Integrand size = 28, antiderivative size = 90 \[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^2 \left (1+x^3+x^4\right )} \, dx=\frac {3 \sqrt [3]{1+x^4}}{x}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1+x^4}}\right )-\log \left (x+\sqrt [3]{1+x^4}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}\right ) \] Output:
3*(x^4+1)^(1/3)/x+3^(1/2)*arctan(3^(1/2)*x/(-x+2*(x^4+1)^(1/3)))-ln(x+(x^4 +1)^(1/3))+1/2*ln(x^2-x*(x^4+1)^(1/3)+(x^4+1)^(2/3))
Time = 1.01 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^2 \left (1+x^3+x^4\right )} \, dx=\frac {3 \sqrt [3]{1+x^4}}{x}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1+x^4}}\right )-\log \left (x+\sqrt [3]{1+x^4}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}\right ) \] Input:
Integrate[((-3 + x^4)*(1 + x^4)^(1/3))/(x^2*(1 + x^3 + x^4)),x]
Output:
(3*(1 + x^4)^(1/3))/x + Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2*(1 + x^4)^(1/3) )] - Log[x + (1 + x^4)^(1/3)] + Log[x^2 - x*(1 + x^4)^(1/3) + (1 + x^4)^(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.
| \(\displaystyle \int \frac {\left (x^4-3\right ) \sqrt [3]{x^4+1}}{x^2 \left (x^4+x^3+1\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x (4 x+3) \sqrt [3]{x^4+1}}{x^4+x^3+1}-\frac {3 \sqrt [3]{x^4+1}}{x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 \int \frac {x \sqrt [3]{x^4+1}}{x^4+x^3+1}dx+4 \int \frac {x^2 \sqrt [3]{x^4+1}}{x^4+x^3+1}dx+\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},-\frac {1}{4},\frac {3}{4},-x^4\right )}{x}\) |
Input:
Int[((-3 + x^4)*(1 + x^4)^(1/3))/(x^2*(1 + x^3 + x^4)),x]
Output:
$Aborted
Time = 6.06 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.97
| method | result | size |
| pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (x -2 \left (x^{4}+1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 x}\right ) x +\ln \left (\frac {x^{2}-x \left (x^{4}+1\right )^{\frac {1}{3}}+\left (x^{4}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) x -2 \ln \left (\frac {x +\left (x^{4}+1\right )^{\frac {1}{3}}}{x}\right ) x +6 \left (x^{4}+1\right )^{\frac {1}{3}}}{2 x}\) | \(87\) |
| risch | \(\frac {3 \left (x^{4}+1\right )^{\frac {1}{3}}}{x}+\frac {\left (-\ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{7}-x^{8}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{8}+2 x^{4}+1\right )^{\frac {1}{3}} x^{5}-2 \left (x^{8}+2 x^{4}+1\right )^{\frac {1}{3}} x^{5}+2 \left (x^{8}+2 x^{4}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-\left (x^{8}+2 x^{4}+1\right )^{\frac {2}{3}} x^{2}-2 x^{4}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{8}+2 x^{4}+1\right )^{\frac {1}{3}} x -2 \left (x^{8}+2 x^{4}+1\right )^{\frac {1}{3}} x -1}{\left (x^{4}+1\right ) \left (x^{4}+x^{3}+1\right )}\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^{7}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{8}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{7}-x^{8}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{8}+2 x^{4}+1\right )^{\frac {1}{3}} x^{5}+x^{7}+\left (x^{8}+2 x^{4}+1\right )^{\frac {1}{3}} x^{5}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-\left (x^{8}+2 x^{4}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{4}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+2 \left (x^{8}+2 x^{4}+1\right )^{\frac {2}{3}} x^{2}-2 x^{4}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{8}+2 x^{4}+1\right )^{\frac {1}{3}} x +x^{3}+\left (x^{8}+2 x^{4}+1\right )^{\frac {1}{3}} x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1}{\left (x^{4}+1\right ) \left (x^{4}+x^{3}+1\right )}\right )\right ) {\left (\left (x^{4}+1\right )^{2}\right )}^{\frac {1}{3}}}{\left (x^{4}+1\right )^{\frac {2}{3}}}\) | \(477\) |
| trager | \(\frac {3 \left (x^{4}+1\right )^{\frac {1}{3}}}{x}-3 \ln \left (\frac {-126 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{4}+252 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}-27 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{4}+45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x +126 \left (x^{4}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}-72 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}+5 x^{4}+27 \left (x^{4}+1\right )^{\frac {2}{3}} x -15 x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}+5 x^{3}-126 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-27 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+5}{x^{4}+x^{3}+1}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-\ln \left (-\frac {-45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{4}+90 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}+72 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{4}+45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x -81 \left (x^{4}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}-99 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}-28 x^{4}-42 \left (x^{4}+1\right )^{\frac {2}{3}} x -15 x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}+14 x^{3}-45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+72 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-28}{x^{4}+x^{3}+1}\right )+\ln \left (\frac {-126 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{4}+252 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}-27 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{4}+45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x +126 \left (x^{4}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}-72 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}+5 x^{4}+27 \left (x^{4}+1\right )^{\frac {2}{3}} x -15 x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}+5 x^{3}-126 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-27 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+5}{x^{4}+x^{3}+1}\right )\) | \(588\) |
Input:
int((x^4-3)*(x^4+1)^(1/3)/x^2/(x^4+x^3+1),x,method=_RETURNVERBOSE)
Output:
1/2*(2*3^(1/2)*arctan(1/3*(x-2*(x^4+1)^(1/3))*3^(1/2)/x)*x+ln((x^2-x*(x^4+ 1)^(1/3)+(x^4+1)^(2/3))/x^2)*x-2*ln((x+(x^4+1)^(1/3))/x)*x+6*(x^4+1)^(1/3) )/x
Time = 1.63 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.36 \[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^2 \left (1+x^3+x^4\right )} \, dx=\frac {2 \, \sqrt {3} x \arctan \left (\frac {2 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {1}{3}} x^{2} + 2 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{4} + x^{3} + 1\right )}}{3 \, {\left (x^{4} - x^{3} + 1\right )}}\right ) - x \log \left (\frac {x^{4} + x^{3} + 3 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{4} + 1\right )}^{\frac {2}{3}} x + 1}{x^{4} + x^{3} + 1}\right ) + 6 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{2 \, x} \] Input:
integrate((x^4-3)*(x^4+1)^(1/3)/x^2/(x^4+x^3+1),x, algorithm="fricas")
Output:
1/2*(2*sqrt(3)*x*arctan(1/3*(2*sqrt(3)*(x^4 + 1)^(1/3)*x^2 + 2*sqrt(3)*(x^ 4 + 1)^(2/3)*x + sqrt(3)*(x^4 + x^3 + 1))/(x^4 - x^3 + 1)) - x*log((x^4 + x^3 + 3*(x^4 + 1)^(1/3)*x^2 + 3*(x^4 + 1)^(2/3)*x + 1)/(x^4 + x^3 + 1)) + 6*(x^4 + 1)^(1/3))/x
Timed out. \[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^2 \left (1+x^3+x^4\right )} \, dx=\text {Timed out} \] Input:
integrate((x**4-3)*(x**4+1)**(1/3)/x**2/(x**4+x**3+1),x)
Output:
Timed out
\[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^2 \left (1+x^3+x^4\right )} \, dx=\int { \frac {{\left (x^{4} + 1\right )}^{\frac {1}{3}} {\left (x^{4} - 3\right )}}{{\left (x^{4} + x^{3} + 1\right )} x^{2}} \,d x } \] Input:
integrate((x^4-3)*(x^4+1)^(1/3)/x^2/(x^4+x^3+1),x, algorithm="maxima")
Output:
integrate((x^4 + 1)^(1/3)*(x^4 - 3)/((x^4 + x^3 + 1)*x^2), x)
\[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^2 \left (1+x^3+x^4\right )} \, dx=\int { \frac {{\left (x^{4} + 1\right )}^{\frac {1}{3}} {\left (x^{4} - 3\right )}}{{\left (x^{4} + x^{3} + 1\right )} x^{2}} \,d x } \] Input:
integrate((x^4-3)*(x^4+1)^(1/3)/x^2/(x^4+x^3+1),x, algorithm="giac")
Output:
integrate((x^4 + 1)^(1/3)*(x^4 - 3)/((x^4 + x^3 + 1)*x^2), x)
Timed out. \[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^2 \left (1+x^3+x^4\right )} \, dx=\int \frac {{\left (x^4+1\right )}^{1/3}\,\left (x^4-3\right )}{x^2\,\left (x^4+x^3+1\right )} \,d x \] Input:
int(((x^4 + 1)^(1/3)*(x^4 - 3))/(x^2*(x^3 + x^4 + 1)),x)
Output:
int(((x^4 + 1)^(1/3)*(x^4 - 3))/(x^2*(x^3 + x^4 + 1)), x)
\[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^2 \left (1+x^3+x^4\right )} \, dx=\frac {3 \left (x^{4}+1\right )^{\frac {1}{3}}-\left (\int \frac {\left (x^{4}+1\right )^{\frac {1}{3}} x^{5}}{x^{8}+x^{7}+2 x^{4}+x^{3}+1}d x \right ) x +3 \left (\int \frac {\left (x^{4}+1\right )^{\frac {1}{3}} x}{x^{8}+x^{7}+2 x^{4}+x^{3}+1}d x \right ) x}{x} \] Input:
int((x^4-3)*(x^4+1)^(1/3)/x^2/(x^4+x^3+1),x)
Output:
(3*(x**4 + 1)**(1/3) - int(((x**4 + 1)**(1/3)*x**5)/(x**8 + x**7 + 2*x**4 + x**3 + 1),x)*x + 3*int(((x**4 + 1)**(1/3)*x)/(x**8 + x**7 + 2*x**4 + x** 3 + 1),x)*x)/x