3.1378 \(\int \frac{1}{2-3 x^6} \, dx\)

Optimal. Leaf size=180 \[ -\frac{\log \left (\sqrt [3]{3} x^2-\sqrt [6]{6} x+\sqrt [3]{2}\right )}{12\ 2^{5/6} \sqrt [6]{3}}+\frac{\log \left (\sqrt [3]{3} x^2+\sqrt [6]{6} x+\sqrt [3]{2}\right )}{12\ 2^{5/6} \sqrt [6]{3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [6]{6}-2 \sqrt [3]{3} x}{\sqrt [6]{2} 3^{2/3}}\right )}{2\ 2^{5/6} 3^{2/3}}+\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{3} x+\sqrt [6]{6}}{\sqrt [6]{2} 3^{2/3}}\right )}{2\ 2^{5/6} 3^{2/3}}+\frac{\tanh ^{-1}\left (\sqrt [6]{\frac{3}{2}} x\right )}{3\ 2^{5/6} \sqrt [6]{3}} \]

[Out]

-ArcTan[(6^(1/6) - 2*3^(1/3)*x)/(2^(1/6)*3^(2/3))]/(2*2^(5/6)*3^(2/3)) + ArcTan[(6^(1/6) + 2*3^(1/3)*x)/(2^(1/
6)*3^(2/3))]/(2*2^(5/6)*3^(2/3)) + ArcTanh[(3/2)^(1/6)*x]/(3*2^(5/6)*3^(1/6)) - Log[2^(1/3) - 6^(1/6)*x + 3^(1
/3)*x^2]/(12*2^(5/6)*3^(1/6)) + Log[2^(1/3) + 6^(1/6)*x + 3^(1/3)*x^2]/(12*2^(5/6)*3^(1/6))

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Rubi [A]  time = 0.279577, antiderivative size = 167, normalized size of antiderivative = 0.93, number of steps used = 10, number of rules used = 6, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {210, 634, 618, 204, 628, 206} \[ -\frac{\log \left (\sqrt [3]{3} x^2-\sqrt [6]{6} x+\sqrt [3]{2}\right )}{12\ 2^{5/6} \sqrt [6]{3}}+\frac{\log \left (\sqrt [3]{3} x^2+\sqrt [6]{6} x+\sqrt [3]{2}\right )}{12\ 2^{5/6} \sqrt [6]{3}}-\frac{\tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2^{5/6} x}{\sqrt [3]{3}}\right )}{2\ 2^{5/6} 3^{2/3}}+\frac{\tan ^{-1}\left (\frac{2^{5/6} x}{\sqrt [3]{3}}+\frac{1}{\sqrt{3}}\right )}{2\ 2^{5/6} 3^{2/3}}+\frac{\tanh ^{-1}\left (\sqrt [6]{\frac{3}{2}} x\right )}{3\ 2^{5/6} \sqrt [6]{3}} \]

Antiderivative was successfully verified.

[In]

Int[(2 - 3*x^6)^(-1),x]

[Out]

-ArcTan[1/Sqrt[3] - (2^(5/6)*x)/3^(1/3)]/(2*2^(5/6)*3^(2/3)) + ArcTan[1/Sqrt[3] + (2^(5/6)*x)/3^(1/3)]/(2*2^(5
/6)*3^(2/3)) + ArcTanh[(3/2)^(1/6)*x]/(3*2^(5/6)*3^(1/6)) - Log[2^(1/3) - 6^(1/6)*x + 3^(1/3)*x^2]/(12*2^(5/6)
*3^(1/6)) + Log[2^(1/3) + 6^(1/6)*x + 3^(1/3)*x^2]/(12*2^(5/6)*3^(1/6))

Rule 210

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[-(a/b), n]], s = Denominator[Rt[-(a/b
), n]], k, u}, Simp[u = Int[(r - s*Cos[(2*k*Pi)/n]*x)/(r^2 - 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x] + Int[(r +
s*Cos[(2*k*Pi)/n]*x)/(r^2 + 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x]; (2*r^2*Int[1/(r^2 - s^2*x^2), x])/(a*n) +
Dist[(2*r)/(a*n), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && NegQ[a/b]

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]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{2-3 x^6} \, dx &=\frac{\int \frac{\sqrt [6]{2}-\frac{\sqrt [6]{3} x}{2}}{\sqrt [3]{2}-\sqrt [6]{6} x+\sqrt [3]{3} x^2} \, dx}{3\ 2^{5/6}}+\frac{\int \frac{\sqrt [6]{2}+\frac{\sqrt [6]{3} x}{2}}{\sqrt [3]{2}+\sqrt [6]{6} x+\sqrt [3]{3} x^2} \, dx}{3\ 2^{5/6}}+\frac{\int \frac{1}{\sqrt [3]{2}-\sqrt [3]{3} x^2} \, dx}{3\ 2^{2/3}}\\ &=\frac{\tanh ^{-1}\left (\sqrt [6]{\frac{3}{2}} x\right )}{3\ 2^{5/6} \sqrt [6]{3}}+\frac{\int \frac{1}{\sqrt [3]{2}-\sqrt [6]{6} x+\sqrt [3]{3} x^2} \, dx}{4\ 2^{2/3}}+\frac{\int \frac{1}{\sqrt [3]{2}+\sqrt [6]{6} x+\sqrt [3]{3} x^2} \, dx}{4\ 2^{2/3}}-\frac{\int \frac{-\sqrt [6]{6}+2 \sqrt [3]{3} x}{\sqrt [3]{2}-\sqrt [6]{6} x+\sqrt [3]{3} x^2} \, dx}{12\ 2^{5/6} \sqrt [6]{3}}+\frac{\int \frac{\sqrt [6]{6}+2 \sqrt [3]{3} x}{\sqrt [3]{2}+\sqrt [6]{6} x+\sqrt [3]{3} x^2} \, dx}{12\ 2^{5/6} \sqrt [6]{3}}\\ &=\frac{\tanh ^{-1}\left (\sqrt [6]{\frac{3}{2}} x\right )}{3\ 2^{5/6} \sqrt [6]{3}}-\frac{\log \left (\sqrt [3]{2}-\sqrt [6]{6} x+\sqrt [3]{3} x^2\right )}{12\ 2^{5/6} \sqrt [6]{3}}+\frac{\log \left (\sqrt [3]{2}+\sqrt [6]{6} x+\sqrt [3]{3} x^2\right )}{12\ 2^{5/6} \sqrt [6]{3}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-2^{5/6} \sqrt [6]{3} x\right )}{2\ 2^{5/6} \sqrt [6]{3}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2^{5/6} \sqrt [6]{3} x\right )}{2\ 2^{5/6} \sqrt [6]{3}}\\ &=-\frac{\tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2^{5/6} x}{\sqrt [3]{3}}\right )}{2\ 2^{5/6} 3^{2/3}}+\frac{\tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{2^{5/6} x}{\sqrt [3]{3}}\right )}{2\ 2^{5/6} 3^{2/3}}+\frac{\tanh ^{-1}\left (\sqrt [6]{\frac{3}{2}} x\right )}{3\ 2^{5/6} \sqrt [6]{3}}-\frac{\log \left (\sqrt [3]{2}-\sqrt [6]{6} x+\sqrt [3]{3} x^2\right )}{12\ 2^{5/6} \sqrt [6]{3}}+\frac{\log \left (\sqrt [3]{2}+\sqrt [6]{6} x+\sqrt [3]{3} x^2\right )}{12\ 2^{5/6} \sqrt [6]{3}}\\ \end{align*}

Mathematica [A]  time = 0.0659047, size = 162, normalized size = 0.9 \[ \frac{\sqrt{3} \left (-\log \left (2^{2/3} \sqrt [3]{3} x^2-2^{5/6} \sqrt [6]{3} x+2\right )+\log \left (2^{2/3} \sqrt [3]{3} x^2+2^{5/6} \sqrt [6]{3} x+2\right )-2 \log \left (2-2^{5/6} \sqrt [6]{3} x\right )+2 \log \left (2^{5/6} \sqrt [6]{3} x+2\right )\right )+6 \tan ^{-1}\left (\frac{2^{5/6} x}{\sqrt [3]{3}}+\frac{1}{\sqrt{3}}\right )+6 \tan ^{-1}\left (\frac{2^{5/6} \sqrt [6]{3} x-1}{\sqrt{3}}\right )}{12\ 2^{5/6} 3^{2/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 - 3*x^6)^(-1),x]

[Out]

(6*ArcTan[1/Sqrt[3] + (2^(5/6)*x)/3^(1/3)] + 6*ArcTan[(-1 + 2^(5/6)*3^(1/6)*x)/Sqrt[3]] + Sqrt[3]*(-2*Log[2 -
2^(5/6)*3^(1/6)*x] + 2*Log[2 + 2^(5/6)*3^(1/6)*x] - Log[2 - 2^(5/6)*3^(1/6)*x + 2^(2/3)*3^(1/3)*x^2] + Log[2 +
2^(5/6)*3^(1/6)*x + 2^(2/3)*3^(1/3)*x^2]))/(12*2^(5/6)*3^(2/3))

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Maple [A]  time = 0.161, size = 228, normalized size = 1.3 \begin{align*} -{\frac{{2}^{{\frac{2}{3}}}\sqrt [3]{3}\sqrt{6}\ln \left ( -x\sqrt{6}\sqrt [3]{12}+{12}^{{\frac{2}{3}}}+6\,{x}^{2} \right ) }{144}}-{\frac{\sqrt [6]{2}\sqrt [3]{3}}{36}\arctan \left ( -{\frac{\sqrt{2}\sqrt{6}}{6}}+{\frac{\sqrt{2}{12}^{{\frac{2}{3}}}x}{6}} \right ) }+{\frac{{2}^{{\frac{5}{6}}}{3}^{{\frac{2}{3}}}{12}^{{\frac{2}{3}}}}{108}\arctan \left ( -{\frac{\sqrt{2}\sqrt{6}}{6}}+{\frac{\sqrt{2}{12}^{{\frac{2}{3}}}x}{6}} \right ) }+{\frac{{2}^{{\frac{2}{3}}}\sqrt [3]{3}\sqrt{6}\ln \left ( x\sqrt{6}\sqrt [3]{12}+{12}^{{\frac{2}{3}}}+6\,{x}^{2} \right ) }{144}}-{\frac{\sqrt [6]{2}\sqrt [3]{3}}{36}\arctan \left ({\frac{\sqrt{2}\sqrt{6}}{6}}+{\frac{\sqrt{2}{12}^{{\frac{2}{3}}}x}{6}} \right ) }+{\frac{{2}^{{\frac{5}{6}}}{3}^{{\frac{2}{3}}}{12}^{{\frac{2}{3}}}}{108}\arctan \left ({\frac{\sqrt{2}\sqrt{6}}{6}}+{\frac{\sqrt{2}{12}^{{\frac{2}{3}}}x}{6}} \right ) }-{\frac{\sqrt{6}\sqrt [3]{3}{2}^{{\frac{2}{3}}}\ln \left ( -\sqrt{6}\sqrt [3]{3}{2}^{{\frac{2}{3}}}+6\,x \right ) }{72}}+{\frac{\sqrt{6}\sqrt [3]{3}{2}^{{\frac{2}{3}}}\ln \left ( \sqrt{6}\sqrt [3]{3}{2}^{{\frac{2}{3}}}+6\,x \right ) }{72}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-3*x^6+2),x)

[Out]

-1/144*2^(2/3)*3^(1/3)*6^(1/2)*ln(-x*6^(1/2)*12^(1/3)+12^(2/3)+6*x^2)-1/36*2^(1/6)*3^(1/3)*arctan(-1/6*2^(1/2)
*6^(1/2)+1/6*2^(1/2)*12^(2/3)*x)+1/108*2^(5/6)*3^(2/3)*12^(2/3)*arctan(-1/6*2^(1/2)*6^(1/2)+1/6*2^(1/2)*12^(2/
3)*x)+1/144*2^(2/3)*3^(1/3)*6^(1/2)*ln(x*6^(1/2)*12^(1/3)+12^(2/3)+6*x^2)-1/36*2^(1/6)*3^(1/3)*arctan(1/6*2^(1
/2)*6^(1/2)+1/6*2^(1/2)*12^(2/3)*x)+1/108*2^(5/6)*3^(2/3)*12^(2/3)*arctan(1/6*2^(1/2)*6^(1/2)+1/6*2^(1/2)*12^(
2/3)*x)-1/72*6^(1/2)*3^(1/3)*2^(2/3)*ln(-6^(1/2)*3^(1/3)*2^(2/3)+6*x)+1/72*6^(1/2)*3^(1/3)*2^(2/3)*ln(6^(1/2)*
3^(1/3)*2^(2/3)+6*x)

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Maxima [A]  time = 1.5812, size = 302, normalized size = 1.68 \begin{align*} \frac{1}{12} \cdot 3^{\frac{2}{3}} 2^{\frac{1}{6}} \left (\frac{1}{3}\right )^{\frac{1}{3}} \arctan \left (\frac{1}{2} \cdot 3^{\frac{1}{3}} 2^{\frac{5}{6}} \left (\frac{1}{3}\right )^{\frac{2}{3}}{\left (2 \, x + \left (\frac{1}{3}\right )^{\frac{1}{3}} \left (\sqrt{3} \sqrt{2}\right )^{\frac{1}{3}}\right )}\right ) + \frac{1}{12} \cdot 3^{\frac{2}{3}} 2^{\frac{1}{6}} \left (\frac{1}{3}\right )^{\frac{1}{3}} \arctan \left (\frac{1}{2} \cdot 3^{\frac{1}{3}} 2^{\frac{5}{6}} \left (\frac{1}{3}\right )^{\frac{2}{3}}{\left (2 \, x - \left (\frac{1}{3}\right )^{\frac{1}{3}} \left (\sqrt{3} \sqrt{2}\right )^{\frac{1}{3}}\right )}\right ) + \frac{1}{24} \cdot 3^{\frac{1}{6}} 2^{\frac{1}{6}} \left (\frac{1}{3}\right )^{\frac{1}{3}} \log \left (x^{2} + \left (\frac{1}{3}\right )^{\frac{1}{3}} \left (\sqrt{3} \sqrt{2}\right )^{\frac{1}{3}} x + \left (\frac{1}{3}\right )^{\frac{2}{3}} \left (\sqrt{3} \sqrt{2}\right )^{\frac{2}{3}}\right ) - \frac{1}{24} \cdot 3^{\frac{1}{6}} 2^{\frac{1}{6}} \left (\frac{1}{3}\right )^{\frac{1}{3}} \log \left (x^{2} - \left (\frac{1}{3}\right )^{\frac{1}{3}} \left (\sqrt{3} \sqrt{2}\right )^{\frac{1}{3}} x + \left (\frac{1}{3}\right )^{\frac{2}{3}} \left (\sqrt{3} \sqrt{2}\right )^{\frac{2}{3}}\right ) + \frac{1}{12} \cdot 3^{\frac{1}{6}} 2^{\frac{1}{6}} \left (\frac{1}{3}\right )^{\frac{1}{3}} \log \left (x + \left (\frac{1}{3}\right )^{\frac{1}{3}} \left (\sqrt{3} \sqrt{2}\right )^{\frac{1}{3}}\right ) - \frac{1}{12} \cdot 3^{\frac{1}{6}} 2^{\frac{1}{6}} \left (\frac{1}{3}\right )^{\frac{1}{3}} \log \left (x - \left (\frac{1}{3}\right )^{\frac{1}{3}} \left (\sqrt{3} \sqrt{2}\right )^{\frac{1}{3}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x^6+2),x, algorithm="maxima")

[Out]

1/12*3^(2/3)*2^(1/6)*(1/3)^(1/3)*arctan(1/2*3^(1/3)*2^(5/6)*(1/3)^(2/3)*(2*x + (1/3)^(1/3)*(sqrt(3)*sqrt(2))^(
1/3))) + 1/12*3^(2/3)*2^(1/6)*(1/3)^(1/3)*arctan(1/2*3^(1/3)*2^(5/6)*(1/3)^(2/3)*(2*x - (1/3)^(1/3)*(sqrt(3)*s
qrt(2))^(1/3))) + 1/24*3^(1/6)*2^(1/6)*(1/3)^(1/3)*log(x^2 + (1/3)^(1/3)*(sqrt(3)*sqrt(2))^(1/3)*x + (1/3)^(2/
3)*(sqrt(3)*sqrt(2))^(2/3)) - 1/24*3^(1/6)*2^(1/6)*(1/3)^(1/3)*log(x^2 - (1/3)^(1/3)*(sqrt(3)*sqrt(2))^(1/3)*x
+ (1/3)^(2/3)*(sqrt(3)*sqrt(2))^(2/3)) + 1/12*3^(1/6)*2^(1/6)*(1/3)^(1/3)*log(x + (1/3)^(1/3)*(sqrt(3)*sqrt(2
))^(1/3)) - 1/12*3^(1/6)*2^(1/6)*(1/3)^(1/3)*log(x - (1/3)^(1/3)*(sqrt(3)*sqrt(2))^(1/3))

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Fricas [A]  time = 1.56167, size = 594, normalized size = 3.3 \begin{align*} -\frac{1}{288} \cdot 96^{\frac{5}{6}} \sqrt{3} \arctan \left (-\frac{1}{3} \cdot 96^{\frac{1}{6}} \sqrt{3} x + \frac{1}{12} \cdot 96^{\frac{1}{6}} \sqrt{48 \, x^{2} + 96^{\frac{5}{6}} x + 8 \cdot 12^{\frac{2}{3}}} - \frac{1}{3} \, \sqrt{3}\right ) - \frac{1}{288} \cdot 96^{\frac{5}{6}} \sqrt{3} \arctan \left (-\frac{1}{3} \cdot 96^{\frac{1}{6}} \sqrt{3} x + \frac{1}{12} \cdot 96^{\frac{1}{6}} \sqrt{48 \, x^{2} - 96^{\frac{5}{6}} x + 8 \cdot 12^{\frac{2}{3}}} + \frac{1}{3} \, \sqrt{3}\right ) + \frac{1}{1152} \cdot 96^{\frac{5}{6}} \log \left (48 \, x^{2} + 96^{\frac{5}{6}} x + 8 \cdot 12^{\frac{2}{3}}\right ) - \frac{1}{1152} \cdot 96^{\frac{5}{6}} \log \left (48 \, x^{2} - 96^{\frac{5}{6}} x + 8 \cdot 12^{\frac{2}{3}}\right ) + \frac{1}{576} \cdot 96^{\frac{5}{6}} \log \left (48 \, x + 96^{\frac{5}{6}}\right ) - \frac{1}{576} \cdot 96^{\frac{5}{6}} \log \left (48 \, x - 96^{\frac{5}{6}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x^6+2),x, algorithm="fricas")

[Out]

-1/288*96^(5/6)*sqrt(3)*arctan(-1/3*96^(1/6)*sqrt(3)*x + 1/12*96^(1/6)*sqrt(48*x^2 + 96^(5/6)*x + 8*12^(2/3))
- 1/3*sqrt(3)) - 1/288*96^(5/6)*sqrt(3)*arctan(-1/3*96^(1/6)*sqrt(3)*x + 1/12*96^(1/6)*sqrt(48*x^2 - 96^(5/6)*
x + 8*12^(2/3)) + 1/3*sqrt(3)) + 1/1152*96^(5/6)*log(48*x^2 + 96^(5/6)*x + 8*12^(2/3)) - 1/1152*96^(5/6)*log(4
8*x^2 - 96^(5/6)*x + 8*12^(2/3)) + 1/576*96^(5/6)*log(48*x + 96^(5/6)) - 1/576*96^(5/6)*log(48*x - 96^(5/6))

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Sympy [A]  time = 0.541419, size = 15, normalized size = 0.08 \begin{align*} - \operatorname{RootSum}{\left (4478976 t^{6} - 1, \left ( t \mapsto t \log{\left (- 12 t + x \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x**6+2),x)

[Out]

-RootSum(4478976*_t**6 - 1, Lambda(_t, _t*log(-12*_t + x)))

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x^6+2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError