Optimal. Leaf size=18 \[ \frac{1}{5} \tanh ^{-1}\left (\frac{x^5}{\sqrt{x^{10}-2}}\right ) \]
[Out]
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Rubi [A] time = 0.0192185, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{1}{5} \tanh ^{-1}\left (\frac{x^5}{\sqrt{x^{10}-2}}\right ) \]
Antiderivative was successfully verified.
[In] Int[x^4/Sqrt[-2 + x^10],x]
[Out]
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Rubi in Sympy [A] time = 1.28651, size = 14, normalized size = 0.78 \[ \frac{\operatorname{atanh}{\left (\frac{x^{5}}{\sqrt{x^{10} - 2}} \right )}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4/(x**10-2)**(1/2),x)
[Out]
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Mathematica [B] time = 0.00651485, size = 42, normalized size = 2.33 \[ \frac{1}{10} \log \left (\frac{x^5}{\sqrt{x^{10}-2}}+1\right )-\frac{1}{10} \log \left (1-\frac{x^5}{\sqrt{x^{10}-2}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^4/Sqrt[-2 + x^10],x]
[Out]
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Maple [C] time = 0.068, size = 34, normalized size = 1.9 \[{\frac{1}{5}\sqrt{-{\it signum} \left ( -1+{\frac{{x}^{10}}{2}} \right ) }\arcsin \left ({\frac{{x}^{5}\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{{\it signum} \left ( -1+{\frac{{x}^{10}}{2}} \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4/(x^10-2)^(1/2),x)
[Out]
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Maxima [A] time = 1.39161, size = 45, normalized size = 2.5 \[ \frac{1}{10} \, \log \left (\frac{\sqrt{x^{10} - 2}}{x^{5}} + 1\right ) - \frac{1}{10} \, \log \left (\frac{\sqrt{x^{10} - 2}}{x^{5}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/sqrt(x^10 - 2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.23782, size = 22, normalized size = 1.22 \[ -\frac{1}{5} \, \log \left (-x^{5} + \sqrt{x^{10} - 2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/sqrt(x^10 - 2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.76178, size = 34, normalized size = 1.89 \[ \begin{cases} \frac{\operatorname{acosh}{\left (\frac{\sqrt{2} x^{5}}{2} \right )}}{5} & \text{for}\: \frac{\left |{x^{10}}\right |}{2} > 1 \\- \frac{i \operatorname{asin}{\left (\frac{\sqrt{2} x^{5}}{2} \right )}}{5} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4/(x**10-2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.22727, size = 23, normalized size = 1.28 \[ -\frac{1}{5} \,{\rm ln}\left ({\left | -x^{5} + \sqrt{x^{10} - 2} \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/sqrt(x^10 - 2),x, algorithm="giac")
[Out]