3.902 \(\int \frac{1}{x^3 (1-x^4)^{3/2}} \, dx\)

Optimal. Leaf size=34 \[ \frac{x^2}{\sqrt{1-x^4}}-\frac{1}{2 x^2 \sqrt{1-x^4}} \]

[Out]

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

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Rubi [A]  time = 0.0071093, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {271, 264} \[ \frac{x^2}{\sqrt{1-x^4}}-\frac{1}{2 x^2 \sqrt{1-x^4}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(1 - x^4)^(3/2)),x]

[Out]

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

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{x^3 \left (1-x^4\right )^{3/2}} \, dx &=-\frac{1}{2 x^2 \sqrt{1-x^4}}+2 \int \frac{x}{\left (1-x^4\right )^{3/2}} \, dx\\ &=-\frac{1}{2 x^2 \sqrt{1-x^4}}+\frac{x^2}{\sqrt{1-x^4}}\\ \end{align*}

Mathematica [A]  time = 0.0046232, size = 25, normalized size = 0.74 \[ -\frac{1-2 x^4}{2 x^2 \sqrt{1-x^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(1 - x^4)^(3/2)),x]

[Out]

-(1 - 2*x^4)/(2*x^2*Sqrt[1 - x^4])

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Maple [A]  time = 0.004, size = 33, normalized size = 1. \begin{align*} -{\frac{ \left ( -1+x \right ) \left ( 1+x \right ) \left ({x}^{2}+1 \right ) \left ( 2\,{x}^{4}-1 \right ) }{2\,{x}^{2}} \left ( -{x}^{4}+1 \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(-x^4+1)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.05482, size = 39, normalized size = 1.15 \begin{align*} \frac{x^{2}}{2 \, \sqrt{-x^{4} + 1}} - \frac{\sqrt{-x^{4} + 1}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(-x^4+1)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.53477, size = 61, normalized size = 1.79 \begin{align*} -\frac{{\left (2 \, x^{4} - 1\right )} \sqrt{-x^{4} + 1}}{2 \,{\left (x^{6} - x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(-x^4+1)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.974441, size = 90, normalized size = 2.65 \begin{align*} \begin{cases} - \frac{2 i x^{4} \sqrt{x^{4} - 1}}{2 x^{6} - 2 x^{2}} + \frac{i \sqrt{x^{4} - 1}}{2 x^{6} - 2 x^{2}} & \text{for}\: \left |{x^{4}}\right | > 1 \\- \frac{2 x^{4} \sqrt{1 - x^{4}}}{2 x^{6} - 2 x^{2}} + \frac{\sqrt{1 - x^{4}}}{2 x^{6} - 2 x^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(-x**4+1)**(3/2),x)

[Out]

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

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Giac [A]  time = 1.20161, size = 42, normalized size = 1.24 \begin{align*} -\frac{\sqrt{-x^{4} + 1} x^{2}}{2 \,{\left (x^{4} - 1\right )}} - \frac{1}{2} \, \sqrt{\frac{1}{x^{4}} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(-x^4+1)^(3/2),x, algorithm="giac")

[Out]

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