqrt( 
1/10*sqrt(5) + 1/10)/(x^5 - x^3 - x)) - 1/2*sqrt(1/10*sqrt(5) - 1/10)*arct 
an(((x^4 - x^2)^(3/4)*(sqrt(5)*(2*x^2 - 1) - 5) + (5*x^4 - 5*x^2 + sqrt(5) 
*(x^4 - 3*x^2))*(x^4 - x^2)^(1/4))*sqrt(1/10*sqrt(5) - 1/10)/(x^5 - x^3 - 
x)) + 1/4*sqrt(1/10*sqrt(5) + 1/10)*log((2*(x^4 - x^2)^(3/4)*(448*x^2 + sq 
rt(5)*(86*x^2 + 181) - 9) + (45*x^5 - 1855*x^3 - sqrt(5)*(905*x^5 - 923*x^ 
3 + 9*x) - 2*sqrt(x^4 - x^2)*(905*x^3 - sqrt(5)*(9*x^3 - 457*x) - 475*x) + 
 905*x)*sqrt(1/10*sqrt(5) + 1/10) - 2*(9*x^4 - 457*x^2 - sqrt(5)*(181*x^4 
- 95*x^2))*(x^4 - x^2)^(1/4))/(x^5 - x^3 - x)) - 1/4*sqrt(1/10*sqrt(5) + 1 
/10)*log((2*(x^4 - x^2)^(3/4)*(448*x^2 + sqrt(5)*(86*x^2 + 181) - 9) - (45 
*x^5 - 1855*x^3 - sqrt(5)*(905*x^5 - 923*x^3 + 9*x) - 2*sqrt(x^4 - x^2)*(9 
05*x^3 - sqrt(5)*(9*x^3 - 457*x) - 475*x) + 905*x)*sqrt(1/10*sqrt(5) + 1/1 
0) - 2*(9*x^4 - 457*x^2 - sqrt(5)*(181*x^4 - 95*x^2))*(x^4 - x^2)^(1/4))/( 
x^5 - x^3 - x)) - 1/4*sqrt(1/10*sqrt(5) - 1/10)*log((2*(x^4 - x^2)^(3/4)*( 
448*x^2 - sqrt(5)*(86*x^2 + 181) - 9) + (45*x^5 - 1855*x^3 + sqrt(5)*(905* 
x^5 - 923*x^3 + 9*x) + 2*sqrt(x^4 - x^2)*(905*x^3 + sqrt(5)*(9*x^3 - 457*x 
) - 475*x) + 905*x)*sqrt(1/10*sqrt(5) - 1/10) + 2*(9*x^4 - 457*x^2 + sqrt( 
5)*(181*x^4 - 95*x^2))*(x^4 - x^2)^(1/4))/(x^5 - x^3 - x)) + 1/4*sqrt(1/10 
*sqrt(5) - 1/10)*log((2*(x^4 - x^2)^(3/4)*(448*x^2 - sqrt(5)*(86*x^2 + ...
 

Sympy [F]

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

integrate((x**4-1)*(x**4-x**2)**(1/4)/(x**4-x**2-1),x)
 

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.96 \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{-x^2+x^4}}{-1-x^2+x^4} \, dx=\frac {1}{2} \, x^{2} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} - \frac {1}{10} \, \sqrt {10 \, \sqrt {5} - 10} \arctan \left (\frac {{\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}\right ) - \frac {1}{10} \, \sqrt {10 \, \sqrt {5} + 10} \arctan \left (\frac {{\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}}{\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}}\right ) - \frac {1}{20} \, \sqrt {10 \, \sqrt {5} - 10} \log \left (\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{20} \, \sqrt {10 \, \sqrt {5} - 10} \log \left (\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} - {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{20} \, \sqrt {10 \, \sqrt {5} + 10} \log \left (\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{20} \, \sqrt {10 \, \sqrt {5} + 10} \log \left ({\left | -\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} \right |}\right ) + \frac {3}{4} \, \arctan \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {3}{8} \, \log \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {3}{8} \, \log \left (-{\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 1\right ) \] Input:

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

Output:

1/2*x^2*(-1/x^2 + 1)^(1/4) - 1/10*sqrt(10*sqrt(5) - 10)*arctan((-1/x^2 + 1 
)^(1/4)/sqrt(1/2*sqrt(5) + 1/2)) - 1/10*sqrt(10*sqrt(5) + 10)*arctan((-1/x 
^2 + 1)^(1/4)/sqrt(1/2*sqrt(5) - 1/2)) - 1/20*sqrt(10*sqrt(5) - 10)*log(sq 
rt(1/2*sqrt(5) + 1/2) + (-1/x^2 + 1)^(1/4)) + 1/20*sqrt(10*sqrt(5) - 10)*l 
og(sqrt(1/2*sqrt(5) + 1/2) - (-1/x^2 + 1)^(1/4)) - 1/20*sqrt(10*sqrt(5) + 
10)*log(sqrt(1/2*sqrt(5) - 1/2) + (-1/x^2 + 1)^(1/4)) + 1/20*sqrt(10*sqrt( 
5) + 10)*log(abs(-sqrt(1/2*sqrt(5) - 1/2) + (-1/x^2 + 1)^(1/4))) + 3/4*arc 
tan((-1/x^2 + 1)^(1/4)) + 3/8*log((-1/x^2 + 1)^(1/4) + 1) - 3/8*log(-(-1/x 
^2 + 1)^(1/4) + 1)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*sqrt(x)*(x**2 - 1)**(1/4)*x + 3*int((sqrt(x)*(x**2 - 1)**(1/4)*x**4)/(x 
**6 - 2*x**4 + 1),x) - 3*int((sqrt(x)*(x**2 - 1)**(1/4)*x**2)/(x**6 - 2*x* 
*4 + 1),x) + int((sqrt(x)*(x**2 - 1)**(1/4))/(x**6 - 2*x**4 + 1),x))/4