Optimal. Leaf size=179 \[ \frac{1}{4} x \sqrt{x^6+2}-\frac{x \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{4 \sqrt [3]{2} \sqrt [4]{3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}} \]
[Out]
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Rubi [A] time = 0.114422, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{1}{4} x \sqrt{x^6+2}-\frac{x \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{4 \sqrt [3]{2} \sqrt [4]{3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}} \]
Antiderivative was successfully verified.
[In] Int[x^6/Sqrt[2 + x^6],x]
[Out]
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Rubi in Sympy [A] time = 4.32574, size = 153, normalized size = 0.85 \[ - \frac{3^{\frac{3}{4}} x \sqrt{\frac{2 \sqrt [3]{2} x^{4} - 2 \cdot 2^{\frac{2}{3}} x^{2} + 4}{\left (x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}\right )^{2}}} \left (x^{2} + \sqrt [3]{2}\right ) F\left (\operatorname{acos}{\left (\frac{x^{2} \left (- \sqrt{3} + 1\right ) + \sqrt [3]{2}}{x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{24 \sqrt{\frac{x^{2} \left (x^{2} + \sqrt [3]{2}\right )}{\left (x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}\right )^{2}}} \sqrt{x^{6} + 2}} + \frac{x \sqrt{x^{6} + 2}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**6/(x**6+2)**(1/2),x)
[Out]
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Mathematica [A] time = 1.24848, size = 173, normalized size = 0.97 \[ \frac{x \left (6 \left (x^6+2\right )-\frac{2^{2/3} 3^{3/4} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{2}-\left (-1+\sqrt{3}\right ) x^2}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{\sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}}}\right )}{24 \sqrt{x^6+2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^6/Sqrt[2 + x^6],x]
[Out]
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Maple [C] time = 0.046, size = 29, normalized size = 0.2 \[{\frac{x}{4}\sqrt{{x}^{6}+2}}-{\frac{x\sqrt{2}}{4}{\mbox{$_2$F$_1$}({\frac{1}{6}},{\frac{1}{2}};\,{\frac{7}{6}};\,-{\frac{{x}^{6}}{2}})}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^6/(x^6+2)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{\sqrt{x^{6} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^6/sqrt(x^6 + 2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{6}}{\sqrt{x^{6} + 2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^6/sqrt(x^6 + 2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.26904, size = 36, normalized size = 0.2 \[ \frac{\sqrt{2} x^{7} \Gamma \left (\frac{7}{6}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{7}{6} \\ \frac{13}{6} \end{matrix}\middle |{\frac{x^{6} e^{i \pi }}{2}} \right )}}{12 \Gamma \left (\frac{13}{6}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**6/(x**6+2)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{\sqrt{x^{6} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^6/sqrt(x^6 + 2),x, algorithm="giac")
[Out]