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

Optimal. Leaf size=9 \[ \log (x)-\log (x+1) \]

[Out]

Log[x] - Log[1 + x]

________________________________________________________________________________________

Rubi [A]  time = 0.0009658, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {36, 29, 31} \[ \log (x)-\log (x+1) \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(1 + x)),x]

[Out]

Log[x] - Log[1 + 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 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

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

Mathematica [A]  time = 0.0020154, size = 9, normalized size = 1. \[ \log (x)-\log (x+1) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(1 + x)),x]

[Out]

Log[x] - Log[1 + x]

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 10, normalized size = 1.1 \begin{align*} \ln \left ( x \right ) -\ln \left ( 1+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(x)-ln(1+x)

________________________________________________________________________________________

Maxima [A]  time = 0.949721, size = 12, normalized size = 1.33 \begin{align*} -\log \left (x + 1\right ) + \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x + 1) + log(x)

________________________________________________________________________________________

Fricas [A]  time = 1.55904, size = 30, normalized size = 3.33 \begin{align*} -\log \left (x + 1\right ) + \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x + 1) + log(x)

________________________________________________________________________________________

Sympy [A]  time = 0.085563, size = 7, normalized size = 0.78 \begin{align*} \log{\left (x \right )} - \log{\left (x + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x) - log(x + 1)

________________________________________________________________________________________

Giac [A]  time = 1.07695, size = 15, normalized size = 1.67 \begin{align*} -\log \left ({\left | x + 1 \right |}\right ) + \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(abs(x + 1)) + log(abs(x))