3.11.24 \(\int \frac {1}{x^3 \sqrt [3]{1-x^2} (3+x^2)^2} \, dx\) [1024]

Optimal. Leaf size=183 \[ -\frac {5 \left (1-x^2\right )^{2/3}}{72 \left (3+x^2\right )}-\frac {\left (1-x^2\right )^{2/3}}{6 x^2 \left (3+x^2\right )}+\frac {\tan ^{-1}\left (\frac {1+\sqrt [3]{2-2 x^2}}{\sqrt {3}}\right )}{8\ 2^{2/3} \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {1+2 \sqrt [3]{1-x^2}}{\sqrt {3}}\right )}{18 \sqrt {3}}+\frac {\log (x)}{54}-\frac {\log \left (3+x^2\right )}{48\ 2^{2/3}}-\frac {1}{36} \log \left (1-\sqrt [3]{1-x^2}\right )+\frac {\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{16\ 2^{2/3}} \]

[Out]

-5/72*(-x^2+1)^(2/3)/(x^2+3)-1/6*(-x^2+1)^(2/3)/x^2/(x^2+3)+1/54*ln(x)-1/96*ln(x^2+3)*2^(1/3)-1/36*ln(1-(-x^2+
1)^(1/3))+1/32*ln(2^(2/3)-(-x^2+1)^(1/3))*2^(1/3)+1/48*arctan(1/3*(1+(-2*x^2+2)^(1/3))*3^(1/2))*3^(1/2)*2^(1/3
)-1/54*arctan(1/3*(1+2*(-x^2+1)^(1/3))*3^(1/2))*3^(1/2)

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Rubi [A]
time = 0.09, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {457, 105, 156, 162, 57, 632, 210, 31, 631} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt [3]{2-2 x^2}+1}{\sqrt {3}}\right )}{8\ 2^{2/3} \sqrt {3}}-\frac {\text {ArcTan}\left (\frac {2 \sqrt [3]{1-x^2}+1}{\sqrt {3}}\right )}{18 \sqrt {3}}-\frac {\left (1-x^2\right )^{2/3}}{6 x^2 \left (x^2+3\right )}-\frac {5 \left (1-x^2\right )^{2/3}}{72 \left (x^2+3\right )}-\frac {\log \left (x^2+3\right )}{48\ 2^{2/3}}-\frac {1}{36} \log \left (1-\sqrt [3]{1-x^2}\right )+\frac {\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{16\ 2^{2/3}}+\frac {\log (x)}{54} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*(1 - x^2)^(2/3))/(72*(3 + x^2)) - (1 - x^2)^(2/3)/(6*x^2*(3 + x^2)) + ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[
3]]/(8*2^(2/3)*Sqrt[3]) - ArcTan[(1 + 2*(1 - x^2)^(1/3))/Sqrt[3]]/(18*Sqrt[3]) + Log[x]/54 - Log[3 + x^2]/(48*
2^(2/3)) - Log[1 - (1 - x^2)^(1/3)]/36 + Log[2^(2/3) - (1 - x^2)^(1/3)]/(16*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 156

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> 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] + 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*
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 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*} \int \frac {1}{x^3 \sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} x^2 (3+x)^2} \, dx,x,x^2\right )\\ &=-\frac {\left (1-x^2\right )^{2/3}}{6 x^2 \left (3+x^2\right )}-\frac {1}{6} \text {Subst}\left (\int \frac {1-\frac {4 x}{3}}{\sqrt [3]{1-x} x (3+x)^2} \, dx,x,x^2\right )\\ &=-\frac {5 \left (1-x^2\right )^{2/3}}{72 \left (3+x^2\right )}-\frac {\left (1-x^2\right )^{2/3}}{6 x^2 \left (3+x^2\right )}-\frac {1}{72} \text {Subst}\left (\int \frac {4-\frac {5 x}{3}}{\sqrt [3]{1-x} x (3+x)} \, dx,x,x^2\right )\\ &=-\frac {5 \left (1-x^2\right )^{2/3}}{72 \left (3+x^2\right )}-\frac {\left (1-x^2\right )^{2/3}}{6 x^2 \left (3+x^2\right )}-\frac {1}{54} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} x} \, dx,x,x^2\right )+\frac {1}{24} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} (3+x)} \, dx,x,x^2\right )\\ &=-\frac {5 \left (1-x^2\right )^{2/3}}{72 \left (3+x^2\right )}-\frac {\left (1-x^2\right )^{2/3}}{6 x^2 \left (3+x^2\right )}+\frac {\log (x)}{54}-\frac {\log \left (3+x^2\right )}{48\ 2^{2/3}}+\frac {1}{36} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1-x^2}\right )-\frac {1}{36} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1-x^2}\right )+\frac {1}{16} \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 {\text {Subst}\left (\int \frac {1}{2^{2/3}-x} \, dx,x,\sqrt [3]{1-x^2}\right )}{16\ 2^{2/3}}\\ &=-\frac {5 \left (1-x^2\right )^{2/3}}{72 \left (3+x^2\right )}-\frac {\left (1-x^2\right )^{2/3}}{6 x^2 \left (3+x^2\right )}+\frac {\log (x)}{54}-\frac {\log \left (3+x^2\right )}{48\ 2^{2/3}}-\frac {1}{36} \log \left (1-\sqrt [3]{1-x^2}\right )+\frac {\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{16\ 2^{2/3}}+\frac {1}{18} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1-x^2}\right )-\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\sqrt [3]{2-2 x^2}\right )}{8\ 2^{2/3}}\\ &=-\frac {5 \left (1-x^2\right )^{2/3}}{72 \left (3+x^2\right )}-\frac {\left (1-x^2\right )^{2/3}}{6 x^2 \left (3+x^2\right )}+\frac {\tan ^{-1}\left (\frac {1+\sqrt [3]{2-2 x^2}}{\sqrt {3}}\right )}{8\ 2^{2/3} \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {1+2 \sqrt [3]{1-x^2}}{\sqrt {3}}\right )}{18 \sqrt {3}}+\frac {\log (x)}{54}-\frac {\log \left (3+x^2\right )}{48\ 2^{2/3}}-\frac {1}{36} \log \left (1-\sqrt [3]{1-x^2}\right )+\frac {\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{16\ 2^{2/3}}\\ \end {align*}

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Mathematica [A]
time = 0.36, size = 194, normalized size = 1.06 \begin {gather*} \frac {1}{864} \left (-\frac {12 \left (1-x^2\right )^{2/3} \left (12+5 x^2\right )}{x^2 \left (3+x^2\right )}+18 \sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {1+\sqrt [3]{2-2 x^2}}{\sqrt {3}}\right )-16 \sqrt {3} \tan ^{-1}\left (\frac {1+2 \sqrt [3]{1-x^2}}{\sqrt {3}}\right )+18 \sqrt [3]{2} \log \left (-2+\sqrt [3]{2-2 x^2}\right )-9 \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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-12*(1 - x^2)^(2/3)*(12 + 5*x^2))/(x^2*(3 + x^2)) + 18*2^(1/3)*Sqrt[3]*ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3
]] - 16*Sqrt[3]*ArcTan[(1 + 2*(1 - x^2)^(1/3))/Sqrt[3]] + 18*2^(1/3)*Log[-2 + (2 - 2*x^2)^(1/3)] - 9*2^(1/3)*L
og[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)])/864

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{3} \left (-x^{2}+1\right )^{\frac {1}{3}} \left (x^{2}+3\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(-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^3), x)

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Fricas [A]
time = 0.49, size = 238, normalized size = 1.30 \begin {gather*} \frac {36 \cdot 4^{\frac {1}{6}} \sqrt {3} {\left (x^{4} + 3 \, x^{2}\right )} \arctan \left (\frac {1}{6} \cdot 4^{\frac {1}{6}} {\left (4^{\frac {1}{3}} \sqrt {3} + 2 \, \sqrt {3} {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right )}\right ) - 9 \cdot 4^{\frac {2}{3}} {\left (x^{4} + 3 \, x^{2}\right )} \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 ) + 18 \cdot 4^{\frac {2}{3}} {\left (x^{4} + 3 \, x^{2}\right )} \log \left (-4^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right ) - 32 \, \sqrt {3} {\left (x^{4} + 3 \, x^{2}\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^{4} + 3 \, x^{2}\right )} \log \left ({\left (-x^{2} + 1\right )}^{\frac {2}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) - 32 \, {\left (x^{4} + 3 \, x^{2}\right )} \log \left ({\left (-x^{2} + 1\right )}^{\frac {1}{3}} - 1\right ) - 24 \, {\left (5 \, x^{2} + 12\right )} {\left (-x^{2} + 1\right )}^{\frac {2}{3}}}{1728 \, {\left (x^{4} + 3 \, x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1728*(36*4^(1/6)*sqrt(3)*(x^4 + 3*x^2)*arctan(1/6*4^(1/6)*(4^(1/3)*sqrt(3) + 2*sqrt(3)*(-x^2 + 1)^(1/3))) -
9*4^(2/3)*(x^4 + 3*x^2)*log(4^(2/3) + 4^(1/3)*(-x^2 + 1)^(1/3) + (-x^2 + 1)^(2/3)) + 18*4^(2/3)*(x^4 + 3*x^2)*
log(-4^(1/3) + (-x^2 + 1)^(1/3)) - 32*sqrt(3)*(x^4 + 3*x^2)*arctan(2/3*sqrt(3)*(-x^2 + 1)^(1/3) + 1/3*sqrt(3))
+ 16*(x^4 + 3*x^2)*log((-x^2 + 1)^(2/3) + (-x^2 + 1)^(1/3) + 1) - 32*(x^4 + 3*x^2)*log((-x^2 + 1)^(1/3) - 1)
- 24*(5*x^2 + 12)*(-x^2 + 1)^(2/3))/(x^4 + 3*x^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{3} \sqrt [3]{- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 3\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.08, size = 190, normalized size = 1.04 \begin {gather*} \frac {1}{96} \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}{192} \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}{96} \cdot 4^{\frac {2}{3}} \log \left (4^{\frac {1}{3}} - {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right ) - \frac {1}{54} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {5 \, {\left (-x^{2} + 1\right )}^{\frac {5}{3}} - 17 \, {\left (-x^{2} + 1\right )}^{\frac {2}{3}}}{72 \, {\left ({\left (x^{2} - 1\right )}^{2} + 5 \, x^{2} - 1\right )}} + \frac {1}{108} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac {2}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{54} \, \log \left (-{\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96*4^(2/3)*sqrt(3)*arctan(1/12*4^(2/3)*sqrt(3)*(4^(1/3) + 2*(-x^2 + 1)^(1/3))) - 1/192*4^(2/3)*log(4^(2/3) +
4^(1/3)*(-x^2 + 1)^(1/3) + (-x^2 + 1)^(2/3)) + 1/96*4^(2/3)*log(4^(1/3) - (-x^2 + 1)^(1/3)) - 1/54*sqrt(3)*ar
ctan(1/3*sqrt(3)*(2*(-x^2 + 1)^(1/3) + 1)) + 1/72*(5*(-x^2 + 1)^(5/3) - 17*(-x^2 + 1)^(2/3))/((x^2 - 1)^2 + 5*
x^2 - 1) + 1/108*log((-x^2 + 1)^(2/3) + (-x^2 + 1)^(1/3) + 1) - 1/54*log(-(-x^2 + 1)^(1/3) + 1)

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Mupad [B]
time = 0.54, size = 409, normalized size = 2.23 \begin {gather*} \frac {2^{1/3}\,\ln \left (\frac {2^{2/3}\,\left (\frac {2^{1/3}\,\left (\frac {10935\,2^{2/3}}{64}-\frac {9099\,{\left (1-x^2\right )}^{1/3}}{64}\right )}{48}-\frac {665}{128}\right )}{2304}+\frac {{\left (1-x^2\right )}^{1/3}}{576}\right )}{48}-\frac {\ln \left (\frac {985\,{\left (1-x^2\right )}^{1/3}}{373248}-\frac {985}{373248}\right )}{54}+\ln \left ({\left (\frac {1}{108}+\frac {\sqrt {3}\,1{}\mathrm {i}}{108}\right )}^2\,\left (\left (\frac {1}{108}+\frac {\sqrt {3}\,1{}\mathrm {i}}{108}\right )\,\left (393660\,{\left (\frac {1}{108}+\frac {\sqrt {3}\,1{}\mathrm {i}}{108}\right )}^2-\frac {9099\,{\left (1-x^2\right )}^{1/3}}{64}\right )-\frac {665}{128}\right )+\frac {{\left (1-x^2\right )}^{1/3}}{576}\right )\,\left (\frac {1}{108}+\frac {\sqrt {3}\,1{}\mathrm {i}}{108}\right )-\ln \left (\frac {{\left (1-x^2\right )}^{1/3}}{576}-{\left (-\frac {1}{108}+\frac {\sqrt {3}\,1{}\mathrm {i}}{108}\right )}^2\,\left (\left (-\frac {1}{108}+\frac {\sqrt {3}\,1{}\mathrm {i}}{108}\right )\,\left (393660\,{\left (-\frac {1}{108}+\frac {\sqrt {3}\,1{}\mathrm {i}}{108}\right )}^2-\frac {9099\,{\left (1-x^2\right )}^{1/3}}{64}\right )+\frac {665}{128}\right )\right )\,\left (-\frac {1}{108}+\frac {\sqrt {3}\,1{}\mathrm {i}}{108}\right )-\frac {\frac {17\,{\left (1-x^2\right )}^{2/3}}{72}-\frac {5\,{\left (1-x^2\right )}^{5/3}}{72}}{{\left (x^2-1\right )}^2+5\,x^2-1}+\frac {2^{1/3}\,\ln \left (\frac {{\left (1-x^2\right )}^{1/3}}{576}+\frac {2^{2/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (\frac {2^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (\frac {10935\,2^{2/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{256}-\frac {9099\,{\left (1-x^2\right )}^{1/3}}{64}\right )}{96}-\frac {665}{128}\right )}{9216}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{96}-\frac {2^{1/3}\,\ln \left (\frac {{\left (1-x^2\right )}^{1/3}}{576}-\frac {2^{2/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (\frac {2^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (\frac {10935\,2^{2/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{256}-\frac {9099\,{\left (1-x^2\right )}^{1/3}}{64}\right )}{96}+\frac {665}{128}\right )}{9216}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{96} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2^(1/3)*log((2^(2/3)*((2^(1/3)*((10935*2^(2/3))/64 - (9099*(1 - x^2)^(1/3))/64))/48 - 665/128))/2304 + (1 - x
^2)^(1/3)/576))/48 - log((985*(1 - x^2)^(1/3))/373248 - 985/373248)/54 + log(((3^(1/2)*1i)/108 + 1/108)^2*(((3
^(1/2)*1i)/108 + 1/108)*(393660*((3^(1/2)*1i)/108 + 1/108)^2 - (9099*(1 - x^2)^(1/3))/64) - 665/128) + (1 - x^
2)^(1/3)/576)*((3^(1/2)*1i)/108 + 1/108) - log((1 - x^2)^(1/3)/576 - ((3^(1/2)*1i)/108 - 1/108)^2*(((3^(1/2)*1
i)/108 - 1/108)*(393660*((3^(1/2)*1i)/108 - 1/108)^2 - (9099*(1 - x^2)^(1/3))/64) + 665/128))*((3^(1/2)*1i)/10
8 - 1/108) - ((17*(1 - x^2)^(2/3))/72 - (5*(1 - x^2)^(5/3))/72)/((x^2 - 1)^2 + 5*x^2 - 1) + (2^(1/3)*log((1 -
x^2)^(1/3)/576 + (2^(2/3)*(3^(1/2)*1i - 1)^2*((2^(1/3)*(3^(1/2)*1i - 1)*((10935*2^(2/3)*(3^(1/2)*1i - 1)^2)/25
6 - (9099*(1 - x^2)^(1/3))/64))/96 - 665/128))/9216)*(3^(1/2)*1i - 1))/96 - (2^(1/3)*log((1 - x^2)^(1/3)/576 -
(2^(2/3)*(3^(1/2)*1i + 1)^2*((2^(1/3)*(3^(1/2)*1i + 1)*((10935*2^(2/3)*(3^(1/2)*1i + 1)^2)/256 - (9099*(1 - x
^2)^(1/3))/64))/96 + 665/128))/9216)*(3^(1/2)*1i + 1))/96

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