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

Optimal. Leaf size=97 \[ -\frac{\log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{8 \sqrt [4]{6}}+\frac{\log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{8 \sqrt [4]{6}}-\frac{\tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac{\tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4 \sqrt [4]{6}} \]

[Out]

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

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Rubi [A]  time = 0.0652577, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {211, 1165, 628, 1162, 617, 204} \[ -\frac{\log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{8 \sqrt [4]{6}}+\frac{\log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{8 \sqrt [4]{6}}-\frac{\tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac{\tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4 \sqrt [4]{6}} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x^4)^(-1),x]

[Out]

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

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

Mathematica [A]  time = 0.0130733, size = 77, normalized size = 0.79 \[ \frac{-\log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )+\log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )-2 \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2 \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{8 \sqrt [4]{6}} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x^4)^(-1),x]

[Out]

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

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Maple [A]  time = 0.002, size = 111, normalized size = 1.1 \begin{align*}{\frac{\sqrt{3}\sqrt [4]{6}\sqrt{2}}{24}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) }+{\frac{\sqrt{3}\sqrt [4]{6}\sqrt{2}}{24}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }+{\frac{\sqrt{3}\sqrt [4]{6}\sqrt{2}}{48}\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),x)

[Out]

1/24*3^(1/2)*6^(1/4)*2^(1/2)*arctan(1/6*2^(1/2)*3^(1/2)*6^(3/4)*x+1)+1/24*3^(1/2)*6^(1/4)*2^(1/2)*arctan(1/6*2
^(1/2)*3^(1/2)*6^(3/4)*x-1)+1/48*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 = 1.55897, size = 163, normalized size = 1.68 \begin{align*} \frac{1}{24} \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}{24} \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}{48} \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}{48} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*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))) + 1/24*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))) + 1/48*3^(3/4)*2^(3/4)*log(sqrt(3)*x^2 + 3^(1/4)*2^(3/4)
*x + sqrt(2)) - 1/48*3^(3/4)*2^(3/4)*log(sqrt(3)*x^2 - 3^(1/4)*2^(3/4)*x + sqrt(2))

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Fricas [B]  time = 1.83039, size = 562, normalized size = 5.79 \begin{align*} -\frac{1}{48} \cdot 24^{\frac{3}{4}} \sqrt{2} \arctan \left (\frac{1}{12} \cdot 24^{\frac{1}{4}} \sqrt{3} \sqrt{2} \sqrt{24^{\frac{3}{4}} \sqrt{2} x + 12 \, x^{2} + 4 \, \sqrt{6}} - \frac{1}{2} \cdot 24^{\frac{1}{4}} \sqrt{2} x - 1\right ) - \frac{1}{48} \cdot 24^{\frac{3}{4}} \sqrt{2} \arctan \left (\frac{1}{12} \cdot 24^{\frac{1}{4}} \sqrt{3} \sqrt{2} \sqrt{-24^{\frac{3}{4}} \sqrt{2} x + 12 \, x^{2} + 4 \, \sqrt{6}} - \frac{1}{2} \cdot 24^{\frac{1}{4}} \sqrt{2} x + 1\right ) + \frac{1}{192} \cdot 24^{\frac{3}{4}} \sqrt{2} \log \left (24^{\frac{3}{4}} \sqrt{2} x + 12 \, x^{2} + 4 \, \sqrt{6}\right ) - \frac{1}{192} \cdot 24^{\frac{3}{4}} \sqrt{2} \log \left (-24^{\frac{3}{4}} \sqrt{2} x + 12 \, x^{2} + 4 \, \sqrt{6}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*24^(3/4)*sqrt(2)*arctan(1/12*24^(1/4)*sqrt(3)*sqrt(2)*sqrt(24^(3/4)*sqrt(2)*x + 12*x^2 + 4*sqrt(6)) - 1/
2*24^(1/4)*sqrt(2)*x - 1) - 1/48*24^(3/4)*sqrt(2)*arctan(1/12*24^(1/4)*sqrt(3)*sqrt(2)*sqrt(-24^(3/4)*sqrt(2)*
x + 12*x^2 + 4*sqrt(6)) - 1/2*24^(1/4)*sqrt(2)*x + 1) + 1/192*24^(3/4)*sqrt(2)*log(24^(3/4)*sqrt(2)*x + 12*x^2
+ 4*sqrt(6)) - 1/192*24^(3/4)*sqrt(2)*log(-24^(3/4)*sqrt(2)*x + 12*x^2 + 4*sqrt(6))

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Sympy [A]  time = 0.503145, size = 87, normalized size = 0.9 \begin{align*} - \frac{6^{\frac{3}{4}} \log{\left (x^{2} - \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{48} + \frac{6^{\frac{3}{4}} \log{\left (x^{2} + \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{48} + \frac{6^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{24} + \frac{6^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{24} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6**(3/4)*log(x**2 - 6**(3/4)*x/3 + sqrt(6)/3)/48 + 6**(3/4)*log(x**2 + 6**(3/4)*x/3 + sqrt(6)/3)/48 + 6**(3/4
)*atan(6**(1/4)*x - 1)/24 + 6**(3/4)*atan(6**(1/4)*x + 1)/24

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Giac [A]  time = 1.18262, size = 128, normalized size = 1.32 \begin{align*} \frac{1}{24} \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}{24} \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}{48} \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}{48} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*6^(3/4)*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x + sqrt(2)*(2/3)^(1/4))) + 1/24*6^(3/4)*arctan(3/4*sqrt(2)*(2/
3)^(3/4)*(2*x - sqrt(2)*(2/3)^(1/4))) + 1/48*6^(3/4)*log(x^2 + sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3)) - 1/48*6^(3/
4)*log(x^2 - sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3))