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

Optimal. Leaf size=41 \[ \frac{1}{12} \log \left (3 x^4-2 x^2+1\right )-\frac{\tan ^{-1}\left (\frac{1-3 x^2}{\sqrt{2}}\right )}{6 \sqrt{2}} \]

[Out]

-ArcTan[(1 - 3*x^2)/Sqrt[2]]/(6*Sqrt[2]) + Log[1 - 2*x^2 + 3*x^4]/12

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Rubi [A]  time = 0.0403192, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {1114, 634, 618, 204, 628} \[ \frac{1}{12} \log \left (3 x^4-2 x^2+1\right )-\frac{\tan ^{-1}\left (\frac{1-3 x^2}{\sqrt{2}}\right )}{6 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(1 - 2*x^2 + 3*x^4),x]

[Out]

-ArcTan[(1 - 3*x^2)/Sqrt[2]]/(6*Sqrt[2]) + Log[1 - 2*x^2 + 3*x^4]/12

Rule 1114

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

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{x^3}{1-2 x^2+3 x^4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{1-2 x+3 x^2} \, dx,x,x^2\right )\\ &=\frac{1}{12} \operatorname{Subst}\left (\int \frac{-2+6 x}{1-2 x+3 x^2} \, dx,x,x^2\right )+\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{1-2 x+3 x^2} \, dx,x,x^2\right )\\ &=\frac{1}{12} \log \left (1-2 x^2+3 x^4\right )-\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{-8-x^2} \, dx,x,2 \left (-1+3 x^2\right )\right )\\ &=-\frac{\tan ^{-1}\left (\frac{1-3 x^2}{\sqrt{2}}\right )}{6 \sqrt{2}}+\frac{1}{12} \log \left (1-2 x^2+3 x^4\right )\\ \end{align*}

Mathematica [A]  time = 0.0096231, size = 38, normalized size = 0.93 \[ \frac{1}{12} \left (\log \left (3 x^4-2 x^2+1\right )+\sqrt{2} \tan ^{-1}\left (\frac{3 x^2-1}{\sqrt{2}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(1 - 2*x^2 + 3*x^4),x]

[Out]

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

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Maple [A]  time = 0.003, size = 35, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( 3\,{x}^{4}-2\,{x}^{2}+1 \right ) }{12}}+{\frac{\sqrt{2}}{12}\arctan \left ({\frac{ \left ( 6\,{x}^{2}-2 \right ) \sqrt{2}}{4}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*ln(3*x^4-2*x^2+1)+1/12*2^(1/2)*arctan(1/4*(6*x^2-2)*2^(1/2))

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Maxima [A]  time = 1.40538, size = 46, normalized size = 1.12 \begin{align*} \frac{1}{12} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x^{2} - 1\right )}\right ) + \frac{1}{12} \, \log \left (3 \, x^{4} - 2 \, x^{2} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.97331, size = 103, normalized size = 2.51 \begin{align*} \frac{1}{12} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x^{2} - 1\right )}\right ) + \frac{1}{12} \, \log \left (3 \, x^{4} - 2 \, x^{2} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.109402, size = 42, normalized size = 1.02 \begin{align*} \frac{\log{\left (x^{4} - \frac{2 x^{2}}{3} + \frac{1}{3} \right )}}{12} + \frac{\sqrt{2} \operatorname{atan}{\left (\frac{3 \sqrt{2} x^{2}}{2} - \frac{\sqrt{2}}{2} \right )}}{12} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.04273, size = 46, normalized size = 1.12 \begin{align*} \frac{1}{12} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x^{2} - 1\right )}\right ) + \frac{1}{12} \, \log \left (3 \, x^{4} - 2 \, x^{2} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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