3.113 \(\int \frac{x^3+x^4}{1+x^2} \, dx\)

Optimal. Leaf size=30 \[ \frac{x^3}{3}+\frac{x^2}{2}-\frac{1}{2} \log \left (x^2+1\right )-x+\tan ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.0595498, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{x^3}{3}+\frac{x^2}{2}-\frac{1}{2} \log \left (x^2+1\right )-x+\tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(x^3 + x^4)/(1 + x^2),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x^{3}}{3} - x - \frac{\log{\left (x^{2} + 1 \right )}}{2} + \operatorname{atan}{\left (x \right )} + \int x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**4+x**3)/(x**2+1),x)

[Out]

x**3/3 - x - log(x**2 + 1)/2 + atan(x) + Integral(x, x)

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Mathematica [A]  time = 0.0094795, size = 30, normalized size = 1. \[ \frac{x^3}{3}+\frac{x^2}{2}-\frac{1}{2} \log \left (x^2+1\right )-x+\tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Integrate[(x^3 + x^4)/(1 + x^2),x]

[Out]

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

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Maple [A]  time = 0.004, size = 25, normalized size = 0.8 \[ -x+{\frac{{x}^{2}}{2}}+{\frac{{x}^{3}}{3}}+\arctan \left ( x \right ) -{\frac{\ln \left ({x}^{2}+1 \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^4+x^3)/(x^2+1),x)

[Out]

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

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Maxima [A]  time = 1.49592, size = 32, normalized size = 1.07 \[ \frac{1}{3} \, x^{3} + \frac{1}{2} \, x^{2} - x + \arctan \left (x\right ) - \frac{1}{2} \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + x^3)/(x^2 + 1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.22549, size = 32, normalized size = 1.07 \[ \frac{1}{3} \, x^{3} + \frac{1}{2} \, x^{2} - x + \arctan \left (x\right ) - \frac{1}{2} \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + x^3)/(x^2 + 1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.090556, size = 22, normalized size = 0.73 \[ \frac{x^{3}}{3} + \frac{x^{2}}{2} - x - \frac{\log{\left (x^{2} + 1 \right )}}{2} + \operatorname{atan}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**4+x**3)/(x**2+1),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.221957, size = 32, normalized size = 1.07 \[ \frac{1}{3} \, x^{3} + \frac{1}{2} \, x^{2} - x + \arctan \left (x\right ) - \frac{1}{2} \,{\rm ln}\left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + x^3)/(x^2 + 1),x, algorithm="giac")

[Out]

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