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

Optimal. Leaf size=53 \[ -\frac{3 \sqrt{1-x^4}}{4 x^4}+\frac{1}{2 x^4 \sqrt{1-x^4}}-\frac{3}{4} \tanh ^{-1}\left (\sqrt{1-x^4}\right ) \]

[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 = 0.0607043, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{3 \sqrt{1-x^4}}{4 x^4}+\frac{1}{2 x^4 \sqrt{1-x^4}}-\frac{3}{4} \tanh ^{-1}\left (\sqrt{1-x^4}\right ) \]

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 = 5.87212, size = 42, normalized size = 0.79 \[ - \frac{3 \operatorname{atanh}{\left (\sqrt{- x^{4} + 1} \right )}}{4} - \frac{3 \sqrt{- x^{4} + 1}}{4 x^{4}} + \frac{1}{2 x^{4} \sqrt{- x^{4} + 1}} \]

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 = 0.0909456, size = 41, normalized size = 0.77 \[ \frac{1}{4} \left (\frac{3 x^4-1}{x^4 \sqrt{1-x^4}}-3 \tanh ^{-1}\left (\sqrt{1-x^4}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.024, size = 82, normalized size = 1.6 \[ -{\frac{1}{4\,{x}^{4}}\sqrt{-{x}^{4}+1}}-{\frac{3}{4}{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{4}+1}}} \right ) }+{\frac{1}{4\,{x}^{2}+4}\sqrt{- \left ({x}^{2}+1 \right ) ^{2}+2+2\,{x}^{2}}}-{\frac{1}{4\,{x}^{2}-4}\sqrt{- \left ({x}^{2}-1 \right ) ^{2}-2\,{x}^{2}+2}} \]

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

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Maxima [A]  time = 1.4393, size = 82, normalized size = 1.55 \[ -\frac{3 \, x^{4} - 1}{4 \,{\left ({\left (-x^{4} + 1\right )}^{\frac{3}{2}} - \sqrt{-x^{4} + 1}\right )}} - \frac{3}{8} \, \log \left (\sqrt{-x^{4} + 1} + 1\right ) + \frac{3}{8} \, \log \left (\sqrt{-x^{4} + 1} - 1\right ) \]

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 = 0.283338, size = 99, normalized size = 1.87 \[ -\frac{3 \, \sqrt{-x^{4} + 1} x^{4} \log \left (\sqrt{-x^{4} + 1} + 1\right ) - 3 \, \sqrt{-x^{4} + 1} x^{4} \log \left (\sqrt{-x^{4} + 1} - 1\right ) - 6 \, x^{4} + 2}{8 \, \sqrt{-x^{4} + 1} x^{4}} \]

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(sq
rt(-x^4 + 1) - 1) - 6*x^4 + 2)/(sqrt(-x^4 + 1)*x^4)

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Sympy [A]  time = 8.95723, size = 95, normalized size = 1.79 \[ \begin{cases} - \frac{3 \operatorname{acosh}{\left (\frac{1}{x^{2}} \right )}}{4} + \frac{3}{4 x^{2} \sqrt{-1 + \frac{1}{x^{4}}}} - \frac{1}{4 x^{6} \sqrt{-1 + \frac{1}{x^{4}}}} & \text{for}\: \left |{\frac{1}{x^{4}}}\right | > 1 \\\frac{3 i \operatorname{asin}{\left (\frac{1}{x^{2}} \right )}}{4} - \frac{3 i}{4 x^{2} \sqrt{1 - \frac{1}{x^{4}}}} + \frac{i}{4 x^{6} \sqrt{1 - \frac{1}{x^{4}}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-3*acosh(x**(-2))/4 + 3/(4*x**2*sqrt(-1 + x**(-4))) - 1/(4*x**6*sqrt(
-1 + x**(-4))), Abs(x**(-4)) > 1), (3*I*asin(x**(-2))/4 - 3*I/(4*x**2*sqrt(1 - 1
/x**4)) + I/(4*x**6*sqrt(1 - 1/x**4)), True))

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GIAC/XCAS [A]  time = 0.216231, size = 85, normalized size = 1.6 \[ -\frac{3 \, x^{4} - 1}{4 \,{\left ({\left (-x^{4} + 1\right )}^{\frac{3}{2}} - \sqrt{-x^{4} + 1}\right )}} - \frac{3}{8} \,{\rm ln}\left (\sqrt{-x^{4} + 1} + 1\right ) + \frac{3}{8} \,{\rm ln}\left (-\sqrt{-x^{4} + 1} + 1\right ) \]

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)