\(\int \frac {1}{(2-4 x+3 x^3)^2} \, dx\) [20]

Optimal result

Integrand size = 12, antiderivative size = 731 \[ \int \frac {1}{\left (2-4 x+3 x^3\right )^2} \, dx=\frac {18 \sqrt [3]{9-\sqrt {17}}}{17 \left (4+\left (9-\sqrt {17}\right )^{2/3}+3 \sqrt [3]{9-\sqrt {17}} x\right )}+\frac {9 \sqrt [3]{-17+9 \sqrt {17}} \left (8-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}\right )}{2\ 17^{2/3} \left (4-\left (9-\sqrt {17}\right )^{2/3}\right ) \left (\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}+3 x\right ) \left (4-\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}-\left (9-\sqrt {17}\right )^{2/3}+\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}-9 x^2\right )}-\frac {12 \left (9-\sqrt {17}\right )^{5/3} \left (585-97 \sqrt {17}+4 \left (65-9 \sqrt {17}\right ) \sqrt [3]{9-\sqrt {17}}\right ) \sqrt {\frac {6}{49-9 \sqrt {17}+8 \left (9-\sqrt {17}\right )^{2/3}-4 \left (9-\sqrt {17}\right )^{4/3}}} \arctan \left (\frac {9-\sqrt {17}+4 \sqrt [3]{9-\sqrt {17}}-6 \left (9-\sqrt {17}\right )^{2/3} x}{\sqrt {6 \left (49-9 \sqrt {17}+8 \left (9-\sqrt {17}\right )^{2/3}-4 \left (9-\sqrt {17}\right )^{4/3}\right )}}\right )}{\left (4-\left (9-\sqrt {17}\right )^{2/3}\right )^2 \left (16+4 \left (9-\sqrt {17}\right )^{2/3}+\left (9-\sqrt {17}\right )^{4/3}\right )^3}+\frac {6 \left (9-\sqrt {17}\right )^{5/3} \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) \log \left (4+\left (9-\sqrt {17}\right )^{2/3}+3 \sqrt [3]{9-\sqrt {17}} x\right )}{\left (16+4 \left (9-\sqrt {17}\right )^{2/3}+\left (9-\sqrt {17}\right )^{4/3}\right )^3}+\frac {6 \left (9-\sqrt {17}\right )^{7/3} \left (8-9 \sqrt [3]{9-\sqrt {17}}+2 \left (9-\sqrt {17}\right )^{2/3}\right ) \log \left (16-4 \left (9-\sqrt {17}\right )^{2/3}+\left (9-\sqrt {17}\right )^{4/3}-3 \left (9-\sqrt {17}+4 \sqrt [3]{9-\sqrt {17}}\right ) x+9 \left (9-\sqrt {17}\right )^{2/3} x^2\right )}{\left (4-\left (9-\sqrt {17}\right )^{2/3}\right )^2 \left (16+4 \left (9-\sqrt {17}\right )^{2/3}+\left (9-\sqrt {17}\right )^{4/3}\right )^3} \] Output:

18*(9-17^(1/2))^(1/3)/(68+17*(9-17^(1/2))^(2/3)+51*(9-17^(1/2))^(1/3)*x)+9 
/34*(-17+9*17^(1/2))^(1/3)*(8-3*(4+(9-17^(1/2))^(2/3))*x/(9-17^(1/2))^(1/3 
))*17^(1/3)/(4-(9-17^(1/2))^(2/3))/((4+(9-17^(1/2))^(2/3))/(9-17^(1/2))^(1 
/3)+3*x)/(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)-12*(9-17^(1/2))^(5/3)*(585-97*17^(1/2)+4*( 
65-9*17^(1/2))*(9-17^(1/2))^(1/3))*6^(1/2)/(49-9*17^(1/2)+8*(9-17^(1/2))^( 
2/3)-4*(9-17^(1/2))^(4/3))^(1/2)*arctan((9-17^(1/2)+4*(9-17^(1/2))^(1/3)-6 
*(9-17^(1/2))^(2/3)*x)/(294-54*17^(1/2)+48*(9-17^(1/2))^(2/3)-24*(9-17^(1/ 
2))^(4/3))^(1/2))/(4-(9-17^(1/2))^(2/3))^2/(16+4*(9-17^(1/2))^(2/3)+(9-17^ 
(1/2))^(4/3))^3+6*(9-17^(1/2))^(5/3)*(4+(9-17^(1/2))^(2/3))*ln(4+(9-17^(1/ 
2))^(2/3)+3*(9-17^(1/2))^(1/3)*x)/(16+4*(9-17^(1/2))^(2/3)+(9-17^(1/2))^(4 
/3))^3+6*(9-17^(1/2))^(7/3)*(8-9*(9-17^(1/2))^(1/3)+2*(9-17^(1/2))^(2/3))* 
ln(16-4*(9-17^(1/2))^(2/3)+(9-17^(1/2))^(4/3)-3*(9-17^(1/2)+4*(9-17^(1/2)) 
^(1/3))*x+9*(9-17^(1/2))^(2/3)*x^2)/(4-(9-17^(1/2))^(2/3))^2/(16+4*(9-17^( 
1/2))^(2/3)+(9-17^(1/2))^(4/3))^3
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.02 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.11 \[ \int \frac {1}{\left (2-4 x+3 x^3\right )^2} \, dx=\frac {-32+27 x+36 x^2}{34 \left (2-4 x+3 x^3\right )}+\frac {9}{17} \text {RootSum}\left [2-4 \text {$\#$1}+3 \text {$\#$1}^3\&,\frac {3 \log (x-\text {$\#$1})+2 \log (x-\text {$\#$1}) \text {$\#$1}}{-4+9 \text {$\#$1}^2}\&\right ] \] Input:

Integrate[(2 - 4*x + 3*x^3)^(-2),x]
 

Output:

(-32 + 27*x + 36*x^2)/(34*(2 - 4*x + 3*x^3)) + (9*RootSum[2 - 4*#1 + 3*#1^ 
3 & , (3*Log[x - #1] + 2*Log[x - #1]*#1)/(-4 + 9*#1^2) & ])/17
 

Rubi [A] (warning: unable to verify)

Time = 1.71 (sec) , antiderivative size = 808, normalized size of antiderivative = 1.11, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2474, 1165, 27, 1200, 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.

\(\displaystyle \int \frac {1}{\left (3 x^3-4 x+2\right )^2} \, dx\)

\(\Big \downarrow \) 2474

\(\displaystyle 81 \int \frac {1}{\left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )^2 \left (-9 x^2+\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}-\left (9-\sqrt {17}\right )^{2/3}-\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}+4\right )^2}dx\)

\(\Big \downarrow \) 1165

\(\displaystyle 81 \left (\frac {\sqrt [3]{9 \sqrt {17}-17} \left (8-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}\right )}{18\ 17^{2/3} \left (4-\left (9-\sqrt {17}\right )^{2/3}\right ) \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right ) \left (-9 x^2+\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}-\left (9-\sqrt {17}\right )^{2/3}-\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}+4\right )}-\frac {\sqrt [3]{9 \sqrt {17}-17} \int -\frac {486 \left (-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}-\left (9-\sqrt {17}\right )^{2/3}-\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}+4\right )}{\left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )^2 \left (-9 x^2+\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}-\left (9-\sqrt {17}\right )^{2/3}-\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}+4\right )}dx}{1458\ 17^{2/3} \left (4-\left (9-\sqrt {17}\right )^{2/3}\right )}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 81 \left (\frac {\sqrt [3]{9 \sqrt {17}-17} \int \frac {-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}-\left (9-\sqrt {17}\right )^{2/3}-\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}+4}{\left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )^2 \left (-9 x^2+\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}-\left (9-\sqrt {17}\right )^{2/3}-\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}+4\right )}dx}{3\ 17^{2/3} \left (4-\left (9-\sqrt {17}\right )^{2/3}\right )}+\frac {\sqrt [3]{9 \sqrt {17}-17} \left (8-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}\right )}{18\ 17^{2/3} \left (4-\left (9-\sqrt {17}\right )^{2/3}\right ) \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right ) \left (-9 x^2+\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}-\left (9-\sqrt {17}\right )^{2/3}-\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}+4\right )}\right )\)

\(\Big \downarrow \) 1200

\(\displaystyle 81 \left (\frac {\sqrt [3]{9 \sqrt {17}-17} \int \left (\frac {4 \left (-3 \left (441-81 \sqrt {17}-2 \left (49-9 \sqrt {17}\right ) \sqrt [3]{9-\sqrt {17}}-4 \left (9-\sqrt {17}\right )^{5/3}\right ) x-2 \left (196-36 \sqrt {17}-\left (513-89 \sqrt {17}\right ) \sqrt [3]{9-\sqrt {17}}-\left (211-27 \sqrt {17}\right ) \left (9-\sqrt {17}\right )^{2/3}\right )\right )}{3 \left (16+4 \left (9-\sqrt {17}\right )^{2/3}+\left (9-\sqrt {17}\right )^{4/3}\right )^2 \left (9 \left (9-\sqrt {17}\right )^{2/3} x^2-3 \left (9-\sqrt {17}+4 \sqrt [3]{9-\sqrt {17}}\right ) x+\left (9-\sqrt {17}\right )^{4/3}-4 \left (9-\sqrt {17}\right )^{2/3}+16\right )}+\frac {\left (9-\sqrt {17}\right )^{2/3} \left (4-\left (9-\sqrt {17}\right )^{2/3}\right )^2 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right )}{3 \left (16+4 \left (9-\sqrt {17}\right )^{2/3}+\left (9-\sqrt {17}\right )^{4/3}\right )^2 \left (3 \sqrt [3]{9-\sqrt {17}} x+\left (9-\sqrt {17}\right )^{2/3}+4\right )}+\frac {4 \left (9-\sqrt {17}\right )^{4/3}}{\left (-16-4 \left (9-\sqrt {17}\right )^{2/3}-\left (9-\sqrt {17}\right )^{4/3}\right ) \left (3 \sqrt [3]{9-\sqrt {17}} x+\left (9-\sqrt {17}\right )^{2/3}+4\right )^2}\right )dx}{3\ 17^{2/3} \left (4-\left (9-\sqrt {17}\right )^{2/3}\right )}+\frac {\sqrt [3]{9 \sqrt {17}-17} \left (8-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}\right )}{18\ 17^{2/3} \left (4-\left (9-\sqrt {17}\right )^{2/3}\right ) \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right ) \left (-9 x^2+\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}-\left (9-\sqrt {17}\right )^{2/3}-\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}+4\right )}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle 81 \left (\frac {\sqrt [3]{-17+9 \sqrt {17}} \left (8-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}\right )}{18\ 17^{2/3} \left (4-\left (9-\sqrt {17}\right )^{2/3}\right ) \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right ) \left (-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\right )}+\frac {\sqrt [3]{-17+9 \sqrt {17}} \left (-\frac {4 \left (3457-729 \sqrt {17}+4 \left (369-73 \sqrt {17}\right ) \sqrt [3]{9-\sqrt {17}}\right ) \sqrt {\frac {2}{3 \left (49-9 \sqrt {17}+8 \left (9-\sqrt {17}\right )^{2/3}-4 \left (9-\sqrt {17}\right )^{4/3}\right )}} \arctan \left (\frac {-6 \left (9-\sqrt {17}\right )^{2/3} x+4 \sqrt [3]{9-\sqrt {17}}-\sqrt {17}+9}{\sqrt {6 \left (49-9 \sqrt {17}+8 \left (9-\sqrt {17}\right )^{2/3}-4 \left (9-\sqrt {17}\right )^{4/3}\right )}}\right )}{3 \left (9-\sqrt {17}\right )^{2/3} \left (16+4 \left (9-\sqrt {17}\right )^{2/3}+\left (9-\sqrt {17}\right )^{4/3}\right )^2}+\frac {\sqrt [3]{9-\sqrt {17}} \left (4-\left (9-\sqrt {17}\right )^{2/3}\right )^2 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) \log \left (3 \sqrt [3]{9-\sqrt {17}} x+\left (9-\sqrt {17}\right )^{2/3}+4\right )}{9 \left (16+4 \left (9-\sqrt {17}\right )^{2/3}+\left (9-\sqrt {17}\right )^{4/3}\right )^2}-\frac {2 \left (441-81 \sqrt {17}-2 \left (49-9 \sqrt {17}\right ) \sqrt [3]{9-\sqrt {17}}-4 \left (9-\sqrt {17}\right )^{5/3}\right ) \log \left (9 \left (9-\sqrt {17}\right )^{2/3} x^2-3 \left (9-\sqrt {17}+4 \sqrt [3]{9-\sqrt {17}}\right ) x+\left (9-\sqrt {17}\right )^{4/3}-4 \left (9-\sqrt {17}\right )^{2/3}+16\right )}{9 \left (9-\sqrt {17}\right )^{2/3} \left (16+4 \left (9-\sqrt {17}\right )^{2/3}+\left (9-\sqrt {17}\right )^{4/3}\right )^2}+\frac {4 \left (9-\sqrt {17}\right )}{3 \left (16+4 \left (9-\sqrt {17}\right )^{2/3}+\left (9-\sqrt {17}\right )^{4/3}\right ) \left (3 \sqrt [3]{9-\sqrt {17}} x+\left (9-\sqrt {17}\right )^{2/3}+4\right )}\right )}{3\ 17^{2/3} \left (4-\left (9-\sqrt {17}\right )^{2/3}\right )}\right )\)

Input:

Int[(2 - 4*x + 3*x^3)^(-2),x]
 

Output:

81*(((-17 + 9*Sqrt[17])^(1/3)*(8 - (3*(4 + (9 - Sqrt[17])^(2/3))*x)/(9 - S 
qrt[17])^(1/3)))/(18*17^(2/3)*(4 - (9 - Sqrt[17])^(2/3))*((4 + (9 - Sqrt[1 
7])^(2/3))/(9 - Sqrt[17])^(1/3) + 3*x)*(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)) + ((-17 + 9*Sqrt[17])^(1/3)*((4*(9 - Sqrt[17]))/(3*(16 + 4*(9 - S 
qrt[17])^(2/3) + (9 - Sqrt[17])^(4/3))*(4 + (9 - Sqrt[17])^(2/3) + 3*(9 - 
Sqrt[17])^(1/3)*x)) - (4*(3457 - 729*Sqrt[17] + 4*(369 - 73*Sqrt[17])*(9 - 
 Sqrt[17])^(1/3))*Sqrt[2/(3*(49 - 9*Sqrt[17] + 8*(9 - Sqrt[17])^(2/3) - 4* 
(9 - Sqrt[17])^(4/3)))]*ArcTan[(9 - Sqrt[17] + 4*(9 - Sqrt[17])^(1/3) - 6* 
(9 - Sqrt[17])^(2/3)*x)/Sqrt[6*(49 - 9*Sqrt[17] + 8*(9 - Sqrt[17])^(2/3) - 
 4*(9 - Sqrt[17])^(4/3))]])/(3*(9 - Sqrt[17])^(2/3)*(16 + 4*(9 - Sqrt[17]) 
^(2/3) + (9 - Sqrt[17])^(4/3))^2) + ((9 - Sqrt[17])^(1/3)*(4 - (9 - Sqrt[1 
7])^(2/3))^2*(4 + (9 - Sqrt[17])^(2/3))*Log[4 + (9 - Sqrt[17])^(2/3) + 3*( 
9 - Sqrt[17])^(1/3)*x])/(9*(16 + 4*(9 - Sqrt[17])^(2/3) + (9 - Sqrt[17])^( 
4/3))^2) - (2*(441 - 81*Sqrt[17] - 2*(49 - 9*Sqrt[17])*(9 - Sqrt[17])^(1/3 
) - 4*(9 - Sqrt[17])^(5/3))*Log[16 - 4*(9 - Sqrt[17])^(2/3) + (9 - Sqrt[17 
])^(4/3) - 3*(9 - Sqrt[17] + 4*(9 - Sqrt[17])^(1/3))*x + 9*(9 - Sqrt[17])^ 
(2/3)*x^2])/(9*(9 - Sqrt[17])^(2/3)*(16 + 4*(9 - Sqrt[17])^(2/3) + (9 - Sq 
rt[17])^(4/3))^2)))/(3*17^(2/3)*(4 - (9 - Sqrt[17])^(2/3))))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1165

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e) 
*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^ 
2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d 
+ e*x)^m*Simp[b*c*d*e*(2*p - m + 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p 
+ 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x, x]*(a + 
 b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p, -1] 
 && IntQuadraticQ[a, b, c, d, e, m, p, x]
 

rule 1200

Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* 
(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* 
x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In 
tegersQ[n]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2474

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[1/d^(2*p)   Int[Sim 
p[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]] /; FreeQ[{a, b, d}, x] && NeQ[4*b^3 + 27*a^2 
*d, 0] && IntegerQ[p]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.06 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.08

method	result	size
default	\(\frac {\frac {6}{17} x^{2}+\frac {9}{34} x -\frac {16}{51}}{x^{3}-\frac {4}{3} x +\frac {2}{3}}+\frac {9 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (3 \textit {\_Z}^{3}-4 \textit {\_Z} +2\right )}{\sum }\frac {\left (2 \textit {\_R} +3\right ) \ln \left (x -\textit {\_R} \right )}{9 \textit {\_R}^{2}-4}\right )}{17}\)	\(60\)
risch	\(\frac {\frac {6}{17} x^{2}+\frac {9}{34} x -\frac {16}{51}}{x^{3}-\frac {4}{3} x +\frac {2}{3}}+\frac {9 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (3 \textit {\_Z}^{3}-4 \textit {\_Z} +2\right )}{\sum }\frac {\left (2 \textit {\_R} +3\right ) \ln \left (x -\textit {\_R} \right )}{9 \textit {\_R}^{2}-4}\right )}{17}\)	\(60\)

Input:

int(1/(3*x^3-4*x+2)^2,x,method=_RETURNVERBOSE)
 

Output:

(6/17*x^2+9/34*x-16/51)/(x^3-4/3*x+2/3)+9/17*sum((2*_R+3)/(9*_R^2-4)*ln(x- 
_R),_R=RootOf(3*_Z^3-4*_Z+2))
 

Fricas [A] (verification not implemented)

Time = 1.44 (sec) , antiderivative size = 580, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\left (2-4 x+3 x^3\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(1/(3*x^3-4*x+2)^2,x, algorithm="fricas")
 

Output:

1/68*(6*(3*x^3 - 4*x + 2)*((1/289)^(1/3)*(19009*sqrt(17) + 2601)^(1/3) - 1 
2104*(1/289)^(2/3)/(19009*sqrt(17) + 2601)^(1/3))*log(2754*((1/289)^(1/3)* 
(19009*sqrt(17) + 2601)^(1/3) - 12104*(1/289)^(2/3)/(19009*sqrt(17) + 2601 
)^(1/3))^2 + 171081*x + 3315*(1/289)^(1/3)*(19009*sqrt(17) + 2601)^(1/3) - 
 40124760*(1/289)^(2/3)/(19009*sqrt(17) + 2601)^(1/3) + 230688) + 72*x^2 - 
 (3*(3*x^3 - 4*x + 2)*((1/289)^(1/3)*(19009*sqrt(17) + 2601)^(1/3) - 12104 
*(1/289)^(2/3)/(19009*sqrt(17) + 2601)^(1/3)) + 2*(3*x^3 - 4*x + 2)*sqrt(- 
27/4*((1/289)^(1/3)*(19009*sqrt(17) + 2601)^(1/3) - 12104*(1/289)^(2/3)/(1 
9009*sqrt(17) + 2601)^(1/3))^2 - 19224/17))*log(-1377*((1/289)^(1/3)*(1900 
9*sqrt(17) + 2601)^(1/3) - 12104*(1/289)^(2/3)/(19009*sqrt(17) + 2601)^(1/ 
3))^2 + 17*sqrt(-27/4*((1/289)^(1/3)*(19009*sqrt(17) + 2601)^(1/3) - 12104 
*(1/289)^(2/3)/(19009*sqrt(17) + 2601)^(1/3))^2 - 19224/17)*(54*(1/289)^(1 
/3)*(19009*sqrt(17) + 2601)^(1/3) - 653616*(1/289)^(2/3)/(19009*sqrt(17) + 
 2601)^(1/3) - 65) + 171081*x - 3315/2*(1/289)^(1/3)*(19009*sqrt(17) + 260 
1)^(1/3) + 20062380*(1/289)^(2/3)/(19009*sqrt(17) + 2601)^(1/3) - 115344) 
- (3*(3*x^3 - 4*x + 2)*((1/289)^(1/3)*(19009*sqrt(17) + 2601)^(1/3) - 1210 
4*(1/289)^(2/3)/(19009*sqrt(17) + 2601)^(1/3)) - 2*(3*x^3 - 4*x + 2)*sqrt( 
-27/4*((1/289)^(1/3)*(19009*sqrt(17) + 2601)^(1/3) - 12104*(1/289)^(2/3)/( 
19009*sqrt(17) + 2601)^(1/3))^2 - 19224/17))*log(-1377*((1/289)^(1/3)*(190 
09*sqrt(17) + 2601)^(1/3) - 12104*(1/289)^(2/3)/(19009*sqrt(17) + 2601)...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6446 vs. \(2 (581) = 1162\).

Time = 3.40 (sec) , antiderivative size = 6446, normalized size of antiderivative = 8.82 \[ \int \frac {1}{\left (2-4 x+3 x^3\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(1/(3*x**3-4*x+2)**2,x)
 

Output:

(36*x**2 + 27*x - 32)/(102*x**3 - 136*x + 68) + (-1602/(4913*(243/39304 + 
513243*sqrt(17)/11358856)**(1/3)) + (243/39304 + 513243*sqrt(17)/11358856) 
**(1/3))*log(x - 23140/(323153*(243/39304 + 513243*sqrt(17)/11358856)**(1/ 
3)) + 39304*(-1602/(4913*(243/39304 + 513243*sqrt(17)/11358856)**(1/3)) + 
(243/39304 + 513243*sqrt(17)/11358856)**(1/3))**2/19009 + 37570*(243/39304 
 + 513243*sqrt(17)/11358856)**(1/3)/171081 + 25632/19009) + (-(6561/314432 
 + 13857561*sqrt(17)/90870848)**(1/3)/3 + 2403/(9826*(6561/314432 + 138575 
61*sqrt(17)/90870848)**(1/3)))*log(x**2 + x*(1271130371503785*sqrt(17)*(26 
01 + 19009*sqrt(17))**(1/3)/(-49286640322110723*17**(2/3) - 26827948069052 
859*17**(1/6) + 753096818123532*(2601 + 19009*sqrt(17))**(1/3) + 184001599 
7483820*sqrt(17)*(2601 + 19009*sqrt(17))**(1/3)) + 1462759815861713*(2601 
+ 19009*sqrt(17))**(1/3)/(-49286640322110723*17**(2/3) - 26827948069052859 
*17**(1/6) + 753096818123532*(2601 + 19009*sqrt(17))**(1/3) + 184001599748 
3820*sqrt(17)*(2601 + 19009*sqrt(17))**(1/3)) + 7297771911926384*17**(1/6) 
/(-49286640322110723*17**(2/3) - 26827948069052859*17**(1/6) + 75309681812 
3532*(2601 + 19009*sqrt(17))**(1/3) + 1840015997483820*sqrt(17)*(2601 + 19 
009*sqrt(17))**(1/3)) - 1410293666898*17**(1/3)*(2601 + 19009*sqrt(17))**( 
2/3)/(-49286640322110723*17**(2/3) - 26827948069052859*17**(1/6) + 7530968 
18123532*(2601 + 19009*sqrt(17))**(1/3) + 1840015997483820*sqrt(17)*(2601 
+ 19009*sqrt(17))**(1/3)) - 3445722841730*17**(5/6)*(2601 + 19009*sqrt(...
 

Maxima [F]

\[ \int \frac {1}{\left (2-4 x+3 x^3\right )^2} \, dx=\int { \frac {1}{{\left (3 \, x^{3} - 4 \, x + 2\right )}^{2}} \,d x } \] Input:

integrate(1/(3*x^3-4*x+2)^2,x, algorithm="maxima")
 

Output:

1/34*(36*x^2 + 27*x - 32)/(3*x^3 - 4*x + 2) + 9/17*integrate((2*x + 3)/(3* 
x^3 - 4*x + 2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{\left (2-4 x+3 x^3\right )^2} \, dx=\text {Exception raised: NotImplementedError} \] Input:

integrate(1/(3*x^3-4*x+2)^2,x, algorithm="giac")
 

Output:

Exception raised: NotImplementedError >> unable to parse Giac output: (36* 
sageVARx^2+27*sageVARx-32)*1/34/(3*sageVARx^3-4*sageVARx+2)+((1/531441*roo 
tof([[-3,0,14580,0,-11337408],[1,0,-5832,0,8503056,0,975725676]])+27)/(9*( 
1/9565938*rootof([[
 

Mupad [B] (verification not implemented)

Time = 12.58 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.11 \[ \int \frac {1}{\left (2-4 x+3 x^3\right )^2} \, dx=\frac {\frac {6\,x^2}{17}+\frac {9\,x}{34}-\frac {16}{51}}{x^3-\frac {4\,x}{3}+\frac {2}{3}}+\left (\sum _{k=1}^3\ln \left (\frac {36\,x}{289}+\mathrm {root}\left (z^3+\frac {4806\,z}{4913}-\frac {243}{19652},z,k\right )\,\left (\frac {27\,x}{17}-\mathrm {root}\left (z^3+\frac {4806\,z}{4913}-\frac {243}{19652},z,k\right )\,\left (8\,x-6\right )+\frac {8}{17}\right )+\frac {54}{289}\right )\,\mathrm {root}\left (z^3+\frac {4806\,z}{4913}-\frac {243}{19652},z,k\right )\right ) \] Input:

int(1/(3*x^3 - 4*x + 2)^2,x)
 

Output:

((9*x)/34 + (6*x^2)/17 - 16/51)/(x^3 - (4*x)/3 + 2/3) + symsum(log((36*x)/ 
289 + root(z^3 + (4806*z)/4913 - 243/19652, z, k)*((27*x)/17 - root(z^3 + 
(4806*z)/4913 - 243/19652, z, k)*(8*x - 6) + 8/17) + 54/289)*root(z^3 + (4 
806*z)/4913 - 243/19652, z, k), k, 1, 3)
 

Reduce [F]

\[ \int \frac {1}{\left (2-4 x+3 x^3\right )^2} \, dx=\int \frac {1}{9 x^{6}-24 x^{4}+12 x^{3}+16 x^{2}-16 x +4}d x \] Input:

int(1/(3*x^3-4*x+2)^2,x)
 

Output:

int(1/(9*x**6 - 24*x**4 + 12*x**3 + 16*x**2 - 16*x + 4),x)