3.363 \(\int \frac{x^4}{\sqrt{-2+x^{10}}} \, dx\)

Optimal. Leaf size=18 \[ \frac{1}{5} \tanh ^{-1}\left (\frac{x^5}{\sqrt{x^{10}-2}}\right ) \]

[Out]

ArcTanh[x^5/Sqrt[-2 + x^10]]/5

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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]

ArcTanh[x^5/Sqrt[-2 + x^10]]/5

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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]

atanh(x**5/sqrt(x**10 - 2))/5

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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]

-Log[1 - x^5/Sqrt[-2 + x^10]]/10 + Log[1 + x^5/Sqrt[-2 + x^10]]/10

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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]

1/5/signum(-1+1/2*x^10)^(1/2)*(-signum(-1+1/2*x^10))^(1/2)*arcsin(1/2*x^5*2^(1/2
))

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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]

1/10*log(sqrt(x^10 - 2)/x^5 + 1) - 1/10*log(sqrt(x^10 - 2)/x^5 - 1)

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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]

-1/5*log(-x^5 + sqrt(x^10 - 2))

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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]

Piecewise((acosh(sqrt(2)*x**5/2)/5, Abs(x**10)/2 > 1), (-I*asin(sqrt(2)*x**5/2)/
5, True))

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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]

-1/5*ln(abs(-x^5 + sqrt(x^10 - 2)))