3.881 \(\int \frac {1}{x^3 \sqrt {1-x^4}} \, dx\)

Optimal. Leaf size=18 \[ -\frac {\sqrt {1-x^4}}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {264} \[ -\frac {\sqrt {1-x^4}}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*Sqrt[1 - x^4]),x]

[Out]

-Sqrt[1 - x^4]/(2*x^2)

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 \sqrt {1-x^4}} \, dx &=-\frac {\sqrt {1-x^4}}{2 x^2}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 18, normalized size = 1.00 \[ -\frac {\sqrt {1-x^4}}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*Sqrt[1 - x^4]),x]

[Out]

-1/2*Sqrt[1 - x^4]/x^2

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fricas [A]  time = 0.71, size = 14, normalized size = 0.78 \[ -\frac {\sqrt {-x^{4} + 1}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(-x^4 + 1)/x^2

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giac [B]  time = 0.17, size = 35, normalized size = 1.94 \[ \frac {x^{2}}{4 \, {\left (\sqrt {-x^{4} + 1} - 1\right )}} - \frac {\sqrt {-x^{4} + 1} - 1}{4 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 26, normalized size = 1.44 \[ \frac {\left (x -1\right ) \left (x +1\right ) \left (x^{2}+1\right )}{2 \sqrt {-x^{4}+1}\, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.33, size = 14, normalized size = 0.78 \[ -\frac {\sqrt {-x^{4} + 1}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(-x^4 + 1)/x^2

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mupad [B]  time = 1.12, size = 14, normalized size = 0.78 \[ -\frac {\sqrt {1-x^4}}{2\,x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(1 - x^4)^(1/2)),x)

[Out]

-(1 - x^4)^(1/2)/(2*x^2)

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sympy [A]  time = 1.18, size = 34, normalized size = 1.89 \[ \begin {cases} - \frac {i \sqrt {x^{4} - 1}}{2 x^{2}} & \text {for}\: \left |{x^{4}}\right | > 1 \\- \frac {\sqrt {1 - x^{4}}}{2 x^{2}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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