Optimal. Leaf size=129 \[ -\frac{x}{12 \left (3 x^4+2\right )}-\frac{\log \left (\sqrt{3} x^2-2^{3/4} \sqrt [4]{3} x+\sqrt{2}\right )}{96 \sqrt [4]{6}}+\frac{\log \left (\sqrt{3} x^2+2^{3/4} \sqrt [4]{3} x+\sqrt{2}\right )}{96 \sqrt [4]{6}}-\frac{\tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{48 \sqrt [4]{6}}+\frac{\tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{48 \sqrt [4]{6}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.146405, antiderivative size = 111, normalized size of antiderivative = 0.86, number of steps used = 10, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538 \[ -\frac{x}{12 \left (3 x^4+2\right )}-\frac{\log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{96 \sqrt [4]{6}}+\frac{\log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{96 \sqrt [4]{6}}-\frac{\tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{48 \sqrt [4]{6}}+\frac{\tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{48 \sqrt [4]{6}} \]
Antiderivative was successfully verified.
[In] Int[x^4/(2 + 3*x^4)^2,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 19.6939, size = 92, normalized size = 0.71 \[ - \frac{x}{12 \left (3 x^{4} + 2\right )} - \frac{6^{\frac{3}{4}} \log{\left (3 x^{2} - 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{576} + \frac{6^{\frac{3}{4}} \log{\left (3 x^{2} + 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{576} + \frac{6^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{288} + \frac{6^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{288} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4/(3*x**4+2)**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.166623, size = 105, normalized size = 0.81 \[ \frac{1}{576} \left (-\frac{48 x}{3 x^4+2}-6^{3/4} \log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )+6^{3/4} \log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )-2\ 6^{3/4} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2\ 6^{3/4} \tan ^{-1}\left (\sqrt [4]{6} x+1\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^4/(2 + 3*x^4)^2,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.011, size = 121, normalized size = 0.9 \[ -{\frac{x}{36} \left ({x}^{4}+{\frac{2}{3}} \right ) ^{-1}}+{\frac{\sqrt{3}\sqrt [4]{6}\sqrt{2}}{288}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }+{\frac{\sqrt{3}\sqrt [4]{6}\sqrt{2}}{576}\ln \left ({1 \left ({x}^{2}+{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) \left ({x}^{2}-{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) ^{-1}} \right ) }+{\frac{\sqrt{3}\sqrt [4]{6}\sqrt{2}}{288}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4/(3*x^4+2)^2,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.58328, size = 180, normalized size = 1.4 \[ \frac{1}{288} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x + 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{1}{288} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x - 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{1}{576} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} \log \left (\sqrt{3} x^{2} + 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) - \frac{1}{576} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} \log \left (\sqrt{3} x^{2} - 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) - \frac{x}{12 \,{\left (3 \, x^{4} + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(3*x^4 + 2)^2,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.240542, size = 277, normalized size = 2.15 \[ -\frac{24^{\frac{3}{4}}{\left (4 \, \sqrt{2}{\left (3 \, x^{4} + 2\right )} \arctan \left (\frac{2}{24^{\frac{1}{4}} \sqrt{2} \sqrt{\frac{1}{6}} \sqrt{\sqrt{6}{\left (\sqrt{6} x^{2} + 24^{\frac{1}{4}} \sqrt{2} x + 2\right )}} + 24^{\frac{1}{4}} \sqrt{2} x + 2}\right ) + 4 \, \sqrt{2}{\left (3 \, x^{4} + 2\right )} \arctan \left (\frac{2}{24^{\frac{1}{4}} \sqrt{2} \sqrt{\frac{1}{6}} \sqrt{\sqrt{6}{\left (\sqrt{6} x^{2} - 24^{\frac{1}{4}} \sqrt{2} x + 2\right )}} + 24^{\frac{1}{4}} \sqrt{2} x - 2}\right ) - \sqrt{2}{\left (3 \, x^{4} + 2\right )} \log \left (2 \, \sqrt{6} x^{2} + 2 \cdot 24^{\frac{1}{4}} \sqrt{2} x + 4\right ) + \sqrt{2}{\left (3 \, x^{4} + 2\right )} \log \left (2 \, \sqrt{6} x^{2} - 2 \cdot 24^{\frac{1}{4}} \sqrt{2} x + 4\right ) + 8 \cdot 24^{\frac{1}{4}} x\right )}}{2304 \,{\left (3 \, x^{4} + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(3*x^4 + 2)^2,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 1.70627, size = 95, normalized size = 0.74 \[ - \frac{x}{36 x^{4} + 24} - \frac{6^{\frac{3}{4}} \log{\left (x^{2} - \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{288} + \frac{6^{\frac{3}{4}} \log{\left (x^{2} + \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{288} + \frac{6^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{144} + \frac{6^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{144} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4/(3*x**4+2)**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.228689, size = 144, normalized size = 1.12 \[ \frac{1}{288} \cdot 6^{\frac{3}{4}} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{288} \cdot 6^{\frac{3}{4}} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{576} \cdot 6^{\frac{3}{4}}{\rm ln}\left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) - \frac{1}{576} \cdot 6^{\frac{3}{4}}{\rm ln}\left (x^{2} - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) - \frac{x}{12 \,{\left (3 \, x^{4} + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(3*x^4 + 2)^2,x, algorithm="giac")
[Out]