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

Optimal. Leaf size=136 \[ \frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^6+x^2}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^6+x^2}}{\sqrt {x^6+x^2}-x^2}\right )}{2 \sqrt {2}}-\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^6+x^2}}\right )+\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {x^6+x^2}}{\sqrt {2}}}{x \sqrt [4]{x^6+x^2}}\right )}{2 \sqrt {2}} \]

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Rubi [C]  time = 0.43, antiderivative size = 163, normalized size of antiderivative = 1.20, number of steps used = 7, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2056, 6728, 466, 510} \begin {gather*} \frac {2 \left (-\sqrt {3}+i\right ) x \sqrt [4]{x^6+x^2} F_1\left (\frac {3}{8};-\frac {1}{4},1;\frac {11}{8};-x^4,-\frac {2 x^4}{1-i \sqrt {3}}\right )}{3 \left (\sqrt {3}+i\right ) \sqrt [4]{x^4+1}}+\frac {2 \left (\sqrt {3}+i\right ) x \sqrt [4]{x^6+x^2} F_1\left (\frac {3}{8};-\frac {1}{4},1;\frac {11}{8};-x^4,-\frac {2 x^4}{1+i \sqrt {3}}\right )}{3 \left (-\sqrt {3}+i\right ) \sqrt [4]{x^4+1}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*(I - Sqrt[3])*x*(x^2 + x^6)^(1/4)*AppellF1[3/8, -1/4, 1, 11/8, -x^4, (-2*x^4)/(1 - I*Sqrt[3])])/(3*(I + Sqr
t[3])*(1 + x^4)^(1/4)) + (2*(I + Sqrt[3])*x*(x^2 + x^6)^(1/4)*AppellF1[3/8, -1/4, 1, 11/8, -x^4, (-2*x^4)/(1 +
I*Sqrt[3])])/(3*(I - Sqrt[3])*(1 + x^4)^(1/4))

Rule 466

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 510

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)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), 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 2056

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 6728

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*} \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^4+x^8} \, dx &=\frac {\sqrt [4]{x^2+x^6} \int \frac {\sqrt {x} \left (-1+x^4\right ) \sqrt [4]{1+x^4}}{1+x^4+x^8} \, dx}{\sqrt {x} \sqrt [4]{1+x^4}}\\ &=\frac {\sqrt [4]{x^2+x^6} \int \left (\frac {\left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^4}}{1-i \sqrt {3}+2 x^4}+\frac {\left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^4}}{1+i \sqrt {3}+2 x^4}\right ) \, dx}{\sqrt {x} \sqrt [4]{1+x^4}}\\ &=\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [4]{x^2+x^6}\right ) \int \frac {\sqrt {x} \sqrt [4]{1+x^4}}{1+i \sqrt {3}+2 x^4} \, dx}{\sqrt {x} \sqrt [4]{1+x^4}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [4]{x^2+x^6}\right ) \int \frac {\sqrt {x} \sqrt [4]{1+x^4}}{1-i \sqrt {3}+2 x^4} \, dx}{\sqrt {x} \sqrt [4]{1+x^4}}\\ &=\frac {\left (2 \left (1-i \sqrt {3}\right ) \sqrt [4]{x^2+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{1+x^8}}{1+i \sqrt {3}+2 x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{1+x^4}}+\frac {\left (2 \left (1+i \sqrt {3}\right ) \sqrt [4]{x^2+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{1+x^8}}{1-i \sqrt {3}+2 x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{1+x^4}}\\ &=\frac {2 \left (i-\sqrt {3}\right ) x \sqrt [4]{x^2+x^6} F_1\left (\frac {3}{8};-\frac {1}{4},1;\frac {11}{8};-x^4,-\frac {2 x^4}{1-i \sqrt {3}}\right )}{3 \left (i+\sqrt {3}\right ) \sqrt [4]{1+x^4}}+\frac {2 \left (i+\sqrt {3}\right ) x \sqrt [4]{x^2+x^6} F_1\left (\frac {3}{8};-\frac {1}{4},1;\frac {11}{8};-x^4,-\frac {2 x^4}{1+i \sqrt {3}}\right )}{3 \left (i-\sqrt {3}\right ) \sqrt [4]{1+x^4}}\\ \end {align*}

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Mathematica [F]  time = 0.10, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^4+x^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.00, size = 136, normalized size = 1.00 \begin {gather*} \frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^2+x^6}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^2+x^6}}{-x^2+\sqrt {x^2+x^6}}\right )}{2 \sqrt {2}}-\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^2+x^6}}\right )+\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {x^2+x^6}}{\sqrt {2}}}{x \sqrt [4]{x^2+x^6}}\right )}{2 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[x/(x^2 + x^6)^(1/4)]/2 - ArcTan[(Sqrt[2]*x*(x^2 + x^6)^(1/4))/(-x^2 + Sqrt[x^2 + x^6])]/(2*Sqrt[2]) - A
rcTanh[x/(x^2 + x^6)^(1/4)]/2 + ArcTanh[(x^2/Sqrt[2] + Sqrt[x^2 + x^6]/Sqrt[2])/(x*(x^2 + x^6)^(1/4))]/(2*Sqrt
[2])

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fricas [B]  time = 12.59, size = 780, normalized size = 5.74 \begin {gather*} -\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {x^{9} + 2 \, x^{7} + 3 \, x^{5} + 2 \, x^{3} + 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} {\left (x^{4} - 3 \, x^{2} + 1\right )} + 2 \, \sqrt {2} {\left (3 \, x^{6} - x^{4} + 3 \, x^{2}\right )} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} + 4 \, \sqrt {x^{6} + x^{2}} {\left (x^{5} + x^{3} + x\right )} + {\left (16 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} x^{2} + 2 \, \sqrt {2} \sqrt {x^{6} + x^{2}} {\left (x^{5} - 3 \, x^{3} + x\right )} + \sqrt {2} {\left (x^{9} - 8 \, x^{7} + x^{5} - 8 \, x^{3} + x\right )} + 4 \, {\left (x^{6} + x^{4} + x^{2}\right )} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{5} + x^{3} + 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {x^{6} + x^{2}} x + 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} + x}{x^{5} + x^{3} + x}} + x}{x^{9} - 14 \, x^{7} + 3 \, x^{5} - 14 \, x^{3} + x}\right ) + \frac {1}{4} \, \sqrt {2} \arctan \left (\frac {x^{9} + 2 \, x^{7} + 3 \, x^{5} + 2 \, x^{3} - 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} {\left (x^{4} - 3 \, x^{2} + 1\right )} - 2 \, \sqrt {2} {\left (3 \, x^{6} - x^{4} + 3 \, x^{2}\right )} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} + 4 \, \sqrt {x^{6} + x^{2}} {\left (x^{5} + x^{3} + x\right )} + {\left (16 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} x^{2} - 2 \, \sqrt {2} \sqrt {x^{6} + x^{2}} {\left (x^{5} - 3 \, x^{3} + x\right )} - \sqrt {2} {\left (x^{9} - 8 \, x^{7} + x^{5} - 8 \, x^{3} + x\right )} + 4 \, {\left (x^{6} + x^{4} + x^{2}\right )} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{5} + x^{3} - 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {x^{6} + x^{2}} x - 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} + x}{x^{5} + x^{3} + x}} + x}{x^{9} - 14 \, x^{7} + 3 \, x^{5} - 14 \, x^{3} + x}\right ) + \frac {1}{16} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{5} + x^{3} + 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {x^{6} + x^{2}} x + 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} + x\right )}}{x^{5} + x^{3} + x}\right ) - \frac {1}{16} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{5} + x^{3} - 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {x^{6} + x^{2}} x - 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} + x\right )}}{x^{5} + x^{3} + x}\right ) + \frac {1}{4} \, \arctan \left (\frac {2 \, {\left ({\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}\right )}}{x^{5} - x^{3} + x}\right ) + \frac {1}{4} \, \log \left (-\frac {x^{5} + x^{3} - 2 \, {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2 \, \sqrt {x^{6} + x^{2}} x + x - 2 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} - x^{3} + x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(2)*arctan((x^9 + 2*x^7 + 3*x^5 + 2*x^3 + 2*sqrt(2)*(x^6 + x^2)^(3/4)*(x^4 - 3*x^2 + 1) + 2*sqrt(2)*(
3*x^6 - x^4 + 3*x^2)*(x^6 + x^2)^(1/4) + 4*sqrt(x^6 + x^2)*(x^5 + x^3 + x) + (16*(x^6 + x^2)^(3/4)*x^2 + 2*sqr
t(2)*sqrt(x^6 + x^2)*(x^5 - 3*x^3 + x) + sqrt(2)*(x^9 - 8*x^7 + x^5 - 8*x^3 + x) + 4*(x^6 + x^4 + x^2)*(x^6 +
x^2)^(1/4))*sqrt((x^5 + x^3 + 2*sqrt(2)*(x^6 + x^2)^(1/4)*x^2 + 4*sqrt(x^6 + x^2)*x + 2*sqrt(2)*(x^6 + x^2)^(3
/4) + x)/(x^5 + x^3 + x)) + x)/(x^9 - 14*x^7 + 3*x^5 - 14*x^3 + x)) + 1/4*sqrt(2)*arctan((x^9 + 2*x^7 + 3*x^5
+ 2*x^3 - 2*sqrt(2)*(x^6 + x^2)^(3/4)*(x^4 - 3*x^2 + 1) - 2*sqrt(2)*(3*x^6 - x^4 + 3*x^2)*(x^6 + x^2)^(1/4) +
4*sqrt(x^6 + x^2)*(x^5 + x^3 + x) + (16*(x^6 + x^2)^(3/4)*x^2 - 2*sqrt(2)*sqrt(x^6 + x^2)*(x^5 - 3*x^3 + x) -
sqrt(2)*(x^9 - 8*x^7 + x^5 - 8*x^3 + x) + 4*(x^6 + x^4 + x^2)*(x^6 + x^2)^(1/4))*sqrt((x^5 + x^3 - 2*sqrt(2)*(
x^6 + x^2)^(1/4)*x^2 + 4*sqrt(x^6 + x^2)*x - 2*sqrt(2)*(x^6 + x^2)^(3/4) + x)/(x^5 + x^3 + x)) + x)/(x^9 - 14*
x^7 + 3*x^5 - 14*x^3 + x)) + 1/16*sqrt(2)*log(4*(x^5 + x^3 + 2*sqrt(2)*(x^6 + x^2)^(1/4)*x^2 + 4*sqrt(x^6 + x^
2)*x + 2*sqrt(2)*(x^6 + x^2)^(3/4) + x)/(x^5 + x^3 + x)) - 1/16*sqrt(2)*log(4*(x^5 + x^3 - 2*sqrt(2)*(x^6 + x^
2)^(1/4)*x^2 + 4*sqrt(x^6 + x^2)*x - 2*sqrt(2)*(x^6 + x^2)^(3/4) + x)/(x^5 + x^3 + x)) + 1/4*arctan(2*((x^6 +
x^2)^(1/4)*x^2 + (x^6 + x^2)^(3/4))/(x^5 - x^3 + x)) + 1/4*log(-(x^5 + x^3 - 2*(x^6 + x^2)^(1/4)*x^2 + 2*sqrt(
x^6 + x^2)*x + x - 2*(x^6 + x^2)^(3/4))/(x^5 - x^3 + x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}}{x^{8} + x^{4} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 10.25, size = 490, normalized size = 3.60

method	result	size
trager	\(-\frac {\ln \left (-\frac {x^{5}+2 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}+2 \sqrt {x^{6}+x^{2}}\, x +2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} x^{2}+x^{3}+x}{\left (x^{4}-x^{2}+1\right ) x}\right )}{4}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{5}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{6}+x^{2}}\, x -\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}-2 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}+2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} x^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x}{\left (x^{4}-x^{2}+1\right ) x}\right )}{4}-\frac {\RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{5}-\RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}-2 \RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \sqrt {x^{6}+x^{2}}\, x +2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+\RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \RootOf \left (\textit {\_Z}^{2}+1\right ) x +2 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}}{\left (x^{2}+x +1\right ) x \left (x^{2}-x +1\right )}\right )}{4}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) x^{5}+2 \RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{6}+x^{2}}\, x -2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+\RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) x^{3}+2 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}-\RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) x}{\left (x^{2}+x +1\right ) x \left (x^{2}-x +1\right )}\right )}{4}\)	\(490\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*ln(-(x^5+2*(x^6+x^2)^(3/4)+2*(x^6+x^2)^(1/2)*x+2*(x^6+x^2)^(1/4)*x^2+x^3+x)/(x^4-x^2+1)/x)+1/4*RootOf(_Z^
2+1)*ln((-RootOf(_Z^2+1)*x^5+2*RootOf(_Z^2+1)*(x^6+x^2)^(1/2)*x-RootOf(_Z^2+1)*x^3-2*(x^6+x^2)^(3/4)+2*(x^6+x^
2)^(1/4)*x^2-RootOf(_Z^2+1)*x)/(x^4-x^2+1)/x)-1/4*RootOf(_Z^2-RootOf(_Z^2+1))*ln(-(RootOf(_Z^2-RootOf(_Z^2+1))
*RootOf(_Z^2+1)*x^5-RootOf(_Z^2-RootOf(_Z^2+1))*RootOf(_Z^2+1)*x^3-2*RootOf(_Z^2-RootOf(_Z^2+1))*(x^6+x^2)^(1/
2)*x+2*(x^6+x^2)^(1/4)*RootOf(_Z^2+1)*x^2+RootOf(_Z^2-RootOf(_Z^2+1))*RootOf(_Z^2+1)*x+2*(x^6+x^2)^(3/4))/(x^2
+x+1)/x/(x^2-x+1))+1/4*RootOf(_Z^2+1)*RootOf(_Z^2-RootOf(_Z^2+1))*ln(-(-RootOf(_Z^2-RootOf(_Z^2+1))*x^5+2*Root
Of(_Z^2-RootOf(_Z^2+1))*RootOf(_Z^2+1)*(x^6+x^2)^(1/2)*x-2*(x^6+x^2)^(1/4)*RootOf(_Z^2+1)*x^2+RootOf(_Z^2-Root
Of(_Z^2+1))*x^3+2*(x^6+x^2)^(3/4)-RootOf(_Z^2-RootOf(_Z^2+1))*x)/(x^2+x+1)/x/(x^2-x+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}}{x^{8} + x^{4} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^6+x^2\right )}^{1/4}\,\left (x^4-1\right )}{x^8+x^4+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^{4} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right ) \left (x^{4} - x^{2} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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