\(\int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x^2} \, dx\) [130]

Optimal result

Integrand size = 14, antiderivative size = 676 \[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x^2} \, dx=a \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}}-\frac {1}{4} \sqrt {2+\sqrt {2}} a \arctan \left (\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}}{\sqrt {2+\sqrt {2}}}\right )-\frac {1}{4} \sqrt {2-\sqrt {2}} a \arctan \left (\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}}{\sqrt {2-\sqrt {2}}}\right )+\frac {1}{4} \sqrt {2+\sqrt {2}} a \arctan \left (\frac {\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{4} \sqrt {2-\sqrt {2}} a \arctan \left (\frac {\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}}{\sqrt {2-\sqrt {2}}}\right )+\frac {1}{8} \sqrt {2-\sqrt {2}} a \log \left (1+\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )-\frac {1}{8} \sqrt {2-\sqrt {2}} a \log \left (1+\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )+\frac {1}{8} \sqrt {2+\sqrt {2}} a \log \left (1+\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )-\frac {1}{8} \sqrt {2+\sqrt {2}} a \log \left (1+\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right ) \]

[Out]

a*(1-1/a/x)^(7/8)*(1+1/a/x)^(1/8)-1/4*a*arctan((-2*(1-1/a/x)^(1/8)/(1+1/a/x)^(1/8)+(2+2^(1/2))^(1/2))/(2-2^(1/
2))^(1/2))*(2-2^(1/2))^(1/2)+1/4*a*arctan((2*(1-1/a/x)^(1/8)/(1+1/a/x)^(1/8)+(2+2^(1/2))^(1/2))/(2-2^(1/2))^(1
/2))*(2-2^(1/2))^(1/2)+1/8*a*ln(1+(1-1/a/x)^(1/4)/(1+1/a/x)^(1/4)-(1-1/a/x)^(1/8)*(2-2^(1/2))^(1/2)/(1+1/a/x)^
(1/8))*(2-2^(1/2))^(1/2)-1/8*a*ln(1+(1-1/a/x)^(1/4)/(1+1/a/x)^(1/4)+(1-1/a/x)^(1/8)*(2-2^(1/2))^(1/2)/(1+1/a/x
)^(1/8))*(2-2^(1/2))^(1/2)-1/4*a*arctan((-2*(1-1/a/x)^(1/8)/(1+1/a/x)^(1/8)+(2-2^(1/2))^(1/2))/(2+2^(1/2))^(1/
2))*(2+2^(1/2))^(1/2)+1/4*a*arctan((2*(1-1/a/x)^(1/8)/(1+1/a/x)^(1/8)+(2-2^(1/2))^(1/2))/(2+2^(1/2))^(1/2))*(2
+2^(1/2))^(1/2)+1/8*a*ln(1+(1-1/a/x)^(1/4)/(1+1/a/x)^(1/4)-(1-1/a/x)^(1/8)*(2+2^(1/2))^(1/2)/(1+1/a/x)^(1/8))*
(2+2^(1/2))^(1/2)-1/8*a*ln(1+(1-1/a/x)^(1/4)/(1+1/a/x)^(1/4)+(1-1/a/x)^(1/8)*(2+2^(1/2))^(1/2)/(1+1/a/x)^(1/8)
)*(2+2^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 676, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {6306, 52, 65, 338, 305, 1136, 1183, 648, 632, 210, 642} \[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x^2} \, dx=-\frac {1}{4} \sqrt {2+\sqrt {2}} a \arctan \left (\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{\frac {1}{a x}+1}}}{\sqrt {2+\sqrt {2}}}\right )-\frac {1}{4} \sqrt {2-\sqrt {2}} a \arctan \left (\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{\frac {1}{a x}+1}}}{\sqrt {2-\sqrt {2}}}\right )+\frac {1}{4} \sqrt {2+\sqrt {2}} a \arctan \left (\frac {\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{\frac {1}{a x}+1}}+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{4} \sqrt {2-\sqrt {2}} a \arctan \left (\frac {\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{\frac {1}{a x}+1}}+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )+a \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}+\frac {1}{8} \sqrt {2-\sqrt {2}} a \log \left (\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{\frac {1}{a x}+1}}+1\right )-\frac {1}{8} \sqrt {2-\sqrt {2}} a \log \left (\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{\frac {1}{a x}+1}}+1\right )+\frac {1}{8} \sqrt {2+\sqrt {2}} a \log \left (\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{\frac {1}{a x}+1}}+1\right )-\frac {1}{8} \sqrt {2+\sqrt {2}} a \log \left (\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{\frac {1}{a x}+1}}+1\right ) \]

[In]

Int[E^(ArcCoth[a*x]/4)/x^2,x]

[Out]

a*(1 - 1/(a*x))^(7/8)*(1 + 1/(a*x))^(1/8) - (Sqrt[2 + Sqrt[2]]*a*ArcTan[(Sqrt[2 - Sqrt[2]] - (2*(1 - 1/(a*x))^
(1/8))/(1 + 1/(a*x))^(1/8))/Sqrt[2 + Sqrt[2]]])/4 - (Sqrt[2 - Sqrt[2]]*a*ArcTan[(Sqrt[2 + Sqrt[2]] - (2*(1 - 1
/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8))/Sqrt[2 - Sqrt[2]]])/4 + (Sqrt[2 + Sqrt[2]]*a*ArcTan[(Sqrt[2 - Sqrt[2]] + (
2*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8))/Sqrt[2 + Sqrt[2]]])/4 + (Sqrt[2 - Sqrt[2]]*a*ArcTan[(Sqrt[2 + Sqrt
[2]] + (2*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8))/Sqrt[2 - Sqrt[2]]])/4 + (Sqrt[2 - Sqrt[2]]*a*Log[1 + (1 -
1/(a*x))^(1/4)/(1 + 1/(a*x))^(1/4) - (Sqrt[2 - Sqrt[2]]*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8)])/8 - (Sqrt[2
- Sqrt[2]]*a*Log[1 + (1 - 1/(a*x))^(1/4)/(1 + 1/(a*x))^(1/4) + (Sqrt[2 - Sqrt[2]]*(1 - 1/(a*x))^(1/8))/(1 + 1
/(a*x))^(1/8)])/8 + (Sqrt[2 + Sqrt[2]]*a*Log[1 + (1 - 1/(a*x))^(1/4)/(1 + 1/(a*x))^(1/4) - (Sqrt[2 + Sqrt[2]]*
(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8)])/8 - (Sqrt[2 + Sqrt[2]]*a*Log[1 + (1 - 1/(a*x))^(1/4)/(1 + 1/(a*x))^
(1/4) + (Sqrt[2 + Sqrt[2]]*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8)])/8

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 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 305

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{r = Numerator[Rt[a/b, 4]], s = Denominator[Rt[a/b,
4]]}, Dist[s^3/(2*Sqrt[2]*b*r), Int[x^(m - n/4)/(r^2 - Sqrt[2]*r*s*x^(n/4) + s^2*x^(n/2)), x], x] - Dist[s^3/
(2*Sqrt[2]*b*r), Int[x^(m - n/4)/(r^2 + Sqrt[2]*r*s*x^(n/4) + s^2*x^(n/2)), x], x]] /; FreeQ[{a, b}, x] && IGt
Q[n/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && GtQ[a/b, 0]

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 1136

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[d^3*(d*x)^(m - 3)*((a + b*
x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 1))), x] - Dist[d^4/(c*(m + 4*p + 1)), Int[(d*x)^(m - 4)*Simp[a*(m - 3) + b
*(m + 2*p - 1)*x^2, x]*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && Gt
Q[m, 3] && NeQ[m + 4*p + 1, 0] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

Rule 1183

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r =
Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(
d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && NegQ[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 {\sqrt [8]{1+\frac {x}{a}}}{\sqrt [8]{1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = a \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}}-\frac {1}{4} \text {Subst}\left (\int \frac {1}{\sqrt [8]{1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{7/8}} \, dx,x,\frac {1}{x}\right ) \\ & = a \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}}+(2 a) \text {Subst}\left (\int \frac {x^6}{\left (2-x^8\right )^{7/8}} \, dx,x,\sqrt [8]{1-\frac {1}{a x}}\right ) \\ & = a \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}}+(2 a) \text {Subst}\left (\int \frac {x^6}{1+x^8} \, dx,x,\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right ) \\ & = a \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}}+\frac {a \text {Subst}\left (\int \frac {x^4}{1-\sqrt {2} x^2+x^4} \, dx,x,\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )}{\sqrt {2}}-\frac {a \text {Subst}\left (\int \frac {x^4}{1+\sqrt {2} x^2+x^4} \, dx,x,\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )}{\sqrt {2}} \\ & = a \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}}-\frac {a \text {Subst}\left (\int \frac {1-\sqrt {2} x^2}{1-\sqrt {2} x^2+x^4} \, dx,x,\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )}{\sqrt {2}}+\frac {a \text {Subst}\left (\int \frac {1+\sqrt {2} x^2}{1+\sqrt {2} x^2+x^4} \, dx,x,\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )}{\sqrt {2}} \\ & = a \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}}+\frac {a \text {Subst}\left (\int \frac {\sqrt {2-\sqrt {2}}-\left (1-\sqrt {2}\right ) x}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )}{2 \sqrt {2 \left (2-\sqrt {2}\right )}}+\frac {a \text {Subst}\left (\int \frac {\sqrt {2-\sqrt {2}}+\left (1-\sqrt {2}\right ) x}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )}{2 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {a \text {Subst}\left (\int \frac {\sqrt {2+\sqrt {2}}-\left (1+\sqrt {2}\right ) x}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )}{2 \sqrt {2 \left (2+\sqrt {2}\right )}}-\frac {a \text {Subst}\left (\int \frac {\sqrt {2+\sqrt {2}}+\left (1+\sqrt {2}\right ) x}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )}{2 \sqrt {2 \left (2+\sqrt {2}\right )}} \\ & = a \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}}+\frac {1}{4} \left (\sqrt {\frac {1}{2} \left (3-2 \sqrt {2}\right )} a\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )+\frac {1}{4} \left (\sqrt {\frac {1}{2} \left (3-2 \sqrt {2}\right )} a\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )+\frac {1}{8} \left (\sqrt {2-\sqrt {2}} a\right ) \text {Subst}\left (\int \frac {-\sqrt {2-\sqrt {2}}+2 x}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )-\frac {1}{8} \left (\sqrt {2-\sqrt {2}} a\right ) \text {Subst}\left (\int \frac {\sqrt {2-\sqrt {2}}+2 x}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )+\frac {1}{8} \left (\sqrt {2+\sqrt {2}} a\right ) \text {Subst}\left (\int \frac {-\sqrt {2+\sqrt {2}}+2 x}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )-\frac {1}{8} \left (\sqrt {2+\sqrt {2}} a\right ) \text {Subst}\left (\int \frac {\sqrt {2+\sqrt {2}}+2 x}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )+\frac {1}{4} \left (\sqrt {\frac {1}{2} \left (3+2 \sqrt {2}\right )} a\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )+\frac {1}{4} \left (\sqrt {\frac {1}{2} \left (3+2 \sqrt {2}\right )} a\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right ) \\ & = a \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}}+\frac {1}{8} \sqrt {2-\sqrt {2}} a \log \left (1+\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )-\frac {1}{8} \sqrt {2-\sqrt {2}} a \log \left (1+\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )+\frac {1}{8} \sqrt {2+\sqrt {2}} a \log \left (1+\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )-\frac {1}{8} \sqrt {2+\sqrt {2}} a \log \left (1+\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )-\frac {1}{2} \left (\sqrt {\frac {1}{2} \left (3-2 \sqrt {2}\right )} a\right ) \text {Subst}\left (\int \frac {1}{-2+\sqrt {2}-x^2} \, dx,x,-\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )-\frac {1}{2} \left (\sqrt {\frac {1}{2} \left (3-2 \sqrt {2}\right )} a\right ) \text {Subst}\left (\int \frac {1}{-2+\sqrt {2}-x^2} \, dx,x,\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )-\frac {1}{2} \left (\sqrt {\frac {1}{2} \left (3+2 \sqrt {2}\right )} a\right ) \text {Subst}\left (\int \frac {1}{-2-\sqrt {2}-x^2} \, dx,x,-\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )-\frac {1}{2} \left (\sqrt {\frac {1}{2} \left (3+2 \sqrt {2}\right )} a\right ) \text {Subst}\left (\int \frac {1}{-2-\sqrt {2}-x^2} \, dx,x,\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right ) \\ & = a \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}}-\frac {1}{4} \sqrt {2+\sqrt {2}} a \arctan \left (\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}}{\sqrt {2+\sqrt {2}}}\right )-\frac {1}{4} \sqrt {2-\sqrt {2}} a \arctan \left (\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}}{\sqrt {2-\sqrt {2}}}\right )+\frac {1}{4} \sqrt {2+\sqrt {2}} a \arctan \left (\frac {\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{4} \sqrt {2-\sqrt {2}} a \arctan \left (\frac {\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}}{\sqrt {2-\sqrt {2}}}\right )+\frac {1}{8} \sqrt {2-\sqrt {2}} a \log \left (1+\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )-\frac {1}{8} \sqrt {2-\sqrt {2}} a \log \left (1+\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )+\frac {1}{8} \sqrt {2+\sqrt {2}} a \log \left (1+\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )-\frac {1}{8} \sqrt {2+\sqrt {2}} a \log \left (1+\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right ) \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.07 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.07 \[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x^2} \, dx=-2 a e^{\frac {1}{4} \coth ^{-1}(a x)} \left (-\frac {1}{1+e^{2 \coth ^{-1}(a x)}}+\operatorname {Hypergeometric2F1}\left (\frac {1}{8},1,\frac {9}{8},-e^{2 \coth ^{-1}(a x)}\right )\right ) \]

[In]

Integrate[E^(ArcCoth[a*x]/4)/x^2,x]

[Out]

-2*a*E^(ArcCoth[a*x]/4)*(-(1 + E^(2*ArcCoth[a*x]))^(-1) + Hypergeometric2F1[1/8, 1, 9/8, -E^(2*ArcCoth[a*x])])

Maple [F]

\[\int \frac {1}{\left (\frac {a x -1}{a x +1}\right )^{\frac {1}{8}} x^{2}}d x\]

[In]

int(1/((a*x-1)/(a*x+1))^(1/8)/x^2,x)

[Out]

int(1/((a*x-1)/(a*x+1))^(1/8)/x^2,x)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.25 (sec) , antiderivative size = 374, normalized size of antiderivative = 0.55 \[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x^2} \, dx=\frac {-\left (i - 1\right ) \, \sqrt {2} \left (-a^{8}\right )^{\frac {1}{8}} x \log \left (2 \, a^{7} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (i + 1\right ) \, \sqrt {2} \left (-a^{8}\right )^{\frac {7}{8}}\right ) + \left (i + 1\right ) \, \sqrt {2} \left (-a^{8}\right )^{\frac {1}{8}} x \log \left (2 \, a^{7} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} - \left (i - 1\right ) \, \sqrt {2} \left (-a^{8}\right )^{\frac {7}{8}}\right ) - \left (i + 1\right ) \, \sqrt {2} \left (-a^{8}\right )^{\frac {1}{8}} x \log \left (2 \, a^{7} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (i - 1\right ) \, \sqrt {2} \left (-a^{8}\right )^{\frac {7}{8}}\right ) + \left (i - 1\right ) \, \sqrt {2} \left (-a^{8}\right )^{\frac {1}{8}} x \log \left (2 \, a^{7} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} - \left (i + 1\right ) \, \sqrt {2} \left (-a^{8}\right )^{\frac {7}{8}}\right ) + 2 \, \left (-a^{8}\right )^{\frac {1}{8}} x \log \left (a^{7} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (-a^{8}\right )^{\frac {7}{8}}\right ) - 2 i \, \left (-a^{8}\right )^{\frac {1}{8}} x \log \left (a^{7} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + i \, \left (-a^{8}\right )^{\frac {7}{8}}\right ) + 2 i \, \left (-a^{8}\right )^{\frac {1}{8}} x \log \left (a^{7} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} - i \, \left (-a^{8}\right )^{\frac {7}{8}}\right ) - 2 \, \left (-a^{8}\right )^{\frac {1}{8}} x \log \left (a^{7} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} - \left (-a^{8}\right )^{\frac {7}{8}}\right ) + 8 \, {\left (a x + 1\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{8}}}{8 \, x} \]

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/8)/x^2,x, algorithm="fricas")

[Out]

1/8*(-(I - 1)*sqrt(2)*(-a^8)^(1/8)*x*log(2*a^7*((a*x - 1)/(a*x + 1))^(1/8) + (I + 1)*sqrt(2)*(-a^8)^(7/8)) + (
I + 1)*sqrt(2)*(-a^8)^(1/8)*x*log(2*a^7*((a*x - 1)/(a*x + 1))^(1/8) - (I - 1)*sqrt(2)*(-a^8)^(7/8)) - (I + 1)*
sqrt(2)*(-a^8)^(1/8)*x*log(2*a^7*((a*x - 1)/(a*x + 1))^(1/8) + (I - 1)*sqrt(2)*(-a^8)^(7/8)) + (I - 1)*sqrt(2)
*(-a^8)^(1/8)*x*log(2*a^7*((a*x - 1)/(a*x + 1))^(1/8) - (I + 1)*sqrt(2)*(-a^8)^(7/8)) + 2*(-a^8)^(1/8)*x*log(a
^7*((a*x - 1)/(a*x + 1))^(1/8) + (-a^8)^(7/8)) - 2*I*(-a^8)^(1/8)*x*log(a^7*((a*x - 1)/(a*x + 1))^(1/8) + I*(-
a^8)^(7/8)) + 2*I*(-a^8)^(1/8)*x*log(a^7*((a*x - 1)/(a*x + 1))^(1/8) - I*(-a^8)^(7/8)) - 2*(-a^8)^(1/8)*x*log(
a^7*((a*x - 1)/(a*x + 1))^(1/8) - (-a^8)^(7/8)) + 8*(a*x + 1)*((a*x - 1)/(a*x + 1))^(7/8))/x

Sympy [F]

\[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x^2} \, dx=\int \frac {1}{x^{2} \sqrt [8]{\frac {a x - 1}{a x + 1}}}\, dx \]

[In]

integrate(1/((a*x-1)/(a*x+1))**(1/8)/x**2,x)

[Out]

Integral(1/(x**2*((a*x - 1)/(a*x + 1))**(1/8)), x)

Maxima [F]

\[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x^2} \, dx=\int { \frac {1}{x^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}} \,d x } \]

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/8)/x^2,x, algorithm="maxima")

[Out]

integrate(1/(x^2*((a*x - 1)/(a*x + 1))^(1/8)), x)

Giac [A] (verification not implemented)

none

Time = 0.46 (sec) , antiderivative size = 432, normalized size of antiderivative = 0.64 \[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x^2} \, dx=\frac {1}{8} \, {\left (2 \, \sqrt {-\sqrt {2} + 2} \arctan \left (\frac {\sqrt {\sqrt {2} + 2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}}{\sqrt {-\sqrt {2} + 2}}\right ) + 2 \, \sqrt {-\sqrt {2} + 2} \arctan \left (-\frac {\sqrt {\sqrt {2} + 2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}}{\sqrt {-\sqrt {2} + 2}}\right ) + 2 \, \sqrt {\sqrt {2} + 2} \arctan \left (\frac {\sqrt {-\sqrt {2} + 2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}}{\sqrt {\sqrt {2} + 2}}\right ) + 2 \, \sqrt {\sqrt {2} + 2} \arctan \left (-\frac {\sqrt {-\sqrt {2} + 2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}}{\sqrt {\sqrt {2} + 2}}\right ) - \sqrt {\sqrt {2} + 2} \log \left (\sqrt {\sqrt {2} + 2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) + \sqrt {\sqrt {2} + 2} \log \left (-\sqrt {\sqrt {2} + 2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) - \sqrt {-\sqrt {2} + 2} \log \left (\sqrt {-\sqrt {2} + 2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) + \sqrt {-\sqrt {2} + 2} \log \left (-\sqrt {-\sqrt {2} + 2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) + \frac {16 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{8}}}{\frac {a x - 1}{a x + 1} + 1}\right )} a \]

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/8)/x^2,x, algorithm="giac")

[Out]

1/8*(2*sqrt(-sqrt(2) + 2)*arctan((sqrt(sqrt(2) + 2) + 2*((a*x - 1)/(a*x + 1))^(1/8))/sqrt(-sqrt(2) + 2)) + 2*s
qrt(-sqrt(2) + 2)*arctan(-(sqrt(sqrt(2) + 2) - 2*((a*x - 1)/(a*x + 1))^(1/8))/sqrt(-sqrt(2) + 2)) + 2*sqrt(sqr
t(2) + 2)*arctan((sqrt(-sqrt(2) + 2) + 2*((a*x - 1)/(a*x + 1))^(1/8))/sqrt(sqrt(2) + 2)) + 2*sqrt(sqrt(2) + 2)
*arctan(-(sqrt(-sqrt(2) + 2) - 2*((a*x - 1)/(a*x + 1))^(1/8))/sqrt(sqrt(2) + 2)) - sqrt(sqrt(2) + 2)*log(sqrt(
sqrt(2) + 2)*((a*x - 1)/(a*x + 1))^(1/8) + ((a*x - 1)/(a*x + 1))^(1/4) + 1) + sqrt(sqrt(2) + 2)*log(-sqrt(sqrt
(2) + 2)*((a*x - 1)/(a*x + 1))^(1/8) + ((a*x - 1)/(a*x + 1))^(1/4) + 1) - sqrt(-sqrt(2) + 2)*log(sqrt(-sqrt(2)
+ 2)*((a*x - 1)/(a*x + 1))^(1/8) + ((a*x - 1)/(a*x + 1))^(1/4) + 1) + sqrt(-sqrt(2) + 2)*log(-sqrt(-sqrt(2) +
2)*((a*x - 1)/(a*x + 1))^(1/8) + ((a*x - 1)/(a*x + 1))^(1/4) + 1) + 16*((a*x - 1)/(a*x + 1))^(7/8)/((a*x - 1)
/(a*x + 1) + 1))*a

Mupad [B] (verification not implemented)

Time = 4.13 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.24 \[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x^2} \, dx=\frac {{\left (-1\right )}^{1/8}\,a\,\mathrm {atan}\left ({\left (-1\right )}^{1/8}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\right )}{2}+\frac {{\left (-1\right )}^{1/8}\,a\,\mathrm {atan}\left ({\left (-1\right )}^{1/8}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}+\frac {2\,a\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/8}}{\frac {a\,x-1}{a\,x+1}+1}+{\left (-1\right )}^{1/8}\,\sqrt {2}\,a\,\mathrm {atan}\left ({\left (-1\right )}^{1/8}\,\sqrt {2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{4}-\frac {1}{4}{}\mathrm {i}\right )+{\left (-1\right )}^{1/8}\,\sqrt {2}\,a\,\mathrm {atan}\left ({\left (-1\right )}^{1/8}\,\sqrt {2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{4}+\frac {1}{4}{}\mathrm {i}\right ) \]

[In]

int(1/(x^2*((a*x - 1)/(a*x + 1))^(1/8)),x)

[Out]

((-1)^(1/8)*a*atan((-1)^(1/8)*((a*x - 1)/(a*x + 1))^(1/8)))/2 + ((-1)^(1/8)*a*atan((-1)^(1/8)*((a*x - 1)/(a*x
+ 1))^(1/8)*1i)*1i)/2 + (2*a*((a*x - 1)/(a*x + 1))^(7/8))/((a*x - 1)/(a*x + 1) + 1) + (-1)^(1/8)*2^(1/2)*a*ata
n((-1)^(1/8)*2^(1/2)*((a*x - 1)/(a*x + 1))^(1/8)*(1/2 - 1i/2))*(1/4 - 1i/4) + (-1)^(1/8)*2^(1/2)*a*atan((-1)^(
1/8)*2^(1/2)*((a*x - 1)/(a*x + 1))^(1/8)*(1/2 + 1i/2))*(1/4 + 1i/4)