### 3.127 $$\int e^{\frac{1}{4} \coth ^{-1}(a x)} x \, dx$$

Optimal. Leaf size=392 $-\frac{\log \left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}+1\right )}{32 \sqrt{2} a^2}+\frac{\log \left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}+\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}+1\right )}{32 \sqrt{2} a^2}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{16 \sqrt{2} a^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}+1\right )}{16 \sqrt{2} a^2}+\frac{\tan ^{-1}\left (\frac{\sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{16 a^2}+\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{16 a^2}+\frac{1}{2} x^2 \left (1-\frac{1}{a x}\right )^{7/8} \left (\frac{1}{a x}+1\right )^{9/8}+\frac{x \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{\frac{1}{a x}+1}}{8 a}$

[Out]

((1 - 1/(a*x))^(7/8)*(1 + 1/(a*x))^(1/8)*x)/(8*a) + ((1 - 1/(a*x))^(7/8)*(1 + 1/(a*x))^(9/8)*x^2)/2 - ArcTan[1
- (Sqrt[2]*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8)]/(16*Sqrt[2]*a^2) + ArcTan[1 + (Sqrt[2]*(1 + 1/(a*x))^(1/
8))/(1 - 1/(a*x))^(1/8)]/(16*Sqrt[2]*a^2) + ArcTan[(1 + 1/(a*x))^(1/8)/(1 - 1/(a*x))^(1/8)]/(16*a^2) + ArcTanh
[(1 + 1/(a*x))^(1/8)/(1 - 1/(a*x))^(1/8)]/(16*a^2) - 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)]/(32*Sqrt[2]*a^2) + 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)]/(32*Sqrt[2]*a^2)

________________________________________________________________________________________

Rubi [A]  time = 0.247184, antiderivative size = 392, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 14, integrand size = 12, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 1.167, Rules used = {6171, 96, 94, 93, 214, 212, 206, 203, 211, 1165, 628, 1162, 617, 204} $-\frac{\log \left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}+1\right )}{32 \sqrt{2} a^2}+\frac{\log \left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}+\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}+1\right )}{32 \sqrt{2} a^2}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{16 \sqrt{2} a^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}+1\right )}{16 \sqrt{2} a^2}+\frac{\tan ^{-1}\left (\frac{\sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{16 a^2}+\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{16 a^2}+\frac{1}{2} x^2 \left (1-\frac{1}{a x}\right )^{7/8} \left (\frac{1}{a x}+1\right )^{9/8}+\frac{x \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{\frac{1}{a x}+1}}{8 a}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((1 - 1/(a*x))^(7/8)*(1 + 1/(a*x))^(1/8)*x)/(8*a) + ((1 - 1/(a*x))^(7/8)*(1 + 1/(a*x))^(9/8)*x^2)/2 - ArcTan[1
- (Sqrt[2]*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8)]/(16*Sqrt[2]*a^2) + ArcTan[1 + (Sqrt[2]*(1 + 1/(a*x))^(1/
8))/(1 - 1/(a*x))^(1/8)]/(16*Sqrt[2]*a^2) + ArcTan[(1 + 1/(a*x))^(1/8)/(1 - 1/(a*x))^(1/8)]/(16*a^2) + ArcTanh
[(1 + 1/(a*x))^(1/8)/(1 - 1/(a*x))^(1/8)]/(16*a^2) - 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)]/(32*Sqrt[2]*a^2) + 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)]/(32*Sqrt[2]*a^2)

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 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(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[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

Rule 94

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[(n*(d*e - c*f))/((m + 1)*(b*e - a*
f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] &&  !(SumSimplerQ[p, 1] &&  !SumSimplerQ[m, 1])

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 \, dx &=-\operatorname{Subst}\left (\int \frac{\sqrt [8]{1+\frac{x}{a}}}{x^3 \sqrt [8]{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{2} \left (1-\frac{1}{a x}\right )^{7/8} \left (1+\frac{1}{a x}\right )^{9/8} x^2-\frac{\operatorname{Subst}\left (\int \frac{\sqrt [8]{1+\frac{x}{a}}}{x^2 \sqrt [8]{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{8 a}\\ &=\frac{\left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x}{8 a}+\frac{1}{2} \left (1-\frac{1}{a x}\right )^{7/8} \left (1+\frac{1}{a x}\right )^{9/8} x^2-\frac{\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 )}{32 a^2}\\ &=\frac{\left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x}{8 a}+\frac{1}{2} \left (1-\frac{1}{a x}\right )^{7/8} \left (1+\frac{1}{a x}\right )^{9/8} x^2-\frac{\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 )}{4 a^2}\\ &=\frac{\left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x}{8 a}+\frac{1}{2} \left (1-\frac{1}{a x}\right )^{7/8} \left (1+\frac{1}{a x}\right )^{9/8} x^2+\frac{\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 )}{8 a^2}+\frac{\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 )}{8 a^2}\\ &=\frac{\left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x}{8 a}+\frac{1}{2} \left (1-\frac{1}{a x}\right )^{7/8} \left (1+\frac{1}{a x}\right )^{9/8} x^2+\frac{\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 )}{16 a^2}+\frac{\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 )}{16 a^2}+\frac{\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 )}{16 a^2}+\frac{\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 )}{16 a^2}\\ &=\frac{\left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x}{8 a}+\frac{1}{2} \left (1-\frac{1}{a x}\right )^{7/8} \left (1+\frac{1}{a x}\right )^{9/8} x^2+\frac{\tan ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{16 a^2}+\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{16 a^2}+\frac{\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 )}{32 a^2}+\frac{\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 )}{32 a^2}-\frac{\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 )}{32 \sqrt{2} a^2}-\frac{\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 )}{32 \sqrt{2} a^2}\\ &=\frac{\left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x}{8 a}+\frac{1}{2} \left (1-\frac{1}{a x}\right )^{7/8} \left (1+\frac{1}{a x}\right )^{9/8} x^2+\frac{\tan ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{16 a^2}+\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{16 a^2}-\frac{\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 )}{32 \sqrt{2} a^2}+\frac{\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 )}{32 \sqrt{2} a^2}+\frac{\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 )}{16 \sqrt{2} a^2}-\frac{\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 )}{16 \sqrt{2} a^2}\\ &=\frac{\left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x}{8 a}+\frac{1}{2} \left (1-\frac{1}{a x}\right )^{7/8} \left (1+\frac{1}{a x}\right )^{9/8} x^2-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{16 \sqrt{2} a^2}+\frac{\tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{16 \sqrt{2} a^2}+\frac{\tan ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{16 a^2}+\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{16 a^2}-\frac{\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 )}{32 \sqrt{2} a^2}+\frac{\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 )}{32 \sqrt{2} a^2}\\ \end{align*}

Mathematica [A]  time = 0.724106, size = 319, normalized size = 0.81 $\frac{\frac{6}{e^{\frac{1}{4} \coth ^{-1}(a x)}-1}+\frac{6}{e^{\frac{1}{4} \coth ^{-1}(a x)}+1}-\frac{12 e^{\frac{1}{4} \coth ^{-1}(a x)}}{e^{\frac{1}{2} \coth ^{-1}(a x)}+1}-\frac{40 e^{\frac{1}{4} \coth ^{-1}(a x)}}{e^{\coth ^{-1}(a x)}+1}+\frac{2}{\left (e^{\frac{1}{4} \coth ^{-1}(a x)}-1\right )^2}-\frac{2}{\left (e^{\frac{1}{4} \coth ^{-1}(a x)}+1\right )^2}+\frac{8 e^{\frac{1}{4} \coth ^{-1}(a x)}}{\left (e^{\frac{1}{2} \coth ^{-1}(a x)}+1\right )^2}+\frac{32 e^{\frac{1}{4} \coth ^{-1}(a x)}}{\left (e^{\coth ^{-1}(a x)}+1\right )^2}-\sqrt{2} \log \left (-\sqrt{2} e^{\frac{1}{4} \coth ^{-1}(a x)}+e^{\frac{1}{2} \coth ^{-1}(a x)}+1\right )+\sqrt{2} \log \left (\sqrt{2} e^{\frac{1}{4} \coth ^{-1}(a x)}+e^{\frac{1}{2} \coth ^{-1}(a x)}+1\right )+4 \tan ^{-1}\left (e^{\frac{1}{4} \coth ^{-1}(a x)}\right )-2 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} e^{\frac{1}{4} \coth ^{-1}(a x)}\right )+2 \sqrt{2} \tan ^{-1}\left (\sqrt{2} e^{\frac{1}{4} \coth ^{-1}(a x)}+1\right )+4 \tanh ^{-1}\left (e^{\frac{1}{4} \coth ^{-1}(a x)}\right )}{64 a^2}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2/(-1 + E^(ArcCoth[a*x]/4))^2 + 6/(-1 + E^(ArcCoth[a*x]/4)) - 2/(1 + E^(ArcCoth[a*x]/4))^2 + 6/(1 + E^(ArcCot
h[a*x]/4)) + (8*E^(ArcCoth[a*x]/4))/(1 + E^(ArcCoth[a*x]/2))^2 - (12*E^(ArcCoth[a*x]/4))/(1 + E^(ArcCoth[a*x]/
2)) + (32*E^(ArcCoth[a*x]/4))/(1 + E^ArcCoth[a*x])^2 - (40*E^(ArcCoth[a*x]/4))/(1 + E^ArcCoth[a*x]) + 4*ArcTan
[E^(ArcCoth[a*x]/4)] - 2*Sqrt[2]*ArcTan[1 - Sqrt[2]*E^(ArcCoth[a*x]/4)] + 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*E^(ArcC
oth[a*x]/4)] + 4*ArcTanh[E^(ArcCoth[a*x]/4)] - Sqrt[2]*Log[1 - Sqrt[2]*E^(ArcCoth[a*x]/4) + E^(ArcCoth[a*x]/2)
] + Sqrt[2]*Log[1 + Sqrt[2]*E^(ArcCoth[a*x]/4) + E^(ArcCoth[a*x]/2)])/(64*a^2)

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Maple [F]  time = 0.139, size = 0, normalized size = 0. \begin{align*} \int{x{\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,x)

[Out]

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

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Maxima [A]  time = 1.55323, size = 410, normalized size = 1.05 \begin{align*} \frac{1}{64} \, a{\left (\frac{16 \,{\left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{15}{8}} - 9 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{8}}\right )}}{\frac{2 \,{\left (a x - 1\right )} a^{3}}{a x + 1} - \frac{{\left (a x - 1\right )}^{2} a^{3}}{{\left (a x + 1\right )}^{2}} - a^{3}} - \frac{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 )}{a^{3}} - \frac{4 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}\right )}{a^{3}} + \frac{2 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} + 1\right )}{a^{3}} - \frac{2 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} - 1\right )}{a^{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,x, algorithm="maxima")

[Out]

1/64*a*(16*(((a*x - 1)/(a*x + 1))^(15/8) - 9*((a*x - 1)/(a*x + 1))^(7/8))/(2*(a*x - 1)*a^3/(a*x + 1) - (a*x -
1)^2*a^3/(a*x + 1)^2 - a^3) - (2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*((a*x - 1)/(a*x + 1))^(1/8))) + 2*sqr
t(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^3 - 4*arctan(((a*x - 1)/(a*x + 1))^(1/8))/a^3 + 2*log(((a*x - 1)/(a*x + 1))^(1/8) + 1)/a
^3 - 2*log(((a*x - 1)/(a*x + 1))^(1/8) - 1)/a^3)

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Fricas [A]  time = 1.73388, size = 1269, normalized size = 3.24 \begin{align*} \frac{4 \, \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \arctan \left (\sqrt{2} \sqrt{\sqrt{2} a^{6} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} \frac{1}{a^{8}}^{\frac{3}{4}} + a^{4} \sqrt{\frac{1}{a^{8}}} + \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} - \sqrt{2} a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} \frac{1}{a^{8}}^{\frac{1}{4}} - 1\right ) + 4 \, \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \arctan \left (\sqrt{2} \sqrt{-\sqrt{2} a^{6} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} \frac{1}{a^{8}}^{\frac{3}{4}} + a^{4} \sqrt{\frac{1}{a^{8}}} + \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} - \sqrt{2} a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} \frac{1}{a^{8}}^{\frac{1}{4}} + 1\right ) + \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \log \left (\sqrt{2} a^{6} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} \frac{1}{a^{8}}^{\frac{3}{4}} + a^{4} \sqrt{\frac{1}{a^{8}}} + \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right ) - \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \log \left (-\sqrt{2} a^{6} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} \frac{1}{a^{8}}^{\frac{3}{4}} + a^{4} \sqrt{\frac{1}{a^{8}}} + \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right ) + 8 \,{\left (4 \, a^{2} x^{2} + 9 \, a x + 5\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{8}} - 4 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}\right ) + 2 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} + 1\right ) - 2 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} - 1\right )}{64 \, a^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(4*sqrt(2)*a^2*(a^(-8))^(1/4)*arctan(sqrt(2)*sqrt(sqrt(2)*a^6*((a*x - 1)/(a*x + 1))^(1/8)*(a^(-8))^(3/4)
+ a^4*sqrt(a^(-8)) + ((a*x - 1)/(a*x + 1))^(1/4))*a^2*(a^(-8))^(1/4) - sqrt(2)*a^2*((a*x - 1)/(a*x + 1))^(1/8)
*(a^(-8))^(1/4) - 1) + 4*sqrt(2)*a^2*(a^(-8))^(1/4)*arctan(sqrt(2)*sqrt(-sqrt(2)*a^6*((a*x - 1)/(a*x + 1))^(1/
8)*(a^(-8))^(3/4) + a^4*sqrt(a^(-8)) + ((a*x - 1)/(a*x + 1))^(1/4))*a^2*(a^(-8))^(1/4) - sqrt(2)*a^2*((a*x - 1
)/(a*x + 1))^(1/8)*(a^(-8))^(1/4) + 1) + sqrt(2)*a^2*(a^(-8))^(1/4)*log(sqrt(2)*a^6*((a*x - 1)/(a*x + 1))^(1/8
)*(a^(-8))^(3/4) + a^4*sqrt(a^(-8)) + ((a*x - 1)/(a*x + 1))^(1/4)) - sqrt(2)*a^2*(a^(-8))^(1/4)*log(-sqrt(2)*a
^6*((a*x - 1)/(a*x + 1))^(1/8)*(a^(-8))^(3/4) + a^4*sqrt(a^(-8)) + ((a*x - 1)/(a*x + 1))^(1/4)) + 8*(4*a^2*x^2
+ 9*a*x + 5)*((a*x - 1)/(a*x + 1))^(7/8) - 4*arctan(((a*x - 1)/(a*x + 1))^(1/8)) + 2*log(((a*x - 1)/(a*x + 1)
)^(1/8) + 1) - 2*log(((a*x - 1)/(a*x + 1))^(1/8) - 1))/a^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt [8]{\frac{a x - 1}{a x + 1}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.20437, size = 405, normalized size = 1.03 \begin{align*} -\frac{1}{64} \, a{\left (\frac{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 )}{a^{3}} + \frac{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 )}{a^{3}} - \frac{\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^{3}} + \frac{\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^{3}} + \frac{4 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}\right )}{a^{3}} - \frac{2 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} + 1\right )}{a^{3}} + \frac{2 \, \log \left ({\left | \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} - 1 \right |}\right )}{a^{3}} + \frac{16 \,{\left (\frac{{\left (a x - 1\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{8}}}{a x + 1} - 9 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{8}}\right )}}{a^{3}{\left (\frac{a x - 1}{a x + 1} - 1\right )}^{2}}\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,x, algorithm="giac")

[Out]

-1/64*a*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*((a*x - 1)/(a*x + 1))^(1/8)))/a^3 + 2*sqrt(2)*arctan(-1/2*s
qrt(2)*(sqrt(2) - 2*((a*x - 1)/(a*x + 1))^(1/8)))/a^3 - sqrt(2)*log(sqrt(2)*((a*x - 1)/(a*x + 1))^(1/8) + ((a*
x - 1)/(a*x + 1))^(1/4) + 1)/a^3 + sqrt(2)*log(-sqrt(2)*((a*x - 1)/(a*x + 1))^(1/8) + ((a*x - 1)/(a*x + 1))^(1
/4) + 1)/a^3 + 4*arctan(((a*x - 1)/(a*x + 1))^(1/8))/a^3 - 2*log(((a*x - 1)/(a*x + 1))^(1/8) + 1)/a^3 + 2*log(
abs(((a*x - 1)/(a*x + 1))^(1/8) - 1))/a^3 + 16*((a*x - 1)*((a*x - 1)/(a*x + 1))^(7/8)/(a*x + 1) - 9*((a*x - 1)
/(a*x + 1))^(7/8))/(a^3*((a*x - 1)/(a*x + 1) - 1)^2))