\(\int \frac {(1+x^2) \sqrt [3]{-1-x^2+x^4+x^6}}{x} \, dx\) [2916]

Optimal result

Integrand size = 26, antiderivative size = 330 \[ \int \frac {\left (1+x^2\right ) \sqrt [3]{-1-x^2+x^4+x^6}}{x} \, dx=\frac {\left (-1+x^2\right )^{2/3} \left (1+x^2\right )^{4/3} \left (\frac {1}{2} \sqrt [3]{-1+x^2} \left (1+x^2\right )^{2/3}+\frac {1}{12} \sqrt [3]{-1+x^2} \left (-2 \left (1+x^2\right )^{2/3}+3 \left (1+x^2\right )^{5/3}\right )+\frac {1}{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{1+x^2}}{2 \sqrt [3]{-1+x^2}-\sqrt [3]{1+x^2}}\right )+\frac {\arctan \left (\frac {\sqrt {3} \sqrt [3]{1+x^2}}{2 \sqrt [3]{-1+x^2}+\sqrt [3]{1+x^2}}\right )}{6 \sqrt {3}}+\frac {1}{18} \log \left (\sqrt [3]{-1+x^2}-\sqrt [3]{1+x^2}\right )-\frac {1}{2} \log \left (\sqrt [3]{-1+x^2}+\sqrt [3]{1+x^2}\right )+\frac {1}{4} \log \left (\left (-1+x^2\right )^{2/3}-\sqrt [3]{-1+x^2} \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right )-\frac {1}{36} \log \left (\left (-1+x^2\right )^{2/3}+\sqrt [3]{-1+x^2} \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right )\right )}{\left (\left (-1+x^2\right ) \left (1+x^2\right )^2\right )^{2/3}} \] Output:

(x^2-1)^(2/3)*(x^2+1)^(4/3)*(1/2*(x^2-1)^(1/3)*(x^2+1)^(2/3)+1/12*(x^2-1)^ 
(1/3)*(-2*(x^2+1)^(2/3)+3*(x^2+1)^(5/3))+1/2*3^(1/2)*arctan(3^(1/2)*(x^2+1 
)^(1/3)/(2*(x^2-1)^(1/3)-(x^2+1)^(1/3)))+1/18*arctan(3^(1/2)*(x^2+1)^(1/3) 
/(2*(x^2-1)^(1/3)+(x^2+1)^(1/3)))*3^(1/2)+1/18*ln((x^2-1)^(1/3)-(x^2+1)^(1 
/3))-1/2*ln((x^2-1)^(1/3)+(x^2+1)^(1/3))+1/4*ln((x^2-1)^(2/3)-(x^2-1)^(1/3 
)*(x^2+1)^(1/3)+(x^2+1)^(2/3))-1/36*ln((x^2-1)^(2/3)+(x^2-1)^(1/3)*(x^2+1) 
^(1/3)+(x^2+1)^(2/3)))/((x^2-1)*(x^2+1)^2)^(2/3)
 

Mathematica [A] (verified)

Time = 0.69 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.92 \[ \int \frac {\left (1+x^2\right ) \sqrt [3]{-1-x^2+x^4+x^6}}{x} \, dx=\frac {\left (-1+x^2\right )^{2/3} \left (1+x^2\right )^{4/3} \left (21 \sqrt [3]{-1+x^2} \left (1+x^2\right )^{2/3}+9 x^2 \sqrt [3]{-1+x^2} \left (1+x^2\right )^{2/3}+18 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-1+x^2}}{\sqrt [3]{-1+x^2}-2 \sqrt [3]{1+x^2}}\right )-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-1+x^2}}{\sqrt [3]{-1+x^2}+2 \sqrt [3]{1+x^2}}\right )+2 \log \left (-\sqrt [3]{-1+x^2}+\sqrt [3]{1+x^2}\right )-18 \log \left (\sqrt [3]{-1+x^2}+\sqrt [3]{1+x^2}\right )+9 \log \left (\left (-1+x^2\right )^{2/3}-\sqrt [3]{-1+x^2} \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right )-\log \left (\left (-1+x^2\right )^{2/3}+\sqrt [3]{-1+x^2} \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right )\right )}{36 \left (\left (-1+x^2\right ) \left (1+x^2\right )^2\right )^{2/3}} \] Input:

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

Output:

((-1 + x^2)^(2/3)*(1 + x^2)^(4/3)*(21*(-1 + x^2)^(1/3)*(1 + x^2)^(2/3) + 9 
*x^2*(-1 + x^2)^(1/3)*(1 + x^2)^(2/3) + 18*Sqrt[3]*ArcTan[(Sqrt[3]*(-1 + x 
^2)^(1/3))/((-1 + x^2)^(1/3) - 2*(1 + x^2)^(1/3))] - 2*Sqrt[3]*ArcTan[(Sqr 
t[3]*(-1 + x^2)^(1/3))/((-1 + x^2)^(1/3) + 2*(1 + x^2)^(1/3))] + 2*Log[-(- 
1 + x^2)^(1/3) + (1 + x^2)^(1/3)] - 18*Log[(-1 + x^2)^(1/3) + (1 + x^2)^(1 
/3)] + 9*Log[(-1 + x^2)^(2/3) - (-1 + x^2)^(1/3)*(1 + x^2)^(1/3) + (1 + x^ 
2)^(2/3)] - Log[(-1 + x^2)^(2/3) + (-1 + x^2)^(1/3)*(1 + x^2)^(1/3) + (1 + 
 x^2)^(2/3)]))/(36*((-1 + x^2)*(1 + x^2)^2)^(2/3))
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.79, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {7282, 7270, 112, 27, 173, 90, 72, 102}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 7282

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

\(\Big \downarrow \) 7270

\(\displaystyle \frac {\sqrt [3]{-\left (\left (1-x^2\right ) \left (x^2+1\right )^2\right )} \int \frac {\sqrt [3]{1-x^2} \left (x^2+1\right )^{5/3}}{x^2}dx^2}{2 \sqrt [3]{1-x^2} \left (x^2+1\right )^{2/3}}\)

\(\Big \downarrow \) 112

\(\displaystyle \frac {\sqrt [3]{-\left (\left (1-x^2\right ) \left (x^2+1\right )^2\right )} \left (\frac {1}{2} \sqrt [3]{1-x^2} \left (x^2+1\right )^{5/3}-\frac {1}{2} \int -\frac {2 \left (3-2 x^2\right ) \left (x^2+1\right )^{2/3}}{3 x^2 \left (1-x^2\right )^{2/3}}dx^2\right )}{2 \sqrt [3]{1-x^2} \left (x^2+1\right )^{2/3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt [3]{-\left (\left (1-x^2\right ) \left (x^2+1\right )^2\right )} \left (\frac {1}{3} \int \frac {\left (3-2 x^2\right ) \left (x^2+1\right )^{2/3}}{x^2 \left (1-x^2\right )^{2/3}}dx^2+\frac {1}{2} \sqrt [3]{1-x^2} \left (x^2+1\right )^{5/3}\right )}{2 \sqrt [3]{1-x^2} \left (x^2+1\right )^{2/3}}\)

\(\Big \downarrow \) 173

\(\displaystyle \frac {\sqrt [3]{-\left (\left (1-x^2\right ) \left (x^2+1\right )^2\right )} \left (\frac {1}{3} \left (3 \int \frac {1}{x^2 \left (1-x^2\right )^{2/3} \sqrt [3]{x^2+1}}dx^2+\int \frac {1-2 x^2}{\left (1-x^2\right )^{2/3} \sqrt [3]{x^2+1}}dx^2\right )+\frac {1}{2} \sqrt [3]{1-x^2} \left (x^2+1\right )^{5/3}\right )}{2 \sqrt [3]{1-x^2} \left (x^2+1\right )^{2/3}}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {\sqrt [3]{-\left (\left (1-x^2\right ) \left (x^2+1\right )^2\right )} \left (\frac {1}{3} \left (\frac {1}{3} \int \frac {1}{\left (1-x^2\right )^{2/3} \sqrt [3]{x^2+1}}dx^2+3 \int \frac {1}{x^2 \left (1-x^2\right )^{2/3} \sqrt [3]{x^2+1}}dx^2+2 \sqrt [3]{1-x^2} \left (x^2+1\right )^{2/3}\right )+\frac {1}{2} \sqrt [3]{1-x^2} \left (x^2+1\right )^{5/3}\right )}{2 \sqrt [3]{1-x^2} \left (x^2+1\right )^{2/3}}\)

\(\Big \downarrow \) 72

\(\displaystyle \frac {\sqrt [3]{-\left (\left (1-x^2\right ) \left (x^2+1\right )^2\right )} \left (\frac {1}{3} \left (3 \int \frac {1}{x^2 \left (1-x^2\right )^{2/3} \sqrt [3]{x^2+1}}dx^2+\frac {1}{3} \left (-\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x^2+1}}{\sqrt {3} \sqrt [3]{1-x^2}}\right )-\frac {1}{2} \log \left (1-x^2\right )-\frac {3}{2} \log \left (\frac {\sqrt [3]{x^2+1}}{\sqrt [3]{1-x^2}}+1\right )\right )+2 \sqrt [3]{1-x^2} \left (x^2+1\right )^{2/3}\right )+\frac {1}{2} \sqrt [3]{1-x^2} \left (x^2+1\right )^{5/3}\right )}{2 \sqrt [3]{1-x^2} \left (x^2+1\right )^{2/3}}\)

\(\Big \downarrow \) 102

\(\displaystyle \frac {\sqrt [3]{-\left (\left (1-x^2\right ) \left (x^2+1\right )^2\right )} \left (\frac {1}{3} \left (3 \left (\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x^2+1}}{\sqrt {3} \sqrt [3]{1-x^2}}+\frac {1}{\sqrt {3}}\right )-\frac {\log \left (x^2\right )}{2}+\frac {3}{2} \log \left (\sqrt [3]{x^2+1}-\sqrt [3]{1-x^2}\right )\right )+\frac {1}{3} \left (-\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x^2+1}}{\sqrt {3} \sqrt [3]{1-x^2}}\right )-\frac {1}{2} \log \left (1-x^2\right )-\frac {3}{2} \log \left (\frac {\sqrt [3]{x^2+1}}{\sqrt [3]{1-x^2}}+1\right )\right )+2 \sqrt [3]{1-x^2} \left (x^2+1\right )^{2/3}\right )+\frac {1}{2} \sqrt [3]{1-x^2} \left (x^2+1\right )^{5/3}\right )}{2 \sqrt [3]{1-x^2} \left (x^2+1\right )^{2/3}}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 72

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

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 102

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)*((e_.) + (f_.) 
*(x_))), x_] :> With[{q = Rt[(d*e - c*f)/(b*e - a*f), 3]}, Simp[(-Sqrt[3])* 
q*(ArcTan[1/Sqrt[3] + 2*q*((a + b*x)^(1/3)/(Sqrt[3]*(c + d*x)^(1/3)))]/(d*e 
 - c*f)), x] + (Simp[q*(Log[e + f*x]/(2*(d*e - c*f))), x] - Simp[3*q*(Log[q 
*(a + b*x)^(1/3) - (c + d*x)^(1/3)]/(2*(d*e - c*f))), x])] /; FreeQ[{a, b, 
c, d, e, f}, x]
 

rule 112

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^m*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + 
p + 1))), x] - Simp[1/(f*(m + n + p + 1))   Int[(a + b*x)^(m - 1)*(c + d*x) 
^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a 
*f) + b*n*(d*e - c*f))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && 
GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (IntegersQ[2*m, 2*n, 2*p 
] || (IntegersQ[m, n + p] || IntegersQ[p, m + n]))
 

rule 173

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((g_.) + (h_.)*(x_ 
)))/((e_.) + (f_.)*(x_)), x_] :> Simp[(f*g - e*h)*((c*f - d*e)^(m + n + 1)/ 
f^(m + n + 2))   Int[(a + b*x)^m/((c + d*x)^(m + 1)*(e + f*x)), x], x] + Si 
mp[1/f^(m + n + 2)   Int[((a + b*x)^m/(c + d*x)^(m + 1))*ExpandToSum[(f^(m 
+ n + 2)*(c + d*x)^(m + n + 1)*(g + h*x) - (f*g - e*h)*(c*f - d*e)^(m + n + 
 1))/(e + f*x), x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IGtQ[m 
+ n + 1, 0] && (LtQ[m, 0] || SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])
 

rule 7270

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Simp[a^IntPart[p 
]*((a*v^m*w^n)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p])))   Int[u*v 
^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !Free 
Q[v, x] &&  !FreeQ[w, x]
 

rule 7282

Int[(u_)/(x_), x_Symbol] :> With[{lst = PowerVariableExpn[u, 0, x]}, Simp[1 
/lst[[2]]   Subst[Int[NormalizeIntegrand[Simplify[lst[[1]]/x], x], x], x, ( 
lst[[3]]*x)^lst[[2]]], x] /;  !FalseQ[lst] && NeQ[lst[[2]], 0]] /; NonsumQ[ 
u] &&  !RationalFunctionQ[u, x]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 39.63 (sec) , antiderivative size = 6304, normalized size of antiderivative = 19.10

method	result	size
trager	\(\text {Expression too large to display}\)	\(6304\)

Input:

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

Output:

result too large to display
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.92 \[ \int \frac {\left (1+x^2\right ) \sqrt [3]{-1-x^2+x^4+x^6}}{x} \, dx=-\frac {1}{18} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (x^{2} + 1\right )} + 2 \, \sqrt {3} {\left (x^{6} + x^{4} - x^{2} - 1\right )}^{\frac {1}{3}}}{3 \, {\left (x^{2} + 1\right )}}\right ) - \frac {1}{2} \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (x^{2} + 1\right )} - 2 \, \sqrt {3} {\left (x^{6} + x^{4} - x^{2} - 1\right )}^{\frac {1}{3}}}{3 \, {\left (x^{2} + 1\right )}}\right ) + \frac {1}{12} \, {\left (x^{6} + x^{4} - x^{2} - 1\right )}^{\frac {1}{3}} {\left (3 \, x^{2} + 7\right )} - \frac {1}{36} \, \log \left (\frac {x^{4} + 2 \, x^{2} + {\left (x^{6} + x^{4} - x^{2} - 1\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )} + {\left (x^{6} + x^{4} - x^{2} - 1\right )}^{\frac {2}{3}} + 1}{x^{4} + 2 \, x^{2} + 1}\right ) + \frac {1}{4} \, \log \left (\frac {x^{4} + 2 \, x^{2} - {\left (x^{6} + x^{4} - x^{2} - 1\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )} + {\left (x^{6} + x^{4} - x^{2} - 1\right )}^{\frac {2}{3}} + 1}{x^{4} + 2 \, x^{2} + 1}\right ) - \frac {1}{2} \, \log \left (\frac {x^{2} + {\left (x^{6} + x^{4} - x^{2} - 1\right )}^{\frac {1}{3}} + 1}{x^{2} + 1}\right ) + \frac {1}{18} \, \log \left (-\frac {x^{2} - {\left (x^{6} + x^{4} - x^{2} - 1\right )}^{\frac {1}{3}} + 1}{x^{2} + 1}\right ) \] Input:

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

Output:

-1/18*sqrt(3)*arctan(1/3*(sqrt(3)*(x^2 + 1) + 2*sqrt(3)*(x^6 + x^4 - x^2 - 
 1)^(1/3))/(x^2 + 1)) - 1/2*sqrt(3)*arctan(-1/3*(sqrt(3)*(x^2 + 1) - 2*sqr 
t(3)*(x^6 + x^4 - x^2 - 1)^(1/3))/(x^2 + 1)) + 1/12*(x^6 + x^4 - x^2 - 1)^ 
(1/3)*(3*x^2 + 7) - 1/36*log((x^4 + 2*x^2 + (x^6 + x^4 - x^2 - 1)^(1/3)*(x 
^2 + 1) + (x^6 + x^4 - x^2 - 1)^(2/3) + 1)/(x^4 + 2*x^2 + 1)) + 1/4*log((x 
^4 + 2*x^2 - (x^6 + x^4 - x^2 - 1)^(1/3)*(x^2 + 1) + (x^6 + x^4 - x^2 - 1) 
^(2/3) + 1)/(x^4 + 2*x^2 + 1)) - 1/2*log((x^2 + (x^6 + x^4 - x^2 - 1)^(1/3 
) + 1)/(x^2 + 1)) + 1/18*log(-(x^2 - (x^6 + x^4 - x^2 - 1)^(1/3) + 1)/(x^2 
 + 1))
 

Sympy [F]

\[ \int \frac {\left (1+x^2\right ) \sqrt [3]{-1-x^2+x^4+x^6}}{x} \, dx=\int \frac {\sqrt [3]{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )^{2}} \left (x^{2} + 1\right )}{x}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {\left (1+x^2\right ) \sqrt [3]{-1-x^2+x^4+x^6}}{x} \, dx=\int { \frac {{\left (x^{6} + x^{4} - x^{2} - 1\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )}}{x} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {\left (1+x^2\right ) \sqrt [3]{-1-x^2+x^4+x^6}}{x} \, dx=\int { \frac {{\left (x^{6} + x^{4} - x^{2} - 1\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )}}{x} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (1+x^2\right ) \sqrt [3]{-1-x^2+x^4+x^6}}{x} \, dx=\int \frac {\left (x^2+1\right )\,{\left (x^6+x^4-x^2-1\right )}^{1/3}}{x} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\left (1+x^2\right ) \sqrt [3]{-1-x^2+x^4+x^6}}{x} \, dx=\frac {\left (x^{6}+x^{4}-x^{2}-1\right )^{\frac {1}{3}} x^{2}}{4}+\frac {3 \left (x^{6}+x^{4}-x^{2}-1\right )^{\frac {1}{3}}}{4}-\left (\int \frac {\left (x^{6}+x^{4}-x^{2}-1\right )^{\frac {1}{3}}}{x^{5}-x}d x \right )-\frac {\left (\int \frac {\left (x^{6}+x^{4}-x^{2}-1\right )^{\frac {1}{3}} x^{3}}{x^{4}-1}d x \right )}{3} \] Input:

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

Output:

(3*(x**6 + x**4 - x**2 - 1)**(1/3)*x**2 + 9*(x**6 + x**4 - x**2 - 1)**(1/3 
) - 12*int((x**6 + x**4 - x**2 - 1)**(1/3)/(x**5 - x),x) - 4*int(((x**6 + 
x**4 - x**2 - 1)**(1/3)*x**3)/(x**4 - 1),x))/12