Integrand size = 36, antiderivative size = 288 \[ \int \frac {\sqrt {-2+3 x^4} \left (-1+3 x^4\right )}{x^3 \left (1-3 x^4+3 x^8\right )} \, dx=\frac {\sqrt {-2+3 x^4}}{2 x^2}+\frac {1}{2} \sqrt {3} \text {RootSum}\left [4+4 \text {$\#$1}-2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-2 \log \left (\sqrt {3} x^2+\sqrt {-2+3 x^4}-\text {$\#$1}\right )-\log \left (\sqrt {3} x^2+\sqrt {-2+3 x^4}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (\sqrt {3} x^2+\sqrt {-2+3 x^4}-\text {$\#$1}\right ) \text {$\#$1}^2}{2-3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ]-\frac {1}{2} \sqrt {3} \text {RootSum}\left [4-4 \text {$\#$1}+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-2 \log \left (\sqrt {3} x^2+\sqrt {-2+3 x^4}-\text {$\#$1}\right )+\log \left (\sqrt {3} x^2+\sqrt {-2+3 x^4}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (\sqrt {3} x^2+\sqrt {-2+3 x^4}-\text {$\#$1}\right ) \text {$\#$1}^2}{-2+3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \]
Time = 0.28 (sec) , antiderivative size = 284, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt {-2+3 x^4} \left (-1+3 x^4\right )}{x^3 \left (1-3 x^4+3 x^8\right )} \, dx=\frac {1}{2} \left (\frac {\sqrt {-2+3 x^4}}{x^2}+\sqrt {3} \text {RootSum}\left [4+4 \text {$\#$1}-2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-2 \log \left (\sqrt {3} x^2+\sqrt {-2+3 x^4}-\text {$\#$1}\right )-\log \left (\sqrt {3} x^2+\sqrt {-2+3 x^4}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (\sqrt {3} x^2+\sqrt {-2+3 x^4}-\text {$\#$1}\right ) \text {$\#$1}^2}{2-3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ]-\sqrt {3} \text {RootSum}\left [4-4 \text {$\#$1}+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-2 \log \left (\sqrt {3} x^2+\sqrt {-2+3 x^4}-\text {$\#$1}\right )+\log \left (\sqrt {3} x^2+\sqrt {-2+3 x^4}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (\sqrt {3} x^2+\sqrt {-2+3 x^4}-\text {$\#$1}\right ) \text {$\#$1}^2}{-2+3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ]\right ) \]
(Sqrt[-2 + 3*x^4]/x^2 + Sqrt[3]*RootSum[4 + 4*#1 - 2*#1^3 + #1^4 & , (-2*L og[Sqrt[3]*x^2 + Sqrt[-2 + 3*x^4] - #1] - Log[Sqrt[3]*x^2 + Sqrt[-2 + 3*x^ 4] - #1]*#1 + Log[Sqrt[3]*x^2 + Sqrt[-2 + 3*x^4] - #1]*#1^2)/(2 - 3*#1^2 + 2*#1^3) & ] - Sqrt[3]*RootSum[4 - 4*#1 + 2*#1^3 + #1^4 & , (-2*Log[Sqrt[3 ]*x^2 + Sqrt[-2 + 3*x^4] - #1] + Log[Sqrt[3]*x^2 + Sqrt[-2 + 3*x^4] - #1]* #1 + Log[Sqrt[3]*x^2 + Sqrt[-2 + 3*x^4] - #1]*#1^2)/(-2 + 3*#1^2 + 2*#1^3) & ])/2
Result contains complex when optimal does not.
Time = 0.93 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.68, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {7279, 2009}
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.
| \(\displaystyle \int \frac {\sqrt {3 x^4-2} \left (3 x^4-1\right )}{x^3 \left (3 x^8-3 x^4+1\right )} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {3 x^5 \sqrt {3 x^4-2}}{3 x^8-3 x^4+1}-\frac {\sqrt {3 x^4-2}}{x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (-\sqrt {3}+i\right ) \sqrt {\frac {3 \left (-\sqrt {3}+3 i\right )}{\sqrt {3}+i}} \arctan \left (\frac {x^2}{\sqrt {\frac {-\sqrt {3}+3 i}{3 \left (\sqrt {3}+i\right )}} \sqrt {3 x^4-2}}\right )}{2 \left (-\sqrt {3}+3 i\right )}-\frac {\left (\sqrt {3}+i\right ) \text {arctanh}\left (\frac {\sqrt {-\frac {3 \left (-\sqrt {3}+i\right )}{\sqrt {3}+3 i}} x^2}{\sqrt {3 x^4-2}}\right )}{2 \sqrt {\frac {2}{3} \left (3+i \sqrt {3}\right )}}+\frac {\sqrt {3 x^4-2}}{2 x^2}\) |
Sqrt[-2 + 3*x^4]/(2*x^2) - ((I - Sqrt[3])*Sqrt[(3*(3*I - Sqrt[3]))/(I + Sq rt[3])]*ArcTan[x^2/(Sqrt[(3*I - Sqrt[3])/(3*(I + Sqrt[3]))]*Sqrt[-2 + 3*x^ 4])])/(2*(3*I - Sqrt[3])) - ((I + Sqrt[3])*ArcTanh[(Sqrt[(-3*(I - Sqrt[3]) )/(3*I + Sqrt[3])]*x^2)/Sqrt[-2 + 3*x^4]])/(2*Sqrt[(2*(3 + I*Sqrt[3]))/3])
3.29.34.3.1 Defintions of rubi rules used
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 7.01 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.85
| method | result | size |
| risch | \(\frac {\sqrt {3 x^{4}-2}}{2 x^{2}}-\frac {\sqrt {2}\, \left (-\ln \left (\frac {-3^{\frac {1}{4}} \sqrt {3 x^{4}-2}\, \sqrt {2}\, x^{2}+\sqrt {3}\, x^{4}+3 x^{4}-2}{3^{\frac {1}{4}} \sqrt {3 x^{4}-2}\, \sqrt {2}\, x^{2}+\sqrt {3}\, x^{4}+3 x^{4}-2}\right ) \sqrt {3}+\ln \left (\frac {3^{\frac {1}{4}} \sqrt {3 x^{4}-2}\, \sqrt {2}\, x^{2}+\sqrt {3}\, x^{4}+3 x^{4}-2}{-3^{\frac {1}{4}} \sqrt {3 x^{4}-2}\, \sqrt {2}\, x^{2}+\sqrt {3}\, x^{4}+3 x^{4}-2}\right )+\left (-2 \sqrt {3}+2\right ) \arctan \left (\frac {\sqrt {2}\, 3^{\frac {3}{4}} \sqrt {3 x^{4}-2}-3 x^{2}}{3 x^{2}}\right )+\left (-2 \sqrt {3}+2\right ) \arctan \left (\frac {\sqrt {2}\, 3^{\frac {3}{4}} \sqrt {3 x^{4}-2}+3 x^{2}}{3 x^{2}}\right )\right ) 3^{\frac {1}{4}}}{16}\) | \(244\) |
| elliptic | \(\frac {\sqrt {3 x^{4}-2}}{2 x^{2}}-\frac {\sqrt {2}\, \left (-\ln \left (\frac {-3^{\frac {1}{4}} \sqrt {3 x^{4}-2}\, \sqrt {2}\, x^{2}+\sqrt {3}\, x^{4}+3 x^{4}-2}{3^{\frac {1}{4}} \sqrt {3 x^{4}-2}\, \sqrt {2}\, x^{2}+\sqrt {3}\, x^{4}+3 x^{4}-2}\right ) \sqrt {3}+\ln \left (\frac {3^{\frac {1}{4}} \sqrt {3 x^{4}-2}\, \sqrt {2}\, x^{2}+\sqrt {3}\, x^{4}+3 x^{4}-2}{-3^{\frac {1}{4}} \sqrt {3 x^{4}-2}\, \sqrt {2}\, x^{2}+\sqrt {3}\, x^{4}+3 x^{4}-2}\right )+\left (-2 \sqrt {3}+2\right ) \arctan \left (\frac {\sqrt {2}\, 3^{\frac {3}{4}} \sqrt {3 x^{4}-2}-3 x^{2}}{3 x^{2}}\right )+\left (-2 \sqrt {3}+2\right ) \arctan \left (\frac {\sqrt {2}\, 3^{\frac {3}{4}} \sqrt {3 x^{4}-2}+3 x^{2}}{3 x^{2}}\right )\right ) 3^{\frac {1}{4}}}{16}\) | \(244\) |
| default | \(-\frac {\left (3 x^{4}-2\right )^{\frac {3}{2}}}{4 x^{2}}+\frac {3 x^{2} \sqrt {3 x^{4}-2}}{4}-\frac {\ln \left (\sqrt {3}\, x^{2}+\sqrt {3 x^{4}-2}\right ) \sqrt {3}}{2}+\frac {\ln \left (\frac {-3^{\frac {1}{4}} \sqrt {3 x^{4}-2}\, \sqrt {2}\, x^{2}+\sqrt {3}\, x^{4}+3 x^{4}-2}{3^{\frac {1}{4}} \sqrt {3 x^{4}-2}\, \sqrt {2}\, x^{2}+\sqrt {3}\, x^{4}+3 x^{4}-2}\right ) \sqrt {2}\, 3^{\frac {3}{4}}}{16}-\frac {\ln \left (\frac {3^{\frac {1}{4}} \sqrt {3 x^{4}-2}\, \sqrt {2}\, x^{2}+\sqrt {3}\, x^{4}+3 x^{4}-2}{-3^{\frac {1}{4}} \sqrt {3 x^{4}-2}\, \sqrt {2}\, x^{2}+\sqrt {3}\, x^{4}+3 x^{4}-2}\right ) 3^{\frac {1}{4}} \sqrt {2}}{16}-\frac {\sqrt {2}\, \left (3^{\frac {1}{4}}-3^{\frac {3}{4}}\right ) \arctan \left (\frac {\sqrt {2}\, 3^{\frac {3}{4}} \sqrt {3 x^{4}-2}-3 x^{2}}{3 x^{2}}\right )}{8}-\frac {\sqrt {2}\, \left (3^{\frac {1}{4}}-3^{\frac {3}{4}}\right ) \arctan \left (\frac {\sqrt {2}\, 3^{\frac {3}{4}} \sqrt {3 x^{4}-2}+3 x^{2}}{3 x^{2}}\right )}{8}+\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {3}\, \sqrt {3 x^{4}-2}}{3 x^{2}}\right )}{2}\) | \(318\) |
| pseudoelliptic | \(-\frac {\sqrt {3}\, \left (-\frac {\ln \left (\frac {-3^{\frac {1}{4}} \sqrt {3 x^{4}-2}\, \sqrt {2}\, x^{2}+\sqrt {3}\, x^{4}+3 x^{4}-2}{3^{\frac {1}{4}} \sqrt {3 x^{4}-2}\, \sqrt {2}\, x^{2}+\sqrt {3}\, x^{4}+3 x^{4}-2}\right ) 3^{\frac {1}{4}} \sqrt {2}\, x^{2}}{2}+\frac {\ln \left (\frac {3^{\frac {1}{4}} \sqrt {3 x^{4}-2}\, \sqrt {2}\, x^{2}+\sqrt {3}\, x^{4}+3 x^{4}-2}{-3^{\frac {1}{4}} \sqrt {3 x^{4}-2}\, \sqrt {2}\, x^{2}+\sqrt {3}\, x^{4}+3 x^{4}-2}\right ) \sqrt {2}\, 3^{\frac {3}{4}} x^{2}}{6}-x^{2} \left (3^{\frac {1}{4}}-\frac {3^{\frac {3}{4}}}{3}\right ) \sqrt {2}\, \arctan \left (\frac {\left (3^{\frac {1}{4}} \sqrt {3 x^{4}-2}\, \sqrt {2}+\sqrt {3}\, x^{2}\right ) \sqrt {3}}{3 x^{2}}\right )+2 \ln \left (\frac {\sqrt {3 x^{4}-2}-\sqrt {3}\, x^{2}}{x^{2}}\right ) x^{2}-2 \ln \left (\frac {\sqrt {3}\, x^{2}+\sqrt {3 x^{4}-2}}{x^{2}}\right ) x^{2}+4 \ln \left (\sqrt {3}\, x^{2}+\sqrt {3 x^{4}-2}\right ) x^{2}-\frac {4 \sqrt {3}\, \sqrt {3 x^{4}-2}}{3}+x^{2} \left (3^{\frac {1}{4}}-\frac {3^{\frac {3}{4}}}{3}\right ) \arctan \left (\frac {\left (-3^{\frac {1}{4}} \sqrt {3 x^{4}-2}\, \sqrt {2}+\sqrt {3}\, x^{2}\right ) \sqrt {3}}{3 x^{2}}\right ) \sqrt {2}\right )}{8 x^{2}}\) | \(366\) |
| trager | \(\text {Expression too large to display}\) | \(730\) |
1/2/x^2*(3*x^4-2)^(1/2)-1/16*2^(1/2)*(-ln((-3^(1/4)*(3*x^4-2)^(1/2)*2^(1/2 )*x^2+3^(1/2)*x^4+3*x^4-2)/(3^(1/4)*(3*x^4-2)^(1/2)*2^(1/2)*x^2+3^(1/2)*x^ 4+3*x^4-2))*3^(1/2)+ln((3^(1/4)*(3*x^4-2)^(1/2)*2^(1/2)*x^2+3^(1/2)*x^4+3* x^4-2)/(-3^(1/4)*(3*x^4-2)^(1/2)*2^(1/2)*x^2+3^(1/2)*x^4+3*x^4-2))+(-2*3^( 1/2)+2)*arctan(1/3*(2^(1/2)*3^(3/4)*(3*x^4-2)^(1/2)-3*x^2)/x^2)+(-2*3^(1/2 )+2)*arctan(1/3*(2^(1/2)*3^(3/4)*(3*x^4-2)^(1/2)+3*x^2)/x^2))*3^(1/4)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.27 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {-2+3 x^4} \left (-1+3 x^4\right )}{x^3 \left (1-3 x^4+3 x^8\right )} \, dx=-\frac {\sqrt {2} x^{2} \sqrt {-\sqrt {-3} + 3} \log \left (\frac {2 \, \sqrt {-3} x^{4} + 6 \, x^{4} + \sqrt {2} \sqrt {3 \, x^{4} - 2} {\left (\sqrt {-3} x^{2} + x^{2}\right )} \sqrt {-\sqrt {-3} + 3} - 4}{x^{4}}\right ) - \sqrt {2} x^{2} \sqrt {-\sqrt {-3} + 3} \log \left (\frac {2 \, \sqrt {-3} x^{4} + 6 \, x^{4} - \sqrt {2} \sqrt {3 \, x^{4} - 2} {\left (\sqrt {-3} x^{2} + x^{2}\right )} \sqrt {-\sqrt {-3} + 3} - 4}{x^{4}}\right ) + \sqrt {2} x^{2} \sqrt {\sqrt {-3} + 3} \log \left (-\frac {2 \, \sqrt {-3} x^{4} - 6 \, x^{4} + \sqrt {2} \sqrt {3 \, x^{4} - 2} {\left (\sqrt {-3} x^{2} - x^{2}\right )} \sqrt {\sqrt {-3} + 3} + 4}{x^{4}}\right ) - \sqrt {2} x^{2} \sqrt {\sqrt {-3} + 3} \log \left (-\frac {2 \, \sqrt {-3} x^{4} - 6 \, x^{4} - \sqrt {2} \sqrt {3 \, x^{4} - 2} {\left (\sqrt {-3} x^{2} - x^{2}\right )} \sqrt {\sqrt {-3} + 3} + 4}{x^{4}}\right ) - 8 \, \sqrt {3 \, x^{4} - 2}}{16 \, x^{2}} \]
-1/16*(sqrt(2)*x^2*sqrt(-sqrt(-3) + 3)*log((2*sqrt(-3)*x^4 + 6*x^4 + sqrt( 2)*sqrt(3*x^4 - 2)*(sqrt(-3)*x^2 + x^2)*sqrt(-sqrt(-3) + 3) - 4)/x^4) - sq rt(2)*x^2*sqrt(-sqrt(-3) + 3)*log((2*sqrt(-3)*x^4 + 6*x^4 - sqrt(2)*sqrt(3 *x^4 - 2)*(sqrt(-3)*x^2 + x^2)*sqrt(-sqrt(-3) + 3) - 4)/x^4) + sqrt(2)*x^2 *sqrt(sqrt(-3) + 3)*log(-(2*sqrt(-3)*x^4 - 6*x^4 + sqrt(2)*sqrt(3*x^4 - 2) *(sqrt(-3)*x^2 - x^2)*sqrt(sqrt(-3) + 3) + 4)/x^4) - sqrt(2)*x^2*sqrt(sqrt (-3) + 3)*log(-(2*sqrt(-3)*x^4 - 6*x^4 - sqrt(2)*sqrt(3*x^4 - 2)*(sqrt(-3) *x^2 - x^2)*sqrt(sqrt(-3) + 3) + 4)/x^4) - 8*sqrt(3*x^4 - 2))/x^2
Timed out. \[ \int \frac {\sqrt {-2+3 x^4} \left (-1+3 x^4\right )}{x^3 \left (1-3 x^4+3 x^8\right )} \, dx=\text {Timed out} \]
Not integrable
Time = 0.32 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.12 \[ \int \frac {\sqrt {-2+3 x^4} \left (-1+3 x^4\right )}{x^3 \left (1-3 x^4+3 x^8\right )} \, dx=\int { \frac {{\left (3 \, x^{4} - 1\right )} \sqrt {3 \, x^{4} - 2}}{{\left (3 \, x^{8} - 3 \, x^{4} + 1\right )} x^{3}} \,d x } \]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.34 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.62 \[ \int \frac {\sqrt {-2+3 x^4} \left (-1+3 x^4\right )}{x^3 \left (1-3 x^4+3 x^8\right )} \, dx=\frac {1}{16} \, \sqrt {3} {\left ({\left (\sqrt {3} x^{2} - \sqrt {3 \, x^{4} - 2}\right )}^{4} - 8 \, {\left (\sqrt {3} x^{2} - \sqrt {3 \, x^{4} - 2}\right )}^{2} + 4\right )} \log \left ({\left | -{\left (\sqrt {3} x^{2} - \sqrt {3 \, x^{4} - 2}\right )}^{8} + 4 \, {\left (\sqrt {3} x^{2} - \sqrt {3 \, x^{4} - 2}\right )}^{6} - 24 \, {\left (\sqrt {3} x^{2} - \sqrt {3 \, x^{4} - 2}\right )}^{4} + 16 \, {\left (\sqrt {3} x^{2} - \sqrt {3 \, x^{4} - 2}\right )}^{2} - 16 \right |}\right ) + \frac {2 \, \sqrt {3}}{{\left (\sqrt {3} x^{2} - \sqrt {3 \, x^{4} - 2}\right )}^{2} + 2} \]
1/16*sqrt(3)*((sqrt(3)*x^2 - sqrt(3*x^4 - 2))^4 - 8*(sqrt(3)*x^2 - sqrt(3* x^4 - 2))^2 + 4)*log(abs(-(sqrt(3)*x^2 - sqrt(3*x^4 - 2))^8 + 4*(sqrt(3)*x ^2 - sqrt(3*x^4 - 2))^6 - 24*(sqrt(3)*x^2 - sqrt(3*x^4 - 2))^4 + 16*(sqrt( 3)*x^2 - sqrt(3*x^4 - 2))^2 - 16)) + 2*sqrt(3)/((sqrt(3)*x^2 - sqrt(3*x^4 - 2))^2 + 2)
Not integrable
Time = 7.41 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.12 \[ \int \frac {\sqrt {-2+3 x^4} \left (-1+3 x^4\right )}{x^3 \left (1-3 x^4+3 x^8\right )} \, dx=\int \frac {\left (3\,x^4-1\right )\,\sqrt {3\,x^4-2}}{x^3\,\left (3\,x^8-3\,x^4+1\right )} \,d x \]