Optimal. Leaf size=581 \[ -\frac{11 x}{648 \left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )}+\frac{11 \left (1-x^2\right )^{2/3}}{648 x}+\frac{11 \tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{216\ 2^{2/3} \sqrt{3}}+\frac{11 \tanh ^{-1}\left (\frac{x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{216\ 2^{2/3}}+\frac{11 \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{324 \sqrt{2} \sqrt [4]{3} \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}-\frac{11 \sqrt{2+\sqrt{3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{432\ 3^{3/4} \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}-\frac{11 \left (1-x^2\right )^{2/3}}{216 x^3}+\frac{\left (1-x^2\right )^{2/3}}{24 \left (x^2+3\right ) x^3}+\frac{11 \tan ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{216\ 2^{2/3} \sqrt{3}}-\frac{11 \tanh ^{-1}(x)}{648\ 2^{2/3}} \]
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Rubi [A] time = 0.974455, antiderivative size = 581, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ -\frac{11 x}{648 \left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )}+\frac{11 \left (1-x^2\right )^{2/3}}{648 x}+\frac{11 \tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{216\ 2^{2/3} \sqrt{3}}+\frac{11 \tanh ^{-1}\left (\frac{x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{216\ 2^{2/3}}+\frac{11 \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{324 \sqrt{2} \sqrt [4]{3} \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}-\frac{11 \sqrt{2+\sqrt{3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{432\ 3^{3/4} \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}-\frac{11 \left (1-x^2\right )^{2/3}}{216 x^3}+\frac{\left (1-x^2\right )^{2/3}}{24 \left (x^2+3\right ) x^3}+\frac{11 \tan ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{216\ 2^{2/3} \sqrt{3}}-\frac{11 \tanh ^{-1}(x)}{648\ 2^{2/3}} \]
Warning: Unable to verify antiderivative.
[In] Int[1/(x^4*(1 - x^2)^(1/3)*(3 + x^2)^2),x]
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Rubi in Sympy [A] time = 7.60229, size = 24, normalized size = 0.04 \[ - \frac{\operatorname{appellf_{1}}{\left (- \frac{3}{2},\frac{1}{3},2,- \frac{1}{2},x^{2},- \frac{x^{2}}{3} \right )}}{27 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**4/(-x**2+1)**(1/3)/(x**2+3)**2,x)
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Mathematica [C] time = 0.406027, size = 246, normalized size = 0.42 \[ \frac{\frac{55 x^6 F_1\left (\frac{3}{2};\frac{1}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right )}{2 x^2 \left (F_1\left (\frac{5}{2};\frac{4}{3},1;\frac{7}{2};x^2,-\frac{x^2}{3}\right )-F_1\left (\frac{5}{2};\frac{1}{3},2;\frac{7}{2};x^2,-\frac{x^2}{3}\right )\right )+15 F_1\left (\frac{3}{2};\frac{1}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right )}+\frac{2079 x^4 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right )}{2 x^2 \left (F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right )-F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};x^2,-\frac{x^2}{3}\right )\right )+9 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right )}-33 x^6+33 x^4+216 x^2-216}{1944 x^3 \sqrt [3]{1-x^2} \left (x^2+3\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/(x^4*(1 - x^2)^(1/3)*(3 + x^2)^2),x]
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Maple [F] time = 0.076, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{4} \left ({x}^{2}+3 \right ) ^{2}}{\frac{1}{\sqrt [3]{-{x}^{2}+1}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^4/(-x^2+1)^(1/3)/(x^2+3)^2,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{2} + 3\right )}^{2}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^2 + 3)^2*(-x^2 + 1)^(1/3)*x^4),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^4),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**4/(-x**2+1)**(1/3)/(x**2+3)**2,x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{2} + 3\right )}^{2}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^2 + 3)^2*(-x^2 + 1)^(1/3)*x^4),x, algorithm="giac")
[Out]