3.1025 \(\int \frac{1}{x^5 \sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx\)

Optimal. Leaf size=208 \[ -\frac{\left (1-x^2\right )^{2/3}}{36 x^2 \left (x^2+3\right )}+\frac{\left (1-x^2\right )^{2/3}}{216 \left (x^2+3\right )}+\frac{13 \log \left (x^2+3\right )}{1296\ 2^{2/3}}+\frac{1}{36} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac{13 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{432\ 2^{2/3}}-\frac{13 \tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{216\ 2^{2/3} \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^2}+1}{\sqrt{3}}\right )}{18 \sqrt{3}}-\frac{\left (1-x^2\right )^{2/3}}{12 x^4 \left (x^2+3\right )}-\frac{\log (x)}{54} \]

[Out]

(1 - x^2)^(2/3)/(216*(3 + x^2)) - (1 - x^2)^(2/3)/(12*x^4*(3 + x^2)) - (1 - x^2)
^(2/3)/(36*x^2*(3 + x^2)) - (13*ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]])/(216*2^
(2/3)*Sqrt[3]) + ArcTan[(1 + 2*(1 - x^2)^(1/3))/Sqrt[3]]/(18*Sqrt[3]) - Log[x]/5
4 + (13*Log[3 + x^2])/(1296*2^(2/3)) + Log[1 - (1 - x^2)^(1/3)]/36 - (13*Log[2^(
2/3) - (1 - x^2)^(1/3)])/(432*2^(2/3))

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Rubi [A]  time = 0.458637, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409 \[ -\frac{\left (1-x^2\right )^{2/3}}{36 x^2 \left (x^2+3\right )}+\frac{\left (1-x^2\right )^{2/3}}{216 \left (x^2+3\right )}+\frac{13 \log \left (x^2+3\right )}{1296\ 2^{2/3}}+\frac{1}{36} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac{13 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{432\ 2^{2/3}}-\frac{13 \tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{216\ 2^{2/3} \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^2}+1}{\sqrt{3}}\right )}{18 \sqrt{3}}-\frac{\left (1-x^2\right )^{2/3}}{12 x^4 \left (x^2+3\right )}-\frac{\log (x)}{54} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*(1 - x^2)^(1/3)*(3 + x^2)^2),x]

[Out]

(1 - x^2)^(2/3)/(216*(3 + x^2)) - (1 - x^2)^(2/3)/(12*x^4*(3 + x^2)) - (1 - x^2)
^(2/3)/(36*x^2*(3 + x^2)) - (13*ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]])/(216*2^
(2/3)*Sqrt[3]) + ArcTan[(1 + 2*(1 - x^2)^(1/3))/Sqrt[3]]/(18*Sqrt[3]) - Log[x]/5
4 + (13*Log[3 + x^2])/(1296*2^(2/3)) + Log[1 - (1 - x^2)^(1/3)]/36 - (13*Log[2^(
2/3) - (1 - x^2)^(1/3)])/(432*2^(2/3))

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Rubi in Sympy [A]  time = 26.9487, size = 177, normalized size = 0.85 \[ - \frac{\log{\left (x^{2} \right )}}{108} + \frac{13 \sqrt [3]{2} \log{\left (x^{2} + 3 \right )}}{2592} + \frac{\log{\left (- \sqrt [3]{- x^{2} + 1} + 1 \right )}}{36} - \frac{13 \sqrt [3]{2} \log{\left (- \sqrt [3]{- x^{2} + 1} + 2^{\frac{2}{3}} \right )}}{864} - \frac{13 \sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{\sqrt [3]{2} \sqrt [3]{- x^{2} + 1}}{3} + \frac{1}{3}\right ) \right )}}{1296} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{- x^{2} + 1}}{3} + \frac{1}{3}\right ) \right )}}{54} + \frac{\left (- x^{2} + 1\right )^{\frac{2}{3}}}{216 x^{2}} - \frac{\left (- x^{2} + 1\right )^{\frac{2}{3}}}{24 x^{2} \left (x^{2} + 3\right )} - \frac{\left (- x^{2} + 1\right )^{\frac{2}{3}}}{12 x^{4} \left (x^{2} + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(-x**2+1)**(1/3)/(x**2+3)**2,x)

[Out]

-log(x**2)/108 + 13*2**(1/3)*log(x**2 + 3)/2592 + log(-(-x**2 + 1)**(1/3) + 1)/3
6 - 13*2**(1/3)*log(-(-x**2 + 1)**(1/3) + 2**(2/3))/864 - 13*2**(1/3)*sqrt(3)*at
an(sqrt(3)*(2**(1/3)*(-x**2 + 1)**(1/3)/3 + 1/3))/1296 + sqrt(3)*atan(sqrt(3)*(2
*(-x**2 + 1)**(1/3)/3 + 1/3))/54 + (-x**2 + 1)**(2/3)/(216*x**2) - (-x**2 + 1)**
(2/3)/(24*x**2*(x**2 + 3)) - (-x**2 + 1)**(2/3)/(12*x**4*(x**2 + 3))

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Mathematica [C]  time = 0.285954, size = 234, normalized size = 1.12 \[ \frac{\frac{2 x^2 F_1\left (1;\frac{1}{3},1;2;x^2,-\frac{x^2}{3}\right )}{\left (x^2+3\right ) \left (x^2 \left (F_1\left (2;\frac{1}{3},2;3;x^2,-\frac{x^2}{3}\right )-F_1\left (2;\frac{4}{3},1;3;x^2,-\frac{x^2}{3}\right )\right )-6 F_1\left (1;\frac{1}{3},1;2;x^2,-\frac{x^2}{3}\right )\right )}-\frac{63 x^2 F_1\left (\frac{4}{3};\frac{1}{3},1;\frac{7}{3};\frac{1}{x^2},-\frac{3}{x^2}\right )}{\left (x^2+3\right ) \left (7 x^2 F_1\left (\frac{4}{3};\frac{1}{3},1;\frac{7}{3};\frac{1}{x^2},-\frac{3}{x^2}\right )-9 F_1\left (\frac{7}{3};\frac{1}{3},2;\frac{10}{3};\frac{1}{x^2},-\frac{3}{x^2}\right )+F_1\left (\frac{7}{3};\frac{4}{3},1;\frac{10}{3};\frac{1}{x^2},-\frac{3}{x^2}\right )\right )}-\frac{x^6-7 x^4-12 x^2+18}{x^6+3 x^4}}{216 \sqrt [3]{1-x^2}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/(x^5*(1 - x^2)^(1/3)*(3 + x^2)^2),x]

[Out]

(-((18 - 12*x^2 - 7*x^4 + x^6)/(3*x^4 + x^6)) + (2*x^2*AppellF1[1, 1/3, 1, 2, x^
2, -x^2/3])/((3 + x^2)*(-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] - AppellF1[2, 4/3, 1, 3, x^2, -x^2/3]))) - (63*x^2*App
ellF1[4/3, 1/3, 1, 7/3, x^(-2), -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])))/(216*(1 - x^2)^(1/3))

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Maple [F]  time = 0.095, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{5} \left ({x}^{2}+3 \right ) ^{2}}{\frac{1}{\sqrt [3]{-{x}^{2}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(-x^2+1)^(1/3)/(x^2+3)^2,x)

[Out]

int(1/x^5/(-x^2+1)^(1/3)/(x^2+3)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{2} + 3\right )}^{2}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 + 3)^2*(-x^2 + 1)^(1/3)*x^5),x, algorithm="maxima")

[Out]

integrate(1/((x^2 + 3)^2*(-x^2 + 1)^(1/3)*x^5), x)

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Fricas [A]  time = 0.240921, size = 383, normalized size = 1.84 \[ -\frac{4^{\frac{2}{3}} \sqrt{3}{\left (13 \, \sqrt{3} \left (-1\right )^{\frac{1}{3}}{\left (x^{6} + 3 \, x^{4}\right )} \log \left (4^{\frac{2}{3}} \left (-1\right )^{\frac{2}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 4^{\frac{1}{3}}{\left (-x^{2} + 1\right )}^{\frac{2}{3}} - 4 \, \left (-1\right )^{\frac{1}{3}}\right ) - 26 \, \sqrt{3} \left (-1\right )^{\frac{1}{3}}{\left (x^{6} + 3 \, x^{4}\right )} \log \left (4^{\frac{2}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} - 4 \, \left (-1\right )^{\frac{2}{3}}\right ) + 12 \cdot 4^{\frac{1}{3}} \sqrt{3}{\left (x^{6} + 3 \, x^{4}\right )} \log \left ({\left (-x^{2} + 1\right )}^{\frac{2}{3}} +{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 1\right ) - 24 \cdot 4^{\frac{1}{3}} \sqrt{3}{\left (x^{6} + 3 \, x^{4}\right )} \log \left ({\left (-x^{2} + 1\right )}^{\frac{1}{3}} - 1\right ) - 6 \cdot 4^{\frac{1}{3}} \sqrt{3}{\left (x^{4} - 6 \, x^{2} - 18\right )}{\left (-x^{2} + 1\right )}^{\frac{2}{3}} - 78 \, \left (-1\right )^{\frac{1}{3}}{\left (x^{6} + 3 \, x^{4}\right )} \arctan \left (-\frac{1}{6} \, \left (-1\right )^{\frac{1}{3}}{\left (4^{\frac{2}{3}} \sqrt{3}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 2 \, \sqrt{3} \left (-1\right )^{\frac{2}{3}}\right )}\right ) - 72 \cdot 4^{\frac{1}{3}}{\left (x^{6} + 3 \, x^{4}\right )} \arctan \left (\frac{2}{3} \, \sqrt{3}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right )\right )}}{15552 \,{\left (x^{6} + 3 \, x^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 + 3)^2*(-x^2 + 1)^(1/3)*x^5),x, algorithm="fricas")

[Out]

-1/15552*4^(2/3)*sqrt(3)*(13*sqrt(3)*(-1)^(1/3)*(x^6 + 3*x^4)*log(4^(2/3)*(-1)^(
2/3)*(-x^2 + 1)^(1/3) + 4^(1/3)*(-x^2 + 1)^(2/3) - 4*(-1)^(1/3)) - 26*sqrt(3)*(-
1)^(1/3)*(x^6 + 3*x^4)*log(4^(2/3)*(-x^2 + 1)^(1/3) - 4*(-1)^(2/3)) + 12*4^(1/3)
*sqrt(3)*(x^6 + 3*x^4)*log((-x^2 + 1)^(2/3) + (-x^2 + 1)^(1/3) + 1) - 24*4^(1/3)
*sqrt(3)*(x^6 + 3*x^4)*log((-x^2 + 1)^(1/3) - 1) - 6*4^(1/3)*sqrt(3)*(x^4 - 6*x^
2 - 18)*(-x^2 + 1)^(2/3) - 78*(-1)^(1/3)*(x^6 + 3*x^4)*arctan(-1/6*(-1)^(1/3)*(4
^(2/3)*sqrt(3)*(-x^2 + 1)^(1/3) + 2*sqrt(3)*(-1)^(2/3))) - 72*4^(1/3)*(x^6 + 3*x
^4)*arctan(2/3*sqrt(3)*(-x^2 + 1)^(1/3) + 1/3*sqrt(3)))/(x^6 + 3*x^4)

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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**5/(-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^5),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError