3.365 \(\int \frac{1+x^3}{\left (13+4 x+x^2\right )^2} \, dx\)

Optimal. Leaf size=45 \[ \frac{47 x+67}{18 \left (x^2+4 x+13\right )}+\frac{1}{2} \log \left (x^2+4 x+13\right )-\frac{61}{54} \tan ^{-1}\left (\frac{x+2}{3}\right ) \]

[Out]

(67 + 47*x)/(18*(13 + 4*x + x^2)) - (61*ArcTan[(2 + x)/3])/54 + Log[13 + 4*x + x
^2]/2

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Rubi [A]  time = 0.0528657, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312 \[ \frac{47 x+67}{18 \left (x^2+4 x+13\right )}+\frac{1}{2} \log \left (x^2+4 x+13\right )-\frac{61}{54} \tan ^{-1}\left (\frac{x+2}{3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 + x^3)/(13 + 4*x + x^2)^2,x]

[Out]

(67 + 47*x)/(18*(13 + 4*x + x^2)) - (61*ArcTan[(2 + x)/3])/54 + Log[13 + 4*x + x
^2]/2

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Rubi in Sympy [A]  time = 13.9818, size = 34, normalized size = 0.76 \[ \frac{0.0138888888888889 \left (188 x + 268\right )}{x^{2} + 4 x + 13} + \frac{\log{\left (x^{2} + 4 x + 13 \right )}}{2} - 1.12962962962963 \operatorname{atan}{\left (\frac{x}{3} + \frac{2}{3} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**3+1)/(x**2+4*x+13)**2,x)

[Out]

0.0138888888888889*(188*x + 268)/(x**2 + 4*x + 13) + log(x**2 + 4*x + 13)/2 - 1.
12962962962963*atan(x/3 + 2/3)

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Mathematica [A]  time = 0.0195404, size = 45, normalized size = 1. \[ \frac{47 x+67}{18 \left (x^2+4 x+13\right )}+\frac{1}{2} \log \left (x^2+4 x+13\right )-\frac{61}{54} \tan ^{-1}\left (\frac{x+2}{3}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x^3)/(13 + 4*x + x^2)^2,x]

[Out]

(67 + 47*x)/(18*(13 + 4*x + x^2)) - (61*ArcTan[(2 + x)/3])/54 + Log[13 + 4*x + x
^2]/2

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Maple [A]  time = 0.009, size = 37, normalized size = 0.8 \[{\frac{1}{{x}^{2}+4\,x+13} \left ({\frac{47\,x}{18}}+{\frac{67}{18}} \right ) }+{\frac{\ln \left ({x}^{2}+4\,x+13 \right ) }{2}}-{\frac{61}{54}\arctan \left ({\frac{2}{3}}+{\frac{x}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^3+1)/(x^2+4*x+13)^2,x)

[Out]

(47/18*x+67/18)/(x^2+4*x+13)+1/2*ln(x^2+4*x+13)-61/54*arctan(2/3+1/3*x)

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Maxima [A]  time = 0.889976, size = 50, normalized size = 1.11 \[ \frac{47 \, x + 67}{18 \,{\left (x^{2} + 4 \, x + 13\right )}} - \frac{61}{54} \, \arctan \left (\frac{1}{3} \, x + \frac{2}{3}\right ) + \frac{1}{2} \, \log \left (x^{2} + 4 \, x + 13\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 + 1)/(x^2 + 4*x + 13)^2,x, algorithm="maxima")

[Out]

1/18*(47*x + 67)/(x^2 + 4*x + 13) - 61/54*arctan(1/3*x + 2/3) + 1/2*log(x^2 + 4*
x + 13)

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Fricas [A]  time = 0.258079, size = 70, normalized size = 1.56 \[ -\frac{61 \,{\left (x^{2} + 4 \, x + 13\right )} \arctan \left (\frac{1}{3} \, x + \frac{2}{3}\right ) - 27 \,{\left (x^{2} + 4 \, x + 13\right )} \log \left (x^{2} + 4 \, x + 13\right ) - 141 \, x - 201}{54 \,{\left (x^{2} + 4 \, x + 13\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 + 1)/(x^2 + 4*x + 13)^2,x, algorithm="fricas")

[Out]

-1/54*(61*(x^2 + 4*x + 13)*arctan(1/3*x + 2/3) - 27*(x^2 + 4*x + 13)*log(x^2 + 4
*x + 13) - 141*x - 201)/(x^2 + 4*x + 13)

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Sympy [A]  time = 0.324539, size = 37, normalized size = 0.82 \[ \frac{47 x + 67}{18 x^{2} + 72 x + 234} + \frac{\log{\left (x^{2} + 4 x + 13 \right )}}{2} - \frac{61 \operatorname{atan}{\left (\frac{x}{3} + \frac{2}{3} \right )}}{54} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**3+1)/(x**2+4*x+13)**2,x)

[Out]

(47*x + 67)/(18*x**2 + 72*x + 234) + log(x**2 + 4*x + 13)/2 - 61*atan(x/3 + 2/3)
/54

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GIAC/XCAS [A]  time = 0.260441, size = 50, normalized size = 1.11 \[ \frac{47 \, x + 67}{18 \,{\left (x^{2} + 4 \, x + 13\right )}} - \frac{61}{54} \, \arctan \left (\frac{1}{3} \, x + \frac{2}{3}\right ) + \frac{1}{2} \,{\rm ln}\left (x^{2} + 4 \, x + 13\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 + 1)/(x^2 + 4*x + 13)^2,x, algorithm="giac")

[Out]

1/18*(47*x + 67)/(x^2 + 4*x + 13) - 61/54*arctan(1/3*x + 2/3) + 1/2*ln(x^2 + 4*x
+ 13)