Integrand size = 14, antiderivative size = 672 \[ \int \frac {1}{\sqrt {2-4 x+3 x^3}} \, dx=\frac {\sqrt [3]{9-\sqrt {17}} \sqrt {\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}+3 x} \sqrt {-4+\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}+\left (9-\sqrt {17}\right )^{2/3}-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}+9 x^2} \sqrt {-\frac {4-\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}-\left (9-\sqrt {17}\right )^{2/3}+\frac {3 \left (9-\sqrt {17}+4 \sqrt [3]{9-\sqrt {17}}\right ) x}{\left (9-\sqrt {17}\right )^{2/3}}-9 x^2}{\left (1+\frac {4+\left (9-\sqrt {17}\right )^{2/3}+3 \sqrt [3]{9-\sqrt {17}} x}{\sqrt {3 \left (16+4 \left (9-\sqrt {17}\right )^{2/3}+\left (9-\sqrt {17}\right )^{4/3}\right )}}\right )^2}} \left (1+\frac {4+\left (9-\sqrt {17}\right )^{2/3}+3 \sqrt [3]{9-\sqrt {17}} x}{\sqrt {3 \left (16+4 \left (9-\sqrt {17}\right )^{2/3}+\left (9-\sqrt {17}\right )^{4/3}\right )}}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [6]{9-\sqrt {17}} \sqrt {\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}+3 x}}{\sqrt [4]{3 \left (16+4 \left (9-\sqrt {17}\right )^{2/3}+\left (9-\sqrt {17}\right )^{4/3}\right )}}\right ),\frac {1}{4} \left (2+\left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) \sqrt {\frac {3}{16+4 \left (9-\sqrt {17}\right )^{2/3}+\left (9-\sqrt {17}\right )^{4/3}}}\right )\right )}{3 \sqrt [4]{\frac {3}{16+4 \left (9-\sqrt {17}\right )^{2/3}+\left (9-\sqrt {17}\right )^{4/3}}} \sqrt {36-4 \sqrt {17}+16 \sqrt [3]{9-\sqrt {17}}+\left (9-\sqrt {17}\right )^{5/3}} \sqrt {-4+\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}+\left (9-\sqrt {17}\right )^{2/3}-\frac {3 \left (9-\sqrt {17}+4 \sqrt [3]{9-\sqrt {17}}\right ) x}{\left (9-\sqrt {17}\right )^{2/3}}+9 x^2} \sqrt {2-4 x+3 x^3}} \] Output:
1/9*(9-17^(1/2))^(1/3)*((4+(9-17^(1/2))^(2/3))/(9-17^(1/2))^(1/3)+3*x)^(1/ 2)*(-4+16/(9-17^(1/2))^(2/3)+(9-17^(1/2))^(2/3)-3*(4+(9-17^(1/2))^(2/3))*x /(9-17^(1/2))^(1/3)+9*x^2)^(1/2)*(-(4-16/(9-17^(1/2))^(2/3)-(9-17^(1/2))^( 2/3)+3*(9-17^(1/2)+4*(9-17^(1/2))^(1/3))*x/(9-17^(1/2))^(2/3)-9*x^2)/(1+(4 +(9-17^(1/2))^(2/3)+3*(9-17^(1/2))^(1/3)*x)/(48+12*(9-17^(1/2))^(2/3)+3*(9 -17^(1/2))^(4/3))^(1/2))^2)^(1/2)*(1+(4+(9-17^(1/2))^(2/3)+3*(9-17^(1/2))^ (1/3)*x)/(48+12*(9-17^(1/2))^(2/3)+3*(9-17^(1/2))^(4/3))^(1/2))*InverseJac obiAM(2*arctan((9-17^(1/2))^(1/6)*((4+(9-17^(1/2))^(2/3))/(9-17^(1/2))^(1/ 3)+3*x)^(1/2)/(48+12*(9-17^(1/2))^(2/3)+3*(9-17^(1/2))^(4/3))^(1/4)),1/2*( 2+(4+(9-17^(1/2))^(2/3))*3^(1/2)/(16+4*(9-17^(1/2))^(2/3)+(9-17^(1/2))^(4/ 3))^(1/2))^(1/2))*3^(3/4)/(1/(16+4*(9-17^(1/2))^(2/3)+(9-17^(1/2))^(4/3))) ^(1/4)/(36-4*17^(1/2)+16*(9-17^(1/2))^(1/3)+(9-17^(1/2))^(5/3))^(1/2)/(-4+ 16/(9-17^(1/2))^(2/3)+(9-17^(1/2))^(2/3)-3*(9-17^(1/2)+4*(9-17^(1/2))^(1/3 ))*x/(9-17^(1/2))^(2/3)+9*x^2)^(1/2)/(3*x^3-4*x+2)^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
Time = 10.13 (sec) , antiderivative size = 359, normalized size of antiderivative = 0.53 \[ \int \frac {1}{\sqrt {2-4 x+3 x^3}} \, dx=\frac {2 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {-x+\text {Root}\left [3 \text {$\#$1}^3-4 \text {$\#$1}+2\&,3\right ]}{-\text {Root}\left [3 \text {$\#$1}^3-4 \text {$\#$1}+2\&,2\right ]+\text {Root}\left [3 \text {$\#$1}^3-4 \text {$\#$1}+2\&,3\right ]}}\right ),\frac {\text {Root}\left [3 \text {$\#$1}^3-4 \text {$\#$1}+2\&,2\right ]-\text {Root}\left [3 \text {$\#$1}^3-4 \text {$\#$1}+2\&,3\right ]}{\text {Root}\left [3 \text {$\#$1}^3-4 \text {$\#$1}+2\&,1\right ]-\text {Root}\left [3 \text {$\#$1}^3-4 \text {$\#$1}+2\&,3\right ]}\right ) \left (x-\text {Root}\left [3 \text {$\#$1}^3-4 \text {$\#$1}+2\&,3\right ]\right ) \sqrt {\frac {x-\text {Root}\left [3 \text {$\#$1}^3-4 \text {$\#$1}+2\&,1\right ]}{-\text {Root}\left [3 \text {$\#$1}^3-4 \text {$\#$1}+2\&,1\right ]+\text {Root}\left [3 \text {$\#$1}^3-4 \text {$\#$1}+2\&,3\right ]}} \sqrt {\frac {x-\text {Root}\left [3 \text {$\#$1}^3-4 \text {$\#$1}+2\&,2\right ]}{-\text {Root}\left [3 \text {$\#$1}^3-4 \text {$\#$1}+2\&,2\right ]+\text {Root}\left [3 \text {$\#$1}^3-4 \text {$\#$1}+2\&,3\right ]}}}{\sqrt {2-4 x+3 x^3} \sqrt {\frac {x-\text {Root}\left [3 \text {$\#$1}^3-4 \text {$\#$1}+2\&,3\right ]}{\text {Root}\left [3 \text {$\#$1}^3-4 \text {$\#$1}+2\&,2\right ]-\text {Root}\left [3 \text {$\#$1}^3-4 \text {$\#$1}+2\&,3\right ]}}} \] Input:
Integrate[1/Sqrt[2 - 4*x + 3*x^3],x]
Output:
(2*EllipticF[ArcSin[Sqrt[(-x + Root[2 - 4*#1 + 3*#1^3 & , 3, 0])/(-Root[2 - 4*#1 + 3*#1^3 & , 2, 0] + Root[2 - 4*#1 + 3*#1^3 & , 3, 0])]], (Root[2 - 4*#1 + 3*#1^3 & , 2, 0] - Root[2 - 4*#1 + 3*#1^3 & , 3, 0])/(Root[2 - 4*# 1 + 3*#1^3 & , 1, 0] - Root[2 - 4*#1 + 3*#1^3 & , 3, 0])]*(x - Root[2 - 4* #1 + 3*#1^3 & , 3, 0])*Sqrt[(x - Root[2 - 4*#1 + 3*#1^3 & , 1, 0])/(-Root[ 2 - 4*#1 + 3*#1^3 & , 1, 0] + Root[2 - 4*#1 + 3*#1^3 & , 3, 0])]*Sqrt[(x - Root[2 - 4*#1 + 3*#1^3 & , 2, 0])/(-Root[2 - 4*#1 + 3*#1^3 & , 2, 0] + Ro ot[2 - 4*#1 + 3*#1^3 & , 3, 0])])/(Sqrt[2 - 4*x + 3*x^3]*Sqrt[(x - Root[2 - 4*#1 + 3*#1^3 & , 3, 0])/(Root[2 - 4*#1 + 3*#1^3 & , 2, 0] - Root[2 - 4* #1 + 3*#1^3 & , 3, 0])])
Result contains complex when optimal does not.
Time = 0.67 (sec) , antiderivative size = 357, normalized size of antiderivative = 0.53, 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.
| \(\displaystyle \int \frac {1}{\sqrt {3 x^3-4 x+2}} \, dx\) |
| \(\Big \downarrow \) 2475 |
| \(\displaystyle \frac {\sqrt {3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}} \sqrt {9 x^2-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}+\left (9-\sqrt {17}\right )^{2/3}+\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}-4} \int \frac {1}{\sqrt {3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}} \sqrt {9 x^2-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}+\left (9-\sqrt {17}\right )^{2/3}+\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}-4}}dx}{\sqrt {3 x^3-4 x+2}}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle -\frac {2 i \sqrt {2} \sqrt [6]{9-\sqrt {17}} \sqrt {\frac {i \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )}{4 \left (-\sqrt {3}+3 i\right )+\left (\sqrt {3}+3 i\right ) \left (9-\sqrt {17}\right )^{2/3}}} \sqrt {9 x^2-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}+\left (9-\sqrt {17}\right )^{2/3}+\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}-4} \int \frac {1}{\sqrt {\frac {i \sqrt [3]{9-\sqrt {17}} \left (\frac {4+i \left (4 \sqrt {3}-\left (i+\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}\right )}{\sqrt [3]{9-\sqrt {17}}}-6 x\right )}{2 \sqrt {3} \left (4-\left (9-\sqrt {17}\right )^{2/3}\right )}+1} \sqrt {\frac {i \sqrt [3]{9-\sqrt {17}} \left (\frac {4+i \left (4 \sqrt {3}-\left (i+\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}\right )}{\sqrt [3]{9-\sqrt {17}}}-6 x\right )}{\sqrt {3} \left (4-4 i \sqrt {3}-\left (1+i \sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}\right )}+1}}d\frac {\sqrt [6]{9-\sqrt {17}} \sqrt {-i \left (\frac {4+i \left (4 \sqrt {3}-\left (i+\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}\right )}{\sqrt [3]{9-\sqrt {17}}}-6 x\right )}}{\sqrt [4]{3} \sqrt {2 \left (4-\left (9-\sqrt {17}\right )^{2/3}\right )}}}{3 \sqrt {3 x^3-4 x+2}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle -\frac {2 i \sqrt {2} \sqrt [6]{9-\sqrt {17}} \sqrt {\frac {i \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )}{4 \left (-\sqrt {3}+3 i\right )+\left (\sqrt {3}+3 i\right ) \left (9-\sqrt {17}\right )^{2/3}}} \sqrt {9 x^2-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}+\left (9-\sqrt {17}\right )^{2/3}+\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}-4} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [6]{9-\sqrt {17}} \sqrt {-i \left (\frac {4+i \left (4 \sqrt {3}-\left (i+\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}\right )}{\sqrt [3]{9-\sqrt {17}}}-6 x\right )}}{\sqrt [4]{3} \sqrt {2 \left (4-\left (9-\sqrt {17}\right )^{2/3}\right )}}\right ),\frac {2 i \left (4-\left (9-\sqrt {17}\right )^{2/3}\right )}{4 \left (i+\sqrt {3}\right )-\left (i-\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}}\right )}{3 \sqrt {3 x^3-4 x+2}}\) |
Input:
Int[1/Sqrt[2 - 4*x + 3*x^3],x]
Output:
(((-2*I)/3)*Sqrt[2]*(9 - Sqrt[17])^(1/6)*Sqrt[(I*((4 + (9 - Sqrt[17])^(2/3 ))/(9 - Sqrt[17])^(1/3) + 3*x))/(4*(3*I - Sqrt[3]) + (3*I + Sqrt[3])*(9 - Sqrt[17])^(2/3))]*Sqrt[-4 + 16/(9 - Sqrt[17])^(2/3) + (9 - Sqrt[17])^(2/3) - (3*(4 + (9 - Sqrt[17])^(2/3))*x)/(9 - Sqrt[17])^(1/3) + 9*x^2]*Elliptic F[ArcSin[((9 - Sqrt[17])^(1/6)*Sqrt[(-I)*((4 + I*(4*Sqrt[3] - (I + Sqrt[3] )*(9 - Sqrt[17])^(2/3)))/(9 - Sqrt[17])^(1/3) - 6*x)])/(3^(1/4)*Sqrt[2*(4 - (9 - Sqrt[17])^(2/3))])], ((2*I)*(4 - (9 - Sqrt[17])^(2/3)))/(4*(I + Sqr t[3]) - (I - Sqrt[3])*(9 - Sqrt[17])^(2/3))])/Sqrt[2 - 4*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 = 0.52 (sec) , antiderivative size = 408, normalized size of antiderivative = 0.61
| method | result | size |
| default | \(\frac {2 i \sqrt {3}\, \left (-\frac {\left (9+\sqrt {17}\right )^{\frac {1}{3}}}{3}+\frac {4}{3 \left (9+\sqrt {17}\right )^{\frac {1}{3}}}\right ) \sqrt {\frac {i \left (x -\frac {\left (9+\sqrt {17}\right )^{\frac {1}{3}}}{6}-\frac {2}{3 \left (9+\sqrt {17}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (9+\sqrt {17}\right )^{\frac {1}{3}}}{3}+\frac {4}{3 \left (9+\sqrt {17}\right )^{\frac {1}{3}}}\right )}{2}\right ) \sqrt {3}}{\frac {\left (9+\sqrt {17}\right )^{\frac {1}{3}}}{3}-\frac {4}{3 \left (9+\sqrt {17}\right )^{\frac {1}{3}}}}}\, \sqrt {\frac {x +\frac {\left (9+\sqrt {17}\right )^{\frac {1}{3}}}{3}+\frac {4}{3 \left (9+\sqrt {17}\right )^{\frac {1}{3}}}}{\frac {\left (9+\sqrt {17}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (9+\sqrt {17}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (9+\sqrt {17}\right )^{\frac {1}{3}}}{3}+\frac {4}{3 \left (9+\sqrt {17}\right )^{\frac {1}{3}}}\right )}{2}}}\, \sqrt {-\frac {i \left (x -\frac {\left (9+\sqrt {17}\right )^{\frac {1}{3}}}{6}-\frac {2}{3 \left (9+\sqrt {17}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (9+\sqrt {17}\right )^{\frac {1}{3}}}{3}+\frac {4}{3 \left (9+\sqrt {17}\right )^{\frac {1}{3}}}\right )}{2}\right ) \sqrt {3}}{\frac {\left (9+\sqrt {17}\right )^{\frac {1}{3}}}{3}-\frac {4}{3 \left (9+\sqrt {17}\right )^{\frac {1}{3}}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x -\frac {\left (9+\sqrt {17}\right )^{\frac {1}{3}}}{6}-\frac {2}{3 \left (9+\sqrt {17}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (9+\sqrt {17}\right )^{\frac {1}{3}}}{3}+\frac {4}{3 \left (9+\sqrt {17}\right )^{\frac {1}{3}}}\right )}{2}\right ) \sqrt {3}}{\frac {\left (9+\sqrt {17}\right )^{\frac {1}{3}}}{3}-\frac {4}{3 \left (9+\sqrt {17}\right )^{\frac {1}{3}}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (\frac {\left (9+\sqrt {17}\right )^{\frac {1}{3}}}{3}-\frac {4}{3 \left (9+\sqrt {17}\right )^{\frac {1}{3}}}\right )}{\frac {\left (9+\sqrt {17}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (9+\sqrt {17}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (9+\sqrt {17}\right )^{\frac {1}{3}}}{3}+\frac {4}{3 \left (9+\sqrt {17}\right )^{\frac {1}{3}}}\right )}{2}}}\right )}{3 \sqrt {3 x^{3}-4 x +2}}\) | \(408\) |
| elliptic | \(\frac {2 i \sqrt {3}\, \left (-\frac {\left (9+\sqrt {17}\right )^{\frac {1}{3}}}{3}+\frac {4}{3 \left (9+\sqrt {17}\right )^{\frac {1}{3}}}\right ) \sqrt {\frac {i \left (x -\frac {\left (9+\sqrt {17}\right )^{\frac {1}{3}}}{6}-\frac {2}{3 \left (9+\sqrt {17}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (9+\sqrt {17}\right )^{\frac {1}{3}}}{3}+\frac {4}{3 \left (9+\sqrt {17}\right )^{\frac {1}{3}}}\right )}{2}\right ) \sqrt {3}}{\frac {\left (9+\sqrt {17}\right )^{\frac {1}{3}}}{3}-\frac {4}{3 \left (9+\sqrt {17}\right )^{\frac {1}{3}}}}}\, \sqrt {\frac {x +\frac {\left (9+\sqrt {17}\right )^{\frac {1}{3}}}{3}+\frac {4}{3 \left (9+\sqrt {17}\right )^{\frac {1}{3}}}}{\frac {\left (9+\sqrt {17}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (9+\sqrt {17}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (9+\sqrt {17}\right )^{\frac {1}{3}}}{3}+\frac {4}{3 \left (9+\sqrt {17}\right )^{\frac {1}{3}}}\right )}{2}}}\, \sqrt {-\frac {i \left (x -\frac {\left (9+\sqrt {17}\right )^{\frac {1}{3}}}{6}-\frac {2}{3 \left (9+\sqrt {17}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (9+\sqrt {17}\right )^{\frac {1}{3}}}{3}+\frac {4}{3 \left (9+\sqrt {17}\right )^{\frac {1}{3}}}\right )}{2}\right ) \sqrt {3}}{\frac {\left (9+\sqrt {17}\right )^{\frac {1}{3}}}{3}-\frac {4}{3 \left (9+\sqrt {17}\right )^{\frac {1}{3}}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x -\frac {\left (9+\sqrt {17}\right )^{\frac {1}{3}}}{6}-\frac {2}{3 \left (9+\sqrt {17}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (9+\sqrt {17}\right )^{\frac {1}{3}}}{3}+\frac {4}{3 \left (9+\sqrt {17}\right )^{\frac {1}{3}}}\right )}{2}\right ) \sqrt {3}}{\frac {\left (9+\sqrt {17}\right )^{\frac {1}{3}}}{3}-\frac {4}{3 \left (9+\sqrt {17}\right )^{\frac {1}{3}}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (\frac {\left (9+\sqrt {17}\right )^{\frac {1}{3}}}{3}-\frac {4}{3 \left (9+\sqrt {17}\right )^{\frac {1}{3}}}\right )}{\frac {\left (9+\sqrt {17}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (9+\sqrt {17}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (9+\sqrt {17}\right )^{\frac {1}{3}}}{3}+\frac {4}{3 \left (9+\sqrt {17}\right )^{\frac {1}{3}}}\right )}{2}}}\right )}{3 \sqrt {3 x^{3}-4 x +2}}\) | \(408\) |
Input:
int(1/(3*x^3-4*x+2)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/3*I*3^(1/2)*(-1/3*(9+17^(1/2))^(1/3)+4/3/(9+17^(1/2))^(1/3))*(I*(x-1/6*( 9+17^(1/2))^(1/3)-2/3/(9+17^(1/2))^(1/3)+1/2*I*3^(1/2)*(-1/3*(9+17^(1/2))^ (1/3)+4/3/(9+17^(1/2))^(1/3)))*3^(1/2)/(1/3*(9+17^(1/2))^(1/3)-4/3/(9+17^( 1/2))^(1/3)))^(1/2)*((x+1/3*(9+17^(1/2))^(1/3)+4/3/(9+17^(1/2))^(1/3))/(1/ 2*(9+17^(1/2))^(1/3)+2/(9+17^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/3*(9+17^(1/2)) ^(1/3)+4/3/(9+17^(1/2))^(1/3))))^(1/2)*(-I*(x-1/6*(9+17^(1/2))^(1/3)-2/3/( 9+17^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/3*(9+17^(1/2))^(1/3)+4/3/(9+17^(1/2))^ (1/3)))*3^(1/2)/(1/3*(9+17^(1/2))^(1/3)-4/3/(9+17^(1/2))^(1/3)))^(1/2)/(3* x^3-4*x+2)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x-1/6*(9+17^(1/2))^(1/3)-2/3/(9 +17^(1/2))^(1/3)+1/2*I*3^(1/2)*(-1/3*(9+17^(1/2))^(1/3)+4/3/(9+17^(1/2))^( 1/3)))*3^(1/2)/(1/3*(9+17^(1/2))^(1/3)-4/3/(9+17^(1/2))^(1/3)))^(1/2),(I*3 ^(1/2)*(1/3*(9+17^(1/2))^(1/3)-4/3/(9+17^(1/2))^(1/3))/(1/2*(9+17^(1/2))^( 1/3)+2/(9+17^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/3*(9+17^(1/2))^(1/3)+4/3/(9+17 ^(1/2))^(1/3))))^(1/2))
Time = 0.07 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.01 \[ \int \frac {1}{\sqrt {2-4 x+3 x^3}} \, dx=\frac {2}{3} \, \sqrt {3} {\rm weierstrassPInverse}\left (\frac {16}{3}, -\frac {8}{3}, x\right ) \] Input:
integrate(1/(3*x^3-4*x+2)^(1/2),x, algorithm="fricas")
Output:
2/3*sqrt(3)*weierstrassPInverse(16/3, -8/3, x)
\[ \int \frac {1}{\sqrt {2-4 x+3 x^3}} \, dx=\int \frac {1}{\sqrt {3 x^{3} - 4 x + 2}}\, dx \] Input:
integrate(1/(3*x**3-4*x+2)**(1/2),x)
Output:
Integral(1/sqrt(3*x**3 - 4*x + 2), x)
\[ \int \frac {1}{\sqrt {2-4 x+3 x^3}} \, dx=\int { \frac {1}{\sqrt {3 \, x^{3} - 4 \, x + 2}} \,d x } \] Input:
integrate(1/(3*x^3-4*x+2)^(1/2),x, algorithm="maxima")
Output:
integrate(1/sqrt(3*x^3 - 4*x + 2), x)
\[ \int \frac {1}{\sqrt {2-4 x+3 x^3}} \, dx=\int { \frac {1}{\sqrt {3 \, x^{3} - 4 \, x + 2}} \,d x } \] Input:
integrate(1/(3*x^3-4*x+2)^(1/2),x, algorithm="giac")
Output:
integrate(1/sqrt(3*x^3 - 4*x + 2), x)
Time = 13.55 (sec) , antiderivative size = 940, normalized size of antiderivative = 1.40 \[ \int \frac {1}{\sqrt {2-4 x+3 x^3}} \, dx=\text {Too large to display} \] Input:
int(1/(3*x^3 - 4*x + 2)^(1/2),x)
Output:
-(2*(-(x - 2/(9*(1/3 - 17^(1/2)/27)^(1/3)) - (1/3 - 17^(1/2)/27)^(1/3)/2 + (3^(1/2)*(4/(9*(1/3 - 17^(1/2)/27)^(1/3)) - (1/3 - 17^(1/2)/27)^(1/3))*1i )/2)/(2/(3*(1/3 - 17^(1/2)/27)^(1/3)) + (3*(1/3 - 17^(1/2)/27)^(1/3))/2 - (3^(1/2)*(4/(9*(1/3 - 17^(1/2)/27)^(1/3)) - (1/3 - 17^(1/2)/27)^(1/3))*1i) /2))^(1/2)*ellipticF(asin((-(x - 2/(9*(1/3 - 17^(1/2)/27)^(1/3)) - (1/3 - 17^(1/2)/27)^(1/3)/2 + (3^(1/2)*(4/(9*(1/3 - 17^(1/2)/27)^(1/3)) - (1/3 - 17^(1/2)/27)^(1/3))*1i)/2)/(2/(3*(1/3 - 17^(1/2)/27)^(1/3)) + (3*(1/3 - 17 ^(1/2)/27)^(1/3))/2 - (3^(1/2)*(4/(9*(1/3 - 17^(1/2)/27)^(1/3)) - (1/3 - 1 7^(1/2)/27)^(1/3))*1i)/2))^(1/2)), (3^(1/2)*(2/(3*(1/3 - 17^(1/2)/27)^(1/3 )) + (3*(1/3 - 17^(1/2)/27)^(1/3))/2 - (3^(1/2)*(4/(9*(1/3 - 17^(1/2)/27)^ (1/3)) - (1/3 - 17^(1/2)/27)^(1/3))*1i)/2)*1i)/(3*(4/(9*(1/3 - 17^(1/2)/27 )^(1/3)) - (1/3 - 17^(1/2)/27)^(1/3))))*((x + 4/(9*(1/3 - 17^(1/2)/27)^(1/ 3)) + (1/3 - 17^(1/2)/27)^(1/3))/(2/(3*(1/3 - 17^(1/2)/27)^(1/3)) + (3*(1/ 3 - 17^(1/2)/27)^(1/3))/2 - (3^(1/2)*(4/(9*(1/3 - 17^(1/2)/27)^(1/3)) - (1 /3 - 17^(1/2)/27)^(1/3))*1i)/2))^(1/2)*(x^3 - (4*x)/3 + 2/3)^(1/2)*(2/(3*( 1/3 - 17^(1/2)/27)^(1/3)) + (3*(1/3 - 17^(1/2)/27)^(1/3))/2 - (3^(1/2)*(4/ (9*(1/3 - 17^(1/2)/27)^(1/3)) - (1/3 - 17^(1/2)/27)^(1/3))*1i)/2)*(-(3^(1/ 2)*(2/(9*(1/3 - 17^(1/2)/27)^(1/3)) - x + (1/3 - 17^(1/2)/27)^(1/3)/2 + (3 ^(1/2)*(4/(9*(1/3 - 17^(1/2)/27)^(1/3)) - (1/3 - 17^(1/2)/27)^(1/3))*1i)/2 )*1i)/(3*(4/(9*(1/3 - 17^(1/2)/27)^(1/3)) - (1/3 - 17^(1/2)/27)^(1/3)))...
\[ \int \frac {1}{\sqrt {2-4 x+3 x^3}} \, dx=\int \frac {\sqrt {3 x^{3}-4 x +2}}{3 x^{3}-4 x +2}d x \] Input:
int(1/(3*x^3-4*x+2)^(1/2),x)
Output:
int(sqrt(3*x**3 - 4*x + 2)/(3*x**3 - 4*x + 2),x)