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

Optimal. Leaf size=16 \[ \frac{1}{x}+2 \log (1-x)-2 \log (x) \]

[Out]

x^(-1) + 2*Log[1 - x] - 2*Log[x]

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Rubi [A]  time = 0.0062877, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {77} \[ \frac{1}{x}+2 \log (1-x)-2 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(1 + x)/((-1 + x)*x^2),x]

[Out]

x^(-1) + 2*Log[1 - x] - 2*Log[x]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

Mathematica [A]  time = 0.002357, size = 16, normalized size = 1. \[ \frac{1}{x}+2 \log (1-x)-2 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)/((-1 + x)*x^2),x]

[Out]

x^(-1) + 2*Log[1 - x] - 2*Log[x]

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Maple [A]  time = 0.006, size = 15, normalized size = 0.9 \begin{align*} 2\,\ln \left ( x-1 \right ) +{x}^{-1}-2\,\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.11391, size = 19, normalized size = 1.19 \begin{align*} \frac{1}{x} + 2 \, \log \left (x - 1\right ) - 2 \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/x + 2*log(x - 1) - 2*log(x)

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Fricas [A]  time = 1.20865, size = 50, normalized size = 3.12 \begin{align*} \frac{2 \, x \log \left (x - 1\right ) - 2 \, x \log \left (x\right ) + 1}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*x*log(x - 1) - 2*x*log(x) + 1)/x

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Sympy [A]  time = 0.09292, size = 14, normalized size = 0.88 \begin{align*} - 2 \log{\left (x \right )} + 2 \log{\left (x - 1 \right )} + \frac{1}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*log(x) + 2*log(x - 1) + 1/x

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Giac [A]  time = 1.08165, size = 22, normalized size = 1.38 \begin{align*} \frac{1}{x} + 2 \, \log \left ({\left | x - 1 \right |}\right ) - 2 \, \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^2,x, algorithm="giac")

[Out]

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