Integrand size = 11, antiderivative size = 142 \[ \int \frac {x}{\left (a+b x^6\right )^2} \, dx=\frac {x^2}{6 a \left (a+b x^6\right )}-\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{9 a^{5/3} \sqrt [3]{b}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{18 a^{5/3} \sqrt [3]{b}} \]
1/6*x^2/a/(b*x^6+a)+1/9*ln(a^(1/3)+b^(1/3)*x^2)/a^(5/3)/b^(1/3)-1/18*ln(a^ (2/3)-a^(1/3)*b^(1/3)*x^2+b^(2/3)*x^4)/a^(5/3)/b^(1/3)-1/9*arctan(1/3*(a^( 1/3)-2*b^(1/3)*x^2)/a^(1/3)*3^(1/2))/a^(5/3)/b^(1/3)*3^(1/2)
Time = 0.11 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.39 \[ \int \frac {x}{\left (a+b x^6\right )^2} \, dx=\frac {\frac {3 a^{2/3} x^2}{a+b x^6}-\frac {2 \sqrt {3} \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{\sqrt [3]{b}}-\frac {2 \sqrt {3} \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{\sqrt [3]{b}}+\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{\sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{\sqrt [3]{b}}}{18 a^{5/3}} \]
((3*a^(2/3)*x^2)/(a + b*x^6) - (2*Sqrt[3]*ArcTan[Sqrt[3] - (2*b^(1/6)*x)/a ^(1/6)])/b^(1/3) - (2*Sqrt[3]*ArcTan[Sqrt[3] + (2*b^(1/6)*x)/a^(1/6)])/b^( 1/3) + (2*Log[a^(1/3) + b^(1/3)*x^2])/b^(1/3) - Log[a^(1/3) - Sqrt[3]*a^(1 /6)*b^(1/6)*x + b^(1/3)*x^2]/b^(1/3) - Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/ 6)*x + b^(1/3)*x^2]/b^(1/3))/(18*a^(5/3))
Time = 0.30 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.04, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.909, Rules used = {807, 749, 750, 16, 1142, 25, 27, 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.
| \(\displaystyle \int \frac {x}{\left (a+b x^6\right )^2} \, dx\) |
| \(\Big \downarrow \) 807 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{\left (b x^6+a\right )^2}dx^2\) |
| \(\Big \downarrow \) 749 |
| \(\displaystyle \frac {1}{2} \left (\frac {2 \int \frac {1}{b x^6+a}dx^2}{3 a}+\frac {x^2}{3 a \left (a+b x^6\right )}\right )\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {1}{2} \left (\frac {2 \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x^2}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{b} x^2+\sqrt [3]{a}}dx^2}{3 a^{2/3}}\right )}{3 a}+\frac {x^2}{3 a \left (a+b x^6\right )}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{2} \left (\frac {2 \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x^2}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {x^2}{3 a \left (a+b x^6\right )}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{2} \left (\frac {2 \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2-\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x^2\right )}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {x^2}{3 a \left (a+b x^6\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {2 \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2+\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x^2\right )}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {x^2}{3 a \left (a+b x^6\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {2 \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2+\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {x^2}{3 a \left (a+b x^6\right )}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{2} \left (\frac {2 \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2+\frac {3 \int \frac {1}{-x^4-3}d\left (1-\frac {2 \sqrt [3]{b} x^2}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {x^2}{3 a \left (a+b x^6\right )}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{2} \left (\frac {2 \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x^2}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {x^2}{3 a \left (a+b x^6\right )}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{2} \left (\frac {2 \left (\frac {-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x^2}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {x^2}{3 a \left (a+b x^6\right )}\right )\) |
(x^2/(3*a*(a + b*x^6)) + (2*(Log[a^(1/3) + b^(1/3)*x^2]/(3*a^(2/3)*b^(1/3) ) + (-((Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*x^2)/a^(1/3))/Sqrt[3]])/b^(1/3)) - Log[a^(2/3) - a^(1/3)*b^(1/3)*x^2 + b^(2/3)*x^4]/(2*b^(1/3)))/(3*a^(2/3))) )/(3*a))/2
3.14.38.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
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_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Simp[(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] && (Inte gerQ[2*p] || Denominator[p + 1/n] < Denominator[p])
Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; FreeQ[{a, b}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
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 = 4.40 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.33
| method | result | size |
| risch | \(\frac {x^{2}}{6 a \left (b \,x^{6}+a \right )}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b \,a^{5}-1\right )}{\sum }\textit {\_R} \ln \left (a^{2} \textit {\_R} +x^{2}\right )\right )}{9}\) | \(47\) |
| default | \(\frac {x^{2}}{6 a \left (b \,x^{6}+a \right )}+\frac {\frac {\ln \left (x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{4}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{2}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{2}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}}{3 a}\) | \(120\) |
Time = 0.29 (sec) , antiderivative size = 407, normalized size of antiderivative = 2.87 \[ \int \frac {x}{\left (a+b x^6\right )^2} \, dx=\left [\frac {3 \, a^{2} b x^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (a b^{2} x^{6} + a^{2} b\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a b x^{6} - 3 \, \left (a^{2} b\right )^{\frac {1}{3}} a x^{2} - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{4} + \left (a^{2} b\right )^{\frac {2}{3}} x^{2} - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{b x^{6} + a}\right ) - {\left (b x^{6} + a\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{4} - \left (a^{2} b\right )^{\frac {2}{3}} x^{2} + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 2 \, {\left (b x^{6} + a\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{18 \, {\left (a^{3} b^{2} x^{6} + a^{4} b\right )}}, \frac {3 \, a^{2} b x^{2} + 6 \, \sqrt {\frac {1}{3}} {\left (a b^{2} x^{6} + a^{2} b\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a^{2} b\right )^{\frac {2}{3}} x^{2} - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{a^{2}}\right ) - {\left (b x^{6} + a\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{4} - \left (a^{2} b\right )^{\frac {2}{3}} x^{2} + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 2 \, {\left (b x^{6} + a\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{18 \, {\left (a^{3} b^{2} x^{6} + a^{4} b\right )}}\right ] \]
[1/18*(3*a^2*b*x^2 + 3*sqrt(1/3)*(a*b^2*x^6 + a^2*b)*sqrt(-(a^2*b)^(1/3)/b )*log((2*a*b*x^6 - 3*(a^2*b)^(1/3)*a*x^2 - a^2 + 3*sqrt(1/3)*(2*a*b*x^4 + (a^2*b)^(2/3)*x^2 - (a^2*b)^(1/3)*a)*sqrt(-(a^2*b)^(1/3)/b))/(b*x^6 + a)) - (b*x^6 + a)*(a^2*b)^(2/3)*log(a*b*x^4 - (a^2*b)^(2/3)*x^2 + (a^2*b)^(1/3 )*a) + 2*(b*x^6 + a)*(a^2*b)^(2/3)*log(a*b*x^2 + (a^2*b)^(2/3)))/(a^3*b^2* x^6 + a^4*b), 1/18*(3*a^2*b*x^2 + 6*sqrt(1/3)*(a*b^2*x^6 + a^2*b)*sqrt((a^ 2*b)^(1/3)/b)*arctan(sqrt(1/3)*(2*(a^2*b)^(2/3)*x^2 - (a^2*b)^(1/3)*a)*sqr t((a^2*b)^(1/3)/b)/a^2) - (b*x^6 + a)*(a^2*b)^(2/3)*log(a*b*x^4 - (a^2*b)^ (2/3)*x^2 + (a^2*b)^(1/3)*a) + 2*(b*x^6 + a)*(a^2*b)^(2/3)*log(a*b*x^2 + ( a^2*b)^(2/3)))/(a^3*b^2*x^6 + a^4*b)]
Time = 0.18 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.29 \[ \int \frac {x}{\left (a+b x^6\right )^2} \, dx=\frac {x^{2}}{6 a^{2} + 6 a b x^{6}} + \operatorname {RootSum} {\left (729 t^{3} a^{5} b - 1, \left ( t \mapsto t \log {\left (9 t a^{2} + x^{2} \right )} \right )\right )} \]
x**2/(6*a**2 + 6*a*b*x**6) + RootSum(729*_t**3*a**5*b - 1, Lambda(_t, _t*l og(9*_t*a**2 + x**2)))
Time = 0.27 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.92 \[ \int \frac {x}{\left (a+b x^6\right )^2} \, dx=\frac {x^{2}}{6 \, {\left (a b x^{6} + a^{2}\right )}} + \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{2} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {\log \left (x^{4} - x^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {\log \left (x^{2} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
1/6*x^2/(a*b*x^6 + a^2) + 1/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^2 - (a/b)^(1 /3))/(a/b)^(1/3))/(a*b*(a/b)^(2/3)) - 1/18*log(x^4 - x^2*(a/b)^(1/3) + (a/ b)^(2/3))/(a*b*(a/b)^(2/3)) + 1/9*log(x^2 + (a/b)^(1/3))/(a*b*(a/b)^(2/3))
Time = 0.28 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.95 \[ \int \frac {x}{\left (a+b x^6\right )^2} \, dx=\frac {x^{2}}{6 \, {\left (b x^{6} + a\right )} a} - \frac {\left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x^{2} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{2}} + \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{2} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{2} b} + \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{4} + x^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{2} b} \]
1/6*x^2/((b*x^6 + a)*a) - 1/9*(-a/b)^(1/3)*log(abs(x^2 - (-a/b)^(1/3)))/a^ 2 + 1/9*sqrt(3)*(-a*b^2)^(1/3)*arctan(1/3*sqrt(3)*(2*x^2 + (-a/b)^(1/3))/( -a/b)^(1/3))/(a^2*b) + 1/18*(-a*b^2)^(1/3)*log(x^4 + x^2*(-a/b)^(1/3) + (- a/b)^(2/3))/(a^2*b)
Time = 5.46 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.90 \[ \int \frac {x}{\left (a+b x^6\right )^2} \, dx=\frac {\ln \left (a^{1/3}+b^{1/3}\,x^2\right )}{9\,a^{5/3}\,b^{1/3}}+\frac {x^2}{6\,a\,\left (b\,x^6+a\right )}+\frac {\ln \left (\frac {16\,b^4\,x^2}{9\,a^3}+\frac {8\,b^{11/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{9\,a^{8/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{18\,a^{5/3}\,b^{1/3}}-\frac {\ln \left (\frac {16\,b^4\,x^2}{9\,a^3}-\frac {8\,b^{11/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{9\,a^{8/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{18\,a^{5/3}\,b^{1/3}} \]
log(a^(1/3) + b^(1/3)*x^2)/(9*a^(5/3)*b^(1/3)) + x^2/(6*a*(a + b*x^6)) + ( log((16*b^4*x^2)/(9*a^3) + (8*b^(11/3)*(3^(1/2)*1i - 1))/(9*a^(8/3)))*(3^( 1/2)*1i - 1))/(18*a^(5/3)*b^(1/3)) - (log((16*b^4*x^2)/(9*a^3) - (8*b^(11/ 3)*(3^(1/2)*1i + 1))/(9*a^(8/3)))*(3^(1/2)*1i + 1))/(18*a^(5/3)*b^(1/3))