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

Optimal. Leaf size=185 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^6+x^2}}\right )}{4 \sqrt [4]{2}}+\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{x^6+x^2}}{\sqrt {2} x^2-\sqrt {x^6+x^2}}\right )}{4\ 2^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^6+x^2}}\right )}{4 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^6+x^2}}{2^{3/4}}}{x \sqrt [4]{x^6+x^2}}\right )}{4\ 2^{3/4}}-\frac {\left (x^6+x^2\right )^{3/4}}{x \left (x^4+1\right )} \]

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Rubi [C]  time = 0.54, antiderivative size = 81, normalized size of antiderivative = 0.44, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2056, 6715, 6725, 245, 1404, 429} \begin {gather*} \frac {2 x \sqrt [4]{x^4+1} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-x^4\right )}{\sqrt [4]{x^6+x^2}}-\frac {4 x \sqrt [4]{x^4+1} F_1\left (\frac {1}{8};1,\frac {5}{4};\frac {9}{8};x^4,-x^4\right )}{\sqrt [4]{x^6+x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-4*x*(1 + x^4)^(1/4)*AppellF1[1/8, 1, 5/4, 9/8, x^4, -x^4])/(x^2 + x^6)^(1/4) + (2*x*(1 + x^4)^(1/4)*Hypergeo
metric2F1[1/8, 1/4, 9/8, -x^4])/(x^2 + x^6)^(1/4)

Rule 245

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, -((b*x^n)/a)],
x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p
] || GtQ[a, 0])

Rule 429

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

Rule 1404

Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((a_) + (c_.)*(x_)^(n2_))^(p_.), x_Symbol] :> Int[(d + e*x^n)^(p + q)*(a/d
+ (c*x^n)/e)^p, x] /; FreeQ[{a, c, d, e, n, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && IntegerQ[p]

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 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rule 6725

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 {1+x^8}{\sqrt [4]{x^2+x^6} \left (-1+x^8\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{1+x^4}\right ) \int \frac {1+x^8}{\sqrt {x} \sqrt [4]{1+x^4} \left (-1+x^8\right )} \, dx}{\sqrt [4]{x^2+x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1+x^{16}}{\sqrt [4]{1+x^8} \left (-1+x^{16}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt [4]{1+x^8}}+\frac {2}{\sqrt [4]{1+x^8} \left (-1+x^{16}\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}+\frac {\left (4 \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^8} \left (-1+x^{16}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}\\ &=\frac {2 x \sqrt [4]{1+x^4} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-x^4\right )}{\sqrt [4]{x^2+x^6}}+\frac {\left (4 \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+x^8\right ) \left (1+x^8\right )^{5/4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}\\ &=-\frac {4 x \sqrt [4]{1+x^4} F_1\left (\frac {1}{8};1,\frac {5}{4};\frac {9}{8};x^4,-x^4\right )}{\sqrt [4]{x^2+x^6}}+\frac {2 x \sqrt [4]{1+x^4} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-x^4\right )}{\sqrt [4]{x^2+x^6}}\\ \end {align*}

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Mathematica [C]  time = 0.37, size = 45, normalized size = 0.24 \begin {gather*} -\frac {x \left (\sqrt [4]{x^4+1} F_1\left (\frac {1}{8};-\frac {3}{4},1;\frac {9}{8};-x^4,x^4\right )+1\right )}{\sqrt [4]{x^6+x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((x*(1 + (1 + x^4)^(1/4)*AppellF1[1/8, -3/4, 1, 9/8, -x^4, x^4]))/(x^2 + x^6)^(1/4))

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IntegrateAlgebraic [A]  time = 0.73, size = 185, normalized size = 1.00 \begin {gather*} -\frac {\left (x^2+x^6\right )^{3/4}}{x \left (1+x^4\right )}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{4 \sqrt [4]{2}}+\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{x^2+x^6}}{\sqrt {2} x^2-\sqrt {x^2+x^6}}\right )}{4\ 2^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{4 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^2+x^6}}{2^{3/4}}}{x \sqrt [4]{x^2+x^6}}\right )}{4\ 2^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((x^2 + x^6)^(3/4)/(x*(1 + x^4))) - ArcTan[(2^(1/4)*x)/(x^2 + x^6)^(1/4)]/(4*2^(1/4)) + ArcTan[(2^(3/4)*x*(x^
2 + x^6)^(1/4))/(Sqrt[2]*x^2 - Sqrt[x^2 + x^6])]/(4*2^(3/4)) - ArcTanh[(2^(1/4)*x)/(x^2 + x^6)^(1/4)]/(4*2^(1/
4)) - ArcTanh[(x^2/2^(1/4) + Sqrt[x^2 + x^6]/2^(3/4))/(x*(x^2 + x^6)^(1/4))]/(4*2^(3/4))

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fricas [B]  time = 54.93, size = 1055, normalized size = 5.70

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32*(4*2^(3/4)*(x^5 + x)*arctan(1/2*(4*2^(3/4)*(x^6 + x^2)^(1/4)*x^2 + 2^(3/4)*(2*2^(3/4)*sqrt(x^6 + x^2)*x
+ 2^(1/4)*(x^5 + 2*x^3 + x)) + 4*2^(1/4)*(x^6 + x^2)^(3/4))/(x^5 - 2*x^3 + x)) + 2^(3/4)*(x^5 + x)*log(-(4*sqr
t(2)*(x^6 + x^2)^(1/4)*x^2 + 2^(3/4)*(x^5 + 2*x^3 + x) + 4*2^(1/4)*sqrt(x^6 + x^2)*x + 4*(x^6 + x^2)^(3/4))/(x
^5 - 2*x^3 + x)) - 2^(3/4)*(x^5 + x)*log(-(4*sqrt(2)*(x^6 + x^2)^(1/4)*x^2 - 2^(3/4)*(x^5 + 2*x^3 + x) - 4*2^(
1/4)*sqrt(x^6 + x^2)*x + 4*(x^6 + x^2)^(3/4))/(x^5 - 2*x^3 + x)) - 4*2^(1/4)*(x^5 + x)*arctan(-1/2*(2*x^9 + 8*
x^7 + 12*x^5 + 8*x^3 + 4*2^(3/4)*(x^6 + x^2)^(3/4)*(x^4 - 6*x^2 + 1) + 8*sqrt(2)*sqrt(x^6 + x^2)*(x^5 + 2*x^3
+ x) - sqrt(2)*(32*sqrt(2)*(x^6 + x^2)^(3/4)*x^2 + 2^(3/4)*(x^9 - 16*x^7 - 2*x^5 - 16*x^3 + x) + 4*2^(1/4)*sqr
t(x^6 + x^2)*(x^5 - 6*x^3 + x) + 8*(x^6 + 2*x^4 + x^2)*(x^6 + x^2)^(1/4))*sqrt((4*2^(3/4)*(x^6 + x^2)^(1/4)*x^
2 + sqrt(2)*(x^5 + 2*x^3 + x) + 8*sqrt(x^6 + x^2)*x + 4*2^(1/4)*(x^6 + x^2)^(3/4))/(x^5 + 2*x^3 + x)) + 8*2^(1
/4)*(3*x^6 - 2*x^4 + 3*x^2)*(x^6 + x^2)^(1/4) + 2*x)/(x^9 - 28*x^7 + 6*x^5 - 28*x^3 + x)) + 4*2^(1/4)*(x^5 + x
)*arctan(-1/2*(2*x^9 + 8*x^7 + 12*x^5 + 8*x^3 - 4*2^(3/4)*(x^6 + x^2)^(3/4)*(x^4 - 6*x^2 + 1) + 8*sqrt(2)*sqrt
(x^6 + x^2)*(x^5 + 2*x^3 + x) - sqrt(2)*(32*sqrt(2)*(x^6 + x^2)^(3/4)*x^2 - 2^(3/4)*(x^9 - 16*x^7 - 2*x^5 - 16
*x^3 + x) - 4*2^(1/4)*sqrt(x^6 + x^2)*(x^5 - 6*x^3 + x) + 8*(x^6 + 2*x^4 + x^2)*(x^6 + x^2)^(1/4))*sqrt(-(4*2^
(3/4)*(x^6 + x^2)^(1/4)*x^2 - sqrt(2)*(x^5 + 2*x^3 + x) - 8*sqrt(x^6 + x^2)*x + 4*2^(1/4)*(x^6 + x^2)^(3/4))/(
x^5 + 2*x^3 + x)) - 8*2^(1/4)*(3*x^6 - 2*x^4 + 3*x^2)*(x^6 + x^2)^(1/4) + 2*x)/(x^9 - 28*x^7 + 6*x^5 - 28*x^3
+ x)) + 2^(1/4)*(x^5 + x)*log(8*(4*2^(3/4)*(x^6 + x^2)^(1/4)*x^2 + sqrt(2)*(x^5 + 2*x^3 + x) + 8*sqrt(x^6 + x^
2)*x + 4*2^(1/4)*(x^6 + x^2)^(3/4))/(x^5 + 2*x^3 + x)) - 2^(1/4)*(x^5 + x)*log(-8*(4*2^(3/4)*(x^6 + x^2)^(1/4)
*x^2 - sqrt(2)*(x^5 + 2*x^3 + x) - 8*sqrt(x^6 + x^2)*x + 4*2^(1/4)*(x^6 + x^2)^(3/4))/(x^5 + 2*x^3 + x)) + 32*
(x^6 + x^2)^(3/4))/(x^5 + x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 43.15, size = 652, normalized size = 3.52

method	result	size
trager	\(-\frac {\left (x^{6}+x^{2}\right )^{\frac {3}{4}}}{x \left (x^{4}+1\right )}-\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} \sqrt {x^{6}+x^{2}}\, x -\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{5}+2 \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} \left (x^{6}+x^{2}\right )^{\frac {1}{4}} x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{3}-4 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x}{\left (-1+x \right )^{2} \left (1+x \right )^{2} x}\right )}{16}-\frac {\RootOf \left (\textit {\_Z}^{4}-8\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}-8\right )^{3} \sqrt {x^{6}+x^{2}}\, x +\RootOf \left (\textit {\_Z}^{4}-8\right ) x^{5}+2 \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} \left (x^{6}+x^{2}\right )^{\frac {1}{4}} x^{2}+2 \RootOf \left (\textit {\_Z}^{4}-8\right ) x^{3}+4 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}+\RootOf \left (\textit {\_Z}^{4}-8\right ) x}{\left (-1+x \right )^{2} \left (1+x \right )^{2} x}\right )}{16}-\frac {\ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{2}+2 \RootOf \left (\textit {\_Z}^{4}-8\right ) \left (x^{6}+x^{2}\right )^{\frac {1}{4}} x +2 \sqrt {x^{6}+x^{2}}}{x \left (x^{2}+1\right )}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{3}}{32}-\frac {\ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{2}+2 \RootOf \left (\textit {\_Z}^{4}-8\right ) \left (x^{6}+x^{2}\right )^{\frac {1}{4}} x +2 \sqrt {x^{6}+x^{2}}}{x \left (x^{2}+1\right )}\right ) \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{2}}{32}+\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} \ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}-8\right )^{3} x^{5}-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{5}-2 \RootOf \left (\textit {\_Z}^{4}-8\right )^{3} x^{3}+2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{3}+8 \RootOf \left (\textit {\_Z}^{4}-8\right ) \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{3} x -\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x +8 \sqrt {x^{6}+x^{2}}\, \RootOf \left (\textit {\_Z}^{4}-8\right ) x +8 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \sqrt {x^{6}+x^{2}}\, x +16 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}}{x \left (x^{2}+1\right )^{2}}\right )}{32}\)	\(652\)
risch	\(-\frac {x}{\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}+\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} \sqrt {x^{6}+x^{2}}\, x -\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x^{5}+2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{2}+2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x^{3}+4 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x}{x \left (x^{2}+1\right )^{2}}\right )}{16}+\frac {\RootOf \left (\textit {\_Z}^{4}+8\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+8\right )^{3} \sqrt {x^{6}+x^{2}}\, x -\RootOf \left (\textit {\_Z}^{4}+8\right ) x^{5}-2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{2}+2 \RootOf \left (\textit {\_Z}^{4}+8\right ) x^{3}+4 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}-\RootOf \left (\textit {\_Z}^{4}+8\right ) x}{x \left (x^{2}+1\right )^{2}}\right )}{16}-\frac {\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{2}-2 \RootOf \left (\textit {\_Z}^{4}+8\right ) \left (x^{6}+x^{2}\right )^{\frac {1}{4}} x +2 \sqrt {x^{6}+x^{2}}}{x \left (-1+x \right ) \left (1+x \right )}\right ) \RootOf \left (\textit {\_Z}^{4}+8\right )^{3}}{32}-\frac {\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{2}-2 \RootOf \left (\textit {\_Z}^{4}+8\right ) \left (x^{6}+x^{2}\right )^{\frac {1}{4}} x +2 \sqrt {x^{6}+x^{2}}}{x \left (-1+x \right ) \left (1+x \right )}\right ) \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right )}{32}+\frac {\RootOf \left (\textit {\_Z}^{4}+8\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x^{5}-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{5}+2 \RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x^{3}-2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{3}+8 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+8\right ) \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x^{2}-8 \sqrt {x^{6}+x^{2}}\, \RootOf \left (\textit {\_Z}^{4}+8\right ) x -8 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \sqrt {x^{6}+x^{2}}\, x +\RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x -\RootOf \left (\textit {\_Z}^{4}+8\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x +16 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}}{\left (-1+x \right )^{2} \left (1+x \right )^{2} x}\right )}{32}\)	\(654\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x^6+x^2)^(3/4)/x/(x^4+1)-1/16*RootOf(_Z^2+RootOf(_Z^4-8)^2)*ln(-(RootOf(_Z^2+RootOf(_Z^4-8)^2)*RootOf(_Z^4-8
)^2*(x^6+x^2)^(1/2)*x-RootOf(_Z^2+RootOf(_Z^4-8)^2)*x^5+2*RootOf(_Z^4-8)^2*(x^6+x^2)^(1/4)*x^2-2*RootOf(_Z^2+R
ootOf(_Z^4-8)^2)*x^3-4*(x^6+x^2)^(3/4)-RootOf(_Z^2+RootOf(_Z^4-8)^2)*x)/(-1+x)^2/(1+x)^2/x)-1/16*RootOf(_Z^4-8
)*ln((RootOf(_Z^4-8)^3*(x^6+x^2)^(1/2)*x+RootOf(_Z^4-8)*x^5+2*RootOf(_Z^4-8)^2*(x^6+x^2)^(1/4)*x^2+2*RootOf(_Z
^4-8)*x^3+4*(x^6+x^2)^(3/4)+RootOf(_Z^4-8)*x)/(-1+x)^2/(1+x)^2/x)-1/32*ln((RootOf(_Z^4-8)^2*x^2+2*RootOf(_Z^4-
8)*(x^6+x^2)^(1/4)*x+2*(x^6+x^2)^(1/2))/x/(x^2+1))*RootOf(_Z^4-8)^3-1/32*ln((RootOf(_Z^4-8)^2*x^2+2*RootOf(_Z^
4-8)*(x^6+x^2)^(1/4)*x+2*(x^6+x^2)^(1/2))/x/(x^2+1))*RootOf(_Z^2+RootOf(_Z^4-8)^2)*RootOf(_Z^4-8)^2+1/32*RootO
f(_Z^2+RootOf(_Z^4-8)^2)*RootOf(_Z^4-8)^2*ln((RootOf(_Z^4-8)^3*x^5-RootOf(_Z^2+RootOf(_Z^4-8)^2)*RootOf(_Z^4-8
)^2*x^5-2*RootOf(_Z^4-8)^3*x^3+2*RootOf(_Z^2+RootOf(_Z^4-8)^2)*RootOf(_Z^4-8)^2*x^3+8*RootOf(_Z^4-8)*(x^6+x^2)
^(1/4)*RootOf(_Z^2+RootOf(_Z^4-8)^2)*x^2+RootOf(_Z^4-8)^3*x-RootOf(_Z^2+RootOf(_Z^4-8)^2)*RootOf(_Z^4-8)^2*x+8
*(x^6+x^2)^(1/2)*RootOf(_Z^4-8)*x+8*RootOf(_Z^2+RootOf(_Z^4-8)^2)*(x^6+x^2)^(1/2)*x+16*(x^6+x^2)^(3/4))/x/(x^2
+1)^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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