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

Optimal result

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

[Out]

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))

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.69, number of steps used = 25, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.387, Rules used = {2081, 6860, 285, 335, 338, 304, 209, 212, 1283, 1532, 1542, 508} \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{-x^2+x^4}}{-1-x^2+x^4} \, dx=-\frac {3 \sqrt [4]{x^4-x^2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{4 \sqrt [4]{x^2-1} \sqrt {x}}+\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt [4]{x^4-x^2} \arctan \left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{\sqrt {5} \sqrt [4]{x^2-1} \sqrt {x}}+\frac {\sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \sqrt [4]{x^4-x^2} \arctan \left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{\sqrt {5} \sqrt [4]{x^2-1} \sqrt {x}}+\frac {3 \sqrt [4]{x^4-x^2} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{4 \sqrt [4]{x^2-1} \sqrt {x}}-\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt [4]{x^4-x^2} \text {arctanh}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{\sqrt {5} \sqrt [4]{x^2-1} \sqrt {x}}-\frac {\sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \sqrt [4]{x^4-x^2} \text {arctanh}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{\sqrt {5} \sqrt [4]{x^2-1} \sqrt {x}}+\frac {1}{2} \sqrt [4]{x^4-x^2} x \]

[In]

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

[Out]

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

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 285

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^p/(c*(m + n
*p + 1))), x] + Dist[a*n*(p/(m + n*p + 1)), Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 304

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}
, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[a/b, 0]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 338

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[
p, -2^(-1)] && IntegersQ[m, p + (m + 1)/n]

Rule 508

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{k = Denominato
r[p]}, Dist[k*(a^(p + (m + 1)/n)/n), Subst[Int[x^(k*((m + 1)/n) - 1)*((c - (b*c - a*d)*x^k)^q/(1 - b*x^k)^(p +
q + (m + 1)/n + 1)), x], x, x^(n/k)/(a + b*x^n)^(1/k)], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && Ration
alQ[m, p] && IntegersQ[p + (m + 1)/n, q] && LtQ[-1, p, 0]

Rule 1283

Int[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With
[{k = Denominator[m]}, Dist[k/f, Subst[Int[x^(k*(m + 1) - 1)*(d + e*(x^(2*k)/f^2))^q*(a + b*(x^(2*k)/f^k) + c*
(x^(4*k)/f^4))^p, x], x, (f*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f, p, q}, x] && NeQ[b^2 - 4*a*c, 0] && Fra
ctionQ[m] && IntegerQ[p]

Rule 1532

Int[(((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^(n_))^(q_))/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbol
] :> Dist[e*(f^n/c), Int[(f*x)^(m - n)*(d + e*x^n)^(q - 1), x], x] - Dist[f^n/c, Int[(f*x)^(m - n)*(d + e*x^n)
^(q - 1)*(Simp[a*e - (c*d - b*e)*x^n, x]/(a + b*x^n + c*x^(2*n))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && E
qQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] &&  !IntegerQ[q] && GtQ[q, 0] && GtQ[m, n - 1] && LeQ[m, 2*n
- 1]

Rule 1542

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_))/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbol]
:> Int[ExpandIntegrand[(d + e*x^n)^q, (f*x)^m/(a + b*x^n + c*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, e, f, q,
n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] &&  !IntegerQ[q] && IntegerQ[m]

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6860

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] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{-x^2+x^4} \int \frac {\sqrt {x} \sqrt [4]{-1+x^2} \left (-1+x^4\right )}{-1-x^2+x^4} \, dx}{\sqrt {x} \sqrt [4]{-1+x^2}} \\ & = \frac {\sqrt [4]{-x^2+x^4} \int \left (\sqrt {x} \sqrt [4]{-1+x^2}+\frac {x^{5/2} \sqrt [4]{-1+x^2}}{-1-x^2+x^4}\right ) \, dx}{\sqrt {x} \sqrt [4]{-1+x^2}} \\ & = \frac {\sqrt [4]{-x^2+x^4} \int \sqrt {x} \sqrt [4]{-1+x^2} \, dx}{\sqrt {x} \sqrt [4]{-1+x^2}}+\frac {\sqrt [4]{-x^2+x^4} \int \frac {x^{5/2} \sqrt [4]{-1+x^2}}{-1-x^2+x^4} \, dx}{\sqrt {x} \sqrt [4]{-1+x^2}} \\ & = \frac {1}{2} x \sqrt [4]{-x^2+x^4}-\frac {\sqrt [4]{-x^2+x^4} \int \frac {\sqrt {x}}{\left (-1+x^2\right )^{3/4}} \, dx}{4 \sqrt {x} \sqrt [4]{-1+x^2}}+\frac {\left (2 \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \frac {x^6 \sqrt [4]{-1+x^4}}{-1-x^4+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{-1+x^2}} \\ & = \frac {1}{2} x \sqrt [4]{-x^2+x^4}-\frac {\sqrt [4]{-x^2+x^4} \text {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt [4]{-1+x^2}}+\frac {\left (2 \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{-1+x^2}}+\frac {\left (2 \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right )^{3/4} \left (-1-x^4+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{-1+x^2}} \\ & = \frac {1}{2} x \sqrt [4]{-x^2+x^4}-\frac {\sqrt [4]{-x^2+x^4} \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \sqrt {x} \sqrt [4]{-1+x^2}}+\frac {\left (2 \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt {x} \sqrt [4]{-1+x^2}}+\frac {\left (2 \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \left (-\frac {2 x^2}{\sqrt {5} \left (1+\sqrt {5}-2 x^4\right ) \left (-1+x^4\right )^{3/4}}-\frac {2 x^2}{\sqrt {5} \left (-1+x^4\right )^{3/4} \left (-1+\sqrt {5}+2 x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{-1+x^2}} \\ & = \frac {1}{2} x \sqrt [4]{-x^2+x^4}-\frac {\sqrt [4]{-x^2+x^4} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{4 \sqrt {x} \sqrt [4]{-1+x^2}}+\frac {\sqrt [4]{-x^2+x^4} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{4 \sqrt {x} \sqrt [4]{-1+x^2}}+\frac {\sqrt [4]{-x^2+x^4} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt {x} \sqrt [4]{-1+x^2}}-\frac {\sqrt [4]{-x^2+x^4} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt {x} \sqrt [4]{-1+x^2}}-\frac {\left (4 \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (1+\sqrt {5}-2 x^4\right ) \left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{\sqrt {5} \sqrt {x} \sqrt [4]{-1+x^2}}-\frac {\left (4 \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right )^{3/4} \left (-1+\sqrt {5}+2 x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {5} \sqrt {x} \sqrt [4]{-1+x^2}} \\ & = \frac {1}{2} x \sqrt [4]{-x^2+x^4}-\frac {3 \sqrt [4]{-x^2+x^4} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{4 \sqrt {x} \sqrt [4]{-1+x^2}}+\frac {3 \sqrt [4]{-x^2+x^4} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{4 \sqrt {x} \sqrt [4]{-1+x^2}}-\frac {\left (4 \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{1+\sqrt {5}-\left (-1+\sqrt {5}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt {5} \sqrt {x} \sqrt [4]{-1+x^2}}-\frac {\left (4 \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{-1+\sqrt {5}-\left (1+\sqrt {5}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt {5} \sqrt {x} \sqrt [4]{-1+x^2}} \\ & = \frac {1}{2} x \sqrt [4]{-x^2+x^4}-\frac {3 \sqrt [4]{-x^2+x^4} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{4 \sqrt {x} \sqrt [4]{-1+x^2}}+\frac {3 \sqrt [4]{-x^2+x^4} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{4 \sqrt {x} \sqrt [4]{-1+x^2}}+\frac {\left (2 \sqrt {\frac {2}{5}} \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {3-\sqrt {5}}-\sqrt {2} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\left (-1-\sqrt {5}\right ) \sqrt {x} \sqrt [4]{-1+x^2}}-\frac {\left (2 \sqrt {\frac {2}{5}} \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {3-\sqrt {5}}+\sqrt {2} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\left (-1-\sqrt {5}\right ) \sqrt {x} \sqrt [4]{-1+x^2}}+\frac {\left (2 \sqrt {\frac {2}{5}} \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {3+\sqrt {5}}-\sqrt {2} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\left (1-\sqrt {5}\right ) \sqrt {x} \sqrt [4]{-1+x^2}}-\frac {\left (2 \sqrt {\frac {2}{5}} \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {3+\sqrt {5}}+\sqrt {2} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\left (1-\sqrt {5}\right ) \sqrt {x} \sqrt [4]{-1+x^2}} \\ & = \frac {1}{2} x \sqrt [4]{-x^2+x^4}-\frac {3 \sqrt [4]{-x^2+x^4} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{4 \sqrt {x} \sqrt [4]{-1+x^2}}+\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt [4]{-x^2+x^4} \arctan \left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt {5} \sqrt {x} \sqrt [4]{-1+x^2}}+\frac {\sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \sqrt [4]{-x^2+x^4} \arctan \left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt {5} \sqrt {x} \sqrt [4]{-1+x^2}}+\frac {3 \sqrt [4]{-x^2+x^4} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{4 \sqrt {x} \sqrt [4]{-1+x^2}}-\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt [4]{-x^2+x^4} \text {arctanh}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt {5} \sqrt {x} \sqrt [4]{-1+x^2}}-\frac {\sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \sqrt [4]{-x^2+x^4} \text {arctanh}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt {5} \sqrt {x} \sqrt [4]{-1+x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.78 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.04 \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{-x^2+x^4}}{-1-x^2+x^4} \, dx=\frac {x^{3/2} \left (-1+x^2\right )^{3/4} \left (10 x^{3/2} \sqrt [4]{-1+x^2}-15 \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )+2 \sqrt {10 \left (1+\sqrt {5}\right )} \arctan \left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )+2 \sqrt {10 \left (-1+\sqrt {5}\right )} \arctan \left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )+15 \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )-2 \sqrt {10 \left (1+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )-2 \sqrt {10 \left (-1+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )\right )}{20 \left (x^2 \left (-1+x^2\right )\right )^{3/4}} \]

[In]

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

[Out]

(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])]*A
rcTanh[(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))

Maple [A] (verified)

Time = 18.22 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.82

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1633 vs. \(2 (187) = 374\).

Time = 23.41 (sec) , antiderivative size = 1633, normalized size of antiderivative = 6.35 \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{-x^2+x^4}}{-1-x^2+x^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/40*sqrt(10)*sqrt(-sqrt(5) + 1)*log((2*sqrt(10)*sqrt(x^4 - x^2)*(905*x^3 + sqrt(5)*(9*x^3 - 457*x) - 475*x)*
sqrt(-sqrt(5) + 1) - sqrt(10)*(45*x^5 - 1855*x^3 + sqrt(5)*(905*x^5 - 923*x^3 + 9*x) + 905*x)*sqrt(-sqrt(5) +
1) + 20*(x^4 - x^2)^(3/4)*(448*x^2 - sqrt(5)*(86*x^2 + 181) - 9) - 20*(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)) + 1/40*sqrt(10)*sqrt(-sqrt(5) + 1)*log(-(2*sqrt(10)*sqrt(x^4 - x^2)
*(905*x^3 + sqrt(5)*(9*x^3 - 457*x) - 475*x)*sqrt(-sqrt(5) + 1) - sqrt(10)*(45*x^5 - 1855*x^3 + sqrt(5)*(905*x
^5 - 923*x^3 + 9*x) + 905*x)*sqrt(-sqrt(5) + 1) - 20*(x^4 - x^2)^(3/4)*(448*x^2 - sqrt(5)*(86*x^2 + 181) - 9)
+ 20*(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)) - 1/40*sqrt(10)*sqrt(-
sqrt(5) - 1)*log((2*sqrt(10)*sqrt(x^4 - x^2)*(905*x^3 - sqrt(5)*(9*x^3 - 457*x) - 475*x)*sqrt(-sqrt(5) - 1) +
sqrt(10)*(45*x^5 - 1855*x^3 - sqrt(5)*(905*x^5 - 923*x^3 + 9*x) + 905*x)*sqrt(-sqrt(5) - 1) + 20*(x^4 - x^2)^(
3/4)*(448*x^2 + sqrt(5)*(86*x^2 + 181) - 9) + 20*(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)) + 1/40*sqrt(10)*sqrt(-sqrt(5) - 1)*log(-(2*sqrt(10)*sqrt(x^4 - x^2)*(905*x^3 - sqrt(5)*(
9*x^3 - 457*x) - 475*x)*sqrt(-sqrt(5) - 1) + sqrt(10)*(45*x^5 - 1855*x^3 - sqrt(5)*(905*x^5 - 923*x^3 + 9*x) +
905*x)*sqrt(-sqrt(5) - 1) - 20*(x^4 - x^2)^(3/4)*(448*x^2 + sqrt(5)*(86*x^2 + 181) - 9) - 20*(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)) - 1/40*sqrt(10)*sqrt(sqrt(5) + 1)*log((20*(
x^4 - x^2)^(3/4)*(448*x^2 + sqrt(5)*(86*x^2 + 181) - 9) + (2*sqrt(10)*sqrt(x^4 - x^2)*(905*x^3 - sqrt(5)*(9*x^
3 - 457*x) - 475*x) - sqrt(10)*(45*x^5 - 1855*x^3 - sqrt(5)*(905*x^5 - 923*x^3 + 9*x) + 905*x))*sqrt(sqrt(5) +
1) - 20*(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)) + 1/40*sqrt(10)*sq
rt(sqrt(5) + 1)*log((20*(x^4 - x^2)^(3/4)*(448*x^2 + sqrt(5)*(86*x^2 + 181) - 9) - (2*sqrt(10)*sqrt(x^4 - x^2)
*(905*x^3 - sqrt(5)*(9*x^3 - 457*x) - 475*x) - sqrt(10)*(45*x^5 - 1855*x^3 - sqrt(5)*(905*x^5 - 923*x^3 + 9*x)
+ 905*x))*sqrt(sqrt(5) + 1) - 20*(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)) - 1/40*sqrt(10)*sqrt(sqrt(5) - 1)*log((20*(x^4 - x^2)^(3/4)*(448*x^2 - sqrt(5)*(86*x^2 + 181) - 9) + (2
*sqrt(10)*sqrt(x^4 - x^2)*(905*x^3 + sqrt(5)*(9*x^3 - 457*x) - 475*x) + sqrt(10)*(45*x^5 - 1855*x^3 + sqrt(5)*
(905*x^5 - 923*x^3 + 9*x) + 905*x))*sqrt(sqrt(5) - 1) + 20*(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)) + 1/40*sqrt(10)*sqrt(sqrt(5) - 1)*log((20*(x^4 - x^2)^(3/4)*(448*x^2 - sqrt(5)
*(86*x^2 + 181) - 9) - (2*sqrt(10)*sqrt(x^4 - x^2)*(905*x^3 + sqrt(5)*(9*x^3 - 457*x) - 475*x) + sqrt(10)*(45*
x^5 - 1855*x^3 + sqrt(5)*(905*x^5 - 923*x^3 + 9*x) + 905*x))*sqrt(sqrt(5) - 1) + 20*(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)) + 1/2*(x^4 - x^2)^(1/4)*x + 3/8*arctan(2*((x^4 - x^2)
^(1/4)*x^2 + (x^4 - x^2)^(3/4))/x) + 3/8*log((2*x^3 + 2*(x^4 - x^2)^(1/4)*x^2 + 2*sqrt(x^4 - x^2)*x - x + 2*(x
^4 - x^2)^(3/4))/x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.96 \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{-x^2+x^4}}{-1-x^2+x^4} \, dx=\frac {1}{2} \, x^{2} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} - \frac {1}{10} \, \sqrt {10 \, \sqrt {5} - 10} \arctan \left (\frac {{\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}\right ) - \frac {1}{10} \, \sqrt {10 \, \sqrt {5} + 10} \arctan \left (\frac {{\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}}{\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}}\right ) - \frac {1}{20} \, \sqrt {10 \, \sqrt {5} - 10} \log \left (\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{20} \, \sqrt {10 \, \sqrt {5} - 10} \log \left (\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} - {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{20} \, \sqrt {10 \, \sqrt {5} + 10} \log \left (\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{20} \, \sqrt {10 \, \sqrt {5} + 10} \log \left ({\left | -\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} \right |}\right ) + \frac {3}{4} \, \arctan \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {3}{8} \, \log \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {3}{8} \, \log \left (-{\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 1\right ) \]

[In]

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

[Out]

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(
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(sqrt(1/2*sqrt(5) - 1/2) + (-1/x^2 + 1)^(1/4)) + 1/20*sqrt(10*s
qrt(5) + 10)*log(abs(-sqrt(1/2*sqrt(5) - 1/2) + (-1/x^2 + 1)^(1/4))) + 3/4*arctan((-1/x^2 + 1)^(1/4)) + 3/8*lo
g((-1/x^2 + 1)^(1/4) + 1) - 3/8*log(-(-1/x^2 + 1)^(1/4) + 1)

Mupad [F(-1)]

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

[In]

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

[Out]

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