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