Optimal. Leaf size=36 \[ \frac{1}{8 (1-x)}-\frac{1}{4 (x+1)}-\frac{1}{8 (x+1)^2}+\frac{3}{8} \tanh ^{-1}(x) \]
[Out]
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Rubi [A] time = 0.0348106, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{1}{8 (1-x)}-\frac{1}{4 (x+1)}-\frac{1}{8 (x+1)^2}+\frac{3}{8} \tanh ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Int[1/((-1 + x)^2*(1 + x)^3),x]
[Out]
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Rubi in Sympy [A] time = 2.92106, size = 27, normalized size = 0.75 \[ \frac{3 \operatorname{atanh}{\left (x \right )}}{8} - \frac{1}{4 \left (x + 1\right )} - \frac{1}{8 \left (x + 1\right )^{2}} + \frac{1}{8 \left (- x + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(-1+x)**2/(1+x)**3,x)
[Out]
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Mathematica [A] time = 0.033627, size = 38, normalized size = 1.06 \[ \frac{1}{16} \left (\frac{-6 x^2-6 x+4}{(x-1) (x+1)^2}-3 \log (x-1)+3 \log (x+1)\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/((-1 + x)^2*(1 + x)^3),x]
[Out]
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Maple [A] time = 0.013, size = 35, normalized size = 1. \[ -{\frac{1}{8\, \left ( 1+x \right ) ^{2}}}-{\frac{1}{4+4\,x}}+{\frac{3\,\ln \left ( 1+x \right ) }{16}}-{\frac{1}{-8+8\,x}}-{\frac{3\,\ln \left ( -1+x \right ) }{16}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(-1+x)^2/(1+x)^3,x)
[Out]
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Maxima [A] time = 1.72926, size = 51, normalized size = 1.42 \[ -\frac{3 \, x^{2} + 3 \, x - 2}{8 \,{\left (x^{3} + x^{2} - x - 1\right )}} + \frac{3}{16} \, \log \left (x + 1\right ) - \frac{3}{16} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x + 1)^3*(x - 1)^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.213652, size = 80, normalized size = 2.22 \[ -\frac{6 \, x^{2} - 3 \,{\left (x^{3} + x^{2} - x - 1\right )} \log \left (x + 1\right ) + 3 \,{\left (x^{3} + x^{2} - x - 1\right )} \log \left (x - 1\right ) + 6 \, x - 4}{16 \,{\left (x^{3} + x^{2} - x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x + 1)^3*(x - 1)^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.154671, size = 41, normalized size = 1.14 \[ - \frac{3 x^{2} + 3 x - 2}{8 x^{3} + 8 x^{2} - 8 x - 8} - \frac{3 \log{\left (x - 1 \right )}}{16} + \frac{3 \log{\left (x + 1 \right )}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(-1+x)**2/(1+x)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.199435, size = 58, normalized size = 1.61 \[ -\frac{1}{8 \,{\left (x - 1\right )}} + \frac{\frac{12}{x - 1} + 5}{32 \,{\left (\frac{2}{x - 1} + 1\right )}^{2}} + \frac{3}{16} \,{\rm ln}\left ({\left | -\frac{2}{x - 1} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x + 1)^3*(x - 1)^2),x, algorithm="giac")
[Out]