3.1028 \(\int \frac{1}{\sqrt [3]{1-x^2} (3+x^2)^2} \, dx\)

Optimal. Leaf size=543 \[ \frac{\left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right ),4 \sqrt{3}-7\right )}{12 \sqrt{2} \sqrt [4]{3} \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}+\frac{\left (1-x^2\right )^{2/3} x}{24 \left (x^2+3\right )}-\frac{x}{24 \left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{8\ 2^{2/3} \sqrt{3}}+\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{8\ 2^{2/3}}-\frac{\sqrt{2+\sqrt{3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{16\ 3^{3/4} \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}+\frac{\tan ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{8\ 2^{2/3} \sqrt{3}}-\frac{\tanh ^{-1}(x)}{24\ 2^{2/3}} \]

[Out]

(x*(1 - x^2)^(2/3))/(24*(3 + x^2)) - x/(24*(1 - Sqrt[3] - (1 - x^2)^(1/3))) + ArcTan[Sqrt[3]/x]/(8*2^(2/3)*Sqr
t[3]) + ArcTan[(Sqrt[3]*(1 - 2^(1/3)*(1 - x^2)^(1/3)))/x]/(8*2^(2/3)*Sqrt[3]) - ArcTanh[x]/(24*2^(2/3)) + ArcT
anh[x/(1 + 2^(1/3)*(1 - x^2)^(1/3))]/(8*2^(2/3)) - (Sqrt[2 + Sqrt[3]]*(1 - (1 - x^2)^(1/3))*Sqrt[(1 + (1 - x^2
)^(1/3) + (1 - x^2)^(2/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))^2]*EllipticE[ArcSin[(1 + Sqrt[3] - (1 - x^2)^(1/3))
/(1 - Sqrt[3] - (1 - x^2)^(1/3))], -7 + 4*Sqrt[3]])/(16*3^(3/4)*x*Sqrt[-((1 - (1 - x^2)^(1/3))/(1 - Sqrt[3] -
(1 - x^2)^(1/3))^2)]) + ((1 - (1 - x^2)^(1/3))*Sqrt[(1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3))/(1 - Sqrt[3] - (1
- x^2)^(1/3))^2]*EllipticF[ArcSin[(1 + Sqrt[3] - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))], -7 + 4*Sqr
t[3]])/(12*Sqrt[2]*3^(1/4)*x*Sqrt[-((1 - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))^2)])

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Rubi [A]  time = 0.229481, antiderivative size = 543, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {414, 530, 235, 304, 219, 1879, 393} \[ \frac{\left (1-x^2\right )^{2/3} x}{24 \left (x^2+3\right )}-\frac{x}{24 \left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{8\ 2^{2/3} \sqrt{3}}+\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{8\ 2^{2/3}}+\frac{\left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{12 \sqrt{2} \sqrt [4]{3} \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}-\frac{\sqrt{2+\sqrt{3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{16\ 3^{3/4} \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}+\frac{\tan ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{8\ 2^{2/3} \sqrt{3}}-\frac{\tanh ^{-1}(x)}{24\ 2^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(1 - x^2)^(2/3))/(24*(3 + x^2)) - x/(24*(1 - Sqrt[3] - (1 - x^2)^(1/3))) + ArcTan[Sqrt[3]/x]/(8*2^(2/3)*Sqr
t[3]) + ArcTan[(Sqrt[3]*(1 - 2^(1/3)*(1 - x^2)^(1/3)))/x]/(8*2^(2/3)*Sqrt[3]) - ArcTanh[x]/(24*2^(2/3)) + ArcT
anh[x/(1 + 2^(1/3)*(1 - x^2)^(1/3))]/(8*2^(2/3)) - (Sqrt[2 + Sqrt[3]]*(1 - (1 - x^2)^(1/3))*Sqrt[(1 + (1 - x^2
)^(1/3) + (1 - x^2)^(2/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))^2]*EllipticE[ArcSin[(1 + Sqrt[3] - (1 - x^2)^(1/3))
/(1 - Sqrt[3] - (1 - x^2)^(1/3))], -7 + 4*Sqrt[3]])/(16*3^(3/4)*x*Sqrt[-((1 - (1 - x^2)^(1/3))/(1 - Sqrt[3] -
(1 - x^2)^(1/3))^2)]) + ((1 - (1 - x^2)^(1/3))*Sqrt[(1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3))/(1 - Sqrt[3] - (1
- x^2)^(1/3))^2]*EllipticF[ArcSin[(1 + Sqrt[3] - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))], -7 + 4*Sqr
t[3]])/(12*Sqrt[2]*3^(1/4)*x*Sqrt[-((1 - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))^2)])

Rule 414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 530

Int[(((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Dist[f/d,
Int[(a + b*x^n)^p, x], x] + Dist[(d*e - c*f)/d, Int[(a + b*x^n)^p/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,
f, p, n}, x]

Rule 235

Int[((a_) + (b_.)*(x_)^2)^(-1/3), x_Symbol] :> Dist[(3*Sqrt[b*x^2])/(2*b*x), Subst[Int[x/Sqrt[-a + x^3], x], x
, (a + b*x^2)^(1/3)], x] /; FreeQ[{a, b}, x]

Rule 304

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, -Dist[(S
qrt[2]*s)/(Sqrt[2 - Sqrt[3]]*r), Int[1/Sqrt[a + b*x^3], x], x] + Dist[1/r, Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a
+ b*x^3], x], x]] /; FreeQ[{a, b}, x] && NegQ[a]

Rule 219

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 - Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]*EllipticF[ArcSin[((1 + Sqrt[3
])*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[-((s*(s + r*x))/((1 - S
qrt[3])*s + r*x)^2)]), x]] /; FreeQ[{a, b}, x] && NegQ[a]

Rule 1879

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Simplify[((1 + Sqrt[3])*d)/c]]
, s = Denom[Simplify[((1 + Sqrt[3])*d)/c]]}, Simp[(2*d*s^3*Sqrt[a + b*x^3])/(a*r^2*((1 - Sqrt[3])*s + r*x)), x
] + Simp[(3^(1/4)*Sqrt[2 + Sqrt[3]]*d*s*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]*Elli
pticE[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]])/(r^2*Sqrt[a + b*x^3]*Sqrt[-((s
*(s + r*x))/((1 - Sqrt[3])*s + r*x)^2)]), x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && EqQ[b*c^3 - 2*(5 + 3*Sqr
t[3])*a*d^3, 0]

Rule 393

Int[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[-(b/a), 2]}, Simp[(q*ArcT
an[Sqrt[3]/(q*x)])/(2*2^(2/3)*Sqrt[3]*a^(1/3)*d), x] + (Simp[(q*ArcTanh[(a^(1/3)*q*x)/(a^(1/3) + 2^(1/3)*(a +
b*x^2)^(1/3))])/(2*2^(2/3)*a^(1/3)*d), x] - Simp[(q*ArcTanh[q*x])/(6*2^(2/3)*a^(1/3)*d), x] + Simp[(q*ArcTan[(
Sqrt[3]*(a^(1/3) - 2^(1/3)*(a + b*x^2)^(1/3)))/(a^(1/3)*q*x)])/(2*2^(2/3)*Sqrt[3]*a^(1/3)*d), x])] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b*c + 3*a*d, 0] && NegQ[b/a]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx &=\frac{x \left (1-x^2\right )^{2/3}}{24 \left (3+x^2\right )}-\frac{1}{24} \int \frac{-7-\frac{x^2}{3}}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx\\ &=\frac{x \left (1-x^2\right )^{2/3}}{24 \left (3+x^2\right )}+\frac{1}{72} \int \frac{1}{\sqrt [3]{1-x^2}} \, dx+\frac{1}{4} \int \frac{1}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx\\ &=\frac{x \left (1-x^2\right )^{2/3}}{24 \left (3+x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{8\ 2^{2/3} \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{8\ 2^{2/3} \sqrt{3}}-\frac{\tanh ^{-1}(x)}{24\ 2^{2/3}}+\frac{\tanh ^{-1}\left (\frac{x}{1+\sqrt [3]{2} \sqrt [3]{1-x^2}}\right )}{8\ 2^{2/3}}-\frac{\sqrt{-x^2} \operatorname{Subst}\left (\int \frac{x}{\sqrt{-1+x^3}} \, dx,x,\sqrt [3]{1-x^2}\right )}{48 x}\\ &=\frac{x \left (1-x^2\right )^{2/3}}{24 \left (3+x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{8\ 2^{2/3} \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{8\ 2^{2/3} \sqrt{3}}-\frac{\tanh ^{-1}(x)}{24\ 2^{2/3}}+\frac{\tanh ^{-1}\left (\frac{x}{1+\sqrt [3]{2} \sqrt [3]{1-x^2}}\right )}{8\ 2^{2/3}}+\frac{\sqrt{-x^2} \operatorname{Subst}\left (\int \frac{1+\sqrt{3}-x}{\sqrt{-1+x^3}} \, dx,x,\sqrt [3]{1-x^2}\right )}{48 x}-\frac{\left (\sqrt{\frac{1}{2} \left (2+\sqrt{3}\right )} \sqrt{-x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x^3}} \, dx,x,\sqrt [3]{1-x^2}\right )}{24 x}\\ &=\frac{x \left (1-x^2\right )^{2/3}}{24 \left (3+x^2\right )}-\frac{x}{24 \left (1-\sqrt{3}-\sqrt [3]{1-x^2}\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{8\ 2^{2/3} \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{8\ 2^{2/3} \sqrt{3}}-\frac{\tanh ^{-1}(x)}{24\ 2^{2/3}}+\frac{\tanh ^{-1}\left (\frac{x}{1+\sqrt [3]{2} \sqrt [3]{1-x^2}}\right )}{8\ 2^{2/3}}-\frac{\sqrt{2+\sqrt{3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{1+\sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}}{\left (1-\sqrt{3}-\sqrt [3]{1-x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{1+\sqrt{3}-\sqrt [3]{1-x^2}}{1-\sqrt{3}-\sqrt [3]{1-x^2}}\right )|-7+4 \sqrt{3}\right )}{16\ 3^{3/4} x \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (1-\sqrt{3}-\sqrt [3]{1-x^2}\right )^2}}}+\frac{\left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{1+\sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}}{\left (1-\sqrt{3}-\sqrt [3]{1-x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{1+\sqrt{3}-\sqrt [3]{1-x^2}}{1-\sqrt{3}-\sqrt [3]{1-x^2}}\right )|-7+4 \sqrt{3}\right )}{12 \sqrt{2} \sqrt [4]{3} x \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (1-\sqrt{3}-\sqrt [3]{1-x^2}\right )^2}}}\\ \end{align*}

Mathematica [C]  time = 0.102374, size = 157, normalized size = 0.29 \[ \frac{1}{648} x \left (x^2 F_1\left (\frac{3}{2};\frac{1}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right )+\frac{27 \left (\frac{63 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right )}{2 x^2 \left (F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right )-F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};x^2,-\frac{x^2}{3}\right )\right )+9 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right )}-x^2+1\right )}{\sqrt [3]{1-x^2} \left (x^2+3\right )}\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x*(x^2*AppellF1[3/2, 1/3, 1, 5/2, x^2, -x^2/3] + (27*(1 - x^2 + (63*AppellF1[1/2, 1/3, 1, 3/2, x^2, -x^2/3])/
(9*AppellF1[1/2, 1/3, 1, 3/2, x^2, -x^2/3] + 2*x^2*(-AppellF1[3/2, 1/3, 2, 5/2, x^2, -x^2/3] + AppellF1[3/2, 4
/3, 1, 5/2, x^2, -x^2/3]))))/((1 - x^2)^(1/3)*(3 + x^2))))/648

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Maple [F]  time = 0.044, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \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(1/(-x^2+1)^(1/3)/(x^2+3)^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (-x^{2} + 1\right )}^{\frac{2}{3}}}{x^{6} + 5 \, x^{4} + 3 \, x^{2} - 9}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(-x^2 + 1)^(2/3)/(x^6 + 5*x^4 + 3*x^2 - 9), x)

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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