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)+2/3*arctan(((-1+x)/x)^(1/6)/(1/x+1)^(1/6))+1/3*arctan(2*((-1+x)/x)^(1/6)/(1/x+1
)^(1/6)-3^(1/2))+1/3*arctan(2*((-1+x)/x)^(1/6)/(1/x+1)^(1/6)+3^(1/2))+1/6*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)-1/6*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 = 0.37, antiderivative size = 233, normalized size of antiderivative = 1.00, 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 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 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 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^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*}

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Mathematica [C]  time = 0.06, size = 39, normalized size = 0.17 \[ -2 e^{\frac {1}{3} \coth ^{-1}(x)} \left (\, _2F_1\left (\frac {1}{6},1;\frac {7}{6};-e^{2 \coth ^{-1}(x)}\right )-\frac {1}{e^{2 \coth ^{-1}(x)}+1}\right ) \]

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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fricas [A]  time = 0.61, size = 223, normalized size = 0.96 \[ -\frac {\sqrt {3} x \log \left (16 \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 16 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 16\right ) - \sqrt {3} x \log \left (-16 \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 16 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 16\right ) + 4 \, x \arctan \left (\sqrt {3} + \frac {1}{2} \, \sqrt {-16 \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 16 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 16} - 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + 4 \, x \arctan \left (-\sqrt {3} + 2 \, \sqrt {\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1} - 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) - 4 \, x \arctan \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) - 6 \, {\left (x + 1\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{6 \, x} \]

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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giac [A]  time = 0.15, size = 152, normalized size = 0.65 \[ -\frac {1}{6} \, \sqrt {3} \log \left (\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \frac {1}{6} \, \sqrt {3} \log \left (-\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \frac {2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{\frac {x - 1}{x + 1} + 1} + \frac {1}{3} \, \arctan \left (\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \frac {1}{3} \, \arctan \left (-\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \frac {2}{3} \, \arctan \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) \]

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]

-1/6*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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maple [C]  time = 7.82, size = 3474, normalized size = 14.91 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-1+x)/x/((-1+x)/(1+x))^(1/6)+(1/1594323*ln(-(-2*RootOf(_Z^4-6561*_Z^2+43046721)^3*x^5-8*RootOf(_Z^4-6561*_Z^2
[phone])^3*x^4-12*RootOf(_Z^4-6561*_Z^2+43046721)^3*x^3+6561*RootOf(_Z^4-6561*_Z^2+43046721)*x^5+26244*RootO
f(_Z^4-6561*_Z^2+43046721)*x^4+39366*RootOf(_Z^4-6561*_Z^2+43046721)*x^3-8*RootOf(_Z^4-6561*_Z^2+43046721)^3*x
^2-2*RootOf(_Z^4-6561*_Z^2+43046721)^3*x+26244*RootOf(_Z^4-6561*_Z^2+43046721)*x^2+6561*RootOf(_Z^4-6561*_Z^2+
43046721)*x+1594323*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(5/6)-3188646*(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)^(2/3)*RootOf(_Z^4-6561*_Z^2+43046721)^3+19683*RootOf(_Z^4-6561*_Z^2+43046721)*(x^6+4*x^5
+5*x^4-5*x^2-4*x-1)^(2/3)+486*RootOf(_Z^4-6561*_Z^2+43046721)^2*(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)*RootOf(_Z^4-6561*_Z^2+43046721)^3-39366*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/3)*RootOf
(_Z^4-6561*_Z^2+43046721)-243*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/6)*RootOf(_Z^4-6561*_Z^2+43046721)^2-3188646*(x
^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/2)*x^2-6377292*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/2)*x+3*(x^6+4*x^5+5*x^4-5*x^2-4
*x-1)^(2/3)*RootOf(_Z^4-6561*_Z^2+43046721)^3*x+3*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/3)*RootOf(_Z^4-6561*_Z^2+43
046721)^3*x^3+486*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/2)*RootOf(_Z^4-6561*_Z^2+43046721)^2*x^2+9*(x^6+4*x^5+5*x^4
-5*x^2-4*x-1)^(1/3)*RootOf(_Z^4-6561*_Z^2+43046721)^3*x^2-243*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/6)*RootOf(_Z^4-
6561*_Z^2+43046721)^2*x^4+19683*RootOf(_Z^4-6561*_Z^2+43046721)*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(2/3)*x+972*Root
Of(_Z^4-6561*_Z^2+43046721)^2*(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)*Root
Of(_Z^4-6561*_Z^2+43046721)^3*x-39366*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/3)*RootOf(_Z^4-6561*_Z^2+43046721)*x^3-
972*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/6)*RootOf(_Z^4-6561*_Z^2+43046721)^2*x^3-118098*(x^6+4*x^5+5*x^4-5*x^2-4*
x-1)^(1/3)*RootOf(_Z^4-6561*_Z^2+43046721)*x^2-1458*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/6)*RootOf(_Z^4-6561*_Z^2+
43046721)^2*x^2-118098*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/3)*RootOf(_Z^4-6561*_Z^2+43046721)*x-972*(x^6+4*x^5+5*
x^4-5*x^2-4*x-1)^(1/6)*RootOf(_Z^4-6561*_Z^2+43046721)^2*x)/(1+x)^4/x)*RootOf(_Z^4-6561*_Z^2+43046721)^3-1/243
*ln(-(-2*RootOf(_Z^4-6561*_Z^2+43046721)^3*x^5-8*RootOf(_Z^4-6561*_Z^2+43046721)^3*x^4-12*RootOf(_Z^4-6561*_Z^
2+43046721)^3*x^3+6561*RootOf(_Z^4-6561*_Z^2+43046721)*x^5+26244*RootOf(_Z^4-6561*_Z^2+43046721)*x^4+39366*Roo
tOf(_Z^4-6561*_Z^2+43046721)*x^3-8*RootOf(_Z^4-6561*_Z^2+43046721)^3*x^2-2*RootOf(_Z^4-6561*_Z^2+43046721)^3*x
+26244*RootOf(_Z^4-6561*_Z^2+43046721)*x^2+6561*RootOf(_Z^4-6561*_Z^2+43046721)*x+1594323*(x^6+4*x^5+5*x^4-5*x
^2-4*x-1)^(5/6)-3188646*(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)^(2/3)*RootOf(_Z^4-
6561*_Z^2+43046721)^3+19683*RootOf(_Z^4-6561*_Z^2+43046721)*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(2/3)+486*RootOf(_Z^
4-6561*_Z^2+43046721)^2*(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)*RootOf(_Z^4-
6561*_Z^2+43046721)^3-39366*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/3)*RootOf(_Z^4-6561*_Z^2+43046721)-243*(x^6+4*x^5
+5*x^4-5*x^2-4*x-1)^(1/6)*RootOf(_Z^4-6561*_Z^2+43046721)^2-3188646*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/2)*x^2-63
77292*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/2)*x+3*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(2/3)*RootOf(_Z^4-6561*_Z^2+430467
21)^3*x+3*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/3)*RootOf(_Z^4-6561*_Z^2+43046721)^3*x^3+486*(x^6+4*x^5+5*x^4-5*x^2
-4*x-1)^(1/2)*RootOf(_Z^4-6561*_Z^2+43046721)^2*x^2+9*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/3)*RootOf(_Z^4-6561*_Z^
2+43046721)^3*x^2-243*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/6)*RootOf(_Z^4-6561*_Z^2+43046721)^2*x^4+19683*RootOf(_
Z^4-6561*_Z^2+43046721)*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(2/3)*x+972*RootOf(_Z^4-6561*_Z^2+43046721)^2*(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)*RootOf(_Z^4-6561*_Z^2+43046721)^3*x-39366*(x
^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/3)*RootOf(_Z^4-6561*_Z^2+43046721)*x^3-972*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/6)*
RootOf(_Z^4-6561*_Z^2+43046721)^2*x^3-118098*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/3)*RootOf(_Z^4-6561*_Z^2+4304672
1)*x^2-1458*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/6)*RootOf(_Z^4-6561*_Z^2+43046721)^2*x^2-118098*(x^6+4*x^5+5*x^4-
5*x^2-4*x-1)^(1/3)*RootOf(_Z^4-6561*_Z^2+43046721)*x-972*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/6)*RootOf(_Z^4-6561*
_Z^2+43046721)^2*x)/(1+x)^4/x)*RootOf(_Z^4-6561*_Z^2+43046721)+1/243*RootOf(_Z^4-6561*_Z^2+43046721)*ln(-(-Roo
tOf(_Z^4-6561*_Z^2+43046721)^3*x^5-4*RootOf(_Z^4-6561*_Z^2+43046721)^3*x^4-6*RootOf(_Z^4-6561*_Z^2+43046721)^3
*x^3-6561*RootOf(_Z^4-6561*_Z^2+43046721)*x^5-26244*RootOf(_Z^4-6561*_Z^2+43046721)*x^4-39366*RootOf(_Z^4-6561
*_Z^2+43046721)*x^3-4*RootOf(_Z^4-6561*_Z^2+43046721)^3*x^2-RootOf(_Z^4-6561*_Z^2+43046721)^3*x-26244*RootOf(_
Z^4-6561*_Z^2+43046721)*x^2-6561*RootOf(_Z^4-6561*_Z^2+43046721)*x-1594323*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/6)
+1594323*(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)*RootOf(_Z^4-6561*_Z^2+43046
721)^3-19683*RootOf(_Z^4-6561*_Z^2+43046721)*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(2/3)-486*RootOf(_Z^4-6561*_Z^2+430
46721)^2*(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)*RootOf(_Z^4-6561*_Z^2+43046
721)^3+39366*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/3)*RootOf(_Z^4-6561*_Z^2+43046721)+243*(x^6+4*x^5+5*x^4-5*x^2-4*
x-1)^(1/6)*RootOf(_Z^4-6561*_Z^2+43046721)^2-6377292*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/6)*x-1594323*(x^6+4*x^5+
5*x^4-5*x^2-4*x-1)^(1/6)*x^4-6377292*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/6)*x^3-9565938*(x^6+4*x^5+5*x^4-5*x^2-4*
x-1)^(1/6)*x^2+6*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(2/3)*RootOf(_Z^4-6561*_Z^2+43046721)^3*x-3*(x^6+4*x^5+5*x^4-5*
x^2-4*x-1)^(1/3)*RootOf(_Z^4-6561*_Z^2+43046721)^3*x^3-486*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/2)*RootOf(_Z^4-656
1*_Z^2+43046721)^2*x^2-9*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/3)*RootOf(_Z^4-6561*_Z^2+43046721)^3*x^2+243*(x^6+4*
x^5+5*x^4-5*x^2-4*x-1)^(1/6)*RootOf(_Z^4-6561*_Z^2+43046721)^2*x^4-19683*RootOf(_Z^4-6561*_Z^2+43046721)*(x^6+
4*x^5+5*x^4-5*x^2-4*x-1)^(2/3)*x-972*RootOf(_Z^4-6561*_Z^2+43046721)^2*(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)*RootOf(_Z^4-6561*_Z^2+43046721)^3*x+39366*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(
1/3)*RootOf(_Z^4-6561*_Z^2+43046721)*x^3+972*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/6)*RootOf(_Z^4-6561*_Z^2+4304672
1)^2*x^3+118098*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/3)*RootOf(_Z^4-6561*_Z^2+43046721)*x^2+1458*(x^6+4*x^5+5*x^4-
5*x^2-4*x-1)^(1/6)*RootOf(_Z^4-6561*_Z^2+43046721)^2*x^2+118098*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/3)*RootOf(_Z^
4-6561*_Z^2+43046721)*x+972*(x^6+4*x^5+5*x^4-5*x^2-4*x-1)^(1/6)*RootOf(_Z^4-6561*_Z^2+43046721)^2*x)/(1+x)^4/x
))/((-1+x)/(1+x))^(1/6)*((-1+x)*(1+x)^5)^(1/6)/(1+x)

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maxima [A]  time = 0.41, size = 152, normalized size = 0.65 \[ -\frac {1}{6} \, \sqrt {3} \log \left (\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \frac {1}{6} \, \sqrt {3} \log \left (-\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \frac {2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{\frac {x - 1}{x + 1} + 1} + \frac {1}{3} \, \arctan \left (\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \frac {1}{3} \, \arctan \left (-\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \frac {2}{3} \, \arctan \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) \]

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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mupad [B]  time = 1.23, size = 109, normalized size = 0.47 \[ \frac {2\,\mathrm {atan}\left ({\left (\frac {x-1}{x+1}\right )}^{1/6}\right )}{3}+\frac {2\,{\left (\frac {x-1}{x+1}\right )}^{5/6}}{\frac {x-1}{x+1}+1}-\mathrm {atan}\left (\frac {64\,{\left (\frac {x-1}{x+1}\right )}^{1/6}}{-32+\sqrt {3}\,32{}\mathrm {i}}\right )\,\left (\frac {1}{3}+\frac {\sqrt {3}\,1{}\mathrm {i}}{3}\right )-\mathrm {atan}\left (\frac {64\,{\left (\frac {x-1}{x+1}\right )}^{1/6}}{32+\sqrt {3}\,32{}\mathrm {i}}\right )\,\left (-\frac {1}{3}+\frac {\sqrt {3}\,1{}\mathrm {i}}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*atan(((x - 1)/(x + 1))^(1/6)))/3 + (2*((x - 1)/(x + 1))^(5/6))/((x - 1)/(x + 1) + 1) - atan((64*((x - 1)/(x
+ 1))^(1/6))/(3^(1/2)*32i - 32))*((3^(1/2)*1i)/3 + 1/3) - atan((64*((x - 1)/(x + 1))^(1/6))/(3^(1/2)*32i + 32
))*((3^(1/2)*1i)/3 - 1/3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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