### 3.162 $$\int \frac{1}{(-1+x) (2+x)} \, dx$$

Optimal. Leaf size=19 $\frac{1}{3} \log (1-x)-\frac{1}{3} \log (x+2)$

[Out]

Log[1 - x]/3 - Log[2 + x]/3

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Rubi [A]  time = 0.0029557, antiderivative size = 19, 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} $\frac{1}{3} \log (1-x)-\frac{1}{3} \log (x+2)$

Antiderivative was successfully veriﬁed.

[In]

Int[1/((-1 + x)*(2 + x)),x]

[Out]

Log[1 - x]/3 - Log[2 + x]/3

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

Mathematica [A]  time = 0.0024767, size = 19, normalized size = 1. $\frac{1}{3} \log (1-x)-\frac{1}{3} \log (x+2)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/((-1 + x)*(2 + x)),x]

[Out]

Log[1 - x]/3 - Log[2 + x]/3

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Maple [A]  time = 0.004, size = 14, normalized size = 0.7 \begin{align*} -{\frac{\ln \left ( 2+x \right ) }{3}}+{\frac{\ln \left ( -1+x \right ) }{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.924871, size = 18, normalized size = 0.95 \begin{align*} -\frac{1}{3} \, \log \left (x + 2\right ) + \frac{1}{3} \, \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]

-1/3*log(x + 2) + 1/3*log(x - 1)

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Fricas [A]  time = 1.75034, size = 46, normalized size = 2.42 \begin{align*} -\frac{1}{3} \, \log \left (x + 2\right ) + \frac{1}{3} \, \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]

-1/3*log(x + 2) + 1/3*log(x - 1)

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Sympy [A]  time = 0.090317, size = 12, normalized size = 0.63 \begin{align*} \frac{\log{\left (x - 1 \right )}}{3} - \frac{\log{\left (x + 2 \right )}}{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x - 1)/3 - log(x + 2)/3

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Giac [A]  time = 1.07272, size = 20, normalized size = 1.05 \begin{align*} -\frac{1}{3} \, \log \left ({\left | x + 2 \right |}\right ) + \frac{1}{3} \, \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]

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