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

Optimal. Leaf size=328 \[ \frac {1}{6} \log \left (x^2+\left (x^4-x^2\right )^{2/3}\right )-\frac {1}{12} i \left (\sqrt {3}-i\right ) \log \left (-i \sqrt {3} x^2-x^2+2 \left (x^4-x^2\right )^{2/3}\right )+\frac {1}{12} i \left (\sqrt {3}+i\right ) \log \left (i \sqrt {3} x^2-x^2+2 \left (x^4-x^2\right )^{2/3}\right )+\frac {\log \left (-2 x^2+2^{2/3} \sqrt {3} \sqrt [3]{x^4-x^2} x-\sqrt [3]{2} \left (x^4-x^2\right )^{2/3}\right )}{24 \sqrt [3]{2}}-\frac {\log \left (2 x^2+\sqrt [3]{2} \left (x^4-x^2\right )^{2/3}\right )}{12 \sqrt [3]{2}}+\frac {\log \left (2 x^2+2^{2/3} \sqrt {3} \sqrt [3]{x^4-x^2} x+\sqrt [3]{2} \left (x^4-x^2\right )^{2/3}\right )}{24 \sqrt [3]{2}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x^2}{\sqrt [3]{2} \left (x^4-x^2\right )^{2/3}-x^2}\right )}{4 \sqrt [3]{2} \sqrt {3}} \]

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Rubi [C]  time = 2.09, antiderivative size = 160, normalized size of antiderivative = 0.49, number of steps used = 55, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2056, 6725, 959, 466, 465, 511, 510} \begin {gather*} \frac {\sqrt [3]{1-x^2} x^4 F_1\left (\frac {5}{3};\frac {1}{3},1;\frac {8}{3};x^2,-x^2\right )}{10 \sqrt [3]{x^4-x^2}}+\frac {\sqrt [3]{1-x^2} 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]{1-x^2} 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}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x^4*(1 - x^2)^(1/3)*AppellF1[5/3, 1/3, 1, 8/3, x^2, -x^2])/(10*(-x^2 + x^4)^(1/3)) + (x^4*(1 - x^2)^(1/3)*App
ellF1[5/3, 1/3, 1, 8/3, x^2, (-1)^(1/3)*x^2])/(10*(-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))

Rule 465

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 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 511

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPa
rt[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 959

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 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 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 {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 {i x^{7/3}}{2 \sqrt [3]{-1+x^2} \left (i-x^3\right )}+\frac {i x^{7/3}}{2 \sqrt [3]{-1+x^2} \left (i+x^3\right )}\right ) \, dx}{\sqrt [3]{-x^2+x^4}}\\ &=\frac {\left (i x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {x^{7/3}}{\sqrt [3]{-1+x^2} \left (i-x^3\right )} \, dx}{2 \sqrt [3]{-x^2+x^4}}+\frac {\left (i x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {x^{7/3}}{\sqrt [3]{-1+x^2} \left (i+x^3\right )} \, dx}{2 \sqrt [3]{-x^2+x^4}}\\ &=\frac {\left (i x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \left (-\frac {(-1)^{2/3} x^{7/3}}{3 \left (\sqrt [6]{-1}-x\right ) \sqrt [3]{-1+x^2}}-\frac {(-1)^{2/3} x^{7/3}}{3 \left (\sqrt [6]{-1}+\sqrt [3]{-1} x\right ) \sqrt [3]{-1+x^2}}-\frac {(-1)^{2/3} x^{7/3}}{3 \left (\sqrt [6]{-1}-(-1)^{2/3} x\right ) \sqrt [3]{-1+x^2}}\right ) \, dx}{2 \sqrt [3]{-x^2+x^4}}+\frac {\left (i x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \left (-\frac {\sqrt [3]{-1} x^{7/3}}{3 \left (-(-1)^{5/6}-x\right ) \sqrt [3]{-1+x^2}}-\frac {\sqrt [3]{-1} x^{7/3}}{3 \left (-(-1)^{5/6}+\sqrt [3]{-1} x\right ) \sqrt [3]{-1+x^2}}-\frac {\sqrt [3]{-1} x^{7/3}}{3 \left (-(-1)^{5/6}-(-1)^{2/3} x\right ) \sqrt [3]{-1+x^2}}\right ) \, dx}{2 \sqrt [3]{-x^2+x^4}}\\ &=\frac {\left (\sqrt [6]{-1} x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {x^{7/3}}{\left (\sqrt [6]{-1}-x\right ) \sqrt [3]{-1+x^2}} \, dx}{6 \sqrt [3]{-x^2+x^4}}+\frac {\left (\sqrt [6]{-1} x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {x^{7/3}}{\left (\sqrt [6]{-1}+\sqrt [3]{-1} x\right ) \sqrt [3]{-1+x^2}} \, dx}{6 \sqrt [3]{-x^2+x^4}}+\frac {\left (\sqrt [6]{-1} x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {x^{7/3}}{\left (\sqrt [6]{-1}-(-1)^{2/3} x\right ) \sqrt [3]{-1+x^2}} \, dx}{6 \sqrt [3]{-x^2+x^4}}-\frac {\left ((-1)^{5/6} x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {x^{7/3}}{\left (-(-1)^{5/6}-x\right ) \sqrt [3]{-1+x^2}} \, dx}{6 \sqrt [3]{-x^2+x^4}}-\frac {\left ((-1)^{5/6} x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {x^{7/3}}{\left (-(-1)^{5/6}+\sqrt [3]{-1} x\right ) \sqrt [3]{-1+x^2}} \, dx}{6 \sqrt [3]{-x^2+x^4}}-\frac {\left ((-1)^{5/6} x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {x^{7/3}}{\left (-(-1)^{5/6}-(-1)^{2/3} x\right ) \sqrt [3]{-1+x^2}} \, dx}{6 \sqrt [3]{-x^2+x^4}}\\ &=\frac {\left (i x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {x^{10/3}}{\sqrt [3]{-1+x^2} \left (-(-1)^{2/3}+\sqrt [3]{-1} x^2\right )} \, dx}{6 \sqrt [3]{-x^2+x^4}}-\frac {\left (i x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {x^{10/3}}{\sqrt [3]{-1+x^2} \left (\sqrt [3]{-1}-(-1)^{2/3} x^2\right )} \, dx}{6 \sqrt [3]{-x^2+x^4}}+\frac {\left (\sqrt [6]{-1} x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {x^{10/3}}{\left (\sqrt [3]{-1}-x^2\right ) \sqrt [3]{-1+x^2}} \, dx}{6 \sqrt [3]{-x^2+x^4}}-\frac {\left (\sqrt [6]{-1} x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {x^{10/3}}{\sqrt [3]{-1+x^2} \left (-(-1)^{2/3}-(-1)^{2/3} x^2\right )} \, dx}{6 \sqrt [3]{-x^2+x^4}}+\frac {\left (\sqrt [3]{-1} x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {x^{7/3}}{\left (\sqrt [3]{-1}-x^2\right ) \sqrt [3]{-1+x^2}} \, dx}{6 \sqrt [3]{-x^2+x^4}}+\frac {\left (\sqrt [3]{-1} x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {x^{7/3}}{\sqrt [3]{-1+x^2} \left (\sqrt [3]{-1}+\sqrt [3]{-1} x^2\right )} \, dx}{6 \sqrt [3]{-x^2+x^4}}+\frac {\left (\sqrt [3]{-1} x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {x^{7/3}}{\sqrt [3]{-1+x^2} \left (\sqrt [3]{-1}-(-1)^{2/3} x^2\right )} \, dx}{6 \sqrt [3]{-x^2+x^4}}-\frac {\left ((-1)^{2/3} x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {x^{7/3}}{\left (-(-1)^{2/3}-x^2\right ) \sqrt [3]{-1+x^2}} \, dx}{6 \sqrt [3]{-x^2+x^4}}-\frac {\left ((-1)^{2/3} x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {x^{7/3}}{\sqrt [3]{-1+x^2} \left (-(-1)^{2/3}+\sqrt [3]{-1} x^2\right )} \, dx}{6 \sqrt [3]{-x^2+x^4}}-\frac {\left ((-1)^{2/3} x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {x^{7/3}}{\sqrt [3]{-1+x^2} \left (-(-1)^{2/3}-(-1)^{2/3} x^2\right )} \, dx}{6 \sqrt [3]{-x^2+x^4}}-\frac {\left ((-1)^{5/6} x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {x^{10/3}}{\left (-(-1)^{2/3}-x^2\right ) \sqrt [3]{-1+x^2}} \, dx}{6 \sqrt [3]{-x^2+x^4}}+\frac {\left ((-1)^{5/6} x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {x^{10/3}}{\sqrt [3]{-1+x^2} \left (\sqrt [3]{-1}+\sqrt [3]{-1} x^2\right )} \, dx}{6 \sqrt [3]{-x^2+x^4}}\\ &=\frac {\left (i x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x^{12}}{\sqrt [3]{-1+x^6} \left (-(-1)^{2/3}+\sqrt [3]{-1} x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{-x^2+x^4}}-\frac {\left (i x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x^{12}}{\sqrt [3]{-1+x^6} \left (\sqrt [3]{-1}-(-1)^{2/3} x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{-x^2+x^4}}+\frac {\left (\sqrt [6]{-1} x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x^{12}}{\left (\sqrt [3]{-1}-x^6\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{-x^2+x^4}}-\frac {\left (\sqrt [6]{-1} x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x^{12}}{\sqrt [3]{-1+x^6} \left (-(-1)^{2/3}-(-1)^{2/3} x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{-x^2+x^4}}+\frac {\left (\sqrt [3]{-1} x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x^9}{\left (\sqrt [3]{-1}-x^6\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{-x^2+x^4}}+\frac {\left (\sqrt [3]{-1} x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x^9}{\sqrt [3]{-1+x^6} \left (\sqrt [3]{-1}+\sqrt [3]{-1} x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{-x^2+x^4}}+\frac {\left (\sqrt [3]{-1} x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x^9}{\sqrt [3]{-1+x^6} \left (\sqrt [3]{-1}-(-1)^{2/3} x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{-x^2+x^4}}-\frac {\left ((-1)^{2/3} x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x^9}{\left (-(-1)^{2/3}-x^6\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{-x^2+x^4}}-\frac {\left ((-1)^{2/3} x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x^9}{\sqrt [3]{-1+x^6} \left (-(-1)^{2/3}+\sqrt [3]{-1} x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{-x^2+x^4}}-\frac {\left ((-1)^{2/3} x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x^9}{\sqrt [3]{-1+x^6} \left (-(-1)^{2/3}-(-1)^{2/3} x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{-x^2+x^4}}-\frac {\left ((-1)^{5/6} x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x^{12}}{\left (-(-1)^{2/3}-x^6\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{-x^2+x^4}}+\frac {\left ((-1)^{5/6} x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x^{12}}{\sqrt [3]{-1+x^6} \left (\sqrt [3]{-1}+\sqrt [3]{-1} x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{-x^2+x^4}}\\ &=\frac {\left (i x^{2/3} \sqrt [3]{1-x^2}\right ) \operatorname {Subst}\left (\int \frac {x^{12}}{\sqrt [3]{1-x^6} \left (-(-1)^{2/3}+\sqrt [3]{-1} x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{-x^2+x^4}}-\frac {\left (i x^{2/3} \sqrt [3]{1-x^2}\right ) \operatorname {Subst}\left (\int \frac {x^{12}}{\sqrt [3]{1-x^6} \left (\sqrt [3]{-1}-(-1)^{2/3} x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{-x^2+x^4}}+\frac {\left (\sqrt [6]{-1} x^{2/3} \sqrt [3]{1-x^2}\right ) \operatorname {Subst}\left (\int \frac {x^{12}}{\sqrt [3]{1-x^6} \left (\sqrt [3]{-1}-x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{-x^2+x^4}}-\frac {\left (\sqrt [6]{-1} x^{2/3} \sqrt [3]{1-x^2}\right ) \operatorname {Subst}\left (\int \frac {x^{12}}{\sqrt [3]{1-x^6} \left (-(-1)^{2/3}-(-1)^{2/3} x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{-x^2+x^4}}-\frac {\left ((-1)^{5/6} x^{2/3} \sqrt [3]{1-x^2}\right ) \operatorname {Subst}\left (\int \frac {x^{12}}{\sqrt [3]{1-x^6} \left (-(-1)^{2/3}-x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{-x^2+x^4}}+\frac {\left ((-1)^{5/6} x^{2/3} \sqrt [3]{1-x^2}\right ) \operatorname {Subst}\left (\int \frac {x^{12}}{\sqrt [3]{1-x^6} \left (\sqrt [3]{-1}+\sqrt [3]{-1} x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{-x^2+x^4}}+\frac {\left (\sqrt [3]{-1} x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (\sqrt [3]{-1}-x^3\right ) \sqrt [3]{-1+x^3}} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{-x^2+x^4}}+\frac {\left (\sqrt [3]{-1} x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [3]{-1+x^3} \left (\sqrt [3]{-1}+\sqrt [3]{-1} x^3\right )} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{-x^2+x^4}}+\frac {\left (\sqrt [3]{-1} x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [3]{-1+x^3} \left (\sqrt [3]{-1}-(-1)^{2/3} x^3\right )} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{-x^2+x^4}}-\frac {\left ((-1)^{2/3} x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (-(-1)^{2/3}-x^3\right ) \sqrt [3]{-1+x^3}} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{-x^2+x^4}}-\frac {\left ((-1)^{2/3} x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [3]{-1+x^3} \left (-(-1)^{2/3}+\sqrt [3]{-1} x^3\right )} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{-x^2+x^4}}-\frac {\left ((-1)^{2/3} x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [3]{-1+x^3} \left (-(-1)^{2/3}-(-1)^{2/3} x^3\right )} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{-x^2+x^4}}\\ &=\frac {\left (\sqrt [3]{-1} x^{2/3} \sqrt [3]{1-x^2}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [3]{1-x^3} \left (\sqrt [3]{-1}-x^3\right )} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{-x^2+x^4}}+\frac {\left (\sqrt [3]{-1} x^{2/3} \sqrt [3]{1-x^2}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [3]{1-x^3} \left (\sqrt [3]{-1}+\sqrt [3]{-1} x^3\right )} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{-x^2+x^4}}+\frac {\left (\sqrt [3]{-1} x^{2/3} \sqrt [3]{1-x^2}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [3]{1-x^3} \left (\sqrt [3]{-1}-(-1)^{2/3} x^3\right )} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{-x^2+x^4}}-\frac {\left ((-1)^{2/3} x^{2/3} \sqrt [3]{1-x^2}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [3]{1-x^3} \left (-(-1)^{2/3}-x^3\right )} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{-x^2+x^4}}-\frac {\left ((-1)^{2/3} x^{2/3} \sqrt [3]{1-x^2}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [3]{1-x^3} \left (-(-1)^{2/3}+\sqrt [3]{-1} x^3\right )} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{-x^2+x^4}}-\frac {\left ((-1)^{2/3} x^{2/3} \sqrt [3]{1-x^2}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [3]{1-x^3} \left (-(-1)^{2/3}-(-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,-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,\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}}\\ \end {align*}

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Mathematica [F]  time = 1.79, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3}{\sqrt [3]{-x^2+x^4} \left (1+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.65, size = 312, normalized size = 0.95 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {3} x^2}{-x^2+2 \left (-x^2+x^4\right )^{2/3}}\right )}{2 \sqrt {3}}+\frac {\tan ^{-1}\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^2+\left (-x^2+x^4\right )^{2/3}\right )+\frac {\log \left (-2 x^2+2^{2/3} \sqrt {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+\sqrt [3]{2} \left (-x^2+x^4\right )^{2/3}\right )}{12 \sqrt [3]{2}}+\frac {\log \left (2 x^2+2^{2/3} \sqrt {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 {1}{12} \log \left (x^4-x^2 \left (-x^2+x^4\right )^{2/3}+\left (-x^2+x^4\right )^{4/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*ArcTan[(Sqrt[3]*x^2)/(-x^2 + 2*(-x^2 + x^4)^(2/3))]/Sqrt[3] + ArcTan[(Sqrt[3]*x^2)/(-x^2 + 2^(1/3)*(-x^2
+ x^4)^(2/3))]/(4*2^(1/3)*Sqrt[3]) + Log[x^2 + (-x^2 + x^4)^(2/3)]/6 + Log[-2*x^2 + 2^(2/3)*Sqrt[3]*x*(-x^2 +
x^4)^(1/3) - 2^(1/3)*(-x^2 + x^4)^(2/3)]/(24*2^(1/3)) - Log[2*x^2 + 2^(1/3)*(-x^2 + x^4)^(2/3)]/(12*2^(1/3)) +
Log[2*x^2 + 2^(2/3)*Sqrt[3]*x*(-x^2 + x^4)^(1/3) + 2^(1/3)*(-x^2 + x^4)^(2/3)]/(24*2^(1/3)) - Log[x^4 - x^2*(
-x^2 + x^4)^(2/3) + (-x^2 + x^4)^(4/3)]/12

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fricas [A]  time = 2.80, size = 470, normalized size = 1.43 \begin {gather*} -\frac {1}{72} \, \sqrt {6} 2^{\frac {1}{6}} \left (-1\right )^{\frac {1}{3}} \arctan \left (\frac {2^{\frac {1}{6}} {\left (24 \, \sqrt {6} 2^{\frac {2}{3}} \left (-1\right )^{\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}} - 12 \, \sqrt {6} \left (-1\right )^{\frac {1}{3}} {\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}} + \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 )}\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}} \left (-1\right )^{\frac {1}{3}} \log \left (-\frac {12 \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{4} - x^{2}\right )}^{\frac {2}{3}} {\left (x^{4} - 4 \, x^{2} + 1\right )} - 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{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}} \left (-1\right )^{\frac {1}{3}} \log \left (-\frac {6 \cdot 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{4} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{2} - 1\right )} - 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{4} + 2 \, 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)*(-1)^(1/3)*arctan(1/6*2^(1/6)*(24*sqrt(6)*2^(2/3)*(-1)^(2/3)*(x^8 - 2*x^6 - 6*x^4 - 2*x^
2 + 1)*(x^4 - x^2)^(2/3) - 12*sqrt(6)*(-1)^(1/3)*(x^10 - 33*x^8 + 110*x^6 - 110*x^4 + 33*x^2 - 1)*(x^4 - x^2)^
(1/3) + sqrt(6)*2^(1/3)*(x^12 + 42*x^10 - 417*x^8 + 812*x^6 - 417*x^4 + 42*x^2 + 1))/(x^12 - 102*x^10 + 447*x^
8 - 628*x^6 + 447*x^4 - 102*x^2 + 1)) - 1/144*2^(2/3)*(-1)^(1/3)*log(-(12*2^(2/3)*(-1)^(1/3)*(x^4 - x^2)^(2/3)
*(x^4 - 4*x^2 + 1) - 2^(1/3)*(-1)^(2/3)*(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)*(-1)^(1/3)*log(-(6*2^(1/3)*(-1)^(2/3)*(x^4
- x^2)^(1/3)*(x^2 - 1) - 2^(2/3)*(-1)^(1/3)*(x^4 + 2*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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{{\left (x^{6} + 1\right )} {\left (x^{4} - x^{2}\right )}^{\frac {1}{3}}}\,{d x} \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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maple [C]  time = 52.74, size = 2469, normalized size = 7.53

method	result	size
trager	\(\text {Expression too large to display}\)	\(2469\)

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]

1/4*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*ln((8763000*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+
4)+36*_Z^2)*RootOf(_Z^3+4)^3*x^4+347184*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)^2*RootOf(_Z^3+4)^
2*x^4-37242750*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)^3*x^2-1475532*RootOf(RootOf
(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)^2*RootOf(_Z^3+4)^2*x^2+50416182*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)-8402697*(x^4-x^2)^(1/3)*RootOf(_Z^3+4)^2*x^2+4019850*(x^4-x^2)
^(1/3)*RootOf(_Z^3+4)*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*x^2-3286125*RootOf(_Z^3+4)*x^4-1301
94*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*x^4+8763000*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4
)+36*_Z^2)*RootOf(_Z^3+4)^3+347184*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)^2*RootOf(_Z^3+4)^2+840
2697*RootOf(_Z^3+4)^2*(x^4-x^2)^(1/3)-4019850*(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)-3651250*RootOf(_Z^3+4)*x^2-144660*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*x^2
-36290688*(x^4-x^2)^(2/3)-3286125*RootOf(_Z^3+4)-130194*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2))/
(x^2+1)^2)+1/24*RootOf(_Z^3+4)*ln(-(57864*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)^
3*x^4+52578000*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)^2*RootOf(_Z^3+4)^2*x^4-245922*RootOf(RootO
f(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)^3*x^2-223456500*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z
^3+4)+36*_Z^2)^2*RootOf(_Z^3+4)^2*x^2+50416182*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)-8402697*(x^4-x^2)^(1/3)*RootOf(_Z^3+4)^2*x^2-54436032*(x^4-x^2)^(1/3)*RootOf(_Z^3+4)*Ro
otOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*x^2-16877*RootOf(_Z^3+4)*x^4-15335250*RootOf(RootOf(_Z^3+4)
^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*x^4+57864*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)^
3+52578000*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)^2*RootOf(_Z^3+4)^2+8402697*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)+188
058*RootOf(_Z^3+4)*x^2+170878500*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*x^2+2679900*(x^4-x^2)^(2
/3)-16877*RootOf(_Z^3+4)-15335250*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2))/(x^2+1)^2)-1/4*RootOf(
RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)^2*ln((-445572*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf
(_Z^3+4)+36*_Z^2)^2*RootOf(_Z^3+4)^4*x^4+1893681*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)^2*RootOf
(_Z^3+4)^4*x^2+469206*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)^2*x^4-1692009*(x^4-x
^2)^(1/3)*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*x^2-445572*RootOf(RootOf(_Z^3+
4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)^2*RootOf(_Z^3+4)^4+1692009*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)-3126621*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+
4)^2*x^2+1692009*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)+700430*
x^4-783690*(x^4-x^2)^(1/3)*x^2+469206*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)^2+78
3690*(x^4-x^2)^(2/3)-840516*x^2+783690*(x^4-x^2)^(1/3)+700430)/(x^4-x^2+1))+1/4*ln(-(445572*RootOf(RootOf(_Z^3
+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)^2*RootOf(_Z^3+4)^4*x^4-1893681*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+
36*_Z^2)^2*RootOf(_Z^3+4)^4*x^2-124890*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)^2*x
^4-1692009*(x^4-x^2)^(1/3)*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*x^2+445572*Ro
otOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)^2*RootOf(_Z^3+4)^4+1692009*RootOf(RootOf(_Z^3+4)^2+6*_Z*Roo
tOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)^2*(x^4-x^2)^(2/3)-601713*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z
^2)*RootOf(_Z^3+4)^2*x^2+1692009*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)-815202*x^4+1911696*(x^4-x^2)^(1/3)*x^2-124890*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*Ro
otOf(_Z^3+4)^2-1911696*(x^4-x^2)^(2/3)+2083294*x^2-1911696*(x^4-x^2)^(1/3)-815202)/(x^4-x^2+1))*RootOf(RootOf(
_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)^2-1/6*ln(-(445572*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z
^3+4)+36*_Z^2)^2*RootOf(_Z^3+4)^4*x^4-1893681*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)^2*RootOf(_Z
^3+4)^4*x^2-124890*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)^2*x^4-1692009*(x^4-x^2)
^(1/3)*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*x^2+445572*RootOf(RootOf(_Z^3+4)^
2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)^2*RootOf(_Z^3+4)^4+1692009*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)-601713*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)^2
*x^2+1692009*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)-815202*x^4+
1911696*(x^4-x^2)^(1/3)*x^2-124890*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)^2-19116
96*(x^4-x^2)^(2/3)+2083294*x^2-1911696*(x^4-x^2)^(1/3)-815202)/(x^4-x^2+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{{\left (x^{6} + 1\right )} {\left (x^{4} - x^{2}\right )}^{\frac {1}{3}}}\,{d x} \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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^3}{\left (x^6+1\right )\,{\left (x^4-x^2\right )}^{1/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 - 1\right ) \left (x + 1\right )} \left (x^{2} + 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**3/(x**4-x**2)**(1/3)/(x**6+1),x)

[Out]

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

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