\(\int \frac {1}{x^2 \sqrt [3]{1-x^2} (3+x^2)} \, dx\) [1409]

Optimal result

Integrand size = 22, antiderivative size = 538 \[ \int \frac {1}{x^2 \sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=-\frac {\left (1-x^2\right )^{2/3}}{3 x}+\frac {x}{3 \left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )}-\frac {\arctan \left (\frac {\sqrt {3}}{x}\right )}{6\ 2^{2/3} \sqrt {3}}-\frac {\arctan \left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{6\ 2^{2/3} \sqrt {3}}+\frac {\text {arctanh}(x)}{18\ 2^{2/3}}-\frac {\text {arctanh}\left (\frac {x}{1+\sqrt [3]{2} \sqrt [3]{1-x^2}}\right )}{6\ 2^{2/3}}+\frac {\sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {1+\sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}} E\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{1-x^2}}{1-\sqrt {3}-\sqrt [3]{1-x^2}}\right )|-7+4 \sqrt {3}\right )}{2\ 3^{3/4} x \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}}}-\frac {\sqrt {2} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {1+\sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{1-x^2}}{1-\sqrt {3}-\sqrt [3]{1-x^2}}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} x \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}}} \] Output:

-1/3*(-x^2+1)^(2/3)/x+x/(3-3*3^(1/2)-3*(-x^2+1)^(1/3))-1/36*arctan(3^(1/2) 
/x)*2^(1/3)*3^(1/2)-1/36*arctan((1-2^(1/3)*(-x^2+1)^(1/3))*3^(1/2)/x)*2^(1 
/3)*3^(1/2)+1/36*arctanh(x)*2^(1/3)-1/12*arctanh(x/(1+2^(1/3)*(-x^2+1)^(1/ 
3)))*2^(1/3)+1/6*3^(1/4)*(1/2*6^(1/2)+1/2*2^(1/2))*(1-(-x^2+1)^(1/3))*((1+ 
(-x^2+1)^(1/3)+(-x^2+1)^(2/3))/(1-3^(1/2)-(-x^2+1)^(1/3))^2)^(1/2)*Ellipti 
cE((1+3^(1/2)-(-x^2+1)^(1/3))/(1-3^(1/2)-(-x^2+1)^(1/3)),2*I-I*3^(1/2))/x/ 
(-(1-(-x^2+1)^(1/3))/(1-3^(1/2)-(-x^2+1)^(1/3))^2)^(1/2)-1/9*2^(1/2)*3^(3/ 
4)*(1-(-x^2+1)^(1/3))*((1+(-x^2+1)^(1/3)+(-x^2+1)^(2/3))/(1-3^(1/2)-(-x^2+ 
1)^(1/3))^2)^(1/2)*EllipticF((1+3^(1/2)-(-x^2+1)^(1/3))/(1-3^(1/2)-(-x^2+1 
)^(1/3)),2*I-I*3^(1/2))/x/(-(1-(-x^2+1)^(1/3))/(1-3^(1/2)-(-x^2+1)^(1/3))^ 
2)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 10.09 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.30 \[ \int \frac {1}{x^2 \sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=-\frac {1}{81} x^3 \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},1,\frac {5}{2},x^2,-\frac {x^2}{3}\right )+\frac {-1+x^2+\frac {18 x^2 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},x^2,-\frac {x^2}{3}\right )}{\left (3+x^2\right ) \left (-9 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},x^2,-\frac {x^2}{3}\right )+2 x^2 \left (\operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},2,\frac {5}{2},x^2,-\frac {x^2}{3}\right )-\operatorname {AppellF1}\left (\frac {3}{2},\frac {4}{3},1,\frac {5}{2},x^2,-\frac {x^2}{3}\right )\right )\right )}}{3 x \sqrt [3]{1-x^2}} \] Input:

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

Output:

-1/81*(x^3*AppellF1[3/2, 1/3, 1, 5/2, x^2, -1/3*x^2]) + (-1 + x^2 + (18*x^ 
2*AppellF1[1/2, 1/3, 1, 3/2, x^2, -1/3*x^2])/((3 + x^2)*(-9*AppellF1[1/2, 
1/3, 1, 3/2, x^2, -1/3*x^2] + 2*x^2*(AppellF1[3/2, 1/3, 2, 5/2, x^2, -1/3* 
x^2] - AppellF1[3/2, 4/3, 1, 5/2, x^2, -1/3*x^2]))))/(3*x*(1 - x^2)^(1/3))
 

Rubi [A] (warning: unable to verify)

Time = 0.49 (sec) , antiderivative size = 591, normalized size of antiderivative = 1.10, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {382, 27, 405, 233, 305, 833, 760, 2418}

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.

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

\(\Big \downarrow \) 382

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

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{9} \int \frac {x^2+6}{\sqrt [3]{1-x^2} \left (x^2+3\right )}dx-\frac {\left (1-x^2\right )^{2/3}}{3 x}\)

\(\Big \downarrow \) 405

\(\displaystyle \frac {1}{9} \left (-\int \frac {1}{\sqrt [3]{1-x^2}}dx-3 \int \frac {1}{\sqrt [3]{1-x^2} \left (x^2+3\right )}dx\right )-\frac {\left (1-x^2\right )^{2/3}}{3 x}\)

\(\Big \downarrow \) 233

\(\displaystyle \frac {1}{9} \left (\frac {3 \sqrt {-x^2} \int \frac {\sqrt [3]{1-x^2}}{\sqrt {-x^2}}d\sqrt [3]{1-x^2}}{2 x}-3 \int \frac {1}{\sqrt [3]{1-x^2} \left (x^2+3\right )}dx\right )-\frac {\left (1-x^2\right )^{2/3}}{3 x}\)

\(\Big \downarrow \) 305

\(\displaystyle \frac {1}{9} \left (\frac {3 \sqrt {-x^2} \int \frac {\sqrt [3]{1-x^2}}{\sqrt {-x^2}}d\sqrt [3]{1-x^2}}{2 x}-3 \left (\frac {\arctan \left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\arctan \left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{2\ 2^{2/3}}-\frac {\text {arctanh}(x)}{6\ 2^{2/3}}\right )\right )-\frac {\left (1-x^2\right )^{2/3}}{3 x}\)

\(\Big \downarrow \) 833

\(\displaystyle \frac {1}{9} \left (\frac {3 \sqrt {-x^2} \left (\left (1+\sqrt {3}\right ) \int \frac {1}{\sqrt {-x^2}}d\sqrt [3]{1-x^2}-\int \frac {-\sqrt [3]{1-x^2}+\sqrt {3}+1}{\sqrt {-x^2}}d\sqrt [3]{1-x^2}\right )}{2 x}-3 \left (\frac {\arctan \left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\arctan \left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{2\ 2^{2/3}}-\frac {\text {arctanh}(x)}{6\ 2^{2/3}}\right )\right )-\frac {\left (1-x^2\right )^{2/3}}{3 x}\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {1}{9} \left (\frac {3 \sqrt {-x^2} \left (-\int \frac {-\sqrt [3]{1-x^2}+\sqrt {3}+1}{\sqrt {-x^2}}d\sqrt [3]{1-x^2}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-x^2}+\sqrt {3}+1}{-\sqrt [3]{1-x^2}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-x^2} \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}}}\right )}{2 x}-3 \left (\frac {\arctan \left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\arctan \left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{2\ 2^{2/3}}-\frac {\text {arctanh}(x)}{6\ 2^{2/3}}\right )\right )-\frac {\left (1-x^2\right )^{2/3}}{3 x}\)

\(\Big \downarrow \) 2418

\(\displaystyle \frac {1}{9} \left (\frac {3 \sqrt {-x^2} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-x^2}+\sqrt {3}+1}{-\sqrt [3]{1-x^2}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-x^2} \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-x^2}+\sqrt {3}+1}{-\sqrt [3]{1-x^2}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-x^2} \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {-x^2}}{-\sqrt [3]{1-x^2}-\sqrt {3}+1}\right )}{2 x}-3 \left (\frac {\arctan \left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\arctan \left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{2\ 2^{2/3}}-\frac {\text {arctanh}(x)}{6\ 2^{2/3}}\right )\right )-\frac {\left (1-x^2\right )^{2/3}}{3 x}\)

Input:

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

Output:

-1/3*(1 - x^2)^(2/3)/x + (-3*(ArcTan[Sqrt[3]/x]/(2*2^(2/3)*Sqrt[3]) + ArcT 
an[(Sqrt[3]*(1 - 2^(1/3)*(1 - x^2)^(1/3)))/x]/(2*2^(2/3)*Sqrt[3]) - ArcTan 
h[x]/(6*2^(2/3)) + ArcTanh[x/(1 + 2^(1/3)*(1 - x^2)^(1/3))]/(2*2^(2/3))) + 
 (3*Sqrt[-x^2]*((-2*Sqrt[-x^2])/(1 - Sqrt[3] - (1 - x^2)^(1/3)) + (3^(1/4) 
*Sqrt[2 + Sqrt[3]]*(1 - (1 - x^2)^(1/3))*Sqrt[(1 + (1 - x^2)^(1/3) + (1 - 
x^2)^(2/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))^2]*EllipticE[ArcSin[(1 + Sqrt[ 
3] - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))], -7 + 4*Sqrt[3]])/( 
Sqrt[-x^2]*Sqrt[-((1 - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))^2) 
]) - (2*Sqrt[2 - Sqrt[3]]*(1 + Sqrt[3])*(1 - (1 - x^2)^(1/3))*Sqrt[(1 + (1 
 - x^2)^(1/3) + (1 - x^2)^(2/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))^2]*Ellipt 
icF[ArcSin[(1 + Sqrt[3] - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3)) 
], -7 + 4*Sqrt[3]])/(3^(1/4)*Sqrt[-x^2]*Sqrt[-((1 - (1 - x^2)^(1/3))/(1 - 
Sqrt[3] - (1 - x^2)^(1/3))^2)])))/(2*x))/9
 

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 233

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

rule 305

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

rule 382

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) 
, x_Symbol] :> Simp[(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/ 
(a*c*e*(m + 1))), x] - Simp[1/(a*c*e^2*(m + 1))   Int[(e*x)^(m + 2)*(a + b* 
x^2)^p*(c + d*x^2)^q*Simp[(b*c + a*d)*(m + 3) + 2*(b*c*p + a*d*q) + b*d*(m 
+ 2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[ 
b*c - a*d, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
 

rule 405

Int[(((a_) + (b_.)*(x_)^2)^(p_)*((e_) + (f_.)*(x_)^2))/((c_) + (d_.)*(x_)^2 
), x_Symbol] :> Simp[f/d   Int[(a + b*x^2)^p, x], x] + Simp[(d*e - c*f)/d 
 Int[(a + b*x^2)^p/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, p}, x]
 

rule 760

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 - Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s 
*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(- 
s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 + Sqrt[3]) 
*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x 
] && NegQ[a]
 

rule 833

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] 
], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 + Sqrt[3]))*(s/r)   Int[1/Sqrt[a + b*x 
^3], x], x] + Simp[1/r   Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x 
]] /; FreeQ[{a, b}, x] && NegQ[a]
 

rule 2418

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N 
umer[Simplify[(1 + Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 + Sqrt[3])*(d/c) 
]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 - Sqrt[3])*s + r*x))), x] + S 
imp[3^(1/4)*Sqrt[2 + Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( 
(1 - Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[(-s)*((s + r*x)/((1 - S 
qrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[ 
3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && 
 EqQ[b*c^3 - 2*(5 + 3*Sqrt[3])*a*d^3, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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