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

Optimal. Leaf size=61 \[ \frac{2^{-n} n (1-x)^{n+1} \, _2F_1\left (n,n+1;n+2;\frac{1-x}{2}\right )}{n+1}-\frac{1}{2} (1-x)^{n+1} (x+1)^{1-n} \]

[Out]

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

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Rubi [A]  time = 0.0126024, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {80, 69} \[ \frac{2^{-n} n (1-x)^{n+1} \, _2F_1\left (n,n+1;n+2;\frac{1-x}{2}\right )}{n+1}-\frac{1}{2} (1-x)^{n+1} (x+1)^{1-n} \]

Antiderivative was successfully verified.

[In]

Int[((1 - x)^n*x)/(1 + x)^n,x]

[Out]

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

Rule 80

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

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 x (1+x)^{-n} \, dx &=-\frac{1}{2} (1-x)^{1+n} (1+x)^{1-n}-n \int (1-x)^n (1+x)^{-n} \, dx\\ &=-\frac{1}{2} (1-x)^{1+n} (1+x)^{1-n}+\frac{2^{-n} 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 = 0.0280958, size = 56, normalized size = 0.92 \[ \frac{1}{2} (1-x)^{n+1} \left (\frac{2^{1-n} n \, _2F_1\left (n,n+1;n+2;\frac{1-x}{2}\right )}{n+1}-(x+1)^{1-n}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - x)^n*x)/(1 + x)^n,x]

[Out]

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

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Maple [F]  time = 0.047, size = 0, normalized size = 0. \begin{align*} \int{\frac{x \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*x/((1+x)^n),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x{\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*x/((1+x)^n),x, algorithm="maxima")

[Out]

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

[Out]

integral(x*(-x + 1)^n/(x + 1)^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x{\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*x/((1+x)^n),x, algorithm="giac")

[Out]

integrate(x*(-x + 1)^n/(x + 1)^n, x)