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

Optimal. Leaf size=233 \[ \sqrt [6]{\frac{1}{x}+1} \left (\frac{x-1}{x}\right )^{5/6}+\frac{\log \left (\frac{\sqrt [3]{\frac{x-1}{x}}}{\sqrt [3]{\frac{1}{x}+1}}-\frac{\sqrt{3} \sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}+1\right )}{2 \sqrt{3}}-\frac{\log \left (\frac{\sqrt [3]{\frac{x-1}{x}}}{\sqrt [3]{\frac{1}{x}+1}}+\frac{\sqrt{3} \sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}+1\right )}{2 \sqrt{3}}-\frac{1}{3} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}\right )+\frac{1}{3} \tan ^{-1}\left (\frac{2 \sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}+\sqrt{3}\right )+\frac{2}{3} \tan ^{-1}\left (\frac{\sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}\right ) \]

[Out]

(1 + x^(-1))^(1/6)*((-1 + x)/x)^(5/6) - ArcTan[Sqrt[3] - (2*((-1 + x)/x)^(1/6))/(1 + x^(-1))^(1/6)]/3 + ArcTan
[Sqrt[3] + (2*((-1 + x)/x)^(1/6))/(1 + x^(-1))^(1/6)]/3 + (2*ArcTan[((-1 + x)/x)^(1/6)/(1 + x^(-1))^(1/6)])/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*Sqrt[3]
)

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Rubi [A]  time = 0.367476, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {6171, 50, 63, 331, 295, 634, 618, 204, 628, 203} \[ \sqrt [6]{\frac{1}{x}+1} \left (\frac{x-1}{x}\right )^{5/6}+\frac{\log \left (\frac{\sqrt [3]{\frac{x-1}{x}}}{\sqrt [3]{\frac{1}{x}+1}}-\frac{\sqrt{3} \sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}+1\right )}{2 \sqrt{3}}-\frac{\log \left (\frac{\sqrt [3]{\frac{x-1}{x}}}{\sqrt [3]{\frac{1}{x}+1}}+\frac{\sqrt{3} \sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}+1\right )}{2 \sqrt{3}}-\frac{1}{3} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}\right )+\frac{1}{3} \tan ^{-1}\left (\frac{2 \sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}+\sqrt{3}\right )+\frac{2}{3} \tan ^{-1}\left (\frac{\sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1 + x^(-1))^(1/6)*((-1 + x)/x)^(5/6) - ArcTan[Sqrt[3] - (2*((-1 + x)/x)^(1/6))/(1 + x^(-1))^(1/6)]/3 + ArcTan
[Sqrt[3] + (2*((-1 + x)/x)^(1/6))/(1 + x^(-1))^(1/6)]/3 + (2*ArcTan[((-1 + x)/x)^(1/6)/(1 + x^(-1))^(1/6)])/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*Sqrt[3]
)

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]

Rule 50

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 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 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 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 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 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 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 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 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])

Rubi steps

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

Mathematica [C]  time = 0.0525046, size = 39, normalized size = 0.17 \[ -2 e^{\frac{1}{3} \coth ^{-1}(x)} \left (\text{Hypergeometric2F1}\left (\frac{1}{6},1,\frac{7}{6},-e^{2 \coth ^{-1}(x)}\right )-\frac{1}{e^{2 \coth ^{-1}(x)}+1}\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-2*E^(ArcCoth[x]/3)*(-(1 + E^(2*ArcCoth[x]))^(-1) + Hypergeometric2F1[1/6, 1, 7/6, -E^(2*ArcCoth[x])])

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Maple [F]  time = 0.08, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}}{\frac{1}{\sqrt [6]{{\frac{-1+x}{1+x}}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.58391, size = 205, normalized size = 0.88 \begin{align*} -\frac{1}{6} \, \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}{6} \, \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{2 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{5}{6}}}{\frac{x - 1}{x + 1} + 1} + \frac{1}{3} \, \arctan \left (\sqrt{3} + 2 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}}\right ) + \frac{1}{3} \, \arctan \left (-\sqrt{3} + 2 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}}\right ) + \frac{2}{3} \, \arctan \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}}\right ) \end{align*}

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]

-1/6*sqrt(3)*log(sqrt(3)*((x - 1)/(x + 1))^(1/6) + ((x - 1)/(x + 1))^(1/3) + 1) + 1/6*sqrt(3)*log(-sqrt(3)*((x
- 1)/(x + 1))^(1/6) + ((x - 1)/(x + 1))^(1/3) + 1) + 2*((x - 1)/(x + 1))^(5/6)/((x - 1)/(x + 1) + 1) + 1/3*ar
ctan(sqrt(3) + 2*((x - 1)/(x + 1))^(1/6)) + 1/3*arctan(-sqrt(3) + 2*((x - 1)/(x + 1))^(1/6)) + 2/3*arctan(((x
- 1)/(x + 1))^(1/6))

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Fricas [A]  time = 1.78059, size = 689, normalized size = 2.96 \begin{align*} -\frac{\sqrt{3} x \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 ) - \sqrt{3} x \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 ) + 4 \, x \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 ) + 4 \, x \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 ) - 4 \, x \arctan \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}}\right ) - 6 \,{\left (x + 1\right )} \left (\frac{x - 1}{x + 1}\right )^{\frac{5}{6}}}{6 \, x} \end{align*}

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]

-1/6*(sqrt(3)*x*log(16*sqrt(3)*((x - 1)/(x + 1))^(1/6) + 16*((x - 1)/(x + 1))^(1/3) + 16) - sqrt(3)*x*log(-16*
sqrt(3)*((x - 1)/(x + 1))^(1/6) + 16*((x - 1)/(x + 1))^(1/3) + 16) + 4*x*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)) + 4*x*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)) - 4*x*ar
ctan(((x - 1)/(x + 1))^(1/6)) - 6*(x + 1)*((x - 1)/(x + 1))^(5/6))/x

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \sqrt [6]{\frac{x - 1}{x + 1}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.385, size = 317, normalized size = 1.36 \begin{align*} \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1\right )}\right ) - \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} - 1\right )}\right ) + \frac{2 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{5}{6}}}{\frac{x - 1}{x + 1} + 1} - \frac{2 \,{\left (2 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}} + \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} - 1\right )}}{3 \,{\left (\left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}} + \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1\right )}{\left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} - 1\right )}} + \frac{2}{3} \, \arctan \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}}\right ) + \frac{5}{18} \, \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}} + \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1\right ) - \frac{1}{2} \, \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}} - \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1\right ) + \frac{4}{9} \, \log \left ({\left | \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} - 1 \right |}\right ) \end{align*}

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]

1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/3) + 1)) - sqrt(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x +
1))^(1/3) - 1)) + 2*((x - 1)/(x + 1))^(5/6)/((x - 1)/(x + 1) + 1) - 2/3*(2*((x - 1)/(x + 1))^(2/3) + ((x - 1)/
(x + 1))^(1/3) - 1)/((((x - 1)/(x + 1))^(2/3) + ((x - 1)/(x + 1))^(1/3) + 1)*(((x - 1)/(x + 1))^(1/3) - 1)) +
2/3*arctan(((x - 1)/(x + 1))^(1/6)) + 5/18*log(((x - 1)/(x + 1))^(2/3) + ((x - 1)/(x + 1))^(1/3) + 1) - 1/2*lo
g(((x - 1)/(x + 1))^(2/3) - ((x - 1)/(x + 1))^(1/3) + 1) + 4/9*log(abs(((x - 1)/(x + 1))^(1/3) - 1))