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]

-1/2*(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)

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Rubi [A]  time = 0.02, antiderivative size = 71, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.02, 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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fricas [F]  time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n} x^{3}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n} x^{3}}\,{d x} \]

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)

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maple [F]  time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {\left (-x +1\right )^{n} \left (x +1\right )^{-n}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n} x^{3}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (1-x\right )}^n}{x^3\,{\left (x+1\right )}^n} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x)^n/(x^3*(x + 1)^n),x)

[Out]

int((1 - x)^n/(x^3*(x + 1)^n), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (1 - x\right )^{n} \left (x + 1\right )^{- n}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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