3.162 \(\int \frac{1}{\sqrt [3]{1-x^2} (9-x^2)} \, dx\)

Optimal. Leaf size=74 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{1-x^2}\right )}{x}\right )}{4 \sqrt{3}}-\frac{1}{12} \tanh ^{-1}\left (\frac{\left (1-\sqrt [3]{1-x^2}\right )^2}{3 x}\right )+\frac{1}{12} \tanh ^{-1}\left (\frac{x}{3}\right ) \]

[Out]

ArcTan[(Sqrt[3]*(1 - (1 - x^2)^(1/3)))/x]/(4*Sqrt[3]) + ArcTanh[x/3]/12 - ArcTanh[(1 - (1 - x^2)^(1/3))^2/(3*x
)]/12

________________________________________________________________________________________

Rubi [A]  time = 0.0112752, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {395} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{1-x^2}\right )}{x}\right )}{4 \sqrt{3}}-\frac{1}{12} \tanh ^{-1}\left (\frac{\left (1-\sqrt [3]{1-x^2}\right )^2}{3 x}\right )+\frac{1}{12} \tanh ^{-1}\left (\frac{x}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - x^2)^(1/3)*(9 - x^2)),x]

[Out]

ArcTan[(Sqrt[3]*(1 - (1 - x^2)^(1/3)))/x]/(4*Sqrt[3]) + ArcTanh[x/3]/12 - ArcTanh[(1 - (1 - x^2)^(1/3))^2/(3*x
)]/12

Rule 395

Int[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[-(b/a), 2]}, -Simp[(q*Arc
Tanh[(q*x)/3])/(12*Rt[a, 3]*d), x] + (Simp[(q*ArcTanh[(Rt[a, 3] - (a + b*x^2)^(1/3))^2/(3*Rt[a, 3]^2*q*x)])/(1
2*Rt[a, 3]*d), x] - Simp[(q*ArcTan[(Sqrt[3]*(Rt[a, 3] - (a + b*x^2)^(1/3)))/(Rt[a, 3]*q*x)])/(4*Sqrt[3]*Rt[a,
3]*d), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b*c - 9*a*d, 0] && NegQ[b/a]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt [3]{1-x^2} \left (9-x^2\right )} \, dx &=\frac{\tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{1-x^2}\right )}{x}\right )}{4 \sqrt{3}}+\frac{1}{12} \tanh ^{-1}\left (\frac{x}{3}\right )-\frac{1}{12} \tanh ^{-1}\left (\frac{\left (1-\sqrt [3]{1-x^2}\right )^2}{3 x}\right )\\ \end{align*}

Mathematica [C]  time = 0.0451872, size = 125, normalized size = 1.69 \[ \frac{\sqrt [3]{\frac{x-1}{x-3}} \sqrt [3]{\frac{x+1}{x-3}} F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};-\frac{4}{x-3},-\frac{2}{x-3}\right )-\sqrt [3]{\frac{x-1}{x+3}} \sqrt [3]{\frac{x+1}{x+3}} F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{2}{x+3},\frac{4}{x+3}\right )}{4 \sqrt [3]{1-x^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((1 - x^2)^(1/3)*(9 - x^2)),x]

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-{x}^{2}+9}{\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+9),x)

[Out]

int(1/(-x^2+1)^(1/3)/(-x^2+9),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{1}{{\left (x^{2} - 9\right )}{\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+9),x, algorithm="maxima")

[Out]

-integrate(1/((x^2 - 9)*(-x^2 + 1)^(1/3)), x)

________________________________________________________________________________________

Fricas [B]  time = 6.55866, size = 749, normalized size = 10.12 \begin{align*} -\frac{1}{36} \, \sqrt{3} \arctan \left (\frac{36 \, \sqrt{3}{\left (x^{4} - 32 \, x^{3} - 42 \, x^{2} + 9\right )}{\left (-x^{2} + 1\right )}^{\frac{2}{3}} + 12 \, \sqrt{3}{\left (x^{5} + 27 \, x^{4} - 210 \, x^{3} - 54 \, x^{2} + 81 \, x + 27\right )}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + \sqrt{3}{\left (x^{6} - 108 \, x^{5} - 567 \, x^{4} + 1080 \, x^{3} + 459 \, x^{2} - 972 \, x - 405\right )}}{3 \,{\left (x^{6} + 108 \, x^{5} - 1647 \, x^{4} - 1080 \, x^{3} + 891 \, x^{2} + 972 \, x + 243\right )}}\right ) - \frac{1}{72} \, \log \left (\frac{x^{3} + 33 \, x^{2} + 18 \,{\left (-x^{2} + 1\right )}^{\frac{2}{3}}{\left (x + 1\right )} - 6 \,{\left (x^{2} + 6 \, x - 3\right )}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} - 9 \, x - 9}{x^{3} + 9 \, x^{2} + 27 \, x + 27}\right ) + \frac{1}{36} \, \log \left (-\frac{x^{3} - 33 \, x^{2} + 18 \,{\left (-x^{2} + 1\right )}^{\frac{2}{3}}{\left (x - 1\right )} + 6 \,{\left (x^{2} - 6 \, x - 3\right )}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} - 9 \, x + 9}{x^{3} + 9 \, x^{2} + 27 \, x + 27}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2+1)^(1/3)/(-x^2+9),x, algorithm="fricas")

[Out]

-1/36*sqrt(3)*arctan(1/3*(36*sqrt(3)*(x^4 - 32*x^3 - 42*x^2 + 9)*(-x^2 + 1)^(2/3) + 12*sqrt(3)*(x^5 + 27*x^4 -
210*x^3 - 54*x^2 + 81*x + 27)*(-x^2 + 1)^(1/3) + sqrt(3)*(x^6 - 108*x^5 - 567*x^4 + 1080*x^3 + 459*x^2 - 972*
x - 405))/(x^6 + 108*x^5 - 1647*x^4 - 1080*x^3 + 891*x^2 + 972*x + 243)) - 1/72*log((x^3 + 33*x^2 + 18*(-x^2 +
1)^(2/3)*(x + 1) - 6*(x^2 + 6*x - 3)*(-x^2 + 1)^(1/3) - 9*x - 9)/(x^3 + 9*x^2 + 27*x + 27)) + 1/36*log(-(x^3
- 33*x^2 + 18*(-x^2 + 1)^(2/3)*(x - 1) + 6*(x^2 - 6*x - 3)*(-x^2 + 1)^(1/3) - 9*x + 9)/(x^3 + 9*x^2 + 27*x + 2
7))

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{1}{x^{2} \sqrt [3]{1 - x^{2}} - 9 \sqrt [3]{1 - x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x**2+1)**(1/3)/(-x**2+9),x)

[Out]

-Integral(1/(x**2*(1 - x**2)**(1/3) - 9*(1 - x**2)**(1/3)), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{1}{{\left (x^{2} - 9\right )}{\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+9),x, algorithm="giac")

[Out]

integrate(-1/((x^2 - 9)*(-x^2 + 1)^(1/3)), x)