3.275 \(\int \frac{1+2 x-x^2+8 x^3+x^4}{(x+x^2) (1+x^3)} \, dx\)

Optimal. Leaf size=44 \[ \log \left (x^2-x+1\right )-\frac{3}{x+1}+\log (x)-2 \log (x+1)-\frac{2 \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

-3/(1 + x) - (2*ArcTan[(1 - 2*x)/Sqrt[3]])/Sqrt[3] + Log[x] - 2*Log[1 + x] + Log[1 - x + x^2]

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Rubi [A]  time = 0.242529, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {1593, 6725, 634, 618, 204, 628} \[ \log \left (x^2-x+1\right )-\frac{3}{x+1}+\log (x)-2 \log (x+1)-\frac{2 \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + 2*x - x^2 + 8*x^3 + x^4)/((x + x^2)*(1 + x^3)),x]

[Out]

-3/(1 + x) - (2*ArcTan[(1 - 2*x)/Sqrt[3]])/Sqrt[3] + Log[x] - 2*Log[1 + x] + Log[1 - x + x^2]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 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{1+2 x-x^2+8 x^3+x^4}{\left (x+x^2\right ) \left (1+x^3\right )} \, dx &=\int \frac{1+2 x-x^2+8 x^3+x^4}{x (1+x) \left (1+x^3\right )} \, dx\\ &=\int \left (\frac{1}{x}+\frac{3}{(1+x)^2}-\frac{2}{1+x}+\frac{2 x}{1-x+x^2}\right ) \, dx\\ &=-\frac{3}{1+x}+\log (x)-2 \log (1+x)+2 \int \frac{x}{1-x+x^2} \, dx\\ &=-\frac{3}{1+x}+\log (x)-2 \log (1+x)+\int \frac{1}{1-x+x^2} \, dx+\int \frac{-1+2 x}{1-x+x^2} \, dx\\ &=-\frac{3}{1+x}+\log (x)-2 \log (1+x)+\log \left (1-x+x^2\right )-2 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 x\right )\\ &=-\frac{3}{1+x}-\frac{2 \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{\sqrt{3}}+\log (x)-2 \log (1+x)+\log \left (1-x+x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0249415, size = 44, normalized size = 1. \[ \log \left (x^2-x+1\right )-\frac{3}{x+1}+\log (x)-2 \log (x+1)+\frac{2 \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + 2*x - x^2 + 8*x^3 + x^4)/((x + x^2)*(1 + x^3)),x]

[Out]

-3/(1 + x) + (2*ArcTan[(-1 + 2*x)/Sqrt[3]])/Sqrt[3] + Log[x] - 2*Log[1 + x] + Log[1 - x + x^2]

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Maple [A]  time = 0.008, size = 42, normalized size = 1. \begin{align*} \ln \left ( x \right ) +\ln \left ({x}^{2}-x+1 \right ) +{\frac{2\,\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }-3\, \left ( 1+x \right ) ^{-1}-2\,\ln \left ( 1+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4+8*x^3-x^2+2*x+1)/(x^2+x)/(x^3+1),x)

[Out]

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

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Maxima [A]  time = 1.48511, size = 55, normalized size = 1.25 \begin{align*} \frac{2}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{3}{x + 1} + \log \left (x^{2} - x + 1\right ) - 2 \, \log \left (x + 1\right ) + \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+8*x^3-x^2+2*x+1)/(x^2+x)/(x^3+1),x, algorithm="maxima")

[Out]

2/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) - 3/(x + 1) + log(x^2 - x + 1) - 2*log(x + 1) + log(x)

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Fricas [A]  time = 1.41565, size = 186, normalized size = 4.23 \begin{align*} \frac{2 \, \sqrt{3}{\left (x + 1\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + 3 \,{\left (x + 1\right )} \log \left (x^{2} - x + 1\right ) - 6 \,{\left (x + 1\right )} \log \left (x + 1\right ) + 3 \,{\left (x + 1\right )} \log \left (x\right ) - 9}{3 \,{\left (x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+8*x^3-x^2+2*x+1)/(x^2+x)/(x^3+1),x, algorithm="fricas")

[Out]

1/3*(2*sqrt(3)*(x + 1)*arctan(1/3*sqrt(3)*(2*x - 1)) + 3*(x + 1)*log(x^2 - x + 1) - 6*(x + 1)*log(x + 1) + 3*(
x + 1)*log(x) - 9)/(x + 1)

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Sympy [A]  time = 0.186507, size = 49, normalized size = 1.11 \begin{align*} \log{\left (x \right )} - 2 \log{\left (x + 1 \right )} + \log{\left (x^{2} - x + 1 \right )} + \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right )}}{3} - \frac{3}{x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4+8*x**3-x**2+2*x+1)/(x**2+x)/(x**3+1),x)

[Out]

log(x) - 2*log(x + 1) + log(x**2 - x + 1) + 2*sqrt(3)*atan(2*sqrt(3)*x/3 - sqrt(3)/3)/3 - 3/(x + 1)

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Giac [A]  time = 1.20969, size = 58, normalized size = 1.32 \begin{align*} \frac{2}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{3}{x + 1} + \log \left (x^{2} - x + 1\right ) - 2 \, \log \left ({\left | x + 1 \right |}\right ) + \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+8*x^3-x^2+2*x+1)/(x^2+x)/(x^3+1),x, algorithm="giac")

[Out]

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