\(\int \frac {e^{\text {sech}^{-1}(a x)}}{x^8} \, dx\) [44]

Optimal result

Integrand size = 10, antiderivative size = 194 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^8} \, dx=\frac {1}{56 a x^8}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}+\frac {\sqrt {1-a x}}{56 a x^8 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{48 x^6 \sqrt {\frac {1}{1+a x}}}+\frac {5 a^3 \sqrt {1-a x}}{192 x^4 \sqrt {\frac {1}{1+a x}}}+\frac {5 a^5 \sqrt {1-a x}}{128 x^2 \sqrt {\frac {1}{1+a x}}}+\frac {5}{128} a^7 \sqrt {\frac {1}{1+a x}} \sqrt {1+a x} \text {arctanh}\left (\sqrt {1-a x} \sqrt {1+a x}\right ) \]

[Out]

1/56/a/x^8-1/7*(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))/x^7+1/56*(-a*x+1)^(1/2)/a/x^8/(1/(a*x+1))^(1/2)+1/48*a*
(-a*x+1)^(1/2)/x^6/(1/(a*x+1))^(1/2)+5/192*a^3*(-a*x+1)^(1/2)/x^4/(1/(a*x+1))^(1/2)+5/128*a^5*(-a*x+1)^(1/2)/x
^2/(1/(a*x+1))^(1/2)+5/128*a^7*arctanh((-a*x+1)^(1/2)*(a*x+1)^(1/2))*(1/(a*x+1))^(1/2)*(a*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6470, 30, 105, 12, 94, 214} \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^8} \, dx=\frac {5}{128} a^7 \sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \text {arctanh}\left (\sqrt {1-a x} \sqrt {a x+1}\right )+\frac {5 a^5 \sqrt {1-a x}}{128 x^2 \sqrt {\frac {1}{a x+1}}}+\frac {5 a^3 \sqrt {1-a x}}{192 x^4 \sqrt {\frac {1}{a x+1}}}+\frac {\sqrt {1-a x}}{56 a x^8 \sqrt {\frac {1}{a x+1}}}+\frac {1}{56 a x^8}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}+\frac {a \sqrt {1-a x}}{48 x^6 \sqrt {\frac {1}{a x+1}}} \]

[In]

Int[E^ArcSech[a*x]/x^8,x]

[Out]

1/(56*a*x^8) - E^ArcSech[a*x]/(7*x^7) + Sqrt[1 - a*x]/(56*a*x^8*Sqrt[(1 + a*x)^(-1)]) + (a*Sqrt[1 - a*x])/(48*
x^6*Sqrt[(1 + a*x)^(-1)]) + (5*a^3*Sqrt[1 - a*x])/(192*x^4*Sqrt[(1 + a*x)^(-1)]) + (5*a^5*Sqrt[1 - a*x])/(128*
x^2*Sqrt[(1 + a*x)^(-1)]) + (5*a^7*Sqrt[(1 + a*x)^(-1)]*Sqrt[1 + a*x]*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/12
8

Rule 12

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 94

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 105

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[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*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

Rule 214

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

Rule 6470

Int[E^ArcSech[(a_.)*(x_)^(p_.)]*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(E^ArcSech[a*x^p]/(m + 1)), x] + (Dist
[p/(a*(m + 1)), Int[x^(m - p), x], x] + Dist[p*(Sqrt[1 + a*x^p]/(a*(m + 1)))*Sqrt[1/(1 + a*x^p)], Int[x^(m - p
)/(Sqrt[1 + a*x^p]*Sqrt[1 - a*x^p]), x], x]) /; FreeQ[{a, m, p}, x] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}-\frac {\int \frac {1}{x^9} \, dx}{7 a}-\frac {\left (\sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {1}{x^9 \sqrt {1-a x} \sqrt {1+a x}} \, dx}{7 a} \\ & = \frac {1}{56 a x^8}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}+\frac {\sqrt {1-a x}}{56 a x^8 \sqrt {\frac {1}{1+a x}}}+\frac {\left (\sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int -\frac {7 a^2}{x^7 \sqrt {1-a x} \sqrt {1+a x}} \, dx}{56 a} \\ & = \frac {1}{56 a x^8}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}+\frac {\sqrt {1-a x}}{56 a x^8 \sqrt {\frac {1}{1+a x}}}-\frac {1}{8} \left (a \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {1}{x^7 \sqrt {1-a x} \sqrt {1+a x}} \, dx \\ & = \frac {1}{56 a x^8}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}+\frac {\sqrt {1-a x}}{56 a x^8 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{48 x^6 \sqrt {\frac {1}{1+a x}}}+\frac {1}{48} \left (a \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int -\frac {5 a^2}{x^5 \sqrt {1-a x} \sqrt {1+a x}} \, dx \\ & = \frac {1}{56 a x^8}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}+\frac {\sqrt {1-a x}}{56 a x^8 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{48 x^6 \sqrt {\frac {1}{1+a x}}}-\frac {1}{48} \left (5 a^3 \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {1}{x^5 \sqrt {1-a x} \sqrt {1+a x}} \, dx \\ & = \frac {1}{56 a x^8}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}+\frac {\sqrt {1-a x}}{56 a x^8 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{48 x^6 \sqrt {\frac {1}{1+a x}}}+\frac {5 a^3 \sqrt {1-a x}}{192 x^4 \sqrt {\frac {1}{1+a x}}}+\frac {1}{192} \left (5 a^3 \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int -\frac {3 a^2}{x^3 \sqrt {1-a x} \sqrt {1+a x}} \, dx \\ & = \frac {1}{56 a x^8}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}+\frac {\sqrt {1-a x}}{56 a x^8 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{48 x^6 \sqrt {\frac {1}{1+a x}}}+\frac {5 a^3 \sqrt {1-a x}}{192 x^4 \sqrt {\frac {1}{1+a x}}}-\frac {1}{64} \left (5 a^5 \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {1}{x^3 \sqrt {1-a x} \sqrt {1+a x}} \, dx \\ & = \frac {1}{56 a x^8}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}+\frac {\sqrt {1-a x}}{56 a x^8 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{48 x^6 \sqrt {\frac {1}{1+a x}}}+\frac {5 a^3 \sqrt {1-a x}}{192 x^4 \sqrt {\frac {1}{1+a x}}}+\frac {5 a^5 \sqrt {1-a x}}{128 x^2 \sqrt {\frac {1}{1+a x}}}-\frac {1}{128} \left (5 a^5 \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {a^2}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx \\ & = \frac {1}{56 a x^8}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}+\frac {\sqrt {1-a x}}{56 a x^8 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{48 x^6 \sqrt {\frac {1}{1+a x}}}+\frac {5 a^3 \sqrt {1-a x}}{192 x^4 \sqrt {\frac {1}{1+a x}}}+\frac {5 a^5 \sqrt {1-a x}}{128 x^2 \sqrt {\frac {1}{1+a x}}}-\frac {1}{128} \left (5 a^7 \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {1}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx \\ & = \frac {1}{56 a x^8}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}+\frac {\sqrt {1-a x}}{56 a x^8 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{48 x^6 \sqrt {\frac {1}{1+a x}}}+\frac {5 a^3 \sqrt {1-a x}}{192 x^4 \sqrt {\frac {1}{1+a x}}}+\frac {5 a^5 \sqrt {1-a x}}{128 x^2 \sqrt {\frac {1}{1+a x}}}+\frac {1}{128} \left (5 a^8 \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \text {Subst}\left (\int \frac {1}{a-a x^2} \, dx,x,\sqrt {1-a x} \sqrt {1+a x}\right ) \\ & = \frac {1}{56 a x^8}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}+\frac {\sqrt {1-a x}}{56 a x^8 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{48 x^6 \sqrt {\frac {1}{1+a x}}}+\frac {5 a^3 \sqrt {1-a x}}{192 x^4 \sqrt {\frac {1}{1+a x}}}+\frac {5 a^5 \sqrt {1-a x}}{128 x^2 \sqrt {\frac {1}{1+a x}}}+\frac {5}{128} a^7 \sqrt {\frac {1}{1+a x}} \sqrt {1+a x} \text {arctanh}\left (\sqrt {1-a x} \sqrt {1+a x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.75 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^8} \, dx=\frac {-48+\sqrt {\frac {1-a x}{1+a x}} \left (-48-48 a x+8 a^2 x^2+8 a^3 x^3+10 a^4 x^4+10 a^5 x^5+15 a^6 x^6+15 a^7 x^7\right )-15 a^8 x^8 \log (x)+15 a^8 x^8 \log \left (1+\sqrt {\frac {1-a x}{1+a x}}+a x \sqrt {\frac {1-a x}{1+a x}}\right )}{384 a x^8} \]

[In]

Integrate[E^ArcSech[a*x]/x^8,x]

[Out]

(-48 + Sqrt[(1 - a*x)/(1 + a*x)]*(-48 - 48*a*x + 8*a^2*x^2 + 8*a^3*x^3 + 10*a^4*x^4 + 10*a^5*x^5 + 15*a^6*x^6
+ 15*a^7*x^7) - 15*a^8*x^8*Log[x] + 15*a^8*x^8*Log[1 + Sqrt[(1 - a*x)/(1 + a*x)] + a*x*Sqrt[(1 - a*x)/(1 + a*x
)]])/(384*a*x^8)

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.78

method	result	size
default	\(\frac {\sqrt {\frac {a x +1}{a x}}\, \sqrt {-\frac {a x -1}{a x}}\, \left (15 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right ) a^{8} x^{8}+15 \sqrt {-a^{2} x^{2}+1}\, a^{6} x^{6}+10 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+8 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}-48 \sqrt {-a^{2} x^{2}+1}\right )}{384 x^{7} \sqrt {-a^{2} x^{2}+1}}-\frac {1}{8 a \,x^{8}}\)	\(152\)

[In]

int((1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))/x^8,x,method=_RETURNVERBOSE)

[Out]

1/384*((a*x+1)/a/x)^(1/2)/x^7*(-(a*x-1)/a/x)^(1/2)*(15*arctanh(1/(-a^2*x^2+1)^(1/2))*a^8*x^8+15*(-a^2*x^2+1)^(
1/2)*a^6*x^6+10*(-a^2*x^2+1)^(1/2)*a^4*x^4+8*a^2*x^2*(-a^2*x^2+1)^(1/2)-48*(-a^2*x^2+1)^(1/2))/(-a^2*x^2+1)^(1
/2)-1/8/a/x^8

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.80 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^8} \, dx=\frac {15 \, a^{8} x^{8} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 1\right ) - 15 \, a^{8} x^{8} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 1\right ) + 2 \, {\left (15 \, a^{7} x^{7} + 10 \, a^{5} x^{5} + 8 \, a^{3} x^{3} - 48 \, a x\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 96}{768 \, a x^{8}} \]

[In]

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

[Out]

1/768*(15*a^8*x^8*log(a*x*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) + 1) - 15*a^8*x^8*log(a*x*sqrt((a*x + 1
)/(a*x))*sqrt(-(a*x - 1)/(a*x)) - 1) + 2*(15*a^7*x^7 + 10*a^5*x^5 + 8*a^3*x^3 - 48*a*x)*sqrt((a*x + 1)/(a*x))*
sqrt(-(a*x - 1)/(a*x)) - 96)/(a*x^8)

Sympy [F]

\[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^8} \, dx=\frac {\int \frac {1}{x^{9}}\, dx + \int \frac {a \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}}{x^{8}}\, dx}{a} \]

[In]

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

[Out]

(Integral(x**(-9), x) + Integral(a*sqrt(-1 + 1/(a*x))*sqrt(1 + 1/(a*x))/x**8, x))/a

Maxima [F]

\[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^8} \, dx=\int { \frac {\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}}{x^{8}} \,d x } \]

[In]

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

[Out]

integrate(sqrt(a*x + 1)*sqrt(-a*x + 1)/x^9, x)/a - 1/8/(a*x^8)

Giac [F]

\[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^8} \, dx=\int { \frac {\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}}{x^{8}} \,d x } \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 43.82 (sec) , antiderivative size = 1155, normalized size of antiderivative = 5.95 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^8} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(5*a^7*atanh(((1/(a*x) - 1)^(1/2) - 1i)/((1/(a*x) + 1)^(1/2) - 1)))/32 - ((1723*a^7*((1/(a*x) - 1)^(1/2) - 1i)
^5)/(96*((1/(a*x) + 1)^(1/2) - 1)^5) - (235*a^7*((1/(a*x) - 1)^(1/2) - 1i)^3)/(96*((1/(a*x) + 1)^(1/2) - 1)^3)
+ (72283*a^7*((1/(a*x) - 1)^(1/2) - 1i)^7)/(32*((1/(a*x) + 1)^(1/2) - 1)^7) + (848801*a^7*((1/(a*x) - 1)^(1/2
) - 1i)^9)/(32*((1/(a*x) + 1)^(1/2) - 1)^9) + (4181067*a^7*((1/(a*x) - 1)^(1/2) - 1i)^11)/(32*((1/(a*x) + 1)^(
1/2) - 1)^11) + (10994181*a^7*((1/(a*x) - 1)^(1/2) - 1i)^13)/(32*((1/(a*x) + 1)^(1/2) - 1)^13) + (17457599*a^7
*((1/(a*x) - 1)^(1/2) - 1i)^15)/(32*((1/(a*x) + 1)^(1/2) - 1)^15) + (17457599*a^7*((1/(a*x) - 1)^(1/2) - 1i)^1
7)/(32*((1/(a*x) + 1)^(1/2) - 1)^17) + (10994181*a^7*((1/(a*x) - 1)^(1/2) - 1i)^19)/(32*((1/(a*x) + 1)^(1/2) -
1)^19) + (4181067*a^7*((1/(a*x) - 1)^(1/2) - 1i)^21)/(32*((1/(a*x) + 1)^(1/2) - 1)^21) + (848801*a^7*((1/(a*x
) - 1)^(1/2) - 1i)^23)/(32*((1/(a*x) + 1)^(1/2) - 1)^23) + (72283*a^7*((1/(a*x) - 1)^(1/2) - 1i)^25)/(32*((1/(
a*x) + 1)^(1/2) - 1)^25) + (1723*a^7*((1/(a*x) - 1)^(1/2) - 1i)^27)/(96*((1/(a*x) + 1)^(1/2) - 1)^27) - (235*a
^7*((1/(a*x) - 1)^(1/2) - 1i)^29)/(96*((1/(a*x) + 1)^(1/2) - 1)^29) + (5*a^7*((1/(a*x) - 1)^(1/2) - 1i)^31)/(3
2*((1/(a*x) + 1)^(1/2) - 1)^31) + (5*a^7*((1/(a*x) - 1)^(1/2) - 1i))/(32*((1/(a*x) + 1)^(1/2) - 1)))/((120*((1
/(a*x) - 1)^(1/2) - 1i)^4)/((1/(a*x) + 1)^(1/2) - 1)^4 - (16*((1/(a*x) - 1)^(1/2) - 1i)^2)/((1/(a*x) + 1)^(1/2
) - 1)^2 - (560*((1/(a*x) - 1)^(1/2) - 1i)^6)/((1/(a*x) + 1)^(1/2) - 1)^6 + (1820*((1/(a*x) - 1)^(1/2) - 1i)^8
)/((1/(a*x) + 1)^(1/2) - 1)^8 - (4368*((1/(a*x) - 1)^(1/2) - 1i)^10)/((1/(a*x) + 1)^(1/2) - 1)^10 + (8008*((1/
(a*x) - 1)^(1/2) - 1i)^12)/((1/(a*x) + 1)^(1/2) - 1)^12 - (11440*((1/(a*x) - 1)^(1/2) - 1i)^14)/((1/(a*x) + 1)
^(1/2) - 1)^14 + (12870*((1/(a*x) - 1)^(1/2) - 1i)^16)/((1/(a*x) + 1)^(1/2) - 1)^16 - (11440*((1/(a*x) - 1)^(1
/2) - 1i)^18)/((1/(a*x) + 1)^(1/2) - 1)^18 + (8008*((1/(a*x) - 1)^(1/2) - 1i)^20)/((1/(a*x) + 1)^(1/2) - 1)^20
- (4368*((1/(a*x) - 1)^(1/2) - 1i)^22)/((1/(a*x) + 1)^(1/2) - 1)^22 + (1820*((1/(a*x) - 1)^(1/2) - 1i)^24)/((
1/(a*x) + 1)^(1/2) - 1)^24 - (560*((1/(a*x) - 1)^(1/2) - 1i)^26)/((1/(a*x) + 1)^(1/2) - 1)^26 + (120*((1/(a*x)
- 1)^(1/2) - 1i)^28)/((1/(a*x) + 1)^(1/2) - 1)^28 - (16*((1/(a*x) - 1)^(1/2) - 1i)^30)/((1/(a*x) + 1)^(1/2) -
1)^30 + ((1/(a*x) - 1)^(1/2) - 1i)^32/((1/(a*x) + 1)^(1/2) - 1)^32 + 1) - 1/(8*a*x^8)