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

Optimal. Leaf size=113 \[ \frac{x^3}{8 \left (3 x^4+2\right )}+\frac{\log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{32\ 6^{3/4}}-\frac{\log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{32\ 6^{3/4}}-\frac{\tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{16\ 6^{3/4}}+\frac{\tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{16\ 6^{3/4}} \]

[Out]

x^3/(8*(2 + 3*x^4)) - ArcTan[1 - 6^(1/4)*x]/(16*6^(3/4)) + ArcTan[1 + 6^(1/4)*x]/(16*6^(3/4)) + Log[Sqrt[6] -
6^(3/4)*x + 3*x^2]/(32*6^(3/4)) - Log[Sqrt[6] + 6^(3/4)*x + 3*x^2]/(32*6^(3/4))

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Rubi [A]  time = 0.0759393, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {290, 297, 1162, 617, 204, 1165, 628} \[ \frac{x^3}{8 \left (3 x^4+2\right )}+\frac{\log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{32\ 6^{3/4}}-\frac{\log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{32\ 6^{3/4}}-\frac{\tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{16\ 6^{3/4}}+\frac{\tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{16\ 6^{3/4}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(2 + 3*x^4)^2,x]

[Out]

x^3/(8*(2 + 3*x^4)) - ArcTan[1 - 6^(1/4)*x]/(16*6^(3/4)) + ArcTan[1 + 6^(1/4)*x]/(16*6^(3/4)) + Log[Sqrt[6] -
6^(3/4)*x + 3*x^2]/(32*6^(3/4)) - Log[Sqrt[6] + 6^(3/4)*x + 3*x^2]/(32*6^(3/4))

Rule 290

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(
a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[
{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 297

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
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 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

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^2}{\left (2+3 x^4\right )^2} \, dx &=\frac{x^3}{8 \left (2+3 x^4\right )}+\frac{1}{8} \int \frac{x^2}{2+3 x^4} \, dx\\ &=\frac{x^3}{8 \left (2+3 x^4\right )}-\frac{\int \frac{\sqrt{2}-\sqrt{3} x^2}{2+3 x^4} \, dx}{16 \sqrt{3}}+\frac{\int \frac{\sqrt{2}+\sqrt{3} x^2}{2+3 x^4} \, dx}{16 \sqrt{3}}\\ &=\frac{x^3}{8 \left (2+3 x^4\right )}+\frac{1}{96} \int \frac{1}{\sqrt{\frac{2}{3}}-\frac{2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx+\frac{1}{96} \int \frac{1}{\sqrt{\frac{2}{3}}+\frac{2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx+\frac{\int \frac{\frac{2^{3/4}}{\sqrt [4]{3}}+2 x}{-\sqrt{\frac{2}{3}}-\frac{2^{3/4} x}{\sqrt [4]{3}}-x^2} \, dx}{32\ 6^{3/4}}+\frac{\int \frac{\frac{2^{3/4}}{\sqrt [4]{3}}-2 x}{-\sqrt{\frac{2}{3}}+\frac{2^{3/4} x}{\sqrt [4]{3}}-x^2} \, dx}{32\ 6^{3/4}}\\ &=\frac{x^3}{8 \left (2+3 x^4\right )}+\frac{\log \left (\sqrt{6}-6^{3/4} x+3 x^2\right )}{32\ 6^{3/4}}-\frac{\log \left (\sqrt{6}+6^{3/4} x+3 x^2\right )}{32\ 6^{3/4}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt [4]{6} x\right )}{16\ 6^{3/4}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt [4]{6} x\right )}{16\ 6^{3/4}}\\ &=\frac{x^3}{8 \left (2+3 x^4\right )}-\frac{\tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{16\ 6^{3/4}}+\frac{\tan ^{-1}\left (1+\sqrt [4]{6} x\right )}{16\ 6^{3/4}}+\frac{\log \left (\sqrt{6}-6^{3/4} x+3 x^2\right )}{32\ 6^{3/4}}-\frac{\log \left (\sqrt{6}+6^{3/4} x+3 x^2\right )}{32\ 6^{3/4}}\\ \end{align*}

Mathematica [A]  time = 0.0693535, size = 107, normalized size = 0.95 \[ \frac{1}{192} \left (\frac{24 x^3}{3 x^4+2}+\sqrt [4]{6} \log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )-\sqrt [4]{6} \log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )-2 \sqrt [4]{6} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2 \sqrt [4]{6} \tan ^{-1}\left (\sqrt [4]{6} x+1\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(2 + 3*x^4)^2,x]

[Out]

((24*x^3)/(2 + 3*x^4) - 2*6^(1/4)*ArcTan[1 - 6^(1/4)*x] + 2*6^(1/4)*ArcTan[1 + 6^(1/4)*x] + 6^(1/4)*Log[2 - 2*
6^(1/4)*x + Sqrt[6]*x^2] - 6^(1/4)*Log[2 + 2*6^(1/4)*x + Sqrt[6]*x^2])/192

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Maple [A]  time = 0.005, size = 125, normalized size = 1.1 \begin{align*}{\frac{{x}^{3}}{24\,{x}^{4}+16}}+{\frac{\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{576}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) }+{\frac{\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{576}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }+{\frac{\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{1152}\ln \left ({ \left ({x}^{2}-{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) \left ({x}^{2}+{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) ^{-1}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x^3/(3*x^4+2)+1/576*3^(1/2)*6^(3/4)*2^(1/2)*arctan(1/6*2^(1/2)*3^(1/2)*6^(3/4)*x+1)+1/576*3^(1/2)*6^(3/4)*
2^(1/2)*arctan(1/6*2^(1/2)*3^(1/2)*6^(3/4)*x-1)+1/1152*3^(1/2)*6^(3/4)*2^(1/2)*ln((x^2-1/3*3^(1/2)*6^(1/4)*x*2
^(1/2)+1/3*6^(1/2))/(x^2+1/3*3^(1/2)*6^(1/4)*x*2^(1/2)+1/3*6^(1/2)))

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Maxima [A]  time = 1.46434, size = 182, normalized size = 1.61 \begin{align*} \frac{x^{3}}{8 \,{\left (3 \, x^{4} + 2\right )}} + \frac{1}{96} \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x + 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{1}{96} \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x - 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) - \frac{1}{192} \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} \log \left (\sqrt{3} x^{2} + 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) + \frac{1}{192} \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} \log \left (\sqrt{3} x^{2} - 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x^3/(3*x^4 + 2) + 1/96*3^(1/4)*2^(1/4)*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x + 3^(1/4)*2^(3/4))) + 1/96*
3^(1/4)*2^(1/4)*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x - 3^(1/4)*2^(3/4))) - 1/192*3^(1/4)*2^(1/4)*log(sqrt(3
)*x^2 + 3^(1/4)*2^(3/4)*x + sqrt(2)) + 1/192*3^(1/4)*2^(1/4)*log(sqrt(3)*x^2 - 3^(1/4)*2^(3/4)*x + sqrt(2))

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Fricas [B]  time = 1.8375, size = 630, normalized size = 5.58 \begin{align*} -\frac{4 \cdot 54^{\frac{3}{4}} \sqrt{2}{\left (3 \, x^{4} + 2\right )} \arctan \left (-\frac{1}{18} \cdot 54^{\frac{3}{4}} \sqrt{2} x + \frac{1}{54} \cdot 54^{\frac{3}{4}} \sqrt{2} \sqrt{9 \, x^{2} + 3 \cdot 54^{\frac{1}{4}} \sqrt{2} x + 3 \, \sqrt{6}} - 1\right ) + 4 \cdot 54^{\frac{3}{4}} \sqrt{2}{\left (3 \, x^{4} + 2\right )} \arctan \left (-\frac{1}{18} \cdot 54^{\frac{3}{4}} \sqrt{2} x + \frac{1}{54} \cdot 54^{\frac{3}{4}} \sqrt{2} \sqrt{9 \, x^{2} - 3 \cdot 54^{\frac{1}{4}} \sqrt{2} x + 3 \, \sqrt{6}} + 1\right ) + 54^{\frac{3}{4}} \sqrt{2}{\left (3 \, x^{4} + 2\right )} \log \left (9 \, x^{2} + 3 \cdot 54^{\frac{1}{4}} \sqrt{2} x + 3 \, \sqrt{6}\right ) - 54^{\frac{3}{4}} \sqrt{2}{\left (3 \, x^{4} + 2\right )} \log \left (9 \, x^{2} - 3 \cdot 54^{\frac{1}{4}} \sqrt{2} x + 3 \, \sqrt{6}\right ) - 432 \, x^{3}}{3456 \,{\left (3 \, x^{4} + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3456*(4*54^(3/4)*sqrt(2)*(3*x^4 + 2)*arctan(-1/18*54^(3/4)*sqrt(2)*x + 1/54*54^(3/4)*sqrt(2)*sqrt(9*x^2 + 3
*54^(1/4)*sqrt(2)*x + 3*sqrt(6)) - 1) + 4*54^(3/4)*sqrt(2)*(3*x^4 + 2)*arctan(-1/18*54^(3/4)*sqrt(2)*x + 1/54*
54^(3/4)*sqrt(2)*sqrt(9*x^2 - 3*54^(1/4)*sqrt(2)*x + 3*sqrt(6)) + 1) + 54^(3/4)*sqrt(2)*(3*x^4 + 2)*log(9*x^2
+ 3*54^(1/4)*sqrt(2)*x + 3*sqrt(6)) - 54^(3/4)*sqrt(2)*(3*x^4 + 2)*log(9*x^2 - 3*54^(1/4)*sqrt(2)*x + 3*sqrt(6
)) - 432*x^3)/(3*x^4 + 2)

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Sympy [A]  time = 0.565553, size = 97, normalized size = 0.86 \begin{align*} \frac{x^{3}}{24 x^{4} + 16} + \frac{\sqrt [4]{6} \log{\left (x^{2} - \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{192} - \frac{\sqrt [4]{6} \log{\left (x^{2} + \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{192} + \frac{\sqrt [4]{6} \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{96} + \frac{\sqrt [4]{6} \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{96} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**3/(24*x**4 + 16) + 6**(1/4)*log(x**2 - 6**(3/4)*x/3 + sqrt(6)/3)/192 - 6**(1/4)*log(x**2 + 6**(3/4)*x/3 + s
qrt(6)/3)/192 + 6**(1/4)*atan(6**(1/4)*x - 1)/96 + 6**(1/4)*atan(6**(1/4)*x + 1)/96

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Giac [A]  time = 1.15869, size = 147, normalized size = 1.3 \begin{align*} \frac{x^{3}}{8 \,{\left (3 \, x^{4} + 2\right )}} + \frac{1}{96} \cdot 6^{\frac{1}{4}} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{96} \cdot 6^{\frac{1}{4}} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) - \frac{1}{192} \cdot 6^{\frac{1}{4}} \log \left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) + \frac{1}{192} \cdot 6^{\frac{1}{4}} \log \left (x^{2} - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x^3/(3*x^4 + 2) + 1/96*6^(1/4)*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x + sqrt(2)*(2/3)^(1/4))) + 1/96*6^(1/4)*
arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x - sqrt(2)*(2/3)^(1/4))) - 1/192*6^(1/4)*log(x^2 + sqrt(2)*(2/3)^(1/4)*x +
sqrt(2/3)) + 1/192*6^(1/4)*log(x^2 - sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3))