Optimal. Leaf size=127 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{2 \sqrt{6}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{2 \sqrt{6}}-\frac{\sqrt{\frac{x^2}{\left (\sqrt{3 x^2-1}+1\right )^2}} \left (\sqrt{3 x^2-1}+1\right ) F\left (2 \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )|\frac{1}{2}\right )}{2 \sqrt{3} x} \]
[Out]
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Rubi [A] time = 0.143443, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{2 \sqrt{6}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{2 \sqrt{6}}-\frac{\sqrt{\frac{x^2}{\left (\sqrt{3 x^2-1}+1\right )^2}} \left (\sqrt{3 x^2-1}+1\right ) F\left (2 \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )|\frac{1}{2}\right )}{2 \sqrt{3} x} \]
Antiderivative was successfully verified.
[In] Int[1/((-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)),x]
[Out]
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Rubi in Sympy [A] time = 49.1533, size = 170, normalized size = 1.34 \[ - \frac{\sqrt{2} x \left (1 - i\right ) \Pi \left (i; \operatorname{asin}{\left (\frac{\sqrt{2} \left (1 + i\right ) \sqrt [4]{3 x^{2} - 1}}{2} \right )}\middle | -1\right )}{2 \sqrt{- i \sqrt{3 x^{2} - 1} + 1} \sqrt{i \sqrt{3 x^{2} - 1} + 1}} - \frac{\sqrt{3} \sqrt{\frac{x^{2}}{\left (\sqrt{3 x^{2} - 1} + 1\right )^{2}}} \left (\sqrt{3 x^{2} - 1} + 1\right ) F\left (2 \operatorname{atan}{\left (\sqrt [4]{3 x^{2} - 1} \right )}\middle | \frac{1}{2}\right )}{12 x} - \frac{\sqrt{6} \sqrt{x^{2}} \operatorname{atanh}{\left (\frac{\sqrt{6} \sqrt [4]{3 x^{2} - 1}}{3 \sqrt{x^{2}}} \right )}}{12 x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(3*x**2-2)/(3*x**2-1)**(3/4),x)
[Out]
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Mathematica [C] time = 0.159596, size = 129, normalized size = 1.02 \[ \frac{2 x F_1\left (\frac{1}{2};\frac{3}{4},1;\frac{3}{2};3 x^2,\frac{3 x^2}{2}\right )}{\left (3 x^2-2\right ) \left (3 x^2-1\right )^{3/4} \left (x^2 \left (2 F_1\left (\frac{3}{2};\frac{3}{4},2;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right )+3 F_1\left (\frac{3}{2};\frac{7}{4},1;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right )\right )+2 F_1\left (\frac{1}{2};\frac{3}{4},1;\frac{3}{2};3 x^2,\frac{3 x^2}{2}\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/((-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)),x]
[Out]
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Maple [F] time = 0.098, size = 0, normalized size = 0. \[ \int{\frac{1}{3\,{x}^{2}-2} \left ( 3\,{x}^{2}-1 \right ) ^{-{\frac{3}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(3*x^2-2)/(3*x^2-1)^(3/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (3 \, x^{2} - 1\right )}^{\frac{3}{4}}{\left (3 \, x^{2} - 2\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((3*x^2 - 1)^(3/4)*(3*x^2 - 2)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (3 \, x^{2} - 1\right )}^{\frac{3}{4}}{\left (3 \, x^{2} - 2\right )}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((3*x^2 - 1)^(3/4)*(3*x^2 - 2)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (3 x^{2} - 2\right ) \left (3 x^{2} - 1\right )^{\frac{3}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(3*x**2-2)/(3*x**2-1)**(3/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (3 \, x^{2} - 1\right )}^{\frac{3}{4}}{\left (3 \, x^{2} - 2\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((3*x^2 - 1)^(3/4)*(3*x^2 - 2)),x, algorithm="giac")
[Out]