Integrand size = 15, antiderivative size = 117 \[ \int \frac {\sqrt {x}}{1+x^{2/3}} \, dx=-6 \sqrt [6]{x}+\frac {6 x^{5/6}}{5}-\frac {3 \arctan \left (1-\sqrt {2} \sqrt [6]{x}\right )}{\sqrt {2}}+\frac {3 \arctan \left (1+\sqrt {2} \sqrt [6]{x}\right )}{\sqrt {2}}-\frac {3 \log \left (1-\sqrt {2} \sqrt [6]{x}+\sqrt [3]{x}\right )}{2 \sqrt {2}}+\frac {3 \log \left (1+\sqrt {2} \sqrt [6]{x}+\sqrt [3]{x}\right )}{2 \sqrt {2}} \]
-6*x^(1/6)+6/5*x^(5/6)+3/2*arctan(-1+x^(1/6)*2^(1/2))*2^(1/2)+3/2*arctan(1 +x^(1/6)*2^(1/2))*2^(1/2)-3/4*ln(1+x^(1/3)-x^(1/6)*2^(1/2))*2^(1/2)+3/4*ln (1+x^(1/3)+x^(1/6)*2^(1/2))*2^(1/2)
Time = 0.12 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt {x}}{1+x^{2/3}} \, dx=\frac {6}{5} \left (-5+x^{2/3}\right ) \sqrt [6]{x}+\frac {3 \arctan \left (\frac {-1+\sqrt [3]{x}}{\sqrt {2} \sqrt [6]{x}}\right )}{\sqrt {2}}+\frac {3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [6]{x}}{1+\sqrt [3]{x}}\right )}{\sqrt {2}} \]
(6*(-5 + x^(2/3))*x^(1/6))/5 + (3*ArcTan[(-1 + x^(1/3))/(Sqrt[2]*x^(1/6))] )/Sqrt[2] + (3*ArcTanh[(Sqrt[2]*x^(1/6))/(1 + x^(1/3))])/Sqrt[2]
Time = 0.32 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.12, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {864, 262, 262, 266, 755, 1476, 1082, 217, 1479, 25, 27, 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.
\(\displaystyle \int \frac {\sqrt {x}}{x^{2/3}+1} \, dx\) |
\(\Big \downarrow \) 864 |
\(\displaystyle 3 \int \frac {x^{7/6}}{x^{2/3}+1}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle 3 \left (\frac {2 x^{5/6}}{5}-\int \frac {\sqrt {x}}{x^{2/3}+1}d\sqrt [3]{x}\right )\) |
\(\Big \downarrow \) 262 |
\(\displaystyle 3 \left (\int \frac {1}{\left (x^{2/3}+1\right ) \sqrt [6]{x}}d\sqrt [3]{x}+\frac {2 x^{5/6}}{5}-2 \sqrt [6]{x}\right )\) |
\(\Big \downarrow \) 266 |
\(\displaystyle 3 \left (2 \int \frac {1}{x^{4/3}+1}d\sqrt [6]{x}+\frac {2 x^{5/6}}{5}-2 \sqrt [6]{x}\right )\) |
\(\Big \downarrow \) 755 |
\(\displaystyle 3 \left (2 \left (\frac {1}{2} \int \frac {1-x^{2/3}}{x^{4/3}+1}d\sqrt [6]{x}+\frac {1}{2} \int \frac {x^{2/3}+1}{x^{4/3}+1}d\sqrt [6]{x}\right )+\frac {2 x^{5/6}}{5}-2 \sqrt [6]{x}\right )\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle 3 \left (2 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{x^{2/3}-\sqrt {2} \sqrt [6]{x}+1}d\sqrt [6]{x}+\frac {1}{2} \int \frac {1}{x^{2/3}+\sqrt {2} \sqrt [6]{x}+1}d\sqrt [6]{x}\right )+\frac {1}{2} \int \frac {1-x^{2/3}}{x^{4/3}+1}d\sqrt [6]{x}\right )+\frac {2 x^{5/6}}{5}-2 \sqrt [6]{x}\right )\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle 3 \left (2 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-x^{2/3}-1}d\left (1-\sqrt {2} \sqrt [6]{x}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-x^{2/3}-1}d\left (\sqrt {2} \sqrt [6]{x}+1\right )}{\sqrt {2}}\right )+\frac {1}{2} \int \frac {1-x^{2/3}}{x^{4/3}+1}d\sqrt [6]{x}\right )+\frac {2 x^{5/6}}{5}-2 \sqrt [6]{x}\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle 3 \left (2 \left (\frac {1}{2} \int \frac {1-x^{2/3}}{x^{4/3}+1}d\sqrt [6]{x}+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt [6]{x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt [6]{x}\right )}{\sqrt {2}}\right )\right )+\frac {2 x^{5/6}}{5}-2 \sqrt [6]{x}\right )\) |
\(\Big \downarrow \) 1479 |
\(\displaystyle 3 \left (2 \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt [6]{x}}{x^{2/3}-\sqrt {2} \sqrt [6]{x}+1}d\sqrt [6]{x}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [6]{x}+1\right )}{x^{2/3}+\sqrt {2} \sqrt [6]{x}+1}d\sqrt [6]{x}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt [6]{x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt [6]{x}\right )}{\sqrt {2}}\right )\right )+\frac {2 x^{5/6}}{5}-2 \sqrt [6]{x}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle 3 \left (2 \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt [6]{x}}{x^{2/3}-\sqrt {2} \sqrt [6]{x}+1}d\sqrt [6]{x}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [6]{x}+1\right )}{x^{2/3}+\sqrt {2} \sqrt [6]{x}+1}d\sqrt [6]{x}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt [6]{x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt [6]{x}\right )}{\sqrt {2}}\right )\right )+\frac {2 x^{5/6}}{5}-2 \sqrt [6]{x}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 3 \left (2 \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt [6]{x}}{x^{2/3}-\sqrt {2} \sqrt [6]{x}+1}d\sqrt [6]{x}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt [6]{x}+1}{x^{2/3}+\sqrt {2} \sqrt [6]{x}+1}d\sqrt [6]{x}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt [6]{x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt [6]{x}\right )}{\sqrt {2}}\right )\right )+\frac {2 x^{5/6}}{5}-2 \sqrt [6]{x}\right )\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle 3 \left (2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt [6]{x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt [6]{x}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (x^{2/3}+\sqrt {2} \sqrt [6]{x}+1\right )}{2 \sqrt {2}}-\frac {\log \left (x^{2/3}-\sqrt {2} \sqrt [6]{x}+1\right )}{2 \sqrt {2}}\right )\right )+\frac {2 x^{5/6}}{5}-2 \sqrt [6]{x}\right )\) |
3*(-2*x^(1/6) + (2*x^(5/6))/5 + 2*((-(ArcTan[1 - Sqrt[2]*x^(1/6)]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*x^(1/6)]/Sqrt[2])/2 + (-1/2*Log[1 - Sqrt[2]*x^(1/6) + x^(2/3)]/Sqrt[2] + Log[1 + Sqrt[2]*x^(1/6) + x^(2/3)]/(2*Sqrt[2]))/2))
3.24.89.3.1 Defintions of rubi rules used
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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[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]]))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denomi nator[n]}, Simp[k Subst[Int[x^(k*(m + 1) - 1)*(a + b*x^(k*n))^p, x], x, x ^(1/k)], x]] /; FreeQ[{a, b, m, p}, x] && FractionQ[n]
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_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 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]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Time = 3.81 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.61
method | result | size |
derivativedivides | \(\frac {6 x^{\frac {5}{6}}}{5}-6 x^{\frac {1}{6}}+\frac {3 \sqrt {2}\, \left (\ln \left (\frac {1+x^{\frac {1}{3}}+x^{\frac {1}{6}} \sqrt {2}}{1+x^{\frac {1}{3}}-x^{\frac {1}{6}} \sqrt {2}}\right )+2 \arctan \left (1+x^{\frac {1}{6}} \sqrt {2}\right )+2 \arctan \left (-1+x^{\frac {1}{6}} \sqrt {2}\right )\right )}{4}\) | \(71\) |
default | \(\frac {6 x^{\frac {5}{6}}}{5}-6 x^{\frac {1}{6}}+\frac {3 \sqrt {2}\, \left (\ln \left (\frac {1+x^{\frac {1}{3}}+x^{\frac {1}{6}} \sqrt {2}}{1+x^{\frac {1}{3}}-x^{\frac {1}{6}} \sqrt {2}}\right )+2 \arctan \left (1+x^{\frac {1}{6}} \sqrt {2}\right )+2 \arctan \left (-1+x^{\frac {1}{6}} \sqrt {2}\right )\right )}{4}\) | \(71\) |
meijerg | \(-\frac {2 x^{\frac {1}{6}} \left (-9 x^{\frac {2}{3}}+45\right )}{15}+\frac {3 x^{\frac {1}{6}} \left (-\frac {\sqrt {2}\, \ln \left (1+x^{\frac {1}{3}}-x^{\frac {1}{6}} \sqrt {2}\right )}{2 x^{\frac {1}{6}}}+\frac {\sqrt {2}\, \arctan \left (\frac {x^{\frac {1}{6}} \sqrt {2}}{2-x^{\frac {1}{6}} \sqrt {2}}\right )}{x^{\frac {1}{6}}}+\frac {\sqrt {2}\, \ln \left (1+x^{\frac {1}{3}}+x^{\frac {1}{6}} \sqrt {2}\right )}{2 x^{\frac {1}{6}}}+\frac {\sqrt {2}\, \arctan \left (\frac {x^{\frac {1}{6}} \sqrt {2}}{2+x^{\frac {1}{6}} \sqrt {2}}\right )}{x^{\frac {1}{6}}}\right )}{2}\) | \(119\) |
6/5*x^(5/6)-6*x^(1/6)+3/4*2^(1/2)*(ln((1+x^(1/3)+x^(1/6)*2^(1/2))/(1+x^(1/ 3)-x^(1/6)*2^(1/2)))+2*arctan(1+x^(1/6)*2^(1/2))+2*arctan(-1+x^(1/6)*2^(1/ 2)))
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {x}}{1+x^{2/3}} \, dx=\left (\frac {3}{4} i + \frac {3}{4}\right ) \, \sqrt {2} \log \left (\left (i + 1\right ) \, \sqrt {2} + 2 \, x^{\frac {1}{6}}\right ) - \left (\frac {3}{4} i - \frac {3}{4}\right ) \, \sqrt {2} \log \left (-\left (i - 1\right ) \, \sqrt {2} + 2 \, x^{\frac {1}{6}}\right ) + \left (\frac {3}{4} i - \frac {3}{4}\right ) \, \sqrt {2} \log \left (\left (i - 1\right ) \, \sqrt {2} + 2 \, x^{\frac {1}{6}}\right ) - \left (\frac {3}{4} i + \frac {3}{4}\right ) \, \sqrt {2} \log \left (-\left (i + 1\right ) \, \sqrt {2} + 2 \, x^{\frac {1}{6}}\right ) + \frac {6}{5} \, x^{\frac {5}{6}} - 6 \, x^{\frac {1}{6}} \]
(3/4*I + 3/4)*sqrt(2)*log((I + 1)*sqrt(2) + 2*x^(1/6)) - (3/4*I - 3/4)*sqr t(2)*log(-(I - 1)*sqrt(2) + 2*x^(1/6)) + (3/4*I - 3/4)*sqrt(2)*log((I - 1) *sqrt(2) + 2*x^(1/6)) - (3/4*I + 3/4)*sqrt(2)*log(-(I + 1)*sqrt(2) + 2*x^( 1/6)) + 6/5*x^(5/6) - 6*x^(1/6)
Result contains complex when optimal does not.
Time = 0.78 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.60 \[ \int \frac {\sqrt {x}}{1+x^{2/3}} \, dx=\frac {27 x^{\frac {5}{6}} \Gamma \left (\frac {9}{4}\right )}{10 \Gamma \left (\frac {13}{4}\right )} - \frac {27 \sqrt [6]{x} \Gamma \left (\frac {9}{4}\right )}{2 \Gamma \left (\frac {13}{4}\right )} - \frac {27 e^{- \frac {i \pi }{4}} \log {\left (- \sqrt [6]{x} e^{\frac {i \pi }{4}} + 1 \right )} \Gamma \left (\frac {9}{4}\right )}{8 \Gamma \left (\frac {13}{4}\right )} + \frac {27 i e^{- \frac {i \pi }{4}} \log {\left (- \sqrt [6]{x} e^{\frac {3 i \pi }{4}} + 1 \right )} \Gamma \left (\frac {9}{4}\right )}{8 \Gamma \left (\frac {13}{4}\right )} + \frac {27 e^{- \frac {i \pi }{4}} \log {\left (- \sqrt [6]{x} e^{\frac {5 i \pi }{4}} + 1 \right )} \Gamma \left (\frac {9}{4}\right )}{8 \Gamma \left (\frac {13}{4}\right )} - \frac {27 i e^{- \frac {i \pi }{4}} \log {\left (- \sqrt [6]{x} e^{\frac {7 i \pi }{4}} + 1 \right )} \Gamma \left (\frac {9}{4}\right )}{8 \Gamma \left (\frac {13}{4}\right )} \]
27*x**(5/6)*gamma(9/4)/(10*gamma(13/4)) - 27*x**(1/6)*gamma(9/4)/(2*gamma( 13/4)) - 27*exp(-I*pi/4)*log(-x**(1/6)*exp_polar(I*pi/4) + 1)*gamma(9/4)/( 8*gamma(13/4)) + 27*I*exp(-I*pi/4)*log(-x**(1/6)*exp_polar(3*I*pi/4) + 1)* gamma(9/4)/(8*gamma(13/4)) + 27*exp(-I*pi/4)*log(-x**(1/6)*exp_polar(5*I*p i/4) + 1)*gamma(9/4)/(8*gamma(13/4)) - 27*I*exp(-I*pi/4)*log(-x**(1/6)*exp _polar(7*I*pi/4) + 1)*gamma(9/4)/(8*gamma(13/4))
Time = 0.29 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {x}}{1+x^{2/3}} \, dx=\frac {3}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, x^{\frac {1}{6}}\right )}\right ) + \frac {3}{2} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, x^{\frac {1}{6}}\right )}\right ) + \frac {3}{4} \, \sqrt {2} \log \left (\sqrt {2} x^{\frac {1}{6}} + x^{\frac {1}{3}} + 1\right ) - \frac {3}{4} \, \sqrt {2} \log \left (-\sqrt {2} x^{\frac {1}{6}} + x^{\frac {1}{3}} + 1\right ) + \frac {6}{5} \, x^{\frac {5}{6}} - 6 \, x^{\frac {1}{6}} \]
3/2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*x^(1/6))) + 3/2*sqrt(2)*arctan (-1/2*sqrt(2)*(sqrt(2) - 2*x^(1/6))) + 3/4*sqrt(2)*log(sqrt(2)*x^(1/6) + x ^(1/3) + 1) - 3/4*sqrt(2)*log(-sqrt(2)*x^(1/6) + x^(1/3) + 1) + 6/5*x^(5/6 ) - 6*x^(1/6)
Time = 0.27 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {x}}{1+x^{2/3}} \, dx=\frac {3}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, x^{\frac {1}{6}}\right )}\right ) + \frac {3}{2} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, x^{\frac {1}{6}}\right )}\right ) + \frac {3}{4} \, \sqrt {2} \log \left (\sqrt {2} x^{\frac {1}{6}} + x^{\frac {1}{3}} + 1\right ) - \frac {3}{4} \, \sqrt {2} \log \left (-\sqrt {2} x^{\frac {1}{6}} + x^{\frac {1}{3}} + 1\right ) + \frac {6}{5} \, x^{\frac {5}{6}} - 6 \, x^{\frac {1}{6}} \]
3/2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*x^(1/6))) + 3/2*sqrt(2)*arctan (-1/2*sqrt(2)*(sqrt(2) - 2*x^(1/6))) + 3/4*sqrt(2)*log(sqrt(2)*x^(1/6) + x ^(1/3) + 1) - 3/4*sqrt(2)*log(-sqrt(2)*x^(1/6) + x^(1/3) + 1) + 6/5*x^(5/6 ) - 6*x^(1/6)
Time = 0.09 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.40 \[ \int \frac {\sqrt {x}}{1+x^{2/3}} \, dx=\frac {6\,x^{5/6}}{5}-6\,x^{1/6}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x^{1/6}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {3}{2}+\frac {3}{2}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x^{1/6}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {3}{2}-\frac {3}{2}{}\mathrm {i}\right ) \]