\(\int \frac {(1-x)^n (1+x)^{-n}}{x^3} \, dx\) [984]

Optimal result

Integrand size = 18, antiderivative size = 71 \[ \int \frac {(1-x)^n (1+x)^{-n}}{x^3} \, dx=-\frac {(1-x)^{1+n} (1+x)^{1-n}}{2 x^2}+\frac {2 n (1-x)^{1+n} (1+x)^{-1-n} \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,\frac {1-x}{1+x}\right )}{1+n} \]

[Out]

-1/2*(1-x)^(1+n)*(1+x)^(1-n)/x^2+2*n*(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 = 71, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {98, 133} \[ \int \frac {(1-x)^n (1+x)^{-n}}{x^3} \, dx=\frac {2 n (1-x)^{n+1} (x+1)^{-n-1} \operatorname {Hypergeometric2F1}\left (2,n+1,n+2,\frac {1-x}{x+1}\right )}{n+1}-\frac {(1-x)^{n+1} (x+1)^{1-n}}{2 x^2} \]

[In]

Int[(1 - x)^n/(x^3*(1 + x)^n),x]

[Out]

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

Rule 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

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 {(1-x)^{1+n} (1+x)^{1-n}}{2 x^2}-n \int \frac {(1-x)^n (1+x)^{-n}}{x^2} \, dx \\ & = -\frac {(1-x)^{1+n} (1+x)^{1-n}}{2 x^2}+\frac {2 n (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 = 0.04 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.93 \[ \int \frac {(1-x)^n (1+x)^{-n}}{x^3} \, dx=\frac {(1-x)^{1+n} (1+x)^{-1-n} \left (-\left ((1+n) (1+x)^2\right )+4 n x^2 \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,\frac {1-x}{1+x}\right )\right )}{2 (1+n) x^2} \]

[In]

Integrate[(1 - x)^n/(x^3*(1 + x)^n),x]

[Out]

((1 - x)^(1 + n)*(1 + x)^(-1 - n)*(-((1 + n)*(1 + x)^2) + 4*n*x^2*Hypergeometric2F1[2, 1 + n, 2 + n, (1 - x)/(
1 + x)]))/(2*(1 + n)*x^2)

Maple [F]

\[\int \frac {\left (1-x \right )^{n} \left (1+x \right )^{-n}}{x^{3}}d x\]

[In]

int((1-x)^n/x^3/((1+x)^n),x)

[Out]

int((1-x)^n/x^3/((1+x)^n),x)

Fricas [F]

\[ \int \frac {(1-x)^n (1+x)^{-n}}{x^3} \, dx=\int { \frac {{\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n} x^{3}} \,d x } \]

[In]

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

[Out]

integral((-x + 1)^n/((x + 1)^n*x^3), x)

Sympy [F]

\[ \int \frac {(1-x)^n (1+x)^{-n}}{x^3} \, dx=\int \frac {\left (1 - x\right )^{n} \left (x + 1\right )^{- n}}{x^{3}}\, dx \]

[In]

integrate((1-x)**n/x**3/((1+x)**n),x)

[Out]

Integral((1 - x)**n/(x**3*(x + 1)**n), x)

Maxima [F]

\[ \int \frac {(1-x)^n (1+x)^{-n}}{x^3} \, dx=\int { \frac {{\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n} x^{3}} \,d x } \]

[In]

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

[Out]

integrate((-x + 1)^n/((x + 1)^n*x^3), x)

Giac [F]

\[ \int \frac {(1-x)^n (1+x)^{-n}}{x^3} \, dx=\int { \frac {{\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n} x^{3}} \,d x } \]

[In]

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

[Out]

integrate((-x + 1)^n/((x + 1)^n*x^3), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(1-x)^n (1+x)^{-n}}{x^3} \, dx=\int \frac {{\left (1-x\right )}^n}{x^3\,{\left (x+1\right )}^n} \,d x \]

[In]

int((1 - x)^n/(x^3*(x + 1)^n),x)

[Out]

int((1 - x)^n/(x^3*(x + 1)^n), x)