Optimal. Leaf size=563 \[ \frac{x}{8 \left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )}-\frac{\left (1-x^2\right )^{2/3}}{8 x}+\frac{\left (1-x^2\right )^{2/3}}{24 \left (x^2+3\right ) x}-\frac{7 \tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{72\ 2^{2/3} \sqrt{3}}-\frac{7 \tanh ^{-1}\left (\frac{x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{72\ 2^{2/3}}-\frac{\left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{4 \sqrt{2} \sqrt [4]{3} \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}+\frac{\sqrt [4]{3} \sqrt{2+\sqrt{3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{16 \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}-\frac{7 \tan ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{72\ 2^{2/3} \sqrt{3}}+\frac{7 \tanh ^{-1}(x)}{216\ 2^{2/3}} \]
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Rubi [A] time = 0.803593, antiderivative size = 563, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ \frac{x}{8 \left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )}-\frac{\left (1-x^2\right )^{2/3}}{8 x}+\frac{\left (1-x^2\right )^{2/3}}{24 \left (x^2+3\right ) x}-\frac{7 \tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{72\ 2^{2/3} \sqrt{3}}-\frac{7 \tanh ^{-1}\left (\frac{x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{72\ 2^{2/3}}-\frac{\left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{4 \sqrt{2} \sqrt [4]{3} \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}+\frac{\sqrt [4]{3} \sqrt{2+\sqrt{3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{16 \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}-\frac{7 \tan ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{72\ 2^{2/3} \sqrt{3}}+\frac{7 \tanh ^{-1}(x)}{216\ 2^{2/3}} \]
Warning: Unable to verify antiderivative.
[In] Int[1/(x^2*(1 - x^2)^(1/3)*(3 + x^2)^2),x]
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Rubi in Sympy [A] time = 7.82815, size = 20, normalized size = 0.04 \[ - \frac{\operatorname{appellf_{1}}{\left (- \frac{1}{2},\frac{1}{3},2,\frac{1}{2},x^{2},- \frac{x^{2}}{3} \right )}}{9 x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(-x**2+1)**(1/3)/(x**2+3)**2,x)
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Mathematica [C] time = 0.279969, size = 241, normalized size = 0.43 \[ \frac{\frac{69 x^2 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right )}{2 x^2 \left (F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};x^2,-\frac{x^2}{3}\right )-F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right )\right )-9 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right )}+\frac{5 x^4 F_1\left (\frac{3}{2};\frac{1}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right )}{2 x^2 \left (F_1\left (\frac{5}{2};\frac{1}{3},2;\frac{7}{2};x^2,-\frac{x^2}{3}\right )-F_1\left (\frac{5}{2};\frac{4}{3},1;\frac{7}{2};x^2,-\frac{x^2}{3}\right )\right )-15 F_1\left (\frac{3}{2};\frac{1}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right )}+3 x^4+5 x^2-8}{24 x \sqrt [3]{1-x^2} \left (x^2+3\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/(x^2*(1 - x^2)^(1/3)*(3 + x^2)^2),x]
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Maple [F] time = 0.079, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2} \left ({x}^{2}+3 \right ) ^{2}}{\frac{1}{\sqrt [3]{-{x}^{2}+1}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(-x^2+1)^(1/3)/(x^2+3)^2,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{2} + 3\right )}^{2}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^2 + 3)^2*(-x^2 + 1)^(1/3)*x^2),x, algorithm="maxima")
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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)^2*(-x^2 + 1)^(1/3)*x^2),x, algorithm="fricas")
[Out]
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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**2/(-x**2+1)**(1/3)/(x**2+3)**2,x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{2} + 3\right )}^{2}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^2 + 3)^2*(-x^2 + 1)^(1/3)*x^2),x, algorithm="giac")
[Out]