### 3.130 $$\int \frac{e^{\frac{1}{4} \coth ^{-1}(a x)}}{x^2} \, dx$$

Optimal. Leaf size=676 $a \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{\frac{1}{a x}+1}+\frac{1}{8} \sqrt{2-\sqrt{2}} a \log \left (\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}-\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{\frac{1}{a x}+1}}+1\right )-\frac{1}{8} \sqrt{2-\sqrt{2}} a \log \left (\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{\frac{1}{a x}+1}}+1\right )+\frac{1}{8} \sqrt{2+\sqrt{2}} a \log \left (\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}-\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{\frac{1}{a x}+1}}+1\right )-\frac{1}{8} \sqrt{2+\sqrt{2}} a \log \left (\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{\frac{1}{a x}+1}}+1\right )-\frac{1}{4} \sqrt{2+\sqrt{2}} a \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-\frac{2 \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{\frac{1}{a x}+1}}}{\sqrt{2+\sqrt{2}}}\right )-\frac{1}{4} \sqrt{2-\sqrt{2}} a \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}-\frac{2 \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{\frac{1}{a x}+1}}}{\sqrt{2-\sqrt{2}}}\right )+\frac{1}{4} \sqrt{2+\sqrt{2}} a \tan ^{-1}\left (\frac{\frac{2 \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{\frac{1}{a x}+1}}+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right )+\frac{1}{4} \sqrt{2-\sqrt{2}} a \tan ^{-1}\left (\frac{\frac{2 \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{\frac{1}{a x}+1}}+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right )$

[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

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Rubi [A]  time = 0.605244, antiderivative size = 676, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 11, integrand size = 14, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.786, Rules used = {6171, 50, 63, 331, 299, 1122, 1169, 634, 618, 204, 628} $a \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{\frac{1}{a x}+1}+\frac{1}{8} \sqrt{2-\sqrt{2}} a \log \left (\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}-\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{\frac{1}{a x}+1}}+1\right )-\frac{1}{8} \sqrt{2-\sqrt{2}} a \log \left (\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{\frac{1}{a x}+1}}+1\right )+\frac{1}{8} \sqrt{2+\sqrt{2}} a \log \left (\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}-\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{\frac{1}{a x}+1}}+1\right )-\frac{1}{8} \sqrt{2+\sqrt{2}} a \log \left (\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{\frac{1}{a x}+1}}+1\right )-\frac{1}{4} \sqrt{2+\sqrt{2}} a \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-\frac{2 \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{\frac{1}{a x}+1}}}{\sqrt{2+\sqrt{2}}}\right )-\frac{1}{4} \sqrt{2-\sqrt{2}} a \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}-\frac{2 \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{\frac{1}{a x}+1}}}{\sqrt{2-\sqrt{2}}}\right )+\frac{1}{4} \sqrt{2+\sqrt{2}} a \tan ^{-1}\left (\frac{\frac{2 \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{\frac{1}{a x}+1}}+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right )+\frac{1}{4} \sqrt{2-\sqrt{2}} a \tan ^{-1}\left (\frac{\frac{2 \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{\frac{1}{a x}+1}}+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right )$

Antiderivative was successfully veriﬁed.

[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 6171

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

Rule 50

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

Rule 63

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

Rule 331

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

Rule 299

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 1122

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 1169

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 634

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

Rule 618

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

Rule 204

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

Rule 628

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

Rubi steps

\begin{align*} \int \frac{e^{\frac{1}{4} \coth ^{-1}(a x)}}{x^2} \, dx &=-\operatorname{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} \operatorname{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) \operatorname{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) \operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 \tan ^{-1}\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 \tan ^{-1}\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 \tan ^{-1}\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 \tan ^{-1}\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]  time = 0.0608243, size = 46, normalized size = 0.07 $-2 a e^{\frac{1}{4} \coth ^{-1}(a x)} \left (\text{Hypergeometric2F1}\left (\frac{1}{8},1,\frac{9}{8},-e^{2 \coth ^{-1}(a x)}\right )-\frac{1}{e^{2 \coth ^{-1}(a x)}+1}\right )$

Warning: Unable to verify antiderivative.

[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])])

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Maple [F]  time = 0.148, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}}{\frac{1}{\sqrt [8]{{\frac{ax-1}{ax+1}}}}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B]  time = 2.19607, size = 7880, normalized size = 11.66 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32*(8*(a^8)^(1/8)*x*sqrt(-sqrt(2) + 2)*arctan(-(2*(a^8)^(1/8)*a^7*((a*x - 1)/(a*x + 1))^(1/8) + (a^8*(sqrt(
2) + 2) - a^8)*sqrt(-sqrt(2) + 2) - 2*sqrt(a^14*((a*x - 1)/(a*x + 1))^(1/4) + (a^8)^(3/4)*a^8 + (a^8)^(7/8)*(a
^7*(sqrt(2) + 2) - a^7)*sqrt(-sqrt(2) + 2)*((a*x - 1)/(a*x + 1))^(1/8))*(a^8)^(1/8))/(a^8*(sqrt(2) + 2)^(3/2)
- 3*a^8*sqrt(sqrt(2) + 2))) + 8*(a^8)^(1/8)*x*sqrt(-sqrt(2) + 2)*arctan(-(2*(a^8)^(1/8)*a^7*((a*x - 1)/(a*x +
1))^(1/8) - (a^8*(sqrt(2) + 2) - a^8)*sqrt(-sqrt(2) + 2) - 2*sqrt(a^14*((a*x - 1)/(a*x + 1))^(1/4) + (a^8)^(3/
4)*a^8 - (a^8)^(7/8)*(a^7*(sqrt(2) + 2) - a^7)*sqrt(-sqrt(2) + 2)*((a*x - 1)/(a*x + 1))^(1/8))*(a^8)^(1/8))/(a
^8*(sqrt(2) + 2)^(3/2) - 3*a^8*sqrt(sqrt(2) + 2))) + 8*(a^8)^(1/8)*x*sqrt(sqrt(2) + 2)*arctan(-(a^8*(sqrt(2) +
2)^(3/2) - 3*a^8*sqrt(sqrt(2) + 2) + 2*(a^8)^(1/8)*a^7*((a*x - 1)/(a*x + 1))^(1/8) - 2*sqrt(a^14*((a*x - 1)/(
a*x + 1))^(1/4) + (a^8)^(3/4)*a^8 + (a^7*(sqrt(2) + 2)^(3/2) - 3*a^7*sqrt(sqrt(2) + 2))*(a^8)^(7/8)*((a*x - 1)
/(a*x + 1))^(1/8))*(a^8)^(1/8))/((a^8*(sqrt(2) + 2) - a^8)*sqrt(-sqrt(2) + 2))) + 8*(a^8)^(1/8)*x*sqrt(sqrt(2)
+ 2)*arctan((a^8*(sqrt(2) + 2)^(3/2) - 3*a^8*sqrt(sqrt(2) + 2) - 2*(a^8)^(1/8)*a^7*((a*x - 1)/(a*x + 1))^(1/8
) + 2*sqrt(a^14*((a*x - 1)/(a*x + 1))^(1/4) + (a^8)^(3/4)*a^8 - (a^7*(sqrt(2) + 2)^(3/2) - 3*a^7*sqrt(sqrt(2)
+ 2))*(a^8)^(7/8)*((a*x - 1)/(a*x + 1))^(1/8))*(a^8)^(1/8))/((a^8*(sqrt(2) + 2) - a^8)*sqrt(-sqrt(2) + 2))) +
2*(a^8)^(1/8)*x*sqrt(sqrt(2) + 2)*log(a^14*((a*x - 1)/(a*x + 1))^(1/4) + (a^8)^(3/4)*a^8 + (a^8)^(7/8)*(a^7*(s
qrt(2) + 2) - a^7)*sqrt(-sqrt(2) + 2)*((a*x - 1)/(a*x + 1))^(1/8)) - 2*(a^8)^(1/8)*x*sqrt(sqrt(2) + 2)*log(a^1
4*((a*x - 1)/(a*x + 1))^(1/4) + (a^8)^(3/4)*a^8 - (a^8)^(7/8)*(a^7*(sqrt(2) + 2) - a^7)*sqrt(-sqrt(2) + 2)*((a
*x - 1)/(a*x + 1))^(1/8)) + 2*(a^8)^(1/8)*x*sqrt(-sqrt(2) + 2)*log(a^14*((a*x - 1)/(a*x + 1))^(1/4) + (a^8)^(3
/4)*a^8 + (a^7*(sqrt(2) + 2)^(3/2) - 3*a^7*sqrt(sqrt(2) + 2))*(a^8)^(7/8)*((a*x - 1)/(a*x + 1))^(1/8)) - 2*(a^
8)^(1/8)*x*sqrt(-sqrt(2) + 2)*log(a^14*((a*x - 1)/(a*x + 1))^(1/4) + (a^8)^(3/4)*a^8 - (a^7*(sqrt(2) + 2)^(3/2
) - 3*a^7*sqrt(sqrt(2) + 2))*(a^8)^(7/8)*((a*x - 1)/(a*x + 1))^(1/8)) + 4*(a^8)^(1/8)*(sqrt(2)*x*sqrt(sqrt(2)
+ 2) + sqrt(2)*x*sqrt(-sqrt(2) + 2))*arctan(-(a^8*(sqrt(2) + 2)^(3/2) - 3*a^8*sqrt(sqrt(2) + 2) + 2*sqrt(2)*(a
^8)^(1/8)*a^7*((a*x - 1)/(a*x + 1))^(1/8) - (a^8*(sqrt(2) + 2) - a^8)*sqrt(-sqrt(2) + 2) - sqrt(2)*sqrt(4*a^14
*((a*x - 1)/(a*x + 1))^(1/4) + 4*(a^8)^(3/4)*a^8 + 2*(sqrt(2)*a^7*(sqrt(2) + 2)^(3/2) - 3*sqrt(2)*a^7*sqrt(sqr
t(2) + 2) - (sqrt(2)*a^7*(sqrt(2) + 2) - sqrt(2)*a^7)*sqrt(-sqrt(2) + 2))*(a^8)^(7/8)*((a*x - 1)/(a*x + 1))^(1
/8))*(a^8)^(1/8))/(a^8*(sqrt(2) + 2)^(3/2) - 3*a^8*sqrt(sqrt(2) + 2) + (a^8*(sqrt(2) + 2) - a^8)*sqrt(-sqrt(2)
+ 2))) + 4*(a^8)^(1/8)*(sqrt(2)*x*sqrt(sqrt(2) + 2) + sqrt(2)*x*sqrt(-sqrt(2) + 2))*arctan((a^8*(sqrt(2) + 2)
^(3/2) - 3*a^8*sqrt(sqrt(2) + 2) - 2*sqrt(2)*(a^8)^(1/8)*a^7*((a*x - 1)/(a*x + 1))^(1/8) - (a^8*(sqrt(2) + 2)
- a^8)*sqrt(-sqrt(2) + 2) + sqrt(2)*sqrt(4*a^14*((a*x - 1)/(a*x + 1))^(1/4) + 4*(a^8)^(3/4)*a^8 - 2*(sqrt(2)*a
^7*(sqrt(2) + 2)^(3/2) - 3*sqrt(2)*a^7*sqrt(sqrt(2) + 2) - (sqrt(2)*a^7*(sqrt(2) + 2) - sqrt(2)*a^7)*sqrt(-sqr
t(2) + 2))*(a^8)^(7/8)*((a*x - 1)/(a*x + 1))^(1/8))*(a^8)^(1/8))/(a^8*(sqrt(2) + 2)^(3/2) - 3*a^8*sqrt(sqrt(2)
+ 2) + (a^8*(sqrt(2) + 2) - a^8)*sqrt(-sqrt(2) + 2))) + 4*(a^8)^(1/8)*(sqrt(2)*x*sqrt(sqrt(2) + 2) - sqrt(2)*
x*sqrt(-sqrt(2) + 2))*arctan((a^8*(sqrt(2) + 2)^(3/2) - 3*a^8*sqrt(sqrt(2) + 2) + 2*sqrt(2)*(a^8)^(1/8)*a^7*((
a*x - 1)/(a*x + 1))^(1/8) + (a^8*(sqrt(2) + 2) - a^8)*sqrt(-sqrt(2) + 2) - sqrt(2)*sqrt(4*a^14*((a*x - 1)/(a*x
+ 1))^(1/4) + 4*(a^8)^(3/4)*a^8 + 2*(sqrt(2)*a^7*(sqrt(2) + 2)^(3/2) - 3*sqrt(2)*a^7*sqrt(sqrt(2) + 2) + (sqr
t(2)*a^7*(sqrt(2) + 2) - sqrt(2)*a^7)*sqrt(-sqrt(2) + 2))*(a^8)^(7/8)*((a*x - 1)/(a*x + 1))^(1/8))*(a^8)^(1/8)
)/(a^8*(sqrt(2) + 2)^(3/2) - 3*a^8*sqrt(sqrt(2) + 2) - (a^8*(sqrt(2) + 2) - a^8)*sqrt(-sqrt(2) + 2))) + 4*(a^8
)^(1/8)*(sqrt(2)*x*sqrt(sqrt(2) + 2) - sqrt(2)*x*sqrt(-sqrt(2) + 2))*arctan(-(a^8*(sqrt(2) + 2)^(3/2) - 3*a^8*
sqrt(sqrt(2) + 2) - 2*sqrt(2)*(a^8)^(1/8)*a^7*((a*x - 1)/(a*x + 1))^(1/8) + (a^8*(sqrt(2) + 2) - a^8)*sqrt(-sq
rt(2) + 2) + sqrt(2)*sqrt(4*a^14*((a*x - 1)/(a*x + 1))^(1/4) + 4*(a^8)^(3/4)*a^8 - 2*(sqrt(2)*a^7*(sqrt(2) + 2
)^(3/2) - 3*sqrt(2)*a^7*sqrt(sqrt(2) + 2) + (sqrt(2)*a^7*(sqrt(2) + 2) - sqrt(2)*a^7)*sqrt(-sqrt(2) + 2))*(a^8
)^(7/8)*((a*x - 1)/(a*x + 1))^(1/8))*(a^8)^(1/8))/(a^8*(sqrt(2) + 2)^(3/2) - 3*a^8*sqrt(sqrt(2) + 2) - (a^8*(s
qrt(2) + 2) - a^8)*sqrt(-sqrt(2) + 2))) + (a^8)^(1/8)*(sqrt(2)*x*sqrt(sqrt(2) + 2) + sqrt(2)*x*sqrt(-sqrt(2) +
2))*log(4*a^14*((a*x - 1)/(a*x + 1))^(1/4) + 4*(a^8)^(3/4)*a^8 + 2*(sqrt(2)*a^7*(sqrt(2) + 2)^(3/2) - 3*sqrt(
2)*a^7*sqrt(sqrt(2) + 2) + (sqrt(2)*a^7*(sqrt(2) + 2) - sqrt(2)*a^7)*sqrt(-sqrt(2) + 2))*(a^8)^(7/8)*((a*x - 1
)/(a*x + 1))^(1/8)) - (a^8)^(1/8)*(sqrt(2)*x*sqrt(sqrt(2) + 2) + sqrt(2)*x*sqrt(-sqrt(2) + 2))*log(4*a^14*((a*
x - 1)/(a*x + 1))^(1/4) + 4*(a^8)^(3/4)*a^8 - 2*(sqrt(2)*a^7*(sqrt(2) + 2)^(3/2) - 3*sqrt(2)*a^7*sqrt(sqrt(2)
+ 2) + (sqrt(2)*a^7*(sqrt(2) + 2) - sqrt(2)*a^7)*sqrt(-sqrt(2) + 2))*(a^8)^(7/8)*((a*x - 1)/(a*x + 1))^(1/8))
- (a^8)^(1/8)*(sqrt(2)*x*sqrt(sqrt(2) + 2) - sqrt(2)*x*sqrt(-sqrt(2) + 2))*log(4*a^14*((a*x - 1)/(a*x + 1))^(1
/4) + 4*(a^8)^(3/4)*a^8 + 2*(sqrt(2)*a^7*(sqrt(2) + 2)^(3/2) - 3*sqrt(2)*a^7*sqrt(sqrt(2) + 2) - (sqrt(2)*a^7*
(sqrt(2) + 2) - sqrt(2)*a^7)*sqrt(-sqrt(2) + 2))*(a^8)^(7/8)*((a*x - 1)/(a*x + 1))^(1/8)) + (a^8)^(1/8)*(sqrt(
2)*x*sqrt(sqrt(2) + 2) - sqrt(2)*x*sqrt(-sqrt(2) + 2))*log(4*a^14*((a*x - 1)/(a*x + 1))^(1/4) + 4*(a^8)^(3/4)*
a^8 - 2*(sqrt(2)*a^7*(sqrt(2) + 2)^(3/2) - 3*sqrt(2)*a^7*sqrt(sqrt(2) + 2) - (sqrt(2)*a^7*(sqrt(2) + 2) - sqrt
(2)*a^7)*sqrt(-sqrt(2) + 2))*(a^8)^(7/8)*((a*x - 1)/(a*x + 1))^(1/8)) - 32*(a*x + 1)*((a*x - 1)/(a*x + 1))^(7/
8))/x

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.2168, size = 583, normalized size = 0.86 \begin{align*} \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 \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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