3.920 \(\int \frac{x^5}{\sqrt{1+x^4}} \, dx\)

Optimal. Leaf size=25 \[ \frac{1}{4} x^2 \sqrt{x^4+1}-\frac{1}{4} \sinh ^{-1}\left (x^2\right ) \]

[Out]

(x^2*Sqrt[1 + x^4])/4 - ArcSinh[x^2]/4

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Rubi [A]  time = 0.0317372, antiderivative size = 25, 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 \[ \frac{1}{4} x^2 \sqrt{x^4+1}-\frac{1}{4} \sinh ^{-1}\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^5/Sqrt[1 + x^4],x]

[Out]

(x^2*Sqrt[1 + x^4])/4 - ArcSinh[x^2]/4

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Rubi in Sympy [A]  time = 4.45925, size = 19, normalized size = 0.76 \[ \frac{x^{2} \sqrt{x^{4} + 1}}{4} - \frac{\operatorname{asinh}{\left (x^{2} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(x**4+1)**(1/2),x)

[Out]

x**2*sqrt(x**4 + 1)/4 - asinh(x**2)/4

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Mathematica [A]  time = 0.0141762, size = 25, normalized size = 1. \[ \frac{1}{4} x^2 \sqrt{x^4+1}-\frac{1}{4} \sinh ^{-1}\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/Sqrt[1 + x^4],x]

[Out]

(x^2*Sqrt[1 + x^4])/4 - ArcSinh[x^2]/4

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Maple [A]  time = 0.014, size = 20, normalized size = 0.8 \[ -{\frac{{\it Arcsinh} \left ({x}^{2} \right ) }{4}}+{\frac{{x}^{2}}{4}\sqrt{{x}^{4}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(x^4+1)^(1/2),x)

[Out]

-1/4*arcsinh(x^2)+1/4*x^2*(x^4+1)^(1/2)

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Maxima [A]  time = 1.44208, size = 78, normalized size = 3.12 \[ \frac{\sqrt{x^{4} + 1}}{4 \, x^{2}{\left (\frac{x^{4} + 1}{x^{4}} - 1\right )}} - \frac{1}{8} \, \log \left (\frac{\sqrt{x^{4} + 1}}{x^{2}} + 1\right ) + \frac{1}{8} \, \log \left (\frac{\sqrt{x^{4} + 1}}{x^{2}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/sqrt(x^4 + 1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.258263, size = 117, normalized size = 4.68 \[ -\frac{2 \, x^{8} + 2 \, x^{4} -{\left (2 \, x^{4} - 2 \, \sqrt{x^{4} + 1} x^{2} + 1\right )} \log \left (-x^{2} + \sqrt{x^{4} + 1}\right ) -{\left (2 \, x^{6} + x^{2}\right )} \sqrt{x^{4} + 1}}{4 \,{\left (2 \, x^{4} - 2 \, \sqrt{x^{4} + 1} x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/sqrt(x^4 + 1),x, algorithm="fricas")

[Out]

-1/4*(2*x^8 + 2*x^4 - (2*x^4 - 2*sqrt(x^4 + 1)*x^2 + 1)*log(-x^2 + sqrt(x^4 + 1)
) - (2*x^6 + x^2)*sqrt(x^4 + 1))/(2*x^4 - 2*sqrt(x^4 + 1)*x^2 + 1)

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Sympy [A]  time = 5.86892, size = 19, normalized size = 0.76 \[ \frac{x^{2} \sqrt{x^{4} + 1}}{4} - \frac{\operatorname{asinh}{\left (x^{2} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(x**4+1)**(1/2),x)

[Out]

x**2*sqrt(x**4 + 1)/4 - asinh(x**2)/4

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GIAC/XCAS [A]  time = 0.222529, size = 39, normalized size = 1.56 \[ \frac{1}{4} \, \sqrt{x^{4} + 1} x^{2} + \frac{1}{4} \,{\rm ln}\left (-x^{2} + \sqrt{x^{4} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/sqrt(x^4 + 1),x, algorithm="giac")

[Out]

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