3.1516 \(\int \frac{x^3}{\sqrt{1+x^8}} \, dx\)

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

[Out]

ArcSinh[x^4]/4

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Rubi [A]  time = 0.015852, antiderivative size = 8, 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 \[ \frac{1}{4} \sinh ^{-1}\left (x^4\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^3/Sqrt[1 + x^8],x]

[Out]

ArcSinh[x^4]/4

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Rubi in Sympy [A]  time = 2.56121, size = 5, normalized size = 0.62 \[ \frac{\operatorname{asinh}{\left (x^{4} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

asinh(x**4)/4

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Mathematica [A]  time = 0.00765015, size = 8, normalized size = 1. \[ \frac{1}{4} \sinh ^{-1}\left (x^4\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/Sqrt[1 + x^8],x]

[Out]

ArcSinh[x^4]/4

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Maple [A]  time = 0.023, size = 7, normalized size = 0.9 \[{\frac{{\it Arcsinh} \left ({x}^{4} \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(x^8+1)^(1/2),x)

[Out]

1/4*arcsinh(x^4)

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Maxima [A]  time = 1.44127, size = 45, normalized size = 5.62 \[ \frac{1}{8} \, \log \left (\frac{\sqrt{x^{8} + 1}}{x^{4}} + 1\right ) - \frac{1}{8} \, \log \left (\frac{\sqrt{x^{8} + 1}}{x^{4}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/sqrt(x^8 + 1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.222058, size = 22, normalized size = 2.75 \[ -\frac{1}{4} \, \log \left (-x^{4} + \sqrt{x^{8} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/sqrt(x^8 + 1),x, algorithm="fricas")

[Out]

-1/4*log(-x^4 + sqrt(x^8 + 1))

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Sympy [A]  time = 3.32458, size = 5, normalized size = 0.62 \[ \frac{\operatorname{asinh}{\left (x^{4} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

asinh(x**4)/4

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GIAC/XCAS [A]  time = 0.234437, size = 22, normalized size = 2.75 \[ -\frac{1}{4} \,{\rm ln}\left (-x^{4} + \sqrt{x^{8} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/sqrt(x^8 + 1),x, algorithm="giac")

[Out]

-1/4*ln(-x^4 + sqrt(x^8 + 1))