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

Optimal. Leaf size=11 \[ \log (x+1)-\log (x+2) \]

[Out]

Log[1 + x] - Log[2 + x]

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Rubi [A]  time = 0.0014919, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {36, 31} \[ \log (x+1)-\log (x+2) \]

Antiderivative was successfully verified.

[In]

Int[1/((1 + x)*(2 + x)),x]

[Out]

Log[1 + x] - Log[2 + x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

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

Mathematica [A]  time = 0.0028611, size = 11, normalized size = 1. \[ \log (x+1)-\log (x+2) \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 + x)*(2 + x)),x]

[Out]

Log[1 + x] - Log[2 + x]

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Maple [A]  time = 0.004, size = 12, normalized size = 1.1 \begin{align*} \ln \left ( 1+x \right ) -\ln \left ( 2+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(1+x)-ln(2+x)

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Maxima [A]  time = 0.92628, size = 15, normalized size = 1.36 \begin{align*} -\log \left (x + 2\right ) + \log \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x + 2) + log(x + 1)

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Fricas [A]  time = 1.91272, size = 35, normalized size = 3.18 \begin{align*} -\log \left (x + 2\right ) + \log \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x + 2) + log(x + 1)

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Sympy [A]  time = 0.090856, size = 8, normalized size = 0.73 \begin{align*} \log{\left (x + 1 \right )} - \log{\left (x + 2 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + 1) - log(x + 2)

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Giac [A]  time = 1.06296, size = 18, normalized size = 1.64 \begin{align*} -\log \left ({\left | x + 2 \right |}\right ) + \log \left ({\left | x + 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(abs(x + 2)) + log(abs(x + 1))