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

Optimal. Leaf size=234 \[ -\frac {1}{2} \log \left (\sqrt [3]{x^3-x}-x\right )-\frac {\log \left (2^{2/3} \sqrt [3]{x^3-x}-2 x\right )}{2^{2/3}}-\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-x}+x}\right )-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^3-x}+x}\right )}{2^{2/3}}-\frac {3 \sqrt [3]{x^3-x} \left (5 x^2-1\right )}{4 x^3}+\frac {1}{4} \log \left (\sqrt [3]{x^3-x} x+\left (x^3-x\right )^{2/3}+x^2\right )+\frac {\log \left (2^{2/3} \sqrt [3]{x^3-x} x+\sqrt [3]{2} \left (x^3-x\right )^{2/3}+2 x^2\right )}{2\ 2^{2/3}} \]

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Rubi [C]  time = 0.80, antiderivative size = 316, normalized size of antiderivative = 1.35, number of steps used = 18, number of rules used = 17, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.630, Rules used = {2056, 6725, 264, 277, 329, 275, 331, 292, 31, 634, 618, 204, 628, 466, 465, 511, 510} \begin {gather*} \frac {3 \sqrt [3]{x^3-x} \left (\left (1-3 x^2\right ) x^2 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {2 x^2}{1-x^2}\right )-3 \left (x^4+x^2\right ) \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};-\frac {2 x^2}{1-x^2}\right )+3 x^4-4 x^2+1\right )}{8 x^3 \left (1-x^2\right )}-\frac {3 \sqrt [3]{x^3-x}}{2 x}+\frac {3 \sqrt [3]{x^3-x} \left (1-x^2\right )}{8 x^3}-\frac {\sqrt [3]{x^3-x} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{x^2-1}}\right )}{2 \sqrt [3]{x} \sqrt [3]{x^2-1}}+\frac {\sqrt [3]{x^3-x} \log \left (\frac {x^{4/3}}{\left (x^2-1\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{x^2-1}}+1\right )}{4 \sqrt [3]{x} \sqrt [3]{x^2-1}}-\frac {\sqrt {3} \sqrt [3]{x^3-x} \tan ^{-1}\left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{x} \sqrt [3]{x^2-1}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-3*(-x + x^3)^(1/3))/(2*x) + (3*(1 - x^2)*(-x + x^3)^(1/3))/(8*x^3) - (Sqrt[3]*(-x + x^3)^(1/3)*ArcTan[(1 + (
2*x^(2/3))/(-1 + x^2)^(1/3))/Sqrt[3]])/(2*x^(1/3)*(-1 + x^2)^(1/3)) + (3*(-x + x^3)^(1/3)*(1 - 4*x^2 + 3*x^4 +
x^2*(1 - 3*x^2)*Hypergeometric2F1[2/3, 1, 5/3, (-2*x^2)/(1 - x^2)] - 3*(x^2 + x^4)*Hypergeometric2F1[2/3, 2,
5/3, (-2*x^2)/(1 - x^2)]))/(8*x^3*(1 - x^2)) - ((-x + x^3)^(1/3)*Log[1 - x^(2/3)/(-1 + x^2)^(1/3)])/(2*x^(1/3)
*(-1 + x^2)^(1/3)) + ((-x + x^3)^(1/3)*Log[1 + x^(4/3)/(-1 + x^2)^(2/3) + x^(2/3)/(-1 + x^2)^(1/3)])/(4*x^(1/3
)*(-1 + x^2)^(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 204

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

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 275

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

Rule 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 292

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> -Dist[(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(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 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 331

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[
p, -2^(-1)] && IntegersQ[m, p + (m + 1)/n]

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 618

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

Rule 628

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 634

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 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 {\sqrt [3]{-x+x^3} \left (-2+x^4\right )}{x^4 \left (1+x^2\right )} \, dx &=\frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{-1+x^2} \left (-2+x^4\right )}{x^{11/3} \left (1+x^2\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=\frac {\sqrt [3]{-x+x^3} \int \left (-\frac {\sqrt [3]{-1+x^2}}{x^{11/3}}+\frac {\sqrt [3]{-1+x^2}}{x^{5/3}}-\frac {\sqrt [3]{-1+x^2}}{x^{11/3} \left (1+x^2\right )}\right ) \, dx}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=-\frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{-1+x^2}}{x^{11/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{-1+x^2}}{x^{5/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}-\frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{-1+x^2}}{x^{11/3} \left (1+x^2\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=-\frac {3 \sqrt [3]{-x+x^3}}{2 x}+\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}+\frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{x}}{\left (-1+x^2\right )^{2/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}-\frac {\left (3 \sqrt [3]{-x+x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1+x^6}}{x^9 \left (1+x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=-\frac {3 \sqrt [3]{-x+x^3}}{2 x}+\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}-\frac {\left (3 \sqrt [3]{-x+x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1+x^3}}{x^5 \left (1+x^3\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {\left (3 \sqrt [3]{-x+x^3}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (-1+x^6\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=-\frac {3 \sqrt [3]{-x+x^3}}{2 x}+\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}-\frac {\left (3 \sqrt [3]{-x+x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1-x^3}}{x^5 \left (1+x^3\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1-x^2}}+\frac {\left (3 \sqrt [3]{-x+x^3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (-1+x^3\right )^{2/3}} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=-\frac {3 \sqrt [3]{-x+x^3}}{2 x}+\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}+\frac {3 \sqrt [3]{-x+x^3} \left (1-4 x^2+3 x^4+x^2 \left (1-3 x^2\right ) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {2 x^2}{1-x^2}\right )-3 \left (x^2+x^4\right ) \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};-\frac {2 x^2}{1-x^2}\right )\right )}{8 x^3 \left (1-x^2\right )}+\frac {\left (3 \sqrt [3]{-x+x^3}\right ) \operatorname {Subst}\left (\int \frac {x}{1-x^3} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{2 \sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=-\frac {3 \sqrt [3]{-x+x^3}}{2 x}+\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}+\frac {3 \sqrt [3]{-x+x^3} \left (1-4 x^2+3 x^4+x^2 \left (1-3 x^2\right ) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {2 x^2}{1-x^2}\right )-3 \left (x^2+x^4\right ) \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};-\frac {2 x^2}{1-x^2}\right )\right )}{8 x^3 \left (1-x^2\right )}+\frac {\sqrt [3]{-x+x^3} \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{2 \sqrt [3]{x} \sqrt [3]{-1+x^2}}-\frac {\sqrt [3]{-x+x^3} \operatorname {Subst}\left (\int \frac {1-x}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{2 \sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=-\frac {3 \sqrt [3]{-x+x^3}}{2 x}+\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}+\frac {3 \sqrt [3]{-x+x^3} \left (1-4 x^2+3 x^4+x^2 \left (1-3 x^2\right ) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {2 x^2}{1-x^2}\right )-3 \left (x^2+x^4\right ) \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};-\frac {2 x^2}{1-x^2}\right )\right )}{8 x^3 \left (1-x^2\right )}-\frac {\sqrt [3]{-x+x^3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{2 \sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {\sqrt [3]{-x+x^3} \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{4 \sqrt [3]{x} \sqrt [3]{-1+x^2}}-\frac {\left (3 \sqrt [3]{-x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{4 \sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=-\frac {3 \sqrt [3]{-x+x^3}}{2 x}+\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}+\frac {3 \sqrt [3]{-x+x^3} \left (1-4 x^2+3 x^4+x^2 \left (1-3 x^2\right ) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {2 x^2}{1-x^2}\right )-3 \left (x^2+x^4\right ) \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};-\frac {2 x^2}{1-x^2}\right )\right )}{8 x^3 \left (1-x^2\right )}-\frac {\sqrt [3]{-x+x^3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{2 \sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {\sqrt [3]{-x+x^3} \log \left (1+\frac {x^{4/3}}{\left (-1+x^2\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{4 \sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {\left (3 \sqrt [3]{-x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{2 \sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=-\frac {3 \sqrt [3]{-x+x^3}}{2 x}+\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}-\frac {\sqrt {3} \sqrt [3]{-x+x^3} \tan ^{-1}\left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{2 \sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {3 \sqrt [3]{-x+x^3} \left (1-4 x^2+3 x^4+x^2 \left (1-3 x^2\right ) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {2 x^2}{1-x^2}\right )-3 \left (x^2+x^4\right ) \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};-\frac {2 x^2}{1-x^2}\right )\right )}{8 x^3 \left (1-x^2\right )}-\frac {\sqrt [3]{-x+x^3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{2 \sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {\sqrt [3]{-x+x^3} \log \left (1+\frac {x^{4/3}}{\left (-1+x^2\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{4 \sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ \end {align*}

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Mathematica [C]  time = 0.12, size = 137, normalized size = 0.59 \begin {gather*} -\frac {3 \sqrt [3]{x \left (x^2-1\right )} \left (\sqrt [3]{1-x^2} \left (\left (3 x^2-1\right ) x^2 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {2 x^2}{x^2-1}\right )+3 \left (x^4+x^2\right ) \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {2 x^2}{x^2-1}\right )-4 x^4+6 x^2-2\right )-4 x^2 \left (x^2-1\right ) \, _2F_1\left (-\frac {1}{3},-\frac {1}{3};\frac {2}{3};x^2\right )\right )}{8 x^3 \left (1-x^2\right )^{4/3}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-3*(x*(-1 + x^2))^(1/3)*(-4*x^2*(-1 + x^2)*Hypergeometric2F1[-1/3, -1/3, 2/3, x^2] + (1 - x^2)^(1/3)*(-2 + 6*
x^2 - 4*x^4 + x^2*(-1 + 3*x^2)*Hypergeometric2F1[2/3, 1, 5/3, (2*x^2)/(-1 + x^2)] + 3*(x^2 + x^4)*Hypergeometr
ic2F1[2/3, 2, 5/3, (2*x^2)/(-1 + x^2)])))/(8*x^3*(1 - x^2)^(4/3))

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IntegrateAlgebraic [A]  time = 0.57, size = 234, normalized size = 1.00 \begin {gather*} -\frac {3 \left (-1+5 x^2\right ) \sqrt [3]{-x+x^3}}{4 x^3}-\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x+x^3}}\right )-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-x+x^3}}\right )}{2^{2/3}}-\frac {1}{2} \log \left (-x+\sqrt [3]{-x+x^3}\right )-\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{-x+x^3}\right )}{2^{2/3}}+\frac {1}{4} \log \left (x^2+x \sqrt [3]{-x+x^3}+\left (-x+x^3\right )^{2/3}\right )+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-x+x^3}+\sqrt [3]{2} \left (-x+x^3\right )^{2/3}\right )}{2\ 2^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(-1 + 5*x^2)*(-x + x^3)^(1/3))/(4*x^3) - (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-x + x^3)^(1/3))])/2 - (Sqrt[
3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(-x + x^3)^(1/3))])/2^(2/3) - Log[-x + (-x + x^3)^(1/3)]/2 - Log[-2*x + 2^(
2/3)*(-x + x^3)^(1/3)]/2^(2/3) + Log[x^2 + x*(-x + x^3)^(1/3) + (-x + x^3)^(2/3)]/4 + Log[2*x^2 + 2^(2/3)*x*(-
x + x^3)^(1/3) + 2^(1/3)*(-x + x^3)^(2/3)]/(2*2^(2/3))

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fricas [B]  time = 8.18, size = 387, normalized size = 1.65 \begin {gather*} -\frac {4 \cdot 4^{\frac {1}{6}} \sqrt {3} \left (-1\right )^{\frac {1}{3}} x^{3} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (6 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (19 \, x^{5} - 16 \, x^{3} + x\right )} {\left (x^{3} - x\right )}^{\frac {1}{3}} + 12 \, \left (-1\right )^{\frac {1}{3}} {\left (5 \, x^{4} + 4 \, x^{2} - 1\right )} {\left (x^{3} - x\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (71 \, x^{6} - 111 \, x^{4} + 33 \, x^{2} - 1\right )}\right )}}{6 \, {\left (109 \, x^{6} - 105 \, x^{4} + 3 \, x^{2} + 1\right )}}\right ) + 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{3} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{3} - x\right )}^{\frac {2}{3}} {\left (5 \, x^{2} - 1\right )} - 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (19 \, x^{4} - 16 \, x^{2} + 1\right )} + 24 \, {\left (2 \, x^{3} - x\right )} {\left (x^{3} - x\right )}^{\frac {1}{3}}}{x^{4} + 2 \, x^{2} + 1}\right ) - 2 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{3} \log \left (-\frac {3 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{3} - x\right )}^{\frac {1}{3}} x + 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{2} + 1\right )} + 6 \, {\left (x^{3} - x\right )}^{\frac {2}{3}}}{x^{2} + 1}\right ) + 12 \, \sqrt {3} x^{3} \arctan \left (-\frac {44032959556 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (16754327161 \, x^{2} - 2707204793\right )} - 10524305234 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {2}{3}}}{81835897185 \, x^{2} - 1102302937}\right ) + 6 \, x^{3} \log \left (-3 \, {\left (x^{3} - x\right )}^{\frac {1}{3}} x + 3 \, {\left (x^{3} - x\right )}^{\frac {2}{3}} + 1\right ) + 18 \, {\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (5 \, x^{2} - 1\right )}}{24 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(4*4^(1/6)*sqrt(3)*(-1)^(1/3)*x^3*arctan(1/6*4^(1/6)*sqrt(3)*(6*4^(2/3)*(-1)^(2/3)*(19*x^5 - 16*x^3 + x)
*(x^3 - x)^(1/3) + 12*(-1)^(1/3)*(5*x^4 + 4*x^2 - 1)*(x^3 - x)^(2/3) + 4^(1/3)*(71*x^6 - 111*x^4 + 33*x^2 - 1)
)/(109*x^6 - 105*x^4 + 3*x^2 + 1)) + 4^(2/3)*(-1)^(1/3)*x^3*log((6*4^(1/3)*(-1)^(2/3)*(x^3 - x)^(2/3)*(5*x^2 -
1) - 4^(2/3)*(-1)^(1/3)*(19*x^4 - 16*x^2 + 1) + 24*(2*x^3 - x)*(x^3 - x)^(1/3))/(x^4 + 2*x^2 + 1)) - 2*4^(2/3
)*(-1)^(1/3)*x^3*log(-(3*4^(2/3)*(-1)^(1/3)*(x^3 - x)^(1/3)*x + 4^(1/3)*(-1)^(2/3)*(x^2 + 1) + 6*(x^3 - x)^(2/
3))/(x^2 + 1)) + 12*sqrt(3)*x^3*arctan(-(44032959556*sqrt(3)*(x^3 - x)^(1/3)*x + sqrt(3)*(16754327161*x^2 - 27
07204793) - 10524305234*sqrt(3)*(x^3 - x)^(2/3))/(81835897185*x^2 - 1102302937)) + 6*x^3*log(-3*(x^3 - x)^(1/3
)*x + 3*(x^3 - x)^(2/3) + 1) + 18*(x^3 - x)^(1/3)*(5*x^2 - 1))/x^3

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giac [A]  time = 0.50, size = 171, normalized size = 0.73 \begin {gather*} \frac {1}{2} \, \sqrt {3} 2^{\frac {1}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}}\right )}\right ) + \frac {1}{2} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {3}{4} \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {4}{3}} + \frac {1}{4} \cdot 2^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}}\right ) - \frac {1}{2} \cdot 2^{\frac {1}{3}} \log \left ({\left | -2^{\frac {1}{3}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} \right |}\right ) - 3 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + \frac {1}{4} \, \log \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{2} \, \log \left ({\left | {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(3)*2^(1/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3) + 2*(-1/x^2 + 1)^(1/3))) + 1/2*sqrt(3)*arctan(1/3*sqrt
(3)*(2*(-1/x^2 + 1)^(1/3) + 1)) - 3/4*(-1/x^2 + 1)^(4/3) + 1/4*2^(1/3)*log(2^(2/3) + 2^(1/3)*(-1/x^2 + 1)^(1/3
) + (-1/x^2 + 1)^(2/3)) - 1/2*2^(1/3)*log(abs(-2^(1/3) + (-1/x^2 + 1)^(1/3))) - 3*(-1/x^2 + 1)^(1/3) + 1/4*log
((-1/x^2 + 1)^(2/3) + (-1/x^2 + 1)^(1/3) + 1) - 1/2*log(abs((-1/x^2 + 1)^(1/3) - 1))

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{3}-x \right )^{\frac {1}{3}} \left (x^{4}-2\right )}{x^{4} \left (x^{2}+1\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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