Optimal. Leaf size=264 \[ \frac{9 \sqrt [4]{3 x^2-1} x}{2 \left (\sqrt{3 x^2-1}+1\right )}-\frac{3 \left (3 x^2-1\right )^{3/4}}{2 x}-\frac{3}{8} \sqrt{\frac{3}{2}} \tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )-\frac{3}{8} \sqrt{\frac{3}{2}} \tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )+\frac{3 \sqrt{3} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-1}+1\right )^2}} \left (\sqrt{3 x^2-1}+1\right ) F\left (2 \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )|\frac{1}{2}\right )}{4 x}-\frac{3 \sqrt{3} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-1}+1\right )^2}} \left (\sqrt{3 x^2-1}+1\right ) E\left (2 \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )|\frac{1}{2}\right )}{2 x}-\frac{\left (3 x^2-1\right )^{3/4}}{6 x^3} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.522526, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{9 \sqrt [4]{3 x^2-1} x}{2 \left (\sqrt{3 x^2-1}+1\right )}-\frac{3 \left (3 x^2-1\right )^{3/4}}{2 x}-\frac{3}{8} \sqrt{\frac{3}{2}} \tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )-\frac{3}{8} \sqrt{\frac{3}{2}} \tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )+\frac{3 \sqrt{3} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-1}+1\right )^2}} \left (\sqrt{3 x^2-1}+1\right ) F\left (2 \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )|\frac{1}{2}\right )}{4 x}-\frac{3 \sqrt{3} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-1}+1\right )^2}} \left (\sqrt{3 x^2-1}+1\right ) E\left (2 \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )|\frac{1}{2}\right )}{2 x}-\frac{\left (3 x^2-1\right )^{3/4}}{6 x^3} \]
Antiderivative was successfully verified.
[In] Int[1/(x^4*(-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 23.8747, size = 46, normalized size = 0.17 \[ - \frac{\left (3 x^{2} - 1\right )^{\frac{3}{4}} \operatorname{appellf_{1}}{\left (- \frac{3}{2},\frac{1}{4},1,- \frac{1}{2},3 x^{2},\frac{3 x^{2}}{2} \right )}}{6 x^{3} \left (- 3 x^{2} + 1\right )^{\frac{3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**4/(3*x**2-2)/(3*x**2-1)**(1/4),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.228995, size = 148, normalized size = 0.56 \[ \frac{1}{2} \left (3 x^2-1\right )^{3/4} \left (\frac{9 x F_1\left (\frac{1}{2};-\frac{3}{4},1;\frac{3}{2};3 x^2,\frac{3 x^2}{2}\right )}{\left (3 x^2-2\right ) \left (x^2 \left (2 F_1\left (\frac{3}{2};-\frac{3}{4},2;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right )-3 F_1\left (\frac{3}{2};\frac{1}{4},1;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right )\right )+2 F_1\left (\frac{1}{2};-\frac{3}{4},1;\frac{3}{2};3 x^2,\frac{3 x^2}{2}\right )\right )}-\frac{9 x^2+1}{3 x^3}\right ) \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/(x^4*(-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.131, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{4} \left ( 3\,{x}^{2}-2 \right ) }{\frac{1}{\sqrt [4]{3\,{x}^{2}-1}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^4/(3*x^2-2)/(3*x^2-1)^(1/4),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}{\left (3 \, x^{2} - 2\right )} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((3*x^2 - 1)^(1/4)*(3*x^2 - 2)*x^4),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (3 \, x^{6} - 2 \, x^{4}\right )}{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((3*x^2 - 1)^(1/4)*(3*x^2 - 2)*x^4),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{4} \left (3 x^{2} - 2\right ) \sqrt [4]{3 x^{2} - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**4/(3*x**2-2)/(3*x**2-1)**(1/4),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}{\left (3 \, x^{2} - 2\right )} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((3*x^2 - 1)^(1/4)*(3*x^2 - 2)*x^4),x, algorithm="giac")
[Out]