\(\int e^{\frac {2}{3} \coth ^{-1}(x)} x^2 \, dx\) [128]

Optimal result

Integrand size = 12, antiderivative size = 157 \[ \int e^{\frac {2}{3} \coth ^{-1}(x)} x^2 \, dx=\frac {14}{27} \left (1-\frac {1}{x}\right )^{2/3} \sqrt [3]{1+\frac {1}{x}} x+\frac {4}{9} \left (1-\frac {1}{x}\right )^{2/3} \sqrt [3]{1+\frac {1}{x}} x^2+\frac {1}{3} \left (1-\frac {1}{x}\right )^{2/3} \sqrt [3]{1+\frac {1}{x}} x^3-\frac {22 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1-\frac {1}{x}}}{\sqrt {3} \sqrt [3]{1+\frac {1}{x}}}\right )}{27 \sqrt {3}}-\frac {11}{27} \log \left (\sqrt [3]{1-\frac {1}{x}}-\sqrt [3]{1+\frac {1}{x}}\right )-\frac {11 \log (x)}{81} \] Output:

14/27*(1-1/x)^(2/3)*(1+1/x)^(1/3)*x+4/9*(1-1/x)^(2/3)*(1+1/x)^(1/3)*x^2+1/ 
3*(1-1/x)^(2/3)*(1+1/x)^(1/3)*x^3-22/81*arctan(1/3*3^(1/2)+2/3*(1-1/x)^(1/ 
3)*3^(1/2)/(1+1/x)^(1/3))*3^(1/2)-11/27*ln((1-1/x)^(1/3)-(1+1/x)^(1/3))-11 
/81*ln(x)
 

Mathematica [A] (verified)

Time = 5.34 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.20 \[ \int e^{\frac {2}{3} \coth ^{-1}(x)} x^2 \, dx=\frac {1}{81} \left (\frac {216 e^{\frac {2}{3} \coth ^{-1}(x)}}{\left (-1+e^{2 \coth ^{-1}(x)}\right )^3}+\frac {360 e^{\frac {2}{3} \coth ^{-1}(x)}}{\left (-1+e^{2 \coth ^{-1}(x)}\right )^2}+\frac {210 e^{\frac {2}{3} \coth ^{-1}(x)}}{-1+e^{2 \coth ^{-1}(x)}}+22 \sqrt {3} \arctan \left (\frac {-1+2 e^{\frac {1}{3} \coth ^{-1}(x)}}{\sqrt {3}}\right )-22 \sqrt {3} \arctan \left (\frac {1+2 e^{\frac {1}{3} \coth ^{-1}(x)}}{\sqrt {3}}\right )-22 \log \left (1-e^{\frac {1}{3} \coth ^{-1}(x)}\right )-22 \log \left (1+e^{\frac {1}{3} \coth ^{-1}(x)}\right )+11 \log \left (1-e^{\frac {1}{3} \coth ^{-1}(x)}+e^{\frac {2}{3} \coth ^{-1}(x)}\right )+11 \log \left (1+e^{\frac {1}{3} \coth ^{-1}(x)}+e^{\frac {2}{3} \coth ^{-1}(x)}\right )\right ) \] Input:

Integrate[E^((2*ArcCoth[x])/3)*x^2,x]
 

Output:

((216*E^((2*ArcCoth[x])/3))/(-1 + E^(2*ArcCoth[x]))^3 + (360*E^((2*ArcCoth 
[x])/3))/(-1 + E^(2*ArcCoth[x]))^2 + (210*E^((2*ArcCoth[x])/3))/(-1 + E^(2 
*ArcCoth[x])) + 22*Sqrt[3]*ArcTan[(-1 + 2*E^(ArcCoth[x]/3))/Sqrt[3]] - 22* 
Sqrt[3]*ArcTan[(1 + 2*E^(ArcCoth[x]/3))/Sqrt[3]] - 22*Log[1 - E^(ArcCoth[x 
]/3)] - 22*Log[1 + E^(ArcCoth[x]/3)] + 11*Log[1 - E^(ArcCoth[x]/3) + E^((2 
*ArcCoth[x])/3)] + 11*Log[1 + E^(ArcCoth[x]/3) + E^((2*ArcCoth[x])/3)])/81
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6721, 110, 27, 168, 27, 168, 27, 102}

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.

\(\displaystyle \int x^2 e^{\frac {2}{3} \coth ^{-1}(x)} \, dx\)

\(\Big \downarrow \) 6721

\(\displaystyle -\int \frac {\sqrt [3]{1+\frac {1}{x}} x^4}{\sqrt [3]{1-\frac {1}{x}}}d\frac {1}{x}\)

\(\Big \downarrow \) 110

\(\displaystyle \frac {1}{3} \left (1-\frac {1}{x}\right )^{2/3} \sqrt [3]{\frac {1}{x}+1} x^3-\frac {1}{3} \int \frac {2 \left (4+\frac {3}{x}\right ) x^3}{3 \sqrt [3]{1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{2/3}}d\frac {1}{x}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{3} \left (1-\frac {1}{x}\right )^{2/3} \sqrt [3]{\frac {1}{x}+1} x^3-\frac {2}{9} \left (-\frac {1}{2} \int -\frac {2 \left (7+\frac {6}{x}\right ) x^2}{3 \sqrt [3]{1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{2/3}}d\frac {1}{x}-2 \left (1-\frac {1}{x}\right )^{2/3} \sqrt [3]{\frac {1}{x}+1} x^2\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (1-\frac {1}{x}\right )^{2/3} \sqrt [3]{\frac {1}{x}+1} x^3-\frac {2}{9} \left (\frac {1}{3} \int \frac {\left (7+\frac {6}{x}\right ) x^2}{\sqrt [3]{1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{2/3}}d\frac {1}{x}-2 \left (1-\frac {1}{x}\right )^{2/3} \sqrt [3]{\frac {1}{x}+1} x^2\right )\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{3} \left (1-\frac {1}{x}\right )^{2/3} \sqrt [3]{\frac {1}{x}+1} x^3-\frac {2}{9} \left (\frac {1}{3} \left (-\int -\frac {11 x}{3 \sqrt [3]{1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{2/3}}d\frac {1}{x}-7 \left (1-\frac {1}{x}\right )^{2/3} \sqrt [3]{\frac {1}{x}+1} x\right )-2 \left (1-\frac {1}{x}\right )^{2/3} \sqrt [3]{\frac {1}{x}+1} x^2\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (1-\frac {1}{x}\right )^{2/3} \sqrt [3]{\frac {1}{x}+1} x^3-\frac {2}{9} \left (\frac {1}{3} \left (\frac {11}{3} \int \frac {x}{\sqrt [3]{1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{2/3}}d\frac {1}{x}-7 \left (1-\frac {1}{x}\right )^{2/3} \sqrt [3]{\frac {1}{x}+1} x\right )-2 \left (1-\frac {1}{x}\right )^{2/3} \sqrt [3]{\frac {1}{x}+1} x^2\right )\)

\(\Big \downarrow \) 102

\(\displaystyle \frac {1}{3} \left (1-\frac {1}{x}\right )^{2/3} \sqrt [3]{\frac {1}{x}+1} x^3-\frac {2}{9} \left (\frac {1}{3} \left (\frac {11}{3} \left (\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{1-\frac {1}{x}}}{\sqrt {3} \sqrt [3]{\frac {1}{x}+1}}+\frac {1}{\sqrt {3}}\right )+\frac {3}{2} \log \left (\sqrt [3]{1-\frac {1}{x}}-\sqrt [3]{\frac {1}{x}+1}\right )-\frac {1}{2} \log \left (\frac {1}{x}\right )\right )-7 \left (1-\frac {1}{x}\right )^{2/3} \sqrt [3]{\frac {1}{x}+1} x\right )-2 \left (1-\frac {1}{x}\right )^{2/3} \sqrt [3]{\frac {1}{x}+1} x^2\right )\)

Input:

Int[E^((2*ArcCoth[x])/3)*x^2,x]
 

Output:

((1 - x^(-1))^(2/3)*(1 + x^(-1))^(1/3)*x^3)/3 - (2*(-2*(1 - x^(-1))^(2/3)* 
(1 + x^(-1))^(1/3)*x^2 + (-7*(1 - x^(-1))^(2/3)*(1 + x^(-1))^(1/3)*x + (11 
*(Sqrt[3]*ArcTan[1/Sqrt[3] + (2*(1 - x^(-1))^(1/3))/(Sqrt[3]*(1 + x^(-1))^ 
(1/3))] + (3*Log[(1 - x^(-1))^(1/3) - (1 + x^(-1))^(1/3)])/2 - Log[x^(-1)] 
/2))/3)/3))/9
 

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 102

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)*((e_.) + (f_.) 
*(x_))), x_] :> With[{q = Rt[(d*e - c*f)/(b*e - a*f), 3]}, Simp[(-Sqrt[3])* 
q*(ArcTan[1/Sqrt[3] + 2*q*((a + b*x)^(1/3)/(Sqrt[3]*(c + d*x)^(1/3)))]/(d*e 
 - c*f)), x] + (Simp[q*(Log[e + f*x]/(2*(d*e - c*f))), x] - Simp[3*q*(Log[q 
*(a + b*x)^(1/3) - (c + d*x)^(1/3)]/(2*(d*e - c*f))), x])] /; FreeQ[{a, b, 
c, d, e, f}, x]
 

rule 110

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 
1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f))   Int[(a + b*x)^(m + 1) 
*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + 
p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt 
Q[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, 
 m + n])
 

rule 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
 

rule 6721

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(x_)^(m_.), x_Symbol] :> -Subst[Int[(1 + x 
/a)^(n/2)/(x^(m + 2)*(1 - x/a)^(n/2)), x], x, 1/x] /; FreeQ[{a, n}, x] && 
!IntegerQ[n] && IntegerQ[m]
 

Maple [C] (verified)

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

Time = 0.60 (sec) , antiderivative size = 616, normalized size of antiderivative = 3.92

method	result	size
risch	\(\frac {\left (9 x^{2}+12 x +14\right ) \left (x -1\right )}{27 \left (\frac {x -1}{1+x}\right )^{\frac {1}{3}}}+\frac {\left (\frac {22 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -2 x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+2}{1+x}\right )}{81}-\frac {22 \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x +2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-3 \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x -x^{2}-3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-2 x -1}{1+x}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{81}+\frac {22 \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x +2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-3 \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x -x^{2}-3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-2 x -1}{1+x}\right )}{81}\right ) \left (\left (x -1\right ) \left (1+x \right )^{2}\right )^{\frac {1}{3}}}{\left (\frac {x -1}{1+x}\right )^{\frac {1}{3}} \left (1+x \right )}\)	\(616\)
trager	\(\frac {\left (1+x \right ) \left (9 x^{2}+12 x +14\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}}{27}+\frac {22 \ln \left (9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}} x +18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x +9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}} x -3 \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}} x +9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -3 \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}-3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}} x -3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}}-x -1\right )}{81}-\frac {22 \ln \left (9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}} x +18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x +9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}} x -3 \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}} x +9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -3 \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}-3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}} x -3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}}-x -1\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )}{27}+\frac {22 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}} x +18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x -9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}} x -9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}}-15 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x +3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+2 x -2\right )}{27}\)	\(668\)

Input:

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

Output:

1/27*(9*x^2+12*x+14)*(x-1)/((x-1)/(1+x))^(1/3)+(22/81*RootOf(_Z^2-_Z+1)*ln 
(-(-2*RootOf(_Z^2-_Z+1)^2*x^2+3*RootOf(_Z^2-_Z+1)*(x^3+x^2-x-1)^(2/3)+3*Ro 
otOf(_Z^2-_Z+1)*(x^3+x^2-x-1)^(1/3)*x-2*RootOf(_Z^2-_Z+1)^2*x+5*RootOf(_Z^ 
2-_Z+1)*x^2+3*RootOf(_Z^2-_Z+1)*(x^3+x^2-x-1)^(1/3)+4*RootOf(_Z^2-_Z+1)*x- 
2*x^2-RootOf(_Z^2-_Z+1)+2)/(1+x))-22/81*ln((2*RootOf(_Z^2-_Z+1)^2*x^2+3*Ro 
otOf(_Z^2-_Z+1)*(x^3+x^2-x-1)^(2/3)+3*RootOf(_Z^2-_Z+1)*(x^3+x^2-x-1)^(1/3 
)*x+2*RootOf(_Z^2-_Z+1)^2*x+RootOf(_Z^2-_Z+1)*x^2-3*(x^3+x^2-x-1)^(2/3)+3* 
RootOf(_Z^2-_Z+1)*(x^3+x^2-x-1)^(1/3)-3*(x^3+x^2-x-1)^(1/3)*x-x^2-3*(x^3+x 
^2-x-1)^(1/3)-RootOf(_Z^2-_Z+1)-2*x-1)/(1+x))*RootOf(_Z^2-_Z+1)+22/81*ln(( 
2*RootOf(_Z^2-_Z+1)^2*x^2+3*RootOf(_Z^2-_Z+1)*(x^3+x^2-x-1)^(2/3)+3*RootOf 
(_Z^2-_Z+1)*(x^3+x^2-x-1)^(1/3)*x+2*RootOf(_Z^2-_Z+1)^2*x+RootOf(_Z^2-_Z+1 
)*x^2-3*(x^3+x^2-x-1)^(2/3)+3*RootOf(_Z^2-_Z+1)*(x^3+x^2-x-1)^(1/3)-3*(x^3 
+x^2-x-1)^(1/3)*x-x^2-3*(x^3+x^2-x-1)^(1/3)-RootOf(_Z^2-_Z+1)-2*x-1)/(1+x) 
))/((x-1)/(1+x))^(1/3)*((x-1)*(1+x)^2)^(1/3)/(1+x)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.64 \[ \int e^{\frac {2}{3} \coth ^{-1}(x)} x^2 \, dx=\frac {1}{27} \, {\left (9 \, x^{3} + 21 \, x^{2} + 26 \, x + 14\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} - \frac {22}{81} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + \frac {11}{81} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) - \frac {22}{81} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - 1\right ) \] Input:

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

Output:

1/27*(9*x^3 + 21*x^2 + 26*x + 14)*((x - 1)/(x + 1))^(2/3) - 22/81*sqrt(3)* 
arctan(2/3*sqrt(3)*((x - 1)/(x + 1))^(1/3) + 1/3*sqrt(3)) + 11/81*log(((x 
- 1)/(x + 1))^(2/3) + ((x - 1)/(x + 1))^(1/3) + 1) - 22/81*log(((x - 1)/(x 
 + 1))^(1/3) - 1)
 

Sympy [F]

\[ \int e^{\frac {2}{3} \coth ^{-1}(x)} x^2 \, dx=\int \frac {x^{2}}{\sqrt [3]{\frac {x - 1}{x + 1}}}\, dx \] Input:

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

Output:

Integral(x**2/((x - 1)/(x + 1))**(1/3), x)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.95 \[ \int e^{\frac {2}{3} \coth ^{-1}(x)} x^2 \, dx=-\frac {22}{81} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right )}\right ) - \frac {2 \, {\left (11 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {8}{3}} - 10 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{3}} + 35 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}\right )}}{27 \, {\left (\frac {3 \, {\left (x - 1\right )}}{x + 1} - \frac {3 \, {\left (x - 1\right )}^{2}}{{\left (x + 1\right )}^{2}} + \frac {{\left (x - 1\right )}^{3}}{{\left (x + 1\right )}^{3}} - 1\right )}} + \frac {11}{81} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) - \frac {22}{81} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - 1\right ) \] Input:

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

Output:

-22/81*sqrt(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/3) + 1)) - 2/27* 
(11*((x - 1)/(x + 1))^(8/3) - 10*((x - 1)/(x + 1))^(5/3) + 35*((x - 1)/(x 
+ 1))^(2/3))/(3*(x - 1)/(x + 1) - 3*(x - 1)^2/(x + 1)^2 + (x - 1)^3/(x + 1 
)^3 - 1) + 11/81*log(((x - 1)/(x + 1))^(2/3) + ((x - 1)/(x + 1))^(1/3) + 1 
) - 22/81*log(((x - 1)/(x + 1))^(1/3) - 1)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.92 \[ \int e^{\frac {2}{3} \coth ^{-1}(x)} x^2 \, dx=-\frac {22}{81} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right )}\right ) + \frac {2 \, {\left (\frac {10 \, {\left (x - 1\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}}{x + 1} - \frac {11 \, {\left (x - 1\right )}^{2} \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}}{{\left (x + 1\right )}^{2}} - 35 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}\right )}}{27 \, {\left (\frac {x - 1}{x + 1} - 1\right )}^{3}} + \frac {11}{81} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) - \frac {22}{81} \, \log \left ({\left | \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - 1 \right |}\right ) \] Input:

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

Output:

-22/81*sqrt(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/3) + 1)) + 2/27* 
(10*(x - 1)*((x - 1)/(x + 1))^(2/3)/(x + 1) - 11*(x - 1)^2*((x - 1)/(x + 1 
))^(2/3)/(x + 1)^2 - 35*((x - 1)/(x + 1))^(2/3))/((x - 1)/(x + 1) - 1)^3 + 
 11/81*log(((x - 1)/(x + 1))^(2/3) + ((x - 1)/(x + 1))^(1/3) + 1) - 22/81* 
log(abs(((x - 1)/(x + 1))^(1/3) - 1))
 

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.09 \[ \int e^{\frac {2}{3} \coth ^{-1}(x)} x^2 \, dx=-\frac {22\,\ln \left (\frac {484\,{\left (\frac {x-1}{x+1}\right )}^{1/3}}{729}-\frac {484}{729}\right )}{81}-\frac {\frac {70\,{\left (\frac {x-1}{x+1}\right )}^{2/3}}{27}-\frac {20\,{\left (\frac {x-1}{x+1}\right )}^{5/3}}{27}+\frac {22\,{\left (\frac {x-1}{x+1}\right )}^{8/3}}{27}}{\frac {3\,\left (x-1\right )}{x+1}-\frac {3\,{\left (x-1\right )}^2}{{\left (x+1\right )}^2}+\frac {{\left (x-1\right )}^3}{{\left (x+1\right )}^3}-1}-\ln \left (\frac {484\,{\left (\frac {x-1}{x+1}\right )}^{1/3}}{729}-9\,{\left (-\frac {11}{81}+\frac {\sqrt {3}\,11{}\mathrm {i}}{81}\right )}^2\right )\,\left (-\frac {11}{81}+\frac {\sqrt {3}\,11{}\mathrm {i}}{81}\right )+\ln \left (\frac {484\,{\left (\frac {x-1}{x+1}\right )}^{1/3}}{729}-9\,{\left (\frac {11}{81}+\frac {\sqrt {3}\,11{}\mathrm {i}}{81}\right )}^2\right )\,\left (\frac {11}{81}+\frac {\sqrt {3}\,11{}\mathrm {i}}{81}\right ) \] Input:

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

Output:

log((484*((x - 1)/(x + 1))^(1/3))/729 - 9*((3^(1/2)*11i)/81 + 11/81)^2)*(( 
3^(1/2)*11i)/81 + 11/81) - ((70*((x - 1)/(x + 1))^(2/3))/27 - (20*((x - 1) 
/(x + 1))^(5/3))/27 + (22*((x - 1)/(x + 1))^(8/3))/27)/((3*(x - 1))/(x + 1 
) - (3*(x - 1)^2)/(x + 1)^2 + (x - 1)^3/(x + 1)^3 - 1) - log((484*((x - 1) 
/(x + 1))^(1/3))/729 - 9*((3^(1/2)*11i)/81 - 11/81)^2)*((3^(1/2)*11i)/81 - 
 11/81) - (22*log((484*((x - 1)/(x + 1))^(1/3))/729 - 484/729))/81
 

Reduce [F]

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

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

Output:

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