3.1511 \(\int x^3 \sqrt{-2+x^8} \, dx\)

Optimal. Leaf size=35 \[ \frac{1}{8} x^4 \sqrt{x^8-2}-\frac{1}{4} \tanh ^{-1}\left (\frac{x^4}{\sqrt{x^8-2}}\right ) \]

[Out]

(x^4*Sqrt[-2 + x^8])/8 - ArcTanh[x^4/Sqrt[-2 + x^8]]/4

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Rubi [A]  time = 0.0366006, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{1}{8} x^4 \sqrt{x^8-2}-\frac{1}{4} \tanh ^{-1}\left (\frac{x^4}{\sqrt{x^8-2}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^3*Sqrt[-2 + x^8],x]

[Out]

(x^4*Sqrt[-2 + x^8])/8 - ArcTanh[x^4/Sqrt[-2 + x^8]]/4

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Rubi in Sympy [A]  time = 2.89497, size = 27, normalized size = 0.77 \[ \frac{x^{4} \sqrt{x^{8} - 2}}{8} - \frac{\operatorname{atanh}{\left (\frac{x^{4}}{\sqrt{x^{8} - 2}} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(x**8-2)**(1/2),x)

[Out]

x**4*sqrt(x**8 - 2)/8 - atanh(x**4/sqrt(x**8 - 2))/4

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Mathematica [A]  time = 0.0154046, size = 35, normalized size = 1. \[ \frac{1}{8} x^4 \sqrt{x^8-2}-\frac{1}{4} \log \left (\sqrt{x^8-2}+x^4\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*Sqrt[-2 + x^8],x]

[Out]

(x^4*Sqrt[-2 + x^8])/8 - Log[x^4 + Sqrt[-2 + x^8]]/4

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Maple [C]  time = 0.069, size = 47, normalized size = 1.3 \[{\frac{{x}^{4}}{8}\sqrt{{x}^{8}-2}}-{\frac{1}{4}\sqrt{-{\it signum} \left ( -1+{\frac{{x}^{8}}{2}} \right ) }\arcsin \left ({\frac{{x}^{4}\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{{\it signum} \left ( -1+{\frac{{x}^{8}}{2}} \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(x^8-2)^(1/2),x)

[Out]

1/8*x^4*(x^8-2)^(1/2)-1/4/signum(-1+1/2*x^8)^(1/2)*(-signum(-1+1/2*x^8))^(1/2)*a
rcsin(1/2*x^4*2^(1/2))

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Maxima [A]  time = 1.43431, size = 78, normalized size = 2.23 \[ -\frac{\sqrt{x^{8} - 2}}{4 \, x^{4}{\left (\frac{x^{8} - 2}{x^{8}} - 1\right )}} - \frac{1}{8} \, \log \left (\frac{\sqrt{x^{8} - 2}}{x^{4}} + 1\right ) + \frac{1}{8} \, \log \left (\frac{\sqrt{x^{8} - 2}}{x^{4}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^8 - 2)*x^3,x, algorithm="maxima")

[Out]

-1/4*sqrt(x^8 - 2)/(x^4*((x^8 - 2)/x^8 - 1)) - 1/8*log(sqrt(x^8 - 2)/x^4 + 1) +
1/8*log(sqrt(x^8 - 2)/x^4 - 1)

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Fricas [A]  time = 0.223891, size = 109, normalized size = 3.11 \[ -\frac{x^{16} - 2 \, x^{8} - 2 \,{\left (x^{8} - \sqrt{x^{8} - 2} x^{4} - 1\right )} \log \left (-x^{4} + \sqrt{x^{8} - 2}\right ) -{\left (x^{12} - x^{4}\right )} \sqrt{x^{8} - 2}}{8 \,{\left (x^{8} - \sqrt{x^{8} - 2} x^{4} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^8 - 2)*x^3,x, algorithm="fricas")

[Out]

-1/8*(x^16 - 2*x^8 - 2*(x^8 - sqrt(x^8 - 2)*x^4 - 1)*log(-x^4 + sqrt(x^8 - 2)) -
(x^12 - x^4)*sqrt(x^8 - 2))/(x^8 - sqrt(x^8 - 2)*x^4 - 1)

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Sympy [A]  time = 5.52243, size = 90, normalized size = 2.57 \[ \begin{cases} \frac{x^{12}}{8 \sqrt{x^{8} - 2}} - \frac{x^{4}}{4 \sqrt{x^{8} - 2}} - \frac{\operatorname{acosh}{\left (\frac{\sqrt{2} x^{4}}{2} \right )}}{4} & \text{for}\: \frac{\left |{x^{8}}\right |}{2} > 1 \\- \frac{i x^{12}}{8 \sqrt{- x^{8} + 2}} + \frac{i x^{4}}{4 \sqrt{- x^{8} + 2}} + \frac{i \operatorname{asin}{\left (\frac{\sqrt{2} x^{4}}{2} \right )}}{4} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(x**8-2)**(1/2),x)

[Out]

Piecewise((x**12/(8*sqrt(x**8 - 2)) - x**4/(4*sqrt(x**8 - 2)) - acosh(sqrt(2)*x*
*4/2)/4, Abs(x**8)/2 > 1), (-I*x**12/(8*sqrt(-x**8 + 2)) + I*x**4/(4*sqrt(-x**8
+ 2)) + I*asin(sqrt(2)*x**4/2)/4, True))

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GIAC/XCAS [A]  time = 0.225305, size = 39, normalized size = 1.11 \[ \frac{1}{8} \, \sqrt{x^{8} - 2} x^{4} + \frac{1}{4} \,{\rm ln}\left (x^{4} - \sqrt{x^{8} - 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^8 - 2)*x^3,x, algorithm="giac")

[Out]

1/8*sqrt(x^8 - 2)*x^4 + 1/4*ln(x^4 - sqrt(x^8 - 2))