Integrand size = 22, antiderivative size = 280 \[ \int \frac {\sqrt [3]{1-x^3}}{1-x+x^2} \, dx=\frac {\sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} (-1+x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3}}+\frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\arctan \left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log \left (-3 (-1+x) \left (1-x+x^2\right )\right )}{2\ 2^{2/3}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}+\frac {3 \log \left (-\sqrt [3]{2} (-1+x)+\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}+\frac {1}{2} \log \left (x+\sqrt [3]{1-x^3}\right )-\frac {\log \left (\sqrt [3]{2} x+\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}} \]
-1/4*ln(-3*(-1+x)*(x^2-x+1))*2^(1/3)+1/4*ln(2^(1/3)-(-x^3+1)^(1/3))*2^(1/3 )+3/4*ln(-2^(1/3)*(-1+x)+(-x^3+1)^(1/3))*2^(1/3)+1/2*ln(x+(-x^3+1)^(1/3))- 1/4*ln(2^(1/3)*x+(-x^3+1)^(1/3))*2^(1/3)+1/3*arctan(1/3*(1-2*x/(-x^3+1)^(1 /3))*3^(1/2))*3^(1/2)-1/6*2^(1/3)*arctan(1/3*(1-2*2^(1/3)*x/(-x^3+1)^(1/3) )*3^(1/2))*3^(1/2)-1/6*2^(1/3)*arctan(1/3*(1+2^(2/3)*(-x^3+1)^(1/3))*3^(1/ 2))*3^(1/2)+1/2*arctan(1/3*(1+2*2^(1/3)*(-1+x)/(-x^3+1)^(1/3))*3^(1/2))*3^ (1/2)*2^(1/3)
\[ \int \frac {\sqrt [3]{1-x^3}}{1-x+x^2} \, dx=\int \frac {\sqrt [3]{1-x^3}}{1-x+x^2} \, dx \]
Time = 0.55 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.46, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2583, 2009}
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.
| \(\displaystyle \int \frac {\sqrt [3]{1-x^3}}{x^2-x+1} \, dx\) |
| \(\Big \downarrow \) 2583 |
| \(\displaystyle \int \left (\frac {\sqrt [3]{1-x^3} x}{x^3+1}+\frac {\sqrt [3]{1-x^3}}{x^3+1}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt [3]{2} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\arctan \left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\sqrt [3]{2} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\log \left (x^3+1\right )}{3\ 2^{2/3}}+\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}-\frac {\log \left (\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}+\frac {1}{3} \sqrt [3]{2} \log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )-\frac {\log \left (\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}\right )}{6\ 2^{2/3}}+\frac {1}{2} \log \left (-\sqrt [3]{1-x^3}-x\right )-\frac {\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{2^{2/3}}\) |
(2^(1/3)*ArcTan[(1 - (2*2^(1/3)*(1 - x))/(1 - x^3)^(1/3))/Sqrt[3]])/Sqrt[3 ] + ArcTan[(1 + (2^(1/3)*(1 - x))/(1 - x^3)^(1/3))/Sqrt[3]]/(2^(2/3)*Sqrt[ 3]) + ArcTan[(1 - (2*x)/(1 - x^3)^(1/3))/Sqrt[3]]/Sqrt[3] - (2^(1/3)*ArcTa n[(1 - (2*2^(1/3)*x)/(1 - x^3)^(1/3))/Sqrt[3]])/Sqrt[3] + Log[1 + x^3]/(3* 2^(2/3)) + Log[2^(2/3) - (1 - x)/(1 - x^3)^(1/3)]/(3*2^(2/3)) - Log[1 + (2 ^(2/3)*(1 - x)^2)/(1 - x^3)^(2/3) - (2^(1/3)*(1 - x))/(1 - x^3)^(1/3)]/(3* 2^(2/3)) + (2^(1/3)*Log[1 + (2^(1/3)*(1 - x))/(1 - x^3)^(1/3)])/3 - Log[2* 2^(1/3) + (1 - x)^2/(1 - x^3)^(2/3) + (2^(2/3)*(1 - x))/(1 - x^3)^(1/3)]/( 6*2^(2/3)) + Log[-x - (1 - x^3)^(1/3)]/2 - Log[-(2^(1/3)*x) - (1 - x^3)^(1 /3)]/2^(2/3)
3.1.59.3.1 Defintions of rubi rules used
Int[(Px_.)*((c_) + (d_.)*(x_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^3)^(p _.), x_Symbol] :> Simp[1/c^q Int[ExpandIntegrand[(c^3 - d^3*x^3)^q*(a + b *x^3)^p, Px/(c - d*x)^q, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && Poly Q[Px, x] && EqQ[d^2 - c*e, 0] && ILtQ[q, 0] && RationalQ[p] && EqQ[Denomina tor[p], 3]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 21.32 (sec) , antiderivative size = 1410, normalized size of antiderivative = 5.04
1/3*ln(RootOf(_Z^2+_Z+1)^2*x^3-3*RootOf(_Z^2+_Z+1)*(-x^3+1)^(2/3)*x+3*Root Of(_Z^2+_Z+1)*(-x^3+1)^(1/3)*x^2-2*RootOf(_Z^2+_Z+1)*x^3+x^3+RootOf(_Z^2+_ Z+1)-1)*RootOf(_Z^2+_Z+1)-1/3*RootOf(_Z^2+_Z+1)*ln(RootOf(_Z^2+_Z+1)^2*x^3 +3*RootOf(_Z^2+_Z+1)*(-x^3+1)^(2/3)*x-3*RootOf(_Z^2+_Z+1)*(-x^3+1)^(1/3)*x ^2+4*RootOf(_Z^2+_Z+1)*x^3+3*x*(-x^3+1)^(2/3)-3*x^2*(-x^3+1)^(1/3)+4*x^3-R ootOf(_Z^2+_Z+1)-2)-1/3*ln(RootOf(_Z^2+_Z+1)^2*x^3+3*RootOf(_Z^2+_Z+1)*(-x ^3+1)^(2/3)*x-3*RootOf(_Z^2+_Z+1)*(-x^3+1)^(1/3)*x^2+4*RootOf(_Z^2+_Z+1)*x ^3+3*x*(-x^3+1)^(2/3)-3*x^2*(-x^3+1)^(1/3)+4*x^3-RootOf(_Z^2+_Z+1)-2)-1/18 *ln(-(12*(-x^3+1)^(1/3)*RootOf(_Z^3-324*RootOf(_Z^2+_Z+1)-162)*RootOf(_Z^2 +_Z+1)*x^3+RootOf(_Z^3-324*RootOf(_Z^2+_Z+1)-162)^2*x^4-36*(-x^3+1)^(1/3)* RootOf(_Z^3-324*RootOf(_Z^2+_Z+1)-162)*RootOf(_Z^2+_Z+1)*x^2+6*(-x^3+1)^(1 /3)*RootOf(_Z^3-324*RootOf(_Z^2+_Z+1)-162)*x^3+2*x^3*RootOf(_Z^3-324*RootO f(_Z^2+_Z+1)-162)^2+108*(-x^3+1)^(2/3)*x^2+12*(-x^3+1)^(1/3)*RootOf(_Z^3-3 24*RootOf(_Z^2+_Z+1)-162)*RootOf(_Z^2+_Z+1)*x-18*(-x^3+1)^(1/3)*RootOf(_Z^ 3-324*RootOf(_Z^2+_Z+1)-162)*x^2-RootOf(_Z^3-324*RootOf(_Z^2+_Z+1)-162)^2* x^2-108*x*(-x^3+1)^(2/3)+6*(-x^3+1)^(1/3)*RootOf(_Z^3-324*RootOf(_Z^2+_Z+1 )-162)*x-2*RootOf(_Z^3-324*RootOf(_Z^2+_Z+1)-162)^2*x+RootOf(_Z^3-324*Root Of(_Z^2+_Z+1)-162)^2)/(x^2-x+1)^2)*RootOf(_Z^3-324*RootOf(_Z^2+_Z+1)-162)* RootOf(_Z^2+_Z+1)-1/18*ln(-(12*(-x^3+1)^(1/3)*RootOf(_Z^3-324*RootOf(_Z^2+ _Z+1)-162)*RootOf(_Z^2+_Z+1)*x^3+RootOf(_Z^3-324*RootOf(_Z^2+_Z+1)-162)...
Result contains complex when optimal does not.
Time = 5.35 (sec) , antiderivative size = 3880, normalized size of antiderivative = 13.86 \[ \int \frac {\sqrt [3]{1-x^3}}{1-x+x^2} \, dx=\text {Too large to display} \]
1/72*sqrt(3)*(sqrt(-3)*(-4)^(1/6) + (-4)^(1/6))*log(3*(6*sqrt(-3)*(-4)^(1/ 3)*(26655*x^12 - 185476*x^11 + 155872*x^10 + 361508*x^9 - 363117*x^8 - 876 12*x^7 - 197936*x^6 + 492924*x^5 - 182367*x^4 - 39436*x^3 + 18254*x^2 + 36 52*x - 1278) + 48*(11866*x^10 - 16425*x^9 - 125794*x^8 + 251931*x^7 - 7118 7*x^6 - 79049*x^5 - 2745*x^4 + 52032*x^3 - 20629*x^2 - sqrt(3)*(3104*I*x^1 0 - 43815*I*x^9 + 84520*I*x^8 + 11329*I*x^7 - 92013*I*x^6 + 5291*I*x^5 + 5 3855*I*x^4 - 20262*I*x^3 - 2009*I*x^2 + 1278*I*x) + 2008*x)*(-x^3 + 1)^(2/ 3) + sqrt(3)*(sqrt(-3)*(-4)^(5/6)*(31397*x^12 + 113940*x^11 - 831396*x^10 + 973364*x^9 - 140709*x^8 + 407484*x^7 - 1009896*x^6 + 313212*x^5 + 248121 *x^4 - 75940*x^3 - 48198*x^2 + 19716*x - 2008) - (-4)^(5/6)*(31397*x^12 + 113940*x^11 - 831396*x^10 + 973364*x^9 - 140709*x^8 + 407484*x^7 - 1009896 *x^6 + 313212*x^5 + 248121*x^4 - 75940*x^3 - 48198*x^2 + 19716*x - 2008)) - 6*(-4)^(1/3)*(26655*x^12 - 185476*x^11 + 155872*x^10 + 361508*x^9 - 3631 17*x^8 - 87612*x^7 - 197936*x^6 + 492924*x^5 - 182367*x^4 - 39436*x^3 + 18 254*x^2 + 3652*x - 1278) - 6*(-x^3 + 1)^(1/3)*(sqrt(-3)*(-4)^(2/3)*(1459*x ^11 + 94937*x^10 - 314364*x^9 + 204807*x^8 + 73586*x^7 + 103515*x^6 - 2639 73*x^5 + 67714*x^4 + 54774*x^3 - 25376*x^2 + 2008*x) + (-4)^(2/3)*(1459*x^ 11 + 94937*x^10 - 314364*x^9 + 204807*x^8 + 73586*x^7 + 103515*x^6 - 26397 3*x^5 + 67714*x^4 + 54774*x^3 - 25376*x^2 + 2008*x) + 2*sqrt(3)*(sqrt(-3)* (-4)^(1/6)*(12049*x^11 - 48557*x^10 - 31048*x^9 + 203745*x^8 - 117748*x...
\[ \int \frac {\sqrt [3]{1-x^3}}{1-x+x^2} \, dx=\int \frac {\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )}}{x^{2} - x + 1}\, dx \]
\[ \int \frac {\sqrt [3]{1-x^3}}{1-x+x^2} \, dx=\int { \frac {{\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{2} - x + 1} \,d x } \]
\[ \int \frac {\sqrt [3]{1-x^3}}{1-x+x^2} \, dx=\int { \frac {{\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{2} - x + 1} \,d x } \]
Timed out. \[ \int \frac {\sqrt [3]{1-x^3}}{1-x+x^2} \, dx=\int \frac {{\left (1-x^3\right )}^{1/3}}{x^2-x+1} \,d x \]