3.105 \(\int \frac{2+x^2+x^3}{x \left (-1+x^2\right )^2} \, dx\)

Optimal. Leaf size=39 \[ \frac{x+3}{2 \left (1-x^2\right )}-\frac{3}{4} \log (1-x)+2 \log (x)-\frac{5}{4} \log (x+1) \]

[Out]

(3 + x)/(2*(1 - x^2)) - (3*Log[1 - x])/4 + 2*Log[x] - (5*Log[1 + x])/4

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Rubi [A]  time = 0.0671782, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{x+3}{2 \left (1-x^2\right )}-\frac{3}{4} \log (1-x)+2 \log (x)-\frac{5}{4} \log (x+1) \]

Antiderivative was successfully verified.

[In]  Int[(2 + x^2 + x^3)/(x*(-1 + x^2)^2),x]

[Out]

(3 + x)/(2*(1 - x^2)) - (3*Log[1 - x])/4 + 2*Log[x] - (5*Log[1 + x])/4

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Rubi in Sympy [A]  time = 6.45137, size = 31, normalized size = 0.79 \[ \frac{x \left (1 + \frac{3}{x}\right )}{2 \left (- x^{2} + 1\right )} - \log{\left (x \right )} + \frac{3 \log{\left (- x + 1 \right )}}{4} + \frac{\log{\left (x + 1 \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**3+x**2+2)/x/(x**2-1)**2,x)

[Out]

x*(1 + 3/x)/(2*(-x**2 + 1)) - log(x) + 3*log(-x + 1)/4 + log(x + 1)/4

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Mathematica [A]  time = 0.0280024, size = 47, normalized size = 1.21 \[ \frac{1}{4} \left (-\frac{4}{x^2-1}-4 \log \left (1-x^2\right )-\frac{2}{x-1}+\log (1-x)+8 \log (x)-\log (x+1)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + x^2 + x^3)/(x*(-1 + x^2)^2),x]

[Out]

(-2/(-1 + x) - 4/(-1 + x^2) + Log[1 - x] + 8*Log[x] - Log[1 + x] - 4*Log[1 - x^2
])/4

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Maple [A]  time = 0.016, size = 32, normalized size = 0.8 \[{\frac{1}{2\,x+2}}-{\frac{5\,\ln \left ( 1+x \right ) }{4}}+2\,\ln \left ( x \right ) - \left ( -1+x \right ) ^{-1}-{\frac{3\,\ln \left ( -1+x \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^3+x^2+2)/x/(x^2-1)^2,x)

[Out]

1/2/(1+x)-5/4*ln(1+x)+2*ln(x)-1/(-1+x)-3/4*ln(-1+x)

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Maxima [A]  time = 1.3695, size = 39, normalized size = 1. \[ -\frac{x + 3}{2 \,{\left (x^{2} - 1\right )}} - \frac{5}{4} \, \log \left (x + 1\right ) - \frac{3}{4} \, \log \left (x - 1\right ) + 2 \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 + x^2 + 2)/((x^2 - 1)^2*x),x, algorithm="maxima")

[Out]

-1/2*(x + 3)/(x^2 - 1) - 5/4*log(x + 1) - 3/4*log(x - 1) + 2*log(x)

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Fricas [A]  time = 0.202227, size = 61, normalized size = 1.56 \[ -\frac{5 \,{\left (x^{2} - 1\right )} \log \left (x + 1\right ) + 3 \,{\left (x^{2} - 1\right )} \log \left (x - 1\right ) - 8 \,{\left (x^{2} - 1\right )} \log \left (x\right ) + 2 \, x + 6}{4 \,{\left (x^{2} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 + x^2 + 2)/((x^2 - 1)^2*x),x, algorithm="fricas")

[Out]

-1/4*(5*(x^2 - 1)*log(x + 1) + 3*(x^2 - 1)*log(x - 1) - 8*(x^2 - 1)*log(x) + 2*x
+ 6)/(x^2 - 1)

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Sympy [A]  time = 0.172215, size = 31, normalized size = 0.79 \[ - \frac{x + 3}{2 x^{2} - 2} + 2 \log{\left (x \right )} - \frac{3 \log{\left (x - 1 \right )}}{4} - \frac{5 \log{\left (x + 1 \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**3+x**2+2)/x/(x**2-1)**2,x)

[Out]

-(x + 3)/(2*x**2 - 2) + 2*log(x) - 3*log(x - 1)/4 - 5*log(x + 1)/4

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GIAC/XCAS [A]  time = 0.19948, size = 47, normalized size = 1.21 \[ -\frac{x + 3}{2 \,{\left (x + 1\right )}{\left (x - 1\right )}} - \frac{5}{4} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{3}{4} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) + 2 \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 + x^2 + 2)/((x^2 - 1)^2*x),x, algorithm="giac")

[Out]

-1/2*(x + 3)/((x + 1)*(x - 1)) - 5/4*ln(abs(x + 1)) - 3/4*ln(abs(x - 1)) + 2*ln(
abs(x))