Integrand size = 14, antiderivative size = 323 \[ \int \frac {1}{\sqrt {2-6 x+3 x^3}} \, dx=-\frac {2^{3/4} \sqrt {3 x-2 \sqrt {6} \cos \left (\frac {1}{6} \left (\pi +2 \arcsin \left (\frac {\sqrt {\frac {3}{2}}}{2}\right )\right )\right )} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{2} \sqrt {3 \left (\cos \left (\frac {1}{3} \arcsin \left (\frac {\sqrt {\frac {3}{2}}}{2}\right )\right )+\sqrt {3} \sin \left (\frac {1}{3} \arcsin \left (\frac {\sqrt {\frac {3}{2}}}{2}\right )\right )\right )}}{\sqrt {3 x+2 \sqrt {6} \sin \left (\frac {1}{3} \left (\pi +\arcsin \left (\frac {\sqrt {\frac {3}{2}}}{2}\right )\right )\right )}}\right ),\frac {\sqrt {3} \cos \left (\frac {1}{3} \arcsin \left (\frac {\sqrt {\frac {3}{2}}}{2}\right )\right )}{\sin \left (\frac {1}{3} \arcsin \left (\frac {\sqrt {\frac {3}{2}}}{2}\right )\right )+\sin \left (\frac {1}{3} \left (\pi +\arcsin \left (\frac {\sqrt {\frac {3}{2}}}{2}\right )\right )\right )}\right ) \sqrt {3 x-2 \sqrt {6} \sin \left (\frac {1}{3} \arcsin \left (\frac {\sqrt {\frac {3}{2}}}{2}\right )\right )} \sqrt {3 x+2 \sqrt {6} \sin \left (\frac {1}{3} \left (\pi +\arcsin \left (\frac {\sqrt {\frac {3}{2}}}{2}\right )\right )\right )}}{3 \sqrt {2-6 x+3 x^3} \sqrt {3 \left (\cos \left (\frac {1}{3} \arcsin \left (\frac {\sqrt {\frac {3}{2}}}{2}\right )\right )+\sqrt {3} \sin \left (\frac {1}{3} \arcsin \left (\frac {\sqrt {\frac {3}{2}}}{2}\right )\right )\right )}} \] Output:
-1/3*2^(3/4)*(3*x-2*6^(1/2)*cos(1/6*Pi+1/3*arcsin(1/4*6^(1/2))))^(1/2)*Ell ipticF(2^(1/4)*(3*cos(1/3*arcsin(1/4*6^(1/2)))+3*3^(1/2)*sin(1/3*arcsin(1/ 4*6^(1/2))))^(1/2)/(3*x+2*6^(1/2)*sin(1/3*Pi+1/3*arcsin(1/4*6^(1/2))))^(1/ 2),(3^(1/2)*cos(1/3*arcsin(1/4*6^(1/2)))/(sin(1/3*arcsin(1/4*6^(1/2)))+sin (1/3*Pi+1/3*arcsin(1/4*6^(1/2)))))^(1/2))*(3*x-2*6^(1/2)*sin(1/3*arcsin(1/ 4*6^(1/2))))^(1/2)*(3*x+2*6^(1/2)*sin(1/3*Pi+1/3*arcsin(1/4*6^(1/2))))^(1/ 2)/(3*x^3-6*x+2)^(1/2)/(3*cos(1/3*arcsin(1/4*6^(1/2)))+3*3^(1/2)*sin(1/3*a rcsin(1/4*6^(1/2))))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
Time = 10.10 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\sqrt {2-6 x+3 x^3}} \, dx=-\frac {2 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {x-\text {Root}\left [3 \text {$\#$1}^3-6 \text {$\#$1}+2\&,3\right ]}{\text {Root}\left [3 \text {$\#$1}^3-6 \text {$\#$1}+2\&,2\right ]-\text {Root}\left [3 \text {$\#$1}^3-6 \text {$\#$1}+2\&,3\right ]}}\right ),\frac {\text {Root}\left [3 \text {$\#$1}^3-6 \text {$\#$1}+2\&,2\right ]-\text {Root}\left [3 \text {$\#$1}^3-6 \text {$\#$1}+2\&,3\right ]}{\text {Root}\left [3 \text {$\#$1}^3-6 \text {$\#$1}+2\&,1\right ]-\text {Root}\left [3 \text {$\#$1}^3-6 \text {$\#$1}+2\&,3\right ]}\right ) \sqrt {x-\text {Root}\left [3 \text {$\#$1}^3-6 \text {$\#$1}+2\&,1\right ]} \sqrt {\left (x-\text {Root}\left [3 \text {$\#$1}^3-6 \text {$\#$1}+2\&,2\right ]\right ) \left (-x+\text {Root}\left [3 \text {$\#$1}^3-6 \text {$\#$1}+2\&,3\right ]\right )}}{\sqrt {2-6 x+3 x^3} \sqrt {-\text {Root}\left [3 \text {$\#$1}^3-6 \text {$\#$1}+2\&,1\right ]+\text {Root}\left [3 \text {$\#$1}^3-6 \text {$\#$1}+2\&,3\right ]}} \] Input:
Integrate[1/Sqrt[2 - 6*x + 3*x^3],x]
Output:
(-2*EllipticF[ArcSin[Sqrt[(x - Root[2 - 6*#1 + 3*#1^3 & , 3, 0])/(Root[2 - 6*#1 + 3*#1^3 & , 2, 0] - Root[2 - 6*#1 + 3*#1^3 & , 3, 0])]], (Root[2 - 6*#1 + 3*#1^3 & , 2, 0] - Root[2 - 6*#1 + 3*#1^3 & , 3, 0])/(Root[2 - 6*#1 + 3*#1^3 & , 1, 0] - Root[2 - 6*#1 + 3*#1^3 & , 3, 0])]*Sqrt[x - Root[2 - 6*#1 + 3*#1^3 & , 1, 0]]*Sqrt[(x - Root[2 - 6*#1 + 3*#1^3 & , 2, 0])*(-x + Root[2 - 6*#1 + 3*#1^3 & , 3, 0])])/(Sqrt[2 - 6*x + 3*x^3]*Sqrt[-Root[2 - 6*#1 + 3*#1^3 & , 1, 0] + Root[2 - 6*#1 + 3*#1^3 & , 3, 0]])
Result contains complex when optimal does not.
Time = 1.71 (sec) , antiderivative size = 813, normalized size of antiderivative = 2.52, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2475, 1172, 321}
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.
| \(\displaystyle \int \frac {1}{\sqrt {3 x^3-6 x+2}} \, dx\) |
| \(\Big \downarrow \) 2475 |
| \(\displaystyle \frac {\sqrt {3 x+\sqrt [3]{9-3 i \sqrt {15}}+\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}} \sqrt {9 x^2-3 \left (\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}+\sqrt [3]{9-3 i \sqrt {15}}\right ) x+\left (9-3 i \sqrt {15}\right )^{2/3}+\frac {12 \sqrt [3]{3}}{\left (3-i \sqrt {15}\right )^{2/3}}-6} \int \frac {1}{\sqrt {3 x+\sqrt [3]{9-3 i \sqrt {15}}+\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}} \sqrt {9 x^2-3 \left (\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}+\sqrt [3]{9-3 i \sqrt {15}}\right ) x+\left (9-3 i \sqrt {15}\right )^{2/3}+\frac {12 \sqrt [3]{3}}{\left (3-i \sqrt {15}\right )^{2/3}}-6}}dx}{\sqrt {3 x^3-6 x+2}}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {2 \sqrt {\frac {2}{3} \left (-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}\right )} \sqrt {\frac {\sqrt [3]{3-i \sqrt {15}} \left (3 x+\sqrt [3]{9-3 i \sqrt {15}}+\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}\right )}{2\ 3^{2/3}+\sqrt [3]{3} \left (3-i \sqrt {15}\right )^{2/3}-\sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}} \sqrt {-\frac {\left (3-i \sqrt {15}\right )^{2/3} \left (-9 x^2+3 \left (\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}+\sqrt [3]{9-3 i \sqrt {15}}\right ) x-\left (9-3 i \sqrt {15}\right )^{2/3}-\frac {12 \sqrt [3]{3}}{\left (3-i \sqrt {15}\right )^{2/3}}+6\right )}{4 \sqrt [3]{3}+\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}-4 \left (3-i \sqrt {15}\right )^{2/3}}} \int \frac {1}{\sqrt {\frac {\sqrt [3]{3-i \sqrt {15}} \left (\frac {2\ 3^{2/3}+\sqrt [3]{3} \left (3-i \sqrt {15}\right )^{2/3}-3 \sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}{\sqrt [3]{3-i \sqrt {15}}}-6 x\right )}{6 \sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}+1} \sqrt {1-\frac {\sqrt [3]{3-i \sqrt {15}} \left (\frac {2\ 3^{2/3}+\sqrt [3]{3} \left (3-i \sqrt {15}\right )^{2/3}-3 \sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}{\sqrt [3]{3-i \sqrt {15}}}-6 x\right )}{3 \left (2\ 3^{2/3}+\sqrt [3]{3} \left (3-i \sqrt {15}\right )^{2/3}-\sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}\right )}}}d\frac {\sqrt {-\frac {\sqrt [3]{3-i \sqrt {15}} \left (\frac {2\ 3^{2/3}+\sqrt [3]{3} \left (3-i \sqrt {15}\right )^{2/3}-3 \sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}{\sqrt [3]{3-i \sqrt {15}}}-6 x\right )}{\sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}}}{\sqrt {6}}}{3 \sqrt [3]{3-i \sqrt {15}} \sqrt {3 x^3-6 x+2}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {2 \sqrt {\frac {2}{3} \left (-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}\right )} \sqrt {\frac {\sqrt [3]{3-i \sqrt {15}} \left (3 x+\sqrt [3]{9-3 i \sqrt {15}}+\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}\right )}{2\ 3^{2/3}+\sqrt [3]{3} \left (3-i \sqrt {15}\right )^{2/3}-\sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}} \sqrt {-\frac {\left (3-i \sqrt {15}\right )^{2/3} \left (-9 x^2+3 \left (\frac {2\ 3^{2/3}}{\sqrt [3]{3-i \sqrt {15}}}+\sqrt [3]{9-3 i \sqrt {15}}\right ) x-\left (9-3 i \sqrt {15}\right )^{2/3}-\frac {12 \sqrt [3]{3}}{\left (3-i \sqrt {15}\right )^{2/3}}+6\right )}{4 \sqrt [3]{3}+\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}-4 \left (3-i \sqrt {15}\right )^{2/3}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-\frac {\sqrt [3]{3-i \sqrt {15}} \left (\frac {2\ 3^{2/3}+\sqrt [3]{3} \left (3-i \sqrt {15}\right )^{2/3}-3 \sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}{\sqrt [3]{3-i \sqrt {15}}}-6 x\right )}{\sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}}}{\sqrt {6}}\right ),-\frac {2 \sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}{2\ 3^{2/3}+\sqrt [3]{3} \left (3-i \sqrt {15}\right )^{2/3}-\sqrt {-4 \sqrt [3]{3}-\left (3^{2/3}-i \sqrt [6]{3} \sqrt {5}\right ) \sqrt [3]{3-i \sqrt {15}}+4 \left (3-i \sqrt {15}\right )^{2/3}}}\right )}{3 \sqrt [3]{3-i \sqrt {15}} \sqrt {3 x^3-6 x+2}}\) |
Input:
Int[1/Sqrt[2 - 6*x + 3*x^3],x]
Output:
(2*Sqrt[(2*(-4*3^(1/3) - (3^(2/3) - I*3^(1/6)*Sqrt[5])*(3 - I*Sqrt[15])^(1 /3) + 4*(3 - I*Sqrt[15])^(2/3)))/3]*Sqrt[((3 - I*Sqrt[15])^(1/3)*((2*3^(2/ 3))/(3 - I*Sqrt[15])^(1/3) + (9 - (3*I)*Sqrt[15])^(1/3) + 3*x))/(2*3^(2/3) + 3^(1/3)*(3 - I*Sqrt[15])^(2/3) - Sqrt[-4*3^(1/3) - (3^(2/3) - I*3^(1/6) *Sqrt[5])*(3 - I*Sqrt[15])^(1/3) + 4*(3 - I*Sqrt[15])^(2/3)])]*Sqrt[-(((3 - I*Sqrt[15])^(2/3)*(6 - (12*3^(1/3))/(3 - I*Sqrt[15])^(2/3) - (9 - (3*I)* Sqrt[15])^(2/3) + 3*((2*3^(2/3))/(3 - I*Sqrt[15])^(1/3) + (9 - (3*I)*Sqrt[ 15])^(1/3))*x - 9*x^2))/(4*3^(1/3) + (3^(2/3) - I*3^(1/6)*Sqrt[5])*(3 - I* Sqrt[15])^(1/3) - 4*(3 - I*Sqrt[15])^(2/3)))]*EllipticF[ArcSin[Sqrt[-(((3 - I*Sqrt[15])^(1/3)*((2*3^(2/3) + 3^(1/3)*(3 - I*Sqrt[15])^(2/3) - 3*Sqrt[ -4*3^(1/3) - (3^(2/3) - I*3^(1/6)*Sqrt[5])*(3 - I*Sqrt[15])^(1/3) + 4*(3 - I*Sqrt[15])^(2/3)])/(3 - I*Sqrt[15])^(1/3) - 6*x))/Sqrt[-4*3^(1/3) - (3^( 2/3) - I*3^(1/6)*Sqrt[5])*(3 - I*Sqrt[15])^(1/3) + 4*(3 - I*Sqrt[15])^(2/3 )])]/Sqrt[6]], (-2*Sqrt[-4*3^(1/3) - (3^(2/3) - I*3^(1/6)*Sqrt[5])*(3 - I* Sqrt[15])^(1/3) + 4*(3 - I*Sqrt[15])^(2/3)])/(2*3^(2/3) + 3^(1/3)*(3 - I*S qrt[15])^(2/3) - Sqrt[-4*3^(1/3) - (3^(2/3) - I*3^(1/6)*Sqrt[5])*(3 - I*Sq rt[15])^(1/3) + 4*(3 - I*Sqrt[15])^(2/3)])])/(3*(3 - I*Sqrt[15])^(1/3)*Sqr t[2 - 6*x + 3*x^3])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 )/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e *Rt[b^2 - 4*a*c, 2])))^m)) Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e }, x] && EqQ[m^2, 1/4]
Int[((a_.) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_Symbol] :> With[{r = Rt[-9* a*d^2 + Sqrt[3]*d*Sqrt[4*b^3*d + 27*a^2*d^2], 3]}, Simp[(a + b*x + d*x^3)^p /(Simp[18^(1/3)*b*(d/(3*r)) - r/18^(1/3) + d*x, x]^p*Simp[b*(d/3) + 12^(1/3 )*b^2*(d^2/(3*r^2)) + r^2/(3*12^(1/3)) - d*(2^(1/3)*b*(d/(3^(1/3)*r)) - r/1 8^(1/3))*x + d^2*x^2, x]^p) Int[Simp[18^(1/3)*b*(d/(3*r)) - r/18^(1/3) + d*x, x]^p*Simp[b*(d/3) + 12^(1/3)*b^2*(d^2/(3*r^2)) + r^2/(3*12^(1/3)) - d* (2^(1/3)*b*(d/(3^(1/3)*r)) - r/18^(1/3))*x + d^2*x^2, x]^p, x], x]] /; Free Q[{a, b, d, p}, x] && NeQ[4*b^3 + 27*a^2*d, 0] && !IntegerQ[p]
Result contains complex when optimal does not.
Time = 0.41 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.93
| method | result | size |
| default | \(\frac {\sqrt {6}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {15}}{3}\right )}{3}\right ) \sqrt {\frac {\left (x +\frac {\sqrt {6}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {15}}{3}\right )}{3}\right ) \sqrt {3}}{3}-\frac {\sqrt {6}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {15}}{3}\right )}{3}\right )}{3}\right ) \sqrt {6}\, \sqrt {3}}{\sin \left (\frac {\arctan \left (\frac {\sqrt {15}}{3}\right )}{3}\right )}}\, \sqrt {\frac {x +\frac {2 \sqrt {6}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {15}}{3}\right )}{3}\right )}{3}}{-\frac {\sqrt {6}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {15}}{3}\right )}{3}\right ) \sqrt {3}}{3}+\sqrt {6}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {15}}{3}\right )}{3}\right )}}\, \sqrt {-\frac {3 \left (x -\frac {\sqrt {6}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {15}}{3}\right )}{3}\right ) \sqrt {3}}{3}-\frac {\sqrt {6}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {15}}{3}\right )}{3}\right )}{3}\right ) \sqrt {6}\, \sqrt {3}}{\sin \left (\frac {\arctan \left (\frac {\sqrt {15}}{3}\right )}{3}\right )}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {\left (x +\frac {\sqrt {6}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {15}}{3}\right )}{3}\right ) \sqrt {3}}{3}-\frac {\sqrt {6}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {15}}{3}\right )}{3}\right )}{3}\right ) \sqrt {6}\, \sqrt {3}}{\sin \left (\frac {\arctan \left (\frac {\sqrt {15}}{3}\right )}{3}\right )}}}{6}, \frac {i \sqrt {6}\, \sqrt {\frac {\sqrt {6}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {15}}{3}\right )}{3}\right ) \sqrt {3}}{-\frac {\sqrt {6}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {15}}{3}\right )}{3}\right ) \sqrt {3}}{3}+\sqrt {6}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {15}}{3}\right )}{3}\right )}}}{3}\right )}{9 \sqrt {3 x^{3}-6 x +2}}\) | \(302\) |
| elliptic | \(\frac {\sqrt {6}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {15}}{3}\right )}{3}\right ) \sqrt {\frac {\left (x +\frac {\sqrt {6}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {15}}{3}\right )}{3}\right ) \sqrt {3}}{3}-\frac {\sqrt {6}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {15}}{3}\right )}{3}\right )}{3}\right ) \sqrt {6}\, \sqrt {3}}{\sin \left (\frac {\arctan \left (\frac {\sqrt {15}}{3}\right )}{3}\right )}}\, \sqrt {\frac {x +\frac {2 \sqrt {6}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {15}}{3}\right )}{3}\right )}{3}}{-\frac {\sqrt {6}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {15}}{3}\right )}{3}\right ) \sqrt {3}}{3}+\sqrt {6}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {15}}{3}\right )}{3}\right )}}\, \sqrt {-\frac {3 \left (x -\frac {\sqrt {6}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {15}}{3}\right )}{3}\right ) \sqrt {3}}{3}-\frac {\sqrt {6}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {15}}{3}\right )}{3}\right )}{3}\right ) \sqrt {6}\, \sqrt {3}}{\sin \left (\frac {\arctan \left (\frac {\sqrt {15}}{3}\right )}{3}\right )}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {\left (x +\frac {\sqrt {6}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {15}}{3}\right )}{3}\right ) \sqrt {3}}{3}-\frac {\sqrt {6}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {15}}{3}\right )}{3}\right )}{3}\right ) \sqrt {6}\, \sqrt {3}}{\sin \left (\frac {\arctan \left (\frac {\sqrt {15}}{3}\right )}{3}\right )}}}{6}, \frac {i \sqrt {6}\, \sqrt {\frac {\sqrt {6}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {15}}{3}\right )}{3}\right ) \sqrt {3}}{-\frac {\sqrt {6}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {15}}{3}\right )}{3}\right ) \sqrt {3}}{3}+\sqrt {6}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {15}}{3}\right )}{3}\right )}}}{3}\right )}{9 \sqrt {3 x^{3}-6 x +2}}\) | \(302\) |
Input:
int(1/(3*x^3-6*x+2)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/9*6^(1/2)*sin(1/3*arctan(1/3*15^(1/2)))*((x+1/3*6^(1/2)*sin(1/3*arctan(1 /3*15^(1/2)))*3^(1/2)-1/3*6^(1/2)*cos(1/3*arctan(1/3*15^(1/2))))*6^(1/2)/s in(1/3*arctan(1/3*15^(1/2)))*3^(1/2))^(1/2)*((x+2/3*6^(1/2)*cos(1/3*arctan (1/3*15^(1/2))))/(-1/3*6^(1/2)*sin(1/3*arctan(1/3*15^(1/2)))*3^(1/2)+6^(1/ 2)*cos(1/3*arctan(1/3*15^(1/2)))))^(1/2)*(-3*(x-1/3*6^(1/2)*sin(1/3*arctan (1/3*15^(1/2)))*3^(1/2)-1/3*6^(1/2)*cos(1/3*arctan(1/3*15^(1/2))))*6^(1/2) /sin(1/3*arctan(1/3*15^(1/2)))*3^(1/2))^(1/2)/(3*x^3-6*x+2)^(1/2)*Elliptic F(1/6*3^(1/2)*((x+1/3*6^(1/2)*sin(1/3*arctan(1/3*15^(1/2)))*3^(1/2)-1/3*6^ (1/2)*cos(1/3*arctan(1/3*15^(1/2))))*6^(1/2)/sin(1/3*arctan(1/3*15^(1/2))) *3^(1/2))^(1/2),1/3*I*6^(1/2)*(6^(1/2)*sin(1/3*arctan(1/3*15^(1/2)))*3^(1/ 2)/(-1/3*6^(1/2)*sin(1/3*arctan(1/3*15^(1/2)))*3^(1/2)+6^(1/2)*cos(1/3*arc tan(1/3*15^(1/2)))))^(1/2))
Time = 0.07 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.03 \[ \int \frac {1}{\sqrt {2-6 x+3 x^3}} \, dx=\frac {2}{3} \, \sqrt {3} {\rm weierstrassPInverse}\left (8, -\frac {8}{3}, x\right ) \] Input:
integrate(1/(3*x^3-6*x+2)^(1/2),x, algorithm="fricas")
Output:
2/3*sqrt(3)*weierstrassPInverse(8, -8/3, x)
\[ \int \frac {1}{\sqrt {2-6 x+3 x^3}} \, dx=\int \frac {1}{\sqrt {3 x^{3} - 6 x + 2}}\, dx \] Input:
integrate(1/(3*x**3-6*x+2)**(1/2),x)
Output:
Integral(1/sqrt(3*x**3 - 6*x + 2), x)
\[ \int \frac {1}{\sqrt {2-6 x+3 x^3}} \, dx=\int { \frac {1}{\sqrt {3 \, x^{3} - 6 \, x + 2}} \,d x } \] Input:
integrate(1/(3*x^3-6*x+2)^(1/2),x, algorithm="maxima")
Output:
integrate(1/sqrt(3*x^3 - 6*x + 2), x)
\[ \int \frac {1}{\sqrt {2-6 x+3 x^3}} \, dx=\int { \frac {1}{\sqrt {3 \, x^{3} - 6 \, x + 2}} \,d x } \] Input:
integrate(1/(3*x^3-6*x+2)^(1/2),x, algorithm="giac")
Output:
integrate(1/sqrt(3*x^3 - 6*x + 2), x)
Time = 12.97 (sec) , antiderivative size = 946, normalized size of antiderivative = 2.93 \[ \int \frac {1}{\sqrt {2-6 x+3 x^3}} \, dx=\text {Too large to display} \] Input:
int(1/(3*x^3 - 6*x + 2)^(1/2),x)
Output:
(2*ellipticF(asin(((x + 1/(3*((15^(1/2)*1i)/9 - 1/3)^(1/3)) + ((15^(1/2)*1 i)/9 - 1/3)^(1/3)/2 - 3^(1/2)*(1/(3*((15^(1/2)*1i)/9 - 1/3)^(1/3)) - ((15^ (1/2)*1i)/9 - 1/3)^(1/3)/2)*1i)/(1/((15^(1/2)*1i)/9 - 1/3)^(1/3) + (3*((15 ^(1/2)*1i)/9 - 1/3)^(1/3))/2 - 3^(1/2)*(1/(3*((15^(1/2)*1i)/9 - 1/3)^(1/3) ) - ((15^(1/2)*1i)/9 - 1/3)^(1/3)/2)*1i))^(1/2)), (3^(1/2)*(1/(3*((15^(1/2 )*1i)/9 - 1/3)^(1/3)) + ((15^(1/2)*1i)/9 - 1/3)^(1/3)/2 - 3^(1/2)*(1/(9*(( 15^(1/2)*1i)/9 - 1/3)^(1/3)) - ((15^(1/2)*1i)/9 - 1/3)^(1/3)/6)*1i)*1i)/(2 /(3*((15^(1/2)*1i)/9 - 1/3)^(1/3)) - ((15^(1/2)*1i)/9 - 1/3)^(1/3)))*((2/( 3*((15^(1/2)*1i)/9 - 1/3)^(1/3)) - x + ((15^(1/2)*1i)/9 - 1/3)^(1/3))/(1/( (15^(1/2)*1i)/9 - 1/3)^(1/3) + (3*((15^(1/2)*1i)/9 - 1/3)^(1/3))/2 - 3^(1/ 2)*(1/(3*((15^(1/2)*1i)/9 - 1/3)^(1/3)) - ((15^(1/2)*1i)/9 - 1/3)^(1/3)/2) *1i))^(1/2)*(x^3 - 2*x + 2/3)^(1/2)*(1/((15^(1/2)*1i)/9 - 1/3)^(1/3) + (3* ((15^(1/2)*1i)/9 - 1/3)^(1/3))/2 - 3^(1/2)*(1/(3*((15^(1/2)*1i)/9 - 1/3)^( 1/3)) - ((15^(1/2)*1i)/9 - 1/3)^(1/3)/2)*1i)^(1/2)*(-(3^(1/2)*(x/3 + 1/(9* ((15^(1/2)*1i)/9 - 1/3)^(1/3)) + ((15^(1/2)*1i)/9 - 1/3)^(1/3)/6 + 3^(1/2) *(1/(9*((15^(1/2)*1i)/9 - 1/3)^(1/3)) - ((15^(1/2)*1i)/9 - 1/3)^(1/3)/6)*1 i)*1i)/(2/(3*((15^(1/2)*1i)/9 - 1/3)^(1/3)) - ((15^(1/2)*1i)/9 - 1/3)^(1/3 )))^(1/2)*(x + 1/(3*((15^(1/2)*1i)/9 - 1/3)^(1/3)) + ((15^(1/2)*1i)/9 - 1/ 3)^(1/3)/2 - 3^(1/2)*(1/(3*((15^(1/2)*1i)/9 - 1/3)^(1/3)) - ((15^(1/2)*1i) /9 - 1/3)^(1/3)/2)*1i)^(1/2))/((x^3 - x*((2/(3*((15^(1/2)*1i)/9 - 1/3)^...
\[ \int \frac {1}{\sqrt {2-6 x+3 x^3}} \, dx=\int \frac {\sqrt {3 x^{3}-6 x +2}}{3 x^{3}-6 x +2}d x \] Input:
int(1/(3*x^3-6*x+2)^(1/2),x)
Output:
int(sqrt(3*x**3 - 6*x + 2)/(3*x**3 - 6*x + 2),x)