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

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Rubi [A]  time = 0.13, antiderivative size = 43, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.02, 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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fricas [A]  time = 0.42, size = 40, normalized size = 0.93 \[ -\frac {4 \, {\left (x - 1\right )} \log \left (x^{2} - x + 1\right ) + {\left (x - 1\right )} \log \left (x + 1\right ) - 9 \, {\left (x - 1\right )} \log \left (x - 1\right ) + 6}{12 \, {\left (x - 1\right )}} \]

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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giac [A]  time = 1.05, size = 36, normalized size = 0.84 \[ -\frac {1}{2 \, {\left (x - 1\right )}} - \frac {1}{3} \, \log \left (\frac {1}{x - 1} + \frac {1}{{\left (x - 1\right )}^{2}} + 1\right ) - \frac {1}{12} \, \log \left ({\left | -\frac {2}{x - 1} - 1 \right |}\right ) \]

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))

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maple [A]  time = 0.01, size = 32, normalized size = 0.74 \[ \frac {3 \ln \left (x -1\right )}{4}-\frac {\ln \left (x +1\right )}{12}-\frac {\ln \left (x^{2}-x +1\right )}{3}-\frac {1}{2 \left (x -1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.95, size = 31, normalized size = 0.72 \[ -\frac {1}{2 \, {\left (x - 1\right )}} - \frac {1}{3} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{12} \, \log \left (x + 1\right ) + \frac {3}{4} \, \log \left (x - 1\right ) \]

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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mupad [B]  time = 0.04, size = 33, normalized size = 0.77 \[ \frac {3\,\ln \left (x-1\right )}{4}-\frac {\ln \left (x+1\right )}{12}-\frac {\ln \left (x^2-x+1\right )}{3}-\frac {1}{2\,\left (x-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/((x^3 + 1)*(x - 1)^2),x)

[Out]

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

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sympy [A]  time = 0.16, size = 31, normalized size = 0.72 \[ \frac {3 \log {\left (x - 1 \right )}}{4} - \frac {\log {\left (x + 1 \right )}}{12} - \frac {\log {\left (x^{2} - x + 1 \right )}}{3} - \frac {1}{2 x - 2} \]

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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