3.981 \(\int (1-x)^n (1+x)^{-n} \, dx\)

Optimal. Leaf size=38 \[ -\frac{2^{-n} (1-x)^{n+1} \, _2F_1\left (n,n+1;n+2;\frac{1-x}{2}\right )}{n+1} \]

[Out]

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

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Rubi [A]  time = 0.0061471, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {69} \[ -\frac{2^{-n} (1-x)^{n+1} \, _2F_1\left (n,n+1;n+2;\frac{1-x}{2}\right )}{n+1} \]

Antiderivative was successfully verified.

[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 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), 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*} \int (1-x)^n (1+x)^{-n} \, dx &=-\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]  time = 0.0052931, size = 38, normalized size = 1. \[ -\frac{2^{-n} (1-x)^{n+1} \, _2F_1\left (n,n+1;n+2;\frac{1-x}{2}\right )}{n+1} \]

Antiderivative was successfully verified.

[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)))

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Maple [F]  time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( 1-x \right ) ^{n}}{ \left ( 1+x \right ) ^{n}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-x + 1)^n/(x + 1)^n, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((-x + 1)^n/(x + 1)^n, x)

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Sympy [C]  time = 59.6141, size = 42, normalized size = 1.11 \begin{align*} \frac{2^{- n} \left (x - 1\right ) \left (x - 1\right )^{n} 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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-x + 1)^n/(x + 1)^n, x)