3.443 \(\int \frac{1}{x^4 \sqrt{1+x^3}} \, dx\)

Optimal. Leaf size=31 \[ \frac{1}{3} \tanh ^{-1}\left (\sqrt{x^3+1}\right )-\frac{\sqrt{x^3+1}}{3 x^3} \]

[Out]

-Sqrt[1 + x^3]/(3*x^3) + ArcTanh[Sqrt[1 + x^3]]/3

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Rubi [A]  time = 0.0376508, antiderivative size = 31, 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 \[ \frac{1}{3} \tanh ^{-1}\left (\sqrt{x^3+1}\right )-\frac{\sqrt{x^3+1}}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*Sqrt[1 + x^3]),x]

[Out]

-Sqrt[1 + x^3]/(3*x^3) + ArcTanh[Sqrt[1 + x^3]]/3

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Rubi in Sympy [A]  time = 4.13414, size = 24, normalized size = 0.77 \[ \frac{\operatorname{atanh}{\left (\sqrt{x^{3} + 1} \right )}}{3} - \frac{\sqrt{x^{3} + 1}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atanh(sqrt(x**3 + 1))/3 - sqrt(x**3 + 1)/(3*x**3)

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Mathematica [A]  time = 0.0282817, size = 31, normalized size = 1. \[ \frac{1}{3} \tanh ^{-1}\left (\sqrt{x^3+1}\right )-\frac{\sqrt{x^3+1}}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*Sqrt[1 + x^3]),x]

[Out]

-Sqrt[1 + x^3]/(3*x^3) + ArcTanh[Sqrt[1 + x^3]]/3

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Maple [A]  time = 0.029, size = 24, normalized size = 0.8 \[{\frac{1}{3}{\it Artanh} \left ( \sqrt{{x}^{3}+1} \right ) }-{\frac{1}{3\,{x}^{3}}\sqrt{{x}^{3}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(x^3+1)^(1/2),x)

[Out]

1/3*arctanh((x^3+1)^(1/2))-1/3*(x^3+1)^(1/2)/x^3

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Maxima [A]  time = 1.44629, size = 50, normalized size = 1.61 \[ -\frac{\sqrt{x^{3} + 1}}{3 \, x^{3}} + \frac{1}{6} \, \log \left (\sqrt{x^{3} + 1} + 1\right ) - \frac{1}{6} \, \log \left (\sqrt{x^{3} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*sqrt(x^3 + 1)/x^3 + 1/6*log(sqrt(x^3 + 1) + 1) - 1/6*log(sqrt(x^3 + 1) - 1)

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Fricas [A]  time = 0.228784, size = 59, normalized size = 1.9 \[ \frac{x^{3} \log \left (\sqrt{x^{3} + 1} + 1\right ) - x^{3} \log \left (\sqrt{x^{3} + 1} - 1\right ) - 2 \, \sqrt{x^{3} + 1}}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(x^3*log(sqrt(x^3 + 1) + 1) - x^3*log(sqrt(x^3 + 1) - 1) - 2*sqrt(x^3 + 1))/
x^3

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Sympy [A]  time = 6.32317, size = 26, normalized size = 0.84 \[ \frac{\operatorname{asinh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} - \frac{\sqrt{1 + \frac{1}{x^{3}}}}{3 x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

asinh(x**(-3/2))/3 - sqrt(1 + x**(-3))/(3*x**(3/2))

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GIAC/XCAS [A]  time = 0.217323, size = 51, normalized size = 1.65 \[ -\frac{\sqrt{x^{3} + 1}}{3 \, x^{3}} + \frac{1}{6} \,{\rm ln}\left (\sqrt{x^{3} + 1} + 1\right ) - \frac{1}{6} \,{\rm ln}\left ({\left | \sqrt{x^{3} + 1} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*sqrt(x^3 + 1)/x^3 + 1/6*ln(sqrt(x^3 + 1) + 1) - 1/6*ln(abs(sqrt(x^3 + 1) -
1))