\(\int (1-x)^n (1+x)^{-n} \, dx\) [981]

Optimal result

Integrand size = 15, antiderivative size = 38 \[ \int (1-x)^n (1+x)^{-n} \, dx=-\frac {2^{-n} (1-x)^{1+n} \operatorname {Hypergeometric2F1}\left (n,1+n,2+n,\frac {1-x}{2}\right )}{1+n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {71} \[ \int (1-x)^n (1+x)^{-n} \, dx=-\frac {2^{-n} (1-x)^{n+1} \operatorname {Hypergeometric2F1}\left (n,n+1,n+2,\frac {1-x}{2}\right )}{n+1} \]

[In]

Int[(1 - x)^n/(1 + x)^n,x]

[Out]

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

Rule 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-d/(b*c - a*d), 0]))

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int (1-x)^n (1+x)^{-n} \, dx=-\frac {2^{-n} (1-x)^{1+n} \operatorname {Hypergeometric2F1}\left (n,1+n,2+n,\frac {1-x}{2}\right )}{1+n} \]

[In]

Integrate[(1 - x)^n/(1 + x)^n,x]

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral((-x + 1)^n/(x + 1)^n, x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 9.51 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.08 \[ \int (1-x)^n (1+x)^{-n} \, dx=\frac {2^{- n} \left (x - 1\right )^{n + 1} e^{i \pi n} \Gamma \left (n + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} n, n + 1 \\ n + 2 \end {matrix}\middle | {\frac {\left (x - 1\right ) e^{i \pi }}{2}} \right )}}{\Gamma \left (n + 2\right )} \]

[In]

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

[Out]

(x - 1)**(n + 1)*exp(I*pi*n)*gamma(n + 1)*hyper((n, n + 1), (n + 2,), (x - 1)*exp_polar(I*pi)/2)/(2**n*gamma(n
+ 2))

Maxima [F]

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

[In]

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

[Out]

integrate((-x + 1)^n/(x + 1)^n, x)

Giac [F]

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

[In]

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

[Out]

integrate((-x + 1)^n/(x + 1)^n, x)

Mupad [F(-1)]

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

[In]

int((1 - x)^n/(x + 1)^n,x)

[Out]

int((1 - x)^n/(x + 1)^n, x)