\(\int (a \text {csch}^4(x))^{5/2} \, dx\) [43]

Optimal result

Integrand size = 10, antiderivative size = 118 \[ \int \left (a \text {csch}^4(x)\right )^{5/2} \, dx=\frac {4}{3} a^2 \cosh ^2(x) \coth (x) \sqrt {a \text {csch}^4(x)}-\frac {6}{5} a^2 \cosh ^2(x) \coth ^3(x) \sqrt {a \text {csch}^4(x)}+\frac {4}{7} a^2 \cosh ^2(x) \coth ^5(x) \sqrt {a \text {csch}^4(x)}-\frac {1}{9} a^2 \cosh ^2(x) \coth ^7(x) \sqrt {a \text {csch}^4(x)}-a^2 \cosh (x) \sqrt {a \text {csch}^4(x)} \sinh (x) \]

[Out]

4/3*a^2*cosh(x)^2*coth(x)*(a*csch(x)^4)^(1/2)-6/5*a^2*cosh(x)^2*coth(x)^3*(a*csch(x)^4)^(1/2)+4/7*a^2*cosh(x)^
2*coth(x)^5*(a*csch(x)^4)^(1/2)-1/9*a^2*cosh(x)^2*coth(x)^7*(a*csch(x)^4)^(1/2)-a^2*cosh(x)*sinh(x)*(a*csch(x)
^4)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4208, 3852} \[ \int \left (a \text {csch}^4(x)\right )^{5/2} \, dx=-\frac {1}{9} a^2 \cosh ^2(x) \coth ^7(x) \sqrt {a \text {csch}^4(x)}+\frac {4}{7} a^2 \cosh ^2(x) \coth ^5(x) \sqrt {a \text {csch}^4(x)}-\frac {6}{5} a^2 \cosh ^2(x) \coth ^3(x) \sqrt {a \text {csch}^4(x)}+\frac {4}{3} a^2 \cosh ^2(x) \coth (x) \sqrt {a \text {csch}^4(x)}-a^2 \sinh (x) \cosh (x) \sqrt {a \text {csch}^4(x)} \]

[In]

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

[Out]

(4*a^2*Cosh[x]^2*Coth[x]*Sqrt[a*Csch[x]^4])/3 - (6*a^2*Cosh[x]^2*Coth[x]^3*Sqrt[a*Csch[x]^4])/5 + (4*a^2*Cosh[
x]^2*Coth[x]^5*Sqrt[a*Csch[x]^4])/7 - (a^2*Cosh[x]^2*Coth[x]^7*Sqrt[a*Csch[x]^4])/9 - a^2*Cosh[x]*Sqrt[a*Csch[
x]^4]*Sinh[x]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 4208

Int[((b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Dist[b^IntPart[p]*((b*(c*Sec[e + f*x])^n)^
FracPart[p]/(c*Sec[e + f*x])^(n*FracPart[p])), Int[(c*Sec[e + f*x])^(n*p), x], x] /; FreeQ[{b, c, e, f, n, p},
x] &&  !IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \left (a^2 \sqrt {a \text {csch}^4(x)} \sinh ^2(x)\right ) \int \text {csch}^{10}(x) \, dx \\ & = -\left (\left (i a^2 \sqrt {a \text {csch}^4(x)} \sinh ^2(x)\right ) \text {Subst}\left (\int \left (1+4 x^2+6 x^4+4 x^6+x^8\right ) \, dx,x,-i \coth (x)\right )\right ) \\ & = \frac {4}{3} a^2 \cosh ^2(x) \coth (x) \sqrt {a \text {csch}^4(x)}-\frac {6}{5} a^2 \cosh ^2(x) \coth ^3(x) \sqrt {a \text {csch}^4(x)}+\frac {4}{7} a^2 \cosh ^2(x) \coth ^5(x) \sqrt {a \text {csch}^4(x)}-\frac {1}{9} a^2 \cosh ^2(x) \coth ^7(x) \sqrt {a \text {csch}^4(x)}-a^2 \cosh (x) \sqrt {a \text {csch}^4(x)} \sinh (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.40 \[ \int \left (a \text {csch}^4(x)\right )^{5/2} \, dx=-\frac {1}{315} a^2 \cosh (x) \sqrt {a \text {csch}^4(x)} \left (128-64 \text {csch}^2(x)+48 \text {csch}^4(x)-40 \text {csch}^6(x)+35 \text {csch}^8(x)\right ) \sinh (x) \]

[In]

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

[Out]

-1/315*(a^2*Cosh[x]*Sqrt[a*Csch[x]^4]*(128 - 64*Csch[x]^2 + 48*Csch[x]^4 - 40*Csch[x]^6 + 35*Csch[x]^8)*Sinh[x
])

Maple [A] (verified)

Time = 0.76 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.51

method	result	size
risch	\(-\frac {256 a^{2} {\mathrm e}^{-2 x} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}-1\right )^{4}}}\, \left (126 \,{\mathrm e}^{8 x}-84 \,{\mathrm e}^{6 x}+36 \,{\mathrm e}^{4 x}-9 \,{\mathrm e}^{2 x}+1\right )}{315 \left ({\mathrm e}^{2 x}-1\right )^{7}}\)	\(60\)

[In]

int((a*csch(x)^4)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-256/315*a^2*exp(-2*x)/(exp(2*x)-1)^7*(a*exp(4*x)/(exp(2*x)-1)^4)^(1/2)*(126*exp(8*x)-84*exp(6*x)+36*exp(4*x)-
9*exp(2*x)+1)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1493 vs. \(2 (100) = 200\).

Time = 0.29 (sec) , antiderivative size = 1493, normalized size of antiderivative = 12.65 \[ \int \left (a \text {csch}^4(x)\right )^{5/2} \, dx=\text {Too large to display} \]

[In]

integrate((a*csch(x)^4)^(5/2),x, algorithm="fricas")

[Out]

-256/315*(126*a^2*cosh(x)^8 + 126*(a^2*e^(4*x) - 2*a^2*e^(2*x) + a^2)*sinh(x)^8 - 84*a^2*cosh(x)^6 + 1008*(a^2
*cosh(x)*e^(4*x) - 2*a^2*cosh(x)*e^(2*x) + a^2*cosh(x))*sinh(x)^7 + 84*(42*a^2*cosh(x)^2 - a^2 + (42*a^2*cosh(
x)^2 - a^2)*e^(4*x) - 2*(42*a^2*cosh(x)^2 - a^2)*e^(2*x))*sinh(x)^6 + 36*a^2*cosh(x)^4 + 504*(14*a^2*cosh(x)^3
- a^2*cosh(x) + (14*a^2*cosh(x)^3 - a^2*cosh(x))*e^(4*x) - 2*(14*a^2*cosh(x)^3 - a^2*cosh(x))*e^(2*x))*sinh(x
)^5 + 36*(245*a^2*cosh(x)^4 - 35*a^2*cosh(x)^2 + a^2 + (245*a^2*cosh(x)^4 - 35*a^2*cosh(x)^2 + a^2)*e^(4*x) -
2*(245*a^2*cosh(x)^4 - 35*a^2*cosh(x)^2 + a^2)*e^(2*x))*sinh(x)^4 - 9*a^2*cosh(x)^2 + 48*(147*a^2*cosh(x)^5 -
35*a^2*cosh(x)^3 + 3*a^2*cosh(x) + (147*a^2*cosh(x)^5 - 35*a^2*cosh(x)^3 + 3*a^2*cosh(x))*e^(4*x) - 2*(147*a^2
*cosh(x)^5 - 35*a^2*cosh(x)^3 + 3*a^2*cosh(x))*e^(2*x))*sinh(x)^3 + 9*(392*a^2*cosh(x)^6 - 140*a^2*cosh(x)^4 +
24*a^2*cosh(x)^2 - a^2 + (392*a^2*cosh(x)^6 - 140*a^2*cosh(x)^4 + 24*a^2*cosh(x)^2 - a^2)*e^(4*x) - 2*(392*a^
2*cosh(x)^6 - 140*a^2*cosh(x)^4 + 24*a^2*cosh(x)^2 - a^2)*e^(2*x))*sinh(x)^2 + a^2 + (126*a^2*cosh(x)^8 - 84*a
^2*cosh(x)^6 + 36*a^2*cosh(x)^4 - 9*a^2*cosh(x)^2 + a^2)*e^(4*x) - 2*(126*a^2*cosh(x)^8 - 84*a^2*cosh(x)^6 + 3
6*a^2*cosh(x)^4 - 9*a^2*cosh(x)^2 + a^2)*e^(2*x) + 18*(56*a^2*cosh(x)^7 - 28*a^2*cosh(x)^5 + 8*a^2*cosh(x)^3 -
a^2*cosh(x) + (56*a^2*cosh(x)^7 - 28*a^2*cosh(x)^5 + 8*a^2*cosh(x)^3 - a^2*cosh(x))*e^(4*x) - 2*(56*a^2*cosh(
x)^7 - 28*a^2*cosh(x)^5 + 8*a^2*cosh(x)^3 - a^2*cosh(x))*e^(2*x))*sinh(x))*sqrt(a/(e^(8*x) - 4*e^(6*x) + 6*e^(
4*x) - 4*e^(2*x) + 1))*e^(2*x)/(18*cosh(x)*e^(2*x)*sinh(x)^17 + e^(2*x)*sinh(x)^18 + 9*(17*cosh(x)^2 - 1)*e^(2
*x)*sinh(x)^16 + 48*(17*cosh(x)^3 - 3*cosh(x))*e^(2*x)*sinh(x)^15 + 36*(85*cosh(x)^4 - 30*cosh(x)^2 + 1)*e^(2*
x)*sinh(x)^14 + 504*(17*cosh(x)^5 - 10*cosh(x)^3 + cosh(x))*e^(2*x)*sinh(x)^13 + 84*(221*cosh(x)^6 - 195*cosh(
x)^4 + 39*cosh(x)^2 - 1)*e^(2*x)*sinh(x)^12 + 144*(221*cosh(x)^7 - 273*cosh(x)^5 + 91*cosh(x)^3 - 7*cosh(x))*e
^(2*x)*sinh(x)^11 + 18*(2431*cosh(x)^8 - 4004*cosh(x)^6 + 2002*cosh(x)^4 - 308*cosh(x)^2 + 7)*e^(2*x)*sinh(x)^
10 + 4*(12155*cosh(x)^9 - 25740*cosh(x)^7 + 18018*cosh(x)^5 - 4620*cosh(x)^3 + 315*cosh(x))*e^(2*x)*sinh(x)^9
+ 18*(2431*cosh(x)^10 - 6435*cosh(x)^8 + 6006*cosh(x)^6 - 2310*cosh(x)^4 + 315*cosh(x)^2 - 7)*e^(2*x)*sinh(x)^
8 + 144*(221*cosh(x)^11 - 715*cosh(x)^9 + 858*cosh(x)^7 - 462*cosh(x)^5 + 105*cosh(x)^3 - 7*cosh(x))*e^(2*x)*s
inh(x)^7 + 84*(221*cosh(x)^12 - 858*cosh(x)^10 + 1287*cosh(x)^8 - 924*cosh(x)^6 + 315*cosh(x)^4 - 42*cosh(x)^2
+ 1)*e^(2*x)*sinh(x)^6 + 504*(17*cosh(x)^13 - 78*cosh(x)^11 + 143*cosh(x)^9 - 132*cosh(x)^7 + 63*cosh(x)^5 -
14*cosh(x)^3 + cosh(x))*e^(2*x)*sinh(x)^5 + 36*(85*cosh(x)^14 - 455*cosh(x)^12 + 1001*cosh(x)^10 - 1155*cosh(x
)^8 + 735*cosh(x)^6 - 245*cosh(x)^4 + 35*cosh(x)^2 - 1)*e^(2*x)*sinh(x)^4 + 48*(17*cosh(x)^15 - 105*cosh(x)^13
+ 273*cosh(x)^11 - 385*cosh(x)^9 + 315*cosh(x)^7 - 147*cosh(x)^5 + 35*cosh(x)^3 - 3*cosh(x))*e^(2*x)*sinh(x)^
3 + 9*(17*cosh(x)^16 - 120*cosh(x)^14 + 364*cosh(x)^12 - 616*cosh(x)^10 + 630*cosh(x)^8 - 392*cosh(x)^6 + 140*
cosh(x)^4 - 24*cosh(x)^2 + 1)*e^(2*x)*sinh(x)^2 + 18*(cosh(x)^17 - 8*cosh(x)^15 + 28*cosh(x)^13 - 56*cosh(x)^1
1 + 70*cosh(x)^9 - 56*cosh(x)^7 + 28*cosh(x)^5 - 8*cosh(x)^3 + cosh(x))*e^(2*x)*sinh(x) + (cosh(x)^18 - 9*cosh
(x)^16 + 36*cosh(x)^14 - 84*cosh(x)^12 + 126*cosh(x)^10 - 126*cosh(x)^8 + 84*cosh(x)^6 - 36*cosh(x)^4 + 9*cosh
(x)^2 - 1)*e^(2*x))

Sympy [F]

\[ \int \left (a \text {csch}^4(x)\right )^{5/2} \, dx=\int \left (a \operatorname {csch}^{4}{\left (x \right )}\right )^{\frac {5}{2}}\, dx \]

[In]

integrate((a*csch(x)**4)**(5/2),x)

[Out]

Integral((a*csch(x)**4)**(5/2), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 322 vs. \(2 (100) = 200\).

Time = 0.27 (sec) , antiderivative size = 322, normalized size of antiderivative = 2.73 \[ \int \left (a \text {csch}^4(x)\right )^{5/2} \, dx=-\frac {256 \, a^{\frac {5}{2}} e^{\left (-2 \, x\right )}}{35 \, {\left (9 \, e^{\left (-2 \, x\right )} - 36 \, e^{\left (-4 \, x\right )} + 84 \, e^{\left (-6 \, x\right )} - 126 \, e^{\left (-8 \, x\right )} + 126 \, e^{\left (-10 \, x\right )} - 84 \, e^{\left (-12 \, x\right )} + 36 \, e^{\left (-14 \, x\right )} - 9 \, e^{\left (-16 \, x\right )} + e^{\left (-18 \, x\right )} - 1\right )}} + \frac {1024 \, a^{\frac {5}{2}} e^{\left (-4 \, x\right )}}{35 \, {\left (9 \, e^{\left (-2 \, x\right )} - 36 \, e^{\left (-4 \, x\right )} + 84 \, e^{\left (-6 \, x\right )} - 126 \, e^{\left (-8 \, x\right )} + 126 \, e^{\left (-10 \, x\right )} - 84 \, e^{\left (-12 \, x\right )} + 36 \, e^{\left (-14 \, x\right )} - 9 \, e^{\left (-16 \, x\right )} + e^{\left (-18 \, x\right )} - 1\right )}} - \frac {1024 \, a^{\frac {5}{2}} e^{\left (-6 \, x\right )}}{15 \, {\left (9 \, e^{\left (-2 \, x\right )} - 36 \, e^{\left (-4 \, x\right )} + 84 \, e^{\left (-6 \, x\right )} - 126 \, e^{\left (-8 \, x\right )} + 126 \, e^{\left (-10 \, x\right )} - 84 \, e^{\left (-12 \, x\right )} + 36 \, e^{\left (-14 \, x\right )} - 9 \, e^{\left (-16 \, x\right )} + e^{\left (-18 \, x\right )} - 1\right )}} + \frac {512 \, a^{\frac {5}{2}} e^{\left (-8 \, x\right )}}{5 \, {\left (9 \, e^{\left (-2 \, x\right )} - 36 \, e^{\left (-4 \, x\right )} + 84 \, e^{\left (-6 \, x\right )} - 126 \, e^{\left (-8 \, x\right )} + 126 \, e^{\left (-10 \, x\right )} - 84 \, e^{\left (-12 \, x\right )} + 36 \, e^{\left (-14 \, x\right )} - 9 \, e^{\left (-16 \, x\right )} + e^{\left (-18 \, x\right )} - 1\right )}} + \frac {256 \, a^{\frac {5}{2}}}{315 \, {\left (9 \, e^{\left (-2 \, x\right )} - 36 \, e^{\left (-4 \, x\right )} + 84 \, e^{\left (-6 \, x\right )} - 126 \, e^{\left (-8 \, x\right )} + 126 \, e^{\left (-10 \, x\right )} - 84 \, e^{\left (-12 \, x\right )} + 36 \, e^{\left (-14 \, x\right )} - 9 \, e^{\left (-16 \, x\right )} + e^{\left (-18 \, x\right )} - 1\right )}} \]

[In]

integrate((a*csch(x)^4)^(5/2),x, algorithm="maxima")

[Out]

-256/35*a^(5/2)*e^(-2*x)/(9*e^(-2*x) - 36*e^(-4*x) + 84*e^(-6*x) - 126*e^(-8*x) + 126*e^(-10*x) - 84*e^(-12*x)
+ 36*e^(-14*x) - 9*e^(-16*x) + e^(-18*x) - 1) + 1024/35*a^(5/2)*e^(-4*x)/(9*e^(-2*x) - 36*e^(-4*x) + 84*e^(-6
*x) - 126*e^(-8*x) + 126*e^(-10*x) - 84*e^(-12*x) + 36*e^(-14*x) - 9*e^(-16*x) + e^(-18*x) - 1) - 1024/15*a^(5
/2)*e^(-6*x)/(9*e^(-2*x) - 36*e^(-4*x) + 84*e^(-6*x) - 126*e^(-8*x) + 126*e^(-10*x) - 84*e^(-12*x) + 36*e^(-14
*x) - 9*e^(-16*x) + e^(-18*x) - 1) + 512/5*a^(5/2)*e^(-8*x)/(9*e^(-2*x) - 36*e^(-4*x) + 84*e^(-6*x) - 126*e^(-
8*x) + 126*e^(-10*x) - 84*e^(-12*x) + 36*e^(-14*x) - 9*e^(-16*x) + e^(-18*x) - 1) + 256/315*a^(5/2)/(9*e^(-2*x
) - 36*e^(-4*x) + 84*e^(-6*x) - 126*e^(-8*x) + 126*e^(-10*x) - 84*e^(-12*x) + 36*e^(-14*x) - 9*e^(-16*x) + e^(
-18*x) - 1)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.33 \[ \int \left (a \text {csch}^4(x)\right )^{5/2} \, dx=-\frac {256 \, a^{\frac {5}{2}} {\left (126 \, e^{\left (8 \, x\right )} - 84 \, e^{\left (6 \, x\right )} + 36 \, e^{\left (4 \, x\right )} - 9 \, e^{\left (2 \, x\right )} + 1\right )}}{315 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{9}} \]

[In]

integrate((a*csch(x)^4)^(5/2),x, algorithm="giac")

[Out]

-256/315*a^(5/2)*(126*e^(8*x) - 84*e^(6*x) + 36*e^(4*x) - 9*e^(2*x) + 1)/(e^(2*x) - 1)^9

Mupad [B] (verification not implemented)

Time = 2.16 (sec) , antiderivative size = 356, normalized size of antiderivative = 3.02 \[ \int \left (a \text {csch}^4(x)\right )^{5/2} \, dx=-\frac {128\,a^2\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{5\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^5\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}-\frac {256\,a^2\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{3\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^6\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}-\frac {768\,a^2\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{7\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^7\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}-\frac {64\,a^2\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{{\left ({\mathrm {e}}^{2\,x}-1\right )}^8\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}-\frac {128\,a^2\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{9\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^9\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )} \]

[In]

int((a/sinh(x)^4)^(5/2),x)

[Out]

- (128*a^2*(a/(exp(-x)/2 - exp(x)/2)^4)^(1/2)*(6*exp(4*x) - 4*exp(2*x) - 4*exp(6*x) + exp(8*x) + 1))/(5*(exp(2
*x) - 1)^5*(exp(2*x) - 2*exp(4*x) + exp(6*x))) - (256*a^2*(a/(exp(-x)/2 - exp(x)/2)^4)^(1/2)*(6*exp(4*x) - 4*e
xp(2*x) - 4*exp(6*x) + exp(8*x) + 1))/(3*(exp(2*x) - 1)^6*(exp(2*x) - 2*exp(4*x) + exp(6*x))) - (768*a^2*(a/(e
xp(-x)/2 - exp(x)/2)^4)^(1/2)*(6*exp(4*x) - 4*exp(2*x) - 4*exp(6*x) + exp(8*x) + 1))/(7*(exp(2*x) - 1)^7*(exp(
2*x) - 2*exp(4*x) + exp(6*x))) - (64*a^2*(a/(exp(-x)/2 - exp(x)/2)^4)^(1/2)*(6*exp(4*x) - 4*exp(2*x) - 4*exp(6
*x) + exp(8*x) + 1))/((exp(2*x) - 1)^8*(exp(2*x) - 2*exp(4*x) + exp(6*x))) - (128*a^2*(a/(exp(-x)/2 - exp(x)/2
)^4)^(1/2)*(6*exp(4*x) - 4*exp(2*x) - 4*exp(6*x) + exp(8*x) + 1))/(9*(exp(2*x) - 1)^9*(exp(2*x) - 2*exp(4*x) +
exp(6*x)))