\(\int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x^4} \, dx\) [119]

Optimal result

Integrand size = 12, antiderivative size = 287 \[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x^4} \, dx=\frac {19}{54} \sqrt [6]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{5/6}+\frac {1}{18} \left (1+\frac {1}{x}\right )^{7/6} \left (\frac {-1+x}{x}\right )^{5/6}+\frac {\left (1+\frac {1}{x}\right )^{7/6} \left (\frac {-1+x}{x}\right )^{5/6}}{3 x}-\frac {19}{162} \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\frac {19}{162} \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\frac {19}{81} \arctan \left (\frac {\sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\frac {19 \log \left (1-\frac {\sqrt {3} \sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}+\frac {\sqrt [3]{\frac {-1+x}{x}}}{\sqrt [3]{1+\frac {1}{x}}}\right )}{108 \sqrt {3}}-\frac {19 \log \left (1+\frac {\sqrt {3} \sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}+\frac {\sqrt [3]{\frac {-1+x}{x}}}{\sqrt [3]{1+\frac {1}{x}}}\right )}{108 \sqrt {3}} \]

[Out]

19/54*(1+1/x)^(1/6)*((-1+x)/x)^(5/6)+1/18*(1+1/x)^(7/6)*((-1+x)/x)^(5/6)+1/3*(1+1/x)^(7/6)*((-1+x)/x)^(5/6)/x+
19/81*arctan(((-1+x)/x)^(1/6)/(1+1/x)^(1/6))+19/162*arctan(2*((-1+x)/x)^(1/6)/(1+1/x)^(1/6)-3^(1/2))+19/162*ar
ctan(2*((-1+x)/x)^(1/6)/(1+1/x)^(1/6)+3^(1/2))+19/324*ln(1+((-1+x)/x)^(1/3)/(1+1/x)^(1/3)-((-1+x)/x)^(1/6)*3^(
1/2)/(1+1/x)^(1/6))*3^(1/2)-19/324*ln(1+((-1+x)/x)^(1/3)/(1+1/x)^(1/3)+((-1+x)/x)^(1/6)*3^(1/2)/(1+1/x)^(1/6))
*3^(1/2)

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6306, 92, 81, 52, 65, 338, 301, 648, 632, 210, 642, 209} \[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x^4} \, dx=-\frac {19}{162} \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{\frac {x-1}{x}}}{\sqrt [6]{\frac {1}{x}+1}}\right )+\frac {19}{162} \arctan \left (\frac {2 \sqrt [6]{\frac {x-1}{x}}}{\sqrt [6]{\frac {1}{x}+1}}+\sqrt {3}\right )+\frac {19}{81} \arctan \left (\frac {\sqrt [6]{\frac {x-1}{x}}}{\sqrt [6]{\frac {1}{x}+1}}\right )+\frac {1}{18} \left (\frac {x-1}{x}\right )^{5/6} \left (\frac {1}{x}+1\right )^{7/6}+\frac {\left (\frac {x-1}{x}\right )^{5/6} \left (\frac {1}{x}+1\right )^{7/6}}{3 x}+\frac {19}{54} \left (\frac {x-1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1}+\frac {19 \log \left (\frac {\sqrt [3]{\frac {x-1}{x}}}{\sqrt [3]{\frac {1}{x}+1}}-\frac {\sqrt {3} \sqrt [6]{\frac {x-1}{x}}}{\sqrt [6]{\frac {1}{x}+1}}+1\right )}{108 \sqrt {3}}-\frac {19 \log \left (\frac {\sqrt [3]{\frac {x-1}{x}}}{\sqrt [3]{\frac {1}{x}+1}}+\frac {\sqrt {3} \sqrt [6]{\frac {x-1}{x}}}{\sqrt [6]{\frac {1}{x}+1}}+1\right )}{108 \sqrt {3}} \]

[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 52

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

Rule 65

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

Rule 81

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

Rule 92

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 209

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

Rule 210

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

Rule 301

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

Rule 338

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

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 6306

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(x_)^(m_.), x_Symbol] :> -Subst[Int[(1 + x/a)^(n/2)/(x^(m + 2)*(1 - x/a)^(n/2
)), x], x, 1/x] /; FreeQ[{a, n}, x] &&  !IntegerQ[n] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {x^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} \text {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} \text {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} \text {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} \text {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} \text {Subst}\left (\int \frac {x^4}{1+x^6} \, dx,x,\frac {\sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right ) \\ & = \frac {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} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\frac {19}{81} \text {Subst}\left (\int \frac {-\frac {1}{2}+\frac {\sqrt {3} x}{2}}{1-\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\frac {19}{81} \text {Subst}\left (\int \frac {-\frac {1}{2}-\frac {\sqrt {3} x}{2}}{1+\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right ) \\ & = \frac {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} \arctan \left (\frac {\sqrt [6]{-\frac {1-x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\frac {19}{324} \text {Subst}\left (\int \frac {1}{1-\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\frac {19}{324} \text {Subst}\left (\int \frac {1}{1+\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\frac {19 \text {Subst}\left (\int \frac {-\sqrt {3}+2 x}{1-\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )}{108 \sqrt {3}}-\frac {19 \text {Subst}\left (\int \frac {\sqrt {3}+2 x}{1+\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )}{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} \arctan \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} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,-\sqrt {3}+\frac {2 \sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )-\frac {19}{162} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {3}+\frac {2 \sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right ) \\ & = \frac {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} \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{-\frac {1-x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\frac {19}{162} \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{-\frac {1-x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\frac {19}{81} \arctan \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*}

Mathematica [C] (verified)

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

Time = 0.23 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.46 \[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x^4} \, dx=\frac {1}{486} \left (\frac {18 e^{\frac {1}{3} \coth ^{-1}(x)} \left (19+8 e^{2 \coth ^{-1}(x)}+61 e^{4 \coth ^{-1}(x)}\right )}{\left (1+e^{2 \coth ^{-1}(x)}\right )^3}-114 \arctan \left (e^{\frac {1}{3} \coth ^{-1}(x)}\right )-19 \text {RootSum}\left [1-\text {$\#$1}^2+\text {$\#$1}^4\&,\frac {-2 \coth ^{-1}(x)+6 \log \left (e^{\frac {1}{3} \coth ^{-1}(x)}-\text {$\#$1}\right )+\coth ^{-1}(x) \text {$\#$1}^2-3 \log \left (e^{\frac {1}{3} \coth ^{-1}(x)}-\text {$\#$1}\right ) \text {$\#$1}^2}{-\text {$\#$1}+2 \text {$\#$1}^3}\&\right ]\right ) \]

[In]

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

Maple [C] (verified)

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

Time = 14.23 (sec) , antiderivative size = 1498, normalized size of antiderivative = 5.22

method	result	size
trager	\(\text {Expression too large to display}\)	\(1498\)
risch	\(\text {Expression too large to display}\)	\(3004\)

[In]

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

[Out]

1/54*(1+x)*(22*x^2+21*x+18)/x^3*(-(1-x)/(1+x))^(5/6)+19/54*RootOf(81*_Z^4-9*_Z^2+1)*ln((54*RootOf(81*_Z^4-9*_Z
^2+1)^3*(-(1-x)/(1+x))^(2/3)*x+54*RootOf(81*_Z^4-9*_Z^2+1)^3*(-(1-x)/(1+x))^(2/3)+(-(1-x)/(1+x))^(5/6)*x-3*Roo
tOf(81*_Z^4-9*_Z^2+1)*(-(1-x)/(1+x))^(2/3)*x-18*x*RootOf(81*_Z^4-9*_Z^2+1)^2*(-(1-x)/(1+x))^(1/2)-27*RootOf(81
*_Z^4-9*_Z^2+1)^3*(-(1-x)/(1+x))^(1/3)*x+(-(1-x)/(1+x))^(5/6)-3*RootOf(81*_Z^4-9*_Z^2+1)*(-(1-x)/(1+x))^(2/3)-
18*RootOf(81*_Z^4-9*_Z^2+1)^2*(-(1-x)/(1+x))^(1/2)-27*RootOf(81*_Z^4-9*_Z^2+1)^3*(-(1-x)/(1+x))^(1/3)+6*RootOf
(81*_Z^4-9*_Z^2+1)*(-(1-x)/(1+x))^(1/3)*x+9*RootOf(81*_Z^4-9*_Z^2+1)^2*(-(1-x)/(1+x))^(1/6)*x-9*RootOf(81*_Z^4
-9*_Z^2+1)^3*x+6*RootOf(81*_Z^4-9*_Z^2+1)*(-(1-x)/(1+x))^(1/3)+9*RootOf(81*_Z^4-9*_Z^2+1)^2*(-(1-x)/(1+x))^(1/
6)-(-(1-x)/(1+x))^(1/6)*x-RootOf(81*_Z^4-9*_Z^2+1)*x-(-(1-x)/(1+x))^(1/6))/x)+19/6*RootOf(81*_Z^4-9*_Z^2+1)^3*
ln((27*RootOf(81*_Z^4-9*_Z^2+1)^3*(-(1-x)/(1+x))^(2/3)*x+27*RootOf(81*_Z^4-9*_Z^2+1)^3*(-(1-x)/(1+x))^(2/3)+(-
(1-x)/(1+x))^(5/6)*x+3*RootOf(81*_Z^4-9*_Z^2+1)*(-(1-x)/(1+x))^(2/3)*x+18*x*RootOf(81*_Z^4-9*_Z^2+1)^2*(-(1-x)
/(1+x))^(1/2)+27*RootOf(81*_Z^4-9*_Z^2+1)^3*(-(1-x)/(1+x))^(1/3)*x+(-(1-x)/(1+x))^(5/6)+3*RootOf(81*_Z^4-9*_Z^
2+1)*(-(1-x)/(1+x))^(2/3)+18*RootOf(81*_Z^4-9*_Z^2+1)^2*(-(1-x)/(1+x))^(1/2)+27*RootOf(81*_Z^4-9*_Z^2+1)^3*(-(
1-x)/(1+x))^(1/3)-2*(-(1-x)/(1+x))^(1/2)*x-6*RootOf(81*_Z^4-9*_Z^2+1)*(-(1-x)/(1+x))^(1/3)*x-9*RootOf(81*_Z^4-
9*_Z^2+1)^2*(-(1-x)/(1+x))^(1/6)*x-18*RootOf(81*_Z^4-9*_Z^2+1)^3*x-2*(-(1-x)/(1+x))^(1/2)-6*RootOf(81*_Z^4-9*_
Z^2+1)*(-(1-x)/(1+x))^(1/3)-9*RootOf(81*_Z^4-9*_Z^2+1)^2*(-(1-x)/(1+x))^(1/6)+RootOf(81*_Z^4-9*_Z^2+1)*x)/x)-1
9/54*RootOf(81*_Z^4-9*_Z^2+1)*ln((27*RootOf(81*_Z^4-9*_Z^2+1)^3*(-(1-x)/(1+x))^(2/3)*x+27*RootOf(81*_Z^4-9*_Z^
2+1)^3*(-(1-x)/(1+x))^(2/3)+(-(1-x)/(1+x))^(5/6)*x+3*RootOf(81*_Z^4-9*_Z^2+1)*(-(1-x)/(1+x))^(2/3)*x+18*x*Root
Of(81*_Z^4-9*_Z^2+1)^2*(-(1-x)/(1+x))^(1/2)+27*RootOf(81*_Z^4-9*_Z^2+1)^3*(-(1-x)/(1+x))^(1/3)*x+(-(1-x)/(1+x)
)^(5/6)+3*RootOf(81*_Z^4-9*_Z^2+1)*(-(1-x)/(1+x))^(2/3)+18*RootOf(81*_Z^4-9*_Z^2+1)^2*(-(1-x)/(1+x))^(1/2)+27*
RootOf(81*_Z^4-9*_Z^2+1)^3*(-(1-x)/(1+x))^(1/3)-2*(-(1-x)/(1+x))^(1/2)*x-6*RootOf(81*_Z^4-9*_Z^2+1)*(-(1-x)/(1
+x))^(1/3)*x-9*RootOf(81*_Z^4-9*_Z^2+1)^2*(-(1-x)/(1+x))^(1/6)*x-18*RootOf(81*_Z^4-9*_Z^2+1)^3*x-2*(-(1-x)/(1+
x))^(1/2)-6*RootOf(81*_Z^4-9*_Z^2+1)*(-(1-x)/(1+x))^(1/3)-9*RootOf(81*_Z^4-9*_Z^2+1)^2*(-(1-x)/(1+x))^(1/6)+Ro
otOf(81*_Z^4-9*_Z^2+1)*x)/x)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.25 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.94 \[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x^4} \, dx=\frac {\sqrt {2} x^{3} \sqrt {361 i \, \sqrt {3} + 361} \log \left ({\left (i \, \sqrt {3} \sqrt {2} - \sqrt {2}\right )} \sqrt {361 i \, \sqrt {3} + 361} + 76 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) - \sqrt {2} x^{3} \sqrt {361 i \, \sqrt {3} + 361} \log \left ({\left (-i \, \sqrt {3} \sqrt {2} + \sqrt {2}\right )} \sqrt {361 i \, \sqrt {3} + 361} + 76 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) - \sqrt {2} x^{3} \sqrt {-361 i \, \sqrt {3} + 361} \log \left ({\left (i \, \sqrt {3} \sqrt {2} + \sqrt {2}\right )} \sqrt {-361 i \, \sqrt {3} + 361} + 76 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \sqrt {2} x^{3} \sqrt {-361 i \, \sqrt {3} + 361} \log \left ({\left (-i \, \sqrt {3} \sqrt {2} - \sqrt {2}\right )} \sqrt {-361 i \, \sqrt {3} + 361} + 76 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + 76 \, x^{3} \arctan \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + 6 \, {\left (22 \, x^{3} + 43 \, x^{2} + 39 \, x + 18\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{324 \, x^{3}} \]

[In]

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

[Out]

1/324*(sqrt(2)*x^3*sqrt(361*I*sqrt(3) + 361)*log((I*sqrt(3)*sqrt(2) - sqrt(2))*sqrt(361*I*sqrt(3) + 361) + 76*
((x - 1)/(x + 1))^(1/6)) - sqrt(2)*x^3*sqrt(361*I*sqrt(3) + 361)*log((-I*sqrt(3)*sqrt(2) + sqrt(2))*sqrt(361*I
*sqrt(3) + 361) + 76*((x - 1)/(x + 1))^(1/6)) - sqrt(2)*x^3*sqrt(-361*I*sqrt(3) + 361)*log((I*sqrt(3)*sqrt(2)
+ sqrt(2))*sqrt(-361*I*sqrt(3) + 361) + 76*((x - 1)/(x + 1))^(1/6)) + sqrt(2)*x^3*sqrt(-361*I*sqrt(3) + 361)*l
og((-I*sqrt(3)*sqrt(2) - sqrt(2))*sqrt(-361*I*sqrt(3) + 361) + 76*((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

Sympy [F]

\[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x^4} \, dx=\int \frac {1}{x^{4} \sqrt [6]{\frac {x - 1}{x + 1}}}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.71 \[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x^4} \, dx=-\frac {19}{324} \, \sqrt {3} \log \left (\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \frac {19}{324} \, \sqrt {3} \log \left (-\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \frac {19 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {17}{6}} + 8 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {11}{6}} + 61 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{27 \, {\left (\frac {3 \, {\left (x - 1\right )}}{x + 1} + \frac {3 \, {\left (x - 1\right )}^{2}}{{\left (x + 1\right )}^{2}} + \frac {{\left (x - 1\right )}^{3}}{{\left (x + 1\right )}^{3}} + 1\right )}} + \frac {19}{162} \, \arctan \left (\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \frac {19}{162} \, \arctan \left (-\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \frac {19}{81} \, \arctan \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) \]

[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))

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.69 \[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x^4} \, dx=-\frac {19}{324} \, \sqrt {3} \log \left (\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \frac {19}{324} \, \sqrt {3} \log \left (-\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \frac {\frac {8 \, {\left (x - 1\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{x + 1} + \frac {19 \, {\left (x - 1\right )}^{2} \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{{\left (x + 1\right )}^{2}} + 61 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{27 \, {\left (\frac {x - 1}{x + 1} + 1\right )}^{3}} + \frac {19}{162} \, \arctan \left (\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \frac {19}{162} \, \arctan \left (-\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \frac {19}{81} \, \arctan \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) \]

[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))

Mupad [B] (verification not implemented)

Time = 4.14 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.56 \[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x^4} \, dx=\frac {19\,\mathrm {atan}\left ({\left (\frac {x-1}{x+1}\right )}^{1/6}\right )}{81}+\frac {\frac {61\,{\left (\frac {x-1}{x+1}\right )}^{5/6}}{27}+\frac {8\,{\left (\frac {x-1}{x+1}\right )}^{11/6}}{27}+\frac {19\,{\left (\frac {x-1}{x+1}\right )}^{17/6}}{27}}{\frac {3\,\left (x-1\right )}{x+1}+\frac {3\,{\left (x-1\right )}^2}{{\left (x+1\right )}^2}+\frac {{\left (x-1\right )}^3}{{\left (x+1\right )}^3}+1}-\mathrm {atan}\left (\frac {4952198\,{\left (\frac {x-1}{x+1}\right )}^{1/6}}{14348907\,\left (-\frac {2476099}{14348907}+\frac {\sqrt {3}\,2476099{}\mathrm {i}}{14348907}\right )}\right )\,\left (\frac {19}{162}+\frac {\sqrt {3}\,19{}\mathrm {i}}{162}\right )-\mathrm {atan}\left (\frac {4952198\,{\left (\frac {x-1}{x+1}\right )}^{1/6}}{14348907\,\left (\frac {2476099}{14348907}+\frac {\sqrt {3}\,2476099{}\mathrm {i}}{14348907}\right )}\right )\,\left (-\frac {19}{162}+\frac {\sqrt {3}\,19{}\mathrm {i}}{162}\right ) \]

[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)