\(\int \frac {\sqrt [4]{-x^2+x^4}}{x^4 (-1+x^4)} \, dx\) [1078]

Optimal result

Integrand size = 24, antiderivative size = 81 \[ \int \frac {\sqrt [4]{-x^2+x^4}}{x^4 \left (-1+x^4\right )} \, dx=-\frac {2 \left (-1+x^2\right ) \sqrt [4]{-x^2+x^4}}{5 x^3}-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^2+x^4}}\right )}{2^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^2+x^4}}\right )}{2^{3/4}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.69, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2081, 1268, 477, 508, 472, 304, 209, 212} \[ \int \frac {\sqrt [4]{-x^2+x^4}}{x^4 \left (-1+x^4\right )} \, dx=-\frac {\sqrt [4]{x^4-x^2} \arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{2^{3/4} \sqrt {x} \sqrt [4]{x^2-1}}+\frac {\sqrt [4]{x^4-x^2} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{2^{3/4} \sqrt {x} \sqrt [4]{x^2-1}}+\frac {2 \sqrt [4]{x^4-x^2} \left (1-x^2\right )}{5 x^3} \]

[In]

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

[Out]

(2*(1 - x^2)*(-x^2 + x^4)^(1/4))/(5*x^3) - ((-x^2 + x^4)^(1/4)*ArcTan[(2^(1/4)*Sqrt[x])/(-1 + x^2)^(1/4)])/(2^
(3/4)*Sqrt[x]*(-1 + x^2)^(1/4)) + ((-x^2 + x^4)^(1/4)*ArcTanh[(2^(1/4)*Sqrt[x])/(-1 + x^2)^(1/4)])/(2^(3/4)*Sq
rt[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 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 472

Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegr
and[(e*x)^m*((a + b*x^n)^p/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n
, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] ||  !RationalQ[m])

Rule 477

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/e^n))^p*(c + d*(x^(k*n)/e^n))^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

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 1268

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[(f*x)^m*(d +
e*x^2)^(q + p)*(a/d + (c/e)*x^2)^p, x] /; FreeQ[{a, c, d, e, f, q, m, q}, x] && EqQ[c*d^2 + a*e^2, 0] && Integ
erQ[p]

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{-x^2+x^4} \int \frac {\sqrt [4]{-1+x^2}}{x^{7/2} \left (-1+x^4\right )} \, dx}{\sqrt {x} \sqrt [4]{-1+x^2}} \\ & = \frac {\sqrt [4]{-x^2+x^4} \int \frac {1}{x^{7/2} \left (-1+x^2\right )^{3/4} \left (1+x^2\right )} \, dx}{\sqrt {x} \sqrt [4]{-1+x^2}} \\ & = \frac {\left (2 \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \frac {1}{x^6 \left (-1+x^4\right )^{3/4} \left (1+x^4\right )} \, 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 {\left (1-x^4\right )^2}{x^6 \left (1-2 x^4\right )} \, 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 {1}{x^6}-\frac {x^2}{-1+2 x^4}\right ) \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt {x} \sqrt [4]{-1+x^2}} \\ & = \frac {2 \left (1-x^2\right ) \sqrt [4]{-x^2+x^4}}{5 x^3}-\frac {\left (2 \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{-1+2 x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt {x} \sqrt [4]{-1+x^2}} \\ & = \frac {2 \left (1-x^2\right ) \sqrt [4]{-x^2+x^4}}{5 x^3}+\frac {\sqrt [4]{-x^2+x^4} \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt {2} \sqrt {x} \sqrt [4]{-1+x^2}}-\frac {\sqrt [4]{-x^2+x^4} \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt {2} \sqrt {x} \sqrt [4]{-1+x^2}} \\ & = \frac {2 \left (1-x^2\right ) \sqrt [4]{-x^2+x^4}}{5 x^3}-\frac {\sqrt [4]{-x^2+x^4} \arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2^{3/4} \sqrt {x} \sqrt [4]{-1+x^2}}+\frac {\sqrt [4]{-x^2+x^4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2^{3/4} \sqrt {x} \sqrt [4]{-1+x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.32 \[ \int \frac {\sqrt [4]{-x^2+x^4}}{x^4 \left (-1+x^4\right )} \, dx=-\frac {\sqrt [4]{x^2 \left (-1+x^2\right )} \left (4 \left (-1+x^2\right )^{5/4}+5 \sqrt [4]{2} x^{5/2} \arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )-5 \sqrt [4]{2} x^{5/2} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )\right )}{10 x^3 \sqrt [4]{-1+x^2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 12.78 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.23

method	result	size
pseudoelliptic	\(\frac {\left (-8 x^{2}+8\right ) \left (x^{4}-x^{2}\right )^{\frac {1}{4}}+5 \,2^{\frac {1}{4}} x^{3} \left (2 \arctan \left (\frac {2^{\frac {3}{4}} \left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{2 x}\right )+\ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{4}-x^{2}\right )^{\frac {1}{4}}}\right )\right )}{20 x^{3}}\)	\(100\)
trager	\(-\frac {2 \left (x^{2}-1\right ) \left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{5 x^{3}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x^{3}+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{4}-x^{2}\right )^{\frac {1}{4}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x +4 \sqrt {x^{4}-x^{2}}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) x +4 \left (x^{4}-x^{2}\right )^{\frac {3}{4}}}{\left (x^{2}+1\right ) x}\right )}{4}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{3}-4 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{4}-x^{2}\right )^{\frac {1}{4}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \sqrt {x^{4}-x^{2}}\, x +4 \left (x^{4}-x^{2}\right )^{\frac {3}{4}}}{\left (x^{2}+1\right ) x}\right )}{4}\)	\(267\)
risch	\(-\frac {2 \left (x^{2} \left (x^{2}-1\right )\right )^{\frac {1}{4}} \left (x^{4}-2 x^{2}+1\right )}{5 x^{3} \left (x^{2}-1\right )}+\frac {\left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} \left (x^{8}-3 x^{6}+3 x^{4}-x^{2}\right )^{\frac {1}{4}} x^{4}+3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{6}-4 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} \left (x^{8}-3 x^{6}+3 x^{4}-x^{2}\right )^{\frac {1}{4}} x^{2}-7 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{4}+2 \left (x^{8}-3 x^{6}+3 x^{4}-x^{2}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3}+5 x^{2} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}+4 \left (x^{8}-3 x^{6}+3 x^{4}-x^{2}\right )^{\frac {3}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )+4 \sqrt {x^{8}-3 x^{6}+3 x^{4}-x^{2}}\, x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}-4 \sqrt {x^{8}-3 x^{6}+3 x^{4}-x^{2}}}{\left (x -1\right )^{2} \left (1+x \right )^{2} \left (x^{2}+1\right )}\right )}{4}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{8}-3 x^{6}+3 x^{4}-x^{2}\right )^{\frac {1}{4}} x^{4}+3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{6}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{8}-3 x^{6}+3 x^{4}-x^{2}\right )^{\frac {1}{4}} x^{2}-7 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{4}+2 \left (x^{8}-3 x^{6}+3 x^{4}-x^{2}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}-4 \left (x^{8}-3 x^{6}+3 x^{4}-x^{2}\right )^{\frac {3}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right )+5 x^{2} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}-4 \sqrt {x^{8}-3 x^{6}+3 x^{4}-x^{2}}\, x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}+4 \sqrt {x^{8}-3 x^{6}+3 x^{4}-x^{2}}}{\left (x -1\right )^{2} \left (1+x \right )^{2} \left (x^{2}+1\right )}\right )}{4}\right ) \left (x^{2} \left (x^{2}-1\right )\right )^{\frac {1}{4}} \left (x^{2} \left (x^{2}-1\right )^{3}\right )^{\frac {1}{4}}}{x \left (x^{2}-1\right )}\)	\(633\)

[In]

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

[Out]

1/20*((-8*x^2+8)*(x^4-x^2)^(1/4)+5*2^(1/4)*x^3*(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^3

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.28 (sec) , antiderivative size = 348, normalized size of antiderivative = 4.30 \[ \int \frac {\sqrt [4]{-x^2+x^4}}{x^4 \left (-1+x^4\right )} \, dx=\frac {5 \cdot 8^{\frac {3}{4}} x^{3} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} x^{2} + 8^{\frac {3}{4}} \sqrt {x^{4} - x^{2}} x + 8^{\frac {1}{4}} {\left (3 \, x^{3} - x\right )} + 4 \, {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}}}{x^{3} + x}\right ) - 5 i \cdot 8^{\frac {3}{4}} x^{3} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} x^{2} + i \cdot 8^{\frac {3}{4}} \sqrt {x^{4} - x^{2}} x - 8^{\frac {1}{4}} {\left (3 i \, x^{3} - i \, x\right )} - 4 \, {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}}}{x^{3} + x}\right ) + 5 i \cdot 8^{\frac {3}{4}} x^{3} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} x^{2} - i \cdot 8^{\frac {3}{4}} \sqrt {x^{4} - x^{2}} x - 8^{\frac {1}{4}} {\left (-3 i \, x^{3} + i \, x\right )} - 4 \, {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}}}{x^{3} + x}\right ) - 5 \cdot 8^{\frac {3}{4}} x^{3} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} x^{2} - 8^{\frac {3}{4}} \sqrt {x^{4} - x^{2}} x - 8^{\frac {1}{4}} {\left (3 \, x^{3} - x\right )} + 4 \, {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}}}{x^{3} + x}\right ) - 64 \, {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} {\left (x^{2} - 1\right )}}{160 \, x^{3}} \]

[In]

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

[Out]

1/160*(5*8^(3/4)*x^3*log((4*sqrt(2)*(x^4 - x^2)^(1/4)*x^2 + 8^(3/4)*sqrt(x^4 - x^2)*x + 8^(1/4)*(3*x^3 - x) +
4*(x^4 - x^2)^(3/4))/(x^3 + x)) - 5*I*8^(3/4)*x^3*log(-(4*sqrt(2)*(x^4 - x^2)^(1/4)*x^2 + I*8^(3/4)*sqrt(x^4 -
x^2)*x - 8^(1/4)*(3*I*x^3 - I*x) - 4*(x^4 - x^2)^(3/4))/(x^3 + x)) + 5*I*8^(3/4)*x^3*log(-(4*sqrt(2)*(x^4 - x
^2)^(1/4)*x^2 - I*8^(3/4)*sqrt(x^4 - x^2)*x - 8^(1/4)*(-3*I*x^3 + I*x) - 4*(x^4 - x^2)^(3/4))/(x^3 + x)) - 5*8
^(3/4)*x^3*log((4*sqrt(2)*(x^4 - x^2)^(1/4)*x^2 - 8^(3/4)*sqrt(x^4 - x^2)*x - 8^(1/4)*(3*x^3 - x) + 4*(x^4 - x
^2)^(3/4))/(x^3 + x)) - 64*(x^4 - x^2)^(1/4)*(x^2 - 1))/x^3

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt [4]{-x^2+x^4}}{x^4 \left (-1+x^4\right )} \, dx=-\frac {2}{5} \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {5}{4}} + \frac {1}{2} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{4} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{4} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} - {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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