3.260 \(\int \frac{1+2 x^2+x^5}{-x+x^3} \, dx\)

Optimal. Leaf size=25 \[ \frac{x^3}{3}+x+2 \log (1-x)-\log (x)+\log (x+1) \]

[Out]

x + x^3/3 + 2*Log[1 - x] - Log[x] + Log[1 + x]

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Rubi [A]  time = 0.0520808, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{x^3}{3}+x+2 \log (1-x)-\log (x)+\log (x+1) \]

Antiderivative was successfully verified.

[In]  Int[(1 + 2*x^2 + x^5)/(-x + x^3),x]

[Out]

x + x^3/3 + 2*Log[1 - x] - Log[x] + Log[1 + x]

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Rubi in Sympy [A]  time = 12.6099, size = 20, normalized size = 0.8 \[ \frac{x^{3}}{3} + x - \log{\left (x \right )} + 2 \log{\left (- x + 1 \right )} + \log{\left (x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**5+2*x**2+1)/(x**3-x),x)

[Out]

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

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Mathematica [A]  time = 0.00868306, size = 25, normalized size = 1. \[ \frac{x^3}{3}+x+2 \log (1-x)-\log (x)+\log (x+1) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + 2*x^2 + x^5)/(-x + x^3),x]

[Out]

x + x^3/3 + 2*Log[1 - x] - Log[x] + Log[1 + x]

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Maple [A]  time = 0.01, size = 22, normalized size = 0.9 \[{\frac{{x}^{3}}{3}}+x+2\,\ln \left ( -1+x \right ) +\ln \left ( 1+x \right ) -\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^5+2*x^2+1)/(x^3-x),x)

[Out]

1/3*x^3+x+2*ln(-1+x)+ln(1+x)-ln(x)

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Maxima [A]  time = 0.823995, size = 28, normalized size = 1.12 \[ \frac{1}{3} \, x^{3} + x + \log \left (x + 1\right ) + 2 \, \log \left (x - 1\right ) - \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^5 + 2*x^2 + 1)/(x^3 - x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.258774, size = 28, normalized size = 1.12 \[ \frac{1}{3} \, x^{3} + x + \log \left (x + 1\right ) + 2 \, \log \left (x - 1\right ) - \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^5 + 2*x^2 + 1)/(x^3 - x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.277487, size = 20, normalized size = 0.8 \[ \frac{x^{3}}{3} + x - \log{\left (x \right )} + 2 \log{\left (x - 1 \right )} + \log{\left (x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**5+2*x**2+1)/(x**3-x),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.260412, size = 32, normalized size = 1.28 \[ \frac{1}{3} \, x^{3} + x +{\rm ln}\left ({\left | x + 1 \right |}\right ) + 2 \,{\rm ln}\left ({\left | x - 1 \right |}\right ) -{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^5 + 2*x^2 + 1)/(x^3 - x),x, algorithm="giac")

[Out]

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