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

Optimal. Leaf size=47 \[ -\frac{3 \sqrt{x^4+1}}{4 x^4}+\frac{1}{2 x^4 \sqrt{x^4+1}}+\frac{3}{4} \tanh ^{-1}\left (\sqrt{x^4+1}\right ) \]

[Out]

1/(2*x^4*Sqrt[1 + x^4]) - (3*Sqrt[1 + x^4])/(4*x^4) + (3*ArcTanh[Sqrt[1 + x^4]])
/4

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Rubi [A]  time = 0.0467639, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{3 \sqrt{x^4+1}}{4 x^4}+\frac{1}{2 x^4 \sqrt{x^4+1}}+\frac{3}{4} \tanh ^{-1}\left (\sqrt{x^4+1}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*(1 + x^4)^(3/2)),x]

[Out]

1/(2*x^4*Sqrt[1 + x^4]) - (3*Sqrt[1 + x^4])/(4*x^4) + (3*ArcTanh[Sqrt[1 + x^4]])
/4

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Rubi in Sympy [A]  time = 4.8724, size = 42, normalized size = 0.89 \[ \frac{3 \operatorname{atanh}{\left (\sqrt{x^{4} + 1} \right )}}{4} - \frac{3 \sqrt{x^{4} + 1}}{4 x^{4}} + \frac{1}{2 x^{4} \sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*atanh(sqrt(x**4 + 1))/4 - 3*sqrt(x**4 + 1)/(4*x**4) + 1/(2*x**4*sqrt(x**4 + 1)
)

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Mathematica [A]  time = 0.061527, size = 38, normalized size = 0.81 \[ \frac{3}{4} \tanh ^{-1}\left (\sqrt{x^4+1}\right )-\frac{3 x^4+1}{4 x^4 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

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

[Out]

-(1 + 3*x^4)/(4*x^4*Sqrt[1 + x^4]) + (3*ArcTanh[Sqrt[1 + x^4]])/4

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Maple [A]  time = 0.016, size = 33, normalized size = 0.7 \[ -{\frac{1}{4\,{x}^{4}}{\frac{1}{\sqrt{{x}^{4}+1}}}}-{\frac{3}{4}{\frac{1}{\sqrt{{x}^{4}+1}}}}+{\frac{3}{4}{\it Artanh} \left ({\frac{1}{\sqrt{{x}^{4}+1}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(x^4+1)^(3/2),x)

[Out]

-1/4/x^4/(x^4+1)^(1/2)-3/4/(x^4+1)^(1/2)+3/4*arctanh(1/(x^4+1)^(1/2))

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Maxima [A]  time = 1.44056, size = 72, normalized size = 1.53 \[ -\frac{3 \, x^{4} + 1}{4 \,{\left ({\left (x^{4} + 1\right )}^{\frac{3}{2}} - \sqrt{x^{4} + 1}\right )}} + \frac{3}{8} \, \log \left (\sqrt{x^{4} + 1} + 1\right ) - \frac{3}{8} \, \log \left (\sqrt{x^{4} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(3*x^4 + 1)/((x^4 + 1)^(3/2) - sqrt(x^4 + 1)) + 3/8*log(sqrt(x^4 + 1) + 1)
- 3/8*log(sqrt(x^4 + 1) - 1)

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Fricas [A]  time = 0.261162, size = 85, normalized size = 1.81 \[ \frac{3 \, \sqrt{x^{4} + 1} x^{4} \log \left (\sqrt{x^{4} + 1} + 1\right ) - 3 \, \sqrt{x^{4} + 1} x^{4} \log \left (\sqrt{x^{4} + 1} - 1\right ) - 6 \, x^{4} - 2}{8 \, \sqrt{x^{4} + 1} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(3*sqrt(x^4 + 1)*x^4*log(sqrt(x^4 + 1) + 1) - 3*sqrt(x^4 + 1)*x^4*log(sqrt(x
^4 + 1) - 1) - 6*x^4 - 2)/(sqrt(x^4 + 1)*x^4)

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Sympy [A]  time = 8.98049, size = 42, normalized size = 0.89 \[ \frac{3 \operatorname{asinh}{\left (\frac{1}{x^{2}} \right )}}{4} - \frac{3}{4 x^{2} \sqrt{1 + \frac{1}{x^{4}}}} - \frac{1}{4 x^{6} \sqrt{1 + \frac{1}{x^{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*asinh(x**(-2))/4 - 3/(4*x**2*sqrt(1 + x**(-4))) - 1/(4*x**6*sqrt(1 + x**(-4)))

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GIAC/XCAS [A]  time = 0.23299, size = 72, normalized size = 1.53 \[ -\frac{3 \, x^{4} + 1}{4 \,{\left ({\left (x^{4} + 1\right )}^{\frac{3}{2}} - \sqrt{x^{4} + 1}\right )}} + \frac{3}{8} \,{\rm ln}\left (\sqrt{x^{4} + 1} + 1\right ) - \frac{3}{8} \,{\rm ln}\left (\sqrt{x^{4} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(3*x^4 + 1)/((x^4 + 1)^(3/2) - sqrt(x^4 + 1)) + 3/8*ln(sqrt(x^4 + 1) + 1) -
3/8*ln(sqrt(x^4 + 1) - 1)