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

Optimal. Leaf size=71 \[ \frac{2 n (1-x)^{n+1} (x+1)^{-n-1} \, _2F_1\left (2,n+1;n+2;\frac{1-x}{x+1}\right )}{n+1}-\frac{(1-x)^{n+1} (x+1)^{1-n}}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0169948, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {96, 131} \[ \frac{2 n (1-x)^{n+1} (x+1)^{-n-1} \, _2F_1\left (2,n+1;n+2;\frac{1-x}{x+1}\right )}{n+1}-\frac{(1-x)^{n+1} (x+1)^{1-n}}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(1 - x)^n/(x^3*(1 + x)^n),x]

[Out]

-((1 - x)^(1 + n)*(1 + x)^(1 - n))/(2*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 96

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 131

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)*Hypergeometric2F1[m + 1, -n, m + 2, -(((d*e - c*f)*(a + b*x))/((b*c - a*d)*(e + f*x))
)])/((m + 1)*(b*e - a*f)^(n + 1)*(e + f*x)^(m + 1)), x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p
+ 2, 0] && ILtQ[n, 0]

Rubi steps

\begin{align*} \int \frac{(1-x)^n (1+x)^{-n}}{x^3} \, dx &=-\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]  time = 0.0218078, size = 66, normalized size = 0.93 \[ \frac{(1-x)^{n+1} (x+1)^{-n-1} \left (4 n x^2 \, _2F_1\left (2,n+1;n+2;\frac{1-x}{x+1}\right )-(n+1) (x+1)^2\right )}{2 (n+1) x^2} \]

Antiderivative was successfully verified.

[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)

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Maple [F]  time = 0.053, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( 1-x \right ) ^{n}}{{x}^{3} \left ( 1+x \right ) ^{n}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((1-x)^n/x^3/((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} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-x + 1)^n/((x + 1)^n*x^3), 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^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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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} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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