\(\int \frac {1}{\sqrt [3]{1-x^2} (3+x^2)^2} \, dx\) [1028]

Optimal result

Integrand size = 19, antiderivative size = 543 \[ \int \frac {1}{\sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx=\frac {x \left (1-x^2\right )^{2/3}}{24 \left (3+x^2\right )}-\frac {x}{24 \left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )}+\frac {\arctan \left (\frac {\sqrt {3}}{x}\right )}{8\ 2^{2/3} \sqrt {3}}+\frac {\arctan \left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{8\ 2^{2/3} \sqrt {3}}-\frac {\text {arctanh}(x)}{24\ 2^{2/3}}+\frac {\text {arctanh}\left (\frac {x}{1+\sqrt [3]{2} \sqrt [3]{1-x^2}}\right )}{8\ 2^{2/3}}-\frac {\sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {1+\sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}} E\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{1-x^2}}{1-\sqrt {3}-\sqrt [3]{1-x^2}}\right )|-7+4 \sqrt {3}\right )}{16\ 3^{3/4} x \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}}}+\frac {\left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {1+\sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{1-x^2}}{1-\sqrt {3}-\sqrt [3]{1-x^2}}\right ),-7+4 \sqrt {3}\right )}{12 \sqrt {2} \sqrt [4]{3} x \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}}} \]

[Out]

1/24*x*(-x^2+1)^(2/3)/(x^2+3)-1/48*arctanh(x)*2^(1/3)+1/16*arctanh(x/(1+2^(1/3)*(-x^2+1)^(1/3)))*2^(1/3)-1/24*
x/(1-(-x^2+1)^(1/3)-3^(1/2))+1/48*arctan(3^(1/2)/x)*2^(1/3)*3^(1/2)+1/48*arctan((1-2^(1/3)*(-x^2+1)^(1/3))*3^(
1/2)/x)*2^(1/3)*3^(1/2)+1/72*3^(3/4)*(1-(-x^2+1)^(1/3))*EllipticF((1-(-x^2+1)^(1/3)+3^(1/2))/(1-(-x^2+1)^(1/3)
-3^(1/2)),2*I-I*3^(1/2))*2^(1/2)*((1+(-x^2+1)^(1/3)+(-x^2+1)^(2/3))/(1-(-x^2+1)^(1/3)-3^(1/2))^2)^(1/2)/x/((-1
+(-x^2+1)^(1/3))/(1-(-x^2+1)^(1/3)-3^(1/2))^2)^(1/2)-1/48*3^(1/4)*(1-(-x^2+1)^(1/3))*EllipticE((1-(-x^2+1)^(1/
3)+3^(1/2))/(1-(-x^2+1)^(1/3)-3^(1/2)),2*I-I*3^(1/2))*((1+(-x^2+1)^(1/3)+(-x^2+1)^(2/3))/(1-(-x^2+1)^(1/3)-3^(
1/2))^2)^(1/2)*(1/2*6^(1/2)+1/2*2^(1/2))/x/((-1+(-x^2+1)^(1/3))/(1-(-x^2+1)^(1/3)-3^(1/2))^2)^(1/2)

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 543, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {425, 544, 241, 310, 225, 1893, 402} \[ \int \frac {1}{\sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx=\frac {\left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-x^2}+\sqrt {3}+1}{-\sqrt [3]{1-x^2}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{12 \sqrt {2} \sqrt [4]{3} \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} x}-\frac {\sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-x^2}+\sqrt {3}+1}{-\sqrt [3]{1-x^2}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{16\ 3^{3/4} \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} x}+\frac {\arctan \left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{8\ 2^{2/3} \sqrt {3}}+\frac {\arctan \left (\frac {\sqrt {3}}{x}\right )}{8\ 2^{2/3} \sqrt {3}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{8\ 2^{2/3}}-\frac {\text {arctanh}(x)}{24\ 2^{2/3}}+\frac {\left (1-x^2\right )^{2/3} x}{24 \left (x^2+3\right )}-\frac {x}{24 \left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )} \]

[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 225

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

Rule 241

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 310

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Dist[(-(
1 + Sqrt[3]))*(s/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 402

Int[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[q*(ArcTan
[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[Sqr
t[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]

Rule 425

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]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

Rule 544

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 1893

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]/(r^2
*Sqrt[a + b*x^3]*Sqrt[(-s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 + Sqrt[3])*s + r*x)/(
(1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && EqQ[b*c^3 - 2*(5 + 3*Sqr
t[3])*a*d^3, 0]

Rubi steps \begin{align*} \text {integral}& = \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} \text {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} \text {Subst}\left (\int \frac {1+\sqrt {3}-x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1-x^2}\right )}{48 x}-\frac {\left (\left (1+\sqrt {3}\right ) \sqrt {-x^2}\right ) \text {Subst}\left (\int \frac {1}{\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 {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] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 10.12 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.29 \[ \int \frac {1}{\sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx=\frac {1}{648} x \left (x^2 \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},1,\frac {5}{2},x^2,-\frac {x^2}{3}\right )+\frac {27 \left (1-x^2+\frac {63 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},x^2,-\frac {x^2}{3}\right )}{9 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},x^2,-\frac {x^2}{3}\right )+2 x^2 \left (-\operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},2,\frac {5}{2},x^2,-\frac {x^2}{3}\right )+\operatorname {AppellF1}\left (\frac {3}{2},\frac {4}{3},1,\frac {5}{2},x^2,-\frac {x^2}{3}\right )\right )}\right )}{\sqrt [3]{1-x^2} \left (3+x^2\right )}\right ) \]

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

Maple [F]

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

[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)

Fricas [F]

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

[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)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

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

[In]

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

[Out]

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