\(\int \frac {1+3 x}{(-1+3 x) \sqrt [3]{-x+x^3}} \, dx\) [10]

Optimal result

Integrand size = 24, antiderivative size = 102 \[ \int \frac {1+3 x}{(-1+3 x) \sqrt [3]{-x+x^3}} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-x+x^3}}{-2+2 x+\sqrt [3]{-x+x^3}}\right )-\log \left (1-x+\sqrt [3]{-x+x^3}\right )+\frac {1}{2} \log \left (1-2 x+x^2+(-1+x) \sqrt [3]{-x+x^3}+\left (-x+x^3\right )^{2/3}\right ) \] Output:

-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))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 15.27 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00 \[ \int \frac {1+3 x}{(-1+3 x) \sqrt [3]{-x+x^3}} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-x+x^3}}{-2+2 x+\sqrt [3]{-x+x^3}}\right )-\log \left (1-x+\sqrt [3]{-x+x^3}\right )+\frac {1}{2} \log \left (1-2 x+x^2+(-1+x) \sqrt [3]{-x+x^3}+\left (-x+x^3\right )^{2/3}\right ) \] Input:

Integrate[(1 + 3*x)/((-1 + 3*x)*(-x + x^3)^(1/3)),x]
 

Output:

-(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
 

Rubi [F]

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.

\(\displaystyle \int \frac {3 x+1}{(3 x-1) \sqrt [3]{x^3-x}} \, dx\)

\(\Big \downarrow \) 2467

\(\displaystyle \frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \int -\frac {3 x+1}{(1-3 x) \sqrt [3]{x} \sqrt [3]{x^2-1}}dx}{\sqrt [3]{x^3-x}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \int \frac {3 x+1}{(1-3 x) \sqrt [3]{x} \sqrt [3]{x^2-1}}dx}{\sqrt [3]{x^3-x}}\)

\(\Big \downarrow \) 2035

\(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^2-1} \int \frac {\sqrt [3]{x} (3 x+1)}{(1-3 x) \sqrt [3]{x^2-1}}d\sqrt [3]{x}}{\sqrt [3]{x^3-x}}\)

\(\Big \downarrow \) 7276

\(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^2-1} \int \left (\frac {2 \sqrt [3]{x}}{(1-3 x) \sqrt [3]{x^2-1}}-\frac {\sqrt [3]{x}}{\sqrt [3]{x^2-1}}\right )d\sqrt [3]{x}}{\sqrt [3]{x^3-x}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^2-1} \left (2 \int \frac {\sqrt [3]{x}}{(1-3 x) \sqrt [3]{x^2-1}}d\sqrt [3]{x}-\frac {\arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{4} \log \left (x^{2/3}-\sqrt [3]{x^2-1}\right )\right )}{\sqrt [3]{x^3-x}}\)

Input:

Int[(1 + 3*x)/((-1 + 3*x)*(-x + x^3)^(1/3)),x]
 

Output:

$Aborted
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.76 (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 )\)	\(607\)

Input:

int((1+3*x)/(-1+3*x)/(x^3-x)^(1/3),x,method=_RETURNVERBOSE)
 

Output:

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*RootOf(_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-135 
8)/(-1+3*x))-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+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 
))*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+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*Root 
Of(_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))
 

Fricas [A] (verification not implemented)

Time = 0.59 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.04 \[ \int \frac {1+3 x}{(-1+3 x) \sqrt [3]{-x+x^3}} \, dx=\sqrt {3} \arctan \left (\frac {612314840 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (x - 1\right )} + \sqrt {3} {\left (1609127381 \, x^{2} + 1235276981 \, x + 124616800\right )} + 2605939922 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {2}{3}}}{2990437623 \, x^{2} + 3108349623 \, x - 39304000}\right ) - \frac {1}{2} \, \log \left (\frac {3 \, {\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (x - 1\right )} + 3 \, x - 3 \, {\left (x^{3} - x\right )}^{\frac {2}{3}} - 1}{3 \, x - 1}\right ) \] Input:

integrate((1+3*x)/(-1+3*x)/(x^3-x)^(1/3),x, algorithm="fricas")
 

Output:

sqrt(3)*arctan((612314840*sqrt(3)*(x^3 - x)^(1/3)*(x - 1) + sqrt(3)*(16091 
27381*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))
 

Sympy [F]

\[ \int \frac {1+3 x}{(-1+3 x) \sqrt [3]{-x+x^3}} \, dx=\int \frac {3 x + 1}{\sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )} \left (3 x - 1\right )}\, dx \] Input:

integrate((1+3*x)/(-1+3*x)/(x**3-x)**(1/3),x)
 

Output:

Integral((3*x + 1)/((x*(x - 1)*(x + 1))**(1/3)*(3*x - 1)), x)
 

Maxima [F]

\[ \int \frac {1+3 x}{(-1+3 x) \sqrt [3]{-x+x^3}} \, dx=\int { \frac {3 \, x + 1}{{\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (3 \, x - 1\right )}} \,d x } \] Input:

integrate((1+3*x)/(-1+3*x)/(x^3-x)^(1/3),x, algorithm="maxima")
 

Output:

integrate((3*x + 1)/((x^3 - x)^(1/3)*(3*x - 1)), x)
 

Giac [F]

\[ \int \frac {1+3 x}{(-1+3 x) \sqrt [3]{-x+x^3}} \, dx=\int { \frac {3 \, x + 1}{{\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (3 \, x - 1\right )}} \,d x } \] Input:

integrate((1+3*x)/(-1+3*x)/(x^3-x)^(1/3),x, algorithm="giac")
 

Output:

integrate((3*x + 1)/((x^3 - x)^(1/3)*(3*x - 1)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1+3 x}{(-1+3 x) \sqrt [3]{-x+x^3}} \, dx=\int \frac {3\,x+1}{{\left (x^3-x\right )}^{1/3}\,\left (3\,x-1\right )} \,d x \] Input:

int((3*x + 1)/((x^3 - x)^(1/3)*(3*x - 1)),x)
 

Output:

int((3*x + 1)/((x^3 - x)^(1/3)*(3*x - 1)), x)
 

Reduce [F]

\[ \int \frac {1+3 x}{(-1+3 x) \sqrt [3]{-x+x^3}} \, dx=3 \left (\int \frac {x}{3 x^{\frac {4}{3}} \left (x^{2}-1\right )^{\frac {1}{3}}-x^{\frac {1}{3}} \left (x^{2}-1\right )^{\frac {1}{3}}}d x \right )+\int \frac {1}{3 x^{\frac {4}{3}} \left (x^{2}-1\right )^{\frac {1}{3}}-x^{\frac {1}{3}} \left (x^{2}-1\right )^{\frac {1}{3}}}d x \] Input:

int((1+3*x)/(-1+3*x)/(x^3-x)^(1/3),x)
 

Output:

3*int(x/(3*x**(1/3)*(x**2 - 1)**(1/3)*x - x**(1/3)*(x**2 - 1)**(1/3)),x) + 
 int(1/(3*x**(1/3)*(x**2 - 1)**(1/3)*x - x**(1/3)*(x**2 - 1)**(1/3)),x)