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

Optimal. Leaf size=131 \[ \frac{x}{8 \left (3 x^4+2\right )}-\frac{3^{3/4} \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{64 \sqrt [4]{2}}+\frac{3^{3/4} \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{64 \sqrt [4]{2}}-\frac{3^{3/4} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{32 \sqrt [4]{2}}+\frac{3^{3/4} \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{32 \sqrt [4]{2}} \]

[Out]

x/(8*(2 + 3*x^4)) - (3^(3/4)*ArcTan[1 - 6^(1/4)*x])/(32*2^(1/4)) + (3^(3/4)*ArcTan[1 + 6^(1/4)*x])/(32*2^(1/4)
) - (3^(3/4)*Log[Sqrt[6] - 6^(3/4)*x + 3*x^2])/(64*2^(1/4)) + (3^(3/4)*Log[Sqrt[6] + 6^(3/4)*x + 3*x^2])/(64*2
^(1/4))

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Rubi [A]  time = 0.0739224, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.778, Rules used = {199, 211, 1165, 628, 1162, 617, 204} \[ \frac{x}{8 \left (3 x^4+2\right )}-\frac{3^{3/4} \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{64 \sqrt [4]{2}}+\frac{3^{3/4} \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{64 \sqrt [4]{2}}-\frac{3^{3/4} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{32 \sqrt [4]{2}}+\frac{3^{3/4} \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{32 \sqrt [4]{2}} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x^4)^(-2),x]

[Out]

x/(8*(2 + 3*x^4)) - (3^(3/4)*ArcTan[1 - 6^(1/4)*x])/(32*2^(1/4)) + (3^(3/4)*ArcTan[1 + 6^(1/4)*x])/(32*2^(1/4)
) - (3^(3/4)*Log[Sqrt[6] - 6^(3/4)*x + 3*x^2])/(64*2^(1/4)) + (3^(3/4)*Log[Sqrt[6] + 6^(3/4)*x + 3*x^2])/(64*2
^(1/4))

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 211

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

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]

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])

Rubi steps

\begin{align*} \int \frac{1}{\left (2+3 x^4\right )^2} \, dx &=\frac{x}{8 \left (2+3 x^4\right )}+\frac{3}{8} \int \frac{1}{2+3 x^4} \, dx\\ &=\frac{x}{8 \left (2+3 x^4\right )}+\frac{3 \int \frac{\sqrt{2}-\sqrt{3} x^2}{2+3 x^4} \, dx}{16 \sqrt{2}}+\frac{3 \int \frac{\sqrt{2}+\sqrt{3} x^2}{2+3 x^4} \, dx}{16 \sqrt{2}}\\ &=\frac{x}{8 \left (2+3 x^4\right )}+\frac{1}{32} \sqrt{\frac{3}{2}} \int \frac{1}{\sqrt{\frac{2}{3}}-\frac{2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx+\frac{1}{32} \sqrt{\frac{3}{2}} \int \frac{1}{\sqrt{\frac{2}{3}}+\frac{2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx-\frac{3^{3/4} \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}{64 \sqrt [4]{2}}-\frac{3^{3/4} \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}{64 \sqrt [4]{2}}\\ &=\frac{x}{8 \left (2+3 x^4\right )}-\frac{3^{3/4} \log \left (\sqrt{6}-6^{3/4} x+3 x^2\right )}{64 \sqrt [4]{2}}+\frac{3^{3/4} \log \left (\sqrt{6}+6^{3/4} x+3 x^2\right )}{64 \sqrt [4]{2}}+\frac{3^{3/4} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt [4]{6} x\right )}{32 \sqrt [4]{2}}-\frac{3^{3/4} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt [4]{6} x\right )}{32 \sqrt [4]{2}}\\ &=\frac{x}{8 \left (2+3 x^4\right )}-\frac{3^{3/4} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{32 \sqrt [4]{2}}+\frac{3^{3/4} \tan ^{-1}\left (1+\sqrt [4]{6} x\right )}{32 \sqrt [4]{2}}-\frac{3^{3/4} \log \left (\sqrt{6}-6^{3/4} x+3 x^2\right )}{64 \sqrt [4]{2}}+\frac{3^{3/4} \log \left (\sqrt{6}+6^{3/4} x+3 x^2\right )}{64 \sqrt [4]{2}}\\ \end{align*}

Mathematica [A]  time = 0.0546884, size = 105, normalized size = 0.8 \[ \frac{1}{128} \left (\frac{16 x}{3 x^4+2}-6^{3/4} \log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )+6^{3/4} \log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )-2\ 6^{3/4} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2\ 6^{3/4} \tan ^{-1}\left (\sqrt [4]{6} x+1\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x^4)^(-2),x]

[Out]

((16*x)/(2 + 3*x^4) - 2*6^(3/4)*ArcTan[1 - 6^(1/4)*x] + 2*6^(3/4)*ArcTan[1 + 6^(1/4)*x] - 6^(3/4)*Log[2 - 2*6^
(1/4)*x + Sqrt[6]*x^2] + 6^(3/4)*Log[2 + 2*6^(1/4)*x + Sqrt[6]*x^2])/128

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Maple [A]  time = 0.004, size = 123, normalized size = 0.9 \begin{align*}{\frac{x}{24\,{x}^{4}+16}}+{\frac{\sqrt{3}\sqrt [4]{6}\sqrt{2}}{64}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) }+{\frac{\sqrt{3}\sqrt [4]{6}\sqrt{2}}{64}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }+{\frac{\sqrt{3}\sqrt [4]{6}\sqrt{2}}{128}\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(1/(3*x^4+2)^2,x)

[Out]

1/8*x/(3*x^4+2)+1/64*3^(1/2)*6^(1/4)*2^(1/2)*arctan(1/6*2^(1/2)*3^(1/2)*6^(3/4)*x+1)+1/64*3^(1/2)*6^(1/4)*2^(1
/2)*arctan(1/6*2^(1/2)*3^(1/2)*6^(3/4)*x-1)+1/128*3^(1/2)*6^(1/4)*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 = 1.53666, size = 180, normalized size = 1.37 \begin{align*} \frac{1}{64} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{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}{64} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{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}{128} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} \log \left (\sqrt{3} x^{2} + 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) - \frac{1}{128} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} \log \left (\sqrt{3} x^{2} - 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) + \frac{x}{8 \,{\left (3 \, x^{4} + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*3^(3/4)*2^(3/4)*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x + 3^(1/4)*2^(3/4))) + 1/64*3^(3/4)*2^(3/4)*arctan
(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x - 3^(1/4)*2^(3/4))) + 1/128*3^(3/4)*2^(3/4)*log(sqrt(3)*x^2 + 3^(1/4)*2^(3/4
)*x + sqrt(2)) - 1/128*3^(3/4)*2^(3/4)*log(sqrt(3)*x^2 - 3^(1/4)*2^(3/4)*x + sqrt(2)) + 1/8*x/(3*x^4 + 2)

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Fricas [B]  time = 1.8388, size = 814, normalized size = 6.21 \begin{align*} -\frac{4 \cdot 27^{\frac{1}{4}} 8^{\frac{3}{4}} \sqrt{2}{\left (3 \, x^{4} + 2\right )} \arctan \left (-\frac{1}{18} \cdot 27^{\frac{3}{4}} 8^{\frac{1}{4}} \sqrt{2} x + \frac{1}{108} \cdot 27^{\frac{3}{4}} 8^{\frac{1}{4}} \sqrt{2} \sqrt{3 \cdot 27^{\frac{1}{4}} 8^{\frac{3}{4}} \sqrt{2} x + 36 \, x^{2} + 12 \, \sqrt{3} \sqrt{2}} - 1\right ) + 4 \cdot 27^{\frac{1}{4}} 8^{\frac{3}{4}} \sqrt{2}{\left (3 \, x^{4} + 2\right )} \arctan \left (-\frac{1}{18} \cdot 27^{\frac{3}{4}} 8^{\frac{1}{4}} \sqrt{2} x + \frac{1}{108} \cdot 27^{\frac{3}{4}} 8^{\frac{1}{4}} \sqrt{2} \sqrt{-3 \cdot 27^{\frac{1}{4}} 8^{\frac{3}{4}} \sqrt{2} x + 36 \, x^{2} + 12 \, \sqrt{3} \sqrt{2}} + 1\right ) - 27^{\frac{1}{4}} 8^{\frac{3}{4}} \sqrt{2}{\left (3 \, x^{4} + 2\right )} \log \left (3 \cdot 27^{\frac{1}{4}} 8^{\frac{3}{4}} \sqrt{2} x + 36 \, x^{2} + 12 \, \sqrt{3} \sqrt{2}\right ) + 27^{\frac{1}{4}} 8^{\frac{3}{4}} \sqrt{2}{\left (3 \, x^{4} + 2\right )} \log \left (-3 \cdot 27^{\frac{1}{4}} 8^{\frac{3}{4}} \sqrt{2} x + 36 \, x^{2} + 12 \, \sqrt{3} \sqrt{2}\right ) - 64 \, x}{512 \,{\left (3 \, x^{4} + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/512*(4*27^(1/4)*8^(3/4)*sqrt(2)*(3*x^4 + 2)*arctan(-1/18*27^(3/4)*8^(1/4)*sqrt(2)*x + 1/108*27^(3/4)*8^(1/4
)*sqrt(2)*sqrt(3*27^(1/4)*8^(3/4)*sqrt(2)*x + 36*x^2 + 12*sqrt(3)*sqrt(2)) - 1) + 4*27^(1/4)*8^(3/4)*sqrt(2)*(
3*x^4 + 2)*arctan(-1/18*27^(3/4)*8^(1/4)*sqrt(2)*x + 1/108*27^(3/4)*8^(1/4)*sqrt(2)*sqrt(-3*27^(1/4)*8^(3/4)*s
qrt(2)*x + 36*x^2 + 12*sqrt(3)*sqrt(2)) + 1) - 27^(1/4)*8^(3/4)*sqrt(2)*(3*x^4 + 2)*log(3*27^(1/4)*8^(3/4)*sqr
t(2)*x + 36*x^2 + 12*sqrt(3)*sqrt(2)) + 27^(1/4)*8^(3/4)*sqrt(2)*(3*x^4 + 2)*log(-3*27^(1/4)*8^(3/4)*sqrt(2)*x
+ 36*x^2 + 12*sqrt(3)*sqrt(2)) - 64*x)/(3*x^4 + 2)

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Sympy [A]  time = 0.530967, size = 95, normalized size = 0.73 \begin{align*} \frac{x}{24 x^{4} + 16} - \frac{6^{\frac{3}{4}} \log{\left (x^{2} - \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{128} + \frac{6^{\frac{3}{4}} \log{\left (x^{2} + \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{128} + \frac{6^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{64} + \frac{6^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{64} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(24*x**4 + 16) - 6**(3/4)*log(x**2 - 6**(3/4)*x/3 + sqrt(6)/3)/128 + 6**(3/4)*log(x**2 + 6**(3/4)*x/3 + sqrt
(6)/3)/128 + 6**(3/4)*atan(6**(1/4)*x - 1)/64 + 6**(3/4)*atan(6**(1/4)*x + 1)/64

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Giac [A]  time = 1.18108, size = 144, normalized size = 1.1 \begin{align*} \frac{1}{64} \cdot 6^{\frac{3}{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}{64} \cdot 6^{\frac{3}{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}{128} \cdot 6^{\frac{3}{4}} \log \left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) - \frac{1}{128} \cdot 6^{\frac{3}{4}} \log \left (x^{2} - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) + \frac{x}{8 \,{\left (3 \, x^{4} + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*6^(3/4)*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x + sqrt(2)*(2/3)^(1/4))) + 1/64*6^(3/4)*arctan(3/4*sqrt(2)*(2/
3)^(3/4)*(2*x - sqrt(2)*(2/3)^(1/4))) + 1/128*6^(3/4)*log(x^2 + sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3)) - 1/128*6^(
3/4)*log(x^2 - sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3)) + 1/8*x/(3*x^4 + 2)