Integrand size = 16, antiderivative size = 111 \[ \int \frac {1}{x \left (2+3 \log ^3(6 x)\right )} \, dx=-\frac {\arctan \left (\frac {1}{\sqrt {3}}-\frac {2^{2/3} \log (6 x)}{\sqrt [6]{3}}\right )}{2^{2/3} 3^{5/6}}+\frac {\log \left (\sqrt [3]{2}+\sqrt [3]{3} \log (6 x)\right )}{3\ 2^{2/3} \sqrt [3]{3}}-\frac {\log \left (2^{2/3}-\sqrt [3]{6} \log (6 x)+3^{2/3} \log ^2(6 x)\right )}{6\ 2^{2/3} \sqrt [3]{3}} \]
1/6*arctan(1/3*2^(2/3)*ln(6*x)*3^(5/6)-1/3*3^(1/2))*2^(1/3)*3^(1/6)+1/18*l n(2^(1/3)+3^(1/3)*ln(6*x))*2^(1/3)*3^(2/3)-1/36*ln(2^(2/3)-6^(1/3)*ln(6*x) +3^(2/3)*ln(6*x)^2)*2^(1/3)*3^(2/3)
Time = 0.07 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x \left (2+3 \log ^3(6 x)\right )} \, dx=\frac {6 \arctan \left (\frac {-1+2^{2/3} \sqrt [3]{3} \log (6 x)}{\sqrt {3}}\right )+\sqrt {3} \left (2 \log \left (2+2^{2/3} \sqrt [3]{3} \log (6 x)\right )-\log \left (2-2^{2/3} \sqrt [3]{3} \log (6 x)+\sqrt [3]{2} 3^{2/3} \log ^2(6 x)\right )\right )}{6\ 2^{2/3} 3^{5/6}} \]
(6*ArcTan[(-1 + 2^(2/3)*3^(1/3)*Log[6*x])/Sqrt[3]] + Sqrt[3]*(2*Log[2 + 2^ (2/3)*3^(1/3)*Log[6*x]] - Log[2 - 2^(2/3)*3^(1/3)*Log[6*x] + 2^(1/3)*3^(2/ 3)*Log[6*x]^2]))/(6*2^(2/3)*3^(5/6))
Time = 0.31 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.02, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3039, 750, 16, 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.
\(\displaystyle \int \frac {1}{x \left (3 \log ^3(6 x)+2\right )} \, dx\) |
\(\Big \downarrow \) 3039 |
\(\displaystyle \int \frac {1}{3 \log ^3(6 x)+2}d\log (6 x)\) |
\(\Big \downarrow \) 750 |
\(\displaystyle \frac {\int \frac {2 \sqrt [3]{2}-\sqrt [3]{3} \log (6 x)}{3^{2/3} \log ^2(6 x)-\sqrt [3]{6} \log (6 x)+2^{2/3}}d\log (6 x)}{3\ 2^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{3} \log (6 x)+\sqrt [3]{2}}d\log (6 x)}{3\ 2^{2/3}}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {\int \frac {2 \sqrt [3]{2}-\sqrt [3]{3} \log (6 x)}{3^{2/3} \log ^2(6 x)-\sqrt [3]{6} \log (6 x)+2^{2/3}}d\log (6 x)}{3\ 2^{2/3}}+\frac {\log \left (\sqrt [3]{3} \log (6 x)+\sqrt [3]{2}\right )}{3\ 2^{2/3} \sqrt [3]{3}}\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {\frac {3 \int \frac {1}{3^{2/3} \log ^2(6 x)-\sqrt [3]{6} \log (6 x)+2^{2/3}}d\log (6 x)}{2^{2/3}}-\frac {\int -\frac {\sqrt [3]{6}-2\ 3^{2/3} \log (6 x)}{3^{2/3} \log ^2(6 x)-\sqrt [3]{6} \log (6 x)+2^{2/3}}d\log (6 x)}{2 \sqrt [3]{3}}}{3\ 2^{2/3}}+\frac {\log \left (\sqrt [3]{3} \log (6 x)+\sqrt [3]{2}\right )}{3\ 2^{2/3} \sqrt [3]{3}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {3 \int \frac {1}{3^{2/3} \log ^2(6 x)-\sqrt [3]{6} \log (6 x)+2^{2/3}}d\log (6 x)}{2^{2/3}}+\frac {\int \frac {\sqrt [3]{6}-2\ 3^{2/3} \log (6 x)}{3^{2/3} \log ^2(6 x)-\sqrt [3]{6} \log (6 x)+2^{2/3}}d\log (6 x)}{2 \sqrt [3]{3}}}{3\ 2^{2/3}}+\frac {\log \left (\sqrt [3]{3} \log (6 x)+\sqrt [3]{2}\right )}{3\ 2^{2/3} \sqrt [3]{3}}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {\frac {\int \frac {\sqrt [3]{6}-2\ 3^{2/3} \log (6 x)}{3^{2/3} \log ^2(6 x)-\sqrt [3]{6} \log (6 x)+2^{2/3}}d\log (6 x)}{2 \sqrt [3]{3}}+3^{2/3} \int \frac {1}{-\left (1-2^{2/3} \sqrt [3]{3} \log (6 x)\right )^2-3}d\left (1-2^{2/3} \sqrt [3]{3} \log (6 x)\right )}{3\ 2^{2/3}}+\frac {\log \left (\sqrt [3]{3} \log (6 x)+\sqrt [3]{2}\right )}{3\ 2^{2/3} \sqrt [3]{3}}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {\frac {\int \frac {\sqrt [3]{6}-2\ 3^{2/3} \log (6 x)}{3^{2/3} \log ^2(6 x)-\sqrt [3]{6} \log (6 x)+2^{2/3}}d\log (6 x)}{2 \sqrt [3]{3}}-\sqrt [6]{3} \arctan \left (\frac {1-2^{2/3} \sqrt [3]{3} \log (6 x)}{\sqrt {3}}\right )}{3\ 2^{2/3}}+\frac {\log \left (\sqrt [3]{3} \log (6 x)+\sqrt [3]{2}\right )}{3\ 2^{2/3} \sqrt [3]{3}}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {-\sqrt [6]{3} \arctan \left (\frac {1-2^{2/3} \sqrt [3]{3} \log (6 x)}{\sqrt {3}}\right )-\frac {\log \left (3^{2/3} \log ^2(6 x)-\sqrt [3]{6} \log (6 x)+2^{2/3}\right )}{2 \sqrt [3]{3}}}{3\ 2^{2/3}}+\frac {\log \left (\sqrt [3]{3} \log (6 x)+\sqrt [3]{2}\right )}{3\ 2^{2/3} \sqrt [3]{3}}\) |
Log[2^(1/3) + 3^(1/3)*Log[6*x]]/(3*2^(2/3)*3^(1/3)) + (-(3^(1/6)*ArcTan[(1 - 2^(2/3)*3^(1/3)*Log[6*x])/Sqrt[3]]) - Log[2^(2/3) - 6^(1/3)*Log[6*x] + 3^(2/3)*Log[6*x]^2]/(2*3^(1/3)))/(3*2^(2/3))
3.2.37.3.1 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_) + (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_)^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[((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]
Int[u_, x_Symbol] :> With[{lst = FunctionOfLog[Cancel[x*u], x]}, Simp[1/lst [[3]] Subst[Int[lst[[1]], x], x, Log[lst[[2]]]], x] /; !FalseQ[lst]] /; NonsumQ[u]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.23 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.21
method | result | size |
risch | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (324 \textit {\_Z}^{3}-1\right )}{\sum }\textit {\_R} \ln \left (\ln \left (6 x \right )+6 \textit {\_R} \right )\) | \(23\) |
derivativedivides | \(\frac {2^{\frac {1}{3}} 3^{\frac {2}{3}} \ln \left (\ln \left (6 x \right )+\frac {2^{\frac {1}{3}} 3^{\frac {2}{3}}}{3}\right )}{18}-\frac {2^{\frac {1}{3}} 3^{\frac {2}{3}} \ln \left (\ln \left (6 x \right )^{2}-\frac {2^{\frac {1}{3}} 3^{\frac {2}{3}} \ln \left (6 x \right )}{3}+\frac {2^{\frac {2}{3}} 3^{\frac {1}{3}}}{3}\right )}{36}+\frac {2^{\frac {1}{3}} 3^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3}\, \left (2^{\frac {2}{3}} 3^{\frac {1}{3}} \ln \left (6 x \right )-1\right )}{3}\right )}{6}\) | \(87\) |
default | \(\frac {2^{\frac {1}{3}} 3^{\frac {2}{3}} \ln \left (\ln \left (6 x \right )+\frac {2^{\frac {1}{3}} 3^{\frac {2}{3}}}{3}\right )}{18}-\frac {2^{\frac {1}{3}} 3^{\frac {2}{3}} \ln \left (\ln \left (6 x \right )^{2}-\frac {2^{\frac {1}{3}} 3^{\frac {2}{3}} \ln \left (6 x \right )}{3}+\frac {2^{\frac {2}{3}} 3^{\frac {1}{3}}}{3}\right )}{36}+\frac {2^{\frac {1}{3}} 3^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3}\, \left (2^{\frac {2}{3}} 3^{\frac {1}{3}} \ln \left (6 x \right )-1\right )}{3}\right )}{6}\) | \(87\) |
Time = 0.32 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.64 \[ \int \frac {1}{x \left (2+3 \log ^3(6 x)\right )} \, dx=-\frac {1}{72} \cdot 12^{\frac {2}{3}} \log \left (6 \, \log \left (6 \, x\right )^{2} - 12^{\frac {2}{3}} \log \left (6 \, x\right ) + 2 \cdot 12^{\frac {1}{3}}\right ) + \frac {1}{36} \cdot 12^{\frac {2}{3}} \log \left (12^{\frac {2}{3}} + 6 \, \log \left (6 \, x\right )\right ) + \frac {1}{6} \cdot 12^{\frac {1}{6}} \arctan \left (\frac {1}{6} \cdot 12^{\frac {1}{6}} {\left (12^{\frac {2}{3}} \log \left (6 \, x\right ) - 12^{\frac {1}{3}}\right )}\right ) \]
-1/72*12^(2/3)*log(6*log(6*x)^2 - 12^(2/3)*log(6*x) + 2*12^(1/3)) + 1/36*1 2^(2/3)*log(12^(2/3) + 6*log(6*x)) + 1/6*12^(1/6)*arctan(1/6*12^(1/6)*(12^ (2/3)*log(6*x) - 12^(1/3)))
Time = 0.09 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.15 \[ \int \frac {1}{x \left (2+3 \log ^3(6 x)\right )} \, dx=\operatorname {RootSum} {\left (324 z^{3} - 1, \left ( i \mapsto i \log {\left (6 i + \log {\left (6 x \right )} \right )} \right )\right )} \]
Time = 0.29 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x \left (2+3 \log ^3(6 x)\right )} \, dx=-\frac {1}{36} \cdot 3^{\frac {2}{3}} 2^{\frac {1}{3}} \log \left (3^{\frac {2}{3}} \log \left (6 \, x\right )^{2} - 3^{\frac {1}{3}} 2^{\frac {1}{3}} \log \left (6 \, x\right ) + 2^{\frac {2}{3}}\right ) + \frac {1}{18} \cdot 3^{\frac {2}{3}} 2^{\frac {1}{3}} \log \left (\frac {1}{3} \cdot 3^{\frac {2}{3}} {\left (3^{\frac {1}{3}} \log \left (6 \, x\right ) + 2^{\frac {1}{3}}\right )}\right ) + \frac {1}{6} \cdot 3^{\frac {1}{6}} 2^{\frac {1}{3}} \arctan \left (\frac {1}{6} \cdot 3^{\frac {1}{6}} 2^{\frac {2}{3}} {\left (2 \cdot 3^{\frac {2}{3}} \log \left (6 \, x\right ) - 3^{\frac {1}{3}} 2^{\frac {1}{3}}\right )}\right ) \]
-1/36*3^(2/3)*2^(1/3)*log(3^(2/3)*log(6*x)^2 - 3^(1/3)*2^(1/3)*log(6*x) + 2^(2/3)) + 1/18*3^(2/3)*2^(1/3)*log(1/3*3^(2/3)*(3^(1/3)*log(6*x) + 2^(1/3 ))) + 1/6*3^(1/6)*2^(1/3)*arctan(1/6*3^(1/6)*2^(2/3)*(2*3^(2/3)*log(6*x) - 3^(1/3)*2^(1/3)))
\[ \int \frac {1}{x \left (2+3 \log ^3(6 x)\right )} \, dx=\int { \frac {1}{{\left (3 \, \log \left (6 \, x\right )^{3} + 2\right )} x} \,d x } \]
Time = 5.95 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x \left (2+3 \log ^3(6 x)\right )} \, dx=\frac {2^{1/3}\,3^{2/3}\,\ln \left (\frac {\ln \left (6\,x\right )}{x^2}+\frac {2^{1/3}\,3^{2/3}}{3\,x^2}\right )}{18}+\frac {2^{1/3}\,3^{2/3}\,\ln \left (\frac {\ln \left (6\,x\right )}{x^2}+\frac {2^{1/3}\,3^{2/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,x^2}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{18}-\frac {2^{1/3}\,3^{2/3}\,\ln \left (\frac {\ln \left (6\,x\right )}{x^2}-\frac {2^{1/3}\,3^{2/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,x^2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{18} \]
(2^(1/3)*3^(2/3)*log(log(6*x)/x^2 + (2^(1/3)*3^(2/3))/(3*x^2)))/18 + (2^(1 /3)*3^(2/3)*log(log(6*x)/x^2 + (2^(1/3)*3^(2/3)*((3^(1/2)*1i)/2 - 1/2))/(3 *x^2))*((3^(1/2)*1i)/2 - 1/2))/18 - (2^(1/3)*3^(2/3)*log(log(6*x)/x^2 - (2 ^(1/3)*3^(2/3)*((3^(1/2)*1i)/2 + 1/2))/(3*x^2))*((3^(1/2)*1i)/2 + 1/2))/18