\(\int \frac {1}{x^3 \sqrt [3]{1-x^2} (3+x^2)} \, dx\) [1404]

Optimal result

Integrand size = 22, antiderivative size = 103 \[ \int \frac {1}{x^3 \sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=-\frac {\left (1-x^2\right )^{2/3}}{6 x^2}+\frac {\arctan \left (\frac {1+\sqrt [3]{2} \sqrt [3]{1-x^2}}{\sqrt {3}}\right )}{6\ 2^{2/3} \sqrt {3}}-\frac {\log \left (3+x^2\right )}{36\ 2^{2/3}}+\frac {\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{12\ 2^{2/3}} \] Output:

-1/6*(-x^2+1)^(2/3)/x^2+1/36*3^(1/2)*arctan(1/3*(1+2^(1/3)*(-x^2+1)^(1/3)) 
*3^(1/2))*2^(1/3)-1/72*ln(x^2+3)*2^(1/3)+1/24*ln(2^(2/3)-(-x^2+1)^(1/3))*2 
^(1/3)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x^3 \sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\frac {1}{72} \left (-\frac {12 \left (1-x^2\right )^{2/3}}{x^2}+2 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {1+\sqrt [3]{2-2 x^2}}{\sqrt {3}}\right )+2 \sqrt [3]{2} \log \left (-2+\sqrt [3]{2-2 x^2}\right )-\sqrt [3]{2} \log \left (4+2 \sqrt [3]{2-2 x^2}+\left (2-2 x^2\right )^{2/3}\right )\right ) \] Input:

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

Output:

((-12*(1 - x^2)^(2/3))/x^2 + 2*2^(1/3)*Sqrt[3]*ArcTan[(1 + (2 - 2*x^2)^(1/ 
3))/Sqrt[3]] + 2*2^(1/3)*Log[-2 + (2 - 2*x^2)^(1/3)] - 2^(1/3)*Log[4 + 2*( 
2 - 2*x^2)^(1/3) + (2 - 2*x^2)^(2/3)])/72
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {354, 114, 27, 67, 16, 1082, 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.

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

\(\Big \downarrow \) 354

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 67

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

Input:

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

Output:

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

Maple [A] (verified)

Time = 11.47 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.38

method	result	size
pseudoelliptic	\(\frac {-2 \sqrt {3}\, 2^{\frac {1}{3}} \arctan \left (\frac {\left (1+2^{\frac {1}{3}} \left (-x^{2}+1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right ) x^{2}-2 \,2^{\frac {1}{3}} \ln \left (\left (-x^{2}+1\right )^{\frac {1}{3}}-2^{\frac {2}{3}}\right ) x^{2}+2^{\frac {1}{3}} \ln \left (\left (-x^{2}+1\right )^{\frac {2}{3}}+2^{\frac {2}{3}} \left (-x^{2}+1\right )^{\frac {1}{3}}+2 \,2^{\frac {1}{3}}\right ) x^{2}+12 \left (-x^{2}+1\right )^{\frac {2}{3}}}{72 \left (-1+\left (-x^{2}+1\right )^{\frac {1}{3}}\right ) \left (1+\left (-x^{2}+1\right )^{\frac {1}{3}}+\left (-x^{2}+1\right )^{\frac {2}{3}}\right )}\)	\(142\)
risch	\(\frac {x^{2}-1}{6 x^{2} \left (-x^{2}+1\right )^{\frac {1}{3}}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right ) \ln \left (-\frac {48 {\operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2}+4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+16 \textit {\_Z}^{2}\right )}^{2} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2} x^{2}+8 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2}+4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+16 \textit {\_Z}^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{3} x^{2}-84 \left (-x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2}+4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+16 \textit {\_Z}^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )-6 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2}+4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+16 \textit {\_Z}^{2}\right ) x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right ) x^{2}-21 \left (-x^{2}+1\right )^{\frac {2}{3}}+126 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2}+4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+16 \textit {\_Z}^{2}\right )+21 \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )}{x^{2}+3}\right )}{36}+\frac {\operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2}+4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+16 \textit {\_Z}^{2}\right ) \ln \left (\frac {64 {\operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2}+4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+16 \textit {\_Z}^{2}\right )}^{2} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2} x^{2}+24 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2}+4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+16 \textit {\_Z}^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{3} x^{2}+168 \left (-x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2}+4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+16 \textit {\_Z}^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+40 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2}+4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+16 \textit {\_Z}^{2}\right ) x^{2}+15 \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right ) x^{2}+42 \left (-x^{2}+1\right )^{\frac {2}{3}}-168 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2}+4 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+16 \textit {\_Z}^{2}\right )-63 \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )}{x^{2}+3}\right )}{9}\)	\(480\)
trager	\(\text {Expression too large to display}\)	\(737\)

Input:

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

Output:

1/72*(-2*3^(1/2)*2^(1/3)*arctan(1/3*(1+2^(1/3)*(-x^2+1)^(1/3))*3^(1/2))*x^ 
2-2*2^(1/3)*ln((-x^2+1)^(1/3)-2^(2/3))*x^2+2^(1/3)*ln((-x^2+1)^(2/3)+2^(2/ 
3)*(-x^2+1)^(1/3)+2*2^(1/3))*x^2+12*(-x^2+1)^(2/3))/(-1+(-x^2+1)^(1/3))/(1 
+(-x^2+1)^(1/3)+(-x^2+1)^(2/3))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^3 \sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\frac {12 \cdot 4^{\frac {1}{6}} \sqrt {\frac {1}{3}} x^{2} \arctan \left (\frac {1}{2} \cdot 4^{\frac {1}{6}} \sqrt {\frac {1}{3}} {\left (4^{\frac {1}{3}} + 2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right )}\right ) - 4^{\frac {2}{3}} x^{2} \log \left (4^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {2}{3}}\right ) + 2 \cdot 4^{\frac {2}{3}} x^{2} \log \left (-4^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right ) - 24 \, {\left (-x^{2} + 1\right )}^{\frac {2}{3}}}{144 \, x^{2}} \] Input:

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

Output:

1/144*(12*4^(1/6)*sqrt(1/3)*x^2*arctan(1/2*4^(1/6)*sqrt(1/3)*(4^(1/3) + 2* 
(-x^2 + 1)^(1/3))) - 4^(2/3)*x^2*log(4^(2/3) + 4^(1/3)*(-x^2 + 1)^(1/3) + 
(-x^2 + 1)^(2/3)) + 2*4^(2/3)*x^2*log(-4^(1/3) + (-x^2 + 1)^(1/3)) - 24*(- 
x^2 + 1)^(2/3))/x^2
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.97 \[ \int \frac {1}{x^3 \sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\frac {1}{72} \cdot 4^{\frac {2}{3}} \sqrt {3} \arctan \left (\frac {1}{12} \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (4^{\frac {1}{3}} + 2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right )}\right ) - \frac {1}{144} \cdot 4^{\frac {2}{3}} \log \left (4^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {2}{3}}\right ) + \frac {1}{72} \cdot 4^{\frac {2}{3}} \log \left (4^{\frac {1}{3}} - {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right ) - \frac {{\left (-x^{2} + 1\right )}^{\frac {2}{3}}}{6 \, x^{2}} \] Input:

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

Output:

1/72*4^(2/3)*sqrt(3)*arctan(1/12*4^(2/3)*sqrt(3)*(4^(1/3) + 2*(-x^2 + 1)^( 
1/3))) - 1/144*4^(2/3)*log(4^(2/3) + 4^(1/3)*(-x^2 + 1)^(1/3) + (-x^2 + 1) 
^(2/3)) + 1/72*4^(2/3)*log(4^(1/3) - (-x^2 + 1)^(1/3)) - 1/6*(-x^2 + 1)^(2 
/3)/x^2
 

Mupad [B] (verification not implemented)

Time = 1.55 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.17 \[ \int \frac {1}{x^3 \sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\frac {2^{1/3}\,\ln \left (\frac {{\left (1-x^2\right )}^{1/3}}{36}-\frac {2^{2/3}}{36}\right )}{36}-\frac {{\left (1-x^2\right )}^{2/3}}{6\,x^2}+\frac {2^{1/3}\,\ln \left (\frac {{\left (1-x^2\right )}^{1/3}}{36}-\frac {2^{2/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{144}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{72}-\frac {2^{1/3}\,\ln \left (\frac {{\left (1-x^2\right )}^{1/3}}{36}-\frac {2^{2/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{144}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{72} \] Input:

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

Output:

(2^(1/3)*log((1 - x^2)^(1/3)/36 - 2^(2/3)/36))/36 - (1 - x^2)^(2/3)/(6*x^2 
) + (2^(1/3)*log((1 - x^2)^(1/3)/36 - (2^(2/3)*(3^(1/2)*1i - 1)^2)/144)*(3 
^(1/2)*1i - 1))/72 - (2^(1/3)*log((1 - x^2)^(1/3)/36 - (2^(2/3)*(3^(1/2)*1 
i + 1)^2)/144)*(3^(1/2)*1i + 1))/72
 

Reduce [F]

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

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

Output:

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