Integrand size = 21, antiderivative size = 107 \[ \int \frac {1}{\left (-1+x^4\right )^2 \sqrt [4]{-x^2+x^4}} \, dx=\frac {\left (-x^2+x^4\right )^{3/4} \left (-85+2 x^2+67 x^4\right )}{80 x \left (-1+x^2\right )^2 \left (1+x^2\right )}+\frac {15 \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^2+x^4}}\right )}{32 \sqrt [4]{2}}+\frac {15 \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^2+x^4}}\right )}{32 \sqrt [4]{2}} \]
1/80*(x^4-x^2)^(3/4)*(67*x^4+2*x^2-85)/x/(x^2-1)^2/(x^2+1)+15/64*arctan(2^ (1/4)*x/(x^4-x^2)^(1/4))*2^(3/4)+15/64*arctanh(2^(1/4)*x/(x^4-x^2)^(1/4))* 2^(3/4)
Time = 0.42 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.24 \[ \int \frac {1}{\left (-1+x^4\right )^2 \sqrt [4]{-x^2+x^4}} \, dx=\frac {x^{5/2} \left (4 \sqrt {x} \left (-85+2 x^2+67 x^4\right )+75\ 2^{3/4} \sqrt [4]{-1+x^2} \left (-1+x^4\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )+75\ 2^{3/4} \sqrt [4]{-1+x^2} \left (-1+x^4\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )\right )}{320 \left (x^2 \left (-1+x^2\right )\right )^{5/4} \left (1+x^2\right )} \]
(x^(5/2)*(4*Sqrt[x]*(-85 + 2*x^2 + 67*x^4) + 75*2^(3/4)*(-1 + x^2)^(1/4)*( -1 + x^4)*ArcTan[(2^(1/4)*Sqrt[x])/(-1 + x^2)^(1/4)] + 75*2^(3/4)*(-1 + x^ 2)^(1/4)*(-1 + x^4)*ArcTanh[(2^(1/4)*Sqrt[x])/(-1 + x^2)^(1/4)]))/(320*(x^ 2*(-1 + x^2))^(5/4)*(1 + x^2))
Time = 0.37 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.55, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {2467, 1388, 368, 931, 1024, 25, 1024, 27, 902, 756, 216, 219}
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.
| \(\displaystyle \int \frac {1}{\left (x^4-1\right )^2 \sqrt [4]{x^4-x^2}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt [4]{x^2-1} \int \frac {1}{\sqrt {x} \sqrt [4]{x^2-1} \left (1-x^4\right )^2}dx}{\sqrt [4]{x^4-x^2}}\) |
| \(\Big \downarrow \) 1388 |
| \(\displaystyle \frac {\sqrt {x} \sqrt [4]{x^2-1} \int \frac {1}{\sqrt {x} \left (-x^2-1\right )^2 \left (x^2-1\right )^{9/4}}dx}{\sqrt [4]{x^4-x^2}}\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int \frac {1}{\left (x^2-1\right )^{9/4} \left (x^2+1\right )^2}d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\) |
| \(\Big \downarrow \) 931 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \left (\frac {1}{8} \int \frac {7-8 x^2}{\left (x^2-1\right )^{9/4} \left (x^2+1\right )}d\sqrt {x}-\frac {\sqrt {x}}{8 \left (x^2-1\right )^{5/4} \left (x^2+1\right )}\right )}{\sqrt [4]{x^4-x^2}}\) |
| \(\Big \downarrow \) 1024 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \left (\frac {1}{8} \left (\frac {1}{10} \int -\frac {71-4 x^2}{\left (x^2-1\right )^{5/4} \left (x^2+1\right )}d\sqrt {x}+\frac {\sqrt {x}}{10 \left (x^2-1\right )^{5/4}}\right )-\frac {\sqrt {x}}{8 \left (x^2-1\right )^{5/4} \left (x^2+1\right )}\right )}{\sqrt [4]{x^4-x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \left (\frac {1}{8} \left (\frac {\sqrt {x}}{10 \left (x^2-1\right )^{5/4}}-\frac {1}{10} \int \frac {71-4 x^2}{\left (x^2-1\right )^{5/4} \left (x^2+1\right )}d\sqrt {x}\right )-\frac {\sqrt {x}}{8 \left (x^2-1\right )^{5/4} \left (x^2+1\right )}\right )}{\sqrt [4]{x^4-x^2}}\) |
| \(\Big \downarrow \) 1024 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \left (\frac {1}{8} \left (\frac {1}{10} \left (\frac {67 \sqrt {x}}{2 \sqrt [4]{x^2-1}}-\frac {1}{2} \int -\frac {75}{\sqrt [4]{x^2-1} \left (x^2+1\right )}d\sqrt {x}\right )+\frac {\sqrt {x}}{10 \left (x^2-1\right )^{5/4}}\right )-\frac {\sqrt {x}}{8 \left (x^2-1\right )^{5/4} \left (x^2+1\right )}\right )}{\sqrt [4]{x^4-x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \left (\frac {1}{8} \left (\frac {1}{10} \left (\frac {75}{2} \int \frac {1}{\sqrt [4]{x^2-1} \left (x^2+1\right )}d\sqrt {x}+\frac {67 \sqrt {x}}{2 \sqrt [4]{x^2-1}}\right )+\frac {\sqrt {x}}{10 \left (x^2-1\right )^{5/4}}\right )-\frac {\sqrt {x}}{8 \left (x^2-1\right )^{5/4} \left (x^2+1\right )}\right )}{\sqrt [4]{x^4-x^2}}\) |
| \(\Big \downarrow \) 902 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \left (\frac {1}{8} \left (\frac {1}{10} \left (\frac {75}{2} \int \frac {1}{1-2 x^2}d\frac {\sqrt {x}}{\sqrt [4]{x^2-1}}+\frac {67 \sqrt {x}}{2 \sqrt [4]{x^2-1}}\right )+\frac {\sqrt {x}}{10 \left (x^2-1\right )^{5/4}}\right )-\frac {\sqrt {x}}{8 \left (x^2-1\right )^{5/4} \left (x^2+1\right )}\right )}{\sqrt [4]{x^4-x^2}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \left (\frac {1}{8} \left (\frac {1}{10} \left (\frac {75}{2} \left (\frac {1}{2} \int \frac {1}{1-\sqrt {2} x}d\frac {\sqrt {x}}{\sqrt [4]{x^2-1}}+\frac {1}{2} \int \frac {1}{\sqrt {2} x+1}d\frac {\sqrt {x}}{\sqrt [4]{x^2-1}}\right )+\frac {67 \sqrt {x}}{2 \sqrt [4]{x^2-1}}\right )+\frac {\sqrt {x}}{10 \left (x^2-1\right )^{5/4}}\right )-\frac {\sqrt {x}}{8 \left (x^2-1\right )^{5/4} \left (x^2+1\right )}\right )}{\sqrt [4]{x^4-x^2}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \left (\frac {1}{8} \left (\frac {1}{10} \left (\frac {75}{2} \left (\frac {1}{2} \int \frac {1}{1-\sqrt {2} x}d\frac {\sqrt {x}}{\sqrt [4]{x^2-1}}+\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{2 \sqrt [4]{2}}\right )+\frac {67 \sqrt {x}}{2 \sqrt [4]{x^2-1}}\right )+\frac {\sqrt {x}}{10 \left (x^2-1\right )^{5/4}}\right )-\frac {\sqrt {x}}{8 \left (x^2-1\right )^{5/4} \left (x^2+1\right )}\right )}{\sqrt [4]{x^4-x^2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \left (\frac {1}{8} \left (\frac {1}{10} \left (\frac {75}{2} \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{2 \sqrt [4]{2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{2 \sqrt [4]{2}}\right )+\frac {67 \sqrt {x}}{2 \sqrt [4]{x^2-1}}\right )+\frac {\sqrt {x}}{10 \left (x^2-1\right )^{5/4}}\right )-\frac {\sqrt {x}}{8 \left (x^2-1\right )^{5/4} \left (x^2+1\right )}\right )}{\sqrt [4]{x^4-x^2}}\) |
(2*Sqrt[x]*(-1 + x^2)^(1/4)*(-1/8*Sqrt[x]/((-1 + x^2)^(5/4)*(1 + x^2)) + ( Sqrt[x]/(10*(-1 + x^2)^(5/4)) + ((67*Sqrt[x])/(2*(-1 + x^2)^(1/4)) + (75*( ArcTan[(2^(1/4)*Sqrt[x])/(-1 + x^2)^(1/4)]/(2*2^(1/4)) + ArcTanh[(2^(1/4)* Sqrt[x])/(-1 + x^2)^(1/4)]/(2*2^(1/4))))/2)/10)/8))/(-x^2 + x^4)^(1/4)
3.16.55.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) , x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*2)/e^2))^p*(c + d*(x^(k*2)/e^2))^q, x], x, (e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && FractionQ[m ] && IntegerQ[p]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Su bst[Int[1/(c - (b*c - a*d)*x^n), x], x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b , c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Simp[1/(a*n*(p + 1)*(b*c - a*d)) Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, n, p, q, x]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f _.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*n*(b*c - a*d)*( p + 1)) Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b *c - a*d)*(p + 1) + d*(b*e - a*f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; Fr eeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]
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]))
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]
Time = 4.28 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.18
| method | result | size |
| pseudoelliptic | \(-\frac {150 \left (\frac {2^{\frac {3}{4}} \left (x^{4}-1\right ) \left (2 \arctan \left (\frac {2^{\frac {3}{4}} \left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{2 x}\right )-\ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{4}-x^{2}\right )^{\frac {1}{4}}}\right )\right ) \left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{2}-\frac {268 x^{5}}{75}-\frac {8 x^{3}}{75}+\frac {68 x}{15}\right )}{\left (x^{4}-x^{2}\right )^{\frac {1}{4}} \left (640 x^{4}-640\right )}\) | \(126\) |
| risch | \(\frac {x \left (67 x^{4}+2 x^{2}-85\right )}{80 \left (x^{2} \left (x^{2}-1\right )\right )^{\frac {1}{4}} \left (x^{2}-1\right ) \left (x^{2}+1\right )}-\frac {15 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) \ln \left (\frac {-\sqrt {x^{4}-x^{2}}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{3} x +2 \left (x^{4}-x^{2}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) x^{3}+4 \left (x^{4}-x^{2}\right )^{\frac {3}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) x}{x \left (x^{2}+1\right )}\right )}{128}-\frac {15 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (\frac {\sqrt {x^{4}-x^{2}}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x -2 \left (x^{4}-x^{2}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{3}+4 \left (x^{4}-x^{2}\right )^{\frac {3}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x}{x \left (x^{2}+1\right )}\right )}{128}\) | \(273\) |
| trager | \(\frac {\left (x^{4}-x^{2}\right )^{\frac {3}{4}} \left (67 x^{4}+2 x^{2}-85\right )}{80 x \left (x^{2}-1\right )^{2} \left (x^{2}+1\right )}-\frac {15 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) \ln \left (\frac {-\sqrt {x^{4}-x^{2}}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{3} x +2 \left (x^{4}-x^{2}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) x^{3}+4 \left (x^{4}-x^{2}\right )^{\frac {3}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) x}{x \left (x^{2}+1\right )}\right )}{128}-\frac {15 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (\frac {\sqrt {x^{4}-x^{2}}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x -2 \left (x^{4}-x^{2}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{3}+4 \left (x^{4}-x^{2}\right )^{\frac {3}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x}{x \left (x^{2}+1\right )}\right )}{128}\) | \(275\) |
-150/(x^4-x^2)^(1/4)*(1/2*2^(3/4)*(x^4-1)*(2*arctan(1/2*2^(3/4)/x*(x^4-x^2 )^(1/4))-ln((-2^(1/4)*x-(x^4-x^2)^(1/4))/(2^(1/4)*x-(x^4-x^2)^(1/4))))*(x^ 4-x^2)^(1/4)-268/75*x^5-8/75*x^3+68/15*x)/(640*x^4-640)
Result contains complex when optimal does not.
Time = 1.34 (sec) , antiderivative size = 426, normalized size of antiderivative = 3.98 \[ \int \frac {1}{\left (-1+x^4\right )^2 \sqrt [4]{-x^2+x^4}} \, dx=\frac {75 \cdot 2^{\frac {3}{4}} {\left (x^{7} - x^{5} - x^{3} + x\right )} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} x^{2} + 2^{\frac {3}{4}} {\left (3 \, x^{3} - x\right )} + 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{4} - x^{2}} x + 4 \, {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}}}{x^{3} + x}\right ) - 75 \cdot 2^{\frac {3}{4}} {\left (x^{7} - x^{5} - x^{3} + x\right )} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} x^{2} - 2^{\frac {3}{4}} {\left (3 \, x^{3} - x\right )} - 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{4} - x^{2}} x + 4 \, {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}}}{x^{3} + x}\right ) - 75 \cdot 2^{\frac {3}{4}} {\left (-i \, x^{7} + i \, x^{5} + i \, x^{3} - i \, x\right )} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} x^{2} - 2^{\frac {3}{4}} {\left (3 i \, x^{3} - i \, x\right )} + 4 i \cdot 2^{\frac {1}{4}} \sqrt {x^{4} - x^{2}} x - 4 \, {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}}}{x^{3} + x}\right ) - 75 \cdot 2^{\frac {3}{4}} {\left (i \, x^{7} - i \, x^{5} - i \, x^{3} + i \, x\right )} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} x^{2} - 2^{\frac {3}{4}} {\left (-3 i \, x^{3} + i \, x\right )} - 4 i \cdot 2^{\frac {1}{4}} \sqrt {x^{4} - x^{2}} x - 4 \, {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}}}{x^{3} + x}\right ) + 16 \, {\left (67 \, x^{4} + 2 \, x^{2} - 85\right )} {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}}}{1280 \, {\left (x^{7} - x^{5} - x^{3} + x\right )}} \]
1/1280*(75*2^(3/4)*(x^7 - x^5 - x^3 + x)*log((4*sqrt(2)*(x^4 - x^2)^(1/4)* x^2 + 2^(3/4)*(3*x^3 - x) + 4*2^(1/4)*sqrt(x^4 - x^2)*x + 4*(x^4 - x^2)^(3 /4))/(x^3 + x)) - 75*2^(3/4)*(x^7 - x^5 - x^3 + x)*log((4*sqrt(2)*(x^4 - x ^2)^(1/4)*x^2 - 2^(3/4)*(3*x^3 - x) - 4*2^(1/4)*sqrt(x^4 - x^2)*x + 4*(x^4 - x^2)^(3/4))/(x^3 + x)) - 75*2^(3/4)*(-I*x^7 + I*x^5 + I*x^3 - I*x)*log( -(4*sqrt(2)*(x^4 - x^2)^(1/4)*x^2 - 2^(3/4)*(3*I*x^3 - I*x) + 4*I*2^(1/4)* sqrt(x^4 - x^2)*x - 4*(x^4 - x^2)^(3/4))/(x^3 + x)) - 75*2^(3/4)*(I*x^7 - I*x^5 - I*x^3 + I*x)*log(-(4*sqrt(2)*(x^4 - x^2)^(1/4)*x^2 - 2^(3/4)*(-3*I *x^3 + I*x) - 4*I*2^(1/4)*sqrt(x^4 - x^2)*x - 4*(x^4 - x^2)^(3/4))/(x^3 + x)) + 16*(67*x^4 + 2*x^2 - 85)*(x^4 - x^2)^(3/4))/(x^7 - x^5 - x^3 + x)
\[ \int \frac {1}{\left (-1+x^4\right )^2 \sqrt [4]{-x^2+x^4}} \, dx=\int \frac {1}{\sqrt [4]{x^{2} \left (x - 1\right ) \left (x + 1\right )} \left (x - 1\right )^{2} \left (x + 1\right )^{2} \left (x^{2} + 1\right )^{2}}\, dx \]
\[ \int \frac {1}{\left (-1+x^4\right )^2 \sqrt [4]{-x^2+x^4}} \, dx=\int { \frac {1}{{\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}^{2}} \,d x } \]
Time = 0.31 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.97 \[ \int \frac {1}{\left (-1+x^4\right )^2 \sqrt [4]{-x^2+x^4}} \, dx=-\frac {15}{64} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {15}{128} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {1}{4}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {15}{128} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {1}{4}} - {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {\frac {10}{x^{2}} - 9}{10 \, {\left (\frac {1}{x^{2}} - 1\right )} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}} - \frac {{\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {3}{4}}}{16 \, {\left (\frac {1}{x^{2}} + 1\right )}} \]
-15/64*2^(3/4)*arctan(1/2*2^(3/4)*(-1/x^2 + 1)^(1/4)) + 15/128*2^(3/4)*log (2^(1/4) + (-1/x^2 + 1)^(1/4)) - 15/128*2^(3/4)*log(2^(1/4) - (-1/x^2 + 1) ^(1/4)) + 1/10*(10/x^2 - 9)/((1/x^2 - 1)*(-1/x^2 + 1)^(1/4)) - 1/16*(-1/x^ 2 + 1)^(3/4)/(1/x^2 + 1)
Timed out. \[ \int \frac {1}{\left (-1+x^4\right )^2 \sqrt [4]{-x^2+x^4}} \, dx=\int \frac {1}{{\left (x^4-1\right )}^2\,{\left (x^4-x^2\right )}^{1/4}} \,d x \]