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

Optimal. Leaf size=7 \[ -\frac{1}{x+1} \]

[Out]

-(1 + x)^(-1)

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Rubi [A]  time = 0.0010104, antiderivative size = 7, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {27, 32} \[ -\frac{1}{x+1} \]

Antiderivative was successfully verified.

[In]

Int[(1 + 2*x + x^2)^(-1),x]

[Out]

-(1 + x)^(-1)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0008212, size = 7, normalized size = 1. \[ -\frac{1}{x+1} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + 2*x + x^2)^(-1),x]

[Out]

-(1 + x)^(-1)

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Maple [A]  time = 0.002, size = 8, normalized size = 1.1 \begin{align*} - \left ( 1+x \right ) ^{-1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2+2*x+1),x)

[Out]

-1/(1+x)

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Maxima [A]  time = 0.924597, size = 9, normalized size = 1.29 \begin{align*} -\frac{1}{x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+2*x+1),x, algorithm="maxima")

[Out]

-1/(x + 1)

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Fricas [A]  time = 2.02132, size = 16, normalized size = 2.29 \begin{align*} -\frac{1}{x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+2*x+1),x, algorithm="fricas")

[Out]

-1/(x + 1)

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Sympy [A]  time = 0.070916, size = 5, normalized size = 0.71 \begin{align*} - \frac{1}{x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**2+2*x+1),x)

[Out]

-1/(x + 1)

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Giac [A]  time = 1.0837, size = 9, normalized size = 1.29 \begin{align*} -\frac{1}{x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(x + 1)