\(\int \frac {(1-x)^n (1+x)^{-n}}{x^2} \, dx\) [983]

Optimal result

Integrand size = 18, antiderivative size = 44 \[ \int \frac {(1-x)^n (1+x)^{-n}}{x^2} \, dx=-\frac {2 (1-x)^{1+n} (1+x)^{-1-n} \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,\frac {1-x}{1+x}\right )}{1+n} \]

[Out]

-2*(1-x)^(1+n)*(1+x)^(-1-n)*hypergeom([2, 1+n],[2+n],(1-x)/(1+x))/(1+n)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {133} \[ \int \frac {(1-x)^n (1+x)^{-n}}{x^2} \, dx=-\frac {2 (1-x)^{n+1} (x+1)^{-n-1} \operatorname {Hypergeometric2F1}\left (2,n+1,n+2,\frac {1-x}{x+1}\right )}{n+1} \]

[In]

Int[(1 - x)^n/(x^2*(1 + x)^n),x]

[Out]

(-2*(1 - x)^(1 + n)*(1 + x)^(-1 - n)*Hypergeometric2F1[2, 1 + n, 2 + n, (1 - x)/(1 + x)])/(1 + n)

Rule 133

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[(b*c - a
*d)^n*((a + b*x)^(m + 1)/((m + 1)*(b*e - a*f)^(n + 1)*(e + f*x)^(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2,
(-(d*e - c*f))*((a + b*x)/((b*c - a*d)*(e + f*x)))], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p
+ 2, 0] && ILtQ[n, 0] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[p, 1]) &&  !ILtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 (1-x)^{1+n} (1+x)^{-1-n} \, _2F_1\left (2,1+n;2+n;\frac {1-x}{1+x}\right )}{1+n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00 \[ \int \frac {(1-x)^n (1+x)^{-n}}{x^2} \, dx=-\frac {2 (1-x)^{1+n} (1+x)^{-1-n} \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,\frac {1-x}{1+x}\right )}{1+n} \]

[In]

Integrate[(1 - x)^n/(x^2*(1 + x)^n),x]

[Out]

(-2*(1 - x)^(1 + n)*(1 + x)^(-1 - n)*Hypergeometric2F1[2, 1 + n, 2 + n, (1 - x)/(1 + x)])/(1 + n)

Maple [F]

\[\int \frac {\left (1-x \right )^{n} \left (1+x \right )^{-n}}{x^{2}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {(1-x)^n (1+x)^{-n}}{x^2} \, dx=\int { \frac {{\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n} x^{2}} \,d x } \]

[In]

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

[Out]

integral((-x + 1)^n/((x + 1)^n*x^2), x)

Sympy [F]

\[ \int \frac {(1-x)^n (1+x)^{-n}}{x^2} \, dx=\int \frac {\left (1 - x\right )^{n} \left (x + 1\right )^{- n}}{x^{2}}\, dx \]

[In]

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

[Out]

Integral((1 - x)**n/(x**2*(x + 1)**n), x)

Maxima [F]

\[ \int \frac {(1-x)^n (1+x)^{-n}}{x^2} \, dx=\int { \frac {{\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n} x^{2}} \,d x } \]

[In]

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

[Out]

integrate((-x + 1)^n/((x + 1)^n*x^2), x)

Giac [F]

\[ \int \frac {(1-x)^n (1+x)^{-n}}{x^2} \, dx=\int { \frac {{\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n} x^{2}} \,d x } \]

[In]

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

[Out]

integrate((-x + 1)^n/((x + 1)^n*x^2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(1-x)^n (1+x)^{-n}}{x^2} \, dx=\int \frac {{\left (1-x\right )}^n}{x^2\,{\left (x+1\right )}^n} \,d x \]

[In]

int((1 - x)^n/(x^2*(x + 1)^n),x)

[Out]

int((1 - x)^n/(x^2*(x + 1)^n), x)