\(\int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^4} \, dx\) [85]

Optimal result

Integrand size = 14, antiderivative size = 385 \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^4} \, dx=-\frac {55}{8} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}-\frac {11}{4} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}-\frac {2 a^3 \left (1+\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{1-\frac {1}{a x}}}-\frac {1}{3} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{9/4}+\frac {55 a^3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{8 \sqrt {2}}-\frac {55 a^3 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{8 \sqrt {2}}-\frac {55 a^3 \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{16 \sqrt {2}}+\frac {55 a^3 \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{16 \sqrt {2}} \]

[Out]

-55/8*a^3*(1-1/a/x)^(3/4)*(1+1/a/x)^(1/4)-11/4*a^3*(1-1/a/x)^(3/4)*(1+1/a/x)^(5/4)-2*a^3*(1+1/a/x)^(9/4)/(1-1/
a/x)^(1/4)-1/3*a^3*(1-1/a/x)^(3/4)*(1+1/a/x)^(9/4)-55/16*a^3*arctan(-1+(1-1/a/x)^(1/4)*2^(1/2)/(1+1/a/x)^(1/4)
)*2^(1/2)-55/16*a^3*arctan(1+(1-1/a/x)^(1/4)*2^(1/2)/(1+1/a/x)^(1/4))*2^(1/2)-55/32*a^3*ln(1-(1-1/a/x)^(1/4)*2
^(1/2)/(1+1/a/x)^(1/4)+(1-1/a/x)^(1/2)/(1+1/a/x)^(1/2))*2^(1/2)+55/32*a^3*ln(1+(1-1/a/x)^(1/4)*2^(1/2)/(1+1/a/
x)^(1/4)+(1-1/a/x)^(1/2)/(1+1/a/x)^(1/2))*2^(1/2)

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {6306, 91, 81, 52, 65, 338, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^4} \, dx=\frac {55 a^3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}\right )}{8 \sqrt {2}}-\frac {55 a^3 \arctan \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{8 \sqrt {2}}-\frac {1}{3} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \left (\frac {1}{a x}+1\right )^{9/4}-\frac {2 a^3 \left (\frac {1}{a x}+1\right )^{9/4}}{\sqrt [4]{1-\frac {1}{a x}}}-\frac {11}{4} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \left (\frac {1}{a x}+1\right )^{5/4}-\frac {55}{8} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}-\frac {55 a^3 \log \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{16 \sqrt {2}}+\frac {55 a^3 \log \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{16 \sqrt {2}} \]

[In]

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

[Out]

(-55*a^3*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4))/8 - (11*a^3*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(5/4))/4 - (2*
a^3*(1 + 1/(a*x))^(9/4))/(1 - 1/(a*x))^(1/4) - (a^3*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(9/4))/3 + (55*a^3*ArcTa
n[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 52

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

Rule 65

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

Rule 81

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 91

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 210

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

Rule 303

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

Rule 338

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

Rule 631

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 642

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

Rule 1176

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 1179

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 6306

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

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {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 ) \text {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 )^{3/4} \left (1+\frac {1}{a x}\right )^{9/4}+\frac {1}{2} \left (11 a^2\right ) \text {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 {11}{4} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}-\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 )^{3/4} \left (1+\frac {1}{a x}\right )^{9/4}+\frac {1}{8} \left (55 a^2\right ) \text {Subst}\left (\int \frac {\sqrt [4]{1+\frac {x}{a}}}{\sqrt [4]{1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {55}{8} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}-\frac {11}{4} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}-\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 )^{3/4} \left (1+\frac {1}{a x}\right )^{9/4}+\frac {1}{16} \left (55 a^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/4}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {55}{8} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}-\frac {11}{4} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}-\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 )^{3/4} \left (1+\frac {1}{a x}\right )^{9/4}-\frac {1}{4} \left (55 a^3\right ) \text {Subst}\left (\int \frac {x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-\frac {1}{a x}}\right ) \\ & = -\frac {55}{8} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}-\frac {11}{4} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}-\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 )^{3/4} \left (1+\frac {1}{a x}\right )^{9/4}-\frac {1}{4} \left (55 a^3\right ) \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right ) \\ & = -\frac {55}{8} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}-\frac {11}{4} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}-\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 )^{3/4} \left (1+\frac {1}{a x}\right )^{9/4}+\frac {1}{8} \left (55 a^3\right ) \text {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 ) \text {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 {55}{8} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}-\frac {11}{4} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}-\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 )^{3/4} \left (1+\frac {1}{a x}\right )^{9/4}-\frac {1}{16} \left (55 a^3\right ) \text {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 ) \text {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 ) \text {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 ) \text {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 {55}{8} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}-\frac {11}{4} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}-\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 )^{3/4} \left (1+\frac {1}{a x}\right )^{9/4}-\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 ) \text {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 ) \text {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 {55}{8} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}-\frac {11}{4} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}-\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 )^{3/4} \left (1+\frac {1}{a x}\right )^{9/4}+\frac {55 a^3 \arctan \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 \arctan \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] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.10 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.27 \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^4} \, dx=a^3 \left (-\frac {e^{\frac {1}{2} \coth ^{-1}(a x)} \left (165+462 e^{2 \coth ^{-1}(a x)}+425 e^{4 \coth ^{-1}(a x)}+96 e^{6 \coth ^{-1}(a x)}\right )}{12 \left (1+e^{2 \coth ^{-1}(a x)}\right )^3}-\frac {55}{32} \text {RootSum}\left [1+\text {$\#$1}^4\&,\frac {\coth ^{-1}(a x)-2 \log \left (e^{\frac {1}{2} \coth ^{-1}(a x)}-\text {$\#$1}\right )}{\text {$\#$1}^3}\&\right ]\right ) \]

[In]

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

[Out]

a^3*(-1/12*(E^(ArcCoth[a*x]/2)*(165 + 462*E^(2*ArcCoth[a*x]) + 425*E^(4*ArcCoth[a*x]) + 96*E^(6*ArcCoth[a*x]))
)/(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)

Maple [F]

\[\int \frac {1}{\left (\frac {a x -1}{a x +1}\right )^{\frac {5}{4}} x^{4}}d x\]

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.70 \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^4} \, dx=-\frac {165 \, \left (-a^{12}\right )^{\frac {1}{4}} {\left (a x^{4} - x^{3}\right )} \log \left (166375 \, a^{9} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 166375 \, \left (-a^{12}\right )^{\frac {3}{4}}\right ) + 165 \, \left (-a^{12}\right )^{\frac {1}{4}} {\left (-i \, a x^{4} + i \, x^{3}\right )} \log \left (166375 \, a^{9} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 166375 i \, \left (-a^{12}\right )^{\frac {3}{4}}\right ) + 165 \, \left (-a^{12}\right )^{\frac {1}{4}} {\left (i \, a x^{4} - i \, x^{3}\right )} \log \left (166375 \, a^{9} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 166375 i \, \left (-a^{12}\right )^{\frac {3}{4}}\right ) - 165 \, \left (-a^{12}\right )^{\frac {1}{4}} {\left (a x^{4} - x^{3}\right )} \log \left (166375 \, a^{9} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 166375 \, \left (-a^{12}\right )^{\frac {3}{4}}\right ) + 2 \, {\left (287 \, a^{4} x^{4} + 226 \, a^{3} x^{3} - 87 \, a^{2} x^{2} - 34 \, a x - 8\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{48 \, {\left (a x^{4} - x^{3}\right )}} \]

[In]

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

[Out]

-1/48*(165*(-a^12)^(1/4)*(a*x^4 - x^3)*log(166375*a^9*((a*x - 1)/(a*x + 1))^(1/4) + 166375*(-a^12)^(3/4)) + 16
5*(-a^12)^(1/4)*(-I*a*x^4 + I*x^3)*log(166375*a^9*((a*x - 1)/(a*x + 1))^(1/4) + 166375*I*(-a^12)^(3/4)) + 165*
(-a^12)^(1/4)*(I*a*x^4 - I*x^3)*log(166375*a^9*((a*x - 1)/(a*x + 1))^(1/4) - 166375*I*(-a^12)^(3/4)) - 165*(-a
^12)^(1/4)*(a*x^4 - x^3)*log(166375*a^9*((a*x - 1)/(a*x + 1))^(1/4) - 166375*(-a^12)^(3/4)) + 2*(287*a^4*x^4 +
226*a^3*x^3 - 87*a^2*x^2 - 34*a*x - 8)*((a*x - 1)/(a*x + 1))^(3/4))/(a*x^4 - x^3)

Sympy [F]

\[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^4} \, dx=\int \frac {1}{x^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}}}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 288, normalized size of antiderivative = 0.75 \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^4} \, dx=-\frac {1}{96} \, {\left (165 \, {\left (2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) - \sqrt {2} \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + \sqrt {2} \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right )\right )} a^{2} + \frac {8 \, {\left (\frac {425 \, {\left (a x - 1\right )} a^{2}}{a x + 1} + \frac {462 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {165 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} + 96 \, a^{2}\right )}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {13}{4}} + 3 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{4}} + 3 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}\right )} a \]

[In]

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

[Out]

-1/96*(165*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*((a*x - 1)/(a*x + 1))^(1/4))) + 2*sqrt(2)*arctan(-1/2*sq
rt(2)*(sqrt(2) - 2*((a*x - 1)/(a*x + 1))^(1/4))) - sqrt(2)*log(sqrt(2)*((a*x - 1)/(a*x + 1))^(1/4) + sqrt((a*x
- 1)/(a*x + 1)) + 1) + sqrt(2)*log(-sqrt(2)*((a*x - 1)/(a*x + 1))^(1/4) + sqrt((a*x - 1)/(a*x + 1)) + 1))*a^2
+ 8*(425*(a*x - 1)*a^2/(a*x + 1) + 462*(a*x - 1)^2*a^2/(a*x + 1)^2 + 165*(a*x - 1)^3*a^2/(a*x + 1)^3 + 96*a^2
)/(((a*x - 1)/(a*x + 1))^(13/4) + 3*((a*x - 1)/(a*x + 1))^(9/4) + 3*((a*x - 1)/(a*x + 1))^(5/4) + ((a*x - 1)/(
a*x + 1))^(1/4)))*a

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.76 \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^4} \, dx=-\frac {1}{96} \, {\left (330 \, \sqrt {2} a^{2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 330 \, \sqrt {2} a^{2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) - 165 \, \sqrt {2} a^{2} \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 165 \, \sqrt {2} a^{2} \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + \frac {768 \, a^{2}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}} + \frac {8 \, {\left (\frac {174 \, {\left (a x - 1\right )} a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{a x + 1} + \frac {69 \, {\left (a x - 1\right )}^{2} a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{{\left (a x + 1\right )}^{2}} + 137 \, a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}\right )}}{{\left (\frac {a x - 1}{a x + 1} + 1\right )}^{3}}\right )} a \]

[In]

integrate(1/((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))^(3/4)/(a
*x + 1) + 69*(a*x - 1)^2*a^2*((a*x - 1)/(a*x + 1))^(3/4)/(a*x + 1)^2 + 137*a^2*((a*x - 1)/(a*x + 1))^(3/4))/((
a*x - 1)/(a*x + 1) + 1)^3)*a

Mupad [B] (verification not implemented)

Time = 4.23 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.49 \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^4} \, dx=\frac {55\,{\left (-1\right )}^{1/4}\,a^3\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{8}-\frac {55\,{\left (-1\right )}^{1/4}\,a^3\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{8}-\frac {8\,a^3+\frac {77\,a^3\,{\left (a\,x-1\right )}^2}{2\,{\left (a\,x+1\right )}^2}+\frac {55\,a^3\,{\left (a\,x-1\right )}^3}{4\,{\left (a\,x+1\right )}^3}+\frac {425\,a^3\,\left (a\,x-1\right )}{12\,\left (a\,x+1\right )}}{{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}+3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/4}+3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/4}+{\left (\frac {a\,x-1}{a\,x+1}\right )}^{13/4}} \]

[In]

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

[Out]

(55*(-1)^(1/4)*a^3*atanh((-1)^(1/4)*((a*x - 1)/(a*x + 1))^(1/4)))/8 - (55*(-1)^(1/4)*a^3*atan((-1)^(1/4)*((a*x
- 1)/(a*x + 1))^(1/4)))/8 - (8*a^3 + (77*a^3*(a*x - 1)^2)/(2*(a*x + 1)^2) + (55*a^3*(a*x - 1)^3)/(4*(a*x + 1)
^3) + (425*a^3*(a*x - 1))/(12*(a*x + 1)))/(((a*x - 1)/(a*x + 1))^(1/4) + 3*((a*x - 1)/(a*x + 1))^(5/4) + 3*((a
*x - 1)/(a*x + 1))^(9/4) + ((a*x - 1)/(a*x + 1))^(13/4))