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

Optimal. Leaf size=402 \[ -\frac {1}{2} \log \left (\frac {\sqrt [3]{\frac {1}{x}+1}}{\sqrt [3]{\frac {x-1}{x}}}-\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}+1\right )+\frac {1}{2} \log \left (\frac {\sqrt [3]{\frac {1}{x}+1}}{\sqrt [3]{\frac {x-1}{x}}}+\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}+1\right )+\frac {1}{2} \sqrt {3} \log \left (\frac {\sqrt [3]{\frac {x-1}{x}}}{\sqrt [3]{\frac {1}{x}+1}}-\frac {\sqrt {3} \sqrt [6]{\frac {x-1}{x}}}{\sqrt [6]{\frac {1}{x}+1}}+1\right )-\frac {1}{2} \sqrt {3} \log \left (\frac {\sqrt [3]{\frac {x-1}{x}}}{\sqrt [3]{\frac {1}{x}+1}}+\frac {\sqrt {3} \sqrt [6]{\frac {x-1}{x}}}{\sqrt [6]{\frac {1}{x}+1}}+1\right )-\sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}}{\sqrt {3}}\right )+\sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}+1}{\sqrt {3}}\right )-\tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{\frac {x-1}{x}}}{\sqrt [6]{\frac {1}{x}+1}}\right )+\tan ^{-1}\left (\frac {2 \sqrt [6]{\frac {x-1}{x}}}{\sqrt [6]{\frac {1}{x}+1}}+\sqrt {3}\right )+2 \tan ^{-1}\left (\frac {\sqrt [6]{\frac {x-1}{x}}}{\sqrt [6]{\frac {1}{x}+1}}\right )+2 \tanh ^{-1}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}\right ) \]

[Out]

2*arctan(((-1+x)/x)^(1/6)/(1/x+1)^(1/6))+arctan(2*((-1+x)/x)^(1/6)/(1/x+1)^(1/6)-3^(1/2))+arctan(2*((-1+x)/x)^
(1/6)/(1/x+1)^(1/6)+3^(1/2))+2*arctanh((1/x+1)^(1/6)/((-1+x)/x)^(1/6))-1/2*ln(1+(1/x+1)^(1/3)/((-1+x)/x)^(1/3)
-(1/x+1)^(1/6)/((-1+x)/x)^(1/6))+1/2*ln(1+(1/x+1)^(1/3)/((-1+x)/x)^(1/3)+(1/x+1)^(1/6)/((-1+x)/x)^(1/6))-arcta
n(1/3*(1-2*(1/x+1)^(1/6)/((-1+x)/x)^(1/6))*3^(1/2))*3^(1/2)+arctan(1/3*(1+2*(1/x+1)^(1/6)/((-1+x)/x)^(1/6))*3^
(1/2))*3^(1/2)+1/2*ln(1+((-1+x)/x)^(1/3)/(1/x+1)^(1/3)-((-1+x)/x)^(1/6)*3^(1/2)/(1/x+1)^(1/6))*3^(1/2)-1/2*ln(
1+((-1+x)/x)^(1/3)/(1/x+1)^(1/3)+((-1+x)/x)^(1/6)*3^(1/2)/(1/x+1)^(1/6))*3^(1/2)

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Rubi [A]  time = 0.53, antiderivative size = 402, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 13, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.083, Rules used = {6171, 105, 63, 331, 295, 634, 618, 204, 628, 203, 93, 210, 206} \[ -\frac {1}{2} \log \left (\frac {\sqrt [3]{\frac {1}{x}+1}}{\sqrt [3]{\frac {x-1}{x}}}-\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}+1\right )+\frac {1}{2} \log \left (\frac {\sqrt [3]{\frac {1}{x}+1}}{\sqrt [3]{\frac {x-1}{x}}}+\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}+1\right )+\frac {1}{2} \sqrt {3} \log \left (\frac {\sqrt [3]{\frac {x-1}{x}}}{\sqrt [3]{\frac {1}{x}+1}}-\frac {\sqrt {3} \sqrt [6]{\frac {x-1}{x}}}{\sqrt [6]{\frac {1}{x}+1}}+1\right )-\frac {1}{2} \sqrt {3} \log \left (\frac {\sqrt [3]{\frac {x-1}{x}}}{\sqrt [3]{\frac {1}{x}+1}}+\frac {\sqrt {3} \sqrt [6]{\frac {x-1}{x}}}{\sqrt [6]{\frac {1}{x}+1}}+1\right )-\sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}}{\sqrt {3}}\right )+\sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}+1}{\sqrt {3}}\right )-\tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{\frac {x-1}{x}}}{\sqrt [6]{\frac {1}{x}+1}}\right )+\tan ^{-1}\left (\frac {2 \sqrt [6]{\frac {x-1}{x}}}{\sqrt [6]{\frac {1}{x}+1}}+\sqrt {3}\right )+2 \tan ^{-1}\left (\frac {\sqrt [6]{\frac {x-1}{x}}}{\sqrt [6]{\frac {1}{x}+1}}\right )+2 \tanh ^{-1}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[3]*ArcTan[(1 - (2*(1 + x^(-1))^(1/6))/((-1 + x)/x)^(1/6))/Sqrt[3]]) + Sqrt[3]*ArcTan[(1 + (2*(1 + x^(-1
))^(1/6))/((-1 + x)/x)^(1/6))/Sqrt[3]] - ArcTan[Sqrt[3] - (2*((-1 + x)/x)^(1/6))/(1 + x^(-1))^(1/6)] + ArcTan[
Sqrt[3] + (2*((-1 + x)/x)^(1/6))/(1 + x^(-1))^(1/6)] + 2*ArcTan[((-1 + x)/x)^(1/6)/(1 + x^(-1))^(1/6)] + 2*Arc
Tanh[(1 + x^(-1))^(1/6)/((-1 + x)/x)^(1/6)] - Log[1 + (1 + x^(-1))^(1/3)/((-1 + x)/x)^(1/3) - (1 + x^(-1))^(1/
6)/((-1 + x)/x)^(1/6)]/2 + Log[1 + (1 + x^(-1))^(1/3)/((-1 + x)/x)^(1/3) + (1 + x^(-1))^(1/6)/((-1 + x)/x)^(1/
6)]/2 + (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)])/2 - (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)])/2

Rule 63

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 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 105

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[b/f, Int[(a
+ b*x)^(m - 1)*(c + d*x)^n, x], x] - Dist[(b*e - a*f)/f, Int[((a + b*x)^(m - 1)*(c + d*x)^n)/(e + f*x), x], x]
/; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[Simplify[m + n + 1], 0] && (GtQ[m, 0] || ( !RationalQ[m] && (Su
mSimplerQ[m, -1] ||  !SumSimplerQ[n, -1])))

Rule 203

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

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 295

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)*Int[1/(r^2 + s^2*x^2), x])/(a*n*s^m) +
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 331

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 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 \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x} \, dx &=-\operatorname {Subst}\left (\int \frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x} x} \, dx,x,\frac {1}{x}\right )\\ &=-\operatorname {Subst}\left (\int \frac {1}{\sqrt [6]{1-x} (1+x)^{5/6}} \, dx,x,\frac {1}{x}\right )-\operatorname {Subst}\left (\int \frac {1}{\sqrt [6]{1-x} x (1+x)^{5/6}} \, dx,x,\frac {1}{x}\right )\\ &=6 \operatorname {Subst}\left (\int \frac {x^4}{\left (2-x^6\right )^{5/6}} \, dx,x,\sqrt [6]{\frac {-1+x}{x}}\right )-6 \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 )\\ &=2 \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 )+2 \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 )+2 \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 )+6 \operatorname {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 )\\ &=2 \tanh ^{-1}\left (\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{-\frac {1-x}{x}}}\right )-\frac {1}{2} \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 {1}{2} \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 {3}{2} \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 {3}{2} \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 )+2 \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+2 \operatorname {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 )+2 \operatorname {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 )\\ &=2 \tan ^{-1}\left (\frac {\sqrt [6]{-\frac {1-x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+2 \tanh ^{-1}\left (\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{-\frac {1-x}{x}}}\right )-\frac {1}{2} \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 {1}{2} \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 {1}{2} \operatorname {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}{2} \operatorname {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 )-3 \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 )-3 \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 {1}{2} \sqrt {3} \operatorname {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 )-\frac {1}{2} \sqrt {3} \operatorname {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 )\\ &=-\sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{-\frac {1-x}{x}}}}{\sqrt {3}}\right )+\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{-\frac {1-x}{x}}}}{\sqrt {3}}\right )+2 \tan ^{-1}\left (\frac {\sqrt [6]{-\frac {1-x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+2 \tanh ^{-1}\left (\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{-\frac {1-x}{x}}}\right )-\frac {1}{2} \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 {1}{2} \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 {1}{2} \sqrt {3} \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 )-\frac {1}{2} \sqrt {3} \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 )-\operatorname {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 )-\operatorname {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 )\\ &=-\sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{-\frac {1-x}{x}}}}{\sqrt {3}}\right )+\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{-\frac {1-x}{x}}}}{\sqrt {3}}\right )-\tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+2 \tan ^{-1}\left (\frac {\sqrt [6]{-\frac {1-x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+2 \tanh ^{-1}\left (\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{-\frac {1-x}{x}}}\right )-\frac {1}{2} \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 {1}{2} \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 {1}{2} \sqrt {3} \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 )-\frac {1}{2} \sqrt {3} \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 )\\ \end {align*}

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Mathematica [C]  time = 0.04, size = 26, normalized size = 0.06 \[ \frac {12}{7} e^{\frac {7}{3} \coth ^{-1}(x)} \, _2F_1\left (\frac {7}{12},1;\frac {19}{12};e^{4 \coth ^{-1}(x)}\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(12*E^((7*ArcCoth[x])/3)*Hypergeometric2F1[7/12, 1, 19/12, E^(4*ArcCoth[x])])/7

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fricas [A]  time = 0.56, size = 340, normalized size = 0.85 \[ -\sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \frac {1}{3} \, \sqrt {3}\right ) - \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - \frac {1}{3} \, \sqrt {3}\right ) - \frac {1}{2} \, \sqrt {3} \log \left (16 \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 16 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 16\right ) + \frac {1}{2} \, \sqrt {3} \log \left (-16 \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 16 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 16\right ) - 2 \, \arctan \left (\sqrt {3} + \frac {1}{2} \, \sqrt {-16 \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 16 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 16} - 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) - 2 \, \arctan \left (-\sqrt {3} + 2 \, \sqrt {\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1} - 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + 2 \, \arctan \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \frac {1}{2} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {1}{2} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) + \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(3)*arctan(2/3*sqrt(3)*((x - 1)/(x + 1))^(1/6) + 1/3*sqrt(3)) - sqrt(3)*arctan(2/3*sqrt(3)*((x - 1)/(x +
1))^(1/6) - 1/3*sqrt(3)) - 1/2*sqrt(3)*log(16*sqrt(3)*((x - 1)/(x + 1))^(1/6) + 16*((x - 1)/(x + 1))^(1/3) + 1
6) + 1/2*sqrt(3)*log(-16*sqrt(3)*((x - 1)/(x + 1))^(1/6) + 16*((x - 1)/(x + 1))^(1/3) + 16) - 2*arctan(sqrt(3)
+ 1/2*sqrt(-16*sqrt(3)*((x - 1)/(x + 1))^(1/6) + 16*((x - 1)/(x + 1))^(1/3) + 16) - 2*((x - 1)/(x + 1))^(1/6)
) - 2*arctan(-sqrt(3) + 2*sqrt(sqrt(3)*((x - 1)/(x + 1))^(1/6) + ((x - 1)/(x + 1))^(1/3) + 1) - 2*((x - 1)/(x
+ 1))^(1/6)) + 2*arctan(((x - 1)/(x + 1))^(1/6)) + 1/2*log(((x - 1)/(x + 1))^(1/3) + ((x - 1)/(x + 1))^(1/6) +
1) - 1/2*log(((x - 1)/(x + 1))^(1/3) - ((x - 1)/(x + 1))^(1/6) + 1) + log(((x - 1)/(x + 1))^(1/6) + 1) - log(
((x - 1)/(x + 1))^(1/6) - 1)

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giac [A]  time = 0.16, size = 261, normalized size = 0.65 \[ -\sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right )}\right ) - \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right )}\right ) - \frac {1}{2} \, \sqrt {3} \log \left (\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \frac {1}{2} \, \sqrt {3} \log \left (-\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \arctan \left (\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \arctan \left (-\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + 2 \, \arctan \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \frac {1}{2} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {1}{2} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) + \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \log \left ({\left | \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/6) + 1)) - sqrt(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))
^(1/6) - 1)) - 1/2*sqrt(3)*log(sqrt(3)*((x - 1)/(x + 1))^(1/6) + ((x - 1)/(x + 1))^(1/3) + 1) + 1/2*sqrt(3)*lo
g(-sqrt(3)*((x - 1)/(x + 1))^(1/6) + ((x - 1)/(x + 1))^(1/3) + 1) + arctan(sqrt(3) + 2*((x - 1)/(x + 1))^(1/6)
) + arctan(-sqrt(3) + 2*((x - 1)/(x + 1))^(1/6)) + 2*arctan(((x - 1)/(x + 1))^(1/6)) + 1/2*log(((x - 1)/(x + 1
))^(1/3) + ((x - 1)/(x + 1))^(1/6) + 1) - 1/2*log(((x - 1)/(x + 1))^(1/3) - ((x - 1)/(x + 1))^(1/6) + 1) + log
(((x - 1)/(x + 1))^(1/6) + 1) - log(abs(((x - 1)/(x + 1))^(1/6) - 1))

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maple [C]  time = 14.62, size = 2320, normalized size = 5.77 \[ \text {Expression too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*ln(-(18*RootOf(_Z^2+1)*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(2/3)*x+18*RootOf(_Z^2+1)*RootOf(9*_Z^2+3*_Z+1)
*(-(1-x)/(1+x))^(2/3)+3*(-(1-x)/(1+x))^(5/6)*x+3*RootOf(_Z^2+1)*(-(1-x)/(1+x))^(2/3)*x+18*RootOf(_Z^2+1)*RootO
f(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(1/3)*x+3*(-(1-x)/(1+x))^(5/6)+3*RootOf(_Z^2+1)*(-(1-x)/(1+x))^(2/3)+18*RootOf
(_Z^2+1)*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(1/3)+6*(-(1-x)/(1+x))^(1/2)*x+3*RootOf(_Z^2+1)*(-(1-x)/(1+x))^(
1/3)*x+6*RootOf(_Z^2+1)*RootOf(9*_Z^2+3*_Z+1)*x+6*(-(1-x)/(1+x))^(1/2)+3*RootOf(_Z^2+1)*(-(1-x)/(1+x))^(1/3)+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)*RootOf(9*_Z^2+3*_Z+1)+3*Roo
tOf(9*_Z^2+3*_Z+1)*ln(9*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(2/3)*x-3*(-(1-x)/(1+x))^(5/6)*x+9*RootOf(9*_Z^2+
3*_Z+1)*(-(1-x)/(1+x))^(2/3)-18*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(1/2)*x-3*(-(1-x)/(1+x))^(5/6)+6*(-(1-x)/
(1+x))^(2/3)*x-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+6
*(-(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)-9*RootOf(9*_Z^2+3
*_Z+1)*(-(1-x)/(1+x))^(1/6)*x-6*(-(1-x)/(1+x))^(1/2)+3*(-(1-x)/(1+x))^(1/3)*x-9*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/
(1+x))^(1/6)+3*(-(1-x)/(1+x))^(1/3)+3*RootOf(9*_Z^2+3*_Z+1)-1)-3*ln(-9*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(2
/3)*x-3*(-(1-x)/(1+x))^(5/6)*x-9*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(2/3)+18*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(
1+x))^(1/2)*x-3*(-(1-x)/(1+x))^(5/6)+3*(-(1-x)/(1+x))^(2/3)*x+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+3*(-(1-x)/(1+x))^(2/3)-18*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(
1/3)+9*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(1/6)*x-3*(-(1-x)/(1+x))^(1/3)*x+9*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(
1+x))^(1/6)-3*(-(1-x)/(1+x))^(1/3)+3*(-(1-x)/(1+x))^(1/6)*x-3*RootOf(9*_Z^2+3*_Z+1)+3*(-(1-x)/(1+x))^(1/6)-2)*
RootOf(9*_Z^2+3*_Z+1)+RootOf(_Z^2+1)*ln(-(9*RootOf(_Z^2+1)*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(2/3)*x+9*Root
Of(_Z^2+1)*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(2/3)+3*(-(1-x)/(1+x))^(5/6)*x+6*RootOf(_Z^2+1)*(-(1-x)/(1+x))
^(2/3)*x-18*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(1/2)*x-18*RootOf(_Z^2+1)*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x)
)^(1/3)*x+3*(-(1-x)/(1+x))^(5/6)+6*RootOf(_Z^2+1)*(-(1-x)/(1+x))^(2/3)-18*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))
^(1/2)-18*RootOf(_Z^2+1)*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(1/3)-6*(-(1-x)/(1+x))^(1/2)*x-3*RootOf(_Z^2+1)*
(-(1-x)/(1+x))^(1/3)*x+9*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(1/6)*x+3*RootOf(_Z^2+1)*RootOf(9*_Z^2+3*_Z+1)*x
-6*(-(1-x)/(1+x))^(1/2)-3*RootOf(_Z^2+1)*(-(1-x)/(1+x))^(1/3)+9*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(1/6)-Roo
tOf(_Z^2+1)*x)/x)-ln(-(18*RootOf(_Z^2+1)*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(2/3)*x+18*RootOf(_Z^2+1)*RootOf
(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(2/3)+3*(-(1-x)/(1+x))^(5/6)*x+3*RootOf(_Z^2+1)*(-(1-x)/(1+x))^(2/3)*x+18*RootO
f(_Z^2+1)*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(1/3)*x+3*(-(1-x)/(1+x))^(5/6)+3*RootOf(_Z^2+1)*(-(1-x)/(1+x))^
(2/3)+18*RootOf(_Z^2+1)*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(1/3)+6*(-(1-x)/(1+x))^(1/2)*x+3*RootOf(_Z^2+1)*(
-(1-x)/(1+x))^(1/3)*x+6*RootOf(_Z^2+1)*RootOf(9*_Z^2+3*_Z+1)*x+6*(-(1-x)/(1+x))^(1/2)+3*RootOf(_Z^2+1)*(-(1-x)
/(1+x))^(1/3)+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)-ln(-9*RootOf
(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(2/3)*x-3*(-(1-x)/(1+x))^(5/6)*x-9*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(2/3)+1
8*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(1/2)*x-3*(-(1-x)/(1+x))^(5/6)+3*(-(1-x)/(1+x))^(2/3)*x+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+3*(-(1-x)/(1+x))^(2/3)-18*RootO
f(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(1/3)+9*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(1/6)*x-3*(-(1-x)/(1+x))^(1/3)*x+
9*RootOf(9*_Z^2+3*_Z+1)*(-(1-x)/(1+x))^(1/6)-3*(-(1-x)/(1+x))^(1/3)+3*(-(1-x)/(1+x))^(1/6)*x-3*RootOf(9*_Z^2+3
*_Z+1)+3*(-(1-x)/(1+x))^(1/6)-2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.27, size = 167, normalized size = 0.42 \[ 2\,\mathrm {atan}\left ({\left (\frac {x-1}{x+1}\right )}^{1/6}\right )-\mathrm {atan}\left ({\left (\frac {x-1}{x+1}\right )}^{1/6}\,1{}\mathrm {i}\right )\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {{\left (\frac {x-1}{x+1}\right )}^{1/6}\,1486016741376{}\mathrm {i}}{-743008370688+\sqrt {3}\,743008370688{}\mathrm {i}}\right )\,\left (\sqrt {3}-\mathrm {i}\right )-\mathrm {atan}\left (\frac {{\left (\frac {x-1}{x+1}\right )}^{1/6}\,1486016741376{}\mathrm {i}}{743008370688+\sqrt {3}\,743008370688{}\mathrm {i}}\right )\,\left (\sqrt {3}+1{}\mathrm {i}\right )-\mathrm {atan}\left (\frac {1486016741376\,{\left (\frac {x-1}{x+1}\right )}^{1/6}}{-743008370688+\sqrt {3}\,743008370688{}\mathrm {i}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )-\mathrm {atan}\left (\frac {1486016741376\,{\left (\frac {x-1}{x+1}\right )}^{1/6}}{743008370688+\sqrt {3}\,743008370688{}\mathrm {i}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*((x - 1)/(x + 1))^(1/6)),x)

[Out]

2*atan(((x - 1)/(x + 1))^(1/6)) - atan(((x - 1)/(x + 1))^(1/6)*1i)*2i - atan((((x - 1)/(x + 1))^(1/6)*14860167
41376i)/(3^(1/2)*743008370688i - 743008370688))*(3^(1/2) - 1i) - atan((((x - 1)/(x + 1))^(1/6)*1486016741376i)
/(3^(1/2)*743008370688i + 743008370688))*(3^(1/2) + 1i) - atan((1486016741376*((x - 1)/(x + 1))^(1/6))/(3^(1/2
)*743008370688i - 743008370688))*(3^(1/2)*1i + 1) - atan((1486016741376*((x - 1)/(x + 1))^(1/6))/(3^(1/2)*7430
08370688i + 743008370688))*(3^(1/2)*1i - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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