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

Optimal. Leaf size=14 \[ 2 \log \left (x^2+1\right )+x-\tan ^{-1}(x) \]

[Out]

x - ArcTan[x] + 2*Log[1 + x^2]

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Rubi [A]  time = 0.0444789, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ 2 \log \left (x^2+1\right )+x-\tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(4*x^2 + x^3)/(x + x^3),x]

[Out]

x - ArcTan[x] + 2*Log[1 + x^2]

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Rubi in Sympy [A]  time = 7.14112, size = 12, normalized size = 0.86 \[ x + 2 \log{\left (x^{2} + 1 \right )} - \operatorname{atan}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x + 2*log(x**2 + 1) - atan(x)

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Mathematica [A]  time = 0.00588897, size = 14, normalized size = 1. \[ 2 \log \left (x^2+1\right )+x-\tan ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

x - ArcTan[x] + 2*Log[1 + x^2]

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Maple [A]  time = 0.003, size = 15, normalized size = 1.1 \[ x-\arctan \left ( x \right ) +2\,\ln \left ({x}^{2}+1 \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^3+4*x^2)/(x^3+x),x)

[Out]

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

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Maxima [A]  time = 0.87066, size = 19, normalized size = 1.36 \[ x - \arctan \left (x\right ) + 2 \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x - arctan(x) + 2*log(x^2 + 1)

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Fricas [A]  time = 0.265371, size = 19, normalized size = 1.36 \[ x - \arctan \left (x\right ) + 2 \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x - arctan(x) + 2*log(x^2 + 1)

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Sympy [A]  time = 0.197064, size = 12, normalized size = 0.86 \[ x + 2 \log{\left (x^{2} + 1 \right )} - \operatorname{atan}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x + 2*log(x**2 + 1) - atan(x)

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GIAC/XCAS [A]  time = 0.261053, size = 19, normalized size = 1.36 \[ x - \arctan \left (x\right ) + 2 \,{\rm ln}\left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x - arctan(x) + 2*ln(x^2 + 1)