\(\int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x^3} \, dx\) [118]

Optimal result

Integrand size = 12, antiderivative size = 260 \[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x^3} \, dx=\frac {1}{6} \sqrt [6]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{5/6}+\frac {1}{2} \left (1+\frac {1}{x}\right )^{7/6} \left (\frac {-1+x}{x}\right )^{5/6}-\frac {1}{18} \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\frac {1}{18} \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\frac {1}{9} \arctan \left (\frac {\sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\frac {\log \left (1-\frac {\sqrt {3} \sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}+\frac {\sqrt [3]{\frac {-1+x}{x}}}{\sqrt [3]{1+\frac {1}{x}}}\right )}{12 \sqrt {3}}-\frac {\log \left (1+\frac {\sqrt {3} \sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}+\frac {\sqrt [3]{\frac {-1+x}{x}}}{\sqrt [3]{1+\frac {1}{x}}}\right )}{12 \sqrt {3}} \]

[Out]

1/6*(1+1/x)^(1/6)*((-1+x)/x)^(5/6)+1/2*(1+1/x)^(7/6)*((-1+x)/x)^(5/6)+1/9*arctan(((-1+x)/x)^(1/6)/(1+1/x)^(1/6
))+1/18*arctan(2*((-1+x)/x)^(1/6)/(1+1/x)^(1/6)-3^(1/2))+1/18*arctan(2*((-1+x)/x)^(1/6)/(1+1/x)^(1/6)+3^(1/2))
+1/36*ln(1+((-1+x)/x)^(1/3)/(1+1/x)^(1/3)-((-1+x)/x)^(1/6)*3^(1/2)/(1+1/x)^(1/6))*3^(1/2)-1/36*ln(1+((-1+x)/x)
^(1/3)/(1+1/x)^(1/3)+((-1+x)/x)^(1/6)*3^(1/2)/(1+1/x)^(1/6))*3^(1/2)

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {6306, 81, 52, 65, 338, 301, 648, 632, 210, 642, 209} \[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x^3} \, dx=-\frac {1}{18} \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{\frac {x-1}{x}}}{\sqrt [6]{\frac {1}{x}+1}}\right )+\frac {1}{18} \arctan \left (\frac {2 \sqrt [6]{\frac {x-1}{x}}}{\sqrt [6]{\frac {1}{x}+1}}+\sqrt {3}\right )+\frac {1}{9} \arctan \left (\frac {\sqrt [6]{\frac {x-1}{x}}}{\sqrt [6]{\frac {1}{x}+1}}\right )+\frac {1}{2} \left (\frac {x-1}{x}\right )^{5/6} \left (\frac {1}{x}+1\right )^{7/6}+\frac {1}{6} \left (\frac {x-1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1}+\frac {\log \left (\frac {\sqrt [3]{\frac {x-1}{x}}}{\sqrt [3]{\frac {1}{x}+1}}-\frac {\sqrt {3} \sqrt [6]{\frac {x-1}{x}}}{\sqrt [6]{\frac {1}{x}+1}}+1\right )}{12 \sqrt {3}}-\frac {\log \left (\frac {\sqrt [3]{\frac {x-1}{x}}}{\sqrt [3]{\frac {1}{x}+1}}+\frac {\sqrt {3} \sqrt [6]{\frac {x-1}{x}}}{\sqrt [6]{\frac {1}{x}+1}}+1\right )}{12 \sqrt {3}} \]

[In]

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

[Out]

((1 + x^(-1))^(1/6)*((-1 + x)/x)^(5/6))/6 + ((1 + x^(-1))^(7/6)*((-1 + x)/x)^(5/6))/2 - ArcTan[Sqrt[3] - (2*((
-1 + x)/x)^(1/6))/(1 + x^(-1))^(1/6)]/18 + ArcTan[Sqrt[3] + (2*((-1 + x)/x)^(1/6))/(1 + x^(-1))^(1/6)]/18 + Ar
cTan[((-1 + x)/x)^(1/6)/(1 + x^(-1))^(1/6)]/9 + Log[1 - (Sqrt[3]*((-1 + x)/x)^(1/6))/(1 + x^(-1))^(1/6) + ((-1
+ x)/x)^(1/3)/(1 + x^(-1))^(1/3)]/(12*Sqrt[3]) - Log[1 + (Sqrt[3]*((-1 + x)/x)^(1/6))/(1 + x^(-1))^(1/6) + ((
-1 + x)/x)^(1/3)/(1 + x^(-1))^(1/3)]/(12*Sqrt[3])

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 81

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

Rule 209

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

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 301

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

Rule 338

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[
p, -2^(-1)] && IntegersQ[m, p + (m + 1)/n]

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 {x \sqrt [6]{1+x}}{\sqrt [6]{1-x}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{2} \left (1+\frac {1}{x}\right )^{7/6} \left (\frac {-1+x}{x}\right )^{5/6}-\frac {1}{6} \text {Subst}\left (\int \frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{6} \sqrt [6]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{5/6}+\frac {1}{2} \left (1+\frac {1}{x}\right )^{7/6} \left (\frac {-1+x}{x}\right )^{5/6}-\frac {1}{18} \text {Subst}\left (\int \frac {1}{\sqrt [6]{1-x} (1+x)^{5/6}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{6} \sqrt [6]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{5/6}+\frac {1}{2} \left (1+\frac {1}{x}\right )^{7/6} \left (\frac {-1+x}{x}\right )^{5/6}+\frac {1}{3} \text {Subst}\left (\int \frac {x^4}{\left (2-x^6\right )^{5/6}} \, dx,x,\sqrt [6]{\frac {-1+x}{x}}\right ) \\ & = \frac {1}{6} \sqrt [6]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{5/6}+\frac {1}{2} \left (1+\frac {1}{x}\right )^{7/6} \left (\frac {-1+x}{x}\right )^{5/6}+\frac {1}{3} \text {Subst}\left (\int \frac {x^4}{1+x^6} \, dx,x,\frac {\sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right ) \\ & = \frac {1}{6} \sqrt [6]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{5/6}+\frac {1}{2} \left (1+\frac {1}{x}\right )^{7/6} \left (\frac {-1+x}{x}\right )^{5/6}+\frac {1}{9} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\frac {1}{9} \text {Subst}\left (\int \frac {-\frac {1}{2}+\frac {\sqrt {3} x}{2}}{1-\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\frac {1}{9} \text {Subst}\left (\int \frac {-\frac {1}{2}-\frac {\sqrt {3} x}{2}}{1+\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right ) \\ & = \frac {1}{6} \sqrt [6]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{5/6}+\frac {1}{2} \left (1+\frac {1}{x}\right )^{7/6} \left (\frac {-1+x}{x}\right )^{5/6}+\frac {1}{9} \arctan \left (\frac {\sqrt [6]{-\frac {1-x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\frac {1}{36} \text {Subst}\left (\int \frac {1}{1-\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\frac {1}{36} \text {Subst}\left (\int \frac {1}{1+\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\frac {\text {Subst}\left (\int \frac {-\sqrt {3}+2 x}{1-\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )}{12 \sqrt {3}}-\frac {\text {Subst}\left (\int \frac {\sqrt {3}+2 x}{1+\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )}{12 \sqrt {3}} \\ & = \frac {1}{6} \sqrt [6]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{5/6}+\frac {1}{2} \left (1+\frac {1}{x}\right )^{7/6} \left (\frac {-1+x}{x}\right )^{5/6}+\frac {1}{9} \arctan \left (\frac {\sqrt [6]{-\frac {1-x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\frac {\log \left (1-\frac {\sqrt {3} \sqrt [6]{-\frac {1-x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}+\frac {\sqrt [3]{-\frac {1-x}{x}}}{\sqrt [3]{1+\frac {1}{x}}}\right )}{12 \sqrt {3}}-\frac {\log \left (1+\frac {\sqrt {3} \sqrt [6]{-\frac {1-x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}+\frac {\sqrt [3]{-\frac {1-x}{x}}}{\sqrt [3]{1+\frac {1}{x}}}\right )}{12 \sqrt {3}}-\frac {1}{18} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,-\sqrt {3}+\frac {2 \sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )-\frac {1}{18} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {3}+\frac {2 \sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right ) \\ & = \frac {1}{6} \sqrt [6]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{5/6}+\frac {1}{2} \left (1+\frac {1}{x}\right )^{7/6} \left (\frac {-1+x}{x}\right )^{5/6}-\frac {1}{18} \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{-\frac {1-x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\frac {1}{18} \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{-\frac {1-x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\frac {1}{9} \arctan \left (\frac {\sqrt [6]{-\frac {1-x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\frac {\log \left (1-\frac {\sqrt {3} \sqrt [6]{-\frac {1-x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}+\frac {\sqrt [3]{-\frac {1-x}{x}}}{\sqrt [3]{1+\frac {1}{x}}}\right )}{12 \sqrt {3}}-\frac {\log \left (1+\frac {\sqrt {3} \sqrt [6]{-\frac {1-x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}+\frac {\sqrt [3]{-\frac {1-x}{x}}}{\sqrt [3]{1+\frac {1}{x}}}\right )}{12 \sqrt {3}} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 0.95 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.48 \[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x^3} \, dx=\frac {1}{54} \left (\frac {18 e^{\frac {1}{3} \coth ^{-1}(x)} \left (1+7 e^{2 \coth ^{-1}(x)}\right )}{\left (1+e^{2 \coth ^{-1}(x)}\right )^2}-6 \arctan \left (e^{\frac {1}{3} \coth ^{-1}(x)}\right )+\text {RootSum}\left [1-\text {$\#$1}^2+\text {$\#$1}^4\&,\frac {2 \coth ^{-1}(x)-6 \log \left (e^{\frac {1}{3} \coth ^{-1}(x)}-\text {$\#$1}\right )-\coth ^{-1}(x) \text {$\#$1}^2+3 \log \left (e^{\frac {1}{3} \coth ^{-1}(x)}-\text {$\#$1}\right ) \text {$\#$1}^2}{-\text {$\#$1}+2 \text {$\#$1}^3}\&\right ]\right ) \]

[In]

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

[Out]

((18*E^(ArcCoth[x]/3)*(1 + 7*E^(2*ArcCoth[x])))/(1 + E^(2*ArcCoth[x]))^2 - 6*ArcTan[E^(ArcCoth[x]/3)] + RootSu
m[1 - #1^2 + #1^4 & , (2*ArcCoth[x] - 6*Log[E^(ArcCoth[x]/3) - #1] - ArcCoth[x]*#1^2 + 3*Log[E^(ArcCoth[x]/3)
- #1]*#1^2)/(-#1 + 2*#1^3) & ])/54

Maple [C] (warning: unable to verify)

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

Time = 15.33 (sec) , antiderivative size = 1262, normalized size of antiderivative = 4.85

method	result	size
trager	\(\text {Expression too large to display}\)	\(1262\)
risch	\(\text {Expression too large to display}\)	\(3478\)

[In]

int(1/((x-1)/(1+x))^(1/6)/x^3,x,method=_RETURNVERBOSE)

[Out]

1/6*(1+x)*(4*x+3)/x^2*(-(1-x)/(1+x))^(5/6)-1/18*RootOf(_Z^2+1)*ln(-(9*RootOf(3*_Z*RootOf(_Z^2+1)+9*_Z^2-1)*(-(
1-x)/(1+x))^(2/3)*x-18*(-(1-x)/(1+x))^(1/2)*RootOf(3*_Z*RootOf(_Z^2+1)+9*_Z^2-1)*RootOf(_Z^2+1)*x-3*(-(1-x)/(1
+x))^(5/6)*x+6*RootOf(_Z^2+1)*(-(1-x)/(1+x))^(2/3)*x+9*RootOf(3*_Z*RootOf(_Z^2+1)+9*_Z^2-1)*(-(1-x)/(1+x))^(2/
3)-18*(-(1-x)/(1+x))^(1/2)*RootOf(3*_Z*RootOf(_Z^2+1)+9*_Z^2-1)*RootOf(_Z^2+1)-3*(-(1-x)/(1+x))^(5/6)+6*RootOf
(_Z^2+1)*(-(1-x)/(1+x))^(2/3)-18*RootOf(3*_Z*RootOf(_Z^2+1)+9*_Z^2-1)*(-(1-x)/(1+x))^(1/3)*x+9*RootOf(3*_Z*Roo
tOf(_Z^2+1)+9*_Z^2-1)*RootOf(_Z^2+1)*(-(1-x)/(1+x))^(1/6)*x+6*(-(1-x)/(1+x))^(1/2)*x-3*RootOf(_Z^2+1)*(-(1-x)/
(1+x))^(1/3)*x-18*RootOf(3*_Z*RootOf(_Z^2+1)+9*_Z^2-1)*(-(1-x)/(1+x))^(1/3)+9*RootOf(3*_Z*RootOf(_Z^2+1)+9*_Z^
2-1)*RootOf(_Z^2+1)*(-(1-x)/(1+x))^(1/6)+6*(-(1-x)/(1+x))^(1/2)-3*RootOf(_Z^2+1)*(-(1-x)/(1+x))^(1/3)+3*RootOf
(3*_Z*RootOf(_Z^2+1)+9*_Z^2-1)*x-RootOf(_Z^2+1)*x)/x)-1/18*ln((18*RootOf(3*_Z*RootOf(_Z^2+1)+9*_Z^2-1)*(-(1-x)
/(1+x))^(2/3)*x+3*(-(1-x)/(1+x))^(5/6)*x+3*RootOf(_Z^2+1)*(-(1-x)/(1+x))^(2/3)*x+18*RootOf(3*_Z*RootOf(_Z^2+1)
+9*_Z^2-1)*(-(1-x)/(1+x))^(2/3)+3*(-(1-x)/(1+x))^(5/6)+3*RootOf(_Z^2+1)*(-(1-x)/(1+x))^(2/3)+18*RootOf(3*_Z*Ro
otOf(_Z^2+1)+9*_Z^2-1)*(-(1-x)/(1+x))^(1/3)*x+6*(-(1-x)/(1+x))^(1/2)*x+3*RootOf(_Z^2+1)*(-(1-x)/(1+x))^(1/3)*x
+18*RootOf(3*_Z*RootOf(_Z^2+1)+9*_Z^2-1)*(-(1-x)/(1+x))^(1/3)+6*(-(1-x)/(1+x))^(1/2)+3*RootOf(_Z^2+1)*(-(1-x)/
(1+x))^(1/3)+6*RootOf(3*_Z*RootOf(_Z^2+1)+9*_Z^2-1)*x+3*(-(1-x)/(1+x))^(1/6)*x+RootOf(_Z^2+1)*x+3*(-(1-x)/(1+x
))^(1/6))/x)*RootOf(_Z^2+1)-1/6*ln((18*RootOf(3*_Z*RootOf(_Z^2+1)+9*_Z^2-1)*(-(1-x)/(1+x))^(2/3)*x+3*(-(1-x)/(
1+x))^(5/6)*x+3*RootOf(_Z^2+1)*(-(1-x)/(1+x))^(2/3)*x+18*RootOf(3*_Z*RootOf(_Z^2+1)+9*_Z^2-1)*(-(1-x)/(1+x))^(
2/3)+3*(-(1-x)/(1+x))^(5/6)+3*RootOf(_Z^2+1)*(-(1-x)/(1+x))^(2/3)+18*RootOf(3*_Z*RootOf(_Z^2+1)+9*_Z^2-1)*(-(1
-x)/(1+x))^(1/3)*x+6*(-(1-x)/(1+x))^(1/2)*x+3*RootOf(_Z^2+1)*(-(1-x)/(1+x))^(1/3)*x+18*RootOf(3*_Z*RootOf(_Z^2
+1)+9*_Z^2-1)*(-(1-x)/(1+x))^(1/3)+6*(-(1-x)/(1+x))^(1/2)+3*RootOf(_Z^2+1)*(-(1-x)/(1+x))^(1/3)+6*RootOf(3*_Z*
RootOf(_Z^2+1)+9*_Z^2-1)*x+3*(-(1-x)/(1+x))^(1/6)*x+RootOf(_Z^2+1)*x+3*(-(1-x)/(1+x))^(1/6))/x)*RootOf(3*_Z*Ro
otOf(_Z^2+1)+9*_Z^2-1)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.02 \[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x^3} \, dx=\frac {\sqrt {2} x^{2} \sqrt {i \, \sqrt {3} + 1} \log \left ({\left (i \, \sqrt {3} \sqrt {2} - \sqrt {2}\right )} \sqrt {i \, \sqrt {3} + 1} + 4 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) - \sqrt {2} x^{2} \sqrt {i \, \sqrt {3} + 1} \log \left ({\left (-i \, \sqrt {3} \sqrt {2} + \sqrt {2}\right )} \sqrt {i \, \sqrt {3} + 1} + 4 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) - \sqrt {2} x^{2} \sqrt {-i \, \sqrt {3} + 1} \log \left ({\left (i \, \sqrt {3} \sqrt {2} + \sqrt {2}\right )} \sqrt {-i \, \sqrt {3} + 1} + 4 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \sqrt {2} x^{2} \sqrt {-i \, \sqrt {3} + 1} \log \left ({\left (-i \, \sqrt {3} \sqrt {2} - \sqrt {2}\right )} \sqrt {-i \, \sqrt {3} + 1} + 4 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + 4 \, x^{2} \arctan \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + 6 \, {\left (4 \, x^{2} + 7 \, x + 3\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{36 \, x^{2}} \]

[In]

integrate(1/((-1+x)/(1+x))^(1/6)/x^3,x, algorithm="fricas")

[Out]

1/36*(sqrt(2)*x^2*sqrt(I*sqrt(3) + 1)*log((I*sqrt(3)*sqrt(2) - sqrt(2))*sqrt(I*sqrt(3) + 1) + 4*((x - 1)/(x +
1))^(1/6)) - sqrt(2)*x^2*sqrt(I*sqrt(3) + 1)*log((-I*sqrt(3)*sqrt(2) + sqrt(2))*sqrt(I*sqrt(3) + 1) + 4*((x -
1)/(x + 1))^(1/6)) - sqrt(2)*x^2*sqrt(-I*sqrt(3) + 1)*log((I*sqrt(3)*sqrt(2) + sqrt(2))*sqrt(-I*sqrt(3) + 1) +
4*((x - 1)/(x + 1))^(1/6)) + sqrt(2)*x^2*sqrt(-I*sqrt(3) + 1)*log((-I*sqrt(3)*sqrt(2) - sqrt(2))*sqrt(-I*sqrt
(3) + 1) + 4*((x - 1)/(x + 1))^(1/6)) + 4*x^2*arctan(((x - 1)/(x + 1))^(1/6)) + 6*(4*x^2 + 7*x + 3)*((x - 1)/(
x + 1))^(5/6))/x^2

Sympy [F]

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

[In]

integrate(1/((-1+x)/(1+x))**(1/6)/x**3,x)

[Out]

Integral(1/(x**3*((x - 1)/(x + 1))**(1/6)), x)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.68 \[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x^3} \, dx=-\frac {1}{36} \, \sqrt {3} \log \left (\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \frac {1}{36} \, \sqrt {3} \log \left (-\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \frac {\left (\frac {x - 1}{x + 1}\right )^{\frac {11}{6}} + 7 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{3 \, {\left (\frac {2 \, {\left (x - 1\right )}}{x + 1} + \frac {{\left (x - 1\right )}^{2}}{{\left (x + 1\right )}^{2}} + 1\right )}} + \frac {1}{18} \, \arctan \left (\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \frac {1}{18} \, \arctan \left (-\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \frac {1}{9} \, \arctan \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) \]

[In]

integrate(1/((-1+x)/(1+x))^(1/6)/x^3,x, algorithm="maxima")

[Out]

-1/36*sqrt(3)*log(sqrt(3)*((x - 1)/(x + 1))^(1/6) + ((x - 1)/(x + 1))^(1/3) + 1) + 1/36*sqrt(3)*log(-sqrt(3)*(
(x - 1)/(x + 1))^(1/6) + ((x - 1)/(x + 1))^(1/3) + 1) + 1/3*(((x - 1)/(x + 1))^(11/6) + 7*((x - 1)/(x + 1))^(5
/6))/(2*(x - 1)/(x + 1) + (x - 1)^2/(x + 1)^2 + 1) + 1/18*arctan(sqrt(3) + 2*((x - 1)/(x + 1))^(1/6)) + 1/18*a
rctan(-sqrt(3) + 2*((x - 1)/(x + 1))^(1/6)) + 1/9*arctan(((x - 1)/(x + 1))^(1/6))

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.67 \[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x^3} \, dx=-\frac {1}{36} \, \sqrt {3} \log \left (\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \frac {1}{36} \, \sqrt {3} \log \left (-\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \frac {\frac {{\left (x - 1\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{x + 1} + 7 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{3 \, {\left (\frac {x - 1}{x + 1} + 1\right )}^{2}} + \frac {1}{18} \, \arctan \left (\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \frac {1}{18} \, \arctan \left (-\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \frac {1}{9} \, \arctan \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) \]

[In]

integrate(1/((-1+x)/(1+x))^(1/6)/x^3,x, algorithm="giac")

[Out]

-1/36*sqrt(3)*log(sqrt(3)*((x - 1)/(x + 1))^(1/6) + ((x - 1)/(x + 1))^(1/3) + 1) + 1/36*sqrt(3)*log(-sqrt(3)*(
(x - 1)/(x + 1))^(1/6) + ((x - 1)/(x + 1))^(1/3) + 1) + 1/3*((x - 1)*((x - 1)/(x + 1))^(5/6)/(x + 1) + 7*((x -
1)/(x + 1))^(5/6))/((x - 1)/(x + 1) + 1)^2 + 1/18*arctan(sqrt(3) + 2*((x - 1)/(x + 1))^(1/6)) + 1/18*arctan(-
sqrt(3) + 2*((x - 1)/(x + 1))^(1/6)) + 1/9*arctan(((x - 1)/(x + 1))^(1/6))

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.52 \[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x^3} \, dx=\frac {\mathrm {atan}\left ({\left (\frac {x-1}{x+1}\right )}^{1/6}\right )}{9}+\frac {\frac {7\,{\left (\frac {x-1}{x+1}\right )}^{5/6}}{3}+\frac {{\left (\frac {x-1}{x+1}\right )}^{11/6}}{3}}{\frac {2\,\left (x-1\right )}{x+1}+\frac {{\left (x-1\right )}^2}{{\left (x+1\right )}^2}+1}-\mathrm {atan}\left (\frac {2\,{\left (\frac {x-1}{x+1}\right )}^{1/6}}{243\,\left (-\frac {1}{243}+\frac {\sqrt {3}\,1{}\mathrm {i}}{243}\right )}\right )\,\left (\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )-\mathrm {atan}\left (\frac {2\,{\left (\frac {x-1}{x+1}\right )}^{1/6}}{243\,\left (\frac {1}{243}+\frac {\sqrt {3}\,1{}\mathrm {i}}{243}\right )}\right )\,\left (-\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right ) \]

[In]

int(1/(x^3*((x - 1)/(x + 1))^(1/6)),x)

[Out]

atan(((x - 1)/(x + 1))^(1/6))/9 + ((7*((x - 1)/(x + 1))^(5/6))/3 + ((x - 1)/(x + 1))^(11/6)/3)/((2*(x - 1))/(x
+ 1) + (x - 1)^2/(x + 1)^2 + 1) - atan((2*((x - 1)/(x + 1))^(1/6))/(243*((3^(1/2)*1i)/243 - 1/243)))*((3^(1/2
)*1i)/18 + 1/18) - atan((2*((x - 1)/(x + 1))^(1/6))/(243*((3^(1/2)*1i)/243 + 1/243)))*((3^(1/2)*1i)/18 - 1/18)