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

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 = 0.0606979, 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 \[ \frac{2 n (1-x)^{n+1} (x+1)^{-n-1} \, _2F_1\left (2,n+1;n+2;\frac{1-x}{x+1}\right )}{n+1}-\frac{(1-x)^{n+1} (x+1)^{1-n}}{2 x^2} \]

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)

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Rubi in Sympy [A]  time = 6.60783, size = 53, normalized size = 0.75 \[ - \frac{2 n \left (- x + 1\right )^{n - 1} \left (x + 1\right )^{- n + 1}{{}_{2}F_{1}\left (\begin{matrix} - n + 1, 2 \\ - n + 2 \end{matrix}\middle |{\frac{- x - 1}{x - 1}} \right )}}{- n + 1} - \frac{\left (- x + 1\right )^{n + 1} \left (x + 1\right )^{- n + 1}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-x)**n/x**3/((1+x)**n),x)

[Out]

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

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Mathematica [C]  time = 0.231674, size = 95, normalized size = 1.34 \[ -\frac{3 (1-x)^n (x+1)^{-n} F_1\left (2;-n,n;3;\frac{1}{x},-\frac{1}{x}\right )}{2 x \left (3 x F_1\left (2;-n,n;3;\frac{1}{x},-\frac{1}{x}\right )-n \left (F_1\left (3;1-n,n;4;\frac{1}{x},-\frac{1}{x}\right )+F_1\left (3;-n,n+1;4;\frac{1}{x},-\frac{1}{x}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(1 - x)^n/(x^3*(1 + x)^n),x]

[Out]

(-3*(1 - x)^n*AppellF1[2, -n, n, 3, x^(-1), -x^(-1)])/(2*x*(1 + x)^n*(3*x*Appell
F1[2, -n, n, 3, x^(-1), -x^(-1)] - n*(AppellF1[3, 1 - n, n, 4, x^(-1), -x^(-1)]
+ AppellF1[3, -n, 1 + n, 4, x^(-1), -x^(-1)])))

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Maple [F]  time = 0.081, size = 0, normalized size = 0. \[ \int{\frac{ \left ( 1-x \right ) ^{n}}{{x}^{3} \left ( 1+x \right ) ^{n}}}\, dx \]

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. \[ \int \frac{{\left (x + 1\right )}^{-n}{\left (-x + 1\right )}^{n}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x + 1)^n/((x + 1)^n*x^3),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. \[{\rm integral}\left (\frac{{\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n} x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x + 1)^n/((x + 1)^n*x^3),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. \[ \text{Timed out} \]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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