### 3.126 $$\int e^{\frac{1}{4} \coth ^{-1}(a x)} x^2 \, dx$$

Optimal. Leaf size=429 $\frac{37 x \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{\frac{1}{a x}+1}}{96 a^2}-\frac{11 \log \left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}+1\right )}{128 \sqrt{2} a^3}+\frac{11 \log \left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}+\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}+1\right )}{128 \sqrt{2} a^3}-\frac{11 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 \sqrt{2} a^3}+\frac{11 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}+1\right )}{64 \sqrt{2} a^3}+\frac{11 \tan ^{-1}\left (\frac{\sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 a^3}+\frac{11 \tanh ^{-1}\left (\frac{\sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 a^3}+\frac{1}{3} x^3 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{\frac{1}{a x}+1}+\frac{3 x^2 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{\frac{1}{a x}+1}}{8 a}$

[Out]

(37*(1 - 1/(a*x))^(7/8)*(1 + 1/(a*x))^(1/8)*x)/(96*a^2) + (3*(1 - 1/(a*x))^(7/8)*(1 + 1/(a*x))^(1/8)*x^2)/(8*a
) + ((1 - 1/(a*x))^(7/8)*(1 + 1/(a*x))^(1/8)*x^3)/3 - (11*ArcTan[1 - (Sqrt[2]*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x
))^(1/8)])/(64*Sqrt[2]*a^3) + (11*ArcTan[1 + (Sqrt[2]*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8)])/(64*Sqrt[2]*a
^3) + (11*ArcTan[(1 + 1/(a*x))^(1/8)/(1 - 1/(a*x))^(1/8)])/(64*a^3) + (11*ArcTanh[(1 + 1/(a*x))^(1/8)/(1 - 1/(
a*x))^(1/8)])/(64*a^3) - (11*Log[1 - (Sqrt[2]*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8) + (1 + 1/(a*x))^(1/4)/(
1 - 1/(a*x))^(1/4)])/(128*Sqrt[2]*a^3) + (11*Log[1 + (Sqrt[2]*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8) + (1 +
1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(128*Sqrt[2]*a^3)

________________________________________________________________________________________

Rubi [A]  time = 0.343278, antiderivative size = 429, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 15, integrand size = 14, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 1.071, Rules used = {6171, 99, 151, 12, 93, 214, 212, 206, 203, 211, 1165, 628, 1162, 617, 204} $\frac{37 x \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{\frac{1}{a x}+1}}{96 a^2}-\frac{11 \log \left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}+1\right )}{128 \sqrt{2} a^3}+\frac{11 \log \left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}+\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}+1\right )}{128 \sqrt{2} a^3}-\frac{11 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 \sqrt{2} a^3}+\frac{11 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}+1\right )}{64 \sqrt{2} a^3}+\frac{11 \tan ^{-1}\left (\frac{\sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 a^3}+\frac{11 \tanh ^{-1}\left (\frac{\sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 a^3}+\frac{1}{3} x^3 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{\frac{1}{a x}+1}+\frac{3 x^2 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{\frac{1}{a x}+1}}{8 a}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(37*(1 - 1/(a*x))^(7/8)*(1 + 1/(a*x))^(1/8)*x)/(96*a^2) + (3*(1 - 1/(a*x))^(7/8)*(1 + 1/(a*x))^(1/8)*x^2)/(8*a
) + ((1 - 1/(a*x))^(7/8)*(1 + 1/(a*x))^(1/8)*x^3)/3 - (11*ArcTan[1 - (Sqrt[2]*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x
))^(1/8)])/(64*Sqrt[2]*a^3) + (11*ArcTan[1 + (Sqrt[2]*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8)])/(64*Sqrt[2]*a
^3) + (11*ArcTan[(1 + 1/(a*x))^(1/8)/(1 - 1/(a*x))^(1/8)])/(64*a^3) + (11*ArcTanh[(1 + 1/(a*x))^(1/8)/(1 - 1/(
a*x))^(1/8)])/(64*a^3) - (11*Log[1 - (Sqrt[2]*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8) + (1 + 1/(a*x))^(1/4)/(
1 - 1/(a*x))^(1/4)])/(128*Sqrt[2]*a^3) + (11*Log[1 + (Sqrt[2]*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8) + (1 +
1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(128*Sqrt[2]*a^3)

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 99

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

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 214

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

Rule 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 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 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

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 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
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])

Rubi steps

\begin{align*} \int e^{\frac{1}{4} \coth ^{-1}(a x)} x^2 \, dx &=-\operatorname{Subst}\left (\int \frac{\sqrt [8]{1+\frac{x}{a}}}{x^4 \sqrt [8]{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{3} \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^3-\frac{1}{3} \operatorname{Subst}\left (\int \frac{\frac{9}{4 a}+\frac{2 x}{a^2}}{x^3 \sqrt [8]{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{7/8}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{3 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^2}{8 a}+\frac{1}{3} \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^3+\frac{1}{6} \operatorname{Subst}\left (\int \frac{-\frac{37}{16 a^2}-\frac{9 x}{4 a^3}}{x^2 \sqrt [8]{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{7/8}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{37 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x}{96 a^2}+\frac{3 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^2}{8 a}+\frac{1}{3} \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^3-\frac{1}{6} \operatorname{Subst}\left (\int \frac{33}{64 a^3 x \sqrt [8]{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{7/8}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{37 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x}{96 a^2}+\frac{3 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^2}{8 a}+\frac{1}{3} \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^3-\frac{11 \operatorname{Subst}\left (\int \frac{1}{x \sqrt [8]{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{7/8}} \, dx,x,\frac{1}{x}\right )}{128 a^3}\\ &=\frac{37 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x}{96 a^2}+\frac{3 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^2}{8 a}+\frac{1}{3} \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^3-\frac{11 \operatorname{Subst}\left (\int \frac{1}{-1+x^8} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{16 a^3}\\ &=\frac{37 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x}{96 a^2}+\frac{3 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^2}{8 a}+\frac{1}{3} \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^3+\frac{11 \operatorname{Subst}\left (\int \frac{1}{1-x^4} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{32 a^3}+\frac{11 \operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{32 a^3}\\ &=\frac{37 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x}{96 a^2}+\frac{3 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^2}{8 a}+\frac{1}{3} \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^3+\frac{11 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 a^3}+\frac{11 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 a^3}+\frac{11 \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 a^3}+\frac{11 \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 a^3}\\ &=\frac{37 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x}{96 a^2}+\frac{3 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^2}{8 a}+\frac{1}{3} \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^3+\frac{11 \tan ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 a^3}+\frac{11 \tanh ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 a^3}+\frac{11 \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{128 a^3}+\frac{11 \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{128 a^3}-\frac{11 \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{128 \sqrt{2} a^3}-\frac{11 \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{128 \sqrt{2} a^3}\\ &=\frac{37 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x}{96 a^2}+\frac{3 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^2}{8 a}+\frac{1}{3} \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^3+\frac{11 \tan ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 a^3}+\frac{11 \tanh ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 a^3}-\frac{11 \log \left (1-\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}+\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 \sqrt{2} a^3}+\frac{11 \log \left (1+\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}+\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 \sqrt{2} a^3}+\frac{11 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 \sqrt{2} a^3}-\frac{11 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 \sqrt{2} a^3}\\ &=\frac{37 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x}{96 a^2}+\frac{3 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^2}{8 a}+\frac{1}{3} \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x^3-\frac{11 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 \sqrt{2} a^3}+\frac{11 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 \sqrt{2} a^3}+\frac{11 \tan ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 a^3}+\frac{11 \tanh ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{64 a^3}-\frac{11 \log \left (1-\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}+\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 \sqrt{2} a^3}+\frac{11 \log \left (1+\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}+\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 \sqrt{2} a^3}\\ \end{align*}

Mathematica [C]  time = 5.30153, size = 167, normalized size = 0.39 $\frac{-33 \text{RootSum}\left [\text{\#1}^4+1\& ,\frac{\coth ^{-1}(a x)-4 \log \left (e^{\frac{1}{4} \coth ^{-1}(a x)}-\text{\#1}\right )}{\text{\#1}^3}\& \right ]-4 \left (-\frac{840 e^{\frac{1}{4} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}-1}-\frac{1600 e^{\frac{1}{4} \coth ^{-1}(a x)}}{\left (e^{2 \coth ^{-1}(a x)}-1\right )^2}-\frac{1024 e^{\frac{1}{4} \coth ^{-1}(a x)}}{\left (e^{2 \coth ^{-1}(a x)}-1\right )^3}+33 \log \left (1-e^{\frac{1}{4} \coth ^{-1}(a x)}\right )-33 \log \left (e^{\frac{1}{4} \coth ^{-1}(a x)}+1\right )-66 \tan ^{-1}\left (e^{\frac{1}{4} \coth ^{-1}(a x)}\right )\right )}{1536 a^3}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-4*((-1024*E^(ArcCoth[a*x]/4))/(-1 + E^(2*ArcCoth[a*x]))^3 - (1600*E^(ArcCoth[a*x]/4))/(-1 + E^(2*ArcCoth[a*x
]))^2 - (840*E^(ArcCoth[a*x]/4))/(-1 + E^(2*ArcCoth[a*x])) - 66*ArcTan[E^(ArcCoth[a*x]/4)] + 33*Log[1 - E^(Arc
Coth[a*x]/4)] - 33*Log[1 + E^(ArcCoth[a*x]/4)]) - 33*RootSum[1 + #1^4 & , (ArcCoth[a*x] - 4*Log[E^(ArcCoth[a*x
]/4) - #1])/#1^3 & ])/(1536*a^3)

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Maple [F]  time = 0.168, size = 0, normalized size = 0. \begin{align*} \int{{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 [A]  time = 1.55936, size = 460, normalized size = 1.07 \begin{align*} -\frac{1}{768} \, a{\left (\frac{16 \,{\left (33 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{23}{8}} - 10 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{15}{8}} + 105 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{8}}\right )}}{\frac{3 \,{\left (a x - 1\right )} a^{4}}{a x + 1} - \frac{3 \,{\left (a x - 1\right )}^{2} a^{4}}{{\left (a x + 1\right )}^{2}} + \frac{{\left (a x - 1\right )}^{3} a^{4}}{{\left (a x + 1\right )}^{3}} - a^{4}} + \frac{33 \,{\left (2 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}\right )}\right ) + 2 \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}\right )}\right ) - \sqrt{2} \log \left (\sqrt{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{2} \log \left (-\sqrt{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 )\right )}}{a^{4}} + \frac{132 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}\right )}{a^{4}} - \frac{66 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} + 1\right )}{a^{4}} + \frac{66 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} - 1\right )}{a^{4}}\right )} \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]

-1/768*a*(16*(33*((a*x - 1)/(a*x + 1))^(23/8) - 10*((a*x - 1)/(a*x + 1))^(15/8) + 105*((a*x - 1)/(a*x + 1))^(7
/8))/(3*(a*x - 1)*a^4/(a*x + 1) - 3*(a*x - 1)^2*a^4/(a*x + 1)^2 + (a*x - 1)^3*a^4/(a*x + 1)^3 - a^4) + 33*(2*s
qrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*((a*x - 1)/(a*x + 1))^(1/8))) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)
- 2*((a*x - 1)/(a*x + 1))^(1/8))) - sqrt(2)*log(sqrt(2)*((a*x - 1)/(a*x + 1))^(1/8) + ((a*x - 1)/(a*x + 1))^(1
/4) + 1) + sqrt(2)*log(-sqrt(2)*((a*x - 1)/(a*x + 1))^(1/8) + ((a*x - 1)/(a*x + 1))^(1/4) + 1))/a^4 + 132*arct
an(((a*x - 1)/(a*x + 1))^(1/8))/a^4 - 66*log(((a*x - 1)/(a*x + 1))^(1/8) + 1)/a^4 + 66*log(((a*x - 1)/(a*x + 1
))^(1/8) - 1)/a^4)

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Fricas [A]  time = 1.91345, size = 1332, normalized size = 3.1 \begin{align*} \frac{132 \, \sqrt{2} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} \arctan \left (\sqrt{2} \sqrt{\sqrt{2} a^{9} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} \frac{1}{a^{12}}^{\frac{3}{4}} + a^{6} \sqrt{\frac{1}{a^{12}}} + \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} - \sqrt{2} a^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} \frac{1}{a^{12}}^{\frac{1}{4}} - 1\right ) + 132 \, \sqrt{2} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} \arctan \left (\sqrt{2} \sqrt{-\sqrt{2} a^{9} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} \frac{1}{a^{12}}^{\frac{3}{4}} + a^{6} \sqrt{\frac{1}{a^{12}}} + \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} - \sqrt{2} a^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} \frac{1}{a^{12}}^{\frac{1}{4}} + 1\right ) + 33 \, \sqrt{2} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} \log \left (\sqrt{2} a^{9} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} \frac{1}{a^{12}}^{\frac{3}{4}} + a^{6} \sqrt{\frac{1}{a^{12}}} + \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right ) - 33 \, \sqrt{2} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} \log \left (-\sqrt{2} a^{9} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} \frac{1}{a^{12}}^{\frac{3}{4}} + a^{6} \sqrt{\frac{1}{a^{12}}} + \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right ) + 8 \,{\left (32 \, a^{3} x^{3} + 68 \, a^{2} x^{2} + 73 \, a x + 37\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{8}} - 132 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}\right ) + 66 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} + 1\right ) - 66 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} - 1\right )}{768 \, a^{3}} \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]

1/768*(132*sqrt(2)*a^3*(a^(-12))^(1/4)*arctan(sqrt(2)*sqrt(sqrt(2)*a^9*((a*x - 1)/(a*x + 1))^(1/8)*(a^(-12))^(
3/4) + a^6*sqrt(a^(-12)) + ((a*x - 1)/(a*x + 1))^(1/4))*a^3*(a^(-12))^(1/4) - sqrt(2)*a^3*((a*x - 1)/(a*x + 1)
)^(1/8)*(a^(-12))^(1/4) - 1) + 132*sqrt(2)*a^3*(a^(-12))^(1/4)*arctan(sqrt(2)*sqrt(-sqrt(2)*a^9*((a*x - 1)/(a*
x + 1))^(1/8)*(a^(-12))^(3/4) + a^6*sqrt(a^(-12)) + ((a*x - 1)/(a*x + 1))^(1/4))*a^3*(a^(-12))^(1/4) - sqrt(2)
*a^3*((a*x - 1)/(a*x + 1))^(1/8)*(a^(-12))^(1/4) + 1) + 33*sqrt(2)*a^3*(a^(-12))^(1/4)*log(sqrt(2)*a^9*((a*x -
1)/(a*x + 1))^(1/8)*(a^(-12))^(3/4) + a^6*sqrt(a^(-12)) + ((a*x - 1)/(a*x + 1))^(1/4)) - 33*sqrt(2)*a^3*(a^(-
12))^(1/4)*log(-sqrt(2)*a^9*((a*x - 1)/(a*x + 1))^(1/8)*(a^(-12))^(3/4) + a^6*sqrt(a^(-12)) + ((a*x - 1)/(a*x
+ 1))^(1/4)) + 8*(32*a^3*x^3 + 68*a^2*x^2 + 73*a*x + 37)*((a*x - 1)/(a*x + 1))^(7/8) - 132*arctan(((a*x - 1)/(
a*x + 1))^(1/8)) + 66*log(((a*x - 1)/(a*x + 1))^(1/8) + 1) - 66*log(((a*x - 1)/(a*x + 1))^(1/8) - 1))/a^3

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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 = 1.22453, size = 450, normalized size = 1.05 \begin{align*} -\frac{1}{768} \, a{\left (\frac{66 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}\right )}\right )}{a^{4}} + \frac{66 \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}\right )}\right )}{a^{4}} - \frac{33 \, \sqrt{2} \log \left (\sqrt{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 )}{a^{4}} + \frac{33 \, \sqrt{2} \log \left (-\sqrt{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 )}{a^{4}} + \frac{132 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}\right )}{a^{4}} - \frac{66 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} + 1\right )}{a^{4}} + \frac{66 \, \log \left ({\left | \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} - 1 \right |}\right )}{a^{4}} - \frac{16 \,{\left (\frac{10 \,{\left (a x - 1\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{8}}}{a x + 1} - \frac{33 \,{\left (a x - 1\right )}^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{8}}}{{\left (a x + 1\right )}^{2}} - 105 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{8}}\right )}}{a^{4}{\left (\frac{a x - 1}{a x + 1} - 1\right )}^{3}}\right )} \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]

-1/768*a*(66*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*((a*x - 1)/(a*x + 1))^(1/8)))/a^4 + 66*sqrt(2)*arctan(-1/
2*sqrt(2)*(sqrt(2) - 2*((a*x - 1)/(a*x + 1))^(1/8)))/a^4 - 33*sqrt(2)*log(sqrt(2)*((a*x - 1)/(a*x + 1))^(1/8)
+ ((a*x - 1)/(a*x + 1))^(1/4) + 1)/a^4 + 33*sqrt(2)*log(-sqrt(2)*((a*x - 1)/(a*x + 1))^(1/8) + ((a*x - 1)/(a*x
+ 1))^(1/4) + 1)/a^4 + 132*arctan(((a*x - 1)/(a*x + 1))^(1/8))/a^4 - 66*log(((a*x - 1)/(a*x + 1))^(1/8) + 1)/
a^4 + 66*log(abs(((a*x - 1)/(a*x + 1))^(1/8) - 1))/a^4 - 16*(10*(a*x - 1)*((a*x - 1)/(a*x + 1))^(7/8)/(a*x + 1
) - 33*(a*x - 1)^2*((a*x - 1)/(a*x + 1))^(7/8)/(a*x + 1)^2 - 105*((a*x - 1)/(a*x + 1))^(7/8))/(a^4*((a*x - 1)/
(a*x + 1) - 1)^3))