### 3.112 $$\int \frac{e^{-\frac{5}{2} \coth ^{-1}(a x)}}{x^4} \, dx$$

Optimal. Leaf size=385 $\frac{1}{3} a^3 \left (1-\frac{1}{a x}\right )^{9/4} \left (\frac{1}{a x}+1\right )^{3/4}+\frac{11}{4} a^3 \left (1-\frac{1}{a x}\right )^{5/4} \left (\frac{1}{a x}+1\right )^{3/4}+\frac{55}{8} a^3 \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}+\frac{2 a^3 \left (1-\frac{1}{a x}\right )^{9/4}}{\sqrt [4]{\frac{1}{a x}+1}}+\frac{55 a^3 \log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{16 \sqrt{2}}-\frac{55 a^3 \log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{16 \sqrt{2}}+\frac{55 a^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}\right )}{8 \sqrt{2}}-\frac{55 a^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{8 \sqrt{2}}$

[Out]

(2*a^3*(1 - 1/(a*x))^(9/4))/(1 + 1/(a*x))^(1/4) + (55*a^3*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4))/8 + (11*a^3
*(1 - 1/(a*x))^(5/4)*(1 + 1/(a*x))^(3/4))/4 + (a^3*(1 - 1/(a*x))^(9/4)*(1 + 1/(a*x))^(3/4))/3 + (55*a^3*ArcTan
[1 - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(8*Sqrt[2]) - (55*a^3*ArcTan[1 + (Sqrt[2]*(1 - 1/(a*x
))^(1/4))/(1 + 1/(a*x))^(1/4)])/(8*Sqrt[2]) + (55*a^3*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] - (Sqrt[2]*(
1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(16*Sqrt[2]) - (55*a^3*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] +
(Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(16*Sqrt[2])

________________________________________________________________________________________

Rubi [A]  time = 0.309019, antiderivative size = 385, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 14, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.857, Rules used = {6171, 89, 80, 50, 63, 240, 211, 1165, 628, 1162, 617, 204} $\frac{1}{3} a^3 \left (1-\frac{1}{a x}\right )^{9/4} \left (\frac{1}{a x}+1\right )^{3/4}+\frac{11}{4} a^3 \left (1-\frac{1}{a x}\right )^{5/4} \left (\frac{1}{a x}+1\right )^{3/4}+\frac{55}{8} a^3 \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}+\frac{2 a^3 \left (1-\frac{1}{a x}\right )^{9/4}}{\sqrt [4]{\frac{1}{a x}+1}}+\frac{55 a^3 \log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{16 \sqrt{2}}-\frac{55 a^3 \log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{16 \sqrt{2}}+\frac{55 a^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}\right )}{8 \sqrt{2}}-\frac{55 a^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{8 \sqrt{2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/(E^((5*ArcCoth[a*x])/2)*x^4),x]

[Out]

(2*a^3*(1 - 1/(a*x))^(9/4))/(1 + 1/(a*x))^(1/4) + (55*a^3*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4))/8 + (11*a^3
*(1 - 1/(a*x))^(5/4)*(1 + 1/(a*x))^(3/4))/4 + (a^3*(1 - 1/(a*x))^(9/4)*(1 + 1/(a*x))^(3/4))/3 + (55*a^3*ArcTan
[1 - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(8*Sqrt[2]) - (55*a^3*ArcTan[1 + (Sqrt[2]*(1 - 1/(a*x
))^(1/4))/(1 + 1/(a*x))^(1/4)])/(8*Sqrt[2]) + (55*a^3*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] - (Sqrt[2]*(
1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(16*Sqrt[2]) - (55*a^3*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] +
(Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(16*Sqrt[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 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 80

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

Rule 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 240

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 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)] && IntegerQ[p
+ 1/n]

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 \frac{e^{-\frac{5}{2} \coth ^{-1}(a x)}}{x^4} \, dx &=-\operatorname{Subst}\left (\int \frac{x^2 \left (1-\frac{x}{a}\right )^{5/4}}{\left (1+\frac{x}{a}\right )^{5/4}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{2 a^3 \left (1-\frac{1}{a x}\right )^{9/4}}{\sqrt [4]{1+\frac{1}{a x}}}-\left (2 a^3\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{5}{2 a}+\frac{x}{2 a^2}\right ) \left (1-\frac{x}{a}\right )^{5/4}}{\sqrt [4]{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{2 a^3 \left (1-\frac{1}{a x}\right )^{9/4}}{\sqrt [4]{1+\frac{1}{a x}}}+\frac{1}{3} a^3 \left (1-\frac{1}{a x}\right )^{9/4} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{1}{2} \left (11 a^2\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{5/4}}{\sqrt [4]{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{2 a^3 \left (1-\frac{1}{a x}\right )^{9/4}}{\sqrt [4]{1+\frac{1}{a x}}}+\frac{11}{4} a^3 \left (1-\frac{1}{a x}\right )^{5/4} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{1}{3} a^3 \left (1-\frac{1}{a x}\right )^{9/4} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{1}{8} \left (55 a^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt [4]{1-\frac{x}{a}}}{\sqrt [4]{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{2 a^3 \left (1-\frac{1}{a x}\right )^{9/4}}{\sqrt [4]{1+\frac{1}{a x}}}+\frac{55}{8} a^3 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{11}{4} a^3 \left (1-\frac{1}{a x}\right )^{5/4} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{1}{3} a^3 \left (1-\frac{1}{a x}\right )^{9/4} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{1}{16} \left (55 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{x}{a}\right )^{3/4} \sqrt [4]{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{2 a^3 \left (1-\frac{1}{a x}\right )^{9/4}}{\sqrt [4]{1+\frac{1}{a x}}}+\frac{55}{8} a^3 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{11}{4} a^3 \left (1-\frac{1}{a x}\right )^{5/4} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{1}{3} a^3 \left (1-\frac{1}{a x}\right )^{9/4} \left (1+\frac{1}{a x}\right )^{3/4}-\frac{1}{4} \left (55 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{2-x^4}} \, dx,x,\sqrt [4]{1-\frac{1}{a x}}\right )\\ &=\frac{2 a^3 \left (1-\frac{1}{a x}\right )^{9/4}}{\sqrt [4]{1+\frac{1}{a x}}}+\frac{55}{8} a^3 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{11}{4} a^3 \left (1-\frac{1}{a x}\right )^{5/4} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{1}{3} a^3 \left (1-\frac{1}{a x}\right )^{9/4} \left (1+\frac{1}{a x}\right )^{3/4}-\frac{1}{4} \left (55 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}\right )\\ &=\frac{2 a^3 \left (1-\frac{1}{a x}\right )^{9/4}}{\sqrt [4]{1+\frac{1}{a x}}}+\frac{55}{8} a^3 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{11}{4} a^3 \left (1-\frac{1}{a x}\right )^{5/4} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{1}{3} a^3 \left (1-\frac{1}{a x}\right )^{9/4} \left (1+\frac{1}{a x}\right )^{3/4}-\frac{1}{8} \left (55 a^3\right ) \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}\right )-\frac{1}{8} \left (55 a^3\right ) \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}\right )\\ &=\frac{2 a^3 \left (1-\frac{1}{a x}\right )^{9/4}}{\sqrt [4]{1+\frac{1}{a x}}}+\frac{55}{8} a^3 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{11}{4} a^3 \left (1-\frac{1}{a x}\right )^{5/4} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{1}{3} a^3 \left (1-\frac{1}{a x}\right )^{9/4} \left (1+\frac{1}{a x}\right )^{3/4}-\frac{1}{16} \left (55 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}\right )-\frac{1}{16} \left (55 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}\right )+\frac{\left (55 a^3\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}\right )}{16 \sqrt{2}}+\frac{\left (55 a^3\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}\right )}{16 \sqrt{2}}\\ &=\frac{2 a^3 \left (1-\frac{1}{a x}\right )^{9/4}}{\sqrt [4]{1+\frac{1}{a x}}}+\frac{55}{8} a^3 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{11}{4} a^3 \left (1-\frac{1}{a x}\right )^{5/4} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{1}{3} a^3 \left (1-\frac{1}{a x}\right )^{9/4} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{55 a^3 \log \left (1+\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{1+\frac{1}{a x}}}-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}\right )}{16 \sqrt{2}}-\frac{55 a^3 \log \left (1+\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{1+\frac{1}{a x}}}+\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}\right )}{16 \sqrt{2}}-\frac{\left (55 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}\right )}{8 \sqrt{2}}+\frac{\left (55 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}\right )}{8 \sqrt{2}}\\ &=\frac{2 a^3 \left (1-\frac{1}{a x}\right )^{9/4}}{\sqrt [4]{1+\frac{1}{a x}}}+\frac{55}{8} a^3 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{11}{4} a^3 \left (1-\frac{1}{a x}\right )^{5/4} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{1}{3} a^3 \left (1-\frac{1}{a x}\right )^{9/4} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{55 a^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}\right )}{8 \sqrt{2}}-\frac{55 a^3 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}\right )}{8 \sqrt{2}}+\frac{55 a^3 \log \left (1+\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{1+\frac{1}{a x}}}-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}\right )}{16 \sqrt{2}}-\frac{55 a^3 \log \left (1+\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{1+\frac{1}{a x}}}+\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}\right )}{16 \sqrt{2}}\\ \end{align*}

Mathematica [C]  time = 0.166763, size = 104, normalized size = 0.27 $a^3 \left (\frac{e^{-\frac{1}{2} \coth ^{-1}(a x)} \left (425 e^{2 \coth ^{-1}(a x)}+462 e^{4 \coth ^{-1}(a x)}+165 e^{6 \coth ^{-1}(a x)}+96\right )}{12 \left (e^{2 \coth ^{-1}(a x)}+1\right )^3}-\frac{55}{32} \text{RootSum}\left [\text{\#1}^4+1\& ,\frac{2 \log \left (e^{-\frac{1}{2} \coth ^{-1}(a x)}-\text{\#1}\right )+\coth ^{-1}(a x)}{\text{\#1}^3}\& \right ]\right )$

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(E^((5*ArcCoth[a*x])/2)*x^4),x]

[Out]

a^3*((96 + 425*E^(2*ArcCoth[a*x]) + 462*E^(4*ArcCoth[a*x]) + 165*E^(6*ArcCoth[a*x]))/(12*E^(ArcCoth[a*x]/2)*(1
+ E^(2*ArcCoth[a*x]))^3) - (55*RootSum[1 + #1^4 & , (ArcCoth[a*x] + 2*Log[E^(-ArcCoth[a*x]/2) - #1])/#1^3 & ]
)/32)

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Maple [F]  time = 0.335, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{5}{4}}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(((a*x-1)/(a*x+1))^(5/4)/x^4,x)

[Out]

int(((a*x-1)/(a*x+1))^(5/4)/x^4,x)

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Maxima [A]  time = 1.52914, size = 401, normalized size = 1.04 \begin{align*} -\frac{1}{96} \,{\left (330 \, \sqrt{2} a^{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}\right ) + 330 \, \sqrt{2} a^{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}\right ) + 165 \, \sqrt{2} a^{2} \log \left (\sqrt{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + \sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 165 \, \sqrt{2} a^{2} \log \left (-\sqrt{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + \sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 768 \, a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - \frac{8 \,{\left (137 \, a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{9}{4}} + 174 \, a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{4}} + 69 \, a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}}{\frac{3 \,{\left (a x - 1\right )}}{a x + 1} + \frac{3 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 1}\right )} a \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(5/4)/x^4,x, algorithm="maxima")

[Out]

-1/96*(330*sqrt(2)*a^2*arctan(1/2*sqrt(2)*(sqrt(2) + 2*((a*x - 1)/(a*x + 1))^(1/4))) + 330*sqrt(2)*a^2*arctan(
-1/2*sqrt(2)*(sqrt(2) - 2*((a*x - 1)/(a*x + 1))^(1/4))) + 165*sqrt(2)*a^2*log(sqrt(2)*((a*x - 1)/(a*x + 1))^(1
/4) + sqrt((a*x - 1)/(a*x + 1)) + 1) - 165*sqrt(2)*a^2*log(-sqrt(2)*((a*x - 1)/(a*x + 1))^(1/4) + sqrt((a*x -
1)/(a*x + 1)) + 1) - 768*a^2*((a*x - 1)/(a*x + 1))^(1/4) - 8*(137*a^2*((a*x - 1)/(a*x + 1))^(9/4) + 174*a^2*((
a*x - 1)/(a*x + 1))^(5/4) + 69*a^2*((a*x - 1)/(a*x + 1))^(1/4))/(3*(a*x - 1)/(a*x + 1) + 3*(a*x - 1)^2/(a*x +
1)^2 + (a*x - 1)^3/(a*x + 1)^3 + 1))*a

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Fricas [A]  time = 1.80943, size = 1143, normalized size = 2.97 \begin{align*} \frac{660 \, \sqrt{2}{\left (a^{12}\right )}^{\frac{1}{4}} x^{3} \arctan \left (-\frac{a^{12} + \sqrt{2}{\left (a^{12}\right )}^{\frac{3}{4}} a^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - \sqrt{2}{\left (a^{12}\right )}^{\frac{3}{4}} \sqrt{a^{6} \sqrt{\frac{a x - 1}{a x + 1}} + \sqrt{2}{\left (a^{12}\right )}^{\frac{1}{4}} a^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + \sqrt{a^{12}}}}{a^{12}}\right ) + 660 \, \sqrt{2}{\left (a^{12}\right )}^{\frac{1}{4}} x^{3} \arctan \left (\frac{a^{12} - \sqrt{2}{\left (a^{12}\right )}^{\frac{3}{4}} a^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + \sqrt{2}{\left (a^{12}\right )}^{\frac{3}{4}} \sqrt{a^{6} \sqrt{\frac{a x - 1}{a x + 1}} - \sqrt{2}{\left (a^{12}\right )}^{\frac{1}{4}} a^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + \sqrt{a^{12}}}}{a^{12}}\right ) - 165 \, \sqrt{2}{\left (a^{12}\right )}^{\frac{1}{4}} x^{3} \log \left (3025 \, a^{6} \sqrt{\frac{a x - 1}{a x + 1}} + 3025 \, \sqrt{2}{\left (a^{12}\right )}^{\frac{1}{4}} a^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 3025 \, \sqrt{a^{12}}\right ) + 165 \, \sqrt{2}{\left (a^{12}\right )}^{\frac{1}{4}} x^{3} \log \left (3025 \, a^{6} \sqrt{\frac{a x - 1}{a x + 1}} - 3025 \, \sqrt{2}{\left (a^{12}\right )}^{\frac{1}{4}} a^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 3025 \, \sqrt{a^{12}}\right ) + 4 \,{\left (287 \, a^{3} x^{3} + 61 \, a^{2} x^{2} - 26 \, a x + 8\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{96 \, x^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(5/4)/x^4,x, algorithm="fricas")

[Out]

1/96*(660*sqrt(2)*(a^12)^(1/4)*x^3*arctan(-(a^12 + sqrt(2)*(a^12)^(3/4)*a^3*((a*x - 1)/(a*x + 1))^(1/4) - sqrt
(2)*(a^12)^(3/4)*sqrt(a^6*sqrt((a*x - 1)/(a*x + 1)) + sqrt(2)*(a^12)^(1/4)*a^3*((a*x - 1)/(a*x + 1))^(1/4) + s
qrt(a^12)))/a^12) + 660*sqrt(2)*(a^12)^(1/4)*x^3*arctan((a^12 - sqrt(2)*(a^12)^(3/4)*a^3*((a*x - 1)/(a*x + 1))
^(1/4) + sqrt(2)*(a^12)^(3/4)*sqrt(a^6*sqrt((a*x - 1)/(a*x + 1)) - sqrt(2)*(a^12)^(1/4)*a^3*((a*x - 1)/(a*x +
1))^(1/4) + sqrt(a^12)))/a^12) - 165*sqrt(2)*(a^12)^(1/4)*x^3*log(3025*a^6*sqrt((a*x - 1)/(a*x + 1)) + 3025*sq
rt(2)*(a^12)^(1/4)*a^3*((a*x - 1)/(a*x + 1))^(1/4) + 3025*sqrt(a^12)) + 165*sqrt(2)*(a^12)^(1/4)*x^3*log(3025*
a^6*sqrt((a*x - 1)/(a*x + 1)) - 3025*sqrt(2)*(a^12)^(1/4)*a^3*((a*x - 1)/(a*x + 1))^(1/4) + 3025*sqrt(a^12)) +
4*(287*a^3*x^3 + 61*a^2*x^2 - 26*a*x + 8)*((a*x - 1)/(a*x + 1))^(1/4))/x^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(((a*x-1)/(a*x+1))**(5/4)/x**4,x)

[Out]

Timed out

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Giac [A]  time = 1.19702, size = 393, normalized size = 1.02 \begin{align*} -\frac{1}{96} \,{\left (330 \, \sqrt{2} a^{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}\right ) + 330 \, \sqrt{2} a^{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}\right ) + 165 \, \sqrt{2} a^{2} \log \left (\sqrt{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + \sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 165 \, \sqrt{2} a^{2} \log \left (-\sqrt{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + \sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 768 \, a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - \frac{8 \,{\left (\frac{174 \,{\left (a x - 1\right )} a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{a x + 1} + \frac{137 \,{\left (a x - 1\right )}^{2} a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{{\left (a x + 1\right )}^{2}} + 69 \, a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}}{{\left (\frac{a x - 1}{a x + 1} + 1\right )}^{3}}\right )} a \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(5/4)/x^4,x, algorithm="giac")

[Out]

-1/96*(330*sqrt(2)*a^2*arctan(1/2*sqrt(2)*(sqrt(2) + 2*((a*x - 1)/(a*x + 1))^(1/4))) + 330*sqrt(2)*a^2*arctan(
-1/2*sqrt(2)*(sqrt(2) - 2*((a*x - 1)/(a*x + 1))^(1/4))) + 165*sqrt(2)*a^2*log(sqrt(2)*((a*x - 1)/(a*x + 1))^(1
/4) + sqrt((a*x - 1)/(a*x + 1)) + 1) - 165*sqrt(2)*a^2*log(-sqrt(2)*((a*x - 1)/(a*x + 1))^(1/4) + sqrt((a*x -
1)/(a*x + 1)) + 1) - 768*a^2*((a*x - 1)/(a*x + 1))^(1/4) - 8*(174*(a*x - 1)*a^2*((a*x - 1)/(a*x + 1))^(1/4)/(a
*x + 1) + 137*(a*x - 1)^2*a^2*((a*x - 1)/(a*x + 1))^(1/4)/(a*x + 1)^2 + 69*a^2*((a*x - 1)/(a*x + 1))^(1/4))/((
a*x - 1)/(a*x + 1) + 1)^3)*a