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

Optimal. Leaf size=6 \[ -\tanh ^{-1}(x+2) \]

[Out]

-ArcTanh[2 + x]

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Rubi [B]  time = 0.0118157, antiderivative size = 17, normalized size of antiderivative = 2.83, number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{1}{2} \log (x+1)-\frac{1}{2} \log (x+3) \]

Antiderivative was successfully verified.

[In]  Int[(3 + 4*x + x^2)^(-1),x]

[Out]

Log[1 + x]/2 - Log[3 + x]/2

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Rubi in Sympy [A]  time = 1.42147, size = 12, normalized size = 2. \[ \frac{\log{\left (x + 1 \right )}}{2} - \frac{\log{\left (x + 3 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x + 1)/2 - log(x + 3)/2

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Mathematica [B]  time = 0.00484806, size = 17, normalized size = 2.83 \[ \frac{1}{2} \log (x+1)-\frac{1}{2} \log (x+3) \]

Antiderivative was successfully verified.

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

[Out]

Log[1 + x]/2 - Log[3 + x]/2

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Maple [B]  time = 0.008, size = 14, normalized size = 2.3 \[{\frac{\ln \left ( 1+x \right ) }{2}}-{\frac{\ln \left ( 3+x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^2+4*x+3),x)

[Out]

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

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Maxima [A]  time = 0.727156, size = 18, normalized size = 3. \[ -\frac{1}{2} \, \log \left (x + 3\right ) + \frac{1}{2} \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*log(x + 3) + 1/2*log(x + 1)

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Fricas [A]  time = 0.225825, size = 18, normalized size = 3. \[ -\frac{1}{2} \, \log \left (x + 3\right ) + \frac{1}{2} \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*log(x + 3) + 1/2*log(x + 1)

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Sympy [A]  time = 0.17887, size = 12, normalized size = 2. \[ \frac{\log{\left (x + 1 \right )}}{2} - \frac{\log{\left (x + 3 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**2+4*x+3),x)

[Out]

log(x + 1)/2 - log(x + 3)/2

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GIAC/XCAS [A]  time = 0.208598, size = 20, normalized size = 3.33 \[ -\frac{1}{2} \,{\rm ln}\left ({\left | x + 3 \right |}\right ) + \frac{1}{2} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*ln(abs(x + 3)) + 1/2*ln(abs(x + 1))