3.423 \(\int \frac{x}{(1-x^2)^5} \, dx\)

Optimal. Leaf size=13 \[ \frac{1}{8 \left (1-x^2\right )^4} \]

[Out]

1/(8*(1 - x^2)^4)

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Rubi [A]  time = 0.0021836, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {261} \[ \frac{1}{8 \left (1-x^2\right )^4} \]

Antiderivative was successfully verified.

[In]

Int[x/(1 - x^2)^5,x]

[Out]

1/(8*(1 - x^2)^4)

Rule 261

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

Rubi steps

\begin{align*} \int \frac{x}{\left (1-x^2\right )^5} \, dx &=\frac{1}{8 \left (1-x^2\right )^4}\\ \end{align*}

Mathematica [A]  time = 0.0024871, size = 11, normalized size = 0.85 \[ \frac{1}{8 \left (x^2-1\right )^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(1 - x^2)^5,x]

[Out]

1/(8*(-1 + x^2)^4)

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Maple [A]  time = 0.002, size = 10, normalized size = 0.8 \begin{align*}{\frac{1}{8\, \left ({x}^{2}-1 \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(-x^2+1)^5,x)

[Out]

1/8/(x^2-1)^4

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Maxima [A]  time = 0.95232, size = 12, normalized size = 0.92 \begin{align*} \frac{1}{8 \,{\left (x^{2} - 1\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x^2+1)^5,x, algorithm="maxima")

[Out]

1/8/(x^2 - 1)^4

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Fricas [B]  time = 0.923981, size = 53, normalized size = 4.08 \begin{align*} \frac{1}{8 \,{\left (x^{8} - 4 \, x^{6} + 6 \, x^{4} - 4 \, x^{2} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x^2+1)^5,x, algorithm="fricas")

[Out]

1/8/(x^8 - 4*x^6 + 6*x^4 - 4*x^2 + 1)

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Sympy [B]  time = 0.120779, size = 22, normalized size = 1.69 \begin{align*} \frac{1}{8 x^{8} - 32 x^{6} + 48 x^{4} - 32 x^{2} + 8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x**2+1)**5,x)

[Out]

1/(8*x**8 - 32*x**6 + 48*x**4 - 32*x**2 + 8)

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Giac [A]  time = 1.12121, size = 12, normalized size = 0.92 \begin{align*} \frac{1}{8 \,{\left (x^{2} - 1\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x^2+1)^5,x, algorithm="giac")

[Out]

1/8/(x^2 - 1)^4