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

Optimal. Leaf size=28 \[ \frac{1}{2 (x+1)}+\frac{1}{4} \log (1-x)+\frac{3}{4} \log (x+1) \]

[Out]

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

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Rubi [A]  time = 0.0111347, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {27, 88} \[ \frac{1}{2 (x+1)}+\frac{1}{4} \log (1-x)+\frac{3}{4} \log (x+1) \]

Antiderivative was successfully verified.

[In]

Int[x^2/((-1 + x)*(1 + 2*x + x^2)),x]

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

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

Mathematica [A]  time = 0.0141945, size = 22, normalized size = 0.79 \[ \frac{1}{4} \left (\frac{2}{x+1}+\log (x-1)+3 \log (x+1)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/((-1 + x)*(1 + 2*x + x^2)),x]

[Out]

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

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Maple [A]  time = 0.008, size = 21, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( x-1 \right ) }{4}}+{\frac{1}{2+2\,x}}+{\frac{3\,\ln \left ( 1+x \right ) }{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.0184, size = 27, normalized size = 0.96 \begin{align*} \frac{1}{2 \,{\left (x + 1\right )}} + \frac{3}{4} \, \log \left (x + 1\right ) + \frac{1}{4} \, \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.72075, size = 82, normalized size = 2.93 \begin{align*} \frac{3 \,{\left (x + 1\right )} \log \left (x + 1\right ) +{\left (x + 1\right )} \log \left (x - 1\right ) + 2}{4 \,{\left (x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.104101, size = 20, normalized size = 0.71 \begin{align*} \frac{\log{\left (x - 1 \right )}}{4} + \frac{3 \log{\left (x + 1 \right )}}{4} + \frac{1}{2 x + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15438, size = 30, normalized size = 1.07 \begin{align*} \frac{1}{2 \,{\left (x + 1\right )}} + \frac{3}{4} \, \log \left ({\left | x + 1 \right |}\right ) + \frac{1}{4} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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