3.31.74 \(\int \frac {\sqrt [4]{-x^2+x^6} (1-x^4+x^8)}{x^4 (1+x^4)} \, dx\) [3074]

Optimal. Leaf size=501 \[ \frac {2 \left (-1+x^4\right ) \sqrt [4]{-x^2+x^6}}{5 x^3}+\frac {3}{4} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \text {ArcTan}\left (\frac {\sqrt {2-\sqrt {2}} x}{-\sqrt {2+\sqrt {2}} x+2^{3/4} \sqrt [4]{-x^2+x^6}}\right )+\frac {3}{4} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \text {ArcTan}\left (\frac {\sqrt {2-\sqrt {2}} x}{\sqrt {2+\sqrt {2}} x+2^{3/4} \sqrt [4]{-x^2+x^6}}\right )-\frac {3}{4} \sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \text {ArcTan}\left (\frac {2^{3/4} \sqrt {2+\sqrt {2}} x \sqrt [4]{-x^2+x^6}}{-2 x^2+\sqrt {2} \sqrt {-x^2+x^6}}\right )-\frac {3}{4} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\frac {\sqrt [4]{2} x^2}{\sqrt {2-\sqrt {2}}}+\frac {\sqrt {-x^2+x^6}}{\sqrt [4]{2} \sqrt {2-\sqrt {2}}}}{x \sqrt [4]{-x^2+x^6}}\right )-\frac {3}{8} \sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \log \left (-2 x^2+2^{3/4} \sqrt {2+\sqrt {2}} x \sqrt [4]{-x^2+x^6}-\sqrt {2} \sqrt {-x^2+x^6}\right )+\frac {3}{8} \sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \log \left (2 \sqrt {2-\sqrt {2}} x^2+2 \sqrt [4]{2} x \sqrt [4]{-x^2+x^6}+\sqrt {4-2 \sqrt {2}} \sqrt {-x^2+x^6}\right ) \]

[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))

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Rubi [C] Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
time = 0.42, antiderivative size = 131, normalized size of antiderivative = 0.26, number of steps used = 14, number of rules used = 10, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {2081, 6857, 283, 335, 372, 371, 285, 477, 525, 524} \begin {gather*} -\frac {6 \sqrt [4]{x^6-x^2} F_1\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 \, _2F_1\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} \end {gather*}

Antiderivative was successfully verified.

[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*} \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+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} F_1\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} F_1\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} \, _2F_1\left (\frac {3}{8},\frac {3}{4};\frac {11}{8};x^4\right )}{5 \sqrt [4]{1-x^4}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.73, size = 203, normalized size = 0.41 \begin {gather*} \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} \text {ArcTan}\left (\frac {\sqrt {-1-i} \sqrt {x}}{\sqrt [4]{-1+x^4}}\right )+15 \sqrt {1-i} x^{5/2} \text {ArcTan}\left (\frac {\sqrt {-1+i} \sqrt {x}}{\sqrt [4]{-1+x^4}}\right )-15 \sqrt {-1+i} x^{5/2} \text {ArcTan}\left (\frac {\sqrt {1-i} \sqrt {x}}{\sqrt [4]{-1+x^4}}\right )-15 \sqrt {-1-i} x^{5/2} \text {ArcTan}\left (\frac {\sqrt {1+i} \sqrt {x}}{\sqrt [4]{-1+x^4}}\right )\right )}{20 x^3 \sqrt [4]{-1+x^4}} \end {gather*}

Antiderivative was successfully verified.

[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))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 141.28, size = 2948, normalized size = 5.88

method	result	size
trager	\(\text {Expression too large to display}\)	\(2948\)
risch	\(\text {Expression too large to display}\)	\(7288\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(x^4-1)*(x^6-x^2)^(1/4)/x^3-3/8*RootOf(_Z^2+16384*RootOf(134217728*_Z^4+16384*_Z^2+1)^2+2)*ln((-8053063680
0*RootOf(_Z^2+16384*RootOf(134217728*_Z^4+16384*_Z^2+1)^2+2)*RootOf(134217728*_Z^4+16384*_Z^2+1)^4*x^5+3019898
88000*RootOf(_Z^2+16384*RootOf(134217728*_Z^4+16384*_Z^2+1)^2+2)*RootOf(134217728*_Z^4+16384*_Z^2+1)^4*x^3-548
12672*RootOf(134217728*_Z^4+16384*_Z^2+1)^2*RootOf(_Z^2+16384*RootOf(134217728*_Z^4+16384*_Z^2+1)^2+2)*x^5+123
109376*RootOf(_Z^2+16384*RootOf(134217728*_Z^4+16384*_Z^2+1)^2+2)*(x^6-x^2)^(1/2)*RootOf(134217728*_Z^4+16384*
_Z^2+1)^2*x+80530636800*RootOf(_Z^2+16384*RootOf(134217728*_Z^4+16384*_Z^2+1)^2+2)*RootOf(134217728*_Z^4+16384
*_Z^2+1)^4*x+249937920*RootOf(134217728*_Z^4+16384*_Z^2+1)^2*RootOf(_Z^2+16384*RootOf(134217728*_Z^4+16384*_Z^
2+1)^2+2)*x^3+3157*RootOf(_Z^2+16384*RootOf(134217728*_Z^4+16384*_Z^2+1)^2+2)*x^5+303038464*(x^6-x^2)^(3/4)*Ro
otOf(134217728*_Z^4+16384*_Z^2+1)^2+179929088*(x^6-x^2)^(1/4)*RootOf(134217728*_Z^4+16384*_Z^2+1)^2*x^2+26010*
RootOf(_Z^2+16384*RootOf(134217728*_Z^4+16384*_Z^2+1)^2+2)*(x^6-x^2)^(1/2)*x+54812672*RootOf(134217728*_Z^4+16
384*_Z^2+1)^2*RootOf(_Z^2+16384*RootOf(134217728*_Z^4+16384*_Z^2+1)^2+2)*x+20746*RootOf(_Z^2+16384*RootOf(1342
17728*_Z^4+16384*_Z^2+1)^2+2)*x^3+10982*(x^6-x^2)^(3/4)-15028*(x^6-x^2)^(1/4)*x^2-3157*RootOf(_Z^2+16384*RootO
f(134217728*_Z^4+16384*_Z^2+1)^2+2)*x)/(16384*x^2*RootOf(134217728*_Z^4+16384*_Z^2+1)^2-65536*RootOf(134217728
*_Z^4+16384*_Z^2+1)^2+5*x^2-3)^2/x)+48*RootOf(134217728*_Z^4+16384*_Z^2+1)*ln(-(-5153960755200*RootOf(13421772
8*_Z^4+16384*_Z^2+1)^5*x^5+19327352832000*RootOf(134217728*_Z^4+16384*_Z^2+1)^5*x^3+2249719808*RootOf(13421772
8*_Z^4+16384*_Z^2+1)^3*x^5-7879000064*RootOf(134217728*_Z^4+16384*_Z^2+1)^3*(x^6-x^2)^(1/2)*x+5153960755200*Ro
otOf(134217728*_Z^4+16384*_Z^2+1)^5*x-11277434880*RootOf(134217728*_Z^4+16384*_Z^2+1)^3*x^3+553472*RootOf(1342
17728*_Z^4+16384*_Z^2+1)*x^5+151519232*(x^6-x^2)^(3/4)*RootOf(134217728*_Z^4+16384*_Z^2+1)^2+89964544*(x^6-x^2
)^(1/4)*RootOf(134217728*_Z^4+16384*_Z^2+1)^2*x^2+702848*RootOf(134217728*_Z^4+16384*_Z^2+1)*(x^6-x^2)^(1/2)*x
-2249719808*RootOf(134217728*_Z^4+16384*_Z^2+1)^3*x-336896*RootOf(134217728*_Z^4+16384*_Z^2+1)*x^3+13005*(x^6-
x^2)^(3/4)+18496*(x^6-x^2)^(1/4)*x^2-553472*RootOf(134217728*_Z^4+16384*_Z^2+1)*x)/(16384*x^2*RootOf(134217728
*_Z^4+16384*_Z^2+1)^2-65536*RootOf(134217728*_Z^4+16384*_Z^2+1)^2-3*x^2-5)^2/x)+6144*ln((-443992244224*RootOf(
_Z^2+16384*RootOf(134217728*_Z^4+16384*_Z^2+1)^2+2)*RootOf(134217728*_Z^4+16384*_Z^2+1)^4*x^5+1664970915840*Ro
otOf(_Z^2+16384*RootOf(134217728*_Z^4+16384*_Z^2+1)^2+2)*RootOf(134217728*_Z^4+16384*_Z^2+1)^4*x^3-929792*Root
Of(134217728*_Z^4+16384*_Z^2+1)^2*RootOf(_Z^2+16384*RootOf(134217728*_Z^4+16384*_Z^2+1)^2+2)*x^5-151519232*Roo
tOf(_Z^2+16384*RootOf(134217728*_Z^4+16384*_Z^2+1)^2+2)*(x^6-x^2)^(1/2)*RootOf(134217728*_Z^4+16384*_Z^2+1)^2*
x+443992244224*RootOf(_Z^2+16384*RootOf(134217728*_Z^4+16384*_Z^2+1)^2+2)*RootOf(134217728*_Z^4+16384*_Z^2+1)^
4*x+248225792*RootOf(134217728*_Z^4+16384*_Z^2+1)^2*RootOf(_Z^2+16384*RootOf(134217728*_Z^4+16384*_Z^2+1)^2+2)
*x^3+1316*RootOf(_Z^2+16384*RootOf(134217728*_Z^4+16384*_Z^2+1)^2+2)*x^5+151519232*(x^6-x^2)^(3/4)*RootOf(1342
17728*_Z^4+16384*_Z^2+1)^2-89964544*(x^6-x^2)^(1/4)*RootOf(134217728*_Z^4+16384*_Z^2+1)^2*x^2-5491*RootOf(_Z^2
+16384*RootOf(134217728*_Z^4+16384*_Z^2+1)^2+2)*(x^6-x^2)^(1/2)*x+929792*RootOf(134217728*_Z^4+16384*_Z^2+1)^2
*RootOf(_Z^2+16384*RootOf(134217728*_Z^4+16384*_Z^2+1)^2+2)*x+8648*RootOf(_Z^2+16384*RootOf(134217728*_Z^4+163
84*_Z^2+1)^2+2)*x^3+5491*(x^6-x^2)^(3/4)+7514*(x^6-x^2)^(1/4)*x^2-1316*RootOf(_Z^2+16384*RootOf(134217728*_Z^4
+16384*_Z^2+1)^2+2)*x)/(16384*x^2*RootOf(134217728*_Z^4+16384*_Z^2+1)^2-65536*RootOf(134217728*_Z^4+16384*_Z^2
+1)^2+5*x^2-3)^2/x)*RootOf(134217728*_Z^4+16384*_Z^2+1)^2*RootOf(_Z^2+16384*RootOf(134217728*_Z^4+16384*_Z^2+1
)^2+2)+3/8*ln((-443992244224*RootOf(_Z^2+16384*RootOf(134217728*_Z^4+16384*_Z^2+1)^2+2)*RootOf(134217728*_Z^4+
16384*_Z^2+1)^4*x^5+1664970915840*RootOf(_Z^2+16384*RootOf(134217728*_Z^4+16384*_Z^2+1)^2+2)*RootOf(134217728*
_Z^4+16384*_Z^2+1)^4*x^3-929792*RootOf(134217728*_Z^4+16384*_Z^2+1)^2*RootOf(_Z^2+16384*RootOf(134217728*_Z^4+
16384*_Z^2+1)^2+2)*x^5-151519232*RootOf(_Z^2+16384*RootOf(134217728*_Z^4+16384*_Z^2+1)^2+2)*(x^6-x^2)^(1/2)*Ro
otOf(134217728*_Z^4+16384*_Z^2+1)^2*x+443992244224*RootOf(_Z^2+16384*RootOf(134217728*_Z^4+16384*_Z^2+1)^2+2)*
RootOf(134217728*_Z^4+16384*_Z^2+1)^4*x+248225792*RootOf(134217728*_Z^4+16384*_Z^2+1)^2*RootOf(_Z^2+16384*Root
Of(134217728*_Z^4+16384*_Z^2+1)^2+2)*x^3+1316*RootOf(_Z^2+16384*RootOf(134217728*_Z^4+16384*_Z^2+1)^2+2)*x^5+1
51519232*(x^6-x^2)^(3/4)*RootOf(134217728*_Z^4+16384*_Z^2+1)^2-89964544*(x^6-x^2)^(1/4)*RootOf(134217728*_Z^4+
16384*_Z^2+1)^2*x^2-5491*RootOf(_Z^2+16384*RootOf(134217728*_Z^4+16384*_Z^2+1)^2+2)*(x^6-x^2)^(1/2)*x+929792*R
ootOf(134217728*_Z^4+16384*_Z^2+1)^2*RootOf(_Z^2+16384*RootOf(134217728*_Z^4+16384*_Z^2+1)^2+2)*x+8648*RootOf(
_Z^2+16384*RootOf(134217728*_Z^4+16384*_Z^2+1)^...

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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