3.119 \(\int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x^4} \, dx\)

Optimal. Leaf size=287 \[ \frac {1}{18} \left (\frac {x-1}{x}\right )^{5/6} \left (\frac {1}{x}+1\right )^{7/6}+\frac {\left (\frac {x-1}{x}\right )^{5/6} \left (\frac {1}{x}+1\right )^{7/6}}{3 x}+\frac {19}{54} \left (\frac {x-1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1}+\frac {19 \log \left (\frac {\sqrt [3]{\frac {x-1}{x}}}{\sqrt [3]{\frac {1}{x}+1}}-\frac {\sqrt {3} \sqrt [6]{\frac {x-1}{x}}}{\sqrt [6]{\frac {1}{x}+1}}+1\right )}{108 \sqrt {3}}-\frac {19 \log \left (\frac {\sqrt [3]{\frac {x-1}{x}}}{\sqrt [3]{\frac {1}{x}+1}}+\frac {\sqrt {3} \sqrt [6]{\frac {x-1}{x}}}{\sqrt [6]{\frac {1}{x}+1}}+1\right )}{108 \sqrt {3}}-\frac {19}{162} \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{\frac {x-1}{x}}}{\sqrt [6]{\frac {1}{x}+1}}\right )+\frac {19}{162} \tan ^{-1}\left (\frac {2 \sqrt [6]{\frac {x-1}{x}}}{\sqrt [6]{\frac {1}{x}+1}}+\sqrt {3}\right )+\frac {19}{81} \tan ^{-1}\left (\frac {\sqrt [6]{\frac {x-1}{x}}}{\sqrt [6]{\frac {1}{x}+1}}\right ) \]

[Out]

19/54*(1/x+1)^(1/6)*((-1+x)/x)^(5/6)+1/18*(1/x+1)^(7/6)*((-1+x)/x)^(5/6)+1/3*(1/x+1)^(7/6)*((-1+x)/x)^(5/6)/x+
19/81*arctan(((-1+x)/x)^(1/6)/(1/x+1)^(1/6))+19/162*arctan(2*((-1+x)/x)^(1/6)/(1/x+1)^(1/6)-3^(1/2))+19/162*ar
ctan(2*((-1+x)/x)^(1/6)/(1/x+1)^(1/6)+3^(1/2))+19/324*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)-19/324*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 = 0.40, antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6171, 90, 80, 50, 63, 331, 295, 634, 618, 204, 628, 203} \[ \frac {1}{18} \left (\frac {x-1}{x}\right )^{5/6} \left (\frac {1}{x}+1\right )^{7/6}+\frac {\left (\frac {x-1}{x}\right )^{5/6} \left (\frac {1}{x}+1\right )^{7/6}}{3 x}+\frac {19}{54} \left (\frac {x-1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1}+\frac {19 \log \left (\frac {\sqrt [3]{\frac {x-1}{x}}}{\sqrt [3]{\frac {1}{x}+1}}-\frac {\sqrt {3} \sqrt [6]{\frac {x-1}{x}}}{\sqrt [6]{\frac {1}{x}+1}}+1\right )}{108 \sqrt {3}}-\frac {19 \log \left (\frac {\sqrt [3]{\frac {x-1}{x}}}{\sqrt [3]{\frac {1}{x}+1}}+\frac {\sqrt {3} \sqrt [6]{\frac {x-1}{x}}}{\sqrt [6]{\frac {1}{x}+1}}+1\right )}{108 \sqrt {3}}-\frac {19}{162} \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{\frac {x-1}{x}}}{\sqrt [6]{\frac {1}{x}+1}}\right )+\frac {19}{162} \tan ^{-1}\left (\frac {2 \sqrt [6]{\frac {x-1}{x}}}{\sqrt [6]{\frac {1}{x}+1}}+\sqrt {3}\right )+\frac {19}{81} \tan ^{-1}\left (\frac {\sqrt [6]{\frac {x-1}{x}}}{\sqrt [6]{\frac {1}{x}+1}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^(ArcCoth[x]/3)/x^4,x]

[Out]

(19*(1 + x^(-1))^(1/6)*((-1 + x)/x)^(5/6))/54 + ((1 + x^(-1))^(7/6)*((-1 + x)/x)^(5/6))/18 + ((1 + x^(-1))^(7/
6)*((-1 + x)/x)^(5/6))/(3*x) - (19*ArcTan[Sqrt[3] - (2*((-1 + x)/x)^(1/6))/(1 + x^(-1))^(1/6)])/162 + (19*ArcT
an[Sqrt[3] + (2*((-1 + x)/x)^(1/6))/(1 + x^(-1))^(1/6)])/162 + (19*ArcTan[((-1 + x)/x)^(1/6)/(1 + x^(-1))^(1/6
)])/81 + (19*Log[1 - (Sqrt[3]*((-1 + x)/x)^(1/6))/(1 + x^(-1))^(1/6) + ((-1 + x)/x)^(1/3)/(1 + x^(-1))^(1/3)])
/(108*Sqrt[3]) - (19*Log[1 + (Sqrt[3]*((-1 + x)/x)^(1/6))/(1 + x^(-1))^(1/6) + ((-1 + x)/x)^(1/3)/(1 + x^(-1))
^(1/3)])/(108*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 80

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 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 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])

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^4} \, dx &=-\operatorname {Subst}\left (\int \frac {x^2 \sqrt [6]{1+x}}{\sqrt [6]{1-x}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {\left (1+\frac {1}{x}\right )^{7/6} \left (\frac {-1+x}{x}\right )^{5/6}}{3 x}+\frac {1}{3} \operatorname {Subst}\left (\int \frac {\left (-1-\frac {x}{3}\right ) \sqrt [6]{1+x}}{\sqrt [6]{1-x}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{18} \left (1+\frac {1}{x}\right )^{7/6} \left (-\frac {1-x}{x}\right )^{5/6}+\frac {\left (1+\frac {1}{x}\right )^{7/6} \left (\frac {-1+x}{x}\right )^{5/6}}{3 x}-\frac {19}{54} \operatorname {Subst}\left (\int \frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {19}{54} \sqrt [6]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{5/6}+\frac {1}{18} \left (1+\frac {1}{x}\right )^{7/6} \left (-\frac {1-x}{x}\right )^{5/6}+\frac {\left (1+\frac {1}{x}\right )^{7/6} \left (\frac {-1+x}{x}\right )^{5/6}}{3 x}-\frac {19}{162} \operatorname {Subst}\left (\int \frac {1}{\sqrt [6]{1-x} (1+x)^{5/6}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {19}{54} \sqrt [6]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{5/6}+\frac {1}{18} \left (1+\frac {1}{x}\right )^{7/6} \left (-\frac {1-x}{x}\right )^{5/6}+\frac {\left (1+\frac {1}{x}\right )^{7/6} \left (\frac {-1+x}{x}\right )^{5/6}}{3 x}+\frac {19}{27} \operatorname {Subst}\left (\int \frac {x^4}{\left (2-x^6\right )^{5/6}} \, dx,x,\sqrt [6]{\frac {-1+x}{x}}\right )\\ &=\frac {19}{54} \sqrt [6]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{5/6}+\frac {1}{18} \left (1+\frac {1}{x}\right )^{7/6} \left (-\frac {1-x}{x}\right )^{5/6}+\frac {\left (1+\frac {1}{x}\right )^{7/6} \left (\frac {-1+x}{x}\right )^{5/6}}{3 x}+\frac {19}{27} \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 )\\ &=\frac {19}{54} \sqrt [6]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{5/6}+\frac {1}{18} \left (1+\frac {1}{x}\right )^{7/6} \left (-\frac {1-x}{x}\right )^{5/6}+\frac {\left (1+\frac {1}{x}\right )^{7/6} \left (\frac {-1+x}{x}\right )^{5/6}}{3 x}+\frac {19}{81} \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 {19}{81} \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 {19}{81} \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 {19}{54} \sqrt [6]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{5/6}+\frac {1}{18} \left (1+\frac {1}{x}\right )^{7/6} \left (-\frac {1-x}{x}\right )^{5/6}+\frac {\left (1+\frac {1}{x}\right )^{7/6} \left (\frac {-1+x}{x}\right )^{5/6}}{3 x}+\frac {19}{81} \tan ^{-1}\left (\frac {\sqrt [6]{-\frac {1-x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\frac {19}{324} \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 {19}{324} \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 {19 \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 )}{108 \sqrt {3}}-\frac {19 \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 )}{108 \sqrt {3}}\\ &=\frac {19}{54} \sqrt [6]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{5/6}+\frac {1}{18} \left (1+\frac {1}{x}\right )^{7/6} \left (-\frac {1-x}{x}\right )^{5/6}+\frac {\left (1+\frac {1}{x}\right )^{7/6} \left (\frac {-1+x}{x}\right )^{5/6}}{3 x}+\frac {19}{81} \tan ^{-1}\left (\frac {\sqrt [6]{-\frac {1-x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\frac {19 \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 )}{108 \sqrt {3}}-\frac {19 \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 )}{108 \sqrt {3}}-\frac {19}{162} \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 {19}{162} \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 {19}{54} \sqrt [6]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{5/6}+\frac {1}{18} \left (1+\frac {1}{x}\right )^{7/6} \left (-\frac {1-x}{x}\right )^{5/6}+\frac {\left (1+\frac {1}{x}\right )^{7/6} \left (\frac {-1+x}{x}\right )^{5/6}}{3 x}-\frac {19}{162} \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{-\frac {1-x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\frac {19}{162} \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{-\frac {1-x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\frac {19}{81} \tan ^{-1}\left (\frac {\sqrt [6]{-\frac {1-x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\frac {19 \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 )}{108 \sqrt {3}}-\frac {19 \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 )}{108 \sqrt {3}}\\ \end {align*}

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Mathematica [C]  time = 0.22, size = 133, normalized size = 0.46 \[ \frac {1}{486} \left (-19 \text {RootSum}\left [\text {$\#$1}^4-\text {$\#$1}^2+1\& ,\frac {\text {$\#$1}^2 \coth ^{-1}(x)-3 \text {$\#$1}^2 \log \left (e^{\frac {1}{3} \coth ^{-1}(x)}-\text {$\#$1}\right )+6 \log \left (e^{\frac {1}{3} \coth ^{-1}(x)}-\text {$\#$1}\right )-2 \coth ^{-1}(x)}{2 \text {$\#$1}^3-\text {$\#$1}}\& \right ]+\frac {18 e^{\frac {1}{3} \coth ^{-1}(x)} \left (8 e^{2 \coth ^{-1}(x)}+61 e^{4 \coth ^{-1}(x)}+19\right )}{\left (e^{2 \coth ^{-1}(x)}+1\right )^3}-114 \tan ^{-1}\left (e^{\frac {1}{3} \coth ^{-1}(x)}\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(ArcCoth[x]/3)/x^4,x]

[Out]

((18*E^(ArcCoth[x]/3)*(19 + 8*E^(2*ArcCoth[x]) + 61*E^(4*ArcCoth[x])))/(1 + E^(2*ArcCoth[x]))^3 - 114*ArcTan[E
^(ArcCoth[x]/3)] - 19*RootSum[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) & ])/486

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fricas [A]  time = 0.42, size = 246, normalized size = 0.86 \[ -\frac {19 \, \sqrt {3} x^{3} \log \left (5776 \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 5776 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 5776\right ) - 19 \, \sqrt {3} x^{3} \log \left (-5776 \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 5776 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 5776\right ) + 76 \, x^{3} \arctan \left (\sqrt {3} + \frac {1}{38} \, \sqrt {-5776 \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 5776 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 5776} - 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + 76 \, x^{3} \arctan \left (-\sqrt {3} + 2 \, \sqrt {\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1} - 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) - 76 \, x^{3} \arctan \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) - 6 \, {\left (22 \, x^{3} + 43 \, x^{2} + 39 \, x + 18\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{324 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((-1+x)/(1+x))^(1/6)/x^4,x, algorithm="fricas")

[Out]

-1/324*(19*sqrt(3)*x^3*log(5776*sqrt(3)*((x - 1)/(x + 1))^(1/6) + 5776*((x - 1)/(x + 1))^(1/3) + 5776) - 19*sq
rt(3)*x^3*log(-5776*sqrt(3)*((x - 1)/(x + 1))^(1/6) + 5776*((x - 1)/(x + 1))^(1/3) + 5776) + 76*x^3*arctan(sqr
t(3) + 1/38*sqrt(-5776*sqrt(3)*((x - 1)/(x + 1))^(1/6) + 5776*((x - 1)/(x + 1))^(1/3) + 5776) - 2*((x - 1)/(x
+ 1))^(1/6)) + 76*x^3*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)) - 76*x^3*arctan(((x - 1)/(x + 1))^(1/6)) - 6*(22*x^3 + 43*x^2 + 39*x + 18)*((x -
1)/(x + 1))^(5/6))/x^3

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giac [A]  time = 0.18, size = 199, normalized size = 0.69 \[ -\frac {19}{324} \, \sqrt {3} \log \left (\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \frac {19}{324} \, \sqrt {3} \log \left (-\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \frac {\frac {8 \, {\left (x - 1\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{x + 1} + \frac {19 \, {\left (x - 1\right )}^{2} \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{{\left (x + 1\right )}^{2}} + 61 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{27 \, {\left (\frac {x - 1}{x + 1} + 1\right )}^{3}} + \frac {19}{162} \, \arctan \left (\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \frac {19}{162} \, \arctan \left (-\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \frac {19}{81} \, \arctan \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((-1+x)/(1+x))^(1/6)/x^4,x, algorithm="giac")

[Out]

-19/324*sqrt(3)*log(sqrt(3)*((x - 1)/(x + 1))^(1/6) + ((x - 1)/(x + 1))^(1/3) + 1) + 19/324*sqrt(3)*log(-sqrt(
3)*((x - 1)/(x + 1))^(1/6) + ((x - 1)/(x + 1))^(1/3) + 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 + 19/162*
arctan(sqrt(3) + 2*((x - 1)/(x + 1))^(1/6)) + 19/162*arctan(-sqrt(3) + 2*((x - 1)/(x + 1))^(1/6)) + 19/81*arct
an(((x - 1)/(x + 1))^(1/6))

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maple [C]  time = 7.22, size = 3484, normalized size = 12.14 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((-1+x)/(1+x))^(1/6)/x^4,x)

[Out]

1/54*(-1+x)*(22*x^2+21*x+18)/x^3/((-1+x)/(1+x))^(1/6)+(19/86093442*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
[phone])*x^5+26244*RootOf(_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+6
561*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+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+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+430
46721)*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^4-6
561*_Z^2+43046721)^3-19/13122*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*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+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+4
3046721)^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^4-656
1*_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^4-656
1*_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)-19/13122*RootOf(_Z
^4-6561*_Z^2+43046721)*ln((-RootOf(_Z^4-6561*_Z^2+43046721)^3*x^5-4*RootOf(_Z^4-6561*_Z^2+43046721)^3*x^4-6*Ro
otOf(_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+43046
721)*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+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+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+95
65938*(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+4304
6721)^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+43046
721)*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^4-656
1*_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 = 205, normalized size = 0.71 \[ -\frac {19}{324} \, \sqrt {3} \log \left (\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \frac {19}{324} \, \sqrt {3} \log \left (-\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \frac {19 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {17}{6}} + 8 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {11}{6}} + 61 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{27 \, {\left (\frac {3 \, {\left (x - 1\right )}}{x + 1} + \frac {3 \, {\left (x - 1\right )}^{2}}{{\left (x + 1\right )}^{2}} + \frac {{\left (x - 1\right )}^{3}}{{\left (x + 1\right )}^{3}} + 1\right )}} + \frac {19}{162} \, \arctan \left (\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \frac {19}{162} \, \arctan \left (-\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \frac {19}{81} \, \arctan \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((-1+x)/(1+x))^(1/6)/x^4,x, algorithm="maxima")

[Out]

-19/324*sqrt(3)*log(sqrt(3)*((x - 1)/(x + 1))^(1/6) + ((x - 1)/(x + 1))^(1/3) + 1) + 19/324*sqrt(3)*log(-sqrt(
3)*((x - 1)/(x + 1))^(1/6) + ((x - 1)/(x + 1))^(1/3) + 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) + 19/162*arctan(sqrt(3) + 2*((x - 1)/(x + 1))^(1/6)) + 19/162*arctan(-sqrt(3) + 2*((x - 1)/(x + 1))^(1/6))
+ 19/81*arctan(((x - 1)/(x + 1))^(1/6))

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mupad [B]  time = 1.26, size = 161, normalized size = 0.56 \[ \frac {19\,\mathrm {atan}\left ({\left (\frac {x-1}{x+1}\right )}^{1/6}\right )}{81}+\frac {\frac {61\,{\left (\frac {x-1}{x+1}\right )}^{5/6}}{27}+\frac {8\,{\left (\frac {x-1}{x+1}\right )}^{11/6}}{27}+\frac {19\,{\left (\frac {x-1}{x+1}\right )}^{17/6}}{27}}{\frac {3\,\left (x-1\right )}{x+1}+\frac {3\,{\left (x-1\right )}^2}{{\left (x+1\right )}^2}+\frac {{\left (x-1\right )}^3}{{\left (x+1\right )}^3}+1}-\mathrm {atan}\left (\frac {4952198\,{\left (\frac {x-1}{x+1}\right )}^{1/6}}{14348907\,\left (-\frac {2476099}{14348907}+\frac {\sqrt {3}\,2476099{}\mathrm {i}}{14348907}\right )}\right )\,\left (\frac {19}{162}+\frac {\sqrt {3}\,19{}\mathrm {i}}{162}\right )-\mathrm {atan}\left (\frac {4952198\,{\left (\frac {x-1}{x+1}\right )}^{1/6}}{14348907\,\left (\frac {2476099}{14348907}+\frac {\sqrt {3}\,2476099{}\mathrm {i}}{14348907}\right )}\right )\,\left (-\frac {19}{162}+\frac {\sqrt {3}\,19{}\mathrm {i}}{162}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*((x - 1)/(x + 1))^(1/6)),x)

[Out]

(19*atan(((x - 1)/(x + 1))^(1/6)))/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) - a
tan((4952198*((x - 1)/(x + 1))^(1/6))/(14348907*((3^(1/2)*2476099i)/14348907 - 2476099/14348907)))*((3^(1/2)*1
9i)/162 + 19/162) - atan((4952198*((x - 1)/(x + 1))^(1/6))/(14348907*((3^(1/2)*2476099i)/14348907 + 2476099/14
348907)))*((3^(1/2)*19i)/162 - 19/162)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{4} \sqrt [6]{\frac {x - 1}{x + 1}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((-1+x)/(1+x))**(1/6)/x**4,x)

[Out]

Integral(1/(x**4*((x - 1)/(x + 1))**(1/6)), x)

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