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

Optimal. Leaf size=187 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^6+x^2}}\right )}{8 \sqrt [4]{2}}+\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{x^6+x^2}}{\sqrt {2} x^2-\sqrt {x^6+x^2}}\right )}{8\ 2^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^6+x^2}}\right )}{8 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^6+x^2}}{2^{3/4}}}{x \sqrt [4]{x^6+x^2}}\right )}{8\ 2^{3/4}}+\frac {\left (x^6+x^2\right )^{3/4}}{2 x \left (x^4+1\right )} \]

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Rubi [C]  time = 0.15, antiderivative size = 46, normalized size of antiderivative = 0.25, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2056, 1479, 466, 510} \begin {gather*} -\frac {2 x^5 \sqrt [4]{x^4+1} F_1\left (\frac {9}{8};1,\frac {5}{4};\frac {17}{8};x^4,-x^4\right )}{9 \sqrt [4]{x^6+x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-2*x^5*(1 + x^4)^(1/4)*AppellF1[9/8, 1, 5/4, 17/8, x^4, -x^4])/(9*(x^2 + x^6)^(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 1479

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_.)*((a_) + (c_.)*(x_)^(n2_))^(p_.), x_Symbol] :> Int[(f*x)^
m*(d + e*x^n)^(q + p)*(a/d + (c*x^n)/e)^p, x] /; FreeQ[{a, c, d, e, f, q, m, 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]

Rubi steps

\begin {align*} \int \frac {x^4}{\sqrt [4]{x^2+x^6} \left (-1+x^8\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{1+x^4}\right ) \int \frac {x^{7/2}}{\sqrt [4]{1+x^4} \left (-1+x^8\right )} \, dx}{\sqrt [4]{x^2+x^6}}\\ &=\frac {\left (\sqrt {x} \sqrt [4]{1+x^4}\right ) \int \frac {x^{7/2}}{\left (-1+x^4\right ) \left (1+x^4\right )^{5/4}} \, dx}{\sqrt [4]{x^2+x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^8}{\left (-1+x^8\right ) \left (1+x^8\right )^{5/4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}\\ &=-\frac {2 x^5 \sqrt [4]{1+x^4} F_1\left (\frac {9}{8};1,\frac {5}{4};\frac {17}{8};x^4,-x^4\right )}{9 \sqrt [4]{x^2+x^6}}\\ \end {align*}

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Mathematica [C]  time = 1.13, size = 48, normalized size = 0.26 \begin {gather*} \frac {x-x \sqrt [4]{x^4+1} F_1\left (\frac {1}{8};-\frac {3}{4},1;\frac {9}{8};-x^4,x^4\right )}{2 \sqrt [4]{x^6+x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.62, size = 187, normalized size = 1.00 \begin {gather*} \frac {\left (x^2+x^6\right )^{3/4}}{2 x \left (1+x^4\right )}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{8 \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 )}{8\ 2^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{8 \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 )}{8\ 2^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2 + x^6)^(3/4)/(2*x*(1 + x^4)) - ArcTan[(2^(1/4)*x)/(x^2 + x^6)^(1/4)]/(8*2^(1/4)) + ArcTan[(2^(3/4)*x*(x^2
+ x^6)^(1/4))/(Sqrt[2]*x^2 - Sqrt[x^2 + x^6])]/(8*2^(3/4)) - ArcTanh[(2^(1/4)*x)/(x^2 + x^6)^(1/4)]/(8*2^(1/4
)) - ArcTanh[(x^2/2^(1/4) + Sqrt[x^2 + x^6]/2^(3/4))/(x*(x^2 + x^6)^(1/4))]/(8*2^(3/4))

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fricas [B]  time = 6.50, size = 765, normalized size = 4.09

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(4*2^(3/4)*(x^5 + x)*arctan(1/2*2^(3/4)*(x^6 + x^2)^(1/4)*(x^4 + 1)/(x^5 + x)) - 2^(3/4)*(x^5 + x)*log((4
*2^(1/4)*(x^6 + x^2)^(1/4)*x^2 + 2*2^(3/4)*(x^6 + x^2)^(3/4) + sqrt(2)*(x^5 + 2*x^3 + x) + 4*sqrt(x^6 + x^2)*x
)/(x^5 - 2*x^3 + x)) + 2^(3/4)*(x^5 + x)*log(-(4*2^(1/4)*(x^6 + x^2)^(1/4)*x^2 + 2*2^(3/4)*(x^6 + x^2)^(3/4) -
sqrt(2)*(x^5 + 2*x^3 + x) - 4*sqrt(x^6 + x^2)*x)/(x^5 - 2*x^3 + x)) - 4*2^(1/4)*(x^5 + x)*arctan(1/2*(4*2^(1/
4)*(x^6 + x^2)^(1/4)*x^2 + sqrt(2)*(2*2^(3/4)*(x^6 + x^2)^(1/4)*x^2 - sqrt(2)*(x^5 + 2*x^3 + x) - 4*sqrt(x^6 +
x^2)*x + 2*2^(1/4)*(x^6 + x^2)^(3/4))*sqrt((x^5 + 2*x^3 + 4*2^(1/4)*(x^6 + x^2)^(1/4)*x^2 + 4*sqrt(2)*sqrt(x^
6 + x^2)*x + 2*2^(3/4)*(x^6 + x^2)^(3/4) + x)/(x^5 + 2*x^3 + x)) + 2*2^(3/4)*(x^6 + x^2)^(3/4))/(x^5 - 2*x^3 +
x)) - 4*2^(1/4)*(x^5 + x)*arctan(1/2*(4*2^(1/4)*(x^6 + x^2)^(1/4)*x^2 + sqrt(2)*(2*2^(3/4)*(x^6 + x^2)^(1/4)*
x^2 + sqrt(2)*(x^5 + 2*x^3 + x) + 4*sqrt(x^6 + x^2)*x + 2*2^(1/4)*(x^6 + x^2)^(3/4))*sqrt((x^5 + 2*x^3 - 4*2^(
1/4)*(x^6 + x^2)^(1/4)*x^2 + 4*sqrt(2)*sqrt(x^6 + x^2)*x - 2*2^(3/4)*(x^6 + x^2)^(3/4) + x)/(x^5 + 2*x^3 + x))
+ 2*2^(3/4)*(x^6 + x^2)^(3/4))/(x^5 - 2*x^3 + x)) - 2^(1/4)*(x^5 + x)*log(2*(x^5 + 2*x^3 + 4*2^(1/4)*(x^6 + x
^2)^(1/4)*x^2 + 4*sqrt(2)*sqrt(x^6 + x^2)*x + 2*2^(3/4)*(x^6 + x^2)^(3/4) + x)/(x^5 + 2*x^3 + x)) + 2^(1/4)*(x
^5 + x)*log(2*(x^5 + 2*x^3 - 4*2^(1/4)*(x^6 + x^2)^(1/4)*x^2 + 4*sqrt(2)*sqrt(x^6 + x^2)*x - 2*2^(3/4)*(x^6 +
x^2)^(3/4) + x)/(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^{4}}{{\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^4/(x^6+x^2)^(1/4)/(x^8-1),x, algorithm="giac")

[Out]

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

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maple [C]  time = 46.01, size = 653, normalized size = 3.49

method	result	size
risch	\(\frac {x}{2 \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}-\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 )}{32}+\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 )}{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 )^{3}}{64}-\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 )}{64}+\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 )}{64}\)	\(653\)
trager	\(\frac {\left (x^{6}+x^{2}\right )^{\frac {3}{4}}}{2 x \left (x^{4}+1\right )}+\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 )}{32}-\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 )}{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}^{4}-8\right )^{3}}{64}-\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}}{64}+\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 )}{64}\)	\(655\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x/(x^2*(x^4+1))^(1/4)-1/32*RootOf(_Z^4+8)*ln(-(-RootOf(_Z^4+8)^3*(x^6+x^2)^(1/2)*x+RootOf(_Z^4+8)*x^5-2*(x
^6+x^2)^(1/4)*RootOf(_Z^4+8)^2*x^2-2*RootOf(_Z^4+8)*x^3+4*(x^6+x^2)^(3/4)+RootOf(_Z^4+8)*x)/x/(x^2+1)^2)+1/32*
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*(x^6+x^2)^(1/4)*RootOf(_Z^4+8)^2*x^2+2*RootOf(_Z^2+RootOf(_Z^4+8)^2)*x^3+4*(x^6+x^2
)^(3/4)-RootOf(_Z^2+RootOf(_Z^4+8)^2)*x)/x/(x^2+1)^2)-1/64*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/(-1+x)/(1+x))*RootOf(_Z^4+8)^3-1/64*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/(-1+x)/(1+x))*RootOf(_Z^4+8)^2*RootOf(_Z^2+RootOf(_Z^4+8)^2)+1/64*RootO
f(_Z^4+8)^2*RootOf(_Z^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*(x^6+x^2)^(1/4)*RootOf(_
Z^4+8)*RootOf(_Z^2+RootOf(_Z^4+8)^2)*x^2-8*(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+RootOf(_Z^4+8)^3*x-RootOf(_Z^4+8)^2*RootOf(_Z^2+RootOf(_Z^4+8)^2)*x+16*(x^6+x^2)^(3/4))/(-1+x)
^2/(1+x)^2/x)

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

[Out]

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

[Out]

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

[Out]

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

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