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

Optimal. Leaf size=32 \[ -\frac {1}{2 (x+1)}+\frac {1}{4} \log (1-x)-\log (x)+\frac {3}{4} \log (x+1) \]

[Out]

-1/2/(1+x)+1/4*ln(1-x)-ln(x)+3/4*ln(1+x)

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {72} \[ -\frac {1}{2 (x+1)}+\frac {1}{4} \log (1-x)-\log (x)+\frac {3}{4} \log (x+1) \]

Antiderivative was successfully verified.

[In]

Int[1/((-1 + x)*x*(1 + x)^2),x]

[Out]

-1/(2*(1 + x)) + Log[1 - x]/4 - Log[x] + (3*Log[1 + x])/4

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 28, normalized size = 0.88 \[ \frac {1}{4} \left (-\frac {2}{x+1}+\log (1-x)-4 \log (x)+3 \log (x+1)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/((-1 + x)*x*(1 + x)^2),x]

[Out]

(-2/(1 + x) + Log[1 - x] - 4*Log[x] + 3*Log[1 + x])/4

________________________________________________________________________________________

fricas [A]  time = 0.44, size = 33, normalized size = 1.03 \[ \frac {3 \, {\left (x + 1\right )} \log \left (x + 1\right ) + {\left (x + 1\right )} \log \left (x - 1\right ) - 4 \, {\left (x + 1\right )} \log \relax (x) - 2}{4 \, {\left (x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(3*(x + 1)*log(x + 1) + (x + 1)*log(x - 1) - 4*(x + 1)*log(x) - 2)/(x + 1)

________________________________________________________________________________________

giac [A]  time = 1.15, size = 34, normalized size = 1.06 \[ -\frac {1}{2 \, {\left (x + 1\right )}} - \log \left ({\left | -\frac {1}{x + 1} + 1 \right |}\right ) + \frac {1}{4} \, \log \left ({\left | -\frac {2}{x + 1} + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(x + 1) - log(abs(-1/(x + 1) + 1)) + 1/4*log(abs(-2/(x + 1) + 1))

________________________________________________________________________________________

maple [A]  time = 0.01, size = 25, normalized size = 0.78 \[ -\ln \relax (x )+\frac {\ln \left (x -1\right )}{4}+\frac {3 \ln \left (x +1\right )}{4}-\frac {1}{2 \left (x +1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(x+1)+3/4*ln(x+1)+1/4*ln(x-1)-ln(x)

________________________________________________________________________________________

maxima [A]  time = 0.45, size = 24, normalized size = 0.75 \[ -\frac {1}{2 \, {\left (x + 1\right )}} + \frac {3}{4} \, \log \left (x + 1\right ) + \frac {1}{4} \, \log \left (x - 1\right ) - \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(x + 1) + 3/4*log(x + 1) + 1/4*log(x - 1) - log(x)

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 26, normalized size = 0.81 \[ \frac {\ln \left (x-1\right )}{4}+\frac {3\,\ln \left (x+1\right )}{4}-\ln \relax (x)-\frac {1}{2\,\left (x+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(x - 1)*(x + 1)^2),x)

[Out]

log(x - 1)/4 + (3*log(x + 1))/4 - log(x) - 1/(2*(x + 1))

________________________________________________________________________________________

sympy [A]  time = 0.15, size = 24, normalized size = 0.75 \[ - \log {\relax (x )} + \frac {\log {\left (x - 1 \right )}}{4} + \frac {3 \log {\left (x + 1 \right )}}{4} - \frac {1}{2 x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x) + log(x - 1)/4 + 3*log(x + 1)/4 - 1/(2*x + 2)

________________________________________________________________________________________