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

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

[Out]

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

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Rubi [A]  time = 0.08, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {1593, 1887, 1874, 31, 634, 618, 204, 628} \[ \frac {x^2}{2}+\frac {1}{2} \log \left (x^2-x+1\right )+\log (x+1)+\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{\sqrt {3}} \]

Antiderivative was successfully verified.

[In]

Int[(2*x^2 + x^4)/(1 + x^3),x]

[Out]

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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 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 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 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 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 1874

Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,
2], q = (a/b)^(1/3)}, Dist[(q*(A - B*q + C*q^2))/(3*a), Int[1/(q + x), x], x] + Dist[q/(3*a), Int[(q*(2*A + B
*q - C*q^2) - (A - B*q - 2*C*q^2)*x)/(q^2 - q*x + x^2), x], x] /; NeQ[a*B^3 - b*A^3, 0] && NeQ[A - B*q + C*q^2
, 0]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2] && GtQ[a/b, 0]

Rule 1887

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^n), x], x] /; FreeQ[{a, b}, x
] && PolyQ[Pq, x] && IntegerQ[n]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 54, normalized size = 1.26 \[ \frac {1}{6} \left (4 \log \left (x^3+1\right )+3 x^2-\log \left (x^2-x+1\right )+2 \log (x+1)-2 \sqrt {3} \tan ^{-1}\left (\frac {2 x-1}{\sqrt {3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(2*x^2 + x^4)/(1 + x^3),x]

[Out]

(3*x^2 - 2*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] + 2*Log[1 + x] - Log[1 - x + x^2] + 4*Log[1 + x^3])/6

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fricas [A]  time = 0.78, size = 37, normalized size = 0.86 \[ \frac {1}{2} \, x^{2} - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{2} \, \log \left (x^{2} - x + 1\right ) + \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.15, size = 38, normalized size = 0.88 \[ \frac {1}{2} \, x^{2} - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{2} \, \log \left (x^{2} - x + 1\right ) + \log \left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 38, normalized size = 0.88 \[ \frac {x^{2}}{2}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{3}+\ln \left (x +1\right )+\frac {\ln \left (x^{2}-x +1\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.86, size = 37, normalized size = 0.86 \[ \frac {1}{2} \, x^{2} - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{2} \, \log \left (x^{2} - x + 1\right ) + \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.10, size = 49, normalized size = 1.14 \[ \ln \left (x+1\right )+\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )-\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )+\frac {x^2}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2 + x^4)/(x^3 + 1),x)

[Out]

log(x + 1) + log(x - (3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/6 + 1/2) - log(x + (3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i
)/6 - 1/2) + x^2/2

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sympy [A]  time = 0.25, size = 44, normalized size = 1.02 \[ \frac {x^{2}}{2} + \log {\left (x + 1 \right )} + \frac {\log {\left (x^{2} - x + 1 \right )}}{2} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3} - \frac {\sqrt {3}}{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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