Integrand size = 30, antiderivative size = 87 \[ \int \frac {\sqrt [3]{-1+x^4} \left (3+x^4\right )}{x^2 \left (-1-x^3+x^4\right )} \, dx=\frac {3 \sqrt [3]{-1+x^4}}{x}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^4}}\right )+\log \left (-x+\sqrt [3]{-1+x^4}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{-1+x^4}+\left (-1+x^4\right )^{2/3}\right ) \]
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 = 0.98 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{-1+x^4} \left (3+x^4\right )}{x^2 \left (-1-x^3+x^4\right )} \, dx=\frac {3 \sqrt [3]{-1+x^4}}{x}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^4}}\right )+\log \left (-x+\sqrt [3]{-1+x^4}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{-1+x^4}+\left (-1+x^4\right )^{2/3}\right ) \]
(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.
| \(\displaystyle \int \frac {\sqrt [3]{x^4-1} \left (x^4+3\right )}{x^2 \left (x^4-x^3-1\right )} \, dx\) |
| \(\Big \downarrow \) 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 |
| \(\displaystyle -3 \int \frac {x \sqrt [3]{x^4-1}}{x^4-x^3-1}dx+4 \int \frac {x^2 \sqrt [3]{x^4-1}}{x^4-x^3-1}dx+\frac {3 \sqrt [3]{x^4-1} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},-\frac {1}{4},\frac {3}{4},x^4\right )}{x \sqrt [3]{1-x^4}}\) |
3.12.82.3.1 Defintions of rubi rules used
Time = 9.94 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.02
| method | result | size |
| pseudoelliptic | \(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{4}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x +2 \ln \left (\frac {-x +\left (x^{4}-1\right )^{\frac {1}{3}}}{x}\right ) x -\ln \left (\frac {x^{2}+x \left (x^{4}-1\right )^{\frac {1}{3}}+\left (x^{4}-1\right )^{\frac {2}{3}}}{x^{2}}\right ) x +6 \left (x^{4}-1\right )^{\frac {1}{3}}}{2 x}\) | \(89\) |
| trager | \(\frac {3 \left (x^{4}-1\right )^{\frac {1}{3}}}{x}-6 \ln \left (-\frac {-1149120 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{4}+2154600 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{3}+160116 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{4}+4118136 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}} x -2586522 \left (x^{4}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{2}-1524150 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{3}+340561 x^{4}+255269 \left (x^{4}-1\right )^{\frac {2}{3}} x -686356 \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}+118456 x^{3}+1149120 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2}-160116 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )-340561}{x^{4}-x^{3}-1}\right ) \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )+\ln \left (\frac {4264416 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{4}-7995780 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{3}-1524150 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{4}+4118136 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}} x -1531614 \left (x^{4}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{2}-1684266 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{3}+59850 x^{4}+431087 \left (x^{4}-1\right )^{\frac {2}{3}} x -686356 \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}+91770 x^{3}-4264416 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2}+1524150 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )-59850}{x^{4}-x^{3}-1}\right )-\ln \left (-\frac {-1149120 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{4}+2154600 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{3}+160116 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{4}+4118136 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}} x -2586522 \left (x^{4}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{2}-1524150 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{3}+340561 x^{4}+255269 \left (x^{4}-1\right )^{\frac {2}{3}} x -686356 \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}+118456 x^{3}+1149120 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2}-160116 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )-340561}{x^{4}-x^{3}-1}\right )\) | \(595\) |
| risch | \(\text {Expression too large to display}\) | \(754\) |
1/2*(-2*3^(1/2)*arctan(1/3*3^(1/2)/x*(x+2*(x^4-1)^(1/3)))*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 = 2.40 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.52 \[ \int \frac {\sqrt [3]{-1+x^4} \left (3+x^4\right )}{x^2 \left (-1-x^3+x^4\right )} \, dx=\frac {2 \, \sqrt {3} x \arctan \left (-\frac {14106128635054532 \, \sqrt {3} {\left (x^{4} - 1\right )}^{\frac {1}{3}} x^{2} - 89654043956484782 \, \sqrt {3} {\left (x^{4} - 1\right )}^{\frac {2}{3}} x - \sqrt {3} {\left (35416555940707109 \, x^{4} + 2357401720008016 \, x^{3} - 35416555940707109\right )}}{3 \, {\left (51678794422160641 \, x^{4} + 201291873609016 \, x^{3} - 51678794422160641\right )}}\right ) + x \log \left (\frac {x^{4} - x^{3} + 3 \, {\left (x^{4} - 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{4} - 1\right )}^{\frac {2}{3}} x - 1}{x^{4} - x^{3} - 1}\right ) + 6 \, {\left (x^{4} - 1\right )}^{\frac {1}{3}}}{2 \, x} \]
1/2*(2*sqrt(3)*x*arctan(-1/3*(14106128635054532*sqrt(3)*(x^4 - 1)^(1/3)*x^ 2 - 89654043956484782*sqrt(3)*(x^4 - 1)^(2/3)*x - sqrt(3)*(354165559407071 09*x^4 + 2357401720008016*x^3 - 35416555940707109))/(51678794422160641*x^4 + 201291873609016*x^3 - 51678794422160641)) + 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. \[ \int \frac {\sqrt [3]{-1+x^4} \left (3+x^4\right )}{x^2 \left (-1-x^3+x^4\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt [3]{-1+x^4} \left (3+x^4\right )}{x^2 \left (-1-x^3+x^4\right )} \, dx=\int { \frac {{\left (x^{4} + 3\right )} {\left (x^{4} - 1\right )}^{\frac {1}{3}}}{{\left (x^{4} - x^{3} - 1\right )} x^{2}} \,d x } \]
\[ \int \frac {\sqrt [3]{-1+x^4} \left (3+x^4\right )}{x^2 \left (-1-x^3+x^4\right )} \, dx=\int { \frac {{\left (x^{4} + 3\right )} {\left (x^{4} - 1\right )}^{\frac {1}{3}}}{{\left (x^{4} - x^{3} - 1\right )} x^{2}} \,d x } \]
Timed out. \[ \int \frac {\sqrt [3]{-1+x^4} \left (3+x^4\right )}{x^2 \left (-1-x^3+x^4\right )} \, dx=\int -\frac {{\left (x^4-1\right )}^{1/3}\,\left (x^4+3\right )}{x^2\,\left (-x^4+x^3+1\right )} \,d x \]