Integrand size = 24, antiderivative size = 102 \[ \int \frac {1+3 x}{(-1+3 x) \sqrt [3]{-x+x^3}} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-x+x^3}}{-2+2 x+\sqrt [3]{-x+x^3}}\right )-\log \left (1-x+\sqrt [3]{-x+x^3}\right )+\frac {1}{2} \log \left (1-2 x+x^2+(-1+x) \sqrt [3]{-x+x^3}+\left (-x+x^3\right )^{2/3}\right ) \]
[Out] -3^(1/2)*arctan(3^(1/2)*(x^3-x)^(1/3)/(-2+2*x+(x^3-x)^(1/3)))-ln(1-x+( x^3-x)^(1/3))+1/2*ln(1-2*x+x^2+(-1+x)*(x^3-x)^(1/3)+(x^3-x)^(2/3))
\[ \int \frac {1+3 x}{(-1+3 x) \sqrt [3]{-x+x^3}} \, dx=\int \frac {1+3 x}{(-1+3 x) \sqrt [3]{-x+x^3}} \, dx \]
[In] Int[(1 + 3*x)/((-1 + 3*x)*(-x + x^3)^(1/3)),x]
[Out] (Sqrt[3]*x^(1/3)*(-1 + x^2)^(1/3)*ArcTan[(1 + (2*x^(2/3))/(-1 + x^2)^( 1/3))/Sqrt[3]])/(2*(-x + x^3)^(1/3)) - (3*x^(1/3)*(-1 + x^2)^(1/3)*Log [x^(2/3) - (-1 + x^2)^(1/3)])/(4*(-x + x^3)^(1/3)) + (6*x^(1/3)*(-1 + x^2)^(1/3)*Defer[Subst][Defer[Int][x/((-1 + 3*x^3)*(-1 + x^6)^(1/3)), x], x, x^(1/3)])/(-x + x^3)^(1/3)
Rubi steps \begin{align*} \text {integral}= \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{\sqrt [3]{-x+x^3}} \int \frac {1+3 x}{\sqrt [3]{x} (-1+3 x) \sqrt [3]{-1+x^2}} \, dx \\ = \frac {3 \sqrt [3]{x} \sqrt [3]{-1+x^2}}{\sqrt [3]{-x+x^3}} \text {Subst}\left (\int \frac {x \left (1+3 x^3\right )}{\left (-1+3 x^3\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right ) \\ = \frac {3 \sqrt [3]{x} \sqrt [3]{-1+x^2}}{\sqrt [3]{-x+x^3}} \text {Subst}\left (\int \left (\frac {x}{\sqrt [3]{-1+x^6}}+\frac {2 x}{\left (-1+3 x^3\right ) \sqrt [3]{-1+x^6}}\right ) \, dx,x,\sqrt [3]{x}\right ) \\ = \frac {3 \sqrt [3]{x} \sqrt [3]{-1+x^2}}{\sqrt [3]{-x+x^3}} \text {Subst}\left (\int \frac {x}{\sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )+\frac {6 \sqrt [3]{x} \sqrt [3]{-1+x^2}}{\sqrt [3]{-x+x^3}} \text {Subst}\left (\int \frac {x}{\left (-1+3 x^3\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right ) \\ = \frac {3 \sqrt [3]{x} \sqrt [3]{-1+x^2}}{2 \sqrt [3]{-x+x^3}} \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^3}} \, dx,x,x^{2/3}\right )+\frac {6 \sqrt [3]{x} \sqrt [3]{-1+x^2}}{\sqrt [3]{-x+x^3}} \text {Subst}\left (\int \frac {x}{\left (-1+3 x^3\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right ) \\ = \frac {\sqrt {3} \sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{2 \sqrt [3]{-x+x^3}}-\frac {3 \sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (x^{2/3}-\sqrt [3]{-1+x^2}\right )}{4 \sqrt [3]{-x+x^3}}+\frac {6 \sqrt [3]{x} \sqrt [3]{-1+x^2}}{\sqrt [3]{-x+x^3}} \text {Subst}\left (\int \frac {x}{\left (-1+3 x^3\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right ) \\ \end{align*}
Time = 15.22 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00 \[ \int \frac {1+3 x}{(-1+3 x) \sqrt [3]{-x+x^3}} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-x+x^3}}{-2+2 x+\sqrt [3]{-x+x^3}}\right )-\log \left (1-x+\sqrt [3]{-x+x^3}\right )+\frac {1}{2} \log \left (1-2 x+x^2+(-1+x) \sqrt [3]{-x+x^3}+\left (-x+x^3\right )^{2/3}\right ) \]
[In] Integrate[(1 + 3*x)/((-1 + 3*x)*(-x + x^3)^(1/3)),x]
[Out] -(Sqrt[3]*ArcTan[(Sqrt[3]*(-x + x^3)^(1/3))/(-2 + 2*x + (-x + x^3)^(1/ 3))]) - Log[1 - x + (-x + x^3)^(1/3)] + Log[1 - 2*x + x^2 + (-1 + x)*( -x + x^3)^(1/3) + (-x + x^3)^(2/3)]/2
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.81 (sec) , antiderivative size = 607, normalized size of antiderivative = 5.95
| method | result | size |
| trager | \(\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\frac {-1415 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+3234 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}+3234 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}} x +3679 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +4649 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-1107 \left (x^{3}-x \right )^{\frac {2}{3}}-3234 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}}-1107 \left (x^{3}-x \right )^{\frac {1}{3}} x -1698 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-4786 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -2522 x^{2}+1107 \left (x^{3}-x \right )^{\frac {1}{3}}+3145 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1552 x -1358}{-1+3 x}\right )-\ln \left (-\frac {1415 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+3234 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}+3234 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}} x -3679 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +1819 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-2127 \left (x^{3}-x \right )^{\frac {2}{3}}-3234 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}}-2127 \left (x^{3}-x \right )^{\frac {1}{3}} x +1698 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+2572 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -712 x^{2}+2127 \left (x^{3}-x \right )^{\frac {1}{3}}-251 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-445 x -89}{-1+3 x}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+\ln \left (-\frac {1415 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+3234 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}+3234 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}} x -3679 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +1819 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-2127 \left (x^{3}-x \right )^{\frac {2}{3}}-3234 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}}-2127 \left (x^{3}-x \right )^{\frac {1}{3}} x +1698 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+2572 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -712 x^{2}+2127 \left (x^{3}-x \right )^{\frac {1}{3}}-251 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-445 x -89}{-1+3 x}\right )\) | \(607\) |
[In] int((1+3*x)/(-1+3*x)/(x^3-x)^(1/3),x,method=_RETURNVERBOSE)
[Out] RootOf(_Z^2-_Z+1)*ln((-1415*RootOf(_Z^2-_Z+1)^2*x^2+3234*RootOf(_Z^2-_ Z+1)*(x^3-x)^(2/3)+3234*RootOf(_Z^2-_Z+1)*(x^3-x)^(1/3)*x+3679*RootOf( _Z^2-_Z+1)^2*x+4649*RootOf(_Z^2-_Z+1)*x^2-1107*(x^3-x)^(2/3)-3234*Root Of(_Z^2-_Z+1)*(x^3-x)^(1/3)-1107*(x^3-x)^(1/3)*x-1698*RootOf(_Z^2-_Z+1 )^2-4786*RootOf(_Z^2-_Z+1)*x-2522*x^2+1107*(x^3-x)^(1/3)+3145*RootOf(_ Z^2-_Z+1)+1552*x-1358)/(-1+3*x))-ln(-(1415*RootOf(_Z^2-_Z+1)^2*x^2+323 4*RootOf(_Z^2-_Z+1)*(x^3-x)^(2/3)+3234*RootOf(_Z^2-_Z+1)*(x^3-x)^(1/3) *x-3679*RootOf(_Z^2-_Z+1)^2*x+1819*RootOf(_Z^2-_Z+1)*x^2-2127*(x^3-x)^ (2/3)-3234*RootOf(_Z^2-_Z+1)*(x^3-x)^(1/3)-2127*(x^3-x)^(1/3)*x+1698*R ootOf(_Z^2-_Z+1)^2+2572*RootOf(_Z^2-_Z+1)*x-712*x^2+2127*(x^3-x)^(1/3) -251*RootOf(_Z^2-_Z+1)-445*x-89)/(-1+3*x))*RootOf(_Z^2-_Z+1)+ln(-(1415 *RootOf(_Z^2-_Z+1)^2*x^2+3234*RootOf(_Z^2-_Z+1)*(x^3-x)^(2/3)+3234*Roo tOf(_Z^2-_Z+1)*(x^3-x)^(1/3)*x-3679*RootOf(_Z^2-_Z+1)^2*x+1819*RootOf( _Z^2-_Z+1)*x^2-2127*(x^3-x)^(2/3)-3234*RootOf(_Z^2-_Z+1)*(x^3-x)^(1/3) -2127*(x^3-x)^(1/3)*x+1698*RootOf(_Z^2-_Z+1)^2+2572*RootOf(_Z^2-_Z+1)* x-712*x^2+2127*(x^3-x)^(1/3)-251*RootOf(_Z^2-_Z+1)-445*x-89)/(-1+3*x))
none
Time = 0.65 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.04 \[ \int \frac {1+3 x}{(-1+3 x) \sqrt [3]{-x+x^3}} \, dx=\sqrt {3} \arctan \left (\frac {612314840 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (x - 1\right )} + \sqrt {3} {\left (1609127381 \, x^{2} + 1235276981 \, x + 124616800\right )} + 2605939922 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {2}{3}}}{2990437623 \, x^{2} + 3108349623 \, x - 39304000}\right ) - \frac {1}{2} \, \log \left (\frac {3 \, {\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (x - 1\right )} + 3 \, x - 3 \, {\left (x^{3} - x\right )}^{\frac {2}{3}} - 1}{3 \, x - 1}\right ) \]
[In] integrate((1+3*x)/(-1+3*x)/(x^3-x)^(1/3),x, algorithm="fricas")
[Out] sqrt(3)*arctan((612314840*sqrt(3)*(x^3 - x)^(1/3)*(x - 1) + sqrt(3)*(1 609127381*x^2 + 1235276981*x + 124616800) + 2605939922*sqrt(3)*(x^3 - x)^(2/3))/(2990437623*x^2 + 3108349623*x - 39304000)) - 1/2*log((3*(x^ 3 - x)^(1/3)*(x - 1) + 3*x - 3*(x^3 - x)^(2/3) - 1)/(3*x - 1))
\[ \int \frac {1+3 x}{(-1+3 x) \sqrt [3]{-x+x^3}} \, dx=\int \frac {3 x + 1}{\sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )} \left (3 x - 1\right )}\, dx \]
[In] integrate((1+3*x)/(-1+3*x)/(x**3-x)**(1/3),x)
[Out] Integral((3*x + 1)/((x*(x - 1)*(x + 1))**(1/3)*(3*x - 1)), x)
\[ \int \frac {1+3 x}{(-1+3 x) \sqrt [3]{-x+x^3}} \, dx=\int { \frac {3 \, x + 1}{{\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (3 \, x - 1\right )}} \,d x } \]
[In] integrate((1+3*x)/(-1+3*x)/(x^3-x)^(1/3),x, algorithm="maxima")
[Out] integrate((3*x + 1)/((x^3 - x)^(1/3)*(3*x - 1)), x)
\[ \int \frac {1+3 x}{(-1+3 x) \sqrt [3]{-x+x^3}} \, dx=\int { \frac {3 \, x + 1}{{\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (3 \, x - 1\right )}} \,d x } \]
[In] integrate((1+3*x)/(-1+3*x)/(x^3-x)^(1/3),x, algorithm="giac")
[Out] integrate((3*x + 1)/((x^3 - x)^(1/3)*(3*x - 1)), x)
Timed out. \[ \int \frac {1+3 x}{(-1+3 x) \sqrt [3]{-x+x^3}} \, dx=\int \frac {3\,x+1}{{\left (x^3-x\right )}^{1/3}\,\left (3\,x-1\right )} \,d x \]
[In] int((3*x + 1)/((x^3 - x)^(1/3)*(3*x - 1)),x)
[Out] int((3*x + 1)/((x^3 - x)^(1/3)*(3*x - 1)), x)