3.20.1 \(\int \frac {-1+x}{x \sqrt [3]{1+2 x+2 x^2+x^3}} \, dx\) [1901]

3.20.1.1 Optimal result

Integrand size = 24, antiderivative size = 132 \[ \int \frac {-1+x}{x \sqrt [3]{1+2 x+2 x^2+x^3}} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{1+2 x+2 x^2+x^3}}{2+2 x+\sqrt [3]{1+2 x+2 x^2+x^3}}\right )-\log \left (-1-x+\sqrt [3]{1+2 x+2 x^2+x^3}\right )+\frac {1}{2} \log \left (1+2 x+x^2+(1+x) \sqrt [3]{1+2 x+2 x^2+x^3}+\left (1+2 x+2 x^2+x^3\right )^{2/3}\right ) \]

output

-3^(1/2)*arctan(3^(1/2)*(x^3+2*x^2+2*x+1)^(1/3)/(2+2*x+(x^3+2*x^2+2*x+1)^( 
1/3)))-ln(-1-x+(x^3+2*x^2+2*x+1)^(1/3))+1/2*ln(1+2*x+x^2+(1+x)*(x^3+2*x^2+ 
2*x+1)^(1/3)+(x^3+2*x^2+2*x+1)^(2/3))
 

3.20.1.2 Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x}{x \sqrt [3]{1+2 x+2 x^2+x^3}} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{1+2 x+2 x^2+x^3}}{2+2 x+\sqrt [3]{1+2 x+2 x^2+x^3}}\right )-\log \left (-1-x+\sqrt [3]{1+2 x+2 x^2+x^3}\right )+\frac {1}{2} \log \left (1+2 x+x^2+(1+x) \sqrt [3]{1+2 x+2 x^2+x^3}+\left (1+2 x+2 x^2+x^3\right )^{2/3}\right ) \]

input

Integrate[(-1 + x)/(x*(1 + 2*x + 2*x^2 + x^3)^(1/3)),x]
 

output

-(Sqrt[3]*ArcTan[(Sqrt[3]*(1 + 2*x + 2*x^2 + x^3)^(1/3))/(2 + 2*x + (1 + 2 
*x + 2*x^2 + x^3)^(1/3))]) - Log[-1 - x + (1 + 2*x + 2*x^2 + x^3)^(1/3)] + 
 Log[1 + 2*x + x^2 + (1 + x)*(1 + 2*x + 2*x^2 + x^3)^(1/3) + (1 + 2*x + 2* 
x^2 + x^3)^(2/3)]/2
 

3.20.1.3 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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 \sqrt [3]{x+1} \sqrt [3]{x^2+x+1} \text {Subst}\left (\int \frac {1}{\left (x-\frac {2}{3}\right ) \sqrt [3]{x+\frac {1}{3}} \sqrt [3]{x^2-\frac {x}{3}+\frac {7}{9}}}dx,x,x+\frac {2}{3}\right )}{\sqrt [3]{(3 x+2)^3+6 (3 x+2)+7}}+\frac {9 (x+1) \sqrt [3]{1-\frac {2 (x+1)}{1-i \sqrt {3}}} \sqrt [3]{1-\frac {2 (x+1)}{1+i \sqrt {3}}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},\frac {1}{3},\frac {5}{3},\frac {2 (x+1)}{1-i \sqrt {3}},\frac {2 (x+1)}{1+i \sqrt {3}}\right )}{2 \sqrt [3]{(3 x+2)^3+6 (3 x+2)+7}}\)

input

Int[(-1 + x)/(x*(1 + 2*x + 2*x^2 + x^3)^(1/3)),x]
 

output

$Aborted
 

3.20.1.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.20.1.4 Maple [C] (verified)

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

Time = 0.79 (sec) , antiderivative size = 455, normalized size of antiderivative = 3.45

method	result	size
trager	\(\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+111 \left (x^{3}+2 x^{2}+2 x +1\right )^{\frac {2}{3}}+111 x \left (x^{3}+2 x^{2}+2 x +1\right )^{\frac {1}{3}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-39 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +110 x^{2}+111 \left (x^{3}+2 x^{2}+2 x +1\right )^{\frac {1}{3}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+187 x +110}{x}\right )-\ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+111 \left (x^{3}+2 x^{2}+2 x +1\right )^{\frac {2}{3}}+111 x \left (x^{3}+2 x^{2}+2 x +1\right )^{\frac {1}{3}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+35 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +110 x^{2}+111 \left (x^{3}+2 x^{2}+2 x +1\right )^{\frac {1}{3}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+150 x +110}{x}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+\ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+111 \left (x^{3}+2 x^{2}+2 x +1\right )^{\frac {2}{3}}+111 x \left (x^{3}+2 x^{2}+2 x +1\right )^{\frac {1}{3}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+35 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +110 x^{2}+111 \left (x^{3}+2 x^{2}+2 x +1\right )^{\frac {1}{3}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+150 x +110}{x}\right )\)	\(455\)

input

int((-1+x)/x/(x^3+2*x^2+2*x+1)^(1/3),x,method=_RETURNVERBOSE)
 

output

RootOf(_Z^2-_Z+1)*ln((-RootOf(_Z^2-_Z+1)^2*x^2+2*RootOf(_Z^2-_Z+1)^2*x+Roo 
tOf(_Z^2-_Z+1)*x^2+111*(x^3+2*x^2+2*x+1)^(2/3)+111*x*(x^3+2*x^2+2*x+1)^(1/ 
3)-RootOf(_Z^2-_Z+1)^2-39*RootOf(_Z^2-_Z+1)*x+110*x^2+111*(x^3+2*x^2+2*x+1 
)^(1/3)+RootOf(_Z^2-_Z+1)+187*x+110)/x)-ln((-RootOf(_Z^2-_Z+1)^2*x^2+2*Roo 
tOf(_Z^2-_Z+1)^2*x+RootOf(_Z^2-_Z+1)*x^2+111*(x^3+2*x^2+2*x+1)^(2/3)+111*x 
*(x^3+2*x^2+2*x+1)^(1/3)-RootOf(_Z^2-_Z+1)^2+35*RootOf(_Z^2-_Z+1)*x+110*x^ 
2+111*(x^3+2*x^2+2*x+1)^(1/3)+RootOf(_Z^2-_Z+1)+150*x+110)/x)*RootOf(_Z^2- 
_Z+1)+ln((-RootOf(_Z^2-_Z+1)^2*x^2+2*RootOf(_Z^2-_Z+1)^2*x+RootOf(_Z^2-_Z+ 
1)*x^2+111*(x^3+2*x^2+2*x+1)^(2/3)+111*x*(x^3+2*x^2+2*x+1)^(1/3)-RootOf(_Z 
^2-_Z+1)^2+35*RootOf(_Z^2-_Z+1)*x+110*x^2+111*(x^3+2*x^2+2*x+1)^(1/3)+Root 
Of(_Z^2-_Z+1)+150*x+110)/x)
 

3.20.1.5 Fricas [A] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.93 \[ \int \frac {-1+x}{x \sqrt [3]{1+2 x+2 x^2+x^3}} \, dx=\sqrt {3} \arctan \left (-\frac {4 \, \sqrt {3} {\left (x^{3} + 2 \, x^{2} + 2 \, x + 1\right )}^{\frac {1}{3}} {\left (x + 1\right )} + \sqrt {3} {\left (x^{2} + x + 1\right )} - 2 \, \sqrt {3} {\left (x^{3} + 2 \, x^{2} + 2 \, x + 1\right )}^{\frac {2}{3}}}{9 \, x^{2} + 17 \, x + 9}\right ) - \frac {1}{2} \, \log \left (-\frac {3 \, {\left (x^{3} + 2 \, x^{2} + 2 \, x + 1\right )}^{\frac {1}{3}} {\left (x + 1\right )} - x - 3 \, {\left (x^{3} + 2 \, x^{2} + 2 \, x + 1\right )}^{\frac {2}{3}}}{x}\right ) \]

input

integrate((-1+x)/x/(x^3+2*x^2+2*x+1)^(1/3),x, algorithm="fricas")
 

output

sqrt(3)*arctan(-(4*sqrt(3)*(x^3 + 2*x^2 + 2*x + 1)^(1/3)*(x + 1) + sqrt(3) 
*(x^2 + x + 1) - 2*sqrt(3)*(x^3 + 2*x^2 + 2*x + 1)^(2/3))/(9*x^2 + 17*x + 
9)) - 1/2*log(-(3*(x^3 + 2*x^2 + 2*x + 1)^(1/3)*(x + 1) - x - 3*(x^3 + 2*x 
^2 + 2*x + 1)^(2/3))/x)
 

3.20.1.6 Sympy [F]

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

input

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

output

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

3.20.1.7 Maxima [F]

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

input

integrate((-1+x)/x/(x^3+2*x^2+2*x+1)^(1/3),x, algorithm="maxima")
 

output

integrate((x - 1)/((x^3 + 2*x^2 + 2*x + 1)^(1/3)*x), x)
 

3.20.1.8 Giac [F]

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

input

integrate((-1+x)/x/(x^3+2*x^2+2*x+1)^(1/3),x, algorithm="giac")
 

output

integrate((x - 1)/((x^3 + 2*x^2 + 2*x + 1)^(1/3)*x), x)
 

3.20.1.9 Mupad [F(-1)]

Timed out. \[ \int \frac {-1+x}{x \sqrt [3]{1+2 x+2 x^2+x^3}} \, dx=\int \frac {x-1}{x\,{\left (x^3+2\,x^2+2\,x+1\right )}^{1/3}} \,d x \]

input

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

output

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