3.1.84 \(\int \frac {e^{-\text {sech}^{-1}(a x)}}{x^4} \, dx\) [84]

Optimal. Leaf size=200 \[ -\frac {a^3}{4 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^2}+\frac {a^3}{4 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )}-\frac {a^3}{2 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^4}+\frac {a^3}{\left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^3}-\frac {a^3}{\left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^2}+\frac {a^3}{2 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )}-\frac {1}{4} a^3 \tanh ^{-1}\left (\sqrt {\frac {1-a x}{1+a x}}\right ) \]

[Out]

-1/4*a^3*arctanh(((-a*x+1)/(a*x+1))^(1/2))-1/4*a^3/(1-((-a*x+1)/(a*x+1))^(1/2))^2+1/4*a^3/(1-((-a*x+1)/(a*x+1)
)^(1/2))-1/2*a^3/(1+((-a*x+1)/(a*x+1))^(1/2))^4+a^3/(1+((-a*x+1)/(a*x+1))^(1/2))^3-a^3/(1+((-a*x+1)/(a*x+1))^(
1/2))^2+1/2*a^3/(1+((-a*x+1)/(a*x+1))^(1/2))

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Rubi [A]
time = 0.33, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6472, 1626, 213} \begin {gather*} \frac {a^3}{4 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )}+\frac {a^3}{2 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )}-\frac {a^3}{4 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^2}-\frac {a^3}{\left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^2}+\frac {a^3}{\left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^3}-\frac {a^3}{2 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^4}-\frac {1}{4} a^3 \tanh ^{-1}\left (\sqrt {\frac {1-a x}{a x+1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(E^ArcSech[a*x]*x^4),x]

[Out]

-1/4*a^3/(1 - Sqrt[(1 - a*x)/(1 + a*x)])^2 + a^3/(4*(1 - Sqrt[(1 - a*x)/(1 + a*x)])) - a^3/(2*(1 + Sqrt[(1 - a
*x)/(1 + a*x)])^4) + a^3/(1 + Sqrt[(1 - a*x)/(1 + a*x)])^3 - a^3/(1 + Sqrt[(1 - a*x)/(1 + a*x)])^2 + a^3/(2*(1
+ Sqrt[(1 - a*x)/(1 + a*x)])) - (a^3*ArcTanh[Sqrt[(1 - a*x)/(1 + a*x)]])/4

Rule 213

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

Rule 1626

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[E
xpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Poly
Q[Px, x] && IntegersQ[m, n]

Rule 6472

Int[E^(ArcSech[u_]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[x^m*(1/u + Sqrt[(1 - u)/(1 + u)] + (1/u)*Sqrt[(1 - u)/(
1 + u)])^n, x] /; FreeQ[m, x] && IntegerQ[n]

Rubi steps

\begin {align*} \int \frac {e^{-\text {sech}^{-1}(a x)}}{x^4} \, dx &=\int \frac {1}{x^4 \left (\frac {1}{a x}+\sqrt {\frac {1-a x}{1+a x}}+\frac {\sqrt {\frac {1-a x}{1+a x}}}{a x}\right )} \, dx\\ &=(4 a) \text {Subst}\left (\int \frac {x \left (a+a x^2\right )^2}{(-1+x)^3 (1+x)^5} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )\\ &=(4 a) \text {Subst}\left (\int \left (\frac {a^2}{8 (-1+x)^3}+\frac {a^2}{16 (-1+x)^2}+\frac {a^2}{2 (1+x)^5}-\frac {3 a^2}{4 (1+x)^4}+\frac {a^2}{2 (1+x)^3}-\frac {a^2}{8 (1+x)^2}+\frac {a^2}{16 \left (-1+x^2\right )}\right ) \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )\\ &=-\frac {a^3}{4 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^2}+\frac {a^3}{4 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )}-\frac {a^3}{2 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^4}+\frac {a^3}{\left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^3}-\frac {a^3}{\left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^2}+\frac {a^3}{2 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )}+\frac {1}{4} a^3 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )\\ &=-\frac {a^3}{4 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^2}+\frac {a^3}{4 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )}-\frac {a^3}{2 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^4}+\frac {a^3}{\left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^3}-\frac {a^3}{\left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^2}+\frac {a^3}{2 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )}-\frac {1}{4} a^3 \tanh ^{-1}\left (\sqrt {\frac {1-a x}{1+a x}}\right )\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 110, normalized size = 0.55 \begin {gather*} -\frac {2+\sqrt {\frac {1-a x}{1+a x}} \left (-2-2 a x+a^2 x^2+a^3 x^3\right )-a^4 x^4 \log (x)+a^4 x^4 \log \left (1+\sqrt {\frac {1-a x}{1+a x}}+a x \sqrt {\frac {1-a x}{1+a x}}\right )}{8 a x^4} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(E^ArcSech[a*x]*x^4),x]

[Out]

-1/8*(2 + Sqrt[(1 - a*x)/(1 + a*x)]*(-2 - 2*a*x + a^2*x^2 + a^3*x^3) - a^4*x^4*Log[x] + a^4*x^4*Log[1 + Sqrt[(
1 - a*x)/(1 + a*x)] + a*x*Sqrt[(1 - a*x)/(1 + a*x)]])/(a*x^4)

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right ) x^{4}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))/x^4,x)

[Out]

int(1/(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))/x^4,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))/x^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.43, size = 138, normalized size = 0.69 \begin {gather*} -\frac {a^{4} x^{4} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 1\right ) - a^{4} x^{4} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 1\right ) + 2 \, {\left (a^{3} x^{3} - 2 \, a x\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 4}{16 \, a x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))/x^4,x, algorithm="fricas")

[Out]

-1/16*(a^4*x^4*log(a*x*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) + 1) - a^4*x^4*log(a*x*sqrt((a*x + 1)/(a*x
))*sqrt(-(a*x - 1)/(a*x)) - 1) + 2*(a^3*x^3 - 2*a*x)*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) + 4)/(a*x^4)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} a \int \frac {1}{a x^{4} \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}} + x^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1/a/x+(1/a/x-1)**(1/2)*(1+1/a/x)**(1/2))/x**4,x)

[Out]

a*Integral(1/(a*x**4*sqrt(-1 + 1/(a*x))*sqrt(1 + 1/(a*x)) + x**3), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))/x^4,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 43.71, size = 1511, normalized size = 7.56 \begin {gather*} -\frac {\frac {a^3\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4\,64{}\mathrm {i}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}+\frac {a^3\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^6\,128{}\mathrm {i}}{3\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^6}+\frac {a^3\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^8\,64{}\mathrm {i}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^8}}{1+\frac {15\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}-\frac {20\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^6}+\frac {15\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^8}-\frac {6\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{10}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{10}}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{12}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{12}}-\frac {6\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}}+\frac {\frac {a^3\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4\,192{}\mathrm {i}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}+\frac {a^3\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^6\,128{}\mathrm {i}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^6}+\frac {a^3\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^8\,192{}\mathrm {i}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^8}}{3\,\left (1+\frac {15\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}-\frac {20\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^6}+\frac {15\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^8}-\frac {6\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{10}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{10}}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{12}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{12}}-\frac {6\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}\right )}-\frac {a^3\,\mathrm {atanh}\left (\frac {\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x}+1}-1}\right )}{2}+\frac {\frac {14\,a^3\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^3}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^3}+\frac {14\,a^3\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^5}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^5}+\frac {2\,a^3\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^7}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^7}+\frac {2\,a^3\,\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}{\sqrt {\frac {1}{a\,x}+1}-1}}{1+\frac {6\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}-\frac {4\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^6}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^8}-\frac {4\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}}+\frac {\frac {23\,a^3\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^3}{2\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^3}+\frac {333\,a^3\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^5}{2\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^5}+\frac {671\,a^3\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^7}{2\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^7}+\frac {671\,a^3\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^9}{2\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^9}+\frac {333\,a^3\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{11}}{2\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{11}}+\frac {23\,a^3\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{13}}{2\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{13}}-\frac {3\,a^3\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{15}}{2\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{15}}-\frac {3\,a^3\,\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}{2\,\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}}{1+\frac {28\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}-\frac {56\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^6}+\frac {70\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^8}-\frac {56\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{10}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{10}}+\frac {28\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{12}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{12}}-\frac {8\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{14}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{14}}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{16}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{16}}-\frac {8\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}}-\frac {1}{4\,a\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*((1/(a*x) - 1)^(1/2)*(1/(a*x) + 1)^(1/2) + 1/(a*x))),x)

[Out]

((a^3*((1/(a*x) - 1)^(1/2) - 1i)^4*192i)/((1/(a*x) + 1)^(1/2) - 1)^4 + (a^3*((1/(a*x) - 1)^(1/2) - 1i)^6*128i)
/((1/(a*x) + 1)^(1/2) - 1)^6 + (a^3*((1/(a*x) - 1)^(1/2) - 1i)^8*192i)/((1/(a*x) + 1)^(1/2) - 1)^8)/(3*((15*((
1/(a*x) - 1)^(1/2) - 1i)^4)/((1/(a*x) + 1)^(1/2) - 1)^4 - (6*((1/(a*x) - 1)^(1/2) - 1i)^2)/((1/(a*x) + 1)^(1/2
) - 1)^2 - (20*((1/(a*x) - 1)^(1/2) - 1i)^6)/((1/(a*x) + 1)^(1/2) - 1)^6 + (15*((1/(a*x) - 1)^(1/2) - 1i)^8)/(
(1/(a*x) + 1)^(1/2) - 1)^8 - (6*((1/(a*x) - 1)^(1/2) - 1i)^10)/((1/(a*x) + 1)^(1/2) - 1)^10 + ((1/(a*x) - 1)^(
1/2) - 1i)^12/((1/(a*x) + 1)^(1/2) - 1)^12 + 1)) - ((a^3*((1/(a*x) - 1)^(1/2) - 1i)^4*64i)/((1/(a*x) + 1)^(1/2
) - 1)^4 + (a^3*((1/(a*x) - 1)^(1/2) - 1i)^6*128i)/(3*((1/(a*x) + 1)^(1/2) - 1)^6) + (a^3*((1/(a*x) - 1)^(1/2)
- 1i)^8*64i)/((1/(a*x) + 1)^(1/2) - 1)^8)/((15*((1/(a*x) - 1)^(1/2) - 1i)^4)/((1/(a*x) + 1)^(1/2) - 1)^4 - (6
*((1/(a*x) - 1)^(1/2) - 1i)^2)/((1/(a*x) + 1)^(1/2) - 1)^2 - (20*((1/(a*x) - 1)^(1/2) - 1i)^6)/((1/(a*x) + 1)^
(1/2) - 1)^6 + (15*((1/(a*x) - 1)^(1/2) - 1i)^8)/((1/(a*x) + 1)^(1/2) - 1)^8 - (6*((1/(a*x) - 1)^(1/2) - 1i)^1
0)/((1/(a*x) + 1)^(1/2) - 1)^10 + ((1/(a*x) - 1)^(1/2) - 1i)^12/((1/(a*x) + 1)^(1/2) - 1)^12 + 1) - (a^3*atanh
(((1/(a*x) - 1)^(1/2) - 1i)/((1/(a*x) + 1)^(1/2) - 1)))/2 + ((14*a^3*((1/(a*x) - 1)^(1/2) - 1i)^3)/((1/(a*x) +
1)^(1/2) - 1)^3 + (14*a^3*((1/(a*x) - 1)^(1/2) - 1i)^5)/((1/(a*x) + 1)^(1/2) - 1)^5 + (2*a^3*((1/(a*x) - 1)^(
1/2) - 1i)^7)/((1/(a*x) + 1)^(1/2) - 1)^7 + (2*a^3*((1/(a*x) - 1)^(1/2) - 1i))/((1/(a*x) + 1)^(1/2) - 1))/((6*
((1/(a*x) - 1)^(1/2) - 1i)^4)/((1/(a*x) + 1)^(1/2) - 1)^4 - (4*((1/(a*x) - 1)^(1/2) - 1i)^2)/((1/(a*x) + 1)^(1
/2) - 1)^2 - (4*((1/(a*x) - 1)^(1/2) - 1i)^6)/((1/(a*x) + 1)^(1/2) - 1)^6 + ((1/(a*x) - 1)^(1/2) - 1i)^8/((1/(
a*x) + 1)^(1/2) - 1)^8 + 1) + ((23*a^3*((1/(a*x) - 1)^(1/2) - 1i)^3)/(2*((1/(a*x) + 1)^(1/2) - 1)^3) + (333*a^
3*((1/(a*x) - 1)^(1/2) - 1i)^5)/(2*((1/(a*x) + 1)^(1/2) - 1)^5) + (671*a^3*((1/(a*x) - 1)^(1/2) - 1i)^7)/(2*((
1/(a*x) + 1)^(1/2) - 1)^7) + (671*a^3*((1/(a*x) - 1)^(1/2) - 1i)^9)/(2*((1/(a*x) + 1)^(1/2) - 1)^9) + (333*a^3
*((1/(a*x) - 1)^(1/2) - 1i)^11)/(2*((1/(a*x) + 1)^(1/2) - 1)^11) + (23*a^3*((1/(a*x) - 1)^(1/2) - 1i)^13)/(2*(
(1/(a*x) + 1)^(1/2) - 1)^13) - (3*a^3*((1/(a*x) - 1)^(1/2) - 1i)^15)/(2*((1/(a*x) + 1)^(1/2) - 1)^15) - (3*a^3
*((1/(a*x) - 1)^(1/2) - 1i))/(2*((1/(a*x) + 1)^(1/2) - 1)))/((28*((1/(a*x) - 1)^(1/2) - 1i)^4)/((1/(a*x) + 1)^
(1/2) - 1)^4 - (8*((1/(a*x) - 1)^(1/2) - 1i)^2)/((1/(a*x) + 1)^(1/2) - 1)^2 - (56*((1/(a*x) - 1)^(1/2) - 1i)^6
)/((1/(a*x) + 1)^(1/2) - 1)^6 + (70*((1/(a*x) - 1)^(1/2) - 1i)^8)/((1/(a*x) + 1)^(1/2) - 1)^8 - (56*((1/(a*x)
- 1)^(1/2) - 1i)^10)/((1/(a*x) + 1)^(1/2) - 1)^10 + (28*((1/(a*x) - 1)^(1/2) - 1i)^12)/((1/(a*x) + 1)^(1/2) -
1)^12 - (8*((1/(a*x) - 1)^(1/2) - 1i)^14)/((1/(a*x) + 1)^(1/2) - 1)^14 + ((1/(a*x) - 1)^(1/2) - 1i)^16/((1/(a*
x) + 1)^(1/2) - 1)^16 + 1) - 1/(4*a*x^4)

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