Integrand size = 30, antiderivative size = 115 \[ \int \frac {\left (3+x^4\right ) \left (-1-x^3+x^4\right )^{2/3}}{x^3 \left (-1+x^4\right )} \, dx=\frac {3 \left (-1-x^3+x^4\right )^{2/3}}{2 x^2}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1-x^3+x^4}}\right )+\log \left (x+\sqrt [3]{-1-x^3+x^4}\right )-\frac {1}{2} \log \left (x^2-x \sqrt [3]{-1-x^3+x^4}+\left (-1-x^3+x^4\right )^{2/3}\right ) \]
3/2*(x^4-x^3-1)^(2/3)/x^2+3^(1/2)*arctan(3^(1/2)*x/(-x+2*(x^4-x^3-1)^(1/3) ))+ln(x+(x^4-x^3-1)^(1/3))-1/2*ln(x^2-x*(x^4-x^3-1)^(1/3)+(x^4-x^3-1)^(2/3 ))
Time = 0.41 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00 \[ \int \frac {\left (3+x^4\right ) \left (-1-x^3+x^4\right )^{2/3}}{x^3 \left (-1+x^4\right )} \, dx=\frac {3 \left (-1-x^3+x^4\right )^{2/3}}{2 x^2}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1-x^3+x^4}}\right )+\log \left (x+\sqrt [3]{-1-x^3+x^4}\right )-\frac {1}{2} \log \left (x^2-x \sqrt [3]{-1-x^3+x^4}+\left (-1-x^3+x^4\right )^{2/3}\right ) \]
(3*(-1 - x^3 + x^4)^(2/3))/(2*x^2) + Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2*(- 1 - x^3 + x^4)^(1/3))] + Log[x + (-1 - x^3 + x^4)^(1/3)] - Log[x^2 - x*(-1 - x^3 + x^4)^(1/3) + (-1 - x^3 + 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.
| \(\displaystyle \int \frac {\left (x^4+3\right ) \left (x^4-x^3-1\right )^{2/3}}{x^3 \left (x^4-1\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {\left (x^4-x^3-1\right )^{2/3}}{x-1}+\frac {\left (x^4-x^3-1\right )^{2/3}}{x+1}-\frac {3 \left (x^4-x^3-1\right )^{2/3}}{x^3}-\frac {2 \left (x^4-x^3-1\right )^{2/3} x}{x^2+1}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \int \frac {\left (x^4-x^3-1\right )^{2/3}}{i-x}dx+\int \frac {\left (x^4-x^3-1\right )^{2/3}}{x-1}dx-3 \int \frac {\left (x^4-x^3-1\right )^{2/3}}{x^3}dx-\int \frac {\left (x^4-x^3-1\right )^{2/3}}{x+i}dx+\int \frac {\left (x^4-x^3-1\right )^{2/3}}{x+1}dx\) |
3.18.10.3.1 Defintions of rubi rules used
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 5.02 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.03
| method | result | size |
| pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x -2 \left (x^{4}-x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{2}-\ln \left (\frac {x^{2}-x \left (x^{4}-x^{3}-1\right )^{\frac {1}{3}}+\left (x^{4}-x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{2}+2 \ln \left (\frac {x +\left (x^{4}-x^{3}-1\right )^{\frac {1}{3}}}{x}\right ) x^{2}+3 \left (x^{4}-x^{3}-1\right )^{\frac {2}{3}}}{2 x^{2}}\) | \(119\) |
| risch | \(\frac {3 \left (x^{4}-x^{3}-1\right )^{\frac {2}{3}}}{2 x^{2}}+\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^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}-x^{3}-1\right )^{\frac {2}{3}} x -3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}-x^{3}-1\right )^{\frac {1}{3}} x^{2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-2 x^{4}+3 \left (x^{4}-x^{3}-1\right )^{\frac {2}{3}} x -3 \left (x^{4}-x^{3}-1\right )^{\frac {1}{3}} x^{2}+4 x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+2}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )-\ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}-x^{3}-1\right )^{\frac {2}{3}} x -3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}-x^{3}-1\right )^{\frac {1}{3}} x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+x^{4}-x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+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^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}-x^{3}-1\right )^{\frac {2}{3}} x -3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}-x^{3}-1\right )^{\frac {1}{3}} x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+x^{4}-x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )\) | \(433\) |
| trager | \(\frac {3 \left (x^{4}-x^{3}-1\right )^{\frac {2}{3}}}{2 x^{2}}-12 \ln \left (\frac {337536 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{4}-632880 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{3}+30972 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{4}-30384 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{4}-x^{3}-1\right )^{\frac {2}{3}} x +30384 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{4}-x^{3}-1\right )^{\frac {1}{3}} x^{2}-83124 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{3}+286 x^{4}+1821 \left (x^{4}-x^{3}-1\right )^{\frac {2}{3}} x -1821 \left (x^{4}-x^{3}-1\right )^{\frac {1}{3}} x^{2}-2574 x^{3}-337536 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}-30972 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-286}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right ) \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+12 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \ln \left (\frac {337536 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{4}-632880 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{3}+25284 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{4}+30384 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{4}-x^{3}-1\right )^{\frac {2}{3}} x -30384 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{4}-x^{3}-1\right )^{\frac {1}{3}} x^{2}-22356 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{3}+49 x^{4}+4353 \left (x^{4}-x^{3}-1\right )^{\frac {2}{3}} x -4353 \left (x^{4}-x^{3}-1\right )^{\frac {1}{3}} x^{2}-42 x^{3}-337536 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}-25284 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-49}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )-\ln \left (\frac {337536 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{4}-632880 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{3}+30972 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{4}-30384 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{4}-x^{3}-1\right )^{\frac {2}{3}} x +30384 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{4}-x^{3}-1\right )^{\frac {1}{3}} x^{2}-83124 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{3}+286 x^{4}+1821 \left (x^{4}-x^{3}-1\right )^{\frac {2}{3}} x -1821 \left (x^{4}-x^{3}-1\right )^{\frac {1}{3}} x^{2}-2574 x^{3}-337536 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}-30972 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-286}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )\) | \(686\) |
1/2*(2*3^(1/2)*arctan(1/3*3^(1/2)*(x-2*(x^4-x^3-1)^(1/3))/x)*x^2-ln((x^2-x *(x^4-x^3-1)^(1/3)+(x^4-x^3-1)^(2/3))/x^2)*x^2+2*ln((x+(x^4-x^3-1)^(1/3))/ x)*x^2+3*(x^4-x^3-1)^(2/3))/x^2
Time = 3.25 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.30 \[ \int \frac {\left (3+x^4\right ) \left (-1-x^3+x^4\right )^{2/3}}{x^3 \left (-1+x^4\right )} \, dx=\frac {2 \, \sqrt {3} x^{2} \arctan \left (\frac {728574532 \, \sqrt {3} {\left (x^{4} - x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 812477430 \, \sqrt {3} {\left (x^{4} - x^{3} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (355231575 \, x^{4} + 41951449 \, x^{3} - 355231575\right )}}{3 \, {\left (447697125 \, x^{4} - 770525981 \, x^{3} - 447697125\right )}}\right ) + x^{2} \log \left (\frac {x^{4} + 3 \, {\left (x^{4} - x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{4} - x^{3} - 1\right )}^{\frac {2}{3}} x - 1}{x^{4} - 1}\right ) + 3 \, {\left (x^{4} - x^{3} - 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} \]
1/2*(2*sqrt(3)*x^2*arctan(1/3*(728574532*sqrt(3)*(x^4 - x^3 - 1)^(1/3)*x^2 + 812477430*sqrt(3)*(x^4 - x^3 - 1)^(2/3)*x + sqrt(3)*(355231575*x^4 + 41 951449*x^3 - 355231575))/(447697125*x^4 - 770525981*x^3 - 447697125)) + x^ 2*log((x^4 + 3*(x^4 - x^3 - 1)^(1/3)*x^2 + 3*(x^4 - x^3 - 1)^(2/3)*x - 1)/ (x^4 - 1)) + 3*(x^4 - x^3 - 1)^(2/3))/x^2
\[ \int \frac {\left (3+x^4\right ) \left (-1-x^3+x^4\right )^{2/3}}{x^3 \left (-1+x^4\right )} \, dx=\int \frac {\left (x^{4} + 3\right ) \left (x^{4} - x^{3} - 1\right )^{\frac {2}{3}}}{x^{3} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \]
\[ \int \frac {\left (3+x^4\right ) \left (-1-x^3+x^4\right )^{2/3}}{x^3 \left (-1+x^4\right )} \, dx=\int { \frac {{\left (x^{4} - x^{3} - 1\right )}^{\frac {2}{3}} {\left (x^{4} + 3\right )}}{{\left (x^{4} - 1\right )} x^{3}} \,d x } \]
\[ \int \frac {\left (3+x^4\right ) \left (-1-x^3+x^4\right )^{2/3}}{x^3 \left (-1+x^4\right )} \, dx=\int { \frac {{\left (x^{4} - x^{3} - 1\right )}^{\frac {2}{3}} {\left (x^{4} + 3\right )}}{{\left (x^{4} - 1\right )} x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (3+x^4\right ) \left (-1-x^3+x^4\right )^{2/3}}{x^3 \left (-1+x^4\right )} \, dx=\int \frac {\left (x^4+3\right )\,{\left (x^4-x^3-1\right )}^{2/3}}{x^3\,\left (x^4-1\right )} \,d x \]