3.30.53 \(\int \frac {x^3}{\sqrt [3]{x^2+x^4} (-1+x^6)} \, dx\) [2953]

Optimal. Leaf size=356 \[ -\frac {\text {ArcTan}\left (\frac {\sqrt {3} x^2}{x^2+2 \left (x^2+x^4\right )^{2/3}}\right )}{2 \sqrt {3}}+\frac {\text {ArcTan}\left (\frac {\sqrt {3} x^2}{x^2+\sqrt [3]{2} \left (x^2+x^4\right )^{2/3}}\right )}{4 \sqrt [3]{2} \sqrt {3}}-\frac {1}{6} \log \left (-x+\sqrt [3]{x^2+x^4}\right )-\frac {1}{6} \log \left (x+\sqrt [3]{x^2+x^4}\right )+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{x^2+x^4}\right )}{12 \sqrt [3]{2}}+\frac {\log \left (2 x+2^{2/3} \sqrt [3]{x^2+x^4}\right )}{12 \sqrt [3]{2}}+\frac {1}{12} \log \left (x^2-x \sqrt [3]{x^2+x^4}+\left (x^2+x^4\right )^{2/3}\right )+\frac {1}{12} \log \left (x^2+x \sqrt [3]{x^2+x^4}+\left (x^2+x^4\right )^{2/3}\right )-\frac {\log \left (-2 x^2+2^{2/3} x \sqrt [3]{x^2+x^4}-\sqrt [3]{2} \left (x^2+x^4\right )^{2/3}\right )}{24 \sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{x^2+x^4}+\sqrt [3]{2} \left (x^2+x^4\right )^{2/3}\right )}{24 \sqrt [3]{2}} \]

[Out]

-1/6*arctan(3^(1/2)*x^2/(x^2+2*(x^4+x^2)^(2/3)))*3^(1/2)+1/24*arctan(3^(1/2)*x^2/(x^2+2^(1/3)*(x^4+x^2)^(2/3))
)*2^(2/3)*3^(1/2)-1/6*ln(-x+(x^4+x^2)^(1/3))-1/6*ln(x+(x^4+x^2)^(1/3))+1/24*ln(-2*x+2^(2/3)*(x^4+x^2)^(1/3))*2
^(2/3)+1/24*ln(2*x+2^(2/3)*(x^4+x^2)^(1/3))*2^(2/3)+1/12*ln(x^2-x*(x^4+x^2)^(1/3)+(x^4+x^2)^(2/3))+1/12*ln(x^2
+x*(x^4+x^2)^(1/3)+(x^4+x^2)^(2/3))-1/48*ln(-2*x^2+2^(2/3)*x*(x^4+x^2)^(1/3)-2^(1/3)*(x^4+x^2)^(2/3))*2^(2/3)-
1/48*ln(2*x^2+2^(2/3)*x*(x^4+x^2)^(1/3)+2^(1/3)*(x^4+x^2)^(2/3))*2^(2/3)

________________________________________________________________________________________

Rubi [C] Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
time = 1.52, antiderivative size = 544, normalized size of antiderivative = 1.53, number of steps used = 61, number of rules used = 16, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {2081, 6857, 973, 477, 476, 494, 371, 502, 2174, 206, 31, 648, 631, 210, 642, 524} \begin {gather*} -\frac {\sqrt [3]{x^2+1} x^4 F_1\left (\frac {5}{3};\frac {1}{3},1;\frac {8}{3};-x^2,-\sqrt [3]{-1} x^2\right )}{10 \sqrt [3]{x^4+x^2}}-\frac {\sqrt [3]{x^2+1} x^4 F_1\left (\frac {5}{3};\frac {1}{3},1;\frac {8}{3};-x^2,(-1)^{2/3} x^2\right )}{10 \sqrt [3]{x^4+x^2}}-\frac {\sqrt [3]{x^2+1} x^{2/3} \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{2} \left (x^{2/3}+1\right )}{\sqrt [3]{x^2+1}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x^4+x^2}}-\frac {\sqrt [3]{x^2+1} x^{2/3} \text {ArcTan}\left (\frac {\frac {\sqrt [3]{2} \left (x^{2/3}+1\right )}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x^4+x^2}}+\frac {\sqrt [3]{x^2+1} x^2 \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};-x^2\right )}{4 \sqrt [3]{x^4+x^2}}-\frac {\sqrt [3]{x^2+1} x^{2/3} \log \left (\left (1-x^{2/3}\right )^2 \left (x^{2/3}+1\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{x^4+x^2}}-\frac {\sqrt [3]{x^2+1} x^{2/3} \log \left (\frac {2^{2/3} \left (x^{2/3}+1\right )^2}{\left (x^2+1\right )^{2/3}}-\frac {\sqrt [3]{2} \left (x^{2/3}+1\right )}{\sqrt [3]{x^2+1}}+1\right )}{12 \sqrt [3]{2} \sqrt [3]{x^4+x^2}}+\frac {\sqrt [3]{x^2+1} x^{2/3} \log \left (\frac {\sqrt [3]{2} \left (x^{2/3}+1\right )}{\sqrt [3]{x^2+1}}+1\right )}{6 \sqrt [3]{2} \sqrt [3]{x^4+x^2}}+\frac {\sqrt [3]{x^2+1} x^{2/3} \log \left (x^{2/3}-2^{2/3} \sqrt [3]{x^2+1}+1\right )}{8 \sqrt [3]{2} \sqrt [3]{x^4+x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/10*(x^4*(1 + x^2)^(1/3)*AppellF1[5/3, 1/3, 1, 8/3, -x^2, -((-1)^(1/3)*x^2)])/(x^2 + x^4)^(1/3) - (x^4*(1 +
x^2)^(1/3)*AppellF1[5/3, 1/3, 1, 8/3, -x^2, (-1)^(2/3)*x^2])/(10*(x^2 + x^4)^(1/3)) - (x^(2/3)*(1 + x^2)^(1/3)
*ArcTan[(1 - (2*2^(1/3)*(1 + x^(2/3)))/(1 + x^2)^(1/3))/Sqrt[3]])/(2*2^(1/3)*Sqrt[3]*(x^2 + x^4)^(1/3)) - (x^(
2/3)*(1 + x^2)^(1/3)*ArcTan[(1 + (2^(1/3)*(1 + x^(2/3)))/(1 + x^2)^(1/3))/Sqrt[3]])/(4*2^(1/3)*Sqrt[3]*(x^2 +
x^4)^(1/3)) + (x^2*(1 + x^2)^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, -x^2])/(4*(x^2 + x^4)^(1/3)) - (x^(2/3)*(1
+ x^2)^(1/3)*Log[(1 - x^(2/3))^2*(1 + x^(2/3))])/(24*2^(1/3)*(x^2 + x^4)^(1/3)) - (x^(2/3)*(1 + x^2)^(1/3)*Lo
g[1 + (2^(2/3)*(1 + x^(2/3))^2)/(1 + x^2)^(2/3) - (2^(1/3)*(1 + x^(2/3)))/(1 + x^2)^(1/3)])/(12*2^(1/3)*(x^2 +
x^4)^(1/3)) + (x^(2/3)*(1 + x^2)^(1/3)*Log[1 + (2^(1/3)*(1 + x^(2/3)))/(1 + x^2)^(1/3)])/(6*2^(1/3)*(x^2 + x^
4)^(1/3)) + (x^(2/3)*(1 + x^2)^(1/3)*Log[1 + x^(2/3) - 2^(2/3)*(1 + x^2)^(1/3)])/(8*2^(1/3)*(x^2 + x^4)^(1/3))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 206

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

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 476

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = GCD[m + 1,
n]}, Dist[1/k, Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k], x] /; k != 1] /;
FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]

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 494

Int[(((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_)^(n_))^(q_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Dist[e^n/b, Int[
(e*x)^(m - n)*(c + d*x^n)^q, x], x] - Dist[a*(e^n/b), Int[(e*x)^(m - n)*((c + d*x^n)^q/(a + b*x^n)), x], x] /;
FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LeQ[n, m, 2*n - 1] && IntBinomialQ[a, b
, c, d, e, m, n, -1, q, x]

Rule 502

Int[(x_)/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[b/a, 3]}, Dist[-q^2/(3
*d), Int[1/((1 - q*x)*(a + b*x^3)^(1/3)), x], x] + Dist[q/d, Subst[Int[1/(1 + 2*a*x^3), x], x, (1 + q*x)/(a +
b*x^3)^(1/3)], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b*c + a*d, 0]

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 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 973

Int[(((g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> Dist[d*((g*x)^n/x^n), In
t[(x^n*(a + c*x^2)^p)/(d^2 - e^2*x^2), x], x] - Dist[e*((g*x)^n/x^n), Int[(x^(n + 1)*(a + c*x^2)^p)/(d^2 - e^2
*x^2), x], x] /; FreeQ[{a, c, d, e, g, n, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] &&  !IntegersQ[n, 2
*p]

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 2174

Int[1/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(1/3)), x_Symbol] :> Simp[Sqrt[3]*(ArcTan[(1 - 2^(1/3)*Rt[b,
3]*((c - d*x)/(d*(a + b*x^3)^(1/3))))/Sqrt[3]]/(2^(4/3)*Rt[b, 3]*c)), x] + (Simp[Log[(c + d*x)^2*(c - d*x)]/(2
^(7/3)*Rt[b, 3]*c), x] - Simp[(3*Log[Rt[b, 3]*(c - d*x) + 2^(2/3)*d*(a + b*x^3)^(1/3)])/(2^(7/3)*Rt[b, 3]*c),
x]) /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^3 + a*d^3, 0]

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 {x^3}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {x^{7/3}}{\sqrt [3]{1+x^2} \left (-1+x^6\right )} \, dx}{\sqrt [3]{x^2+x^4}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \left (-\frac {x^{7/3}}{2 \sqrt [3]{1+x^2} \left (1-x^3\right )}-\frac {x^{7/3}}{2 \sqrt [3]{1+x^2} \left (1+x^3\right )}\right ) \, dx}{\sqrt [3]{x^2+x^4}}\\ &=-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {x^{7/3}}{\sqrt [3]{1+x^2} \left (1-x^3\right )} \, dx}{2 \sqrt [3]{x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {x^{7/3}}{\sqrt [3]{1+x^2} \left (1+x^3\right )} \, dx}{2 \sqrt [3]{x^2+x^4}}\\ &=-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \left (-\frac {x^{7/3}}{3 (-1-x) \sqrt [3]{1+x^2}}-\frac {x^{7/3}}{3 \left (-1+\sqrt [3]{-1} x\right ) \sqrt [3]{1+x^2}}-\frac {x^{7/3}}{3 \left (-1-(-1)^{2/3} x\right ) \sqrt [3]{1+x^2}}\right ) \, dx}{2 \sqrt [3]{x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \left (\frac {x^{7/3}}{3 (1-x) \sqrt [3]{1+x^2}}+\frac {x^{7/3}}{3 \left (1+\sqrt [3]{-1} x\right ) \sqrt [3]{1+x^2}}+\frac {x^{7/3}}{3 \left (1-(-1)^{2/3} x\right ) \sqrt [3]{1+x^2}}\right ) \, dx}{2 \sqrt [3]{x^2+x^4}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {x^{7/3}}{(-1-x) \sqrt [3]{1+x^2}} \, dx}{6 \sqrt [3]{x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {x^{7/3}}{(1-x) \sqrt [3]{1+x^2}} \, dx}{6 \sqrt [3]{x^2+x^4}}+\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {x^{7/3}}{\left (-1+\sqrt [3]{-1} x\right ) \sqrt [3]{1+x^2}} \, dx}{6 \sqrt [3]{x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {x^{7/3}}{\left (1+\sqrt [3]{-1} x\right ) \sqrt [3]{1+x^2}} \, dx}{6 \sqrt [3]{x^2+x^4}}+\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {x^{7/3}}{\left (-1-(-1)^{2/3} x\right ) \sqrt [3]{1+x^2}} \, dx}{6 \sqrt [3]{x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {x^{7/3}}{\left (1-(-1)^{2/3} x\right ) \sqrt [3]{1+x^2}} \, dx}{6 \sqrt [3]{x^2+x^4}}\\ &=-2 \frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {x^{7/3}}{\left (1-x^2\right ) \sqrt [3]{1+x^2}} \, dx}{6 \sqrt [3]{x^2+x^4}}-2 \frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {x^{7/3}}{\sqrt [3]{1+x^2} \left (1+\sqrt [3]{-1} x^2\right )} \, dx}{6 \sqrt [3]{x^2+x^4}}-2 \frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {x^{7/3}}{\sqrt [3]{1+x^2} \left (1-(-1)^{2/3} x^2\right )} \, dx}{6 \sqrt [3]{x^2+x^4}}\\ &=-2 \frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {x^9}{\left (1-x^6\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x^2+x^4}}-2 \frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {x^9}{\sqrt [3]{1+x^6} \left (1+\sqrt [3]{-1} x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x^2+x^4}}-2 \frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {x^9}{\sqrt [3]{1+x^6} \left (1-(-1)^{2/3} x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x^2+x^4}}\\ &=-2 \frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {x^4}{\left (1-x^3\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{x^2+x^4}}-2 \frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt [3]{1+x^3} \left (1+\sqrt [3]{-1} x^3\right )} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{x^2+x^4}}-2 \frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt [3]{1+x^3} \left (1-(-1)^{2/3} x^3\right )} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{x^2+x^4}}\\ &=-\frac {x^4 \sqrt [3]{1+x^2} F_1\left (\frac {5}{3};\frac {1}{3},1;\frac {8}{3};-x^2,-\sqrt [3]{-1} x^2\right )}{10 \sqrt [3]{x^2+x^4}}-\frac {x^4 \sqrt [3]{1+x^2} F_1\left (\frac {5}{3};\frac {1}{3},1;\frac {8}{3};-x^2,(-1)^{2/3} x^2\right )}{10 \sqrt [3]{x^2+x^4}}-\frac {x^4 \sqrt [3]{1+x^2} F_1\left (\frac {5}{3};1,\frac {1}{3};\frac {8}{3};x^2,-x^2\right )}{10 \sqrt [3]{x^2+x^4}}\\ \end {align*}

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Mathematica [A]
time = 0.75, size = 383, normalized size = 1.08 \begin {gather*} -\frac {x^{2/3} \sqrt [3]{1+x^2} \left (8 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \left (1+x^2\right )^{2/3}}\right )-2\ 2^{2/3} \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+\sqrt [3]{2} \left (1+x^2\right )^{2/3}}\right )+8 \log \left (-\sqrt [3]{x}+\sqrt [3]{1+x^2}\right )+8 \log \left (\sqrt [3]{x}+\sqrt [3]{1+x^2}\right )-2\ 2^{2/3} \log \left (-2 \sqrt [3]{x}+2^{2/3} \sqrt [3]{1+x^2}\right )-2\ 2^{2/3} \log \left (2 \sqrt [3]{x}+2^{2/3} \sqrt [3]{1+x^2}\right )-4 \log \left (x^{2/3}-\sqrt [3]{x} \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right )-4 \log \left (x^{2/3}+\sqrt [3]{x} \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right )+2^{2/3} \log \left (-2 x^{2/3}+2^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}-\sqrt [3]{2} \left (1+x^2\right )^{2/3}\right )+2^{2/3} \log \left (2 x^{2/3}+2^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}+\sqrt [3]{2} \left (1+x^2\right )^{2/3}\right )\right )}{48 \sqrt [3]{x^2+x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/48*(x^(2/3)*(1 + x^2)^(1/3)*(8*Sqrt[3]*ArcTan[(Sqrt[3]*x^(2/3))/(x^(2/3) + 2*(1 + x^2)^(2/3))] - 2*2^(2/3)*
Sqrt[3]*ArcTan[(Sqrt[3]*x^(2/3))/(x^(2/3) + 2^(1/3)*(1 + x^2)^(2/3))] + 8*Log[-x^(1/3) + (1 + x^2)^(1/3)] + 8*
Log[x^(1/3) + (1 + x^2)^(1/3)] - 2*2^(2/3)*Log[-2*x^(1/3) + 2^(2/3)*(1 + x^2)^(1/3)] - 2*2^(2/3)*Log[2*x^(1/3)
+ 2^(2/3)*(1 + x^2)^(1/3)] - 4*Log[x^(2/3) - x^(1/3)*(1 + x^2)^(1/3) + (1 + x^2)^(2/3)] - 4*Log[x^(2/3) + x^(
1/3)*(1 + x^2)^(1/3) + (1 + x^2)^(2/3)] + 2^(2/3)*Log[-2*x^(2/3) + 2^(2/3)*x^(1/3)*(1 + x^2)^(1/3) - 2^(1/3)*(
1 + x^2)^(2/3)] + 2^(2/3)*Log[2*x^(2/3) + 2^(2/3)*x^(1/3)*(1 + x^2)^(1/3) + 2^(1/3)*(1 + x^2)^(2/3)]))/(x^2 +
x^4)^(1/3)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 26.61, size = 2435, normalized size = 6.84

method	result	size
trager	\(\text {Expression too large to display}\)	\(2435\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(x^4+x^2)^(1/3)/(x^6-1),x,method=_RETURNVERBOSE)

[Out]

1/24*RootOf(_Z^3-4)*ln(-(3586488*RootOf(_Z^3-4)^3*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*x^4-189
31536*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4)^2*x^4-15242574*RootOf(RootOf(_Z^3-
4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^3*x^2+80459028*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36
*_Z^2)^2*RootOf(_Z^3-4)^2*x^2+104139450*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^2*
(x^4+x^2)^(2/3)-17356575*RootOf(_Z^3-4)^2*(x^4+x^2)^(1/3)*x^2-44008128*RootOf(_Z^3-4)*RootOf(RootOf(_Z^3-4)^2+
6*_Z*RootOf(_Z^3-4)+36*_Z^2)*(x^4+x^2)^(1/3)*x^2+6126917*RootOf(_Z^3-4)*x^4-32341374*RootOf(RootOf(_Z^3-4)^2+6
*_Z*RootOf(_Z^3-4)+36*_Z^2)*x^4+3586488*RootOf(_Z^3-4)^3*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)-
18931536*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4)^2-17356575*RootOf(_Z^3-4)^2*(x^
4+x^2)^(1/3)-44008128*(x^4+x^2)^(1/3)*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)+2181
7802*RootOf(_Z^3-4)*x^2-115166844*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*x^2+40087548*(x^4+x^2)^
(2/3)+6126917*RootOf(_Z^3-4)-32341374*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2))/(1+x)^2/(-1+x)^2)+
1/4*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*ln(-(1051752*RootOf(_Z^3-4)^3*RootOf(RootOf(_Z^3-4)^2
+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*x^4-7172976*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4
)^2*x^4-4469946*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^3*x^2+30485148*RootOf(Root
Of(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4)^2*x^2-34713150*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(
_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^2*(x^4+x^2)^(2/3)+5785525*RootOf(_Z^3-4)^2*(x^4+x^2)^(1/3)*x^2+20043774*RootOf
(_Z^3-4)*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*(x^4+x^2)^(1/3)*x^2-1095575*RootOf(_Z^3-4)*x^4+7
471850*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*x^4+1051752*RootOf(_Z^3-4)^3*RootOf(RootOf(_Z^3-4)
^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)-7172976*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4)^
2+5785525*RootOf(_Z^3-4)^2*(x^4+x^2)^(1/3)[phone]*(x^4+x^2)^(1/3)*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4
)+36*_Z^2)*RootOf(_Z^3-4)-9378122*RootOf(_Z^3-4)*x^2+63959036*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_
Z^2)*x^2-9779584*(x^4+x^2)^(2/3)-1095575*RootOf(_Z^3-4)+7471850*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36
*_Z^2))/(1+x)^2/(-1+x)^2)+1/6*ln(-(1340172*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-
4)^4*x^4-5695731*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4)^4*x^2-5235654*RootOf(Ro
otOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^2*x^4+1340172*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_
Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4)^4-9200025*RootOf(_Z^3-4)^2*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^
2)*(x^4+x^2)^(1/3)*x^2-9200025*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^2*(x^4+x^2)
^(2/3)-19126281*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^2*x^2-9200025*RootOf(RootO
f(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^2*(x^4+x^2)^(1/3)-1451650*x^4-5235654*RootOf(_Z^3-4)^2
*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)-8496*(x^4+x^2)^(1/3)*x^2-8496*(x^4+x^2)^(2/3)-4587214*x^
2-8496*(x^4+x^2)^(1/3)-1451650)/(x^2+x+1)/(x^2-x+1))+1/4*ln(-(1340172*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3
-4)+36*_Z^2)^2*RootOf(_Z^3-4)^4*x^4-5695731*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3
-4)^4*x^2-5235654*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^2*x^4+1340172*RootOf(Roo
tOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4)^4-9200025*RootOf(_Z^3-4)^2*RootOf(RootOf(_Z^3-4)^2
+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*(x^4+x^2)^(1/3)*x^2-9200025*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)
*RootOf(_Z^3-4)^2*(x^4+x^2)^(2/3)-19126281*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)
^2*x^2-9200025*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^2*(x^4+x^2)^(1/3)-1451650*x
^4-5235654*RootOf(_Z^3-4)^2*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)-8496*(x^4+x^2)^(1/3)*x^2-8496
*(x^4+x^2)^(2/3)-4587214*x^2-8496*(x^4+x^2)^(1/3)-1451650)/(x^2+x+1)/(x^2-x+1))*RootOf(_Z^3-4)^2*RootOf(RootOf
(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)-1/4*RootOf(_Z^3-4)^2*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z
^2)*ln(-(1340172*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4)^4*x^4-5695731*RootOf(Ro
otOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4)^4*x^2+7022550*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf
(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^2*x^4+1340172*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_
Z^3-4)^4+9200025*RootOf(_Z^3-4)^2*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*(x^4+x^2)^(1/3)*x^2+920
0025*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^2*(x^4+x^2)^(2/3)[phone]*RootOf(Roo
tOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*Root...

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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^3/(x^4+x^2)^(1/3)/(x^6-1),x, algorithm="maxima")

[Out]

integrate(x^3/((x^6 - 1)*(x^4 + x^2)^(1/3)), x)

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Fricas [A]
time = 1.93, size = 418, normalized size = 1.17 \begin {gather*} -\frac {1}{72} \, \sqrt {6} 2^{\frac {1}{6}} \arctan \left (\frac {2^{\frac {1}{6}} {\left (24 \, \sqrt {6} 2^{\frac {2}{3}} {\left (x^{8} + 2 \, x^{6} - 6 \, x^{4} + 2 \, x^{2} + 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}} + \sqrt {6} 2^{\frac {1}{3}} {\left (x^{12} - 42 \, x^{10} - 417 \, x^{8} - 812 \, x^{6} - 417 \, x^{4} - 42 \, x^{2} + 1\right )} - 12 \, \sqrt {6} {\left (x^{10} + 33 \, x^{8} + 110 \, x^{6} + 110 \, x^{4} + 33 \, x^{2} + 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (x^{12} + 102 \, x^{10} + 447 \, x^{8} + 628 \, x^{6} + 447 \, x^{4} + 102 \, x^{2} + 1\right )}}\right ) - \frac {1}{144} \cdot 2^{\frac {2}{3}} \log \left (\frac {12 \cdot 2^{\frac {2}{3}} {\left (x^{4} + 4 \, x^{2} + 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (x^{8} + 32 \, x^{6} + 78 \, x^{4} + 32 \, x^{2} + 1\right )} + 6 \, {\left (x^{6} + 11 \, x^{4} + 11 \, x^{2} + 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}}}{x^{8} - 4 \, x^{6} + 6 \, x^{4} - 4 \, x^{2} + 1}\right ) + \frac {1}{72} \cdot 2^{\frac {2}{3}} \log \left (-\frac {2^{\frac {2}{3}} {\left (x^{4} - 2 \, x^{2} + 1\right )} - 6 \cdot 2^{\frac {1}{3}} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )} + 12 \, {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}}}{x^{4} - 2 \, x^{2} + 1}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (x^{2} + 1\right )} + 2 \, \sqrt {3} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}}}{3 \, {\left (x^{2} + 1\right )}}\right ) - \frac {1}{12} \, \log \left (\frac {x^{4} + x^{2} - 3 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )} + 3 \, {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}} + 1}{x^{4} + x^{2} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(x^4+x^2)^(1/3)/(x^6-1),x, algorithm="fricas")

[Out]

-1/72*sqrt(6)*2^(1/6)*arctan(1/6*2^(1/6)*(24*sqrt(6)*2^(2/3)*(x^8 + 2*x^6 - 6*x^4 + 2*x^2 + 1)*(x^4 + x^2)^(2/
3) + sqrt(6)*2^(1/3)*(x^12 - 42*x^10 - 417*x^8 - 812*x^6 - 417*x^4 - 42*x^2 + 1) - 12*sqrt(6)*(x^10 + 33*x^8 +
110*x^6 + 110*x^4 + 33*x^2 + 1)*(x^4 + x^2)^(1/3))/(x^12 + 102*x^10 + 447*x^8 + 628*x^6 + 447*x^4 + 102*x^2 +
1)) - 1/144*2^(2/3)*log((12*2^(2/3)*(x^4 + 4*x^2 + 1)*(x^4 + x^2)^(2/3) + 2^(1/3)*(x^8 + 32*x^6 + 78*x^4 + 32
*x^2 + 1) + 6*(x^6 + 11*x^4 + 11*x^2 + 1)*(x^4 + x^2)^(1/3))/(x^8 - 4*x^6 + 6*x^4 - 4*x^2 + 1)) + 1/72*2^(2/3)
*log(-(2^(2/3)*(x^4 - 2*x^2 + 1) - 6*2^(1/3)*(x^4 + x^2)^(1/3)*(x^2 + 1) + 12*(x^4 + x^2)^(2/3))/(x^4 - 2*x^2
+ 1)) - 1/6*sqrt(3)*arctan(1/3*(sqrt(3)*(x^2 + 1) + 2*sqrt(3)*(x^4 + x^2)^(1/3))/(x^2 + 1)) - 1/12*log((x^4 +
x^2 - 3*(x^4 + x^2)^(1/3)*(x^2 + 1) + 3*(x^4 + x^2)^(2/3) + 1)/(x^4 + x^2 + 1))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{\sqrt [3]{x^{2} \left (x^{2} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(x**4+x**2)**(1/3)/(x**6-1),x)

[Out]

Integral(x**3/((x**2*(x**2 + 1))**(1/3)*(x - 1)*(x + 1)*(x**2 - x + 1)*(x**2 + x + 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^3/(x^4+x^2)^(1/3)/(x^6-1),x, algorithm="giac")

[Out]

integrate(x^3/((x^6 - 1)*(x^4 + x^2)^(1/3)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^3}{{\left (x^4+x^2\right )}^{1/3}\,\left (x^6-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/((x^2 + x^4)^(1/3)*(x^6 - 1)),x)

[Out]

int(x^3/((x^2 + x^4)^(1/3)*(x^6 - 1)), x)

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