\(\int \frac {1}{x^5 \sqrt [3]{1-x^2} (3+x^2)} \, dx\) [1405]

Optimal result

Integrand size = 22, antiderivative size = 178 \[ \int \frac {1}{x^5 \sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=-\frac {\left (1-x^2\right )^{2/3}}{12 x^4}-\frac {\left (1-x^2\right )^{2/3}}{18 x^2}+\frac {\arctan \left (\frac {1+2 \sqrt [3]{1-x^2}}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {\arctan \left (\frac {1+\sqrt [3]{2} \sqrt [3]{1-x^2}}{\sqrt {3}}\right )}{18\ 2^{2/3} \sqrt {3}}-\frac {\log (x)}{27}+\frac {\log \left (3+x^2\right )}{108\ 2^{2/3}}+\frac {1}{18} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac {\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{36\ 2^{2/3}} \] Output:

-1/12*(-x^2+1)^(2/3)/x^4-1/18*(-x^2+1)^(2/3)/x^2+1/27*arctan(1/3*(1+2*(-x^ 
2+1)^(1/3))*3^(1/2))*3^(1/2)-1/108*3^(1/2)*arctan(1/3*(1+2^(1/3)*(-x^2+1)^ 
(1/3))*3^(1/2))*2^(1/3)-1/27*ln(x)+1/216*ln(x^2+3)*2^(1/3)+1/18*ln(1-(-x^2 
+1)^(1/3))-1/72*ln(2^(2/3)-(-x^2+1)^(1/3))*2^(1/3)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x^5 \sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=-\frac {18 \left (1-x^2\right )^{2/3}+12 x^2 \left (1-x^2\right )^{2/3}+2 \sqrt [3]{2} \sqrt {3} x^4 \arctan \left (\frac {1+\sqrt [3]{2-2 x^2}}{\sqrt {3}}\right )-8 \sqrt {3} x^4 \arctan \left (\frac {1+2 \sqrt [3]{1-x^2}}{\sqrt {3}}\right )+2 \sqrt [3]{2} x^4 \log \left (-2+\sqrt [3]{2-2 x^2}\right )-\sqrt [3]{2} x^4 \log \left (4+2 \sqrt [3]{2-2 x^2}+\left (2-2 x^2\right )^{2/3}\right )-8 x^4 \log \left (-1+\sqrt [3]{1-x^2}\right )+4 x^4 \log \left (1+\sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}\right )}{216 x^4} \] Input:

Integrate[1/(x^5*(1 - x^2)^(1/3)*(3 + x^2)),x]
 

Output:

-1/216*(18*(1 - x^2)^(2/3) + 12*x^2*(1 - x^2)^(2/3) + 2*2^(1/3)*Sqrt[3]*x^ 
4*ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]] - 8*Sqrt[3]*x^4*ArcTan[(1 + 2*(1 
 - x^2)^(1/3))/Sqrt[3]] + 2*2^(1/3)*x^4*Log[-2 + (2 - 2*x^2)^(1/3)] - 2^(1 
/3)*x^4*Log[4 + 2*(2 - 2*x^2)^(1/3) + (2 - 2*x^2)^(2/3)] - 8*x^4*Log[-1 + 
(1 - x^2)^(1/3)] + 4*x^4*Log[1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3)])/x^4
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.07, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {354, 114, 27, 168, 25, 174, 67, 16, 1082, 217, 1083, 217}

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.

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

\(\Big \downarrow \) 354

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (\frac {1}{9} \int \frac {2 x^2+3}{x^4 \sqrt [3]{1-x^2} \left (x^2+3\right )}dx^2-\frac {\left (1-x^2\right )^{2/3}}{6 x^4}\right )\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{2} \left (\frac {1}{9} \left (-\frac {1}{3} \int -\frac {x^2+6}{x^2 \sqrt [3]{1-x^2} \left (x^2+3\right )}dx^2-\frac {\left (1-x^2\right )^{2/3}}{x^2}\right )-\frac {\left (1-x^2\right )^{2/3}}{6 x^4}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} \left (\frac {1}{9} \left (\frac {1}{3} \int \frac {x^2+6}{x^2 \sqrt [3]{1-x^2} \left (x^2+3\right )}dx^2-\frac {\left (1-x^2\right )^{2/3}}{x^2}\right )-\frac {\left (1-x^2\right )^{2/3}}{6 x^4}\right )\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {1}{2} \left (\frac {1}{9} \left (\frac {1}{3} \left (2 \int \frac {1}{x^2 \sqrt [3]{1-x^2}}dx^2-\int \frac {1}{\sqrt [3]{1-x^2} \left (x^2+3\right )}dx^2\right )-\frac {\left (1-x^2\right )^{2/3}}{x^2}\right )-\frac {\left (1-x^2\right )^{2/3}}{6 x^4}\right )\)

\(\Big \downarrow \) 67

\(\displaystyle \frac {1}{2} \left (\frac {1}{9} \left (\frac {1}{3} \left (\frac {3 \int \frac {1}{2^{2/3}-\sqrt [3]{1-x^2}}d\sqrt [3]{1-x^2}}{2\ 2^{2/3}}-\frac {3}{2} \int \frac {1}{x^4+2^{2/3} \sqrt [3]{1-x^2}+2 \sqrt [3]{2}}d\sqrt [3]{1-x^2}+2 \left (-\frac {3}{2} \int \frac {1}{1-\sqrt [3]{1-x^2}}d\sqrt [3]{1-x^2}+\frac {3}{2} \int \frac {1}{x^4+\sqrt [3]{1-x^2}+1}d\sqrt [3]{1-x^2}-\frac {1}{2} \log \left (x^2\right )\right )+\frac {\log \left (x^2+3\right )}{2\ 2^{2/3}}\right )-\frac {\left (1-x^2\right )^{2/3}}{x^2}\right )-\frac {\left (1-x^2\right )^{2/3}}{6 x^4}\right )\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {1}{2} \left (\frac {1}{9} \left (\frac {1}{3} \left (-\frac {3}{2} \int \frac {1}{x^4+2^{2/3} \sqrt [3]{1-x^2}+2 \sqrt [3]{2}}d\sqrt [3]{1-x^2}+2 \left (\frac {3}{2} \int \frac {1}{x^4+\sqrt [3]{1-x^2}+1}d\sqrt [3]{1-x^2}-\frac {1}{2} \log \left (x^2\right )+\frac {3}{2} \log \left (1-\sqrt [3]{1-x^2}\right )\right )+\frac {\log \left (x^2+3\right )}{2\ 2^{2/3}}-\frac {3 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{2\ 2^{2/3}}\right )-\frac {\left (1-x^2\right )^{2/3}}{x^2}\right )-\frac {\left (1-x^2\right )^{2/3}}{6 x^4}\right )\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {1}{2} \left (\frac {1}{9} \left (\frac {1}{3} \left (\frac {3 \int \frac {1}{-x^4-3}d\left (\sqrt [3]{2} \sqrt [3]{1-x^2}+1\right )}{2^{2/3}}+2 \left (\frac {3}{2} \int \frac {1}{x^4+\sqrt [3]{1-x^2}+1}d\sqrt [3]{1-x^2}-\frac {1}{2} \log \left (x^2\right )+\frac {3}{2} \log \left (1-\sqrt [3]{1-x^2}\right )\right )+\frac {\log \left (x^2+3\right )}{2\ 2^{2/3}}-\frac {3 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{2\ 2^{2/3}}\right )-\frac {\left (1-x^2\right )^{2/3}}{x^2}\right )-\frac {\left (1-x^2\right )^{2/3}}{6 x^4}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{2} \left (\frac {1}{9} \left (\frac {1}{3} \left (2 \left (\frac {3}{2} \int \frac {1}{x^4+\sqrt [3]{1-x^2}+1}d\sqrt [3]{1-x^2}-\frac {1}{2} \log \left (x^2\right )+\frac {3}{2} \log \left (1-\sqrt [3]{1-x^2}\right )\right )-\frac {\sqrt {3} \arctan \left (\frac {\sqrt [3]{2} \sqrt [3]{1-x^2}+1}{\sqrt {3}}\right )}{2^{2/3}}+\frac {\log \left (x^2+3\right )}{2\ 2^{2/3}}-\frac {3 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{2\ 2^{2/3}}\right )-\frac {\left (1-x^2\right )^{2/3}}{x^2}\right )-\frac {\left (1-x^2\right )^{2/3}}{6 x^4}\right )\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {1}{2} \left (\frac {1}{9} \left (\frac {1}{3} \left (2 \left (-3 \int \frac {1}{-x^4-3}d\left (2 \sqrt [3]{1-x^2}+1\right )-\frac {1}{2} \log \left (x^2\right )+\frac {3}{2} \log \left (1-\sqrt [3]{1-x^2}\right )\right )-\frac {\sqrt {3} \arctan \left (\frac {\sqrt [3]{2} \sqrt [3]{1-x^2}+1}{\sqrt {3}}\right )}{2^{2/3}}+\frac {\log \left (x^2+3\right )}{2\ 2^{2/3}}-\frac {3 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{2\ 2^{2/3}}\right )-\frac {\left (1-x^2\right )^{2/3}}{x^2}\right )-\frac {\left (1-x^2\right )^{2/3}}{6 x^4}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{2} \left (\frac {1}{9} \left (\frac {1}{3} \left (-\frac {\sqrt {3} \arctan \left (\frac {\sqrt [3]{2} \sqrt [3]{1-x^2}+1}{\sqrt {3}}\right )}{2^{2/3}}+2 \left (\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{1-x^2}+1}{\sqrt {3}}\right )-\frac {\log \left (x^2\right )}{2}+\frac {3}{2} \log \left (1-\sqrt [3]{1-x^2}\right )\right )+\frac {\log \left (x^2+3\right )}{2\ 2^{2/3}}-\frac {3 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{2\ 2^{2/3}}\right )-\frac {\left (1-x^2\right )^{2/3}}{x^2}\right )-\frac {\left (1-x^2\right )^{2/3}}{6 x^4}\right )\)

Input:

Int[1/(x^5*(1 - x^2)^(1/3)*(3 + x^2)),x]
 

Output:

(-1/6*(1 - x^2)^(2/3)/x^4 + (-((1 - x^2)^(2/3)/x^2) + (-((Sqrt[3]*ArcTan[( 
1 + 2^(1/3)*(1 - x^2)^(1/3))/Sqrt[3]])/2^(2/3)) + Log[3 + x^2]/(2*2^(2/3)) 
 + 2*(Sqrt[3]*ArcTan[(1 + 2*(1 - x^2)^(1/3))/Sqrt[3]] - Log[x^2]/2 + (3*Lo 
g[1 - (1 - x^2)^(1/3)])/2) - (3*Log[2^(2/3) - (1 - x^2)^(1/3)])/(2*2^(2/3) 
))/3)/9)/2
 

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 67

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

rule 114

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e 
 - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
 

rule 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
 

rule 174

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

rule 217

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

rule 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1083

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

Maple [A] (verified)

Time = 2.52 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.27

method	result	size
pseudoelliptic	\(\frac {-2 \sqrt {3}\, 2^{\frac {1}{3}} \arctan \left (\frac {\left (1+2^{\frac {1}{3}} \left (-x^{2}+1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right ) x^{4}+8 \sqrt {3}\, \arctan \left (\frac {\left (1+2 \left (-x^{2}+1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right ) x^{4}-2 \,2^{\frac {1}{3}} \ln \left (\left (-x^{2}+1\right )^{\frac {1}{3}}-2^{\frac {2}{3}}\right ) x^{4}+2^{\frac {1}{3}} \ln \left (\left (-x^{2}+1\right )^{\frac {2}{3}}+2^{\frac {2}{3}} \left (-x^{2}+1\right )^{\frac {1}{3}}+2 \,2^{\frac {1}{3}}\right ) x^{4}+8 \ln \left (-1+\left (-x^{2}+1\right )^{\frac {1}{3}}\right ) x^{4}-4 \ln \left (1+\left (-x^{2}+1\right )^{\frac {1}{3}}+\left (-x^{2}+1\right )^{\frac {2}{3}}\right ) x^{4}-12 \left (-x^{2}+1\right )^{\frac {2}{3}} x^{2}-18 \left (-x^{2}+1\right )^{\frac {2}{3}}}{216 {\left (-1+\left (-x^{2}+1\right )^{\frac {1}{3}}\right )}^{2} {\left (1+\left (-x^{2}+1\right )^{\frac {1}{3}}+\left (-x^{2}+1\right )^{\frac {2}{3}}\right )}^{2}}\)	\(226\)

Input:

int(1/x^5/(-x^2+1)^(1/3)/(x^2+3),x,method=_RETURNVERBOSE)
 

Output:

1/216*(-2*3^(1/2)*2^(1/3)*arctan(1/3*(1+2^(1/3)*(-x^2+1)^(1/3))*3^(1/2))*x 
^4+8*3^(1/2)*arctan(1/3*(1+2*(-x^2+1)^(1/3))*3^(1/2))*x^4-2*2^(1/3)*ln((-x 
^2+1)^(1/3)-2^(2/3))*x^4+2^(1/3)*ln((-x^2+1)^(2/3)+2^(2/3)*(-x^2+1)^(1/3)+ 
2*2^(1/3))*x^4+8*ln(-1+(-x^2+1)^(1/3))*x^4-4*ln(1+(-x^2+1)^(1/3)+(-x^2+1)^ 
(2/3))*x^4-12*(-x^2+1)^(2/3)*x^2-18*(-x^2+1)^(2/3))/(-1+(-x^2+1)^(1/3))^2/ 
(1+(-x^2+1)^(1/3)+(-x^2+1)^(2/3))^2
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x^5 \sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=-\frac {12 \cdot 4^{\frac {1}{6}} \sqrt {\frac {1}{3}} x^{4} \arctan \left (\frac {1}{2} \cdot 4^{\frac {1}{6}} \sqrt {\frac {1}{3}} {\left (4^{\frac {1}{3}} + 2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right )}\right ) - 4^{\frac {2}{3}} x^{4} \log \left (4^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {2}{3}}\right ) + 2 \cdot 4^{\frac {2}{3}} x^{4} \log \left (-4^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right ) - 16 \, \sqrt {3} x^{4} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + 8 \, x^{4} \log \left ({\left (-x^{2} + 1\right )}^{\frac {2}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) - 16 \, x^{4} \log \left ({\left (-x^{2} + 1\right )}^{\frac {1}{3}} - 1\right ) + 12 \, {\left (2 \, x^{2} + 3\right )} {\left (-x^{2} + 1\right )}^{\frac {2}{3}}}{432 \, x^{4}} \] Input:

integrate(1/x^5/(-x^2+1)^(1/3)/(x^2+3),x, algorithm="fricas")
 

Output:

-1/432*(12*4^(1/6)*sqrt(1/3)*x^4*arctan(1/2*4^(1/6)*sqrt(1/3)*(4^(1/3) + 2 
*(-x^2 + 1)^(1/3))) - 4^(2/3)*x^4*log(4^(2/3) + 4^(1/3)*(-x^2 + 1)^(1/3) + 
 (-x^2 + 1)^(2/3)) + 2*4^(2/3)*x^4*log(-4^(1/3) + (-x^2 + 1)^(1/3)) - 16*s 
qrt(3)*x^4*arctan(2/3*sqrt(3)*(-x^2 + 1)^(1/3) + 1/3*sqrt(3)) + 8*x^4*log( 
(-x^2 + 1)^(2/3) + (-x^2 + 1)^(1/3) + 1) - 16*x^4*log((-x^2 + 1)^(1/3) - 1 
) + 12*(2*x^2 + 3)*(-x^2 + 1)^(2/3))/x^4
 

Sympy [F]

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

integrate(1/x**5/(-x**2+1)**(1/3)/(x**2+3),x)
 

Output:

Integral(1/(x**5*(-(x - 1)*(x + 1))**(1/3)*(x**2 + 3)), x)
 

Maxima [F]

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

integrate(1/x^5/(-x^2+1)^(1/3)/(x^2+3),x, algorithm="maxima")
 

Output:

integrate(1/((x^2 + 3)*(-x^2 + 1)^(1/3)*x^5), x)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.99 \[ \int \frac {1}{x^5 \sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=-\frac {1}{216} \cdot 4^{\frac {2}{3}} \sqrt {3} \arctan \left (\frac {1}{12} \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (4^{\frac {1}{3}} + 2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right )}\right ) + \frac {1}{432} \cdot 4^{\frac {2}{3}} \log \left (4^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {2}{3}}\right ) - \frac {1}{216} \cdot 4^{\frac {2}{3}} \log \left (4^{\frac {1}{3}} - {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right ) + \frac {1}{27} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {2 \, {\left (-x^{2} + 1\right )}^{\frac {5}{3}} - 5 \, {\left (-x^{2} + 1\right )}^{\frac {2}{3}}}{36 \, x^{4}} - \frac {1}{54} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac {2}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{27} \, \log \left (-{\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) \] Input:

integrate(1/x^5/(-x^2+1)^(1/3)/(x^2+3),x, algorithm="giac")
 

Output:

-1/216*4^(2/3)*sqrt(3)*arctan(1/12*4^(2/3)*sqrt(3)*(4^(1/3) + 2*(-x^2 + 1) 
^(1/3))) + 1/432*4^(2/3)*log(4^(2/3) + 4^(1/3)*(-x^2 + 1)^(1/3) + (-x^2 + 
1)^(2/3)) - 1/216*4^(2/3)*log(4^(1/3) - (-x^2 + 1)^(1/3)) + 1/27*sqrt(3)*a 
rctan(1/3*sqrt(3)*(2*(-x^2 + 1)^(1/3) + 1)) + 1/36*(2*(-x^2 + 1)^(5/3) - 5 
*(-x^2 + 1)^(2/3))/x^4 - 1/54*log((-x^2 + 1)^(2/3) + (-x^2 + 1)^(1/3) + 1) 
 + 1/27*log(-(-x^2 + 1)^(1/3) + 1)
 

Mupad [B] (verification not implemented)

Time = 1.68 (sec) , antiderivative size = 397, normalized size of antiderivative = 2.23 \[ \int \frac {1}{x^5 \sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx =\text {Too large to display} \] Input:

int(1/(x^5*(1 - x^2)^(1/3)*(x^2 + 3)),x)
 

Output:

log(11/486 - (11*(1 - x^2)^(1/3))/486)/27 - (2^(1/3)*log(- (2^(2/3)*((2^(1 
/3)*((135*2^(2/3))/4 - (1755*(1 - x^2)^(1/3))/4))/108 + 7/2))/11664 - (1 - 
 x^2)^(1/3)/2916))/108 + log(((3^(1/2)*1i)/54 - 1/54)^2*(((3^(1/2)*1i)/54 
- 1/54)*(393660*((3^(1/2)*1i)/54 - 1/54)^2 - (1755*(1 - x^2)^(1/3))/4) - 7 
/2) - (1 - x^2)^(1/3)/2916)*((3^(1/2)*1i)/54 - 1/54) - log(- ((3^(1/2)*1i) 
/54 + 1/54)^2*(((3^(1/2)*1i)/54 + 1/54)*(393660*((3^(1/2)*1i)/54 + 1/54)^2 
 - (1755*(1 - x^2)^(1/3))/4) + 7/2) - (1 - x^2)^(1/3)/2916)*((3^(1/2)*1i)/ 
54 + 1/54) - ((5*(1 - x^2)^(2/3))/36 - (1 - x^2)^(5/3)/18)/((x^2 - 1)^2 + 
2*x^2 - 1) + ((-1)^(1/3)*2^(1/3)*log(((-1)^(2/3)*2^(2/3)*(((-1)^(1/3)*2^(1 
/3)*((135*(-1)^(2/3)*2^(2/3))/4 - (1755*(1 - x^2)^(1/3))/4))/108 - 7/2))/1 
1664 - (1 - x^2)^(1/3)/2916))/108 - ((-1)^(1/3)*2^(1/3)*log(((-1)^(2/3)*2^ 
(2/3)*(3^(1/2)*1i + 1)^2*(((-1)^(1/3)*2^(1/3)*(3^(1/2)*1i + 1)*((1755*(1 - 
 x^2)^(1/3))/4 - (135*(-1)^(2/3)*2^(2/3)*(3^(1/2)*1i + 1)^2)/16))/216 - 7/ 
2))/46656 - (1 - x^2)^(1/3)/2916)*(3^(1/2)*1i + 1))/216
 

Reduce [F]

\[ \int \frac {1}{x^5 \sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\int \frac {1}{\left (-x^{2}+1\right )^{\frac {1}{3}} x^{7}+3 \left (-x^{2}+1\right )^{\frac {1}{3}} x^{5}}d x \] Input:

int(1/x^5/(-x^2+1)^(1/3)/(x^2+3),x)
 

Output:

int(1/(( - x**2 + 1)**(1/3)*x**7 + 3*( - x**2 + 1)**(1/3)*x**5),x)