\(\int e^{\frac {1}{3} \coth ^{-1}(x)} x^2 \, dx\) [113]

Optimal result

Integrand size = 12, antiderivative size = 285 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x^2 \, dx=\frac {11}{27} \sqrt [6]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{5/6} x+\frac {7}{18} \sqrt [6]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{5/6} x^2+\frac {1}{3} \sqrt [6]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{5/6} x^3-\frac {19 \arctan \left (\frac {1-\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}}{\sqrt {3}}\right )}{54 \sqrt {3}}+\frac {19 \arctan \left (\frac {1+\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}}{\sqrt {3}}\right )}{54 \sqrt {3}}+\frac {19}{81} \text {arctanh}\left (\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )-\frac {19}{324} \log \left (1+\frac {\sqrt [3]{1+\frac {1}{x}}}{\sqrt [3]{\frac {-1+x}{x}}}-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )+\frac {19}{324} \log \left (1+\frac {\sqrt [3]{1+\frac {1}{x}}}{\sqrt [3]{\frac {-1+x}{x}}}+\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right ) \]

[Out]

11/27*(1+1/x)^(1/6)*((-1+x)/x)^(5/6)*x+7/18*(1+1/x)^(1/6)*((-1+x)/x)^(5/6)*x^2+1/3*(1+1/x)^(1/6)*((-1+x)/x)^(5
/6)*x^3+19/81*arctanh((1+1/x)^(1/6)/((-1+x)/x)^(1/6))-19/324*ln(1+(1+1/x)^(1/3)/((-1+x)/x)^(1/3)-(1+1/x)^(1/6)
/((-1+x)/x)^(1/6))+19/324*ln(1+(1+1/x)^(1/3)/((-1+x)/x)^(1/3)+(1+1/x)^(1/6)/((-1+x)/x)^(1/6))-19/162*arctan(1/
3*(1-2*(1+1/x)^(1/6)/((-1+x)/x)^(1/6))*3^(1/2))*3^(1/2)+19/162*arctan(1/3*(1+2*(1+1/x)^(1/6)/((-1+x)/x)^(1/6))
*3^(1/2))*3^(1/2)

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {6306, 101, 156, 12, 95, 216, 648, 632, 210, 642, 212} \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x^2 \, dx=-\frac {19 \arctan \left (\frac {1-\frac {2 \sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}}{\sqrt {3}}\right )}{54 \sqrt {3}}+\frac {19 \arctan \left (\frac {\frac {2 \sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}+1}{\sqrt {3}}\right )}{54 \sqrt {3}}+\frac {19}{81} \text {arctanh}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}\right )+\frac {1}{3} \sqrt [6]{\frac {1}{x}+1} \left (\frac {x-1}{x}\right )^{5/6} x^3+\frac {7}{18} \sqrt [6]{\frac {1}{x}+1} \left (\frac {x-1}{x}\right )^{5/6} x^2+\frac {11}{27} \sqrt [6]{\frac {1}{x}+1} \left (\frac {x-1}{x}\right )^{5/6} x-\frac {19}{324} \log \left (\frac {\sqrt [3]{\frac {1}{x}+1}}{\sqrt [3]{\frac {x-1}{x}}}-\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}+1\right )+\frac {19}{324} \log \left (\frac {\sqrt [3]{\frac {1}{x}+1}}{\sqrt [3]{\frac {x-1}{x}}}+\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}+1\right ) \]

[In]

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

[Out]

(11*(1 + x^(-1))^(1/6)*((-1 + x)/x)^(5/6)*x)/27 + (7*(1 + x^(-1))^(1/6)*((-1 + x)/x)^(5/6)*x^2)/18 + ((1 + x^(
-1))^(1/6)*((-1 + x)/x)^(5/6)*x^3)/3 - (19*ArcTan[(1 - (2*(1 + x^(-1))^(1/6))/((-1 + x)/x)^(1/6))/Sqrt[3]])/(5
4*Sqrt[3]) + (19*ArcTan[(1 + (2*(1 + x^(-1))^(1/6))/((-1 + x)/x)^(1/6))/Sqrt[3]])/(54*Sqrt[3]) + (19*ArcTanh[(
1 + x^(-1))^(1/6)/((-1 + x)/x)^(1/6)])/81 - (19*Log[1 + (1 + x^(-1))^(1/3)/((-1 + x)/x)^(1/3) - (1 + x^(-1))^(
1/6)/((-1 + x)/x)^(1/6)])/324 + (19*Log[1 + (1 + x^(-1))^(1/3)/((-1 + x)/x)^(1/3) + (1 + x^(-1))^(1/6)/((-1 +
x)/x)^(1/6)])/324

Rule 12

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

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*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 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 216

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[-a/b, n]], s = Denominator[Rt[-a/b, n
]], k, u}, Simp[u = Int[(r - s*Cos[(2*k*Pi)/n]*x)/(r^2 - 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x] + Int[(r + s*C
os[(2*k*Pi)/n]*x)/(r^2 + 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x]; 2*(r^2/(a*n))*Int[1/(r^2 - s^2*x^2), x] + Dis
t[2*(r/(a*n)), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && NegQ[a/b]

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

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 [6]{1+x}}{\sqrt [6]{1-x} x^4} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{3} \sqrt [6]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{5/6} x^3-\frac {1}{3} \text {Subst}\left (\int \frac {\frac {7}{3}+2 x}{\sqrt [6]{1-x} x^3 (1+x)^{5/6}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {7}{18} \sqrt [6]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{5/6} x^2+\frac {1}{3} \sqrt [6]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{5/6} x^3+\frac {1}{6} \text {Subst}\left (\int \frac {-\frac {22}{9}-\frac {7 x}{3}}{\sqrt [6]{1-x} x^2 (1+x)^{5/6}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {11}{27} \sqrt [6]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{5/6} x+\frac {7}{18} \sqrt [6]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{5/6} x^2+\frac {1}{3} \sqrt [6]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{5/6} x^3-\frac {1}{6} \text {Subst}\left (\int \frac {19}{27 \sqrt [6]{1-x} x (1+x)^{5/6}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {11}{27} \sqrt [6]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{5/6} x+\frac {7}{18} \sqrt [6]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{5/6} x^2+\frac {1}{3} \sqrt [6]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{5/6} x^3-\frac {19}{162} \text {Subst}\left (\int \frac {1}{\sqrt [6]{1-x} x (1+x)^{5/6}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {11}{27} \sqrt [6]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{5/6} x+\frac {7}{18} \sqrt [6]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{5/6} x^2+\frac {1}{3} \sqrt [6]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{5/6} x^3-\frac {19}{27} \text {Subst}\left (\int \frac {1}{-1+x^6} \, dx,x,\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right ) \\ & = \frac {11}{27} \sqrt [6]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{5/6} x+\frac {7}{18} \sqrt [6]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{5/6} x^2+\frac {1}{3} \sqrt [6]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{5/6} x^3+\frac {19}{81} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )+\frac {19}{81} \text {Subst}\left (\int \frac {1-\frac {x}{2}}{1-x+x^2} \, dx,x,\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )+\frac {19}{81} \text {Subst}\left (\int \frac {1+\frac {x}{2}}{1+x+x^2} \, dx,x,\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right ) \\ & = \frac {11}{27} \sqrt [6]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{5/6} x+\frac {7}{18} \sqrt [6]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{5/6} x^2+\frac {1}{3} \sqrt [6]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{5/6} x^3+\frac {19}{81} \text {arctanh}\left (\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{-\frac {1-x}{x}}}\right )-\frac {19}{324} \text {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )+\frac {19}{324} \text {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )+\frac {19}{108} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )+\frac {19}{108} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right ) \\ & = \frac {11}{27} \sqrt [6]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{5/6} x+\frac {7}{18} \sqrt [6]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{5/6} x^2+\frac {1}{3} \sqrt [6]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{5/6} x^3+\frac {19}{81} \text {arctanh}\left (\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{-\frac {1-x}{x}}}\right )-\frac {19}{324} \log \left (1+\frac {\sqrt [3]{1+\frac {1}{x}}}{\sqrt [3]{-\frac {1-x}{x}}}-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{-\frac {1-x}{x}}}\right )+\frac {19}{324} \log \left (1+\frac {\sqrt [3]{1+\frac {1}{x}}}{\sqrt [3]{-\frac {1-x}{x}}}+\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{-\frac {1-x}{x}}}\right )-\frac {19}{54} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )-\frac {19}{54} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right ) \\ & = \frac {11}{27} \sqrt [6]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{5/6} x+\frac {7}{18} \sqrt [6]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{5/6} x^2+\frac {1}{3} \sqrt [6]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{5/6} x^3-\frac {19 \arctan \left (\frac {1-\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{-\frac {1-x}{x}}}}{\sqrt {3}}\right )}{54 \sqrt {3}}+\frac {19 \arctan \left (\frac {1+\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{-\frac {1-x}{x}}}}{\sqrt {3}}\right )}{54 \sqrt {3}}+\frac {19}{81} \text {arctanh}\left (\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{-\frac {1-x}{x}}}\right )-\frac {19}{324} \log \left (1+\frac {\sqrt [3]{1+\frac {1}{x}}}{\sqrt [3]{-\frac {1-x}{x}}}-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{-\frac {1-x}{x}}}\right )+\frac {19}{324} \log \left (1+\frac {\sqrt [3]{1+\frac {1}{x}}}{\sqrt [3]{-\frac {1-x}{x}}}+\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{-\frac {1-x}{x}}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 5.27 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.66 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x^2 \, dx=\frac {1}{324} \left (\frac {864 e^{\frac {1}{3} \coth ^{-1}(x)}}{\left (-1+e^{2 \coth ^{-1}(x)}\right )^3}+\frac {1368 e^{\frac {1}{3} \coth ^{-1}(x)}}{\left (-1+e^{2 \coth ^{-1}(x)}\right )^2}+\frac {732 e^{\frac {1}{3} \coth ^{-1}(x)}}{-1+e^{2 \coth ^{-1}(x)}}+38 \sqrt {3} \arctan \left (\frac {-1+2 e^{\frac {1}{3} \coth ^{-1}(x)}}{\sqrt {3}}\right )+38 \sqrt {3} \arctan \left (\frac {1+2 e^{\frac {1}{3} \coth ^{-1}(x)}}{\sqrt {3}}\right )-38 \log \left (1-e^{\frac {1}{3} \coth ^{-1}(x)}\right )+38 \log \left (1+e^{\frac {1}{3} \coth ^{-1}(x)}\right )-19 \log \left (1-e^{\frac {1}{3} \coth ^{-1}(x)}+e^{\frac {2}{3} \coth ^{-1}(x)}\right )+19 \log \left (1+e^{\frac {1}{3} \coth ^{-1}(x)}+e^{\frac {2}{3} \coth ^{-1}(x)}\right )\right ) \]

[In]

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

[Out]

((864*E^(ArcCoth[x]/3))/(-1 + E^(2*ArcCoth[x]))^3 + (1368*E^(ArcCoth[x]/3))/(-1 + E^(2*ArcCoth[x]))^2 + (732*E
^(ArcCoth[x]/3))/(-1 + E^(2*ArcCoth[x])) + 38*Sqrt[3]*ArcTan[(-1 + 2*E^(ArcCoth[x]/3))/Sqrt[3]] + 38*Sqrt[3]*A
rcTan[(1 + 2*E^(ArcCoth[x]/3))/Sqrt[3]] - 38*Log[1 - E^(ArcCoth[x]/3)] + 38*Log[1 + E^(ArcCoth[x]/3)] - 19*Log
[1 - E^(ArcCoth[x]/3) + E^((2*ArcCoth[x])/3)] + 19*Log[1 + E^(ArcCoth[x]/3) + E^((2*ArcCoth[x])/3)])/324

Maple [C] (verified)

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

Time = 5.14 (sec) , antiderivative size = 705, normalized size of antiderivative = 2.47

method	result	size
trager	\(\frac {\left (1+x \right ) \left (18 x^{2}+21 x +22\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {5}{6}}}{54}-\frac {19 \ln \left (3 \left (-\frac {1-x}{1+x}\right )^{\frac {5}{6}} x +9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}} x +3 \left (-\frac {1-x}{1+x}\right )^{\frac {5}{6}}+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 ) \sqrt {-\frac {1-x}{1+x}}\, x -3 \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}-18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \sqrt {-\frac {1-x}{1+x}}+18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}} x +18 \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 -9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {1}{6}} x +3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {1}{6}}-3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{6}} x -3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{6}}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+2\right )}{162}+\frac {19 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (3 \left (-\frac {1-x}{1+x}\right )^{\frac {5}{6}} x -18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}} x +3 \left (-\frac {1-x}{1+x}\right )^{\frac {5}{6}}-18 \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 -3 \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}-6 \sqrt {-\frac {1-x}{1+x}}\, x +18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}} x -6 \sqrt {-\frac {1-x}{1+x}}+18 \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 +3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}}+3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{6}} x +3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{6}}-6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1\right )}{54}\)	\(705\)
risch	\(\text {Expression too large to display}\)	\(1772\)

[In]

int(1/((x-1)/(1+x))^(1/6)*x^2,x,method=_RETURNVERBOSE)

[Out]

1/54*(1+x)*(18*x^2+21*x+22)*(-(1-x)/(1+x))^(5/6)-19/162*ln(3*(-(1-x)/(1+x))^(5/6)*x+9*RootOf(9*_Z^2+3*_Z+1)*(-
(1-x)/(1+x))^(2/3)*x+3*(-(1-x)/(1+x))^(5/6)+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)*(-(1-x)/(1+x))^(1/2)*x-3*(-(1-x)/(1+x))^(2/3)-18*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(1
+x))^(1/2)+18*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(1/3)*x+18*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(1/3)+3*(-(
1-x)/(1+x))^(1/3)*x-9*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(1/6)*x+3*(-(1-x)/(1+x))^(1/3)-9*RootOf(9*_Z^2+3*_Z
+1)*(-(1-x)/(1+x))^(1/6)-3*(-(1-x)/(1+x))^(1/6)*x-3*(-(1-x)/(1+x))^(1/6)+3*RootOf(9*_Z^2+3*_Z+1)+2)+19/54*Root
Of(9*_Z^2+3*_Z+1)*ln(3*(-(1-x)/(1+x))^(5/6)*x-18*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(2/3)*x+3*(-(1-x)/(1+x))
^(5/6)-18*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(2/3)-3*(-(1-x)/(1+x))^(2/3)*x-3*(-(1-x)/(1+x))^(2/3)-6*(-(1-x)
/(1+x))^(1/2)*x+18*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(1/3)*x-6*(-(1-x)/(1+x))^(1/2)+18*RootOf(9*_Z^2+3*_Z+1
)*(-(1-x)/(1+x))^(1/3)+3*(-(1-x)/(1+x))^(1/3)*x+3*(-(1-x)/(1+x))^(1/3)+3*(-(1-x)/(1+x))^(1/6)*x+3*(-(1-x)/(1+x
))^(1/6)-6*RootOf(9*_Z^2+3*_Z+1)-1)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.61 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x^2 \, dx=\frac {1}{54} \, {\left (18 \, x^{3} + 39 \, x^{2} + 43 \, x + 22\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}} - \frac {19}{162} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \frac {1}{3} \, \sqrt {3}\right ) - \frac {19}{162} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - \frac {1}{3} \, \sqrt {3}\right ) + \frac {19}{324} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {19}{324} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) + \frac {19}{162} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {19}{162} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right ) \]

[In]

integrate(1/((-1+x)/(1+x))^(1/6)*x^2,x, algorithm="fricas")

[Out]

1/54*(18*x^3 + 39*x^2 + 43*x + 22)*((x - 1)/(x + 1))^(5/6) - 19/162*sqrt(3)*arctan(2/3*sqrt(3)*((x - 1)/(x + 1
))^(1/6) + 1/3*sqrt(3)) - 19/162*sqrt(3)*arctan(2/3*sqrt(3)*((x - 1)/(x + 1))^(1/6) - 1/3*sqrt(3)) + 19/324*lo
g(((x - 1)/(x + 1))^(1/3) + ((x - 1)/(x + 1))^(1/6) + 1) - 19/324*log(((x - 1)/(x + 1))^(1/3) - ((x - 1)/(x +
1))^(1/6) + 1) + 19/162*log(((x - 1)/(x + 1))^(1/6) + 1) - 19/162*log(((x - 1)/(x + 1))^(1/6) - 1)

Sympy [F]

\[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x^2 \, dx=\int \frac {x^{2}}{\sqrt [6]{\frac {x - 1}{x + 1}}}\, dx \]

[In]

integrate(1/((-1+x)/(1+x))**(1/6)*x**2,x)

[Out]

Integral(x**2/((x - 1)/(x + 1))**(1/6), x)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.77 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x^2 \, dx=-\frac {19}{162} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right )}\right ) - \frac {19}{162} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right )}\right ) - \frac {19 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {17}{6}} - 8 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {11}{6}} + 61 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{27 \, {\left (\frac {3 \, {\left (x - 1\right )}}{x + 1} - \frac {3 \, {\left (x - 1\right )}^{2}}{{\left (x + 1\right )}^{2}} + \frac {{\left (x - 1\right )}^{3}}{{\left (x + 1\right )}^{3}} - 1\right )}} + \frac {19}{324} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {19}{324} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) + \frac {19}{162} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {19}{162} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right ) \]

[In]

integrate(1/((-1+x)/(1+x))^(1/6)*x^2,x, algorithm="maxima")

[Out]

-19/162*sqrt(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/6) + 1)) - 19/162*sqrt(3)*arctan(1/3*sqrt(3)*(2*((x
- 1)/(x + 1))^(1/6) - 1)) - 1/27*(19*((x - 1)/(x + 1))^(17/6) - 8*((x - 1)/(x + 1))^(11/6) + 61*((x - 1)/(x +
1))^(5/6))/(3*(x - 1)/(x + 1) - 3*(x - 1)^2/(x + 1)^2 + (x - 1)^3/(x + 1)^3 - 1) + 19/324*log(((x - 1)/(x + 1
))^(1/3) + ((x - 1)/(x + 1))^(1/6) + 1) - 19/324*log(((x - 1)/(x + 1))^(1/3) - ((x - 1)/(x + 1))^(1/6) + 1) +
19/162*log(((x - 1)/(x + 1))^(1/6) + 1) - 19/162*log(((x - 1)/(x + 1))^(1/6) - 1)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.75 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x^2 \, dx=-\frac {19}{162} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right )}\right ) - \frac {19}{162} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right )}\right ) + \frac {\frac {8 \, {\left (x - 1\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{x + 1} - \frac {19 \, {\left (x - 1\right )}^{2} \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{{\left (x + 1\right )}^{2}} - 61 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{27 \, {\left (\frac {x - 1}{x + 1} - 1\right )}^{3}} + \frac {19}{324} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {19}{324} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) + \frac {19}{162} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {19}{162} \, \log \left ({\left | \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1 \right |}\right ) \]

[In]

integrate(1/((-1+x)/(1+x))^(1/6)*x^2,x, algorithm="giac")

[Out]

-19/162*sqrt(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/6) + 1)) - 19/162*sqrt(3)*arctan(1/3*sqrt(3)*(2*((x
- 1)/(x + 1))^(1/6) - 1)) + 1/27*(8*(x - 1)*((x - 1)/(x + 1))^(5/6)/(x + 1) - 19*(x - 1)^2*((x - 1)/(x + 1))^
(5/6)/(x + 1)^2 - 61*((x - 1)/(x + 1))^(5/6))/((x - 1)/(x + 1) - 1)^3 + 19/324*log(((x - 1)/(x + 1))^(1/3) + (
(x - 1)/(x + 1))^(1/6) + 1) - 19/324*log(((x - 1)/(x + 1))^(1/3) - ((x - 1)/(x + 1))^(1/6) + 1) + 19/162*log((
(x - 1)/(x + 1))^(1/6) + 1) - 19/162*log(abs(((x - 1)/(x + 1))^(1/6) - 1))

Mupad [B] (verification not implemented)

Time = 4.18 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.59 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x^2 \, dx=-\frac {\mathrm {atan}\left ({\left (\frac {x-1}{x+1}\right )}^{1/6}\,1{}\mathrm {i}\right )\,19{}\mathrm {i}}{81}-\frac {\frac {61\,{\left (\frac {x-1}{x+1}\right )}^{5/6}}{27}-\frac {8\,{\left (\frac {x-1}{x+1}\right )}^{11/6}}{27}+\frac {19\,{\left (\frac {x-1}{x+1}\right )}^{17/6}}{27}}{\frac {3\,\left (x-1\right )}{x+1}-\frac {3\,{\left (x-1\right )}^2}{{\left (x+1\right )}^2}+\frac {{\left (x-1\right )}^3}{{\left (x+1\right )}^3}-1}-\mathrm {atan}\left (\frac {{\left (\frac {x-1}{x+1}\right )}^{1/6}\,4952198{}\mathrm {i}}{14348907\,\left (-\frac {2476099}{14348907}+\frac {\sqrt {3}\,2476099{}\mathrm {i}}{14348907}\right )}\right )\,\left (\frac {19\,\sqrt {3}}{162}-\frac {19}{162}{}\mathrm {i}\right )-\mathrm {atan}\left (\frac {{\left (\frac {x-1}{x+1}\right )}^{1/6}\,4952198{}\mathrm {i}}{14348907\,\left (\frac {2476099}{14348907}+\frac {\sqrt {3}\,2476099{}\mathrm {i}}{14348907}\right )}\right )\,\left (\frac {19\,\sqrt {3}}{162}+\frac {19}{162}{}\mathrm {i}\right ) \]

[In]

int(x^2/((x - 1)/(x + 1))^(1/6),x)

[Out]

- (atan(((x - 1)/(x + 1))^(1/6)*1i)*19i)/81 - ((61*((x - 1)/(x + 1))^(5/6))/27 - (8*((x - 1)/(x + 1))^(11/6))/
27 + (19*((x - 1)/(x + 1))^(17/6))/27)/((3*(x - 1))/(x + 1) - (3*(x - 1)^2)/(x + 1)^2 + (x - 1)^3/(x + 1)^3 -
1) - atan((((x - 1)/(x + 1))^(1/6)*4952198i)/(14348907*((3^(1/2)*2476099i)/14348907 - 2476099/14348907)))*((19
*3^(1/2))/162 - 19i/162) - atan((((x - 1)/(x + 1))^(1/6)*4952198i)/(14348907*((3^(1/2)*2476099i)/14348907 + 24
76099/14348907)))*((19*3^(1/2))/162 + 19i/162)