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

Optimal. Leaf size=11 \[ \log (x)+\frac{1}{2} \tan ^{-1}(2 x) \]

[Out]

ArcTan[2*x]/2 + Log[x]

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Rubi [A]  time = 0.0482499, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \log (x)+\frac{1}{2} \tan ^{-1}(2 x) \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[2*x]/2 + Log[x]

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Rubi in Sympy [A]  time = 7.83613, size = 8, normalized size = 0.73 \[ \log{\left (x \right )} + \frac{\operatorname{atan}{\left (2 x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x) + atan(2*x)/2

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Mathematica [A]  time = 0.00733817, size = 11, normalized size = 1. \[ \log (x)+\frac{1}{2} \tan ^{-1}(2 x) \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[2*x]/2 + Log[x]

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Maple [A]  time = 0.007, size = 10, normalized size = 0.9 \[{\frac{\arctan \left ( 2\,x \right ) }{2}}+\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((4*x^2+x+1)/(4*x^3+x),x)

[Out]

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

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Maxima [A]  time = 0.877128, size = 12, normalized size = 1.09 \[ \frac{1}{2} \, \arctan \left (2 \, x\right ) + \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*arctan(2*x) + log(x)

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Fricas [A]  time = 0.290213, size = 12, normalized size = 1.09 \[ \frac{1}{2} \, \arctan \left (2 \, x\right ) + \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*arctan(2*x) + log(x)

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Sympy [A]  time = 0.39055, size = 8, normalized size = 0.73 \[ \log{\left (x \right )} + \frac{\operatorname{atan}{\left (2 x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x) + atan(2*x)/2

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GIAC/XCAS [A]  time = 0.263086, size = 14, normalized size = 1.27 \[ \frac{1}{2} \, \arctan \left (2 \, x\right ) +{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*arctan(2*x) + ln(abs(x))