\(\int \frac {\sqrt [3]{-x+x^3} (-2+x^4)}{x^4 (1+x^2)} \, dx\) [2639]

Optimal result

Integrand size = 27, antiderivative size = 234 \[ \int \frac {\sqrt [3]{-x+x^3} \left (-2+x^4\right )}{x^4 \left (1+x^2\right )} \, dx=-\frac {3 \left (-1+5 x^2\right ) \sqrt [3]{-x+x^3}}{4 x^3}-\frac {1}{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x+x^3}}\right )-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-x+x^3}}\right )}{2^{2/3}}-\frac {1}{2} \log \left (-x+\sqrt [3]{-x+x^3}\right )-\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{-x+x^3}\right )}{2^{2/3}}+\frac {1}{4} \log \left (x^2+x \sqrt [3]{-x+x^3}+\left (-x+x^3\right )^{2/3}\right )+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-x+x^3}+\sqrt [3]{2} \left (-x+x^3\right )^{2/3}\right )}{2\ 2^{2/3}} \]

[Out]

-3/4*(5*x^2-1)*(x^3-x)^(1/3)/x^3-1/2*arctan(3^(1/2)*x/(x+2*(x^3-x)^(1/3)))*3^(1/2)-1/2*3^(1/2)*arctan(3^(1/2)*
x/(x+2^(2/3)*(x^3-x)^(1/3)))*2^(1/3)-1/2*ln(-x+(x^3-x)^(1/3))-1/2*ln(-2*x+2^(2/3)*(x^3-x)^(1/3))*2^(1/3)+1/4*l
n(x^2+x*(x^3-x)^(1/3)+(x^3-x)^(2/3))+1/4*ln(2*x^2+2^(2/3)*x*(x^3-x)^(1/3)+2^(1/3)*(x^3-x)^(2/3))*2^(1/3)

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.42, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {2081, 6857, 270, 283, 335, 281, 337, 477, 476, 486, 597, 12, 503} \[ \int \frac {\sqrt [3]{-x+x^3} \left (-2+x^4\right )}{x^4 \left (1+x^2\right )} \, dx=-\frac {\sqrt {3} \sqrt [3]{x^3-x} \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{x} \sqrt [3]{x^2-1}}-\frac {\sqrt {3} \sqrt [3]{x^3-x} \arctan \left (\frac {\frac {2 \sqrt [3]{2} x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{2^{2/3} \sqrt [3]{x} \sqrt [3]{x^2-1}}-\frac {27 \sqrt [3]{x^3-x}}{8 x}+\frac {3 \sqrt [3]{x^3-x}}{8 x^3}+\frac {3 \sqrt [3]{x^3-x} \left (1-x^2\right )}{8 x^3}+\frac {\sqrt [3]{x^3-x} \log \left (x^2+1\right )}{2\ 2^{2/3} \sqrt [3]{x} \sqrt [3]{x^2-1}}-\frac {3 \sqrt [3]{x^3-x} \log \left (x^{2/3}-\sqrt [3]{x^2-1}\right )}{4 \sqrt [3]{x} \sqrt [3]{x^2-1}}-\frac {3 \sqrt [3]{x^3-x} \log \left (\sqrt [3]{2} x^{2/3}-\sqrt [3]{x^2-1}\right )}{2\ 2^{2/3} \sqrt [3]{x} \sqrt [3]{x^2-1}} \]

[In]

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

[Out]

(3*(-x + x^3)^(1/3))/(8*x^3) - (27*(-x + x^3)^(1/3))/(8*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)) - (Sqrt[3]*
(-x + x^3)^(1/3)*ArcTan[(1 + (2*2^(1/3)*x^(2/3))/(-1 + x^2)^(1/3))/Sqrt[3]])/(2^(2/3)*x^(1/3)*(-1 + x^2)^(1/3)
) + ((-x + x^3)^(1/3)*Log[1 + x^2])/(2*2^(2/3)*x^(1/3)*(-1 + x^2)^(1/3)) - (3*(-x + x^3)^(1/3)*Log[x^(2/3) - (
-1 + x^2)^(1/3)])/(4*x^(1/3)*(-1 + x^2)^(1/3)) - (3*(-x + x^3)^(1/3)*Log[2^(1/3)*x^(2/3) - (-1 + x^2)^(1/3)])/
(2*2^(2/3)*x^(1/3)*(-1 + x^2)^(1/3))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 270

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 281

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 283

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 335

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 337

Int[(x_)/((a_) + (b_.)*(x_)^3)^(2/3), x_Symbol] :> With[{q = Rt[b, 3]}, Simp[-ArcTan[(1 + 2*q*(x/(a + b*x^3)^(
1/3)))/Sqrt[3]]/(Sqrt[3]*q^2), x] - Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*q^2), x]] /; FreeQ[{a, b}, x]

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 486

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

Rule 503

Int[(x_)/(((a_) + (b_.)*(x_)^3)^(2/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Si
mp[-ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*c*q^2), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/
(2*c*q^2), x] + Simp[Log[c + d*x^3]/(6*c*q^2), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 597

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

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 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*} \text {integral}& = \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 ) \text {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 ) \text {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 ) \text {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}}{8 x^3}-\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 ) \text {Subst}\left (\int \frac {5-3 x^3}{x^2 \left (-1+x^3\right )^{2/3} \left (1+x^3\right )} \, dx,x,x^{2/3}\right )}{8 \sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {\left (3 \sqrt [3]{-x+x^3}\right ) \text {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}}{8 x^3}-\frac {27 \sqrt [3]{-x+x^3}}{8 x}+\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}-\frac {\sqrt {3} \sqrt [3]{-x+x^3} \arctan \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} \log \left (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 ) \text {Subst}\left (\int -\frac {8 x}{\left (-1+x^3\right )^{2/3} \left (1+x^3\right )} \, dx,x,x^{2/3}\right )}{8 \sqrt [3]{x} \sqrt [3]{-1+x^2}} \\ & = \frac {3 \sqrt [3]{-x+x^3}}{8 x^3}-\frac {27 \sqrt [3]{-x+x^3}}{8 x}+\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}-\frac {\sqrt {3} \sqrt [3]{-x+x^3} \arctan \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} \log \left (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 ) \text {Subst}\left (\int \frac {x}{\left (-1+x^3\right )^{2/3} \left (1+x^3\right )} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{-1+x^2}} \\ & = \frac {3 \sqrt [3]{-x+x^3}}{8 x^3}-\frac {27 \sqrt [3]{-x+x^3}}{8 x}+\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}-\frac {\sqrt {3} \sqrt [3]{-x+x^3} \arctan \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 {\sqrt {3} \sqrt [3]{-x+x^3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} x^{2/3}}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {\sqrt [3]{-x+x^3} \log \left (1+x^2\right )}{2\ 2^{2/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}-\frac {3 \sqrt [3]{-x+x^3} \log \left (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} \log \left (\sqrt [3]{2} x^{2/3}-\sqrt [3]{-1+x^2}\right )}{2\ 2^{2/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.38 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.29 \[ \int \frac {\sqrt [3]{-x+x^3} \left (-2+x^4\right )}{x^4 \left (1+x^2\right )} \, dx=\frac {\sqrt [3]{x \left (-1+x^2\right )} \left (3 \sqrt [3]{-1+x^2}-15 x^2 \sqrt [3]{-1+x^2}-2 \sqrt {3} x^{8/3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{-1+x^2}}\right )-2 \sqrt [3]{2} \sqrt {3} x^{8/3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2^{2/3} \sqrt [3]{-1+x^2}}\right )-2 x^{8/3} \log \left (-x^{2/3}+\sqrt [3]{-1+x^2}\right )-2 \sqrt [3]{2} x^{8/3} \log \left (-2 x^{2/3}+2^{2/3} \sqrt [3]{-1+x^2}\right )+x^{8/3} \log \left (x^{4/3}+x^{2/3} \sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}\right )+\sqrt [3]{2} x^{8/3} \log \left (2 x^{4/3}+2^{2/3} x^{2/3} \sqrt [3]{-1+x^2}+\sqrt [3]{2} \left (-1+x^2\right )^{2/3}\right )\right )}{4 x^3 \sqrt [3]{-1+x^2}} \]

[In]

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

[Out]

((x*(-1 + x^2))^(1/3)*(3*(-1 + x^2)^(1/3) - 15*x^2*(-1 + x^2)^(1/3) - 2*Sqrt[3]*x^(8/3)*ArcTan[(Sqrt[3]*x^(2/3
))/(x^(2/3) + 2*(-1 + x^2)^(1/3))] - 2*2^(1/3)*Sqrt[3]*x^(8/3)*ArcTan[(Sqrt[3]*x^(2/3))/(x^(2/3) + 2^(2/3)*(-1
+ x^2)^(1/3))] - 2*x^(8/3)*Log[-x^(2/3) + (-1 + x^2)^(1/3)] - 2*2^(1/3)*x^(8/3)*Log[-2*x^(2/3) + 2^(2/3)*(-1
+ x^2)^(1/3)] + x^(8/3)*Log[x^(4/3) + x^(2/3)*(-1 + x^2)^(1/3) + (-1 + x^2)^(2/3)] + 2^(1/3)*x^(8/3)*Log[2*x^(
4/3) + 2^(2/3)*x^(2/3)*(-1 + x^2)^(1/3) + 2^(1/3)*(-1 + x^2)^(2/3)]))/(4*x^3*(-1 + x^2)^(1/3))

Maple [A] (verified)

Time = 2.70 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.85

method	result	size
pseudoelliptic	\(\frac {\left (-15 x^{2}+3\right ) \left (x^{3}-x \right )^{\frac {1}{3}}+x^{3} \left (\left (2 \arctan \left (\frac {\sqrt {3}\, \left (x +2^{\frac {2}{3}} \left (x^{3}-x \right )^{\frac {1}{3}}\right )}{3 x}\right ) \sqrt {3}+\ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} \left (x^{3}-x \right )^{\frac {1}{3}} x +\left (x^{3}-x \right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-2^{\frac {1}{3}} x +\left (x^{3}-x \right )^{\frac {1}{3}}}{x}\right )\right ) 2^{\frac {1}{3}}+2 \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}-x \right )^{\frac {1}{3}}\right )}{3 x}\right ) \sqrt {3}+\ln \left (\frac {x^{2}+x \left (x^{3}-x \right )^{\frac {1}{3}}+\left (x^{3}-x \right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-x +\left (x^{3}-x \right )^{\frac {1}{3}}}{x}\right )\right )}{4 x^{3}}\)	\(198\)

[In]

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

[Out]

1/4*((-15*x^2+3)*(x^3-x)^(1/3)+x^3*((2*arctan(1/3*3^(1/2)/x*(x+2^(2/3)*(x^3-x)^(1/3)))*3^(1/2)+ln((2^(2/3)*x^2
+2^(1/3)*(x^3-x)^(1/3)*x+(x^3-x)^(2/3))/x^2)-2*ln((-2^(1/3)*x+(x^3-x)^(1/3))/x))*2^(1/3)+2*arctan(1/3*3^(1/2)/
x*(x+2*(x^3-x)^(1/3)))*3^(1/2)+ln((x^2+x*(x^3-x)^(1/3)+(x^3-x)^(2/3))/x^2)-2*ln((-x+(x^3-x)^(1/3))/x)))/x^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 387 vs. \(2 (184) = 368\).

Time = 4.92 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.65 \[ \int \frac {\sqrt [3]{-x+x^3} \left (-2+x^4\right )}{x^4 \left (1+x^2\right )} \, dx=-\frac {4 \cdot 4^{\frac {1}{6}} \sqrt {3} \left (-1\right )^{\frac {1}{3}} x^{3} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (6 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (19 \, x^{5} - 16 \, x^{3} + x\right )} {\left (x^{3} - x\right )}^{\frac {1}{3}} + 12 \, \left (-1\right )^{\frac {1}{3}} {\left (5 \, x^{4} + 4 \, x^{2} - 1\right )} {\left (x^{3} - x\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (71 \, x^{6} - 111 \, x^{4} + 33 \, x^{2} - 1\right )}\right )}}{6 \, {\left (109 \, x^{6} - 105 \, x^{4} + 3 \, x^{2} + 1\right )}}\right ) + 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{3} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{3} - x\right )}^{\frac {2}{3}} {\left (5 \, x^{2} - 1\right )} - 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (19 \, x^{4} - 16 \, x^{2} + 1\right )} + 24 \, {\left (2 \, x^{3} - x\right )} {\left (x^{3} - x\right )}^{\frac {1}{3}}}{x^{4} + 2 \, x^{2} + 1}\right ) - 2 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{3} \log \left (-\frac {3 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{3} - x\right )}^{\frac {1}{3}} x + 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{2} + 1\right )} + 6 \, {\left (x^{3} - x\right )}^{\frac {2}{3}}}{x^{2} + 1}\right ) + 12 \, \sqrt {3} x^{3} \arctan \left (-\frac {44032959556 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (16754327161 \, x^{2} - 2707204793\right )} - 10524305234 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {2}{3}}}{81835897185 \, x^{2} - 1102302937}\right ) + 6 \, x^{3} \log \left (-3 \, {\left (x^{3} - x\right )}^{\frac {1}{3}} x + 3 \, {\left (x^{3} - x\right )}^{\frac {2}{3}} + 1\right ) + 18 \, {\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (5 \, x^{2} - 1\right )}}{24 \, x^{3}} \]

[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

Sympy [F]

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

[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)

Maxima [F]

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

[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)

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt [3]{-x+x^3} \left (-2+x^4\right )}{x^4 \left (1+x^2\right )} \, dx=\frac {1}{2} \, \sqrt {3} 2^{\frac {1}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}}\right )}\right ) + \frac {1}{2} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {3}{4} \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {4}{3}} + \frac {1}{4} \cdot 2^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}}\right ) - \frac {1}{2} \cdot 2^{\frac {1}{3}} \log \left ({\left | -2^{\frac {1}{3}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} \right |}\right ) - 3 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + \frac {1}{4} \, \log \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{2} \, \log \left ({\left | {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]

[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))

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{-x+x^3} \left (-2+x^4\right )}{x^4 \left (1+x^2\right )} \, dx=\int \frac {{\left (x^3-x\right )}^{1/3}\,\left (x^4-2\right )}{x^4\,\left (x^2+1\right )} \,d x \]

[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)