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

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

[Out]

x^2/(2*Sqrt[1 - x^4])

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Rubi [A]  time = 0.0026559, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {264} \[ \frac{x^2}{2 \sqrt{1-x^4}} \]

Antiderivative was successfully verified.

[In]

Int[x/(1 - x^4)^(3/2),x]

[Out]

x^2/(2*Sqrt[1 - x^4])

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

Mathematica [A]  time = 0.0024016, size = 18, normalized size = 1. \[ \frac{x^2}{2 \sqrt{1-x^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(1 - x^4)^(3/2),x]

[Out]

x^2/(2*Sqrt[1 - x^4])

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Maple [A]  time = 0.003, size = 26, normalized size = 1.4 \begin{align*} -{\frac{ \left ( -1+x \right ) \left ( 1+x \right ) \left ({x}^{2}+1 \right ){x}^{2}}{2} \left ( -{x}^{4}+1 \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.991446, size = 19, normalized size = 1.06 \begin{align*} \frac{x^{2}}{2 \, \sqrt{-x^{4} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2/sqrt(-x^4 + 1)

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Fricas [A]  time = 1.54599, size = 47, normalized size = 2.61 \begin{align*} -\frac{\sqrt{-x^{4} + 1} x^{2}}{2 \,{\left (x^{4} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(-x^4 + 1)*x^2/(x^4 - 1)

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Sympy [A]  time = 0.515764, size = 32, normalized size = 1.78 \begin{align*} \begin{cases} - \frac{i x^{2}}{2 \sqrt{x^{4} - 1}} & \text{for}\: \left |{x^{4}}\right | > 1 \\\frac{x^{2}}{2 \sqrt{1 - x^{4}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.17425, size = 28, normalized size = 1.56 \begin{align*} -\frac{\sqrt{-x^{4} + 1} x^{2}}{2 \,{\left (x^{4} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(-x^4 + 1)*x^2/(x^4 - 1)