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

Optimal. Leaf size=285 \[ \frac {1}{3} \sqrt [6]{\frac {1}{x}+1} \left (\frac {x-1}{x}\right )^{5/6} x^3+\frac {7}{18} \sqrt [6]{\frac {1}{x}+1} \left (\frac {x-1}{x}\right )^{5/6} x^2+\frac {11}{27} \sqrt [6]{\frac {1}{x}+1} \left (\frac {x-1}{x}\right )^{5/6} x-\frac {19}{324} \log \left (\frac {\sqrt [3]{\frac {1}{x}+1}}{\sqrt [3]{\frac {x-1}{x}}}-\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}+1\right )+\frac {19}{324} \log \left (\frac {\sqrt [3]{\frac {1}{x}+1}}{\sqrt [3]{\frac {x-1}{x}}}+\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}+1\right )-\frac {19 \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}}{\sqrt {3}}\right )}{54 \sqrt {3}}+\frac {19 \tan ^{-1}\left (\frac {\frac {2 \sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}+1}{\sqrt {3}}\right )}{54 \sqrt {3}}+\frac {19}{81} \tanh ^{-1}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.25, antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 11, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {6171, 99, 151, 12, 93, 210, 634, 618, 204, 628, 206} \[ \frac {1}{3} \sqrt [6]{\frac {1}{x}+1} \left (\frac {x-1}{x}\right )^{5/6} x^3+\frac {7}{18} \sqrt [6]{\frac {1}{x}+1} \left (\frac {x-1}{x}\right )^{5/6} x^2+\frac {11}{27} \sqrt [6]{\frac {1}{x}+1} \left (\frac {x-1}{x}\right )^{5/6} x-\frac {19}{324} \log \left (\frac {\sqrt [3]{\frac {1}{x}+1}}{\sqrt [3]{\frac {x-1}{x}}}-\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}+1\right )+\frac {19}{324} \log \left (\frac {\sqrt [3]{\frac {1}{x}+1}}{\sqrt [3]{\frac {x-1}{x}}}+\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}+1\right )-\frac {19 \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}}{\sqrt {3}}\right )}{54 \sqrt {3}}+\frac {19 \tan ^{-1}\left (\frac {\frac {2 \sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}+1}{\sqrt {3}}\right )}{54 \sqrt {3}}+\frac {19}{81} \tanh ^{-1}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}\right ) \]

Antiderivative was successfully verified.

[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 93

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 99

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 151

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] && LtQ[m, -1] && IntegerQ[m]

Rule 204

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

Rule 206

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

Rule 210

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*Cos[(2*k*Pi)/n]*x)/(r^2 + 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x]; (2*r^2*Int[1/(r^2 - s^2*x^2), x])/(a*n) +
Dist[(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 618

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 628

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 634

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 6171

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*} \int e^{\frac {1}{3} \coth ^{-1}(x)} x^2 \, dx &=-\operatorname {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} \operatorname {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} \operatorname {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} \operatorname {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} \operatorname {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} \operatorname {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} \operatorname {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} \operatorname {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} \operatorname {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} \tanh ^{-1}\left (\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{-\frac {1-x}{x}}}\right )-\frac {19}{324} \operatorname {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} \operatorname {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} \operatorname {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} \operatorname {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} \tanh ^{-1}\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} \operatorname {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} \operatorname {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 \tan ^{-1}\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 \tan ^{-1}\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} \tanh ^{-1}\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*}

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Mathematica [A]  time = 5.29, size = 189, normalized size = 0.66 \[ \frac {1}{324} \left (\frac {732 e^{\frac {1}{3} \coth ^{-1}(x)}}{e^{2 \coth ^{-1}(x)}-1}+\frac {1368 e^{\frac {1}{3} \coth ^{-1}(x)}}{\left (e^{2 \coth ^{-1}(x)}-1\right )^2}+\frac {864 e^{\frac {1}{3} \coth ^{-1}(x)}}{\left (e^{2 \coth ^{-1}(x)}-1\right )^3}-38 \log \left (1-e^{\frac {1}{3} \coth ^{-1}(x)}\right )+38 \log \left (e^{\frac {1}{3} \coth ^{-1}(x)}+1\right )-19 \log \left (-e^{\frac {1}{3} \coth ^{-1}(x)}+e^{\frac {2}{3} \coth ^{-1}(x)}+1\right )+19 \log \left (e^{\frac {1}{3} \coth ^{-1}(x)}+e^{\frac {2}{3} \coth ^{-1}(x)}+1\right )+38 \sqrt {3} \tan ^{-1}\left (\frac {2 e^{\frac {1}{3} \coth ^{-1}(x)}-1}{\sqrt {3}}\right )+38 \sqrt {3} \tan ^{-1}\left (\frac {2 e^{\frac {1}{3} \coth ^{-1}(x)}+1}{\sqrt {3}}\right )\right ) \]

Warning: Unable to verify antiderivative.

[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

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fricas [A]  time = 0.48, size = 173, normalized size = 0.61 \[ \frac {1}{54} \, {\left (18 \, x^{3} + 39 \, x^{2} + 43 \, x + 22\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}} - \frac {19}{162} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \frac {1}{3} \, \sqrt {3}\right ) - \frac {19}{162} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - \frac {1}{3} \, \sqrt {3}\right ) + \frac {19}{324} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {19}{324} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) + \frac {19}{162} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {19}{162} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.19, size = 215, normalized size = 0.75 \[ -\frac {19}{162} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right )}\right ) - \frac {19}{162} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right )}\right ) + \frac {\frac {8 \, {\left (x - 1\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{x + 1} - \frac {19 \, {\left (x - 1\right )}^{2} \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{{\left (x + 1\right )}^{2}} - 61 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{27 \, {\left (\frac {x - 1}{x + 1} - 1\right )}^{3}} + \frac {19}{324} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {19}{324} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) + \frac {19}{162} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {19}{162} \, \log \left ({\left | \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [C]  time = 4.64, size = 2891, normalized size = 10.14 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/54*(18*x^2+21*x+22)*(-1+x)/((-1+x)/(1+x))^(1/6)+(19/162*RootOf(_Z^2-_Z+1)*ln(-(1+4*x+6*x^2+x^4+4*x^3+6*RootO
f(_Z^2-_Z+1)*x^2+RootOf(_Z^2-_Z+1)+RootOf(_Z^2-_Z+1)*x^4+4*RootOf(_Z^2-_Z+1)*x^3+4*RootOf(_Z^2-_Z+1)*x-3*(x^6+
4*x^5+5*x^4-5*x^2-4*x-1)^(5/6)-6*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(2/3)-6*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/2)-3*(
x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/3)+3*RootOf(_Z^2-_Z+1)*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(2/3)*x+6*RootOf(_Z^2-_Z+
1)*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/2)*x^2+6*RootOf(_Z^2-_Z+1)*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/3)*x^3+3*RootO
f(_Z^2-_Z+1)*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/6)*x^4+12*RootOf(_Z^2-_Z+1)*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/2)*
x+18*RootOf(_Z^2-_Z+1)*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/3)*x^2+12*RootOf(_Z^2-_Z+1)*(x^6+4*x^5+5*x^4-5*x^2-4*x
-1)^(1/6)*x^3+18*RootOf(_Z^2-_Z+1)*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/3)*x+18*RootOf(_Z^2-_Z+1)*(x^6+4*x^5+5*x^4
-5*x^2-4*x-1)^(1/6)*x^2+12*RootOf(_Z^2-_Z+1)*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/6)*x+3*RootOf(_Z^2-_Z+1)*(x^6+4*
x^5+5*x^4-5*x^2-4*x-1)^(2/3)-6*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(2/3)*x-6*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/2)*x^2
-3*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/3)*x^3+6*RootOf(_Z^2-_Z+1)*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/2)-12*(x^6+4*x
^5+5*x^4-5*x^2-4*x-1)^(1/2)*x-9*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/3)*x^2+6*RootOf(_Z^2-_Z+1)*(x^6+4*x^5+5*x^4-5
*x^2-4*x-1)^(1/3)-9*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/3)*x+3*RootOf(_Z^2-_Z+1)*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1
/6))/(1+x)^4)-19/162*ln((-2-8*x-12*x^2-2*x^4-8*x^3+6*RootOf(_Z^2-_Z+1)*x^2+RootOf(_Z^2-_Z+1)+RootOf(_Z^2-_Z+1)
*x^4+4*RootOf(_Z^2-_Z+1)*x^3+4*RootOf(_Z^2-_Z+1)*x-3*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/6)+3*(x^6+4*x^5+5*x^4-5*
x^2-4*x-1)^(5/6)+3*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(2/3)-3*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/3)+3*RootOf(_Z^2-_Z+
1)*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(2/3)*x+6*RootOf(_Z^2-_Z+1)*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/2)*x^2+6*RootOf(
_Z^2-_Z+1)*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/3)*x^3+3*RootOf(_Z^2-_Z+1)*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/6)*x^4
+12*RootOf(_Z^2-_Z+1)*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/2)*x+18*RootOf(_Z^2-_Z+1)*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)
^(1/3)*x^2+12*RootOf(_Z^2-_Z+1)*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/6)*x^3+18*RootOf(_Z^2-_Z+1)*(x^6+4*x^5+5*x^4-
5*x^2-4*x-1)^(1/3)*x+18*RootOf(_Z^2-_Z+1)*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/6)*x^2+12*RootOf(_Z^2-_Z+1)*(x^6+4*
x^5+5*x^4-5*x^2-4*x-1)^(1/6)*x+3*RootOf(_Z^2-_Z+1)*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(2/3)+3*(x^6+4*x^5+5*x^4-5*x^
2-4*x-1)^(2/3)*x-3*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/3)*x^3+6*RootOf(_Z^2-_Z+1)*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(
1/2)-9*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/3)*x^2+6*RootOf(_Z^2-_Z+1)*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/3)-9*(x^6+
4*x^5+5*x^4-5*x^2-4*x-1)^(1/3)*x+3*RootOf(_Z^2-_Z+1)*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/6)-12*(x^6+4*x^5+5*x^4-5
*x^2-4*x-1)^(1/6)*x-3*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/6)*x^4-12*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/6)*x^3-18*(x
^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/6)*x^2)/(1+x)^4)*RootOf(_Z^2-_Z+1)+19/162*ln((-2-8*x-12*x^2-2*x^4-8*x^3+6*RootO
f(_Z^2-_Z+1)*x^2+RootOf(_Z^2-_Z+1)+RootOf(_Z^2-_Z+1)*x^4+4*RootOf(_Z^2-_Z+1)*x^3+4*RootOf(_Z^2-_Z+1)*x-3*(x^6+
4*x^5+5*x^4-5*x^2-4*x-1)^(1/6)+3*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(5/6)+3*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(2/3)-3*(
x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/3)+3*RootOf(_Z^2-_Z+1)*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(2/3)*x+6*RootOf(_Z^2-_Z+
1)*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/2)*x^2+6*RootOf(_Z^2-_Z+1)*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/3)*x^3+3*RootO
f(_Z^2-_Z+1)*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/6)*x^4+12*RootOf(_Z^2-_Z+1)*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/2)*
x+18*RootOf(_Z^2-_Z+1)*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/3)*x^2+12*RootOf(_Z^2-_Z+1)*(x^6+4*x^5+5*x^4-5*x^2-4*x
-1)^(1/6)*x^3+18*RootOf(_Z^2-_Z+1)*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/3)*x+18*RootOf(_Z^2-_Z+1)*(x^6+4*x^5+5*x^4
-5*x^2-4*x-1)^(1/6)*x^2+12*RootOf(_Z^2-_Z+1)*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/6)*x+3*RootOf(_Z^2-_Z+1)*(x^6+4*
x^5+5*x^4-5*x^2-4*x-1)^(2/3)+3*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(2/3)*x-3*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/3)*x^3
+6*RootOf(_Z^2-_Z+1)*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/2)-9*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/3)*x^2+6*RootOf(_Z
^2-_Z+1)*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/3)-9*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/3)*x+3*RootOf(_Z^2-_Z+1)*(x^6+
4*x^5+5*x^4-5*x^2-4*x-1)^(1/6)-12*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/6)*x-3*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/6)*
x^4-12*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/6)*x^3-18*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/6)*x^2)/(1+x)^4))/((-1+x)/(
1+x))^(1/6)*((-1+x)*(1+x)^5)^(1/6)/(1+x)

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maxima [A]  time = 0.42, size = 220, normalized size = 0.77 \[ -\frac {19}{162} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right )}\right ) - \frac {19}{162} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right )}\right ) - \frac {19 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {17}{6}} - 8 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {11}{6}} + 61 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{27 \, {\left (\frac {3 \, {\left (x - 1\right )}}{x + 1} - \frac {3 \, {\left (x - 1\right )}^{2}}{{\left (x + 1\right )}^{2}} + \frac {{\left (x - 1\right )}^{3}}{{\left (x + 1\right )}^{3}} - 1\right )}} + \frac {19}{324} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {19}{324} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) + \frac {19}{162} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {19}{162} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.13, size = 168, normalized size = 0.59 \[ -\frac {\mathrm {atan}\left ({\left (\frac {x-1}{x+1}\right )}^{1/6}\,1{}\mathrm {i}\right )\,19{}\mathrm {i}}{81}-\frac {\frac {61\,{\left (\frac {x-1}{x+1}\right )}^{5/6}}{27}-\frac {8\,{\left (\frac {x-1}{x+1}\right )}^{11/6}}{27}+\frac {19\,{\left (\frac {x-1}{x+1}\right )}^{17/6}}{27}}{\frac {3\,\left (x-1\right )}{x+1}-\frac {3\,{\left (x-1\right )}^2}{{\left (x+1\right )}^2}+\frac {{\left (x-1\right )}^3}{{\left (x+1\right )}^3}-1}-\mathrm {atan}\left (\frac {{\left (\frac {x-1}{x+1}\right )}^{1/6}\,4952198{}\mathrm {i}}{14348907\,\left (-\frac {2476099}{14348907}+\frac {\sqrt {3}\,2476099{}\mathrm {i}}{14348907}\right )}\right )\,\left (\frac {19\,\sqrt {3}}{162}-\frac {19}{162}{}\mathrm {i}\right )-\mathrm {atan}\left (\frac {{\left (\frac {x-1}{x+1}\right )}^{1/6}\,4952198{}\mathrm {i}}{14348907\,\left (\frac {2476099}{14348907}+\frac {\sqrt {3}\,2476099{}\mathrm {i}}{14348907}\right )}\right )\,\left (\frac {19\,\sqrt {3}}{162}+\frac {19}{162}{}\mathrm {i}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt [6]{\frac {x - 1}{x + 1}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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