3.941 \(\int \frac{x^5}{\left (1+x^4\right )^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0300038, 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 \[ \frac{1}{2} \sinh ^{-1}\left (x^2\right )-\frac{x^2}{2 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(1 + x^4)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.26764, size = 19, normalized size = 0.76 \[ - \frac{x^{2}}{2 \sqrt{x^{4} + 1}} + \frac{\operatorname{asinh}{\left (x^{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(x**4+1)**(3/2),x)

[Out]

-x**2/(2*sqrt(x**4 + 1)) + asinh(x**2)/2

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Mathematica [A]  time = 0.0218577, size = 25, normalized size = 1. \[ \frac{1}{2} \sinh ^{-1}\left (x^2\right )-\frac{x^2}{2 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(1 + x^4)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.013, size = 20, normalized size = 0.8 \[{\frac{{\it Arcsinh} \left ({x}^{2} \right ) }{2}}-{\frac{{x}^{2}}{2}{\frac{1}{\sqrt{{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(x^4+1)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.44653, size = 61, normalized size = 2.44 \[ -\frac{x^{2}}{2 \, \sqrt{x^{4} + 1}} + \frac{1}{4} \, \log \left (\frac{\sqrt{x^{4} + 1}}{x^{2}} + 1\right ) - \frac{1}{4} \, \log \left (\frac{\sqrt{x^{4} + 1}}{x^{2}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(x^4 + 1)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.264491, size = 74, normalized size = 2.96 \[ -\frac{{\left (x^{4} - \sqrt{x^{4} + 1} x^{2} + 1\right )} \log \left (-x^{2} + \sqrt{x^{4} + 1}\right ) + 1}{2 \,{\left (x^{4} - \sqrt{x^{4} + 1} x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(x^4 + 1)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.81203, size = 19, normalized size = 0.76 \[ - \frac{x^{2}}{2 \sqrt{x^{4} + 1}} + \frac{\operatorname{asinh}{\left (x^{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(x**4+1)**(3/2),x)

[Out]

-x**2/(2*sqrt(x**4 + 1)) + asinh(x**2)/2

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GIAC/XCAS [A]  time = 0.229368, size = 39, normalized size = 1.56 \[ -\frac{x^{2}}{2 \, \sqrt{x^{4} + 1}} - \frac{1}{2} \,{\rm ln}\left (-x^{2} + \sqrt{x^{4} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(x^4 + 1)^(3/2),x, algorithm="giac")

[Out]

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