3.2837 \(\int \frac{2+x}{1+(2+x)^2} \, dx\)

Optimal. Leaf size=12 \[ \frac{1}{2} \log \left ((x+2)^2+1\right ) \]

[Out]

Log[1 + (2 + x)^2]/2

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Rubi [A]  time = 0.00808821, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{1}{2} \log \left ((x+2)^2+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[(2 + x)/(1 + (2 + x)^2),x]

[Out]

Log[1 + (2 + x)^2]/2

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Rubi in Sympy [A]  time = 2.1565, size = 8, normalized size = 0.67 \[ \frac{\log{\left (\left (x + 2\right )^{2} + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log((x + 2)**2 + 1)/2

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Mathematica [A]  time = 0.00372332, size = 12, normalized size = 1. \[ \frac{1}{2} \log \left ((x+2)^2+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

Log[1 + (2 + x)^2]/2

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Maple [A]  time = 0.001, size = 12, normalized size = 1. \[{\frac{\ln \left ({x}^{2}+4\,x+5 \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+x)/(1+(2+x)^2),x)

[Out]

1/2*ln(x^2+4*x+5)

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Maxima [A]  time = 1.37722, size = 14, normalized size = 1.17 \[ \frac{1}{2} \, \log \left ({\left (x + 2\right )}^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*log((x + 2)^2 + 1)

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Fricas [A]  time = 0.205985, size = 15, normalized size = 1.25 \[ \frac{1}{2} \, \log \left (x^{2} + 4 \, x + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*log(x^2 + 4*x + 5)

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Sympy [A]  time = 0.143202, size = 10, normalized size = 0.83 \[ \frac{\log{\left (x^{2} + 4 x + 5 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2+x)/(1+(2+x)**2),x)

[Out]

log(x**2 + 4*x + 5)/2

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GIAC/XCAS [A]  time = 0.215847, size = 15, normalized size = 1.25 \[ \frac{1}{2} \,{\rm ln}\left (x^{2} + 4 \, x + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*ln(x^2 + 4*x + 5)