\(\int e^{\frac {2}{3} \coth ^{-1}(x)} x^2 \, dx\) [120]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6306, 101, 156, 12, 93} \[ \int e^{\frac {2}{3} \coth ^{-1}(x)} x^2 \, dx=-\frac {22 \arctan \left (\frac {2 \sqrt [3]{\frac {x-1}{x}}}{\sqrt {3} \sqrt [3]{\frac {1}{x}+1}}+\frac {1}{\sqrt {3}}\right )}{27 \sqrt {3}}+\frac {1}{3} \sqrt [3]{\frac {1}{x}+1} \left (\frac {x-1}{x}\right )^{2/3} x^3+\frac {4}{9} \sqrt [3]{\frac {1}{x}+1} \left (\frac {x-1}{x}\right )^{2/3} x^2+\frac {14}{27} \sqrt [3]{\frac {1}{x}+1} \left (\frac {x-1}{x}\right )^{2/3} x-\frac {11}{27} \log \left (\sqrt [3]{\frac {1}{x}+1}-\sqrt [3]{\frac {x-1}{x}}\right )-\frac {11 \log (x)}{81} \]

[In]

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

[Out]

(14*(1 + x^(-1))^(1/3)*((-1 + x)/x)^(2/3)*x)/27 + (4*(1 + x^(-1))^(1/3)*((-1 + x)/x)^(2/3)*x^2)/9 + ((1 + x^(-
1))^(1/3)*((-1 + x)/x)^(2/3)*x^3)/3 - (22*ArcTan[1/Sqrt[3] + (2*((-1 + x)/x)^(1/3))/(Sqrt[3]*(1 + x^(-1))^(1/3
))])/(27*Sqrt[3]) - (11*Log[(1 + x^(-1))^(1/3) - ((-1 + x)/x)^(1/3)])/27 - (11*Log[x])/81

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 93

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)*((e_.) + (f_.)*(x_))), x_Symbol] :> 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 101

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

Rule 156

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> 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] + Dist[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 6306

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]

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x} x^4} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{3} \sqrt [3]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{2/3} x^3-\frac {1}{3} \text {Subst}\left (\int \frac {\frac {8}{3}+2 x}{\sqrt [3]{1-x} x^3 (1+x)^{2/3}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {4}{9} \sqrt [3]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{2/3} x^2+\frac {1}{3} \sqrt [3]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{2/3} x^3+\frac {1}{6} \text {Subst}\left (\int \frac {-\frac {28}{9}-\frac {8 x}{3}}{\sqrt [3]{1-x} x^2 (1+x)^{2/3}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {14}{27} \sqrt [3]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{2/3} x+\frac {4}{9} \sqrt [3]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{2/3} x^2+\frac {1}{3} \sqrt [3]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{2/3} x^3-\frac {1}{6} \text {Subst}\left (\int \frac {44}{27 \sqrt [3]{1-x} x (1+x)^{2/3}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {14}{27} \sqrt [3]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{2/3} x+\frac {4}{9} \sqrt [3]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{2/3} x^2+\frac {1}{3} \sqrt [3]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{2/3} x^3-\frac {22}{81} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} x (1+x)^{2/3}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {14}{27} \sqrt [3]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{2/3} x+\frac {4}{9} \sqrt [3]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{2/3} x^2+\frac {1}{3} \sqrt [3]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{2/3} x^3-\frac {22 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-\frac {1-x}{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]{-\frac {1-x}{x}}\right )-\frac {11 \log (x)}{81} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.27 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.20 \[ \int e^{\frac {2}{3} \coth ^{-1}(x)} x^2 \, dx=\frac {1}{81} \left (\frac {216 e^{\frac {2}{3} \coth ^{-1}(x)}}{\left (-1+e^{2 \coth ^{-1}(x)}\right )^3}+\frac {360 e^{\frac {2}{3} \coth ^{-1}(x)}}{\left (-1+e^{2 \coth ^{-1}(x)}\right )^2}+\frac {210 e^{\frac {2}{3} \coth ^{-1}(x)}}{-1+e^{2 \coth ^{-1}(x)}}+22 \sqrt {3} \arctan \left (\frac {-1+2 e^{\frac {1}{3} \coth ^{-1}(x)}}{\sqrt {3}}\right )-22 \sqrt {3} \arctan \left (\frac {1+2 e^{\frac {1}{3} \coth ^{-1}(x)}}{\sqrt {3}}\right )-22 \log \left (1-e^{\frac {1}{3} \coth ^{-1}(x)}\right )-22 \log \left (1+e^{\frac {1}{3} \coth ^{-1}(x)}\right )+11 \log \left (1-e^{\frac {1}{3} \coth ^{-1}(x)}+e^{\frac {2}{3} \coth ^{-1}(x)}\right )+11 \log \left (1+e^{\frac {1}{3} \coth ^{-1}(x)}+e^{\frac {2}{3} \coth ^{-1}(x)}\right )\right ) \]

[In]

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

[Out]

((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]] - 2
2*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

Maple [C] (verified)

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

Time = 0.76 (sec) , antiderivative size = 408, normalized size of antiderivative = 2.60

method	result	size
trager	\(\frac {\left (1+x \right ) \left (9 x^{2}+12 x +14\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}}{27}-\frac {22 \ln \left (-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}} x -9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}} x -36 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x +9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}}-3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}} x +12 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}}-6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-x +1\right )}{81}+\frac {22 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}} x +9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}-3 \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}} x +18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x -3 \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}+3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}} x -15 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x +3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+2 x +2\right )}{27}\)	\(408\)
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 (1+x \right )^{2} \left (x -1\right )\right )^{\frac {1}{3}}}{\left (\frac {x -1}{1+x}\right )^{\frac {1}{3}} \left (1+x \right )}\)	\(616\)

[In]

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

[Out]

1/27*(1+x)*(9*x^2+12*x+14)*(-(1-x)/(1+x))^(2/3)-22/81*ln(-9*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(2/3)*x-9*Roo
tOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(2/3)+9*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(1/3)*x-36*RootOf(9*_Z^2-3*_Z+1
)^2*x+9*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(1/3)-3*(-(1-x)/(1+x))^(1/3)*x+12*RootOf(9*_Z^2-3*_Z+1)*x-3*(-(1-
x)/(1+x))^(1/3)-6*RootOf(9*_Z^2-3*_Z+1)-x+1)+22/27*RootOf(9*_Z^2-3*_Z+1)*ln(9*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1
+x))^(2/3)*x+9*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(2/3)-3*(-(1-x)/(1+x))^(2/3)*x+18*RootOf(9*_Z^2-3*_Z+1)^2*
x-3*(-(1-x)/(1+x))^(2/3)+3*(-(1-x)/(1+x))^(1/3)*x-15*RootOf(9*_Z^2-3*_Z+1)*x+3*(-(1-x)/(1+x))^(1/3)-3*RootOf(9
*_Z^2-3*_Z+1)+2*x+2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.64 \[ \int e^{\frac {2}{3} \coth ^{-1}(x)} x^2 \, dx=\frac {1}{27} \, {\left (9 \, x^{3} + 21 \, x^{2} + 26 \, x + 14\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} - \frac {22}{81} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + \frac {11}{81} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) - \frac {22}{81} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - 1\right ) \]

[In]

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

[Out]

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]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.95 \[ \int e^{\frac {2}{3} \coth ^{-1}(x)} x^2 \, dx=-\frac {22}{81} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right )}\right ) - \frac {2 \, {\left (11 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {8}{3}} - 10 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{3}} + 35 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}\right )}}{27 \, {\left (\frac {3 \, {\left (x - 1\right )}}{x + 1} - \frac {3 \, {\left (x - 1\right )}^{2}}{{\left (x + 1\right )}^{2}} + \frac {{\left (x - 1\right )}^{3}}{{\left (x + 1\right )}^{3}} - 1\right )}} + \frac {11}{81} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) - \frac {22}{81} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - 1\right ) \]

[In]

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

[Out]

-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)

none

Time = 0.29 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.92 \[ \int e^{\frac {2}{3} \coth ^{-1}(x)} x^2 \, dx=-\frac {22}{81} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right )}\right ) + \frac {2 \, {\left (\frac {10 \, {\left (x - 1\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}}{x + 1} - \frac {11 \, {\left (x - 1\right )}^{2} \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}}{{\left (x + 1\right )}^{2}} - 35 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}\right )}}{27 \, {\left (\frac {x - 1}{x + 1} - 1\right )}^{3}} + \frac {11}{81} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) - \frac {22}{81} \, \log \left ({\left | \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - 1 \right |}\right ) \]

[In]

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

[Out]

-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 = 4.05 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.09 \[ \int e^{\frac {2}{3} \coth ^{-1}(x)} x^2 \, dx=-\frac {22\,\ln \left (\frac {484\,{\left (\frac {x-1}{x+1}\right )}^{1/3}}{729}-\frac {484}{729}\right )}{81}-\frac {\frac {70\,{\left (\frac {x-1}{x+1}\right )}^{2/3}}{27}-\frac {20\,{\left (\frac {x-1}{x+1}\right )}^{5/3}}{27}+\frac {22\,{\left (\frac {x-1}{x+1}\right )}^{8/3}}{27}}{\frac {3\,\left (x-1\right )}{x+1}-\frac {3\,{\left (x-1\right )}^2}{{\left (x+1\right )}^2}+\frac {{\left (x-1\right )}^3}{{\left (x+1\right )}^3}-1}-\ln \left (\frac {484\,{\left (\frac {x-1}{x+1}\right )}^{1/3}}{729}-9\,{\left (-\frac {11}{81}+\frac {\sqrt {3}\,11{}\mathrm {i}}{81}\right )}^2\right )\,\left (-\frac {11}{81}+\frac {\sqrt {3}\,11{}\mathrm {i}}{81}\right )+\ln \left (\frac {484\,{\left (\frac {x-1}{x+1}\right )}^{1/3}}{729}-9\,{\left (\frac {11}{81}+\frac {\sqrt {3}\,11{}\mathrm {i}}{81}\right )}^2\right )\,\left (\frac {11}{81}+\frac {\sqrt {3}\,11{}\mathrm {i}}{81}\right ) \]

[In]

int(x^2/((x - 1)/(x + 1))^(1/3),x)

[Out]

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