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))
Time = 0.24 (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
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
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.02 (sec) , antiderivative size = 344, normalized size of antiderivative = 2.61
method | result | size |
trager | \(-\ln \left (\frac {11 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-111 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2 x^{2}+2 x +1\right )^{\frac {1}{3}} x -22 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +100 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+111 \left (x^{3}+2 x^{2}+2 x +1\right )^{\frac {2}{3}}-111 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2 x^{2}+2 x +1\right )^{\frac {1}{3}}+11 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+207 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -100 x^{2}+100 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-170 x -100}{x}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\frac {-10 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-111 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2 x^{2}+2 x +1\right )^{\frac {1}{3}} x +20 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +121 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+111 \left (x^{3}+2 x^{2}+2 x +1\right )^{\frac {2}{3}}-111 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2 x^{2}+2 x +1\right )^{\frac {1}{3}}-10 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+128 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -121 x^{2}+121 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-165 x -121}{x}\right )\) | \(344\) |
Input:
int((-1+x)/x/(x^3+2*x^2+2*x+1)^(1/3),x,method=_RETURNVERBOSE)
Output:
-ln((11*RootOf(_Z^2-_Z+1)^2*x^2-111*RootOf(_Z^2-_Z+1)*(x^3+2*x^2+2*x+1)^(1 /3)*x-22*RootOf(_Z^2-_Z+1)^2*x+100*RootOf(_Z^2-_Z+1)*x^2+111*(x^3+2*x^2+2* x+1)^(2/3)-111*RootOf(_Z^2-_Z+1)*(x^3+2*x^2+2*x+1)^(1/3)+11*RootOf(_Z^2-_Z +1)^2+207*RootOf(_Z^2-_Z+1)*x-100*x^2+100*RootOf(_Z^2-_Z+1)-170*x-100)/x)+ RootOf(_Z^2-_Z+1)*ln((-10*RootOf(_Z^2-_Z+1)^2*x^2-111*RootOf(_Z^2-_Z+1)*(x ^3+2*x^2+2*x+1)^(1/3)*x+20*RootOf(_Z^2-_Z+1)^2*x+121*RootOf(_Z^2-_Z+1)*x^2 +111*(x^3+2*x^2+2*x+1)^(2/3)-111*RootOf(_Z^2-_Z+1)*(x^3+2*x^2+2*x+1)^(1/3) -10*RootOf(_Z^2-_Z+1)^2+128*RootOf(_Z^2-_Z+1)*x-121*x^2+121*RootOf(_Z^2-_Z +1)-165*x-121)/x)
Time = 0.26 (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)
\[ \int \frac {-1+x}{x \sqrt [3]{1+2 x+2 x^2+x^3}} \, dx=\int \frac {x - 1}{x \sqrt [3]{\left (x + 1\right ) \left (x^{2} + x + 1\right )}}\, dx \] 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)
\[ \int \frac {-1+x}{x \sqrt [3]{1+2 x+2 x^2+x^3}} \, dx=\int { \frac {x - 1}{{\left (x^{3} + 2 \, x^{2} + 2 \, x + 1\right )}^{\frac {1}{3}} x} \,d x } \] 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)
\[ \int \frac {-1+x}{x \sqrt [3]{1+2 x+2 x^2+x^3}} \, dx=\int { \frac {x - 1}{{\left (x^{3} + 2 \, x^{2} + 2 \, x + 1\right )}^{\frac {1}{3}} x} \,d x } \] 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)
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)
\[ \int \frac {-1+x}{x \sqrt [3]{1+2 x+2 x^2+x^3}} \, dx=\int \frac {1}{\left (x^{3}+2 x^{2}+2 x +1\right )^{\frac {1}{3}}}d x -\left (\int \frac {1}{\left (x^{3}+2 x^{2}+2 x +1\right )^{\frac {1}{3}} x}d x \right ) \] Input:
int((-1+x)/x/(x^3+2*x^2+2*x+1)^(1/3),x)
Output:
int(1/(x**3 + 2*x**2 + 2*x + 1)**(1/3),x) - int(1/((x**3 + 2*x**2 + 2*x + 1)**(1/3)*x),x)