Integrand size = 9, antiderivative size = 78 \[ \int \frac {1}{-1+2 x^3} \, dx=-\frac {\arctan \left (\frac {1+2 \sqrt [3]{2} x}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log \left (1-\sqrt [3]{2} x\right )}{3 \sqrt [3]{2}}-\frac {\log \left (1+\sqrt [3]{2} x+2^{2/3} x^2\right )}{6 \sqrt [3]{2}} \]
1/6*ln(1-2^(1/3)*x)*2^(2/3)-1/12*ln(1+2^(1/3)*x+2^(2/3)*x^2)*2^(2/3)-1/6*a rctan(1/3*(1+2*2^(1/3)*x)*3^(1/2))*2^(2/3)*3^(1/2)
Time = 0.02 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.85 \[ \int \frac {1}{-1+2 x^3} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{2} x}{\sqrt {3}}\right )-2 \log \left (1-\sqrt [3]{2} x\right )+\log \left (1+\sqrt [3]{2} x+2^{2/3} x^2\right )}{6 \sqrt [3]{2}} \]
-1/6*(2*Sqrt[3]*ArcTan[(1 + 2*2^(1/3)*x)/Sqrt[3]] - 2*Log[1 - 2^(1/3)*x] + Log[1 + 2^(1/3)*x + 2^(2/3)*x^2])/2^(1/3)
Time = 0.23 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.12, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {750, 16, 25, 27, 1142, 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.
\(\displaystyle \int \frac {1}{2 x^3-1} \, dx\) |
\(\Big \downarrow \) 750 |
\(\displaystyle \frac {1}{3} \int -\frac {\sqrt [3]{2} x+2}{2^{2/3} x^2+\sqrt [3]{2} x+1}dx+\frac {1}{3} \int \frac {1}{\sqrt [3]{2} x-1}dx\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {1}{3} \int -\frac {\sqrt [3]{2} x+2}{2^{2/3} x^2+\sqrt [3]{2} x+1}dx+\frac {\log \left (1-\sqrt [3]{2} x\right )}{3 \sqrt [3]{2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\log \left (1-\sqrt [3]{2} x\right )}{3 \sqrt [3]{2}}-\frac {1}{3} \int \frac {\sqrt [3]{2} \left (x+2^{2/3}\right )}{2^{2/3} x^2+\sqrt [3]{2} x+1}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\log \left (1-\sqrt [3]{2} x\right )}{3 \sqrt [3]{2}}-\frac {1}{3} \sqrt [3]{2} \int \frac {x+2^{2/3}}{2^{2/3} x^2+\sqrt [3]{2} x+1}dx\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {\log \left (1-\sqrt [3]{2} x\right )}{3 \sqrt [3]{2}}-\frac {1}{3} \sqrt [3]{2} \left (\frac {3 \int \frac {1}{2^{2/3} x^2+\sqrt [3]{2} x+1}dx}{2 \sqrt [3]{2}}+\frac {\int \frac {\sqrt [3]{2} \left (2 \sqrt [3]{2} x+1\right )}{2^{2/3} x^2+\sqrt [3]{2} x+1}dx}{2\ 2^{2/3}}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\log \left (1-\sqrt [3]{2} x\right )}{3 \sqrt [3]{2}}-\frac {1}{3} \sqrt [3]{2} \left (\frac {3 \int \frac {1}{2^{2/3} x^2+\sqrt [3]{2} x+1}dx}{2 \sqrt [3]{2}}+\frac {\int \frac {2 \sqrt [3]{2} x+1}{2^{2/3} x^2+\sqrt [3]{2} x+1}dx}{2 \sqrt [3]{2}}\right )\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {\log \left (1-\sqrt [3]{2} x\right )}{3 \sqrt [3]{2}}-\frac {1}{3} \sqrt [3]{2} \left (\frac {\int \frac {2 \sqrt [3]{2} x+1}{2^{2/3} x^2+\sqrt [3]{2} x+1}dx}{2 \sqrt [3]{2}}-\frac {3 \int \frac {1}{-\left (2 \sqrt [3]{2} x+1\right )^2-3}d\left (2 \sqrt [3]{2} x+1\right )}{2^{2/3}}\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {\log \left (1-\sqrt [3]{2} x\right )}{3 \sqrt [3]{2}}-\frac {1}{3} \sqrt [3]{2} \left (\frac {\int \frac {2 \sqrt [3]{2} x+1}{2^{2/3} x^2+\sqrt [3]{2} x+1}dx}{2 \sqrt [3]{2}}+\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{2} x+1}{\sqrt {3}}\right )}{2^{2/3}}\right )\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {\log \left (1-\sqrt [3]{2} x\right )}{3 \sqrt [3]{2}}-\frac {1}{3} \sqrt [3]{2} \left (\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{2} x+1}{\sqrt {3}}\right )}{2^{2/3}}+\frac {\log \left (2^{2/3} x^2+\sqrt [3]{2} x+1\right )}{2\ 2^{2/3}}\right )\) |
Log[1 - 2^(1/3)*x]/(3*2^(1/3)) - (2^(1/3)*((Sqrt[3]*ArcTan[(1 + 2*2^(1/3)* x)/Sqrt[3]])/2^(2/3) + Log[1 + 2^(1/3)*x + 2^(2/3)*x^2]/(2*2^(2/3))))/3
3.1.33.3.1 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_)^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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.31
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{3}-1\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{6}\) | \(24\) |
default | \(\frac {2^{\frac {2}{3}} \ln \left (x -\frac {2^{\frac {2}{3}}}{2}\right )}{6}-\frac {2^{\frac {2}{3}} \ln \left (x^{2}+\frac {2^{\frac {2}{3}} x}{2}+\frac {2^{\frac {1}{3}}}{2}\right )}{12}-\frac {\arctan \left (\frac {\left (1+2 \,2^{\frac {1}{3}} x \right ) \sqrt {3}}{3}\right ) 2^{\frac {2}{3}} \sqrt {3}}{6}\) | \(58\) |
meijerg | \(\frac {2^{\frac {2}{3}} x \left (\ln \left (1-2^{\frac {1}{3}} \left (x^{3}\right )^{\frac {1}{3}}\right )-\frac {\ln \left (1+2^{\frac {1}{3}} \left (x^{3}\right )^{\frac {1}{3}}+2^{\frac {2}{3}} \left (x^{3}\right )^{\frac {2}{3}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, 2^{\frac {1}{3}} \left (x^{3}\right )^{\frac {1}{3}}}{2+2^{\frac {1}{3}} \left (x^{3}\right )^{\frac {1}{3}}}\right )\right )}{6 \left (x^{3}\right )^{\frac {1}{3}}}\) | \(82\) |
Time = 0.26 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.81 \[ \int \frac {1}{-1+2 x^3} \, dx=-\frac {1}{6} \, \sqrt {6} 2^{\frac {1}{6}} \arctan \left (\frac {1}{6} \, \sqrt {6} 2^{\frac {1}{6}} {\left (2 \cdot 2^{\frac {2}{3}} x + 2^{\frac {1}{3}}\right )}\right ) - \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (2 \, x^{2} + 2^{\frac {2}{3}} x + 2^{\frac {1}{3}}\right ) + \frac {1}{6} \cdot 2^{\frac {2}{3}} \log \left (2 \, x - 2^{\frac {2}{3}}\right ) \]
-1/6*sqrt(6)*2^(1/6)*arctan(1/6*sqrt(6)*2^(1/6)*(2*2^(2/3)*x + 2^(1/3))) - 1/12*2^(2/3)*log(2*x^2 + 2^(2/3)*x + 2^(1/3)) + 1/6*2^(2/3)*log(2*x - 2^( 2/3))
Time = 0.18 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00 \[ \int \frac {1}{-1+2 x^3} \, dx=\frac {2^{\frac {2}{3}} \log {\left (x - \frac {2^{\frac {2}{3}}}{2} \right )}}{6} - \frac {2^{\frac {2}{3}} \log {\left (x^{2} + \frac {2^{\frac {2}{3}} x}{2} + \frac {\sqrt [3]{2}}{2} \right )}}{12} - \frac {2^{\frac {2}{3}} \sqrt {3} \operatorname {atan}{\left (\frac {2 \cdot \sqrt [3]{2} \sqrt {3} x}{3} + \frac {\sqrt {3}}{3} \right )}}{6} \]
2**(2/3)*log(x - 2**(2/3)/2)/6 - 2**(2/3)*log(x**2 + 2**(2/3)*x/2 + 2**(1/ 3)/2)/12 - 2**(2/3)*sqrt(3)*atan(2*2**(1/3)*sqrt(3)*x/3 + sqrt(3)/3)/6
Time = 0.28 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.85 \[ \int \frac {1}{-1+2 x^3} \, dx=-\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2 \cdot 2^{\frac {2}{3}} x + 2^{\frac {1}{3}}\right )}\right ) - \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} x^{2} + 2^{\frac {1}{3}} x + 1\right ) + \frac {1}{6} \cdot 2^{\frac {2}{3}} \log \left (\frac {1}{2} \cdot 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} x - 1\right )}\right ) \]
-1/6*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2*2^(2/3)*x + 2^(1/3))) - 1/12*2^(2/3)*log(2^(2/3)*x^2 + 2^(1/3)*x + 1) + 1/6*2^(2/3)*log(1/2*2^(2/ 3)*(2^(1/3)*x - 1))
Time = 0.28 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.73 \[ \int \frac {1}{-1+2 x^3} \, dx=-\frac {1}{3} \, \sqrt {3} \left (\frac {1}{2}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {1}{2}\right )^{\frac {2}{3}} {\left (2 \, x + \left (\frac {1}{2}\right )^{\frac {1}{3}}\right )}\right ) - \frac {1}{12} \cdot 4^{\frac {1}{3}} \log \left (x^{2} + \left (\frac {1}{2}\right )^{\frac {1}{3}} x + \left (\frac {1}{2}\right )^{\frac {2}{3}}\right ) + \frac {1}{3} \, \left (\frac {1}{2}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (\frac {1}{2}\right )^{\frac {1}{3}} \right |}\right ) \]
-1/3*sqrt(3)*(1/2)^(1/3)*arctan(2/3*sqrt(3)*(1/2)^(2/3)*(2*x + (1/2)^(1/3) )) - 1/12*4^(1/3)*log(x^2 + (1/2)^(1/3)*x + (1/2)^(2/3)) + 1/3*(1/2)^(1/3) *log(abs(x - (1/2)^(1/3)))
Time = 0.28 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.92 \[ \int \frac {1}{-1+2 x^3} \, dx=\frac {2^{2/3}\,\ln \left (x-\frac {2^{2/3}}{2}\right )}{6}+\frac {2^{2/3}\,\ln \left (x-\frac {2^{2/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{4}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{12}-\frac {2^{2/3}\,\ln \left (x+\frac {2^{2/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{4}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{12} \]
(2^(2/3)*log(x - 2^(2/3)/2))/6 + (2^(2/3)*log(x - (2^(2/3)*(3^(1/2)*1i - 1 ))/4)*(3^(1/2)*1i - 1))/12 - (2^(2/3)*log(x + (2^(2/3)*(3^(1/2)*1i + 1))/4 )*(3^(1/2)*1i + 1))/12
Time = 0.00 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.65 \[ \int \frac {1}{-1+2 x^3} \, dx=\frac {2^{\frac {2}{3}} \left (-2 \sqrt {3}\, \mathit {atan} \left (\frac {2 \,2^{\frac {1}{3}} x +1}{\sqrt {3}}\right )-\mathrm {log}\left (2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} x +1\right )+2 \,\mathrm {log}\left (2^{\frac {1}{3}} x -1\right )\right )}{12} \]