3.1024 \(\int \frac{1}{x^3 \sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx\)

Optimal. Leaf size=183 \[ -\frac{\left (1-x^2\right )^{2/3}}{6 x^2 \left (x^2+3\right )}-\frac{5 \left (1-x^2\right )^{2/3}}{72 \left (x^2+3\right )}-\frac{\log \left (x^2+3\right )}{48\ 2^{2/3}}-\frac{1}{36} \log \left (1-\sqrt [3]{1-x^2}\right )+\frac{\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{16\ 2^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{8\ 2^{2/3} \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^2}+1}{\sqrt{3}}\right )}{18 \sqrt{3}}+\frac{\log (x)}{54} \]

[Out]

(-5*(1 - x^2)^(2/3))/(72*(3 + x^2)) - (1 - x^2)^(2/3)/(6*x^2*(3 + x^2)) + ArcTan
[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]]/(8*2^(2/3)*Sqrt[3]) - ArcTan[(1 + 2*(1 - x^2)^
(1/3))/Sqrt[3]]/(18*Sqrt[3]) + Log[x]/54 - Log[3 + x^2]/(48*2^(2/3)) - Log[1 - (
1 - x^2)^(1/3)]/36 + Log[2^(2/3) - (1 - x^2)^(1/3)]/(16*2^(2/3))

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Rubi [A]  time = 0.387676, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409 \[ -\frac{\left (1-x^2\right )^{2/3}}{6 x^2 \left (x^2+3\right )}-\frac{5 \left (1-x^2\right )^{2/3}}{72 \left (x^2+3\right )}-\frac{\log \left (x^2+3\right )}{48\ 2^{2/3}}-\frac{1}{36} \log \left (1-\sqrt [3]{1-x^2}\right )+\frac{\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{16\ 2^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{8\ 2^{2/3} \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^2}+1}{\sqrt{3}}\right )}{18 \sqrt{3}}+\frac{\log (x)}{54} \]

Antiderivative was successfully verified.

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

[Out]

(-5*(1 - x^2)^(2/3))/(72*(3 + x^2)) - (1 - x^2)^(2/3)/(6*x^2*(3 + x^2)) + ArcTan
[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]]/(8*2^(2/3)*Sqrt[3]) - ArcTan[(1 + 2*(1 - x^2)^
(1/3))/Sqrt[3]]/(18*Sqrt[3]) + Log[x]/54 - Log[3 + x^2]/(48*2^(2/3)) - Log[1 - (
1 - x^2)^(1/3)]/36 + Log[2^(2/3) - (1 - x^2)^(1/3)]/(16*2^(2/3))

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Rubi in Sympy [A]  time = 22.7324, size = 156, normalized size = 0.85 \[ - \frac{5 \left (- x^{2} + 1\right )^{\frac{2}{3}}}{72 \left (x^{2} + 3\right )} + \frac{\log{\left (x^{2} \right )}}{108} - \frac{\sqrt [3]{2} \log{\left (x^{2} + 3 \right )}}{96} - \frac{\log{\left (- \sqrt [3]{- x^{2} + 1} + 1 \right )}}{36} + \frac{\sqrt [3]{2} \log{\left (- \sqrt [3]{- x^{2} + 1} + 2^{\frac{2}{3}} \right )}}{32} + \frac{\sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{\sqrt [3]{2} \sqrt [3]{- x^{2} + 1}}{3} + \frac{1}{3}\right ) \right )}}{48} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{- x^{2} + 1}}{3} + \frac{1}{3}\right ) \right )}}{54} - \frac{\left (- x^{2} + 1\right )^{\frac{2}{3}}}{6 x^{2} \left (x^{2} + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*(-x**2 + 1)**(2/3)/(72*(x**2 + 3)) + log(x**2)/108 - 2**(1/3)*log(x**2 + 3)/9
6 - log(-(-x**2 + 1)**(1/3) + 1)/36 + 2**(1/3)*log(-(-x**2 + 1)**(1/3) + 2**(2/3
))/32 + 2**(1/3)*sqrt(3)*atan(sqrt(3)*(2**(1/3)*(-x**2 + 1)**(1/3)/3 + 1/3))/48
- sqrt(3)*atan(sqrt(3)*(2*(-x**2 + 1)**(1/3)/3 + 1/3))/54 - (-x**2 + 1)**(2/3)/(
6*x**2*(x**2 + 3))

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Mathematica [C]  time = 0.280329, size = 213, normalized size = 1.16 \[ \frac{\frac{10 x^4 F_1\left (1;\frac{1}{3},1;2;x^2,-\frac{x^2}{3}\right )}{x^2 \left (F_1\left (2;\frac{4}{3},1;3;x^2,-\frac{x^2}{3}\right )-F_1\left (2;\frac{1}{3},2;3;x^2,-\frac{x^2}{3}\right )\right )+6 F_1\left (1;\frac{1}{3},1;2;x^2,-\frac{x^2}{3}\right )}+\frac{21 x^4 F_1\left (\frac{4}{3};\frac{1}{3},1;\frac{7}{3};\frac{1}{x^2},-\frac{3}{x^2}\right )}{7 x^2 F_1\left (\frac{4}{3};\frac{1}{3},1;\frac{7}{3};\frac{1}{x^2},-\frac{3}{x^2}\right )-9 F_1\left (\frac{7}{3};\frac{1}{3},2;\frac{10}{3};\frac{1}{x^2},-\frac{3}{x^2}\right )+F_1\left (\frac{7}{3};\frac{4}{3},1;\frac{10}{3};\frac{1}{x^2},-\frac{3}{x^2}\right )}+5 x^4+7 x^2-12}{72 x^2 \sqrt [3]{1-x^2} \left (x^2+3\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-12 + 7*x^2 + 5*x^4 + (10*x^4*AppellF1[1, 1/3, 1, 2, x^2, -x^2/3])/(6*AppellF1[
1, 1/3, 1, 2, x^2, -x^2/3] + x^2*(-AppellF1[2, 1/3, 2, 3, x^2, -x^2/3] + AppellF
1[2, 4/3, 1, 3, x^2, -x^2/3])) + (21*x^4*AppellF1[4/3, 1/3, 1, 7/3, x^(-2), -3/x
^2])/(7*x^2*AppellF1[4/3, 1/3, 1, 7/3, x^(-2), -3/x^2] - 9*AppellF1[7/3, 1/3, 2,
10/3, x^(-2), -3/x^2] + AppellF1[7/3, 4/3, 1, 10/3, x^(-2), -3/x^2]))/(72*x^2*(
1 - x^2)^(1/3)*(3 + x^2))

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Maple [F]  time = 0.087, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3} \left ({x}^{2}+3 \right ) ^{2}}{\frac{1}{\sqrt [3]{-{x}^{2}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.241523, size = 343, normalized size = 1.87 \[ \frac{4^{\frac{2}{3}} \sqrt{3}{\left (4 \cdot 4^{\frac{1}{3}} \sqrt{3}{\left (x^{4} + 3 \, x^{2}\right )} \log \left ({\left (-x^{2} + 1\right )}^{\frac{2}{3}} +{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 1\right ) - 8 \cdot 4^{\frac{1}{3}} \sqrt{3}{\left (x^{4} + 3 \, x^{2}\right )} \log \left ({\left (-x^{2} + 1\right )}^{\frac{1}{3}} - 1\right ) - 6 \cdot 4^{\frac{1}{3}} \sqrt{3}{\left (5 \, x^{2} + 12\right )}{\left (-x^{2} + 1\right )}^{\frac{2}{3}} - 9 \, \sqrt{3}{\left (x^{4} + 3 \, x^{2}\right )} \log \left (4^{\frac{2}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 4^{\frac{1}{3}}{\left (-x^{2} + 1\right )}^{\frac{2}{3}} + 4\right ) + 18 \, \sqrt{3}{\left (x^{4} + 3 \, x^{2}\right )} \log \left (4^{\frac{2}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} - 4\right ) - 24 \cdot 4^{\frac{1}{3}}{\left (x^{4} + 3 \, x^{2}\right )} \arctan \left (\frac{2}{3} \, \sqrt{3}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right ) + 54 \,{\left (x^{4} + 3 \, x^{2}\right )} \arctan \left (\frac{1}{6} \cdot 4^{\frac{2}{3}} \sqrt{3}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right )\right )}}{5184 \,{\left (x^{4} + 3 \, x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5184*4^(2/3)*sqrt(3)*(4*4^(1/3)*sqrt(3)*(x^4 + 3*x^2)*log((-x^2 + 1)^(2/3) + (
-x^2 + 1)^(1/3) + 1) - 8*4^(1/3)*sqrt(3)*(x^4 + 3*x^2)*log((-x^2 + 1)^(1/3) - 1)
- 6*4^(1/3)*sqrt(3)*(5*x^2 + 12)*(-x^2 + 1)^(2/3) - 9*sqrt(3)*(x^4 + 3*x^2)*log
(4^(2/3)*(-x^2 + 1)^(1/3) + 4^(1/3)*(-x^2 + 1)^(2/3) + 4) + 18*sqrt(3)*(x^4 + 3*
x^2)*log(4^(2/3)*(-x^2 + 1)^(1/3) - 4) - 24*4^(1/3)*(x^4 + 3*x^2)*arctan(2/3*sqr
t(3)*(-x^2 + 1)^(1/3) + 1/3*sqrt(3)) + 54*(x^4 + 3*x^2)*arctan(1/6*4^(2/3)*sqrt(
3)*(-x^2 + 1)^(1/3) + 1/3*sqrt(3)))/(x^4 + 3*x^2)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError