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 = 0.0036109, 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, Rules used = {616, 31} \[ \frac{1}{2} \log (x+1)-\frac{1}{2} \log (x+3) \]

Antiderivative was successfully verified.

[In]

Int[(3 + 4*x + x^2)^(-1),x]

[Out]

Log[1 + x]/2 - Log[3 + x]/2

Rule 616

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/Simp
[b/2 - q/2 + c*x, x], x], x] - Dist[c/q, Int[1/Simp[b/2 + q/2 + c*x, x], x], x]] /; FreeQ[{a, b, c}, x] && NeQ
[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c] && PerfectSquareQ[b^2 - 4*a*c]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{1}{3+4 x+x^2} \, dx &=\frac{1}{2} \int \frac{1}{1+x} \, dx-\frac{1}{2} \int \frac{1}{3+x} \, dx\\ &=\frac{1}{2} \log (1+x)-\frac{1}{2} \log (3+x)\\ \end{align*}

Mathematica [B]  time = 0.0025821, 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 = 0.047, size = 14, normalized size = 2.3 \begin{align*}{\frac{\ln \left ( 1+x \right ) }{2}}-{\frac{\ln \left ( 3+x \right ) }{2}} \end{align*}

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 [B]  time = 1.13718, size = 18, normalized size = 3. \begin{align*} -\frac{1}{2} \, \log \left (x + 3\right ) + \frac{1}{2} \, \log \left (x + 1\right ) \end{align*}

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 [B]  time = 2.09493, size = 46, normalized size = 7.67 \begin{align*} -\frac{1}{2} \, \log \left (x + 3\right ) + \frac{1}{2} \, \log \left (x + 1\right ) \end{align*}

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 [B]  time = 0.094988, size = 12, normalized size = 2. \begin{align*} \frac{\log{\left (x + 1 \right )}}{2} - \frac{\log{\left (x + 3 \right )}}{2} \end{align*}

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 [B]  time = 1.26072, size = 20, normalized size = 3.33 \begin{align*} -\frac{1}{2} \, \log \left ({\left | x + 3 \right |}\right ) + \frac{1}{2} \, \log \left ({\left | x + 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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