\(\int \frac {x}{(1-x^2)^5} \, dx\) [423]

Optimal result

Integrand size = 11, antiderivative size = 13 \[ \int \frac {x}{\left (1-x^2\right )^5} \, dx=\frac {1}{8 \left (1-x^2\right )^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {267} \[ \int \frac {x}{\left (1-x^2\right )^5} \, dx=\frac {1}{8 \left (1-x^2\right )^4} \]

[In]

Int[x/(1 - x^2)^5,x]

[Out]

1/(8*(1 - x^2)^4)

Rule 267

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*} \text {integral}& = \frac {1}{8 \left (1-x^2\right )^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {x}{\left (1-x^2\right )^5} \, dx=\frac {1}{8 \left (-1+x^2\right )^4} \]

[In]

Integrate[x/(1 - x^2)^5,x]

[Out]

1/(8*(-1 + x^2)^4)

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77

method	result	size
gosper	\(\frac {1}{8 \left (x^{2}-1\right )^{4}}\)	\(10\)
default	\(\frac {1}{8 \left (x^{2}-1\right )^{4}}\)	\(10\)
norman	\(\frac {1}{8 \left (x^{2}-1\right )^{4}}\)	\(10\)
risch	\(\frac {1}{8 \left (x^{2}-1\right )^{4}}\)	\(10\)
parallelrisch	\(\frac {1}{8 \left (x^{2}-1\right )^{4}}\)	\(10\)
derivativedivides	\(\frac {1}{8 \left (-x^{2}+1\right )^{4}}\)	\(12\)
meijerg	\(\frac {x^{2} \left (-x^{6}+4 x^{4}-6 x^{2}+4\right )}{8 \left (-x^{2}+1\right )^{4}}\)	\(32\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 24 vs. \(2 (9) = 18\).

Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.85 \[ \int \frac {x}{\left (1-x^2\right )^5} \, dx=\frac {1}{8 \, {\left (x^{8} - 4 \, x^{6} + 6 \, x^{4} - 4 \, x^{2} + 1\right )}} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 22 vs. \(2 (8) = 16\).

Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.69 \[ \int \frac {x}{\left (1-x^2\right )^5} \, dx=\frac {1}{8 x^{8} - 32 x^{6} + 48 x^{4} - 32 x^{2} + 8} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \frac {x}{\left (1-x^2\right )^5} \, dx=\frac {1}{8 \, {\left (x^{2} - 1\right )}^{4}} \]

[In]

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

[Out]

1/8/(x^2 - 1)^4

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \frac {x}{\left (1-x^2\right )^5} \, dx=\frac {1}{8 \, {\left (x^{2} - 1\right )}^{4}} \]

[In]

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

[Out]

1/8/(x^2 - 1)^4

Mupad [B] (verification not implemented)

Time = 9.35 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \frac {x}{\left (1-x^2\right )^5} \, dx=\frac {1}{8\,{\left (x^2-1\right )}^4} \]

[In]

int(-x/(x^2 - 1)^5,x)

[Out]

1/(8*(x^2 - 1)^4)