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

Optimal. Leaf size=68 \[ \frac{2^{-n} (1-x)^n \, _2F_1\left (n,n;n+1;\frac{1-x}{2}\right )}{n}-\frac{(1-x)^n (x+1)^{-n} \, _2F_1\left (1,n;n+1;\frac{1-x}{x+1}\right )}{n} \]

[Out]

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

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Rubi [A]  time = 0.0767463, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{2^{-n} (1-x)^n \, _2F_1\left (n,n;n+1;\frac{1-x}{2}\right )}{n}-\frac{(1-x)^n (x+1)^{-n} \, _2F_1\left (1,n;n+1;\frac{1-x}{x+1}\right )}{n} \]

Antiderivative was successfully verified.

[In]  Int[(1 - x)^n/(x*(1 + x)^n),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.95134, size = 61, normalized size = 0.9 \[ - \frac{\left (- x + 1\right )^{n + 1} \left (x + 1\right )^{- n - 1}{{}_{2}F_{1}\left (\begin{matrix} n + 1, 1 \\ n + 2 \end{matrix}\middle |{\frac{x - 1}{- x - 1}} \right )}}{n + 1} - \frac{2^{- n} \left (- x + 1\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} n + 1, n + 1 \\ n + 2 \end{matrix}\middle |{- \frac{x}{2} + \frac{1}{2}} \right )}}{2 \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-x)**n/x/((1+x)**n),x)

[Out]

-(-x + 1)**(n + 1)*(x + 1)**(-n - 1)*hyper((n + 1, 1), (n + 2,), (x - 1)/(-x - 1
))/(n + 1) - 2**(-n)*(-x + 1)**(n + 1)*hyper((n + 1, n + 1), (n + 2,), -x/2 + 1/
2)/(2*(n + 1))

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Mathematica [C]  time = 0.314842, size = 140, normalized size = 2.06 \[ \frac{2 (n+2) (1-x)^{n+1} (x+1)^{-n} F_1\left (n+1;n,1;n+2;\frac{1-x}{2},1-x\right )}{(n+1) x \left ((x-1) \left (2 F_1\left (n+2;n,2;n+3;\frac{1-x}{2},1-x\right )+n F_1\left (n+2;n+1,1;n+3;\frac{1-x}{2},1-x\right )\right )-2 (n+2) F_1\left (n+1;n,1;n+2;\frac{1-x}{2},1-x\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(1 - x)^n/(x*(1 + x)^n),x]

[Out]

(2*(2 + n)*(1 - x)^(1 + n)*AppellF1[1 + n, n, 1, 2 + n, (1 - x)/2, 1 - x])/((1 +
n)*x*(1 + x)^n*(-2*(2 + n)*AppellF1[1 + n, n, 1, 2 + n, (1 - x)/2, 1 - x] + (-1
+ x)*(2*AppellF1[2 + n, n, 2, 3 + n, (1 - x)/2, 1 - x] + n*AppellF1[2 + n, 1 +
n, 1, 3 + n, (1 - x)/2, 1 - x])))

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Maple [F]  time = 0.073, size = 0, normalized size = 0. \[ \int{\frac{ \left ( 1-x \right ) ^{n}}{x \left ( 1+x \right ) ^{n}}}\, dx \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x + 1)^n/((x + 1)^n*x),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x + 1)^n/((x + 1)^n*x),x, algorithm="fricas")

[Out]

integral((-x + 1)^n/((x + 1)^n*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1-x)**n/x/((1+x)**n),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x + 1)^n/((x + 1)^n*x),x, algorithm="giac")

[Out]

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