Integrand size = 18, antiderivative size = 243 \[ \int \frac {-1+x}{(1+x) \sqrt [3]{-1+x^3}} \, dx=\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )}{\sqrt {3}}-\frac {\sqrt {3} \arctan \left (\frac {-\frac {\sqrt [3]{2}}{\sqrt {3}}+\frac {\sqrt [3]{2} x}{\sqrt {3}}+\frac {\sqrt [3]{-1+x^3}}{\sqrt {3}}}{\sqrt [3]{-1+x^3}}\right )}{\sqrt [3]{2}}+\frac {\log \left (-\sqrt [3]{2}+\sqrt [3]{2} x-2 \sqrt [3]{-1+x^3}\right )}{\sqrt [3]{2}}-\frac {1}{3} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{6} \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )-\frac {\log \left (2^{2/3}-2\ 2^{2/3} x+2^{2/3} x^2+\left (-2 \sqrt [3]{2}+2 \sqrt [3]{2} x\right ) \sqrt [3]{-1+x^3}+4 \left (-1+x^3\right )^{2/3}\right )}{2 \sqrt [3]{2}} \]
1/3*arctan(3^(1/2)*x/(x+2*(x^3-1)^(1/3)))*3^(1/2)-1/2*3^(1/2)*arctan((-1/3 *2^(1/3)*3^(1/2)+1/3*2^(1/3)*x*3^(1/2)+1/3*(x^3-1)^(1/3)*3^(1/2))/(x^3-1)^ (1/3))*2^(2/3)+1/2*ln(-2^(1/3)+2^(1/3)*x-2*(x^3-1)^(1/3))*2^(2/3)-1/3*ln(- x+(x^3-1)^(1/3))+1/6*ln(x^2+x*(x^3-1)^(1/3)+(x^3-1)^(2/3))-1/4*ln(2^(2/3)- 2*2^(2/3)*x+2^(2/3)*x^2+(-2*2^(1/3)+2*2^(1/3)*x)*(x^3-1)^(1/3)+4*(x^3-1)^( 2/3))*2^(2/3)
Time = 5.32 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.93 \[ \int \frac {-1+x}{(1+x) \sqrt [3]{-1+x^3}} \, dx=\frac {1}{12} \left (-6 2^{2/3} \sqrt {3} \arctan \left (\frac {-\sqrt [3]{2}+\sqrt [3]{2} x+\sqrt [3]{-1+x^3}}{\sqrt {3} \sqrt [3]{-1+x^3}}\right )+4 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )+6\ 2^{2/3} \log \left (-\sqrt [3]{2}+\sqrt [3]{2} x-2 \sqrt [3]{-1+x^3}\right )-4 \log \left (-x+\sqrt [3]{-1+x^3}\right )+2 \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )-3\ 2^{2/3} \log \left (2^{2/3}-2\ 2^{2/3} x+2^{2/3} x^2+2 \sqrt [3]{2} (-1+x) \sqrt [3]{-1+x^3}+4 \left (-1+x^3\right )^{2/3}\right )\right ) \]
(-6*2^(2/3)*Sqrt[3]*ArcTan[(-2^(1/3) + 2^(1/3)*x + (-1 + x^3)^(1/3))/(Sqrt [3]*(-1 + x^3)^(1/3))] + 4*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^3)^(1 /3))] + 6*2^(2/3)*Log[-2^(1/3) + 2^(1/3)*x - 2*(-1 + x^3)^(1/3)] - 4*Log[- x + (-1 + x^3)^(1/3)] + 2*Log[x^2 + x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)] - 3*2^(2/3)*Log[2^(2/3) - 2*2^(2/3)*x + 2^(2/3)*x^2 + 2*2^(1/3)*(-1 + x)* (-1 + x^3)^(1/3) + 4*(-1 + x^3)^(2/3)])/12
Time = 0.30 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.59, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2577, 769, 2574}
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.
| \(\displaystyle \int \frac {x-1}{(x+1) \sqrt [3]{x^3-1}} \, dx\) |
| \(\Big \downarrow \) 2577 |
| \(\displaystyle \int \frac {1}{\sqrt [3]{x^3-1}}dx-2 \int \frac {1}{(x+1) \sqrt [3]{x^3-1}}dx\) |
| \(\Big \downarrow \) 769 |
| \(\displaystyle -2 \int \frac {1}{(x+1) \sqrt [3]{x^3-1}}dx+\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )\) |
| \(\Big \downarrow \) 2574 |
| \(\displaystyle \frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-2 \left (\frac {\sqrt {3} \arctan \left (\frac {1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2}}-\frac {3 \log \left (2^{2/3} \sqrt [3]{x^3-1}-x+1\right )}{4 \sqrt [3]{2}}+\frac {\log \left ((1-x) (x+1)^2\right )}{4 \sqrt [3]{2}}\right )-\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )\) |
ArcTan[(1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[3]]/Sqrt[3] - Log[-x + (-1 + x^3) ^(1/3)]/2 - 2*((Sqrt[3]*ArcTan[(1 - (2^(1/3)*(1 - x))/(-1 + x^3)^(1/3))/Sq rt[3]])/(2*2^(1/3)) + Log[(1 - x)*(1 + x)^2]/(4*2^(1/3)) - (3*Log[1 - x + 2^(2/3)*(-1 + x^3)^(1/3)])/(4*2^(1/3)))
3.27.85.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]* (x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^ 3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]
Int[1/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(1/3)), x_Symbol] :> Simp[ Sqrt[3]*(ArcTan[(1 - 2^(1/3)*Rt[b, 3]*((c - d*x)/(d*(a + b*x^3)^(1/3))))/Sq rt[3]]/(2^(4/3)*Rt[b, 3]*c)), x] + (Simp[Log[(c + d*x)^2*(c - d*x)]/(2^(7/3 )*Rt[b, 3]*c), x] - Simp[(3*Log[Rt[b, 3]*(c - d*x) + 2^(2/3)*d*(a + b*x^3)^ (1/3)])/(2^(7/3)*Rt[b, 3]*c), x]) /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^3 + a*d^3, 0]
Int[((e_.) + (f_.)*(x_))/(((c_.) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(1/3)) , x_Symbol] :> Simp[f/d Int[1/(a + b*x^3)^(1/3), x], x] + Simp[(d*e - c*f )/d Int[1/((c + d*x)*(a + b*x^3)^(1/3)), x], x] /; FreeQ[{a, b, c, d, e, f}, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 11.90 (sec) , antiderivative size = 1285, normalized size of antiderivative = 5.29
-1/3*ln(-RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-4)^ 4*x^3-6*(x^3-1)^(2/3)*RootOf(_Z^3-4)^2*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_ Z^3-4)+_Z^2)*x+2*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z ^3-4)^2*x^3-4*RootOf(_Z^3-4)^2*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_ Z^2)-24*x^2*(x^3-1)^(1/3)+8*x^3-8)+1/2*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_ Z^3-4)+_Z^2)*ln((2*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf( _Z^3-4)^3*x+5*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^ 3-4)^2*x-4*(x^3-1)^(2/3)*RootOf(_Z^3-4)^2*RootOf(RootOf(_Z^3-4)^2+_Z*RootO f(_Z^3-4)+_Z^2)+4*(x^3-1)^(1/3)*RootOf(_Z^3-4)^2*x-9*RootOf(RootOf(_Z^3-4) ^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)*(x^3-1)^(1/3)*x-4*(x^3-1)^(1/3)* RootOf(_Z^3-4)^2+9*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf( _Z^3-4)*(x^3-1)^(1/3)+14*RootOf(_Z^3-4)*x^2+35*RootOf(RootOf(_Z^3-4)^2+_Z* RootOf(_Z^3-4)+_Z^2)*x^2+12*RootOf(_Z^3-4)*x+30*RootOf(RootOf(_Z^3-4)^2+_Z *RootOf(_Z^3-4)+_Z^2)*x-52*(x^3-1)^(2/3)+14*RootOf(_Z^3-4)+35*RootOf(RootO f(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2))/(1+x)^2)+1/2*RootOf(_Z^3-4)*ln(-(5*Ro otOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^3*x+2*RootOf( RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-4)^2*x-4*(x^3-1)^(2 /3)*RootOf(_Z^3-4)^2*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)+4*(x^ 3-1)^(1/3)*RootOf(_Z^3-4)^2*x+13*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4) +_Z^2)*RootOf(_Z^3-4)*(x^3-1)^(1/3)*x-4*(x^3-1)^(1/3)*RootOf(_Z^3-4)^2-...
Time = 1.62 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.52 \[ \int \frac {-1+x}{(1+x) \sqrt [3]{-1+x^3}} \, dx=\frac {1}{6} \cdot 4^{\frac {1}{3}} \sqrt {3} \arctan \left (\frac {4 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (x^{4} + 2 \, x^{3} + 2 \, x^{2} + 2 \, x + 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 2 \cdot 4^{\frac {1}{3}} \sqrt {3} {\left (5 \, x^{5} - 5 \, x^{4} + 6 \, x^{3} - 6 \, x^{2} + 5 \, x - 5\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} + \sqrt {3} {\left (13 \, x^{6} + 2 \, x^{5} + 19 \, x^{4} - 4 \, x^{3} + 19 \, x^{2} + 2 \, x + 13\right )}}{3 \, {\left (3 \, x^{6} - 18 \, x^{5} - 3 \, x^{4} - 28 \, x^{3} - 3 \, x^{2} - 18 \, x + 3\right )}}\right ) + \frac {1}{3} \, \sqrt {3} \arctan \left (-\frac {4 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 2 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{3} - 1\right )}}{9 \, x^{3} - 1}\right ) - \frac {1}{12} \cdot 4^{\frac {1}{3}} \log \left (\frac {8 \cdot 4^{\frac {1}{3}} {\left (x^{3} - 1\right )}^{\frac {2}{3}} {\left (x^{2} + 1\right )} + 4^{\frac {2}{3}} {\left (5 \, x^{4} + 6 \, x^{2} + 5\right )} + 4 \, {\left (3 \, x^{3} - x^{2} + x - 3\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1}\right ) + \frac {1}{6} \cdot 4^{\frac {1}{3}} \log \left (\frac {4^{\frac {2}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} + 4^{\frac {1}{3}} {\left (x^{2} + 2 \, x + 1\right )} - 4 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2} + 2 \, x + 1}\right ) - \frac {1}{6} \, \log \left (-3 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + 1\right ) \]
1/6*4^(1/3)*sqrt(3)*arctan(1/3*(4*4^(2/3)*sqrt(3)*(x^4 + 2*x^3 + 2*x^2 + 2 *x + 1)*(x^3 - 1)^(2/3) + 2*4^(1/3)*sqrt(3)*(5*x^5 - 5*x^4 + 6*x^3 - 6*x^2 + 5*x - 5)*(x^3 - 1)^(1/3) + sqrt(3)*(13*x^6 + 2*x^5 + 19*x^4 - 4*x^3 + 1 9*x^2 + 2*x + 13))/(3*x^6 - 18*x^5 - 3*x^4 - 28*x^3 - 3*x^2 - 18*x + 3)) + 1/3*sqrt(3)*arctan(-(4*sqrt(3)*(x^3 - 1)^(1/3)*x^2 - 2*sqrt(3)*(x^3 - 1)^ (2/3)*x + sqrt(3)*(x^3 - 1))/(9*x^3 - 1)) - 1/12*4^(1/3)*log((8*4^(1/3)*(x ^3 - 1)^(2/3)*(x^2 + 1) + 4^(2/3)*(5*x^4 + 6*x^2 + 5) + 4*(3*x^3 - x^2 + x - 3)*(x^3 - 1)^(1/3))/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1)) + 1/6*4^(1/3)*log( (4^(2/3)*(x^3 - 1)^(1/3)*(x - 1) + 4^(1/3)*(x^2 + 2*x + 1) - 4*(x^3 - 1)^( 2/3))/(x^2 + 2*x + 1)) - 1/6*log(-3*(x^3 - 1)^(1/3)*x^2 + 3*(x^3 - 1)^(2/3 )*x + 1)
\[ \int \frac {-1+x}{(1+x) \sqrt [3]{-1+x^3}} \, dx=\int \frac {x - 1}{\sqrt [3]{\left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 1\right )}\, dx \]
\[ \int \frac {-1+x}{(1+x) \sqrt [3]{-1+x^3}} \, dx=\int { \frac {x - 1}{{\left (x^{3} - 1\right )}^{\frac {1}{3}} {\left (x + 1\right )}} \,d x } \]
\[ \int \frac {-1+x}{(1+x) \sqrt [3]{-1+x^3}} \, dx=\int { \frac {x - 1}{{\left (x^{3} - 1\right )}^{\frac {1}{3}} {\left (x + 1\right )}} \,d x } \]
Timed out. \[ \int \frac {-1+x}{(1+x) \sqrt [3]{-1+x^3}} \, dx=\int \frac {x-1}{{\left (x^3-1\right )}^{1/3}\,\left (x+1\right )} \,d x \]