3.17 \(\int \frac{x^3}{(-1+x)^2 (1+x^3)} \, dx\)

Optimal. Leaf size=43 \[ -\frac{1}{3} \log \left (x^2-x+1\right )+\frac{1}{2 (1-x)}+\frac{3}{4} \log (1-x)-\frac{1}{12} \log (x+1) \]

[Out]

1/(2*(1 - x)) + (3*Log[1 - x])/4 - Log[1 + x]/12 - Log[1 - x + x^2]/3

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Rubi [A]  time = 0.130239, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6725, 628} \[ -\frac{1}{3} \log \left (x^2-x+1\right )+\frac{1}{2 (1-x)}+\frac{3}{4} \log (1-x)-\frac{1}{12} \log (x+1) \]

Antiderivative was successfully verified.

[In]

Int[x^3/((-1 + x)^2*(1 + x^3)),x]

[Out]

1/(2*(1 - x)) + (3*Log[1 - x])/4 - Log[1 + x]/12 - Log[1 - x + x^2]/3

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{x^3}{(-1+x)^2 \left (1+x^3\right )} \, dx &=\int \left (\frac{1}{2 (-1+x)^2}+\frac{3}{4 (-1+x)}-\frac{1}{12 (1+x)}+\frac{1-2 x}{3 \left (1-x+x^2\right )}\right ) \, dx\\ &=\frac{1}{2 (1-x)}+\frac{3}{4} \log (1-x)-\frac{1}{12} \log (1+x)+\frac{1}{3} \int \frac{1-2 x}{1-x+x^2} \, dx\\ &=\frac{1}{2 (1-x)}+\frac{3}{4} \log (1-x)-\frac{1}{12} \log (1+x)-\frac{1}{3} \log \left (1-x+x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0194789, size = 34, normalized size = 0.79 \[ \frac{1}{12} \left (-\frac{6}{x-1}+9 \log (x-1)-\log (x+1)-4 \log \left ((x-1)^2+x\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/((-1 + x)^2*(1 + x^3)),x]

[Out]

(-6/(-1 + x) + 9*Log[-1 + x] - Log[1 + x] - 4*Log[(-1 + x)^2 + x])/12

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Maple [A]  time = 0.006, size = 32, normalized size = 0.7 \begin{align*} -{\frac{\ln \left ( 1+x \right ) }{12}}-{\frac{\ln \left ({x}^{2}-x+1 \right ) }{3}}-{\frac{1}{2\,x-2}}+{\frac{3\,\ln \left ( -1+x \right ) }{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(-1+x)^2/(x^3+1),x)

[Out]

-1/12*ln(1+x)-1/3*ln(x^2-x+1)-1/2/(-1+x)+3/4*ln(-1+x)

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Maxima [A]  time = 1.4097, size = 42, normalized size = 0.98 \begin{align*} -\frac{1}{2 \,{\left (x - 1\right )}} - \frac{1}{3} \, \log \left (x^{2} - x + 1\right ) - \frac{1}{12} \, \log \left (x + 1\right ) + \frac{3}{4} \, \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(-1+x)^2/(x^3+1),x, algorithm="maxima")

[Out]

-1/2/(x - 1) - 1/3*log(x^2 - x + 1) - 1/12*log(x + 1) + 3/4*log(x - 1)

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Fricas [A]  time = 1.84451, size = 124, normalized size = 2.88 \begin{align*} -\frac{4 \,{\left (x - 1\right )} \log \left (x^{2} - x + 1\right ) +{\left (x - 1\right )} \log \left (x + 1\right ) - 9 \,{\left (x - 1\right )} \log \left (x - 1\right ) + 6}{12 \,{\left (x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(-1+x)^2/(x^3+1),x, algorithm="fricas")

[Out]

-1/12*(4*(x - 1)*log(x^2 - x + 1) + (x - 1)*log(x + 1) - 9*(x - 1)*log(x - 1) + 6)/(x - 1)

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Sympy [A]  time = 0.130973, size = 31, normalized size = 0.72 \begin{align*} \frac{3 \log{\left (x - 1 \right )}}{4} - \frac{\log{\left (x + 1 \right )}}{12} - \frac{\log{\left (x^{2} - x + 1 \right )}}{3} - \frac{1}{2 x - 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(-1+x)**2/(x**3+1),x)

[Out]

3*log(x - 1)/4 - log(x + 1)/12 - log(x**2 - x + 1)/3 - 1/(2*x - 2)

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Giac [A]  time = 1.10436, size = 49, normalized size = 1.14 \begin{align*} -\frac{1}{2 \,{\left (x - 1\right )}} - \frac{1}{3} \, \log \left (\frac{1}{x - 1} + \frac{1}{{\left (x - 1\right )}^{2}} + 1\right ) - \frac{1}{12} \, \log \left ({\left | -\frac{2}{x - 1} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(-1+x)^2/(x^3+1),x, algorithm="giac")

[Out]

-1/2/(x - 1) - 1/3*log(1/(x - 1) + 1/(x - 1)^2 + 1) - 1/12*log(abs(-2/(x - 1) - 1))