\(\int \frac {1}{x^2 \sqrt [3]{1-x^2} (3+x^2)} \, dx\) [1017]

Optimal result

Integrand size = 22, antiderivative size = 538 \[ \int \frac {1}{x^2 \sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=-\frac {\left (1-x^2\right )^{2/3}}{3 x}+\frac {x}{3 \left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )}-\frac {\arctan \left (\frac {\sqrt {3}}{x}\right )}{6\ 2^{2/3} \sqrt {3}}-\frac {\arctan \left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{6\ 2^{2/3} \sqrt {3}}+\frac {\text {arctanh}(x)}{18\ 2^{2/3}}-\frac {\text {arctanh}\left (\frac {x}{1+\sqrt [3]{2} \sqrt [3]{1-x^2}}\right )}{6\ 2^{2/3}}+\frac {\sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {1+\sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}} E\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{1-x^2}}{1-\sqrt {3}-\sqrt [3]{1-x^2}}\right )|-7+4 \sqrt {3}\right )}{2\ 3^{3/4} x \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}}}-\frac {\sqrt {2} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {1+\sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{1-x^2}}{1-\sqrt {3}-\sqrt [3]{1-x^2}}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} x \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}}} \]

[Out]

-1/3*(-x^2+1)^(2/3)/x+1/36*arctanh(x)*2^(1/3)-1/12*arctanh(x/(1+2^(1/3)*(-x^2+1)^(1/3)))*2^(1/3)+1/3*x/(1-(-x^
2+1)^(1/3)-3^(1/2))-1/36*arctan(3^(1/2)/x)*2^(1/3)*3^(1/2)-1/36*arctan((1-2^(1/3)*(-x^2+1)^(1/3))*3^(1/2)/x)*2
^(1/3)*3^(1/2)-1/9*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)+1/6*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 = 0.16 (sec) , antiderivative size = 538, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {491, 544, 241, 310, 225, 1893, 402} \[ \int \frac {1}{x^2 \sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=-\frac {\sqrt {2} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-x^2}+\sqrt {3}+1}{-\sqrt [3]{1-x^2}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} x}+\frac {\sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-x^2}+\sqrt {3}+1}{-\sqrt [3]{1-x^2}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{2\ 3^{3/4} \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} x}-\frac {\arctan \left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{6\ 2^{2/3} \sqrt {3}}-\frac {\arctan \left (\frac {\sqrt {3}}{x}\right )}{6\ 2^{2/3} \sqrt {3}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{6\ 2^{2/3}}+\frac {\text {arctanh}(x)}{18\ 2^{2/3}}+\frac {x}{3 \left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )}-\frac {\left (1-x^2\right )^{2/3}}{3 x} \]

[In]

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

[Out]

-1/3*(1 - x^2)^(2/3)/x + x/(3*(1 - Sqrt[3] - (1 - x^2)^(1/3))) - ArcTan[Sqrt[3]/x]/(6*2^(2/3)*Sqrt[3]) - ArcTa
n[(Sqrt[3]*(1 - 2^(1/3)*(1 - x^2)^(1/3)))/x]/(6*2^(2/3)*Sqrt[3]) + ArcTanh[x]/(18*2^(2/3)) - ArcTanh[x/(1 + 2^
(1/3)*(1 - x^2)^(1/3))]/(6*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]])/(2*3^(3/4)*x*Sqrt[-((1 - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3
))^2)]) - (Sqrt[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*Sqrt[3]])
/(3*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 491

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*e*(m + 1))), x] - Dist[1/(a*c*e^n*(m + 1)), Int[(e*x)^(m +
n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[(b*c + a*d)*(m + n + 1) + n*(b*c*p + a*d*q) + b*d*(m + n*(p + q + 2) + 1)*
x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && IntBino
mialQ[a, b, c, d, e, m, 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 {\left (1-x^2\right )^{2/3}}{3 x}+\frac {1}{3} \int \frac {-2-\frac {x^2}{3}}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx \\ & = -\frac {\left (1-x^2\right )^{2/3}}{3 x}-\frac {1}{9} \int \frac {1}{\sqrt [3]{1-x^2}} \, dx-\frac {1}{3} \int \frac {1}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx \\ & = -\frac {\left (1-x^2\right )^{2/3}}{3 x}-\frac {\tan ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{6\ 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 )}{6\ 2^{2/3} \sqrt {3}}+\frac {\tanh ^{-1}(x)}{18\ 2^{2/3}}-\frac {\tanh ^{-1}\left (\frac {x}{1+\sqrt [3]{2} \sqrt [3]{1-x^2}}\right )}{6\ 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 )}{6 x} \\ & = -\frac {\left (1-x^2\right )^{2/3}}{3 x}-\frac {\tan ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{6\ 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 )}{6\ 2^{2/3} \sqrt {3}}+\frac {\tanh ^{-1}(x)}{18\ 2^{2/3}}-\frac {\tanh ^{-1}\left (\frac {x}{1+\sqrt [3]{2} \sqrt [3]{1-x^2}}\right )}{6\ 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 )}{6 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 )}{6 x} \\ & = -\frac {\left (1-x^2\right )^{2/3}}{3 x}+\frac {x}{3 \left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )}-\frac {\tan ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{6\ 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 )}{6\ 2^{2/3} \sqrt {3}}+\frac {\tanh ^{-1}(x)}{18\ 2^{2/3}}-\frac {\tanh ^{-1}\left (\frac {x}{1+\sqrt [3]{2} \sqrt [3]{1-x^2}}\right )}{6\ 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 )}{2\ 3^{3/4} x \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}}}-\frac {\sqrt {2} \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 )}{3 \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 = 10.09 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.30 \[ \int \frac {1}{x^2 \sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=-\frac {1}{81} x^3 \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},1,\frac {5}{2},x^2,-\frac {x^2}{3}\right )+\frac {-1+x^2+\frac {18 x^2 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},x^2,-\frac {x^2}{3}\right )}{\left (3+x^2\right ) \left (-9 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},x^2,-\frac {x^2}{3}\right )+2 x^2 \left (\operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},2,\frac {5}{2},x^2,-\frac {x^2}{3}\right )-\operatorname {AppellF1}\left (\frac {3}{2},\frac {4}{3},1,\frac {5}{2},x^2,-\frac {x^2}{3}\right )\right )\right )}}{3 x \sqrt [3]{1-x^2}} \]

[In]

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

[Out]

-1/81*(x^3*AppellF1[3/2, 1/3, 1, 5/2, x^2, -1/3*x^2]) + (-1 + x^2 + (18*x^2*AppellF1[1/2, 1/3, 1, 3/2, x^2, -1
/3*x^2])/((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] - AppellF1[3/2, 4/3, 1, 5/2, x^2, -1/3*x^2]))))/(3*x*(1 - x^2)^(1/3))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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