3.1022 \(\int \frac{x}{\sqrt [3]{1-x^2} (3+x^2)^2} \, dx\)

Optimal. Leaf size=101 \[ -\frac{\left (1-x^2\right )^{2/3}}{8 \left (x^2+3\right )}-\frac{\log \left (x^2+3\right )}{48\ 2^{2/3}}+\frac{\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{16\ 2^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{8\ 2^{2/3} \sqrt{3}} \]

[Out]

-(1 - x^2)^(2/3)/(8*(3 + x^2)) + ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]]/(8*2^(2/3)*Sqrt[3]) - Log[3 + x^2]/(4
8*2^(2/3)) + Log[2^(2/3) - (1 - x^2)^(1/3)]/(16*2^(2/3))

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Rubi [A]  time = 0.0630688, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {444, 51, 55, 617, 204, 31} \[ -\frac{\left (1-x^2\right )^{2/3}}{8 \left (x^2+3\right )}-\frac{\log \left (x^2+3\right )}{48\ 2^{2/3}}+\frac{\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{16\ 2^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{8\ 2^{2/3} \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1 - x^2)^(2/3)/(8*(3 + x^2)) + ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]]/(8*2^(2/3)*Sqrt[3]) - Log[3 + x^2]/(4
8*2^(2/3)) + Log[2^(2/3) - (1 - x^2)^(1/3)]/(16*2^(2/3))

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(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] && EqQ[m
- n + 1, 0]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 55

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 617

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 204

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

Rule 31

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

Rubi steps

\begin{align*} \int \frac{x}{\sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{1-x} (3+x)^2} \, dx,x,x^2\right )\\ &=-\frac{\left (1-x^2\right )^{2/3}}{8 \left (3+x^2\right )}+\frac{1}{24} \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{1-x} (3+x)} \, dx,x,x^2\right )\\ &=-\frac{\left (1-x^2\right )^{2/3}}{8 \left (3+x^2\right )}-\frac{\log \left (3+x^2\right )}{48\ 2^{2/3}}+\frac{1}{16} \operatorname{Subst}\left (\int \frac{1}{2 \sqrt [3]{2}+2^{2/3} x+x^2} \, dx,x,\sqrt [3]{1-x^2}\right )-\frac{\operatorname{Subst}\left (\int \frac{1}{2^{2/3}-x} \, dx,x,\sqrt [3]{1-x^2}\right )}{16\ 2^{2/3}}\\ &=-\frac{\left (1-x^2\right )^{2/3}}{8 \left (3+x^2\right )}-\frac{\log \left (3+x^2\right )}{48\ 2^{2/3}}+\frac{\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{16\ 2^{2/3}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\sqrt [3]{2-2 x^2}\right )}{8\ 2^{2/3}}\\ &=-\frac{\left (1-x^2\right )^{2/3}}{8 \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{\log \left (3+x^2\right )}{48\ 2^{2/3}}+\frac{\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{16\ 2^{2/3}}\\ \end{align*}

Mathematica [C]  time = 0.0075964, size = 34, normalized size = 0.34 \[ -\frac{3}{64} \left (1-x^2\right )^{2/3} \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{1}{4} \left (1-x^2\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(1 - x^2)^(2/3)*Hypergeometric2F1[2/3, 2, 5/3, (1 - x^2)/4])/64

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Maple [F]  time = 0.039, size = 0, normalized size = 0. \begin{align*} \int{\frac{x}{ \left ({x}^{2}+3 \right ) ^{2}}{\frac{1}{\sqrt [3]{-{x}^{2}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.48261, size = 140, normalized size = 1.39 \begin{align*} \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{{\left (-x^{2} + 1\right )}^{\frac{2}{3}}}{8 \,{\left (x^{2} + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)) - 1/8*(-x^2 + 1)
^(2/3)/(x^2 + 3)

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Fricas [A]  time = 1.51495, size = 365, normalized size = 3.61 \begin{align*} \frac{4 \cdot 4^{\frac{1}{6}} \sqrt{3}{\left (x^{2} + 3\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 ) - 4^{\frac{2}{3}}{\left (x^{2} + 3\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 ) + 2 \cdot 4^{\frac{2}{3}}{\left (x^{2} + 3\right )} \log \left (-4^{\frac{1}{3}} +{\left (-x^{2} + 1\right )}^{\frac{1}{3}}\right ) - 24 \,{\left (-x^{2} + 1\right )}^{\frac{2}{3}}}{192 \,{\left (x^{2} + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*(4*4^(1/6)*sqrt(3)*(x^2 + 3)*arctan(1/6*4^(1/6)*(4^(1/3)*sqrt(3) + 2*sqrt(3)*(-x^2 + 1)^(1/3))) - 4^(2/3
)*(x^2 + 3)*log(4^(2/3) + 4^(1/3)*(-x^2 + 1)^(1/3) + (-x^2 + 1)^(2/3)) + 2*4^(2/3)*(x^2 + 3)*log(-4^(1/3) + (-
x^2 + 1)^(1/3)) - 24*(-x^2 + 1)^(2/3))/(x^2 + 3)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError