Integrand size = 15, antiderivative size = 244 \[ \int \frac {\sqrt [3]{x}}{1-x^6} \, dx=-\frac {\arctan \left (\frac {1+2 x^{2/3}}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{3} \arctan \left (\left (x^{2/3}-\cos \left (\frac {2 \pi }{9}\right )\right ) \csc \left (\frac {2 \pi }{9}\right )\right ) \cos \left (\frac {\pi }{18}\right )-\frac {1}{6} \log \left (1-x^{2/3}\right )+\frac {1}{12} \log \left (1+x^{2/3}+x^{4/3}\right )-\frac {1}{6} \cos \left (\frac {2 \pi }{9}\right ) \log \left (1+x^{4/3}+2 x^{2/3} \cos \left (\frac {\pi }{9}\right )\right )+\frac {1}{6} \cos \left (\frac {\pi }{9}\right ) \log \left (1+x^{4/3}-2 x^{2/3} \sin \left (\frac {\pi }{18}\right )\right )-\frac {1}{6} \log \left (1+x^{4/3}-2 x^{2/3} \cos \left (\frac {2 \pi }{9}\right )\right ) \sin \left (\frac {\pi }{18}\right )+\frac {1}{3} \arctan \left (\sec \left (\frac {\pi }{18}\right ) \left (x^{2/3}-\sin \left (\frac {\pi }{18}\right )\right )\right ) \sin \left (\frac {\pi }{9}\right )-\frac {1}{3} \arctan \left (\left (x^{2/3}+\cos \left (\frac {\pi }{9}\right )\right ) \csc \left (\frac {\pi }{9}\right )\right ) \sin \left (\frac {2 \pi }{9}\right ) \]
1/3*arctan((x^(2/3)-cos(2/9*Pi))*csc(2/9*Pi))*cos(1/18*Pi)-1/6*ln(1-x^(2/3 ))+1/12*ln(1+x^(2/3)+x^(4/3))-1/6*cos(2/9*Pi)*ln(1+x^(4/3)+2*x^(2/3)*cos(1 /9*Pi))+1/6*cos(1/9*Pi)*ln(1+x^(4/3)-2*x^(2/3)*sin(1/18*Pi))-1/6*ln(1+x^(4 /3)-2*x^(2/3)*cos(2/9*Pi))*sin(1/18*Pi)+1/3*arctan(sec(1/18*Pi)*(x^(2/3)-s in(1/18*Pi)))*sin(1/9*Pi)-1/3*arctan((x^(2/3)+cos(1/9*Pi))*csc(1/9*Pi))*si n(2/9*Pi)-1/6*arctan(1/3*(1+2*x^(2/3))*3^(1/2))*3^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.18 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt [3]{x}}{1-x^6} \, dx=\frac {1}{12} \left (2 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{x}}{\sqrt {3}}\right )+2 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{x}}{\sqrt {3}}\right )-2 \log \left (-1+\sqrt [3]{x}\right )-2 \log \left (1+\sqrt [3]{x}\right )+\log \left (1-\sqrt [3]{x}+x^{2/3}\right )+\log \left (1+\sqrt [3]{x}+x^{2/3}\right )+2 \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {\log \left (\sqrt [3]{x}-\text {$\#$1}\right )+\log \left (\sqrt [3]{x}-\text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ]+2 \text {RootSum}\left [1+\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{x}-\text {$\#$1}\right )+\log \left (\sqrt [3]{x}-\text {$\#$1}\right ) \text {$\#$1}^3}{\text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ]\right ) \]
(2*Sqrt[3]*ArcTan[(1 - 2*x^(1/3))/Sqrt[3]] + 2*Sqrt[3]*ArcTan[(1 + 2*x^(1/ 3))/Sqrt[3]] - 2*Log[-1 + x^(1/3)] - 2*Log[1 + x^(1/3)] + Log[1 - x^(1/3) + x^(2/3)] + Log[1 + x^(1/3) + x^(2/3)] + 2*RootSum[1 - #1^3 + #1^6 & , (L og[x^(1/3) - #1] + Log[x^(1/3) - #1]*#1^3)/(-#1^2 + 2*#1^5) & ] + 2*RootSu m[1 + #1^3 + #1^6 & , (-Log[x^(1/3) - #1] + Log[x^(1/3) - #1]*#1^3)/(#1^2 + 2*#1^5) & ])/12
Time = 0.52 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.18, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.733, Rules used = {851, 807, 823, 16, 25, 27, 1142, 27, 1083, 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.
| \(\displaystyle \int \frac {\sqrt [3]{x}}{1-x^6} \, dx\) |
| \(\Big \downarrow \) 851 |
| \(\displaystyle 3 \int \frac {x}{1-x^6}d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 807 |
| \(\displaystyle \frac {3}{2} \int \frac {x^{2/3}}{1-x^3}dx^{2/3}\) |
| \(\Big \downarrow \) 823 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{9} \int \frac {1}{1-x^{2/3}}dx^{2/3}-\frac {2}{9} \int \frac {1-x^{2/3}}{2 \left (2 x^{2/3}+1\right )}dx^{2/3}-\frac {2}{9} \int \frac {\cos \left (\frac {2 \pi }{9}\right ) x^{2/3}+\cos \left (\frac {\pi }{9}\right )}{2 \cos \left (\frac {\pi }{9}\right ) x^{2/3}+x^{2/3}+1}dx^{2/3}-\frac {2}{9} \int -\frac {\cos \left (\frac {\pi }{9}\right ) x^{2/3}+\sin \left (\frac {\pi }{18}\right )}{-2 \sin \left (\frac {\pi }{18}\right ) x^{2/3}+x^{2/3}+1}dx^{2/3}-\frac {2}{9} \int -\frac {\cos \left (\frac {2 \pi }{9}\right )-x^{2/3} \sin \left (\frac {\pi }{18}\right )}{-2 \cos \left (\frac {2 \pi }{9}\right ) x^{2/3}+x^{2/3}+1}dx^{2/3}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {3}{2} \left (-\frac {2}{9} \int \frac {1-x^{2/3}}{2 \left (2 x^{2/3}+1\right )}dx^{2/3}-\frac {2}{9} \int \frac {\cos \left (\frac {2 \pi }{9}\right ) x^{2/3}+\cos \left (\frac {\pi }{9}\right )}{2 \cos \left (\frac {\pi }{9}\right ) x^{2/3}+x^{2/3}+1}dx^{2/3}-\frac {2}{9} \int -\frac {\cos \left (\frac {\pi }{9}\right ) x^{2/3}+\sin \left (\frac {\pi }{18}\right )}{-2 \sin \left (\frac {\pi }{18}\right ) x^{2/3}+x^{2/3}+1}dx^{2/3}-\frac {2}{9} \int -\frac {\cos \left (\frac {2 \pi }{9}\right )-x^{2/3} \sin \left (\frac {\pi }{18}\right )}{-2 \cos \left (\frac {2 \pi }{9}\right ) x^{2/3}+x^{2/3}+1}dx^{2/3}-\frac {1}{9} \log \left (1-x^{2/3}\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3}{2} \left (-\frac {2}{9} \int \frac {1-x^{2/3}}{2 \left (2 x^{2/3}+1\right )}dx^{2/3}-\frac {2}{9} \int \frac {\cos \left (\frac {2 \pi }{9}\right ) x^{2/3}+\cos \left (\frac {\pi }{9}\right )}{2 \cos \left (\frac {\pi }{9}\right ) x^{2/3}+x^{2/3}+1}dx^{2/3}+\frac {2}{9} \int \frac {\cos \left (\frac {\pi }{9}\right ) x^{2/3}+\sin \left (\frac {\pi }{18}\right )}{-2 \sin \left (\frac {\pi }{18}\right ) x^{2/3}+x^{2/3}+1}dx^{2/3}+\frac {2}{9} \int \frac {\cos \left (\frac {2 \pi }{9}\right )-x^{2/3} \sin \left (\frac {\pi }{18}\right )}{-2 \cos \left (\frac {2 \pi }{9}\right ) x^{2/3}+x^{2/3}+1}dx^{2/3}-\frac {1}{9} \log \left (1-x^{2/3}\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{2} \left (-\frac {1}{9} \int \frac {1-x^{2/3}}{2 x^{2/3}+1}dx^{2/3}-\frac {2}{9} \int \frac {\cos \left (\frac {2 \pi }{9}\right ) x^{2/3}+\cos \left (\frac {\pi }{9}\right )}{2 \cos \left (\frac {\pi }{9}\right ) x^{2/3}+x^{2/3}+1}dx^{2/3}+\frac {2}{9} \int \frac {\cos \left (\frac {\pi }{9}\right ) x^{2/3}+\sin \left (\frac {\pi }{18}\right )}{-2 \sin \left (\frac {\pi }{18}\right ) x^{2/3}+x^{2/3}+1}dx^{2/3}+\frac {2}{9} \int \frac {\cos \left (\frac {2 \pi }{9}\right )-x^{2/3} \sin \left (\frac {\pi }{18}\right )}{-2 \cos \left (\frac {2 \pi }{9}\right ) x^{2/3}+x^{2/3}+1}dx^{2/3}-\frac {1}{9} \log \left (1-x^{2/3}\right )\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{9} \left (\frac {\int 1dx^{2/3}}{2}-\frac {3}{2} \int \frac {1}{2 x^{2/3}+1}dx^{2/3}\right )-\frac {2}{9} \left (\frac {1}{2} \cos \left (\frac {2 \pi }{9}\right ) \int \frac {2 \left (x^{2/3}+\cos \left (\frac {\pi }{9}\right )\right )}{2 \cos \left (\frac {\pi }{9}\right ) x^{2/3}+x^{2/3}+1}dx^{2/3}+\sin \left (\frac {\pi }{9}\right ) \sin \left (\frac {2 \pi }{9}\right ) \int \frac {1}{2 \cos \left (\frac {\pi }{9}\right ) x^{2/3}+x^{2/3}+1}dx^{2/3}\right )+\frac {2}{9} \left (\left (1-\sin \left (\frac {\pi }{18}\right )\right ) \cos \left (\frac {2 \pi }{9}\right ) \int \frac {1}{-2 \cos \left (\frac {2 \pi }{9}\right ) x^{2/3}+x^{2/3}+1}dx^{2/3}-\frac {1}{2} \sin \left (\frac {\pi }{18}\right ) \int \frac {2 \left (x^{2/3}-\cos \left (\frac {2 \pi }{9}\right )\right )}{-2 \cos \left (\frac {2 \pi }{9}\right ) x^{2/3}+x^{2/3}+1}dx^{2/3}\right )+\frac {2}{9} \left (\sin \left (\frac {\pi }{18}\right ) \left (1+\cos \left (\frac {\pi }{9}\right )\right ) \int \frac {1}{-2 \sin \left (\frac {\pi }{18}\right ) x^{2/3}+x^{2/3}+1}dx^{2/3}+\frac {1}{2} \cos \left (\frac {\pi }{9}\right ) \int \frac {2 \left (x^{2/3}-\sin \left (\frac {\pi }{18}\right )\right )}{-2 \sin \left (\frac {\pi }{18}\right ) x^{2/3}+x^{2/3}+1}dx^{2/3}\right )-\frac {1}{9} \log \left (1-x^{2/3}\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{9} \left (\frac {\int 1dx^{2/3}}{2}-\frac {3}{2} \int \frac {1}{2 x^{2/3}+1}dx^{2/3}\right )-\frac {2}{9} \left (\cos \left (\frac {2 \pi }{9}\right ) \int \frac {x^{2/3}+\cos \left (\frac {\pi }{9}\right )}{2 \cos \left (\frac {\pi }{9}\right ) x^{2/3}+x^{2/3}+1}dx^{2/3}+\sin \left (\frac {\pi }{9}\right ) \sin \left (\frac {2 \pi }{9}\right ) \int \frac {1}{2 \cos \left (\frac {\pi }{9}\right ) x^{2/3}+x^{2/3}+1}dx^{2/3}\right )+\frac {2}{9} \left (\left (1-\sin \left (\frac {\pi }{18}\right )\right ) \cos \left (\frac {2 \pi }{9}\right ) \int \frac {1}{-2 \cos \left (\frac {2 \pi }{9}\right ) x^{2/3}+x^{2/3}+1}dx^{2/3}-\sin \left (\frac {\pi }{18}\right ) \int \frac {x^{2/3}-\cos \left (\frac {2 \pi }{9}\right )}{-2 \cos \left (\frac {2 \pi }{9}\right ) x^{2/3}+x^{2/3}+1}dx^{2/3}\right )+\frac {2}{9} \left (\sin \left (\frac {\pi }{18}\right ) \left (1+\cos \left (\frac {\pi }{9}\right )\right ) \int \frac {1}{-2 \sin \left (\frac {\pi }{18}\right ) x^{2/3}+x^{2/3}+1}dx^{2/3}+\cos \left (\frac {\pi }{9}\right ) \int \frac {x^{2/3}-\sin \left (\frac {\pi }{18}\right )}{-2 \sin \left (\frac {\pi }{18}\right ) x^{2/3}+x^{2/3}+1}dx^{2/3}\right )-\frac {1}{9} \log \left (1-x^{2/3}\right )\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{9} \left (\frac {\int 1dx^{2/3}}{2}+3 \int \frac {1}{-2 x^{2/3}-4}d\left (2 x^{2/3}+1\right )\right )+\frac {2}{9} \left (\cos \left (\frac {\pi }{9}\right ) \int \frac {x^{2/3}-\sin \left (\frac {\pi }{18}\right )}{-2 \sin \left (\frac {\pi }{18}\right ) x^{2/3}+x^{2/3}+1}dx^{2/3}-2 \sin \left (\frac {\pi }{18}\right ) \left (1+\cos \left (\frac {\pi }{9}\right )\right ) \int \frac {1}{-2 x^{2/3}+2 \sin \left (\frac {\pi }{18}\right )-4 \cos ^2\left (\frac {\pi }{18}\right )}d\left (2 x^{2/3}-2 \sin \left (\frac {\pi }{18}\right )\right )\right )-\frac {2}{9} \left (\cos \left (\frac {2 \pi }{9}\right ) \int \frac {x^{2/3}+\cos \left (\frac {\pi }{9}\right )}{2 \cos \left (\frac {\pi }{9}\right ) x^{2/3}+x^{2/3}+1}dx^{2/3}-2 \sin \left (\frac {\pi }{9}\right ) \sin \left (\frac {2 \pi }{9}\right ) \int \frac {1}{-2 x^{2/3}-4 \sin ^2\left (\frac {\pi }{9}\right )-2 \cos \left (\frac {\pi }{9}\right )}d\left (2 x^{2/3}+2 \cos \left (\frac {\pi }{9}\right )\right )\right )+\frac {2}{9} \left (-2 \left (1-\sin \left (\frac {\pi }{18}\right )\right ) \cos \left (\frac {2 \pi }{9}\right ) \int \frac {1}{-2 x^{2/3}-4 \sin ^2\left (\frac {2 \pi }{9}\right )+2 \cos \left (\frac {2 \pi }{9}\right )}d\left (2 x^{2/3}-2 \cos \left (\frac {2 \pi }{9}\right )\right )-\sin \left (\frac {\pi }{18}\right ) \int \frac {x^{2/3}-\cos \left (\frac {2 \pi }{9}\right )}{-2 \cos \left (\frac {2 \pi }{9}\right ) x^{2/3}+x^{2/3}+1}dx^{2/3}\right )-\frac {1}{9} \log \left (1-x^{2/3}\right )\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{9} \left (\frac {\int 1dx^{2/3}}{2}-\sqrt {3} \arctan \left (\frac {2 x^{2/3}+1}{\sqrt {3}}\right )\right )-\frac {2}{9} \left (\cos \left (\frac {2 \pi }{9}\right ) \int \frac {x^{2/3}+\cos \left (\frac {\pi }{9}\right )}{2 \cos \left (\frac {\pi }{9}\right ) x^{2/3}+x^{2/3}+1}dx^{2/3}+\sin \left (\frac {2 \pi }{9}\right ) \arctan \left (\frac {1}{2} \csc \left (\frac {\pi }{9}\right ) \left (2 x^{2/3}+2 \cos \left (\frac {\pi }{9}\right )\right )\right )\right )+\frac {2}{9} \left (\left (1-\sin \left (\frac {\pi }{18}\right )\right ) \cot \left (\frac {2 \pi }{9}\right ) \arctan \left (\frac {1}{2} \csc \left (\frac {2 \pi }{9}\right ) \left (2 x^{2/3}-2 \cos \left (\frac {2 \pi }{9}\right )\right )\right )-\sin \left (\frac {\pi }{18}\right ) \int \frac {x^{2/3}-\cos \left (\frac {2 \pi }{9}\right )}{-2 \cos \left (\frac {2 \pi }{9}\right ) x^{2/3}+x^{2/3}+1}dx^{2/3}\right )+\frac {2}{9} \left (\cos \left (\frac {\pi }{9}\right ) \int \frac {x^{2/3}-\sin \left (\frac {\pi }{18}\right )}{-2 \sin \left (\frac {\pi }{18}\right ) x^{2/3}+x^{2/3}+1}dx^{2/3}+\left (1+\cos \left (\frac {\pi }{9}\right )\right ) \tan \left (\frac {\pi }{18}\right ) \arctan \left (\frac {1}{2} \sec \left (\frac {\pi }{18}\right ) \left (2 x^{2/3}-2 \sin \left (\frac {\pi }{18}\right )\right )\right )\right )-\frac {1}{9} \log \left (1-x^{2/3}\right )\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{9} \left (\frac {1}{2} \log \left (2 x^{2/3}+1\right )-\sqrt {3} \arctan \left (\frac {2 x^{2/3}+1}{\sqrt {3}}\right )\right )-\frac {2}{9} \left (\sin \left (\frac {2 \pi }{9}\right ) \arctan \left (\frac {1}{2} \csc \left (\frac {\pi }{9}\right ) \left (2 x^{2/3}+2 \cos \left (\frac {\pi }{9}\right )\right )\right )+\frac {1}{2} \cos \left (\frac {2 \pi }{9}\right ) \log \left (x^{2/3}+2 x^{2/3} \cos \left (\frac {\pi }{9}\right )+1\right )\right )+\frac {2}{9} \left (\left (1+\cos \left (\frac {\pi }{9}\right )\right ) \tan \left (\frac {\pi }{18}\right ) \arctan \left (\frac {1}{2} \sec \left (\frac {\pi }{18}\right ) \left (2 x^{2/3}-2 \sin \left (\frac {\pi }{18}\right )\right )\right )+\frac {1}{2} \cos \left (\frac {\pi }{9}\right ) \log \left (x^{2/3}-2 x^{2/3} \sin \left (\frac {\pi }{18}\right )+1\right )\right )+\frac {2}{9} \left (\left (1-\sin \left (\frac {\pi }{18}\right )\right ) \cot \left (\frac {2 \pi }{9}\right ) \arctan \left (\frac {1}{2} \csc \left (\frac {2 \pi }{9}\right ) \left (2 x^{2/3}-2 \cos \left (\frac {2 \pi }{9}\right )\right )\right )-\frac {1}{2} \sin \left (\frac {\pi }{18}\right ) \log \left (x^{2/3}-2 x^{2/3} \cos \left (\frac {2 \pi }{9}\right )+1\right )\right )-\frac {1}{9} \log \left (1-x^{2/3}\right )\right )\) |
(3*(-1/9*Log[1 - x^(2/3)] + (-(Sqrt[3]*ArcTan[(1 + 2*x^(2/3))/Sqrt[3]]) + Log[1 + 2*x^(2/3)]/2)/9 + (2*(ArcTan[((2*x^(2/3) - 2*Cos[(2*Pi)/9])*Csc[(2 *Pi)/9])/2]*Cot[(2*Pi)/9]*(1 - Sin[Pi/18]) - (Log[1 + x^(2/3) - 2*x^(2/3)* Cos[(2*Pi)/9]]*Sin[Pi/18])/2))/9 - (2*((Cos[(2*Pi)/9]*Log[1 + x^(2/3) + 2* x^(2/3)*Cos[Pi/9]])/2 + ArcTan[((2*x^(2/3) + 2*Cos[Pi/9])*Csc[Pi/9])/2]*Si n[(2*Pi)/9]))/9 + (2*((Cos[Pi/9]*Log[1 + x^(2/3) - 2*x^(2/3)*Sin[Pi/18]])/ 2 + ArcTan[(Sec[Pi/18]*(2*x^(2/3) - 2*Sin[Pi/18]))/2]*(1 + Cos[Pi/9])*Tan[ Pi/18]))/9))/2
3.14.79.3.1 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[(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[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator [Rt[-a/b, n]], s = Denominator[Rt[-a/b, n]], k, u}, Simp[u = Int[(r*Cos[(2* k - 1)*m*(Pi/n)] + s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x]; r^(m + 1)/(a*n*s^m) Int[1/(r - s*x), x] - 2*((-r)^(m + 1)/(a*n*s^m)) Sum[u, {k, 1, (n - 1)/2}], x]] /; FreeQ[{a, b }, x] && IGtQ[(n - 1)/2, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && NegQ[a/b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^ n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{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 = 5.90 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.66
| method | result | size |
| derivativedivides | \(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+\textit {\_Z}^{3}+1\right )}{\sum }\frac {\left (-\textit {\_R}^{3}+1\right ) \ln \left (x^{\frac {1}{3}}-\textit {\_R} \right )}{2 \textit {\_R}^{5}+\textit {\_R}^{2}}\right )}{6}+\frac {\ln \left (x^{\frac {2}{3}}+x^{\frac {1}{3}}+1\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{\frac {1}{3}}+1\right ) \sqrt {3}}{3}\right )}{6}-\frac {\ln \left (x^{\frac {1}{3}}-1\right )}{6}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\left (\textit {\_R}^{3}+1\right ) \ln \left (x^{\frac {1}{3}}-\textit {\_R} \right )}{2 \textit {\_R}^{5}-\textit {\_R}^{2}}\right )}{6}-\frac {\ln \left (1+x^{\frac {1}{3}}\right )}{6}+\frac {\ln \left (x^{\frac {2}{3}}-x^{\frac {1}{3}}+1\right )}{12}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right )}{6}\) | \(162\) |
| default | \(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+\textit {\_Z}^{3}+1\right )}{\sum }\frac {\left (-\textit {\_R}^{3}+1\right ) \ln \left (x^{\frac {1}{3}}-\textit {\_R} \right )}{2 \textit {\_R}^{5}+\textit {\_R}^{2}}\right )}{6}+\frac {\ln \left (x^{\frac {2}{3}}+x^{\frac {1}{3}}+1\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{\frac {1}{3}}+1\right ) \sqrt {3}}{3}\right )}{6}-\frac {\ln \left (x^{\frac {1}{3}}-1\right )}{6}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\left (\textit {\_R}^{3}+1\right ) \ln \left (x^{\frac {1}{3}}-\textit {\_R} \right )}{2 \textit {\_R}^{5}-\textit {\_R}^{2}}\right )}{6}-\frac {\ln \left (1+x^{\frac {1}{3}}\right )}{6}+\frac {\ln \left (x^{\frac {2}{3}}-x^{\frac {1}{3}}+1\right )}{12}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right )}{6}\) | \(162\) |
| meijerg | \(-\frac {x^{\frac {4}{3}} \left (\ln \left (1-\left (x^{6}\right )^{\frac {1}{9}}\right )+\cos \left (\frac {4 \pi }{9}\right ) \ln \left (1-2 \cos \left (\frac {2 \pi }{9}\right ) \left (x^{6}\right )^{\frac {1}{9}}+\left (x^{6}\right )^{\frac {2}{9}}\right )-2 \sin \left (\frac {4 \pi }{9}\right ) \arctan \left (\frac {\sin \left (\frac {2 \pi }{9}\right ) \left (x^{6}\right )^{\frac {1}{9}}}{1-\cos \left (\frac {2 \pi }{9}\right ) \left (x^{6}\right )^{\frac {1}{9}}}\right )-\cos \left (\frac {\pi }{9}\right ) \ln \left (1-2 \cos \left (\frac {4 \pi }{9}\right ) \left (x^{6}\right )^{\frac {1}{9}}+\left (x^{6}\right )^{\frac {2}{9}}\right )-2 \sin \left (\frac {\pi }{9}\right ) \arctan \left (\frac {\sin \left (\frac {4 \pi }{9}\right ) \left (x^{6}\right )^{\frac {1}{9}}}{1-\cos \left (\frac {4 \pi }{9}\right ) \left (x^{6}\right )^{\frac {1}{9}}}\right )-\frac {\ln \left (1+\left (x^{6}\right )^{\frac {1}{9}}+\left (x^{6}\right )^{\frac {2}{9}}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{9}}}{2+\left (x^{6}\right )^{\frac {1}{9}}}\right )+\cos \left (\frac {2 \pi }{9}\right ) \ln \left (1+2 \cos \left (\frac {\pi }{9}\right ) \left (x^{6}\right )^{\frac {1}{9}}+\left (x^{6}\right )^{\frac {2}{9}}\right )+2 \sin \left (\frac {2 \pi }{9}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{9}\right ) \left (x^{6}\right )^{\frac {1}{9}}}{1+\cos \left (\frac {\pi }{9}\right ) \left (x^{6}\right )^{\frac {1}{9}}}\right )\right )}{6 \left (x^{6}\right )^{\frac {2}{9}}}\) | \(231\) |
| trager | \(\text {Expression too large to display}\) | \(1301\) |
-1/6*sum((-_R^3+1)/(2*_R^5+_R^2)*ln(x^(1/3)-_R),_R=RootOf(_Z^6+_Z^3+1))+1/ 12*ln(x^(2/3)+x^(1/3)+1)+1/6*3^(1/2)*arctan(1/3*(2*x^(1/3)+1)*3^(1/2))-1/6 *ln(x^(1/3)-1)+1/6*sum((_R^3+1)/(2*_R^5-_R^2)*ln(x^(1/3)-_R),_R=RootOf(_Z^ 6-_Z^3+1))-1/6*ln(1+x^(1/3))+1/12*ln(x^(2/3)-x^(1/3)+1)-1/6*3^(1/2)*arctan (1/3*(2*x^(1/3)-1)*3^(1/2))
Result contains complex when optimal does not.
Time = 0.53 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.51 \[ \int \frac {\sqrt [3]{x}}{1-x^6} \, dx=-\frac {1}{24} \cdot 2^{\frac {2}{3}} {\left (i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} \log \left ({\left (\sqrt {3} 2^{\frac {1}{3}} {\left (i \, \sqrt {-3} - i\right )} + 2^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )}\right )} {\left (i \, \sqrt {3} + 1\right )}^{\frac {2}{3}} + 8 \, x^{\frac {2}{3}}\right ) + \frac {1}{24} \cdot 2^{\frac {2}{3}} {\left (i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} \log \left ({\left (\sqrt {3} 2^{\frac {1}{3}} {\left (-i \, \sqrt {-3} - i\right )} - 2^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )}\right )} {\left (i \, \sqrt {3} + 1\right )}^{\frac {2}{3}} + 8 \, x^{\frac {2}{3}}\right ) + \frac {1}{24} \cdot 2^{\frac {2}{3}} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} \log \left ({\left (\sqrt {3} 2^{\frac {1}{3}} {\left (i \, \sqrt {-3} + i\right )} - 2^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )}\right )} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {2}{3}} + 8 \, x^{\frac {2}{3}}\right ) - \frac {1}{24} \cdot 2^{\frac {2}{3}} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} \log \left ({\left (\sqrt {3} 2^{\frac {1}{3}} {\left (-i \, \sqrt {-3} + i\right )} + 2^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )}\right )} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {2}{3}} + 8 \, x^{\frac {2}{3}}\right ) + \frac {1}{12} \cdot 2^{\frac {2}{3}} {\left (i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} \log \left ({\left (i \, \sqrt {3} 2^{\frac {1}{3}} + 2^{\frac {1}{3}}\right )} {\left (i \, \sqrt {3} + 1\right )}^{\frac {2}{3}} + 4 \, x^{\frac {2}{3}}\right ) + \frac {1}{12} \cdot 2^{\frac {2}{3}} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} \log \left ({\left (-i \, \sqrt {3} 2^{\frac {1}{3}} + 2^{\frac {1}{3}}\right )} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {2}{3}} + 4 \, x^{\frac {2}{3}}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} x^{\frac {2}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{12} \, \log \left (x^{\frac {4}{3}} + x^{\frac {2}{3}} + 1\right ) - \frac {1}{6} \, \log \left (x^{\frac {2}{3}} - 1\right ) \]
-1/24*2^(2/3)*(I*sqrt(3) + 1)^(1/3)*(sqrt(-3) + 1)*log((sqrt(3)*2^(1/3)*(I *sqrt(-3) - I) + 2^(1/3)*(sqrt(-3) - 1))*(I*sqrt(3) + 1)^(2/3) + 8*x^(2/3) ) + 1/24*2^(2/3)*(I*sqrt(3) + 1)^(1/3)*(sqrt(-3) - 1)*log((sqrt(3)*2^(1/3) *(-I*sqrt(-3) - I) - 2^(1/3)*(sqrt(-3) + 1))*(I*sqrt(3) + 1)^(2/3) + 8*x^( 2/3)) + 1/24*2^(2/3)*(-I*sqrt(3) + 1)^(1/3)*(sqrt(-3) - 1)*log((sqrt(3)*2^ (1/3)*(I*sqrt(-3) + I) - 2^(1/3)*(sqrt(-3) + 1))*(-I*sqrt(3) + 1)^(2/3) + 8*x^(2/3)) - 1/24*2^(2/3)*(-I*sqrt(3) + 1)^(1/3)*(sqrt(-3) + 1)*log((sqrt( 3)*2^(1/3)*(-I*sqrt(-3) + I) + 2^(1/3)*(sqrt(-3) - 1))*(-I*sqrt(3) + 1)^(2 /3) + 8*x^(2/3)) + 1/12*2^(2/3)*(I*sqrt(3) + 1)^(1/3)*log((I*sqrt(3)*2^(1/ 3) + 2^(1/3))*(I*sqrt(3) + 1)^(2/3) + 4*x^(2/3)) + 1/12*2^(2/3)*(-I*sqrt(3 ) + 1)^(1/3)*log((-I*sqrt(3)*2^(1/3) + 2^(1/3))*(-I*sqrt(3) + 1)^(2/3) + 4 *x^(2/3)) - 1/6*sqrt(3)*arctan(2/3*sqrt(3)*x^(2/3) + 1/3*sqrt(3)) + 1/12*l og(x^(4/3) + x^(2/3) + 1) - 1/6*log(x^(2/3) - 1)
Timed out. \[ \int \frac {\sqrt [3]{x}}{1-x^6} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt [3]{x}}{1-x^6} \, dx=\int { -\frac {x^{\frac {1}{3}}}{x^{6} - 1} \,d x } \]
1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^(1/3) + 1)) - 1/6*sqrt(3)*arctan(1/3*s qrt(3)*(2*x^(1/3) - 1)) + integrate(1/6*(x^(4/3) + 2*x^(1/3))/(x^2 + x + 1 ), x) - integrate(1/6*(x^(4/3) - 2*x^(1/3))/(x^2 - x + 1), x) + 1/12*log(x ^(2/3) + x^(1/3) + 1) + 1/12*log(x^(2/3) - x^(1/3) + 1) - 1/6*log(x^(1/3) + 1) - 1/6*log(x^(1/3) - 1)
Time = 0.90 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt [3]{x}}{1-x^6} \, dx=\frac {2}{3} \, \arctan \left (\frac {x^{\frac {2}{3}} - \cos \left (\frac {4}{9} \, \pi \right )}{\sin \left (\frac {4}{9} \, \pi \right )}\right ) \cos \left (\frac {4}{9} \, \pi \right ) \sin \left (\frac {4}{9} \, \pi \right ) + \frac {2}{3} \, \arctan \left (\frac {x^{\frac {2}{3}} - \cos \left (\frac {2}{9} \, \pi \right )}{\sin \left (\frac {2}{9} \, \pi \right )}\right ) \cos \left (\frac {2}{9} \, \pi \right ) \sin \left (\frac {2}{9} \, \pi \right ) - \frac {2}{3} \, \arctan \left (\frac {x^{\frac {2}{3}} + \cos \left (\frac {1}{9} \, \pi \right )}{\sin \left (\frac {1}{9} \, \pi \right )}\right ) \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right ) - \frac {1}{6} \, {\left (\cos \left (\frac {4}{9} \, \pi \right )^{2} - \sin \left (\frac {4}{9} \, \pi \right )^{2}\right )} \log \left (-2 \, x^{\frac {2}{3}} \cos \left (\frac {4}{9} \, \pi \right ) + x^{\frac {4}{3}} + 1\right ) - \frac {1}{6} \, {\left (\cos \left (\frac {2}{9} \, \pi \right )^{2} - \sin \left (\frac {2}{9} \, \pi \right )^{2}\right )} \log \left (-2 \, x^{\frac {2}{3}} \cos \left (\frac {2}{9} \, \pi \right ) + x^{\frac {4}{3}} + 1\right ) - \frac {1}{6} \, {\left (\cos \left (\frac {1}{9} \, \pi \right )^{2} - \sin \left (\frac {1}{9} \, \pi \right )^{2}\right )} \log \left (2 \, x^{\frac {2}{3}} \cos \left (\frac {1}{9} \, \pi \right ) + x^{\frac {4}{3}} + 1\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{\frac {2}{3}} + 1\right )}\right ) + \frac {1}{12} \, \log \left (x^{\frac {4}{3}} + x^{\frac {2}{3}} + 1\right ) - \frac {1}{6} \, \log \left ({\left | x^{\frac {2}{3}} - 1 \right |}\right ) \]
2/3*arctan((x^(2/3) - cos(4/9*pi))/sin(4/9*pi))*cos(4/9*pi)*sin(4/9*pi) + 2/3*arctan((x^(2/3) - cos(2/9*pi))/sin(2/9*pi))*cos(2/9*pi)*sin(2/9*pi) - 2/3*arctan((x^(2/3) + cos(1/9*pi))/sin(1/9*pi))*cos(1/9*pi)*sin(1/9*pi) - 1/6*(cos(4/9*pi)^2 - sin(4/9*pi)^2)*log(-2*x^(2/3)*cos(4/9*pi) + x^(4/3) + 1) - 1/6*(cos(2/9*pi)^2 - sin(2/9*pi)^2)*log(-2*x^(2/3)*cos(2/9*pi) + x^( 4/3) + 1) - 1/6*(cos(1/9*pi)^2 - sin(1/9*pi)^2)*log(2*x^(2/3)*cos(1/9*pi) + x^(4/3) + 1) - 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^(2/3) + 1)) + 1/12*lo g(x^(4/3) + x^(2/3) + 1) - 1/6*log(abs(x^(2/3) - 1))
Time = 5.76 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt [3]{x}}{1-x^6} \, dx=-\frac {\ln \left (43046721\,x^{2/3}-43046721\right )}{6}+\frac {\ln \left (43046721\,x^{2/3}\,{\mathrm {e}}^{\frac {\pi \,4{}\mathrm {i}}{9}}-43046721\right )\,{\mathrm {e}}^{\frac {\pi \,1{}\mathrm {i}}{9}}}{6}+\frac {\ln \left (43046721\,x^{2/3}\,{\mathrm {e}}^{\frac {\pi \,2{}\mathrm {i}}{9}}-43046721\right )\,{\mathrm {e}}^{\frac {\pi \,5{}\mathrm {i}}{9}}}{6}+\frac {\ln \left (-43046721\,x^{2/3}\,{\mathrm {e}}^{\frac {\pi \,1{}\mathrm {i}}{9}}-43046721\right )\,{\mathrm {e}}^{\frac {\pi \,7{}\mathrm {i}}{9}}}{6}-\frac {\ln \left (43046721\,x^{2/3}\,{\mathrm {e}}^{\frac {\pi \,8{}\mathrm {i}}{9}}-43046721\right )\,{\mathrm {e}}^{\frac {\pi \,2{}\mathrm {i}}{9}}}{6}-\frac {\ln \left (-43046721\,x^{2/3}\,{\mathrm {e}}^{\frac {\pi \,7{}\mathrm {i}}{9}}-43046721\right )\,{\mathrm {e}}^{\frac {\pi \,4{}\mathrm {i}}{9}}}{6}-\frac {\ln \left (-43046721\,x^{2/3}\,{\mathrm {e}}^{\frac {\pi \,5{}\mathrm {i}}{9}}-43046721\right )\,{\mathrm {e}}^{\frac {\pi \,8{}\mathrm {i}}{9}}}{6}-\ln \left (55788550416\,x^{2/3}\,{\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )}^4-43046721\right )\,\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )+\ln \left (55788550416\,x^{2/3}\,{\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )}^4-43046721\right )\,\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right ) \]
(log(43046721*x^(2/3)*exp((pi*4i)/9) - 43046721)*exp((pi*1i)/9))/6 - log(4 3046721*x^(2/3) - 43046721)/6 + (log(43046721*x^(2/3)*exp((pi*2i)/9) - 430 46721)*exp((pi*5i)/9))/6 + (log(- 43046721*x^(2/3)*exp((pi*1i)/9) - 430467 21)*exp((pi*7i)/9))/6 - (log(43046721*x^(2/3)*exp((pi*8i)/9) - 43046721)*e xp((pi*2i)/9))/6 - (log(- 43046721*x^(2/3)*exp((pi*7i)/9) - 43046721)*exp( (pi*4i)/9))/6 - (log(- 43046721*x^(2/3)*exp((pi*5i)/9) - 43046721)*exp((pi *8i)/9))/6 - log(55788550416*x^(2/3)*((3^(1/2)*1i)/12 - 1/12)^4 - 43046721 )*((3^(1/2)*1i)/12 - 1/12) + log(55788550416*x^(2/3)*((3^(1/2)*1i)/12 + 1/ 12)^4 - 43046721)*((3^(1/2)*1i)/12 + 1/12)