3.1023 \(\int \frac{1}{x \sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx\)

Optimal. Leaf size=158 \[ \frac{\left (1-x^2\right )^{2/3}}{24 \left (x^2+3\right )}+\frac{5 \log \left (x^2+3\right )}{144\ 2^{2/3}}+\frac{1}{12} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac{5 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{48\ 2^{2/3}}-\frac{5 \tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{24\ 2^{2/3} \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^2}+1}{\sqrt{3}}\right )}{6 \sqrt{3}}-\frac{\log (x)}{18} \]

[Out]

(1 - x^2)^(2/3)/(24*(3 + x^2)) - (5*ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]])/(24
*2^(2/3)*Sqrt[3]) + ArcTan[(1 + 2*(1 - x^2)^(1/3))/Sqrt[3]]/(6*Sqrt[3]) - Log[x]
/18 + (5*Log[3 + x^2])/(144*2^(2/3)) + Log[1 - (1 - x^2)^(1/3)]/12 - (5*Log[2^(2
/3) - (1 - x^2)^(1/3)])/(48*2^(2/3))

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Rubi [A]  time = 0.315347, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ \frac{\left (1-x^2\right )^{2/3}}{24 \left (x^2+3\right )}+\frac{5 \log \left (x^2+3\right )}{144\ 2^{2/3}}+\frac{1}{12} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac{5 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{48\ 2^{2/3}}-\frac{5 \tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{24\ 2^{2/3} \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^2}+1}{\sqrt{3}}\right )}{6 \sqrt{3}}-\frac{\log (x)}{18} \]

Antiderivative was successfully verified.

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

[Out]

(1 - x^2)^(2/3)/(24*(3 + x^2)) - (5*ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]])/(24
*2^(2/3)*Sqrt[3]) + ArcTan[(1 + 2*(1 - x^2)^(1/3))/Sqrt[3]]/(6*Sqrt[3]) - Log[x]
/18 + (5*Log[3 + x^2])/(144*2^(2/3)) + Log[1 - (1 - x^2)^(1/3)]/12 - (5*Log[2^(2
/3) - (1 - x^2)^(1/3)])/(48*2^(2/3))

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Rubi in Sympy [A]  time = 18.459, size = 141, normalized size = 0.89 \[ \frac{\left (- x^{2} + 1\right )^{\frac{2}{3}}}{24 \left (x^{2} + 3\right )} - \frac{\log{\left (x^{2} \right )}}{36} + \frac{5 \sqrt [3]{2} \log{\left (x^{2} + 3 \right )}}{288} + \frac{\log{\left (- \sqrt [3]{- x^{2} + 1} + 1 \right )}}{12} - \frac{5 \sqrt [3]{2} \log{\left (- \sqrt [3]{- x^{2} + 1} + 2^{\frac{2}{3}} \right )}}{96} - \frac{5 \sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{\sqrt [3]{2} \sqrt [3]{- x^{2} + 1}}{3} + \frac{1}{3}\right ) \right )}}{144} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{- x^{2} + 1}}{3} + \frac{1}{3}\right ) \right )}}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-x**2 + 1)**(2/3)/(24*(x**2 + 3)) - log(x**2)/36 + 5*2**(1/3)*log(x**2 + 3)/288
+ log(-(-x**2 + 1)**(1/3) + 1)/12 - 5*2**(1/3)*log(-(-x**2 + 1)**(1/3) + 2**(2/
3))/96 - 5*2**(1/3)*sqrt(3)*atan(sqrt(3)*(2**(1/3)*(-x**2 + 1)**(1/3)/3 + 1/3))/
144 + sqrt(3)*atan(sqrt(3)*(2*(-x**2 + 1)**(1/3)/3 + 1/3))/18

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Mathematica [C]  time = 0.152672, size = 205, normalized size = 1.3 \[ \frac{\frac{2 x^2 F_1\left (1;\frac{1}{3},1;2;x^2,-\frac{x^2}{3}\right )}{x^2 \left (F_1\left (2;\frac{1}{3},2;3;x^2,-\frac{x^2}{3}\right )-F_1\left (2;\frac{4}{3},1;3;x^2,-\frac{x^2}{3}\right )\right )-6 F_1\left (1;\frac{1}{3},1;2;x^2,-\frac{x^2}{3}\right )}-\frac{21 x^2 F_1\left (\frac{4}{3};\frac{1}{3},1;\frac{7}{3};\frac{1}{x^2},-\frac{3}{x^2}\right )}{7 x^2 F_1\left (\frac{4}{3};\frac{1}{3},1;\frac{7}{3};\frac{1}{x^2},-\frac{3}{x^2}\right )-9 F_1\left (\frac{7}{3};\frac{1}{3},2;\frac{10}{3};\frac{1}{x^2},-\frac{3}{x^2}\right )+F_1\left (\frac{7}{3};\frac{4}{3},1;\frac{10}{3};\frac{1}{x^2},-\frac{3}{x^2}\right )}-x^2+1}{24 \sqrt [3]{1-x^2} \left (x^2+3\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(1 - x^2 + (2*x^2*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] - AppellF1[2, 4/3, 1,
3, x^2, -x^2/3])) - (21*x^2*AppellF1[4/3, 1/3, 1, 7/3, x^(-2), -3/x^2])/(7*x^2*A
ppellF1[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]))/(24*(1 - x^2)^(1/3)*(3
+ x^2))

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Maple [F]  time = 0.082, size = 0, normalized size = 0. \[ \int{\frac{1}{x \left ({x}^{2}+3 \right ) ^{2}}{\frac{1}{\sqrt [3]{-{x}^{2}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.237826, size = 332, normalized size = 2.1 \[ -\frac{4^{\frac{2}{3}} \sqrt{3}{\left (5 \, \sqrt{3} \left (-1\right )^{\frac{1}{3}}{\left (x^{2} + 3\right )} \log \left (4^{\frac{2}{3}} \left (-1\right )^{\frac{2}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 4^{\frac{1}{3}}{\left (-x^{2} + 1\right )}^{\frac{2}{3}} - 4 \, \left (-1\right )^{\frac{1}{3}}\right ) - 10 \, \sqrt{3} \left (-1\right )^{\frac{1}{3}}{\left (x^{2} + 3\right )} \log \left (4^{\frac{2}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} - 4 \, \left (-1\right )^{\frac{2}{3}}\right ) + 4 \cdot 4^{\frac{1}{3}} \sqrt{3}{\left (x^{2} + 3\right )} \log \left ({\left (-x^{2} + 1\right )}^{\frac{2}{3}} +{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 1\right ) - 8 \cdot 4^{\frac{1}{3}} \sqrt{3}{\left (x^{2} + 3\right )} \log \left ({\left (-x^{2} + 1\right )}^{\frac{1}{3}} - 1\right ) - 30 \, \left (-1\right )^{\frac{1}{3}}{\left (x^{2} + 3\right )} \arctan \left (-\frac{1}{6} \, \left (-1\right )^{\frac{1}{3}}{\left (4^{\frac{2}{3}} \sqrt{3}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 2 \, \sqrt{3} \left (-1\right )^{\frac{2}{3}}\right )}\right ) - 24 \cdot 4^{\frac{1}{3}}{\left (x^{2} + 3\right )} \arctan \left (\frac{2}{3} \, \sqrt{3}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right ) - 6 \cdot 4^{\frac{1}{3}} \sqrt{3}{\left (-x^{2} + 1\right )}^{\frac{2}{3}}\right )}}{1728 \,{\left (x^{2} + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1728*4^(2/3)*sqrt(3)*(5*sqrt(3)*(-1)^(1/3)*(x^2 + 3)*log(4^(2/3)*(-1)^(2/3)*(
-x^2 + 1)^(1/3) + 4^(1/3)*(-x^2 + 1)^(2/3) - 4*(-1)^(1/3)) - 10*sqrt(3)*(-1)^(1/
3)*(x^2 + 3)*log(4^(2/3)*(-x^2 + 1)^(1/3) - 4*(-1)^(2/3)) + 4*4^(1/3)*sqrt(3)*(x
^2 + 3)*log((-x^2 + 1)^(2/3) + (-x^2 + 1)^(1/3) + 1) - 8*4^(1/3)*sqrt(3)*(x^2 +
3)*log((-x^2 + 1)^(1/3) - 1) - 30*(-1)^(1/3)*(x^2 + 3)*arctan(-1/6*(-1)^(1/3)*(4
^(2/3)*sqrt(3)*(-x^2 + 1)^(1/3) + 2*sqrt(3)*(-1)^(2/3))) - 24*4^(1/3)*(x^2 + 3)*
arctan(2/3*sqrt(3)*(-x^2 + 1)^(1/3) + 1/3*sqrt(3)) - 6*4^(1/3)*sqrt(3)*(-x^2 + 1
)^(2/3))/(x^2 + 3)

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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/(-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),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError