Optimal. Leaf size=105 \[ -\frac{2 \left (2 n^2+1\right ) (1-x)^{n+1} (x+1)^{-n-1} \, _2F_1\left (2,n+1;n+2;\frac{1-x}{x+1}\right )}{3 (n+1)}-\frac{(1-x)^{n+1} (x+1)^{1-n}}{3 x^3}+\frac{n (1-x)^{n+1} (x+1)^{1-n}}{3 x^2} \]
[Out]
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Rubi [A] time = 0.124585, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{2 \left (2 n^2+1\right ) (1-x)^{n+1} (x+1)^{-n-1} \, _2F_1\left (2,n+1;n+2;\frac{1-x}{x+1}\right )}{3 (n+1)}-\frac{(1-x)^{n+1} (x+1)^{1-n}}{3 x^3}+\frac{n (1-x)^{n+1} (x+1)^{1-n}}{3 x^2} \]
Antiderivative was successfully verified.
[In] Int[(1 - x)^n/(x^4*(1 + x)^n),x]
[Out]
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Rubi in Sympy [A] time = 13.9519, size = 80, normalized size = 0.76 \[ \frac{n \left (- x + 1\right )^{n + 1} \left (x + 1\right )^{- n + 1}}{3 x^{2}} - \frac{2 \left (2 n^{2} + 1\right ) \left (- x + 1\right )^{n + 1} \left (x + 1\right )^{- n - 1}{{}_{2}F_{1}\left (\begin{matrix} n + 1, 2 \\ n + 2 \end{matrix}\middle |{\frac{x - 1}{- x - 1}} \right )}}{3 \left (n + 1\right )} - \frac{\left (- x + 1\right )^{n + 1} \left (x + 1\right )^{- n + 1}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-x)**n/x**4/((1+x)**n),x)
[Out]
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Mathematica [C] time = 0.239668, size = 95, normalized size = 0.9 \[ -\frac{4 (1-x)^n (x+1)^{-n} F_1\left (3;-n,n;4;\frac{1}{x},-\frac{1}{x}\right )}{3 x^2 \left (4 x F_1\left (3;-n,n;4;\frac{1}{x},-\frac{1}{x}\right )-n \left (F_1\left (4;1-n,n;5;\frac{1}{x},-\frac{1}{x}\right )+F_1\left (4;-n,n+1;5;\frac{1}{x},-\frac{1}{x}\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(1 - x)^n/(x^4*(1 + x)^n),x]
[Out]
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Maple [F] time = 0.105, size = 0, normalized size = 0. \[ \int{\frac{ \left ( 1-x \right ) ^{n}}{{x}^{4} \left ( 1+x \right ) ^{n}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-x)^n/x^4/((1+x)^n),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (x + 1\right )}^{-n}{\left (-x + 1\right )}^{n}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x + 1)^n/((x + 1)^n*x^4),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n} x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x + 1)^n/((x + 1)^n*x^4),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**4/((1+x)**n),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x + 1)^n/((x + 1)^n*x^4),x, algorithm="giac")
[Out]