\(\int \frac {1}{x \sqrt [3]{1-x^2} (3+x^2)^2} \, dx\) [1023]

Optimal result

Integrand size = 22, antiderivative size = 158 \[ \int \frac {1}{x \sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx=\frac {\left (1-x^2\right )^{2/3}}{24 \left (3+x^2\right )}-\frac {5 \arctan \left (\frac {1+\sqrt [3]{2-2 x^2}}{\sqrt {3}}\right )}{24\ 2^{2/3} \sqrt {3}}+\frac {\arctan \left (\frac {1+2 \sqrt [3]{1-x^2}}{\sqrt {3}}\right )}{6 \sqrt {3}}-\frac {\log (x)}{18}+\frac {5 \log \left (3+x^2\right )}{144\ 2^{2/3}}+\frac {1}{12} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac {5 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{48\ 2^{2/3}} \]

[Out]

1/24*(-x^2+1)^(2/3)/(x^2+3)-1/18*ln(x)+5/288*ln(x^2+3)*2^(1/3)+1/12*ln(1-(-x^2+1)^(1/3))-5/96*ln(2^(2/3)-(-x^2
+1)^(1/3))*2^(1/3)-5/144*arctan(1/3*(1+(-2*x^2+2)^(1/3))*3^(1/2))*3^(1/2)*2^(1/3)+1/18*arctan(1/3*(1+2*(-x^2+1
)^(1/3))*3^(1/2))*3^(1/2)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {457, 105, 162, 57, 632, 210, 31, 631} \[ \int \frac {1}{x \sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx=-\frac {5 \arctan \left (\frac {\sqrt [3]{2-2 x^2}+1}{\sqrt {3}}\right )}{24\ 2^{2/3} \sqrt {3}}+\frac {\arctan \left (\frac {2 \sqrt [3]{1-x^2}+1}{\sqrt {3}}\right )}{6 \sqrt {3}}+\frac {\left (1-x^2\right )^{2/3}}{24 \left (x^2+3\right )}+\frac {5 \log \left (x^2+3\right )}{144\ 2^{2/3}}+\frac {1}{12} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac {5 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{48\ 2^{2/3}}-\frac {\log (x)}{18} \]

[In]

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

[Out]

(1 - x^2)^(2/3)/(24*(3 + x^2)) - (5*ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]])/(24*2^(2/3)*Sqrt[3]) + ArcTan[(1
+ 2*(1 - x^2)^(1/3))/Sqrt[3]]/(6*Sqrt[3]) - Log[x]/18 + (5*Log[3 + x^2])/(144*2^(2/3)) + Log[1 - (1 - x^2)^(1/
3)]/12 - (5*Log[2^(2/3) - (1 - x^2)^(1/3)])/(48*2^(2/3))

Rule 31

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

Rule 57

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, Simp[-L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[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 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> 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] + Dist[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 162

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(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 210

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 457

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

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[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])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} x (3+x)^2} \, dx,x,x^2\right ) \\ & = \frac {\left (1-x^2\right )^{2/3}}{24 \left (3+x^2\right )}+\frac {1}{24} \text {Subst}\left (\int \frac {4-\frac {x}{3}}{\sqrt [3]{1-x} x (3+x)} \, dx,x,x^2\right ) \\ & = \frac {\left (1-x^2\right )^{2/3}}{24 \left (3+x^2\right )}+\frac {1}{18} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} x} \, dx,x,x^2\right )-\frac {5}{72} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} (3+x)} \, dx,x,x^2\right ) \\ & = \frac {\left (1-x^2\right )^{2/3}}{24 \left (3+x^2\right )}-\frac {\log (x)}{18}+\frac {5 \log \left (3+x^2\right )}{144\ 2^{2/3}}-\frac {1}{12} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1-x^2}\right )+\frac {1}{12} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1-x^2}\right )-\frac {5}{48} \text {Subst}\left (\int \frac {1}{2 \sqrt [3]{2}+2^{2/3} x+x^2} \, dx,x,\sqrt [3]{1-x^2}\right )+\frac {5 \text {Subst}\left (\int \frac {1}{2^{2/3}-x} \, dx,x,\sqrt [3]{1-x^2}\right )}{48\ 2^{2/3}} \\ & = \frac {\left (1-x^2\right )^{2/3}}{24 \left (3+x^2\right )}-\frac {\log (x)}{18}+\frac {5 \log \left (3+x^2\right )}{144\ 2^{2/3}}+\frac {1}{12} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac {5 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{48\ 2^{2/3}}-\frac {1}{6} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1-x^2}\right )+\frac {5 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\sqrt [3]{2-2 x^2}\right )}{24\ 2^{2/3}} \\ & = \frac {\left (1-x^2\right )^{2/3}}{24 \left (3+x^2\right )}-\frac {5 \tan ^{-1}\left (\frac {1+\sqrt [3]{2-2 x^2}}{\sqrt {3}}\right )}{24\ 2^{2/3} \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {1+2 \sqrt [3]{1-x^2}}{\sqrt {3}}\right )}{6 \sqrt {3}}-\frac {\log (x)}{18}+\frac {5 \log \left (3+x^2\right )}{144\ 2^{2/3}}+\frac {1}{12} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac {5 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{48\ 2^{2/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.16 \[ \int \frac {1}{x \sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx=\frac {1}{288} \left (\frac {12 \left (1-x^2\right )^{2/3}}{3+x^2}-10 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {1+\sqrt [3]{2-2 x^2}}{\sqrt {3}}\right )+16 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{1-x^2}}{\sqrt {3}}\right )-10 \sqrt [3]{2} \log \left (-2+\sqrt [3]{2-2 x^2}\right )+5 \sqrt [3]{2} \log \left (4+2 \sqrt [3]{2-2 x^2}+\left (2-2 x^2\right )^{2/3}\right )+16 \log \left (-1+\sqrt [3]{1-x^2}\right )-8 \log \left (1+\sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}\right )\right ) \]

[In]

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

[Out]

((12*(1 - x^2)^(2/3))/(3 + x^2) - 10*2^(1/3)*Sqrt[3]*ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]] + 16*Sqrt[3]*ArcT
an[(1 + 2*(1 - x^2)^(1/3))/Sqrt[3]] - 10*2^(1/3)*Log[-2 + (2 - 2*x^2)^(1/3)] + 5*2^(1/3)*Log[4 + 2*(2 - 2*x^2)
^(1/3) + (2 - 2*x^2)^(2/3)] + 16*Log[-1 + (1 - x^2)^(1/3)] - 8*Log[1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3)])/288

Maple [A] (verified)

Time = 4.76 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.37

method	result	size
pseudoelliptic	\(\frac {-5 \left (x^{2}+3\right ) \left (2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (1+2^{\frac {1}{3}} \left (-x^{2}+1\right )^{\frac {1}{3}}\right )}{3}\right )+2 \ln \left (\left (-x^{2}+1\right )^{\frac {1}{3}}-2^{\frac {2}{3}}\right )-\ln \left (\left (-x^{2}+1\right )^{\frac {2}{3}}+2^{\frac {2}{3}} \left (-x^{2}+1\right )^{\frac {1}{3}}+2 \,2^{\frac {1}{3}}\right )\right ) 2^{\frac {1}{3}}+16 \arctan \left (\frac {\left (1+2 \left (-x^{2}+1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right ) \left (x^{2}+3\right ) \sqrt {3}+8 \left (2 \ln \left (-1+\left (-x^{2}+1\right )^{\frac {1}{3}}\right )-\ln \left (1+\left (-x^{2}+1\right )^{\frac {1}{3}}+\left (-x^{2}+1\right )^{\frac {2}{3}}\right )\right ) x^{2}+12 \left (-x^{2}+1\right )^{\frac {2}{3}}+48 \ln \left (-1+\left (-x^{2}+1\right )^{\frac {1}{3}}\right )-24 \ln \left (1+\left (-x^{2}+1\right )^{\frac {1}{3}}+\left (-x^{2}+1\right )^{\frac {2}{3}}\right )}{288 x^{2}+864}\)	\(217\)

[In]

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

[Out]

1/288*(-5*(x^2+3)*(2*3^(1/2)*arctan(1/3*3^(1/2)*(1+2^(1/3)*(-x^2+1)^(1/3)))+2*ln((-x^2+1)^(1/3)-2^(2/3))-ln((-
x^2+1)^(2/3)+2^(2/3)*(-x^2+1)^(1/3)+2*2^(1/3)))*2^(1/3)+16*arctan(1/3*(1+2*(-x^2+1)^(1/3))*3^(1/2))*(x^2+3)*3^
(1/2)+8*(2*ln(-1+(-x^2+1)^(1/3))-ln(1+(-x^2+1)^(1/3)+(-x^2+1)^(2/3)))*x^2+12*(-x^2+1)^(2/3)+48*ln(-1+(-x^2+1)^
(1/3))-24*ln(1+(-x^2+1)^(1/3)+(-x^2+1)^(2/3)))/(x^2+3)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.44 \[ \int \frac {1}{x \sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx=-\frac {20 \cdot 4^{\frac {1}{6}} \sqrt {3} \left (-1\right )^{\frac {1}{3}} {\left (x^{2} + 3\right )} \arctan \left (\frac {1}{6} \cdot 4^{\frac {1}{6}} {\left (2 \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} - 4^{\frac {1}{3}} \sqrt {3}\right )}\right ) + 5 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{2} + 3\right )} \log \left (4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} - 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {2}{3}}\right ) - 10 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{2} + 3\right )} \log \left (-4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right ) - 32 \, \sqrt {3} {\left (x^{2} + 3\right )} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + 16 \, {\left (x^{2} + 3\right )} \log \left ({\left (-x^{2} + 1\right )}^{\frac {2}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) - 32 \, {\left (x^{2} + 3\right )} \log \left ({\left (-x^{2} + 1\right )}^{\frac {1}{3}} - 1\right ) - 24 \, {\left (-x^{2} + 1\right )}^{\frac {2}{3}}}{576 \, {\left (x^{2} + 3\right )}} \]

[In]

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

[Out]

-1/576*(20*4^(1/6)*sqrt(3)*(-1)^(1/3)*(x^2 + 3)*arctan(1/6*4^(1/6)*(2*sqrt(3)*(-1)^(1/3)*(-x^2 + 1)^(1/3) - 4^
(1/3)*sqrt(3))) + 5*4^(2/3)*(-1)^(1/3)*(x^2 + 3)*log(4^(1/3)*(-1)^(2/3)*(-x^2 + 1)^(1/3) - 4^(2/3)*(-1)^(1/3)
+ (-x^2 + 1)^(2/3)) - 10*4^(2/3)*(-1)^(1/3)*(x^2 + 3)*log(-4^(1/3)*(-1)^(2/3) + (-x^2 + 1)^(1/3)) - 32*sqrt(3)
*(x^2 + 3)*arctan(2/3*sqrt(3)*(-x^2 + 1)^(1/3) + 1/3*sqrt(3)) + 16*(x^2 + 3)*log((-x^2 + 1)^(2/3) + (-x^2 + 1)
^(1/3) + 1) - 32*(x^2 + 3)*log((-x^2 + 1)^(1/3) - 1) - 24*(-x^2 + 1)^(2/3))/(x^2 + 3)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.06 \[ \int \frac {1}{x \sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx=-\frac {5}{288} \cdot 4^{\frac {2}{3}} \sqrt {3} \arctan \left (\frac {1}{12} \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (4^{\frac {1}{3}} + 2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right )}\right ) + \frac {5}{576} \cdot 4^{\frac {2}{3}} \log \left (4^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {2}{3}}\right ) - \frac {5}{288} \cdot 4^{\frac {2}{3}} \log \left (4^{\frac {1}{3}} - {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right ) + \frac {1}{18} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {{\left (-x^{2} + 1\right )}^{\frac {2}{3}}}{24 \, {\left (x^{2} + 3\right )}} - \frac {1}{36} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac {2}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{18} \, \log \left (-{\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) \]

[In]

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

[Out]

-5/288*4^(2/3)*sqrt(3)*arctan(1/12*4^(2/3)*sqrt(3)*(4^(1/3) + 2*(-x^2 + 1)^(1/3))) + 5/576*4^(2/3)*log(4^(2/3)
+ 4^(1/3)*(-x^2 + 1)^(1/3) + (-x^2 + 1)^(2/3)) - 5/288*4^(2/3)*log(4^(1/3) - (-x^2 + 1)^(1/3)) + 1/18*sqrt(3)
*arctan(1/3*sqrt(3)*(2*(-x^2 + 1)^(1/3) + 1)) + 1/24*(-x^2 + 1)^(2/3)/(x^2 + 3) - 1/36*log((-x^2 + 1)^(2/3) +
(-x^2 + 1)^(1/3) + 1) + 1/18*log(-(-x^2 + 1)^(1/3) + 1)

Mupad [B] (verification not implemented)

Time = 5.38 (sec) , antiderivative size = 375, normalized size of antiderivative = 2.37 \[ \int \frac {1}{x \sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx=\frac {\ln \left (\frac {127}{512}-\frac {127\,{\left (1-x^2\right )}^{1/3}}{512}\right )}{18}-\frac {5\,2^{1/3}\,\ln \left (-\frac {25\,2^{2/3}\,\left (\frac {5\,2^{1/3}\,\left (\frac {30375\,2^{2/3}}{64}-\frac {68283\,{\left (1-x^2\right )}^{1/3}}{64}\right )}{144}-\frac {1647}{128}\right )}{20736}-\frac {25\,{\left (1-x^2\right )}^{1/3}}{384}\right )}{144}+\ln \left ({\left (-\frac {1}{36}+\frac {\sqrt {3}\,1{}\mathrm {i}}{36}\right )}^2\,\left (\left (-\frac {1}{36}+\frac {\sqrt {3}\,1{}\mathrm {i}}{36}\right )\,\left (393660\,{\left (-\frac {1}{36}+\frac {\sqrt {3}\,1{}\mathrm {i}}{36}\right )}^2-\frac {68283\,{\left (1-x^2\right )}^{1/3}}{64}\right )+\frac {1647}{128}\right )-\frac {25\,{\left (1-x^2\right )}^{1/3}}{384}\right )\,\left (-\frac {1}{36}+\frac {\sqrt {3}\,1{}\mathrm {i}}{36}\right )-\ln \left (-{\left (\frac {1}{36}+\frac {\sqrt {3}\,1{}\mathrm {i}}{36}\right )}^2\,\left (\left (\frac {1}{36}+\frac {\sqrt {3}\,1{}\mathrm {i}}{36}\right )\,\left (393660\,{\left (\frac {1}{36}+\frac {\sqrt {3}\,1{}\mathrm {i}}{36}\right )}^2-\frac {68283\,{\left (1-x^2\right )}^{1/3}}{64}\right )-\frac {1647}{128}\right )-\frac {25\,{\left (1-x^2\right )}^{1/3}}{384}\right )\,\left (\frac {1}{36}+\frac {\sqrt {3}\,1{}\mathrm {i}}{36}\right )+\frac {{\left (1-x^2\right )}^{2/3}}{24\,\left (x^2+3\right )}+\frac {5\,{\left (-1\right )}^{1/3}\,2^{1/3}\,\ln \left (\frac {25\,{\left (-1\right )}^{2/3}\,2^{2/3}\,\left (\frac {5\,{\left (-1\right )}^{1/3}\,2^{1/3}\,\left (\frac {30375\,{\left (-1\right )}^{2/3}\,2^{2/3}}{64}-\frac {68283\,{\left (1-x^2\right )}^{1/3}}{64}\right )}{144}+\frac {1647}{128}\right )}{20736}-\frac {25\,{\left (1-x^2\right )}^{1/3}}{384}\right )}{144}-\frac {5\,{\left (-1\right )}^{1/3}\,2^{1/3}\,\ln \left (-\frac {25\,{\left (1-x^2\right )}^{1/3}}{384}+\frac {25\,{\left (-1\right )}^{2/3}\,2^{2/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (\frac {5\,{\left (-1\right )}^{1/3}\,2^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (\frac {68283\,{\left (1-x^2\right )}^{1/3}}{64}-\frac {30375\,{\left (-1\right )}^{2/3}\,2^{2/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{256}\right )}{288}+\frac {1647}{128}\right )}{82944}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{288} \]

[In]

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

[Out]

log(127/512 - (127*(1 - x^2)^(1/3))/512)/18 - (5*2^(1/3)*log(- (25*2^(2/3)*((5*2^(1/3)*((30375*2^(2/3))/64 - (
68283*(1 - x^2)^(1/3))/64))/144 - 1647/128))/20736 - (25*(1 - x^2)^(1/3))/384))/144 + log(((3^(1/2)*1i)/36 - 1
/36)^2*(((3^(1/2)*1i)/36 - 1/36)*(393660*((3^(1/2)*1i)/36 - 1/36)^2 - (68283*(1 - x^2)^(1/3))/64) + 1647/128)
- (25*(1 - x^2)^(1/3))/384)*((3^(1/2)*1i)/36 - 1/36) - log(- ((3^(1/2)*1i)/36 + 1/36)^2*(((3^(1/2)*1i)/36 + 1/
36)*(393660*((3^(1/2)*1i)/36 + 1/36)^2 - (68283*(1 - x^2)^(1/3))/64) - 1647/128) - (25*(1 - x^2)^(1/3))/384)*(
(3^(1/2)*1i)/36 + 1/36) + (1 - x^2)^(2/3)/(24*(x^2 + 3)) + (5*(-1)^(1/3)*2^(1/3)*log((25*(-1)^(2/3)*2^(2/3)*((
5*(-1)^(1/3)*2^(1/3)*((30375*(-1)^(2/3)*2^(2/3))/64 - (68283*(1 - x^2)^(1/3))/64))/144 + 1647/128))/20736 - (2
5*(1 - x^2)^(1/3))/384))/144 - (5*(-1)^(1/3)*2^(1/3)*log((25*(-1)^(2/3)*2^(2/3)*(3^(1/2)*1i + 1)^2*((5*(-1)^(1
/3)*2^(1/3)*(3^(1/2)*1i + 1)*((68283*(1 - x^2)^(1/3))/64 - (30375*(-1)^(2/3)*2^(2/3)*(3^(1/2)*1i + 1)^2)/256))
/288 + 1647/128))/82944 - (25*(1 - x^2)^(1/3))/384)*(3^(1/2)*1i + 1))/288