Integrand size = 9, antiderivative size = 131 \[ \int \frac {1}{2-3 x^6} \, dx=-\frac {\arctan \left (\frac {1}{\sqrt {3}}-\frac {2^{5/6} x}{\sqrt [3]{3}}\right )}{2\ 2^{5/6} 3^{2/3}}+\frac {\arctan \left (\frac {1}{\sqrt {3}}+\frac {2^{5/6} x}{\sqrt [3]{3}}\right )}{2\ 2^{5/6} 3^{2/3}}+\frac {\text {arctanh}\left (\sqrt [6]{\frac {3}{2}} x\right )}{3\ 2^{5/6} \sqrt [6]{3}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{6} x}{\sqrt [3]{2}+\sqrt [3]{3} x^2}\right )}{6\ 2^{5/6} \sqrt [6]{3}} \] Output:
1/12*arctan(-1/3*3^(1/2)+1/3*2^(5/6)*x*3^(2/3))*2^(1/6)*3^(1/3)+1/12*arcta n(1/3*3^(1/2)+1/3*2^(5/6)*x*3^(2/3))*2^(1/6)*3^(1/3)+1/18*arctanh(1/2*3^(1 /6)*2^(5/6)*x)*3^(5/6)*2^(1/6)+1/36*arctanh(6^(1/6)*x/(2^(1/3)+3^(1/3)*x^2 ))*3^(5/6)*2^(1/6)
Time = 0.05 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.24 \[ \int \frac {1}{2-3 x^6} \, dx=\frac {6 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2^{5/6} x}{\sqrt [3]{3}}\right )+6 \arctan \left (\frac {-1+2^{5/6} \sqrt [6]{3} x}{\sqrt {3}}\right )+\sqrt {3} \left (-2 \log \left (2-2^{5/6} \sqrt [6]{3} x\right )+2 \log \left (2+2^{5/6} \sqrt [6]{3} x\right )-\log \left (2-2^{5/6} \sqrt [6]{3} x+2^{2/3} \sqrt [3]{3} x^2\right )+\log \left (2+2^{5/6} \sqrt [6]{3} x+2^{2/3} \sqrt [3]{3} x^2\right )\right )}{12\ 2^{5/6} 3^{2/3}} \] Input:
Integrate[(2 - 3*x^6)^(-1),x]
Output:
(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))
Time = 0.57 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.27, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.889, Rules used = {754, 27, 219, 1142, 25, 1082, 217, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{2-3 x^6} \, dx\) |
| \(\Big \downarrow \) 754 |
| \(\displaystyle \frac {\int \frac {1}{\sqrt [3]{2}-\sqrt [3]{3} x^2}dx}{3\ 2^{2/3}}+\frac {\int \frac {2 \sqrt [6]{2}-\sqrt [6]{3} x}{2 \left (\sqrt [3]{3} x^2-\sqrt [6]{6} x+\sqrt [3]{2}\right )}dx}{3\ 2^{5/6}}+\frac {\int \frac {\sqrt [6]{3} x+2 \sqrt [6]{2}}{2 \left (\sqrt [3]{3} x^2+\sqrt [6]{6} x+\sqrt [3]{2}\right )}dx}{3\ 2^{5/6}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {1}{\sqrt [3]{2}-\sqrt [3]{3} x^2}dx}{3\ 2^{2/3}}+\frac {\int \frac {2 \sqrt [6]{2}-\sqrt [6]{3} x}{\sqrt [3]{3} x^2-\sqrt [6]{6} x+\sqrt [3]{2}}dx}{6\ 2^{5/6}}+\frac {\int \frac {\sqrt [6]{3} x+2 \sqrt [6]{2}}{\sqrt [3]{3} x^2+\sqrt [6]{6} x+\sqrt [3]{2}}dx}{6\ 2^{5/6}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\int \frac {2 \sqrt [6]{2}-\sqrt [6]{3} x}{\sqrt [3]{3} x^2-\sqrt [6]{6} x+\sqrt [3]{2}}dx}{6\ 2^{5/6}}+\frac {\int \frac {\sqrt [6]{3} x+2 \sqrt [6]{2}}{\sqrt [3]{3} x^2+\sqrt [6]{6} x+\sqrt [3]{2}}dx}{6\ 2^{5/6}}+\frac {\text {arctanh}\left (\sqrt [6]{\frac {3}{2}} x\right )}{3\ 2^{5/6} \sqrt [6]{3}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {3 \int \frac {1}{\sqrt [3]{3} x^2-\sqrt [6]{6} x+\sqrt [3]{2}}dx}{2^{5/6}}-\frac {\int -\frac {\sqrt [6]{6}-2 \sqrt [3]{3} x}{\sqrt [3]{3} x^2-\sqrt [6]{6} x+\sqrt [3]{2}}dx}{2 \sqrt [6]{3}}}{6\ 2^{5/6}}+\frac {\frac {3 \int \frac {1}{\sqrt [3]{3} x^2+\sqrt [6]{6} x+\sqrt [3]{2}}dx}{2^{5/6}}+\frac {\int \frac {2 \sqrt [3]{3} x+\sqrt [6]{6}}{\sqrt [3]{3} x^2+\sqrt [6]{6} x+\sqrt [3]{2}}dx}{2 \sqrt [6]{3}}}{6\ 2^{5/6}}+\frac {\text {arctanh}\left (\sqrt [6]{\frac {3}{2}} x\right )}{3\ 2^{5/6} \sqrt [6]{3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3 \int \frac {1}{\sqrt [3]{3} x^2-\sqrt [6]{6} x+\sqrt [3]{2}}dx}{2^{5/6}}+\frac {\int \frac {\sqrt [6]{6}-2 \sqrt [3]{3} x}{\sqrt [3]{3} x^2-\sqrt [6]{6} x+\sqrt [3]{2}}dx}{2 \sqrt [6]{3}}}{6\ 2^{5/6}}+\frac {\frac {3 \int \frac {1}{\sqrt [3]{3} x^2+\sqrt [6]{6} x+\sqrt [3]{2}}dx}{2^{5/6}}+\frac {\int \frac {2 \sqrt [3]{3} x+\sqrt [6]{6}}{\sqrt [3]{3} x^2+\sqrt [6]{6} x+\sqrt [3]{2}}dx}{2 \sqrt [6]{3}}}{6\ 2^{5/6}}+\frac {\text {arctanh}\left (\sqrt [6]{\frac {3}{2}} x\right )}{3\ 2^{5/6} \sqrt [6]{3}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt [6]{6}-2 \sqrt [3]{3} x}{\sqrt [3]{3} x^2-\sqrt [6]{6} x+\sqrt [3]{2}}dx}{2 \sqrt [6]{3}}+3^{5/6} \int \frac {1}{-\left (1-2^{5/6} \sqrt [6]{3} x\right )^2-3}d\left (1-2^{5/6} \sqrt [6]{3} x\right )}{6\ 2^{5/6}}+\frac {\frac {\int \frac {2 \sqrt [3]{3} x+\sqrt [6]{6}}{\sqrt [3]{3} x^2+\sqrt [6]{6} x+\sqrt [3]{2}}dx}{2 \sqrt [6]{3}}-3^{5/6} \int \frac {1}{-\left (2^{5/6} \sqrt [6]{3} x+1\right )^2-3}d\left (2^{5/6} \sqrt [6]{3} x+1\right )}{6\ 2^{5/6}}+\frac {\text {arctanh}\left (\sqrt [6]{\frac {3}{2}} x\right )}{3\ 2^{5/6} \sqrt [6]{3}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt [6]{6}-2 \sqrt [3]{3} x}{\sqrt [3]{3} x^2-\sqrt [6]{6} x+\sqrt [3]{2}}dx}{2 \sqrt [6]{3}}-\sqrt [3]{3} \arctan \left (\frac {1-2^{5/6} \sqrt [6]{3} x}{\sqrt {3}}\right )}{6\ 2^{5/6}}+\frac {\frac {\int \frac {2 \sqrt [3]{3} x+\sqrt [6]{6}}{\sqrt [3]{3} x^2+\sqrt [6]{6} x+\sqrt [3]{2}}dx}{2 \sqrt [6]{3}}+\sqrt [3]{3} \arctan \left (\frac {2^{5/6} \sqrt [6]{3} x+1}{\sqrt {3}}\right )}{6\ 2^{5/6}}+\frac {\text {arctanh}\left (\sqrt [6]{\frac {3}{2}} x\right )}{3\ 2^{5/6} \sqrt [6]{3}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {-\sqrt [3]{3} \arctan \left (\frac {1-2^{5/6} \sqrt [6]{3} x}{\sqrt {3}}\right )-\frac {\log \left (\sqrt [3]{3} x^2-\sqrt [6]{6} x+\sqrt [3]{2}\right )}{2 \sqrt [6]{3}}}{6\ 2^{5/6}}+\frac {\sqrt [3]{3} \arctan \left (\frac {2^{5/6} \sqrt [6]{3} x+1}{\sqrt {3}}\right )+\frac {\log \left (\sqrt [3]{3} x^2+\sqrt [6]{6} x+\sqrt [3]{2}\right )}{2 \sqrt [6]{3}}}{6\ 2^{5/6}}+\frac {\text {arctanh}\left (\sqrt [6]{\frac {3}{2}} x\right )}{3\ 2^{5/6} \sqrt [6]{3}}\) |
Input:
Int[(2 - 3*x^6)^(-1),x]
Output:
ArcTanh[(3/2)^(1/6)*x]/(3*2^(5/6)*3^(1/6)) + (-(3^(1/3)*ArcTan[(1 - 2^(5/6 )*3^(1/6)*x)/Sqrt[3]]) - Log[2^(1/3) - 6^(1/6)*x + 3^(1/3)*x^2]/(2*3^(1/6) ))/(6*2^(5/6)) + (3^(1/3)*ArcTan[(1 + 2^(5/6)*3^(1/6)*x)/Sqrt[3]] + Log[2^ (1/3) + 6^(1/6)*x + 3^(1/3)*x^2]/(2*3^(1/6)))/(6*2^(5/6))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
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/(a*n)) Int[1/(r^2 - s^2*x^2), x] + 2*(r/(a*n)) Sum[u, {k, 1, (n - 2)/4}], x]] / ; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && NegQ[a/b]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[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])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[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]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.42 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.18
| method | result | size |
| risch | \(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (3 \textit {\_Z}^{6}-2\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}}\right )}{18}\) | \(24\) |
| meijerg | \(-\frac {96^{\frac {5}{6}} x \left (\ln \left (1-\frac {3^{\frac {1}{6}} 2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}{2}\right )-\ln \left (1+\frac {3^{\frac {1}{6}} 2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}{2}\right )+\frac {\ln \left (1-\frac {3^{\frac {1}{6}} 2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}{2}+\frac {3^{\frac {1}{3}} 2^{\frac {2}{3}} \left (x^{6}\right )^{\frac {1}{3}}}{2}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {3^{\frac {2}{3}} 2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}{4-3^{\frac {1}{6}} 2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}\right )-\frac {\ln \left (1+\frac {3^{\frac {1}{6}} 2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}{2}+\frac {3^{\frac {1}{3}} 2^{\frac {2}{3}} \left (x^{6}\right )^{\frac {1}{3}}}{2}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {3^{\frac {2}{3}} 2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}{4+3^{\frac {1}{6}} 2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}\right )\right )}{576 \left (x^{6}\right )^{\frac {1}{6}}}\) | \(181\) |
| default | \(\frac {2^{\frac {2}{3}} 3^{\frac {1}{3}} \sqrt {6}\, \ln \left (x \sqrt {6}\, 12^{\frac {1}{3}}+12^{\frac {2}{3}}+6 x^{2}\right )}{144}-\frac {2^{\frac {1}{6}} 3^{\frac {1}{3}} \arctan \left (\frac {\sqrt {2}\, \sqrt {6}}{6}+\frac {\sqrt {2}\, 12^{\frac {2}{3}} x}{6}\right )}{36}+\frac {2^{\frac {5}{6}} 3^{\frac {2}{3}} 12^{\frac {2}{3}} \arctan \left (\frac {\sqrt {2}\, \sqrt {6}}{6}+\frac {\sqrt {2}\, 12^{\frac {2}{3}} x}{6}\right )}{108}-\frac {\sqrt {6}\, 3^{\frac {1}{3}} 2^{\frac {2}{3}} \ln \left (-\sqrt {6}\, 3^{\frac {1}{3}} 2^{\frac {2}{3}}+6 x \right )}{72}+\frac {\sqrt {6}\, 3^{\frac {1}{3}} 2^{\frac {2}{3}} \ln \left (\sqrt {6}\, 3^{\frac {1}{3}} 2^{\frac {2}{3}}+6 x \right )}{72}-\frac {2^{\frac {2}{3}} 3^{\frac {1}{3}} \sqrt {6}\, \ln \left (-x \sqrt {6}\, 12^{\frac {1}{3}}+12^{\frac {2}{3}}+6 x^{2}\right )}{144}-\frac {2^{\frac {1}{6}} 3^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {2}\, \sqrt {6}}{6}+\frac {\sqrt {2}\, 12^{\frac {2}{3}} x}{6}\right )}{36}+\frac {2^{\frac {5}{6}} 3^{\frac {2}{3}} 12^{\frac {2}{3}} \arctan \left (-\frac {\sqrt {2}\, \sqrt {6}}{6}+\frac {\sqrt {2}\, 12^{\frac {2}{3}} x}{6}\right )}{108}\) | \(228\) |
Input:
int(1/(-3*x^6+2),x,method=_RETURNVERBOSE)
Output:
-1/18*sum(1/_R^5*ln(x-_R),_R=RootOf(3*_Z^6-2))
Time = 0.07 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.97 \[ \int \frac {1}{2-3 x^6} \, dx=\frac {1}{1152} \cdot 96^{\frac {5}{6}} {\left (\sqrt {-3} + 1\right )} \log \left (96^{\frac {5}{6}} {\left (\sqrt {-3} + 1\right )} + 96 \, x\right ) - \frac {1}{1152} \cdot 96^{\frac {5}{6}} {\left (\sqrt {-3} + 1\right )} \log \left (-96^{\frac {5}{6}} {\left (\sqrt {-3} + 1\right )} + 96 \, x\right ) + \frac {1}{1152} \cdot 96^{\frac {5}{6}} {\left (\sqrt {-3} - 1\right )} \log \left (96^{\frac {5}{6}} {\left (\sqrt {-3} - 1\right )} + 96 \, x\right ) - \frac {1}{1152} \cdot 96^{\frac {5}{6}} {\left (\sqrt {-3} - 1\right )} \log \left (-96^{\frac {5}{6}} {\left (\sqrt {-3} - 1\right )} + 96 \, x\right ) + \frac {1}{576} \cdot 96^{\frac {5}{6}} \log \left (48 \, x + 96^{\frac {5}{6}}\right ) - \frac {1}{576} \cdot 96^{\frac {5}{6}} \log \left (48 \, x - 96^{\frac {5}{6}}\right ) \] Input:
integrate(1/(-3*x^6+2),x, algorithm="fricas")
Output:
1/1152*96^(5/6)*(sqrt(-3) + 1)*log(96^(5/6)*(sqrt(-3) + 1) + 96*x) - 1/115 2*96^(5/6)*(sqrt(-3) + 1)*log(-96^(5/6)*(sqrt(-3) + 1) + 96*x) + 1/1152*96 ^(5/6)*(sqrt(-3) - 1)*log(96^(5/6)*(sqrt(-3) - 1) + 96*x) - 1/1152*96^(5/6 )*(sqrt(-3) - 1)*log(-96^(5/6)*(sqrt(-3) - 1) + 96*x) + 1/576*96^(5/6)*log (48*x + 96^(5/6)) - 1/576*96^(5/6)*log(48*x - 96^(5/6))
Time = 0.34 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.11 \[ \int \frac {1}{2-3 x^6} \, dx=- \operatorname {RootSum} {\left (4478976 t^{6} - 1, \left ( t \mapsto t \log {\left (- 12 t + x \right )} \right )\right )} \] Input:
integrate(1/(-3*x**6+2),x)
Output:
-RootSum(4478976*_t**6 - 1, Lambda(_t, _t*log(-12*_t + x)))
Leaf count of result is larger than twice the leaf count of optimal. 224 vs. \(2 (94) = 188\).
Time = 0.12 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.71 \[ \int \frac {1}{2-3 x^6} \, dx=\frac {1}{12} \cdot 3^{\frac {2}{3}} 2^{\frac {1}{6}} \left (\frac {1}{3}\right )^{\frac {1}{3}} \arctan \left (\frac {1}{2} \cdot 3^{\frac {1}{3}} 2^{\frac {5}{6}} \left (\frac {1}{3}\right )^{\frac {2}{3}} {\left (2 \, x + \left (\frac {1}{3}\right )^{\frac {1}{3}} \left (\sqrt {3} \sqrt {2}\right )^{\frac {1}{3}}\right )}\right ) + \frac {1}{12} \cdot 3^{\frac {2}{3}} 2^{\frac {1}{6}} \left (\frac {1}{3}\right )^{\frac {1}{3}} \arctan \left (\frac {1}{2} \cdot 3^{\frac {1}{3}} 2^{\frac {5}{6}} \left (\frac {1}{3}\right )^{\frac {2}{3}} {\left (2 \, x - \left (\frac {1}{3}\right )^{\frac {1}{3}} \left (\sqrt {3} \sqrt {2}\right )^{\frac {1}{3}}\right )}\right ) + \frac {1}{24} \cdot 3^{\frac {1}{6}} 2^{\frac {1}{6}} \left (\frac {1}{3}\right )^{\frac {1}{3}} \log \left (x^{2} + \left (\frac {1}{3}\right )^{\frac {1}{3}} \left (\sqrt {3} \sqrt {2}\right )^{\frac {1}{3}} x + \left (\frac {1}{3}\right )^{\frac {2}{3}} \left (\sqrt {3} \sqrt {2}\right )^{\frac {2}{3}}\right ) - \frac {1}{24} \cdot 3^{\frac {1}{6}} 2^{\frac {1}{6}} \left (\frac {1}{3}\right )^{\frac {1}{3}} \log \left (x^{2} - \left (\frac {1}{3}\right )^{\frac {1}{3}} \left (\sqrt {3} \sqrt {2}\right )^{\frac {1}{3}} x + \left (\frac {1}{3}\right )^{\frac {2}{3}} \left (\sqrt {3} \sqrt {2}\right )^{\frac {2}{3}}\right ) + \frac {1}{12} \cdot 3^{\frac {1}{6}} 2^{\frac {1}{6}} \left (\frac {1}{3}\right )^{\frac {1}{3}} \log \left (x + \left (\frac {1}{3}\right )^{\frac {1}{3}} \left (\sqrt {3} \sqrt {2}\right )^{\frac {1}{3}}\right ) - \frac {1}{12} \cdot 3^{\frac {1}{6}} 2^{\frac {1}{6}} \left (\frac {1}{3}\right )^{\frac {1}{3}} \log \left (x - \left (\frac {1}{3}\right )^{\frac {1}{3}} \left (\sqrt {3} \sqrt {2}\right )^{\frac {1}{3}}\right ) \] Input:
integrate(1/(-3*x^6+2),x, algorithm="maxima")
Output:
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)*sqr t(2))^(1/3))) + 1/24*3^(1/6)*2^(1/6)*(1/3)^(1/3)*log(x^2 + (1/3)^(1/3)*(sq rt(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))
Time = 0.13 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.87 \[ \int \frac {1}{2-3 x^6} \, dx=\frac {1}{12} \, \sqrt {3} \left (\frac {2}{3}\right )^{\frac {1}{6}} \arctan \left (\frac {1}{2} \, \sqrt {3} \left (\frac {2}{3}\right )^{\frac {5}{6}} {\left (2 \, x + \left (\frac {2}{3}\right )^{\frac {1}{6}}\right )}\right ) + \frac {1}{12} \, \sqrt {3} \left (\frac {2}{3}\right )^{\frac {1}{6}} \arctan \left (\frac {1}{2} \, \sqrt {3} \left (\frac {2}{3}\right )^{\frac {5}{6}} {\left (2 \, x - \left (\frac {2}{3}\right )^{\frac {1}{6}}\right )}\right ) + \frac {1}{72} \cdot 486^{\frac {1}{6}} \log \left (x^{2} + \left (\frac {2}{3}\right )^{\frac {1}{6}} x + \left (\frac {2}{3}\right )^{\frac {1}{3}}\right ) - \frac {1}{72} \cdot 486^{\frac {1}{6}} \log \left (x^{2} - \left (\frac {2}{3}\right )^{\frac {1}{6}} x + \left (\frac {2}{3}\right )^{\frac {1}{3}}\right ) + \frac {1}{36} \cdot 486^{\frac {1}{6}} \log \left ({\left | x + \left (\frac {2}{3}\right )^{\frac {1}{6}} \right |}\right ) - \frac {1}{36} \cdot 486^{\frac {1}{6}} \log \left ({\left | x - \left (\frac {2}{3}\right )^{\frac {1}{6}} \right |}\right ) \] Input:
integrate(1/(-3*x^6+2),x, algorithm="giac")
Output:
1/12*sqrt(3)*(2/3)^(1/6)*arctan(1/2*sqrt(3)*(2/3)^(5/6)*(2*x + (2/3)^(1/6) )) + 1/12*sqrt(3)*(2/3)^(1/6)*arctan(1/2*sqrt(3)*(2/3)^(5/6)*(2*x - (2/3)^ (1/6))) + 1/72*486^(1/6)*log(x^2 + (2/3)^(1/6)*x + (2/3)^(1/3)) - 1/72*486 ^(1/6)*log(x^2 - (2/3)^(1/6)*x + (2/3)^(1/3)) + 1/36*486^(1/6)*log(abs(x + (2/3)^(1/6))) - 1/36*486^(1/6)*log(abs(x - (2/3)^(1/6)))
Time = 0.23 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.25 \[ \int \frac {1}{2-3 x^6} \, dx=\frac {{486}^{1/6}\,\mathrm {atanh}\left (\frac {{486}^{5/6}\,x}{162}\right )}{18}-\frac {2^{1/6}\,\mathrm {atanh}\left (\frac {2^{1/6}\,3^{1/3}\,x\,1{}\mathrm {i}}{162\,\left (\frac {2^{1/3}\,3^{2/3}}{486}-\frac {2^{1/3}\,3^{1/6}\,1{}\mathrm {i}}{162}\right )}+\frac {2^{1/6}\,3^{5/6}\,x}{486\,\left (\frac {2^{1/3}\,3^{2/3}}{486}-\frac {2^{1/3}\,3^{1/6}\,1{}\mathrm {i}}{162}\right )}\right )\,\left (3^{5/6}+3^{1/3}\,3{}\mathrm {i}\right )}{36}-\frac {2^{1/6}\,\mathrm {atan}\left (\frac {2^{1/6}\,3^{1/3}\,x}{162\,\left (\frac {2^{1/3}\,3^{2/3}}{486}+\frac {2^{1/3}\,3^{1/6}\,1{}\mathrm {i}}{162}\right )}+\frac {2^{1/6}\,3^{5/6}\,x\,1{}\mathrm {i}}{486\,\left (\frac {2^{1/3}\,3^{2/3}}{486}+\frac {2^{1/3}\,3^{1/6}\,1{}\mathrm {i}}{162}\right )}\right )\,\left (-3^{5/6}+3^{1/3}\,3{}\mathrm {i}\right )\,1{}\mathrm {i}}{36} \] Input:
int(-1/(3*x^6 - 2),x)
Output:
(486^(1/6)*atanh((486^(5/6)*x)/162))/18 - (2^(1/6)*atanh((2^(1/6)*3^(1/3)* x*1i)/(162*((2^(1/3)*3^(2/3))/486 - (2^(1/3)*3^(1/6)*1i)/162)) + (2^(1/6)* 3^(5/6)*x)/(486*((2^(1/3)*3^(2/3))/486 - (2^(1/3)*3^(1/6)*1i)/162)))*(3^(1 /3)*3i + 3^(5/6)))/36 - (2^(1/6)*atan((2^(1/6)*3^(1/3)*x)/(162*((2^(1/3)*3 ^(2/3))/486 + (2^(1/3)*3^(1/6)*1i)/162)) + (2^(1/6)*3^(5/6)*x*1i)/(486*((2 ^(1/3)*3^(2/3))/486 + (2^(1/3)*3^(1/6)*1i)/162)))*(3^(1/3)*3i - 3^(5/6))*1 i)/36
Time = 0.42 (sec) , antiderivative size = 433, normalized size of antiderivative = 3.31 \[ \int \frac {1}{2-3 x^6} \, dx =\text {Too large to display} \] Input:
int(1/(-3*x^6+2),x)
Output:
(3**(1/9)*3**(1/18)*( - 129*6**(2/3)*6**(1/6)*2**(8/9)*atan((2**(1/6)*3**( 1/18) - 2*3**(2/9)*x)/(6**(1/6)*3**(1/3)*3**(1/18))) - 1032*6**(1/6)*2**(5 /9)*3**(2/3)*atan((2**(1/6)*3**(1/18) - 2*3**(2/9)*x)/(6**(1/6)*3**(1/3)*3 **(1/18))) - 774*2**(1/3)*6**(1/6)*2**(2/9)*3**(2/3)*atan((2**(1/6)*3**(1/ 18) - 2*3**(2/9)*x)/(6**(1/6)*3**(1/3)*3**(1/18))) - 129*2**(2/3)*6**(1/6) *2**(8/9)*9**(1/3)*atan((2**(1/6)*3**(1/18) - 2*3**(2/9)*x)/(6**(1/6)*3**( 1/3)*3**(1/18))) - 1032*6**(1/6)*2**(5/9)*9**(1/3)*atan((2**(1/6)*3**(1/18 ) - 2*3**(2/9)*x)/(6**(1/6)*3**(1/3)*3**(1/18))) - 774*18**(1/3)*6**(1/6)* 2**(2/9)*atan((2**(1/6)*3**(1/18) - 2*3**(2/9)*x)/(6**(1/6)*3**(1/3)*3**(1 /18))) - 126*6**(2/3)*6**(1/6)*2**(8/9)*atan((2**(1/6)*3**(1/18) - 2*3**(2 /9)*x)/(6**(1/6)*3**(1/3)*3**(1/18))) - 1008*6**(1/6)*2**(5/9)*3**(2/3)*at an((2**(1/6)*3**(1/18) - 2*3**(2/9)*x)/(6**(1/6)*3**(1/3)*3**(1/18))) - 75 6*2**(1/3)*6**(1/6)*2**(2/9)*3**(2/3)*atan((2**(1/6)*3**(1/18) - 2*3**(2/9 )*x)/(6**(1/6)*3**(1/3)*3**(1/18))) + 516*6**(1/6)*2**(5/9)*3**(2/3)*atan( (2**(1/6)*3**(1/18) + 2*3**(2/9)*x)/(6**(1/6)*3**(1/3)*3**(1/18))) + 1548* 2**(1/3)*6**(1/6)*2**(2/9)*3**(2/3)*atan((2**(1/6)*3**(1/18) + 2*3**(2/9)* x)/(6**(1/6)*3**(1/3)*3**(1/18))) + 516*6**(1/6)*2**(5/9)*9**(1/3)*atan((2 **(1/6)*3**(1/18) + 2*3**(2/9)*x)/(6**(1/6)*3**(1/3)*3**(1/18))) + 1548*18 **(1/3)*6**(1/6)*2**(2/9)*atan((2**(1/6)*3**(1/18) + 2*3**(2/9)*x)/(6**(1/ 6)*3**(1/3)*3**(1/18))) + 504*6**(1/6)*2**(5/9)*3**(2/3)*atan((2**(1/6)...