Optimal. Leaf size=821 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt [3]{2} (2-x)^{2/3}}{\sqrt{3} \sqrt [3]{1-x}}+\frac{1}{\sqrt{3}}\right )}{2 \sqrt [3]{2} \sqrt{3}}+\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{4 \sqrt [3]{2} (3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}}-\frac{\sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{2^{5/6} \sqrt [4]{3} (3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}}+\frac{\log \left (\frac{(2-x)^{2/3}}{2^{2/3}}-\sqrt [3]{1-x}\right )}{4 \sqrt [3]{2}}-\frac{\log (x)}{6 \sqrt [3]{2}}-\frac{(1-x)^{2/3} (2-x)^{2/3}}{2 x}-\frac{\sqrt{(3-2 x)^2} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2}}{2 \sqrt [3]{2} (3-2 x) \sqrt [3]{1-x} \sqrt [3]{2-x} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )}-\frac{(1-x)^{2/3} (2-x)^{2/3}}{4 x^2} \]
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Rubi [A] time = 1.2221, antiderivative size = 821, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt [3]{2} (2-x)^{2/3}}{\sqrt{3} \sqrt [3]{1-x}}+\frac{1}{\sqrt{3}}\right )}{2 \sqrt [3]{2} \sqrt{3}}+\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{4 \sqrt [3]{2} (3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}}-\frac{\sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{2^{5/6} \sqrt [4]{3} (3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}}+\frac{\log \left (\frac{(2-x)^{2/3}}{2^{2/3}}-\sqrt [3]{1-x}\right )}{4 \sqrt [3]{2}}-\frac{\log (x)}{6 \sqrt [3]{2}}-\frac{(1-x)^{2/3} (2-x)^{2/3}}{2 x}-\frac{\sqrt{(3-2 x)^2} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2}}{2 \sqrt [3]{2} (3-2 x) \sqrt [3]{1-x} \sqrt [3]{2-x} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )}-\frac{(1-x)^{2/3} (2-x)^{2/3}}{4 x^2} \]
Warning: Unable to verify antiderivative.
[In] Int[1/((1 - x)^(1/3)*(2 - x)^(1/3)*x^3),x]
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Rubi in Sympy [A] time = 43.5908, size = 731, normalized size = 0.89 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(1-x)**(1/3)/(2-x)**(1/3)/x**3,x)
[Out]
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Mathematica [C] time = 0.350429, size = 225, normalized size = 0.27 \[ \frac{(1-x)^{2/3} \left (\frac{75 x F_1\left (\frac{2}{3};\frac{1}{3},1;\frac{5}{3};x-1,1-x\right )}{(x-1) \left (3 F_1\left (\frac{5}{3};\frac{1}{3},2;\frac{8}{3};x-1,1-x\right )-F_1\left (\frac{5}{3};\frac{4}{3},1;\frac{8}{3};x-1,1-x\right )\right )-5 F_1\left (\frac{2}{3};\frac{1}{3},1;\frac{5}{3};x-1,1-x\right )}+\frac{16 (x-1) x F_1\left (\frac{5}{3};\frac{1}{3},1;\frac{8}{3};x-1,1-x\right )}{(x-1) \left (3 F_1\left (\frac{8}{3};\frac{1}{3},2;\frac{11}{3};x-1,1-x\right )-F_1\left (\frac{8}{3};\frac{4}{3},1;\frac{11}{3};x-1,1-x\right )\right )-8 F_1\left (\frac{5}{3};\frac{1}{3},1;\frac{8}{3};x-1,1-x\right )}+5 (x-2) (2 x+1)\right )}{20 \sqrt [3]{2-x} x^2} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/((1 - x)^(1/3)*(2 - x)^(1/3)*x^3),x]
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Maple [F] time = 0.115, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3}}{\frac{1}{\sqrt [3]{1-x}}}{\frac{1}{\sqrt [3]{2-x}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(1-x)^(1/3)/(2-x)^(1/3)/x^3,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3}{\left (-x + 2\right )}^{\frac{1}{3}}{\left (-x + 1\right )}^{\frac{1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x^3*(-x + 2)^(1/3)*(-x + 1)^(1/3)),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^3*(-x + 2)^(1/3)*(-x + 1)^(1/3)),x, algorithm="fricas")
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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/(1-x)**(1/3)/(2-x)**(1/3)/x**3,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3}{\left (-x + 2\right )}^{\frac{1}{3}}{\left (-x + 1\right )}^{\frac{1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x^3*(-x + 2)^(1/3)*(-x + 1)^(1/3)),x, algorithm="giac")
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