Integrand size = 28, antiderivative size = 90 \[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^2 \left (1+x^3+x^4\right )} \, dx=\frac {3 \sqrt [3]{1+x^4}}{x}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1+x^4}}\right )-\log \left (x+\sqrt [3]{1+x^4}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}\right ) \]
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 = 1.00 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^2 \left (1+x^3+x^4\right )} \, dx=\frac {3 \sqrt [3]{1+x^4}}{x}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1+x^4}}\right )-\log \left (x+\sqrt [3]{1+x^4}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}\right ) \]
(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 |
\(\displaystyle 3 \int \frac {x \sqrt [3]{x^4+1}}{x^4+x^3+1}dx+4 \int \frac {x^2 \sqrt [3]{x^4+1}}{x^4+x^3+1}dx+\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},-\frac {1}{4},\frac {3}{4},-x^4\right )}{x}\) |
3.13.37.3.1 Defintions of rubi rules used
Time = 7.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.97
method | result | size |
pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x -2 \left (x^{4}+1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x -2 \ln \left (\frac {x +\left (x^{4}+1\right )^{\frac {1}{3}}}{x}\right ) x +\ln \left (\frac {x^{2}-x \left (x^{4}+1\right )^{\frac {1}{3}}+\left (x^{4}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) x +6 \left (x^{4}+1\right )^{\frac {1}{3}}}{2 x}\) | \(87\) |
trager | \(\frac {3 \left (x^{4}+1\right )^{\frac {1}{3}}}{x}-\ln \left (\frac {15 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{4}-30 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}+14 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{4}+15 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x -27 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}} x^{2}-13 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}+3 x^{4}+9 \left (x^{4}+1\right )^{\frac {2}{3}} x +14 x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}+3 x^{3}+15 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+14 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+3}{x^{4}+x^{3}+1}\right )+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (-\frac {18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{4}-36 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}-15 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{4}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x +18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}} x^{2}+12 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}+2 x^{4}-6 \left (x^{4}+1\right )^{\frac {2}{3}} x -3 x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}-x^{3}+18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-15 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+2}{x^{4}+x^{3}+1}\right )\) | \(402\) |
risch | \(\text {Expression too large to display}\) | \(778\) |
1/2*(2*3^(1/2)*arctan(1/3*3^(1/2)*(x-2*(x^4+1)^(1/3))/x)*x-2*ln((x+(x^4+1) ^(1/3))/x)*x+ln((x^2-x*(x^4+1)^(1/3)+(x^4+1)^(2/3))/x^2)*x+6*(x^4+1)^(1/3) )/x
Time = 1.74 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.36 \[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^2 \left (1+x^3+x^4\right )} \, dx=\frac {2 \, \sqrt {3} x \arctan \left (\frac {2 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {1}{3}} x^{2} + 2 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{4} + x^{3} + 1\right )}}{3 \, {\left (x^{4} - x^{3} + 1\right )}}\right ) - x \log \left (\frac {x^{4} + x^{3} + 3 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{4} + 1\right )}^{\frac {2}{3}} x + 1}{x^{4} + x^{3} + 1}\right ) + 6 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{2 \, x} \]
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. \[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^2 \left (1+x^3+x^4\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^2 \left (1+x^3+x^4\right )} \, dx=\int { \frac {{\left (x^{4} + 1\right )}^{\frac {1}{3}} {\left (x^{4} - 3\right )}}{{\left (x^{4} + x^{3} + 1\right )} x^{2}} \,d x } \]
\[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^2 \left (1+x^3+x^4\right )} \, dx=\int { \frac {{\left (x^{4} + 1\right )}^{\frac {1}{3}} {\left (x^{4} - 3\right )}}{{\left (x^{4} + x^{3} + 1\right )} x^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^2 \left (1+x^3+x^4\right )} \, dx=\int \frac {{\left (x^4+1\right )}^{1/3}\,\left (x^4-3\right )}{x^2\,\left (x^4+x^3+1\right )} \,d x \]