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

Optimal. Leaf size=18 \[ -\frac{1}{2} \log \left (x^2+1\right )-x+\log (x)+\tan ^{-1}(x) \]

[Out]

-x + ArcTan[x] + Log[x] - Log[1 + x^2]/2

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Rubi [A]  time = 0.0430217, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{1}{2} \log \left (x^2+1\right )-x+\log (x)+\tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(1 - x^3)/(x*(1 + x^2)),x]

[Out]

-x + ArcTan[x] + Log[x] - Log[1 + x^2]/2

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Rubi in Sympy [A]  time = 4.77189, size = 15, normalized size = 0.83 \[ - x + \log{\left (x \right )} - \frac{\log{\left (x^{2} + 1 \right )}}{2} + \operatorname{atan}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00788118, size = 18, normalized size = 1. \[ -\frac{1}{2} \log \left (x^2+1\right )-x+\log (x)+\tan ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

-x + ArcTan[x] + Log[x] - Log[1 + x^2]/2

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Maple [A]  time = 0.007, size = 17, normalized size = 0.9 \[ -x+\arctan \left ( x \right ) +\ln \left ( x \right ) -{\frac{\ln \left ({x}^{2}+1 \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^3+1)/x/(x^2+1),x)

[Out]

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

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Maxima [A]  time = 1.50263, size = 22, normalized size = 1.22 \[ -x + \arctan \left (x\right ) - \frac{1}{2} \, \log \left (x^{2} + 1\right ) + \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.202171, size = 22, normalized size = 1.22 \[ -x + \arctan \left (x\right ) - \frac{1}{2} \, \log \left (x^{2} + 1\right ) + \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.129338, size = 15, normalized size = 0.83 \[ - x + \log{\left (x \right )} - \frac{\log{\left (x^{2} + 1 \right )}}{2} + \operatorname{atan}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.227086, size = 23, normalized size = 1.28 \[ -x + \arctan \left (x\right ) - \frac{1}{2} \,{\rm ln}\left (x^{2} + 1\right ) +{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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