Optimal. Leaf size=122 \[ \frac{x^3}{9}-\frac{\log \left (\sqrt{3} x^2-2^{3/4} \sqrt [4]{3} x+\sqrt{2}\right )}{6\ 6^{3/4}}+\frac{\log \left (\sqrt{3} x^2+2^{3/4} \sqrt [4]{3} x+\sqrt{2}\right )}{6\ 6^{3/4}}+\frac{\tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{3\ 6^{3/4}}-\frac{\tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{3\ 6^{3/4}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.195524, antiderivative size = 104, normalized size of antiderivative = 0.85, number of steps used = 10, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538 \[ \frac{x^3}{9}-\frac{\log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{6\ 6^{3/4}}+\frac{\log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{6\ 6^{3/4}}+\frac{\tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{3\ 6^{3/4}}-\frac{\tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{3\ 6^{3/4}} \]
Antiderivative was successfully verified.
[In] Int[x^6/(2 + 3*x^4),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 20.2006, size = 88, normalized size = 0.72 \[ \frac{x^{3}}{9} - \frac{\sqrt [4]{6} \log{\left (3 x^{2} - 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{36} + \frac{\sqrt [4]{6} \log{\left (3 x^{2} + 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{36} - \frac{\sqrt [4]{6} \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{18} - \frac{\sqrt [4]{6} \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{18} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**6/(3*x**4+2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0533002, size = 98, normalized size = 0.8 \[ \frac{1}{36} \left (4 x^3-\sqrt [4]{6} \log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )+\sqrt [4]{6} \log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )+2 \sqrt [4]{6} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )-2 \sqrt [4]{6} \tan ^{-1}\left (\sqrt [4]{6} x+1\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^6/(2 + 3*x^4),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.008, size = 116, normalized size = 1. \[{\frac{{x}^{3}}{9}}-{\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}}{108}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) }-{\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}}{108}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }-{\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}}{216}\ln \left ({1 \left ({x}^{2}-{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) \left ({x}^{2}+{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) ^{-1}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^6/(3*x^4+2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.60352, size = 170, normalized size = 1.39 \[ \frac{1}{9} \, x^{3} - \frac{1}{18} \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x + 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) - \frac{1}{18} \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x - 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{1}{36} \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} \log \left (\sqrt{3} x^{2} + 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) - \frac{1}{36} \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} \log \left (\sqrt{3} x^{2} - 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^6/(3*x^4 + 2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.240086, size = 235, normalized size = 1.93 \[ \frac{1}{1944} \cdot 54^{\frac{3}{4}}{\left (4 \cdot 54^{\frac{1}{4}} x^{3} + 12 \, \sqrt{2} \arctan \left (\frac{54}{54^{\frac{3}{4}} \sqrt{2} \sqrt{\frac{1}{6}} \sqrt{\sqrt{6}{\left (9 \, \sqrt{6} x^{2} + 54^{\frac{3}{4}} \sqrt{2} x + 18\right )}} + 3 \cdot 54^{\frac{3}{4}} \sqrt{2} x + 54}\right ) + 12 \, \sqrt{2} \arctan \left (\frac{54}{54^{\frac{3}{4}} \sqrt{2} \sqrt{\frac{1}{6}} \sqrt{\sqrt{6}{\left (9 \, \sqrt{6} x^{2} - 54^{\frac{3}{4}} \sqrt{2} x + 18\right )}} + 3 \cdot 54^{\frac{3}{4}} \sqrt{2} x - 54}\right ) + 3 \, \sqrt{2} \log \left (9 \, \sqrt{6} x^{2} + 54^{\frac{3}{4}} \sqrt{2} x + 18\right ) - 3 \, \sqrt{2} \log \left (9 \, \sqrt{6} x^{2} - 54^{\frac{3}{4}} \sqrt{2} x + 18\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^6/(3*x^4 + 2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 1.51627, size = 92, normalized size = 0.75 \[ \frac{x^{3}}{9} - \frac{\sqrt [4]{6} \log{\left (x^{2} - \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{36} + \frac{\sqrt [4]{6} \log{\left (x^{2} + \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{36} - \frac{\sqrt [4]{6} \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{18} - \frac{\sqrt [4]{6} \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{18} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**6/(3*x**4+2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.226364, size = 135, normalized size = 1.11 \[ \frac{1}{9} \, x^{3} - \frac{1}{18} \cdot 6^{\frac{1}{4}} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) - \frac{1}{18} \cdot 6^{\frac{1}{4}} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{36} \cdot 6^{\frac{1}{4}}{\rm ln}\left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) - \frac{1}{36} \cdot 6^{\frac{1}{4}}{\rm ln}\left (x^{2} - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^6/(3*x^4 + 2),x, algorithm="giac")
[Out]