\(\int \frac {\sqrt [4]{-x^2+x^6} (1-x^4+x^8)}{x^4 (1+x^4)} \, dx\) [3075]

Optimal result

Integrand size = 34, antiderivative size = 501 \[ \int \frac {\sqrt [4]{-x^2+x^6} \left (1-x^4+x^8\right )}{x^4 \left (1+x^4\right )} \, dx=\frac {2 \left (-1+x^4\right ) \sqrt [4]{-x^2+x^6}}{5 x^3}+\frac {3}{4} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {2-\sqrt {2}} x}{-\sqrt {2+\sqrt {2}} x+2^{3/4} \sqrt [4]{-x^2+x^6}}\right )+\frac {3}{4} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {2-\sqrt {2}} x}{\sqrt {2+\sqrt {2}} x+2^{3/4} \sqrt [4]{-x^2+x^6}}\right )-\frac {3}{4} \sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \arctan \left (\frac {2^{3/4} \sqrt {2+\sqrt {2}} x \sqrt [4]{-x^2+x^6}}{-2 x^2+\sqrt {2} \sqrt {-x^2+x^6}}\right )-\frac {3}{4} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \text {arctanh}\left (\frac {\frac {\sqrt [4]{2} x^2}{\sqrt {2-\sqrt {2}}}+\frac {\sqrt {-x^2+x^6}}{\sqrt [4]{2} \sqrt {2-\sqrt {2}}}}{x \sqrt [4]{-x^2+x^6}}\right )-\frac {3}{8} \sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \log \left (-2 x^2+2^{3/4} \sqrt {2+\sqrt {2}} x \sqrt [4]{-x^2+x^6}-\sqrt {2} \sqrt {-x^2+x^6}\right )+\frac {3}{8} \sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \log \left (2 \sqrt {2-\sqrt {2}} x^2+2 \sqrt [4]{2} x \sqrt [4]{-x^2+x^6}+\sqrt {4-2 \sqrt {2}} \sqrt {-x^2+x^6}\right ) \]

[Out]

2/5*(x^4-1)*(x^6-x^2)^(1/4)/x^3+3/8*(2+2*2^(1/2))^(1/2)*arctan((2-2^(1/2))^(1/2)*x/(-(2+2^(1/2))^(1/2)*x+2^(3/
4)*(x^6-x^2)^(1/4)))+3/8*(2+2*2^(1/2))^(1/2)*arctan((2-2^(1/2))^(1/2)*x/((2+2^(1/2))^(1/2)*x+2^(3/4)*(x^6-x^2)
^(1/4)))-3/8*(-2+2*2^(1/2))^(1/2)*arctan(2^(3/4)*(2+2^(1/2))^(1/2)*x*(x^6-x^2)^(1/4)/(-2*x^2+2^(1/2)*(x^6-x^2)
^(1/2)))-3/8*(2+2*2^(1/2))^(1/2)*arctanh((2^(1/4)*x^2/(2-2^(1/2))^(1/2)+1/2*(x^6-x^2)^(1/2)*2^(3/4)/(2-2^(1/2)
)^(1/2))/x/(x^6-x^2)^(1/4))-3/16*(-2+2*2^(1/2))^(1/2)*ln(-2*x^2+2^(3/4)*(2+2^(1/2))^(1/2)*x*(x^6-x^2)^(1/4)-2^
(1/2)*(x^6-x^2)^(1/2))+3/16*(-2+2*2^(1/2))^(1/2)*ln(2*(2-2^(1/2))^(1/2)*x^2+2*2^(1/4)*x*(x^6-x^2)^(1/4)+(4-2*2
^(1/2))^(1/2)*(x^6-x^2)^(1/2))

Rubi [C] (verified)

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

Time = 0.29 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.26, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {2081, 6857, 283, 335, 372, 371, 285, 477, 525, 524} \[ \int \frac {\sqrt [4]{-x^2+x^6} \left (1-x^4+x^8\right )}{x^4 \left (1+x^4\right )} \, dx=-\frac {6 \sqrt [4]{x^6-x^2} \operatorname {AppellF1}\left (-\frac {5}{8},-\frac {1}{4},1,\frac {3}{8},x^4,-x^4\right )}{5 \sqrt [4]{1-x^4} x^3}+\frac {4 \sqrt [4]{x^6-x^2} x \operatorname {Hypergeometric2F1}\left (\frac {3}{8},\frac {3}{4},\frac {11}{8},x^4\right )}{5 \sqrt [4]{1-x^4}}+\frac {2}{5} \sqrt [4]{x^6-x^2} x+\frac {4 \sqrt [4]{x^6-x^2}}{5 x^3} \]

[In]

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

[Out]

(4*(-x^2 + x^6)^(1/4))/(5*x^3) + (2*x*(-x^2 + x^6)^(1/4))/5 - (6*(-x^2 + x^6)^(1/4)*AppellF1[-5/8, -1/4, 1, 3/
8, x^4, -x^4])/(5*x^3*(1 - x^4)^(1/4)) + (4*x*(-x^2 + x^6)^(1/4)*Hypergeometric2F1[3/8, 3/4, 11/8, x^4])/(5*(1
- x^4)^(1/4))

Rule 283

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

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 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 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 372

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

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 524

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*
((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 525

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPar
t[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]), Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] &&  !(IntegerQ[
p] || GtQ[a, 0])

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 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{-x^2+x^6} \int \frac {\sqrt [4]{-1+x^4} \left (1-x^4+x^8\right )}{x^{7/2} \left (1+x^4\right )} \, dx}{\sqrt {x} \sqrt [4]{-1+x^4}} \\ & = \frac {\sqrt [4]{-x^2+x^6} \int \left (-\frac {2 \sqrt [4]{-1+x^4}}{x^{7/2}}+\sqrt {x} \sqrt [4]{-1+x^4}+\frac {3 \sqrt [4]{-1+x^4}}{x^{7/2} \left (1+x^4\right )}\right ) \, dx}{\sqrt {x} \sqrt [4]{-1+x^4}} \\ & = \frac {\sqrt [4]{-x^2+x^6} \int \sqrt {x} \sqrt [4]{-1+x^4} \, dx}{\sqrt {x} \sqrt [4]{-1+x^4}}-\frac {\left (2 \sqrt [4]{-x^2+x^6}\right ) \int \frac {\sqrt [4]{-1+x^4}}{x^{7/2}} \, dx}{\sqrt {x} \sqrt [4]{-1+x^4}}+\frac {\left (3 \sqrt [4]{-x^2+x^6}\right ) \int \frac {\sqrt [4]{-1+x^4}}{x^{7/2} \left (1+x^4\right )} \, dx}{\sqrt {x} \sqrt [4]{-1+x^4}} \\ & = \frac {4 \sqrt [4]{-x^2+x^6}}{5 x^3}+\frac {2}{5} x \sqrt [4]{-x^2+x^6}-\frac {\left (2 \sqrt [4]{-x^2+x^6}\right ) \int \frac {\sqrt {x}}{\left (-1+x^4\right )^{3/4}} \, dx}{5 \sqrt {x} \sqrt [4]{-1+x^4}}-\frac {\left (4 \sqrt [4]{-x^2+x^6}\right ) \int \frac {\sqrt {x}}{\left (-1+x^4\right )^{3/4}} \, dx}{5 \sqrt {x} \sqrt [4]{-1+x^4}}+\frac {\left (6 \sqrt [4]{-x^2+x^6}\right ) \text {Subst}\left (\int \frac {\sqrt [4]{-1+x^8}}{x^6 \left (1+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{-1+x^4}} \\ & = \frac {4 \sqrt [4]{-x^2+x^6}}{5 x^3}+\frac {2}{5} x \sqrt [4]{-x^2+x^6}+\frac {\left (6 \sqrt [4]{-x^2+x^6}\right ) \text {Subst}\left (\int \frac {\sqrt [4]{1-x^8}}{x^6 \left (1+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{1-x^4}}-\frac {\left (4 \sqrt [4]{-x^2+x^6}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-1+x^8\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{5 \sqrt {x} \sqrt [4]{-1+x^4}}-\frac {\left (8 \sqrt [4]{-x^2+x^6}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-1+x^8\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{5 \sqrt {x} \sqrt [4]{-1+x^4}} \\ & = \frac {4 \sqrt [4]{-x^2+x^6}}{5 x^3}+\frac {2}{5} x \sqrt [4]{-x^2+x^6}-\frac {6 \sqrt [4]{-x^2+x^6} \operatorname {AppellF1}\left (-\frac {5}{8},-\frac {1}{4},1,\frac {3}{8},x^4,-x^4\right )}{5 x^3 \sqrt [4]{1-x^4}}-\frac {\left (4 \left (1-x^4\right )^{3/4} \sqrt [4]{-x^2+x^6}\right ) \text {Subst}\left (\int \frac {x^2}{\left (1-x^8\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{5 \sqrt {x} \left (-1+x^4\right )}-\frac {\left (8 \left (1-x^4\right )^{3/4} \sqrt [4]{-x^2+x^6}\right ) \text {Subst}\left (\int \frac {x^2}{\left (1-x^8\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{5 \sqrt {x} \left (-1+x^4\right )} \\ & = \frac {4 \sqrt [4]{-x^2+x^6}}{5 x^3}+\frac {2}{5} x \sqrt [4]{-x^2+x^6}-\frac {6 \sqrt [4]{-x^2+x^6} \operatorname {AppellF1}\left (-\frac {5}{8},-\frac {1}{4},1,\frac {3}{8},x^4,-x^4\right )}{5 x^3 \sqrt [4]{1-x^4}}+\frac {4 x \sqrt [4]{-x^2+x^6} \operatorname {Hypergeometric2F1}\left (\frac {3}{8},\frac {3}{4},\frac {11}{8},x^4\right )}{5 \sqrt [4]{1-x^4}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.00 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.41 \[ \int \frac {\sqrt [4]{-x^2+x^6} \left (1-x^4+x^8\right )}{x^4 \left (1+x^4\right )} \, dx=\frac {\sqrt [4]{x^2 \left (-1+x^4\right )} \left (-8 \sqrt [4]{-1+x^4}+8 x^4 \sqrt [4]{-1+x^4}+15 \sqrt {1+i} x^{5/2} \arctan \left (\frac {\sqrt {-1-i} \sqrt {x}}{\sqrt [4]{-1+x^4}}\right )+15 \sqrt {1-i} x^{5/2} \arctan \left (\frac {\sqrt {-1+i} \sqrt {x}}{\sqrt [4]{-1+x^4}}\right )-15 \sqrt {-1+i} x^{5/2} \arctan \left (\frac {\sqrt {1-i} \sqrt {x}}{\sqrt [4]{-1+x^4}}\right )-15 \sqrt {-1-i} x^{5/2} \arctan \left (\frac {\sqrt {1+i} \sqrt {x}}{\sqrt [4]{-1+x^4}}\right )\right )}{20 x^3 \sqrt [4]{-1+x^4}} \]

[In]

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

[Out]

((x^2*(-1 + x^4))^(1/4)*(-8*(-1 + x^4)^(1/4) + 8*x^4*(-1 + x^4)^(1/4) + 15*Sqrt[1 + I]*x^(5/2)*ArcTan[(Sqrt[-1
- I]*Sqrt[x])/(-1 + x^4)^(1/4)] + 15*Sqrt[1 - I]*x^(5/2)*ArcTan[(Sqrt[-1 + I]*Sqrt[x])/(-1 + x^4)^(1/4)] - 15
*Sqrt[-1 + I]*x^(5/2)*ArcTan[(Sqrt[1 - I]*Sqrt[x])/(-1 + x^4)^(1/4)] - 15*Sqrt[-1 - I]*x^(5/2)*ArcTan[(Sqrt[1
+ I]*Sqrt[x])/(-1 + x^4)^(1/4)]))/(20*x^3*(-1 + x^4)^(1/4))

Maple [A] (verified)

Time = 139.44 (sec) , antiderivative size = 407, normalized size of antiderivative = 0.81

method	result	size
pseudoelliptic	\(\frac {32 \left (x^{4}-1\right ) \left (x^{6}-x^{2}\right )^{\frac {1}{4}}+15 x^{3} \left (\left (-\ln \left (\frac {-\sqrt {2+2 \sqrt {2}}\, x \left (x^{6}-x^{2}\right )^{\frac {1}{4}}+\sqrt {2}\, x^{2}+\sqrt {x^{6}-x^{2}}}{x^{2}}\right )-2 \arctan \left (\frac {x \sqrt {-2+2 \sqrt {2}}-2 \left (x^{6}-x^{2}\right )^{\frac {1}{4}}}{\sqrt {2+2 \sqrt {2}}\, x}\right )+\ln \left (\frac {\sqrt {2+2 \sqrt {2}}\, x \left (x^{6}-x^{2}\right )^{\frac {1}{4}}+\sqrt {2}\, x^{2}+\sqrt {x^{6}-x^{2}}}{x^{2}}\right )+2 \arctan \left (\frac {x \sqrt {-2+2 \sqrt {2}}+2 \left (x^{6}-x^{2}\right )^{\frac {1}{4}}}{\sqrt {2+2 \sqrt {2}}\, x}\right )\right ) \sqrt {-2+2 \sqrt {2}}+\sqrt {2+2 \sqrt {2}}\, \left (\ln \left (\frac {\sqrt {2}\, x^{2}-x \sqrt {-2+2 \sqrt {2}}\, \left (x^{6}-x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{6}-x^{2}}}{x^{2}}\right )+2 \arctan \left (\frac {\sqrt {2+2 \sqrt {2}}\, x -2 \left (x^{6}-x^{2}\right )^{\frac {1}{4}}}{x \sqrt {-2+2 \sqrt {2}}}\right )-\ln \left (\frac {x \sqrt {-2+2 \sqrt {2}}\, \left (x^{6}-x^{2}\right )^{\frac {1}{4}}+\sqrt {2}\, x^{2}+\sqrt {x^{6}-x^{2}}}{x^{2}}\right )-2 \arctan \left (\frac {\sqrt {2+2 \sqrt {2}}\, x +2 \left (x^{6}-x^{2}\right )^{\frac {1}{4}}}{x \sqrt {-2+2 \sqrt {2}}}\right )\right )\right )}{80 x^{3}}\)	\(407\)
trager	\(\text {Expression too large to display}\)	\(2952\)
risch	\(\text {Expression too large to display}\)	\(7288\)

[In]

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

[Out]

1/80*(32*(x^4-1)*(x^6-x^2)^(1/4)+15*x^3*((-ln((-(2+2*2^(1/2))^(1/2)*x*(x^6-x^2)^(1/4)+2^(1/2)*x^2+(x^6-x^2)^(1
/2))/x^2)-2*arctan((x*(-2+2*2^(1/2))^(1/2)-2*(x^6-x^2)^(1/4))/(2+2*2^(1/2))^(1/2)/x)+ln(((2+2*2^(1/2))^(1/2)*x
*(x^6-x^2)^(1/4)+2^(1/2)*x^2+(x^6-x^2)^(1/2))/x^2)+2*arctan((x*(-2+2*2^(1/2))^(1/2)+2*(x^6-x^2)^(1/4))/(2+2*2^
(1/2))^(1/2)/x))*(-2+2*2^(1/2))^(1/2)+(2+2*2^(1/2))^(1/2)*(ln((2^(1/2)*x^2-x*(-2+2*2^(1/2))^(1/2)*(x^6-x^2)^(1
/4)+(x^6-x^2)^(1/2))/x^2)+2*arctan(((2+2*2^(1/2))^(1/2)*x-2*(x^6-x^2)^(1/4))/x/(-2+2*2^(1/2))^(1/2))-ln((x*(-2
+2*2^(1/2))^(1/2)*(x^6-x^2)^(1/4)+2^(1/2)*x^2+(x^6-x^2)^(1/2))/x^2)-2*arctan(((2+2*2^(1/2))^(1/2)*x+2*(x^6-x^2
)^(1/4))/x/(-2+2*2^(1/2))^(1/2)))))/x^3

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 22.03 (sec) , antiderivative size = 1108, normalized size of antiderivative = 2.21 \[ \int \frac {\sqrt [4]{-x^2+x^6} \left (1-x^4+x^8\right )}{x^4 \left (1+x^4\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/80*(5*sqrt(-9*I - 9)*x^3*log(-(4*sqrt(-9*I - 9)*sqrt(x^6 - x^2)*((70980218386*I + 35099697147)*x^5 + (701993
94294*I - 141960436772)*x^3 - (70980218386*I + 35099697147)*x) + 12*(x^6 - x^2)^(3/4)*(-(35099697147*I - 70980
218386)*x^4 + (141960436772*I + 70199394294)*x^2 + 35099697147*I - 70980218386) - sqrt(-9*I - 9)*(-(3588052123
9*I + 106079915533)*x^9 - (424319662132*I - 143522084956)*x^7 + (215283127434*I + 636479493198)*x^5 + (4243196
62132*I - 143522084956)*x^3 - (35880521239*I + 106079915533)*x) + 12*((35880521239*I + 106079915533)*x^6 + (21
2159831066*I - 71761042478)*x^4 - (35880521239*I + 106079915533)*x^2)*(x^6 - x^2)^(1/4))/(x^9 + 2*x^5 + x)) +
5*sqrt(-9*I + 9)*x^3*log(-(4*sqrt(-9*I + 9)*sqrt(x^6 - x^2)*(-(70980218386*I - 35099697147)*x^5 - (70199394294
*I + 141960436772)*x^3 + (70980218386*I - 35099697147)*x) + 12*(x^6 - x^2)^(3/4)*((35099697147*I + 70980218386
)*x^4 - (141960436772*I - 70199394294)*x^2 - 35099697147*I - 70980218386) - sqrt(-9*I + 9)*(-(35880521239*I -
106079915533)*x^9 - (424319662132*I + 143522084956)*x^7 + (215283127434*I - 636479493198)*x^5 + (424319662132*
I + 143522084956)*x^3 - (35880521239*I - 106079915533)*x) + 12*(x^6 - x^2)^(1/4)*((35880521239*I - 10607991553
3)*x^6 + (212159831066*I + 71761042478)*x^4 - (35880521239*I - 106079915533)*x^2))/(x^9 + 2*x^5 + x)) - 5*sqrt
(-9*I + 9)*x^3*log(-(4*sqrt(-9*I + 9)*sqrt(x^6 - x^2)*((70980218386*I - 35099697147)*x^5 + (70199394294*I + 14
1960436772)*x^3 - (70980218386*I - 35099697147)*x) + 12*(x^6 - x^2)^(3/4)*((35099697147*I + 70980218386)*x^4 -
(141960436772*I - 70199394294)*x^2 - 35099697147*I - 70980218386) - sqrt(-9*I + 9)*((35880521239*I - 10607991
5533)*x^9 + (424319662132*I + 143522084956)*x^7 - (215283127434*I - 636479493198)*x^5 - (424319662132*I + 1435
22084956)*x^3 + (35880521239*I - 106079915533)*x) + 12*(x^6 - x^2)^(1/4)*((35880521239*I - 106079915533)*x^6 +
(212159831066*I + 71761042478)*x^4 - (35880521239*I - 106079915533)*x^2))/(x^9 + 2*x^5 + x)) - 5*sqrt(-9*I -
9)*x^3*log(-(4*sqrt(-9*I - 9)*sqrt(x^6 - x^2)*(-(70980218386*I + 35099697147)*x^5 - (70199394294*I - 141960436
772)*x^3 + (70980218386*I + 35099697147)*x) + 12*(x^6 - x^2)^(3/4)*(-(35099697147*I - 70980218386)*x^4 + (1419
60436772*I + 70199394294)*x^2 + 35099697147*I - 70980218386) - sqrt(-9*I - 9)*((35880521239*I + 106079915533)*
x^9 + (424319662132*I - 143522084956)*x^7 - (215283127434*I + 636479493198)*x^5 - (424319662132*I - 1435220849
56)*x^3 + (35880521239*I + 106079915533)*x) + 12*((35880521239*I + 106079915533)*x^6 + (212159831066*I - 71761
042478)*x^4 - (35880521239*I + 106079915533)*x^2)*(x^6 - x^2)^(1/4))/(x^9 + 2*x^5 + x)) - 5*sqrt(9*I - 9)*x^3*
log(-(12*(x^6 - x^2)^(3/4)*((35099697147*I + 70980218386)*x^4 - (141960436772*I - 70199394294)*x^2 - 350996971
47*I - 70980218386) - sqrt(9*I - 9)*(-(35880521239*I - 106079915533)*x^9 - (424319662132*I + 143522084956)*x^7
+ (215283127434*I - 636479493198)*x^5 + (424319662132*I + 143522084956)*x^3 - 4*sqrt(x^6 - x^2)*((70980218386
*I - 35099697147)*x^5 + (70199394294*I + 141960436772)*x^3 - (70980218386*I - 35099697147)*x) - (35880521239*I
- 106079915533)*x) + 12*(-(35880521239*I - 106079915533)*x^6 - (212159831066*I + 71761042478)*x^4 + (35880521
239*I - 106079915533)*x^2)*(x^6 - x^2)^(1/4))/(x^9 + 2*x^5 + x)) + 5*sqrt(9*I - 9)*x^3*log(-(12*(x^6 - x^2)^(3
/4)*((35099697147*I + 70980218386)*x^4 - (141960436772*I - 70199394294)*x^2 - 35099697147*I - 70980218386) - s
qrt(9*I - 9)*((35880521239*I - 106079915533)*x^9 + (424319662132*I + 143522084956)*x^7 - (215283127434*I - 636
479493198)*x^5 - (424319662132*I + 143522084956)*x^3 - 4*sqrt(x^6 - x^2)*(-(70980218386*I - 35099697147)*x^5 -
(70199394294*I + 141960436772)*x^3 + (70980218386*I - 35099697147)*x) + (35880521239*I - 106079915533)*x) + 1
2*(-(35880521239*I - 106079915533)*x^6 - (212159831066*I + 71761042478)*x^4 + (35880521239*I - 106079915533)*x
^2)*(x^6 - x^2)^(1/4))/(x^9 + 2*x^5 + x)) + 5*sqrt(9*I + 9)*x^3*log(-(12*(x^6 - x^2)^(3/4)*(-(35099697147*I -
70980218386)*x^4 + (141960436772*I + 70199394294)*x^2 + 35099697147*I - 70980218386) - sqrt(9*I + 9)*((3588052
1239*I + 106079915533)*x^9 + (424319662132*I - 143522084956)*x^7 - (215283127434*I + 636479493198)*x^5 - (4243
19662132*I - 143522084956)*x^3 - 4*sqrt(x^6 - x^2)*((70980218386*I + 35099697147)*x^5 + (70199394294*I - 14196
0436772)*x^3 - (70980218386*I + 35099697147)*x) + (35880521239*I + 106079915533)*x) + 12*(x^6 - x^2)^(1/4)*(-(
35880521239*I + 106079915533)*x^6 - (212159831066*I - 71761042478)*x^4 + (35880521239*I + 106079915533)*x^2))/
(x^9 + 2*x^5 + x)) - 5*sqrt(9*I + 9)*x^3*log(-(12*(x^6 - x^2)^(3/4)*(-(35099697147*I - 70980218386)*x^4 + (141
960436772*I + 70199394294)*x^2 + 35099697147*I - 70980218386) - sqrt(9*I + 9)*(-(35880521239*I + 106079915533)
*x^9 - (424319662132*I - 143522084956)*x^7 + (215283127434*I + 636479493198)*x^5 + (424319662132*I - 143522084
956)*x^3 - 4*sqrt(x^6 - x^2)*(-(70980218386*I + 35099697147)*x^5 - (70199394294*I - 141960436772)*x^3 + (70980
218386*I + 35099697147)*x) - (35880521239*I + 106079915533)*x) + 12*(x^6 - x^2)^(1/4)*(-(35880521239*I + 10607
9915533)*x^6 - (212159831066*I - 71761042478)*x^4 + (35880521239*I + 106079915533)*x^2))/(x^9 + 2*x^5 + x)) +
32*(x^6 - x^2)^(1/4)*(x^4 - 1))/x^3

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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