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

Optimal. Leaf size=38 \[ \frac{1}{4} \log \left (x^2+1\right )-\frac{1}{3} \log \left (x^2-x+1\right )+\frac{1}{6} \log (x+1)+\frac{1}{2} \tan ^{-1}(x) \]

[Out]

ArcTan[x]/2 + Log[1 + x]/6 + Log[1 + x^2]/4 - Log[1 - x + x^2]/3

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Rubi [A]  time = 0.0489702, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{1}{4} \log \left (x^2+1\right )-\frac{1}{3} \log \left (x^2-x+1\right )+\frac{1}{6} \log (x+1)+\frac{1}{2} \tan ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[x]/2 + Log[1 + x]/6 + Log[1 + x^2]/4 - Log[1 - x + x^2]/3

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{5} + x^{3} + x^{2} + 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**5 + x**3 + x**2 + 1), x)

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Mathematica [A]  time = 0.0107985, size = 38, normalized size = 1. \[ \frac{1}{4} \log \left (x^2+1\right )-\frac{1}{3} \log \left (x^2-x+1\right )+\frac{1}{6} \log (x+1)+\frac{1}{2} \tan ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[x]/2 + Log[1 + x]/6 + Log[1 + x^2]/4 - Log[1 - x + x^2]/3

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Maple [A]  time = 0.01, size = 31, normalized size = 0.8 \[{\frac{\arctan \left ( x \right ) }{2}}+{\frac{\ln \left ( 1+x \right ) }{6}}+{\frac{\ln \left ({x}^{2}+1 \right ) }{4}}-{\frac{\ln \left ({x}^{2}-x+1 \right ) }{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^5+x^3+x^2+1),x)

[Out]

1/2*arctan(x)+1/6*ln(1+x)+1/4*ln(x^2+1)-1/3*ln(x^2-x+1)

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Maxima [A]  time = 0.909293, size = 41, normalized size = 1.08 \[ \frac{1}{2} \, \arctan \left (x\right ) - \frac{1}{3} \, \log \left (x^{2} - x + 1\right ) + \frac{1}{4} \, \log \left (x^{2} + 1\right ) + \frac{1}{6} \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.25712, size = 41, normalized size = 1.08 \[ \frac{1}{2} \, \arctan \left (x\right ) - \frac{1}{3} \, \log \left (x^{2} - x + 1\right ) + \frac{1}{4} \, \log \left (x^{2} + 1\right ) + \frac{1}{6} \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.318706, size = 29, normalized size = 0.76 \[ \frac{\log{\left (x + 1 \right )}}{6} + \frac{\log{\left (x^{2} + 1 \right )}}{4} - \frac{\log{\left (x^{2} - x + 1 \right )}}{3} + \frac{\operatorname{atan}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x + 1)/6 + log(x**2 + 1)/4 - log(x**2 - x + 1)/3 + atan(x)/2

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GIAC/XCAS [A]  time = 0.26338, size = 42, normalized size = 1.11 \[ \frac{1}{2} \, \arctan \left (x\right ) - \frac{1}{3} \,{\rm ln}\left (x^{2} - x + 1\right ) + \frac{1}{4} \,{\rm ln}\left (x^{2} + 1\right ) + \frac{1}{6} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*arctan(x) - 1/3*ln(x^2 - x + 1) + 1/4*ln(x^2 + 1) + 1/6*ln(abs(x + 1))