Optimal. Leaf size=239 \[ -\frac{\sqrt{x^8+1}}{5 x^5}+\frac{\sqrt{\frac{\left (x^2+1\right )^2}{x^2}} \sqrt{-\frac{x^8+1}{x^4}} x^3 F\left (\sin ^{-1}\left (\frac{1}{2} \sqrt{-\frac{\sqrt{2} x^4-2 x^2+\sqrt{2}}{x^2}}\right )|-2 \left (1-\sqrt{2}\right )\right )}{10 \sqrt{2+\sqrt{2}} \left (x^2+1\right ) \sqrt{x^8+1}}+\frac{\sqrt{-\frac{\left (1-x^2\right )^2}{x^2}} \sqrt{-\frac{x^8+1}{x^4}} x^3 F\left (\sin ^{-1}\left (\frac{1}{2} \sqrt{\frac{\sqrt{2} x^4+2 x^2+\sqrt{2}}{x^2}}\right )|-2 \left (1-\sqrt{2}\right )\right )}{10 \sqrt{2+\sqrt{2}} \left (1-x^2\right ) \sqrt{x^8+1}} \]
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Rubi [A] time = 0.16191, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{\sqrt{x^8+1}}{5 x^5}+\frac{\sqrt{\frac{\left (x^2+1\right )^2}{x^2}} \sqrt{-\frac{x^8+1}{x^4}} x^3 F\left (\sin ^{-1}\left (\frac{1}{2} \sqrt{-\frac{\sqrt{2} x^4-2 x^2+\sqrt{2}}{x^2}}\right )|-2 \left (1-\sqrt{2}\right )\right )}{10 \sqrt{2+\sqrt{2}} \left (x^2+1\right ) \sqrt{x^8+1}}+\frac{\sqrt{-\frac{\left (1-x^2\right )^2}{x^2}} \sqrt{-\frac{x^8+1}{x^4}} x^3 F\left (\sin ^{-1}\left (\frac{1}{2} \sqrt{\frac{\sqrt{2} x^4+2 x^2+\sqrt{2}}{x^2}}\right )|-2 \left (1-\sqrt{2}\right )\right )}{10 \sqrt{2+\sqrt{2}} \left (1-x^2\right ) \sqrt{x^8+1}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^6*Sqrt[1 + x^8]),x]
[Out]
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Rubi in Sympy [A] time = 13.9414, size = 202, normalized size = 0.85 \[ \frac{x^{3} \sqrt{\frac{- x^{8} - 1}{x^{4}}} \sqrt{\frac{\left (x^{2} + 1\right )^{2}}{x^{2}}} F\left (\operatorname{asin}{\left (\frac{\sqrt{- \frac{\sqrt{2} x^{4} - 2 x^{2} + \sqrt{2}}{x^{2}}}}{2} \right )}\middle | -2 + 2 \sqrt{2}\right )}{10 \sqrt{\sqrt{2} + 2} \left (x^{2} + 1\right ) \sqrt{x^{8} + 1}} + \frac{x^{3} \sqrt{\frac{- x^{8} - 1}{x^{4}}} \sqrt{- \frac{\left (- x^{2} + 1\right )^{2}}{x^{2}}} F\left (\operatorname{asin}{\left (\frac{\sqrt{\frac{\sqrt{2} x^{4} + 2 x^{2} + \sqrt{2}}{x^{2}}}}{2} \right )}\middle | -2 + 2 \sqrt{2}\right )}{10 \sqrt{\sqrt{2} + 2} \left (- x^{2} + 1\right ) \sqrt{x^{8} + 1}} - \frac{\sqrt{x^{8} + 1}}{5 x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**6/(x**8+1)**(1/2),x)
[Out]
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Mathematica [A] time = 1.41224, size = 221, normalized size = 0.92 \[ \frac{-\frac{\sqrt{-\frac{\left (x^2-1\right )^2}{x^2}} \left (x^2+1\right ) \sqrt{-\frac{x^8+1}{x^4}} x^8 F\left (\sin ^{-1}\left (\frac{\sqrt{x^2+\sqrt{2}+\frac{1}{x^2}}}{2^{3/4}}\right )|2 \left (-1+\sqrt{2}\right )\right )}{\sqrt{2+\sqrt{2}}}-\left (x^2-1\right ) \left (2 \left (x^{10}+x^8+x^2+1\right )+\sqrt{x^2+\frac{1}{x^2}+\sqrt{2}} \sqrt{\frac{\left (x^2+1\right )^2 \left (x^4-\sqrt{2} x^2+1\right )}{\left (\sqrt{2}-2\right ) x^4}} x^8 F\left (\sin ^{-1}\left (\frac{\sqrt{x^2+\sqrt{2}+\frac{1}{x^2}}}{2^{3/4}}\right )|-2 \left (1+\sqrt{2}\right )\right )\right )}{10 x^5 \left (x^4-1\right ) \sqrt{x^8+1}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/(x^6*Sqrt[1 + x^8]),x]
[Out]
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Maple [C] time = 0.042, size = 30, normalized size = 0.1 \[ -{\frac{1}{5\,{x}^{5}}\sqrt{{x}^{8}+1}}-{\frac{{x}^{3}}{15}{\mbox{$_2$F$_1$}({\frac{3}{8}},{\frac{1}{2}};\,{\frac{11}{8}};\,-{x}^{8})}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^6/(x^8+1)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{8} + 1} x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x^8 + 1)*x^6),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{x^{8} + 1} x^{6}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x^8 + 1)*x^6),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.80788, size = 32, normalized size = 0.13 \[ \frac{\Gamma \left (- \frac{5}{8}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{8}, \frac{1}{2} \\ \frac{3}{8} \end{matrix}\middle |{x^{8} e^{i \pi }} \right )}}{8 x^{5} \Gamma \left (\frac{3}{8}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**6/(x**8+1)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{8} + 1} x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x^8 + 1)*x^6),x, algorithm="giac")
[Out]