3.1017 \(\int \frac{1}{x^2 \sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx\)

Optimal. Leaf size=538 \[ \frac{x}{3 \left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )}-\frac{\left (1-x^2\right )^{2/3}}{3 x}-\frac{\tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{6\ 2^{2/3} \sqrt{3}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{6\ 2^{2/3}}-\frac{\sqrt{2} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}+\frac{\sqrt{2+\sqrt{3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{2\ 3^{3/4} \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}-\frac{\tan ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{6\ 2^{2/3} \sqrt{3}}+\frac{\tanh ^{-1}(x)}{18\ 2^{2/3}} \]

[Out]

-(1 - x^2)^(2/3)/(3*x) + x/(3*(1 - Sqrt[3] - (1 - x^2)^(1/3))) - ArcTan[Sqrt[3]/
x]/(6*2^(2/3)*Sqrt[3]) - ArcTan[(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)])

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Rubi [A]  time = 0.598849, antiderivative size = 538, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ \frac{x}{3 \left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )}-\frac{\left (1-x^2\right )^{2/3}}{3 x}-\frac{\tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{6\ 2^{2/3} \sqrt{3}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{6\ 2^{2/3}}-\frac{\sqrt{2} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}+\frac{\sqrt{2+\sqrt{3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{2\ 3^{3/4} \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}-\frac{\tan ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{6\ 2^{2/3} \sqrt{3}}+\frac{\tanh ^{-1}(x)}{18\ 2^{2/3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

-(1 - x^2)^(2/3)/(3*x) + x/(3*(1 - Sqrt[3] - (1 - x^2)^(1/3))) - ArcTan[Sqrt[3]/
x]/(6*2^(2/3)*Sqrt[3]) - ArcTan[(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)])

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Rubi in Sympy [A]  time = 8.22013, size = 20, normalized size = 0.04 \[ - \frac{\operatorname{appellf_{1}}{\left (- \frac{1}{2},\frac{1}{3},1,\frac{1}{2},x^{2},- \frac{x^{2}}{3} \right )}}{3 x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(-x**2+1)**(1/3)/(x**2+3),x)

[Out]

-appellf1(-1/2, 1/3, 1, 1/2, x**2, -x**2/3)/(3*x)

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Mathematica [C]  time = 0.274591, size = 243, normalized size = 0.45 \[ \frac{\frac{54 x^2 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right )}{\left (x^2+3\right ) \left (2 x^2 \left (F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};x^2,-\frac{x^2}{3}\right )-F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right )\right )-9 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right )\right )}+\frac{5 x^4 F_1\left (\frac{3}{2};\frac{1}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right )}{\left (x^2+3\right ) \left (2 x^2 \left (F_1\left (\frac{5}{2};\frac{1}{3},2;\frac{7}{2};x^2,-\frac{x^2}{3}\right )-F_1\left (\frac{5}{2};\frac{4}{3},1;\frac{7}{2};x^2,-\frac{x^2}{3}\right )\right )-15 F_1\left (\frac{3}{2};\frac{1}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right )\right )}+3 x^2-3}{9 x \sqrt [3]{1-x^2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-3 + 3*x^2 + (54*x^2*AppellF1[1/2, 1/3, 1, 3/2, x^2, -x^2/3])/((3 + x^2)*(-9*Ap
pellF1[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]))) + (5*x^4*AppellF1[3/2, 1/3,
1, 5/2, x^2, -x^2/3])/((3 + x^2)*(-15*AppellF1[3/2, 1/3, 1, 5/2, x^2, -x^2/3] +
2*x^2*(AppellF1[5/2, 1/3, 2, 7/2, x^2, -x^2/3] - AppellF1[5/2, 4/3, 1, 7/2, x^2,
-x^2/3]))))/(9*x*(1 - x^2)^(1/3))

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Maple [F]  time = 0.069, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2} \left ({x}^{2}+3 \right ) }{\frac{1}{\sqrt [3]{-{x}^{2}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{2} + 3\right )}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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