3.215 \(\int \text{csch}^{11}(c+d x) (a+b \sinh ^4(c+d x))^3 \, dx\)

Optimal. Leaf size=189 \[ \frac{3 a \left (21 a^2+80 a b+128 b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{256 d}-\frac{3 a \left (21 a^2+80 a b+128 b^2\right ) \coth (c+d x) \text{csch}(c+d x)}{256 d}-\frac{a^2 (21 a+80 b) \coth (c+d x) \text{csch}^5(c+d x)}{160 d}+\frac{a^2 (21 a+80 b) \coth (c+d x) \text{csch}^3(c+d x)}{128 d}-\frac{a^3 \coth (c+d x) \text{csch}^9(c+d x)}{10 d}+\frac{9 a^3 \coth (c+d x) \text{csch}^7(c+d x)}{80 d}+\frac{b^3 \cosh (c+d x)}{d} \]

[Out]

(3*a*(21*a^2 + 80*a*b + 128*b^2)*ArcTanh[Cosh[c + d*x]])/(256*d) + (b^3*Cosh[c + d*x])/d - (3*a*(21*a^2 + 80*a
*b + 128*b^2)*Coth[c + d*x]*Csch[c + d*x])/(256*d) + (a^2*(21*a + 80*b)*Coth[c + d*x]*Csch[c + d*x]^3)/(128*d)
- (a^2*(21*a + 80*b)*Coth[c + d*x]*Csch[c + d*x]^5)/(160*d) + (9*a^3*Coth[c + d*x]*Csch[c + d*x]^7)/(80*d) -
(a^3*Coth[c + d*x]*Csch[c + d*x]^9)/(10*d)

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Rubi [A]  time = 0.369044, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {3215, 1157, 1814, 388, 206} \[ \frac{3 a \left (21 a^2+80 a b+128 b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{256 d}-\frac{3 a \left (21 a^2+80 a b+128 b^2\right ) \coth (c+d x) \text{csch}(c+d x)}{256 d}-\frac{a^2 (21 a+80 b) \coth (c+d x) \text{csch}^5(c+d x)}{160 d}+\frac{a^2 (21 a+80 b) \coth (c+d x) \text{csch}^3(c+d x)}{128 d}-\frac{a^3 \coth (c+d x) \text{csch}^9(c+d x)}{10 d}+\frac{9 a^3 \coth (c+d x) \text{csch}^7(c+d x)}{80 d}+\frac{b^3 \cosh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^11*(a + b*Sinh[c + d*x]^4)^3,x]

[Out]

(3*a*(21*a^2 + 80*a*b + 128*b^2)*ArcTanh[Cosh[c + d*x]])/(256*d) + (b^3*Cosh[c + d*x])/d - (3*a*(21*a^2 + 80*a
*b + 128*b^2)*Coth[c + d*x]*Csch[c + d*x])/(256*d) + (a^2*(21*a + 80*b)*Coth[c + d*x]*Csch[c + d*x]^3)/(128*d)
- (a^2*(21*a + 80*b)*Coth[c + d*x]*Csch[c + d*x]^5)/(160*d) + (9*a^3*Coth[c + d*x]*Csch[c + d*x]^7)/(80*d) -
(a^3*Coth[c + d*x]*Csch[c + d*x]^9)/(10*d)

Rule 3215

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, -Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4
)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rule 1157

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,
x], x, 0]}, -Simp[(R*x*(d + e*x^2)^(q + 1))/(2*d*(q + 1)), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*
ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && N
eQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rule 1814

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[((a
*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

Rule 388

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

Rule 206

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

Rubi steps

\begin{align*} \int \text{csch}^{11}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b-2 b x^2+b x^4\right )^3}{\left (1-x^2\right )^6} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{a^3 \coth (c+d x) \text{csch}^9(c+d x)}{10 d}-\frac{\operatorname{Subst}\left (\int \frac{-9 a^3-30 a^2 b-30 a b^2-10 b^3+10 b \left (3 a^2+9 a b+5 b^2\right ) x^2-10 b^2 (9 a+10 b) x^4+10 b^2 (3 a+10 b) x^6-50 b^3 x^8+10 b^3 x^{10}}{\left (1-x^2\right )^5} \, dx,x,\cosh (c+d x)\right )}{10 d}\\ &=\frac{9 a^3 \coth (c+d x) \text{csch}^7(c+d x)}{80 d}-\frac{a^3 \coth (c+d x) \text{csch}^9(c+d x)}{10 d}+\frac{\operatorname{Subst}\left (\int \frac{63 a^3+240 a^2 b+240 a b^2+80 b^3-160 b^2 (3 a+2 b) x^2+240 b^2 (a+2 b) x^4-320 b^3 x^6+80 b^3 x^8}{\left (1-x^2\right )^4} \, dx,x,\cosh (c+d x)\right )}{80 d}\\ &=-\frac{a^2 (21 a+80 b) \coth (c+d x) \text{csch}^5(c+d x)}{160 d}+\frac{9 a^3 \coth (c+d x) \text{csch}^7(c+d x)}{80 d}-\frac{a^3 \coth (c+d x) \text{csch}^9(c+d x)}{10 d}-\frac{\operatorname{Subst}\left (\int \frac{-15 \left (21 a^3+80 a^2 b+96 a b^2+32 b^3\right )+1440 b^2 (a+b) x^2-1440 b^3 x^4+480 b^3 x^6}{\left (1-x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{480 d}\\ &=\frac{a^2 (21 a+80 b) \coth (c+d x) \text{csch}^3(c+d x)}{128 d}-\frac{a^2 (21 a+80 b) \coth (c+d x) \text{csch}^5(c+d x)}{160 d}+\frac{9 a^3 \coth (c+d x) \text{csch}^7(c+d x)}{80 d}-\frac{a^3 \coth (c+d x) \text{csch}^9(c+d x)}{10 d}+\frac{\operatorname{Subst}\left (\int \frac{15 \left (63 a^3+240 a^2 b+384 a b^2+128 b^3\right )-3840 b^3 x^2+1920 b^3 x^4}{\left (1-x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{1920 d}\\ &=-\frac{3 a \left (21 a^2+80 a b+128 b^2\right ) \coth (c+d x) \text{csch}(c+d x)}{256 d}+\frac{a^2 (21 a+80 b) \coth (c+d x) \text{csch}^3(c+d x)}{128 d}-\frac{a^2 (21 a+80 b) \coth (c+d x) \text{csch}^5(c+d x)}{160 d}+\frac{9 a^3 \coth (c+d x) \text{csch}^7(c+d x)}{80 d}-\frac{a^3 \coth (c+d x) \text{csch}^9(c+d x)}{10 d}-\frac{\operatorname{Subst}\left (\int \frac{-15 \left (63 a^3+240 a^2 b+384 a b^2+256 b^3\right )+3840 b^3 x^2}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{3840 d}\\ &=\frac{b^3 \cosh (c+d x)}{d}-\frac{3 a \left (21 a^2+80 a b+128 b^2\right ) \coth (c+d x) \text{csch}(c+d x)}{256 d}+\frac{a^2 (21 a+80 b) \coth (c+d x) \text{csch}^3(c+d x)}{128 d}-\frac{a^2 (21 a+80 b) \coth (c+d x) \text{csch}^5(c+d x)}{160 d}+\frac{9 a^3 \coth (c+d x) \text{csch}^7(c+d x)}{80 d}-\frac{a^3 \coth (c+d x) \text{csch}^9(c+d x)}{10 d}+\frac{\left (3 a \left (21 a^2+80 a b+128 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{256 d}\\ &=\frac{3 a \left (21 a^2+80 a b+128 b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{256 d}+\frac{b^3 \cosh (c+d x)}{d}-\frac{3 a \left (21 a^2+80 a b+128 b^2\right ) \coth (c+d x) \text{csch}(c+d x)}{256 d}+\frac{a^2 (21 a+80 b) \coth (c+d x) \text{csch}^3(c+d x)}{128 d}-\frac{a^2 (21 a+80 b) \coth (c+d x) \text{csch}^5(c+d x)}{160 d}+\frac{9 a^3 \coth (c+d x) \text{csch}^7(c+d x)}{80 d}-\frac{a^3 \coth (c+d x) \text{csch}^9(c+d x)}{10 d}\\ \end{align*}

Mathematica [A]  time = 2.47304, size = 265, normalized size = 1.4 \[ \frac{b^3 \cosh (c+d x)}{d}-\frac{a \left (60 \left (21 a^2+80 a b+128 b^2\right ) \text{csch}^2\left (\frac{1}{2} (c+d x)\right )+60 \left (21 a^2+80 a b+128 b^2\right ) \text{sech}^2\left (\frac{1}{2} (c+d x)\right )+240 \left (21 a^2+80 a b+128 b^2\right ) \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )+2 a^2 \text{csch}^{10}\left (\frac{1}{2} (c+d x)\right )-15 a^2 \text{csch}^8\left (\frac{1}{2} (c+d x)\right )+2 a^2 \text{sech}^{10}\left (\frac{1}{2} (c+d x)\right )+15 a^2 \text{sech}^8\left (\frac{1}{2} (c+d x)\right )+10 a (7 a+16 b) \text{csch}^6\left (\frac{1}{2} (c+d x)\right )-40 a (7 a+24 b) \text{csch}^4\left (\frac{1}{2} (c+d x)\right )+10 a (7 a+16 b) \text{sech}^6\left (\frac{1}{2} (c+d x)\right )+40 a (7 a+24 b) \text{sech}^4\left (\frac{1}{2} (c+d x)\right )\right )}{20480 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^11*(a + b*Sinh[c + d*x]^4)^3,x]

[Out]

(b^3*Cosh[c + d*x])/d - (a*(60*(21*a^2 + 80*a*b + 128*b^2)*Csch[(c + d*x)/2]^2 - 40*a*(7*a + 24*b)*Csch[(c + d
*x)/2]^4 + 10*a*(7*a + 16*b)*Csch[(c + d*x)/2]^6 - 15*a^2*Csch[(c + d*x)/2]^8 + 2*a^2*Csch[(c + d*x)/2]^10 + 2
40*(21*a^2 + 80*a*b + 128*b^2)*Log[Tanh[(c + d*x)/2]] + 60*(21*a^2 + 80*a*b + 128*b^2)*Sech[(c + d*x)/2]^2 + 4
0*a*(7*a + 24*b)*Sech[(c + d*x)/2]^4 + 10*a*(7*a + 16*b)*Sech[(c + d*x)/2]^6 + 15*a^2*Sech[(c + d*x)/2]^8 + 2*
a^2*Sech[(c + d*x)/2]^10))/(20480*d)

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Maple [A]  time = 0.081, size = 166, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ( \left ( -{\frac{ \left ({\rm csch} \left (dx+c\right ) \right ) ^{9}}{10}}+{\frac{9\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{7}}{80}}-{\frac{21\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{5}}{160}}+{\frac{21\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{3}}{128}}-{\frac{63\,{\rm csch} \left (dx+c\right )}{256}} \right ){\rm coth} \left (dx+c\right )+{\frac{63\,{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) }{128}} \right ) +3\,{a}^{2}b \left ( \left ( -1/6\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{5}+{\frac{5\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{3}}{24}}-{\frac{5\,{\rm csch} \left (dx+c\right )}{16}} \right ){\rm coth} \left (dx+c\right )+5/8\,{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) \right ) +3\,a{b}^{2} \left ( -1/2\,{\rm csch} \left (dx+c\right ){\rm coth} \left (dx+c\right )+{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) \right ) +{b}^{3}\cosh \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^11*(a+b*sinh(d*x+c)^4)^3,x)

[Out]

1/d*(a^3*((-1/10*csch(d*x+c)^9+9/80*csch(d*x+c)^7-21/160*csch(d*x+c)^5+21/128*csch(d*x+c)^3-63/256*csch(d*x+c)
)*coth(d*x+c)+63/128*arctanh(exp(d*x+c)))+3*a^2*b*((-1/6*csch(d*x+c)^5+5/24*csch(d*x+c)^3-5/16*csch(d*x+c))*co
th(d*x+c)+5/8*arctanh(exp(d*x+c)))+3*a*b^2*(-1/2*csch(d*x+c)*coth(d*x+c)+arctanh(exp(d*x+c)))+b^3*cosh(d*x+c))

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Maxima [B]  time = 1.14825, size = 774, normalized size = 4.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^11*(a+b*sinh(d*x+c)^4)^3,x, algorithm="maxima")

[Out]

1/2*b^3*(e^(d*x + c)/d + e^(-d*x - c)/d) + 1/1280*a^3*(315*log(e^(-d*x - c) + 1)/d - 315*log(e^(-d*x - c) - 1)
/d + 2*(315*e^(-d*x - c) - 3045*e^(-3*d*x - 3*c) + 13188*e^(-5*d*x - 5*c) - 33660*e^(-7*d*x - 7*c) + 55970*e^(
-9*d*x - 9*c) + 55970*e^(-11*d*x - 11*c) - 33660*e^(-13*d*x - 13*c) + 13188*e^(-15*d*x - 15*c) - 3045*e^(-17*d
*x - 17*c) + 315*e^(-19*d*x - 19*c))/(d*(10*e^(-2*d*x - 2*c) - 45*e^(-4*d*x - 4*c) + 120*e^(-6*d*x - 6*c) - 21
0*e^(-8*d*x - 8*c) + 252*e^(-10*d*x - 10*c) - 210*e^(-12*d*x - 12*c) + 120*e^(-14*d*x - 14*c) - 45*e^(-16*d*x
- 16*c) + 10*e^(-18*d*x - 18*c) - e^(-20*d*x - 20*c) - 1))) + 1/16*a^2*b*(15*log(e^(-d*x - c) + 1)/d - 15*log(
e^(-d*x - c) - 1)/d + 2*(15*e^(-d*x - c) - 85*e^(-3*d*x - 3*c) + 198*e^(-5*d*x - 5*c) + 198*e^(-7*d*x - 7*c) -
85*e^(-9*d*x - 9*c) + 15*e^(-11*d*x - 11*c))/(d*(6*e^(-2*d*x - 2*c) - 15*e^(-4*d*x - 4*c) + 20*e^(-6*d*x - 6*
c) - 15*e^(-8*d*x - 8*c) + 6*e^(-10*d*x - 10*c) - e^(-12*d*x - 12*c) - 1))) + 3/2*a*b^2*(log(e^(-d*x - c) + 1)
/d - log(e^(-d*x - c) - 1)/d + 2*(e^(-d*x - c) + e^(-3*d*x - 3*c))/(d*(2*e^(-2*d*x - 2*c) - e^(-4*d*x - 4*c) -
1)))

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Fricas [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(csch(d*x+c)^11*(a+b*sinh(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

Timed out

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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(csch(d*x+c)**11*(a+b*sinh(d*x+c)**4)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.70407, size = 653, normalized size = 3.46 \begin{align*} \frac{b^{3}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}}{2 \, d} + \frac{3 \,{\left (21 \, a^{3} + 80 \, a^{2} b + 128 \, a b^{2}\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right )}{512 \, d} - \frac{3 \,{\left (21 \, a^{3} + 80 \, a^{2} b + 128 \, a b^{2}\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right )}{512 \, d} - \frac{315 \, a^{3}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{9} + 1200 \, a^{2} b{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{9} + 1920 \, a b^{2}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{9} - 5880 \, a^{3}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} - 22400 \, a^{2} b{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} - 30720 \, a b^{2}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} + 43008 \, a^{3}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} + 163840 \, a^{2} b{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} + 184320 \, a b^{2}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} - 151680 \, a^{3}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 542720 \, a^{2} b{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 491520 \, a b^{2}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 247040 \, a^{3}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 675840 \, a^{2} b{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 491520 \, a b^{2}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}}{640 \,{\left ({\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 4\right )}^{5} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^11*(a+b*sinh(d*x+c)^4)^3,x, algorithm="giac")

[Out]

1/2*b^3*(e^(d*x + c) + e^(-d*x - c))/d + 3/512*(21*a^3 + 80*a^2*b + 128*a*b^2)*log(e^(d*x + c) + e^(-d*x - c)
+ 2)/d - 3/512*(21*a^3 + 80*a^2*b + 128*a*b^2)*log(e^(d*x + c) + e^(-d*x - c) - 2)/d - 1/640*(315*a^3*(e^(d*x
+ c) + e^(-d*x - c))^9 + 1200*a^2*b*(e^(d*x + c) + e^(-d*x - c))^9 + 1920*a*b^2*(e^(d*x + c) + e^(-d*x - c))^9
- 5880*a^3*(e^(d*x + c) + e^(-d*x - c))^7 - 22400*a^2*b*(e^(d*x + c) + e^(-d*x - c))^7 - 30720*a*b^2*(e^(d*x
+ c) + e^(-d*x - c))^7 + 43008*a^3*(e^(d*x + c) + e^(-d*x - c))^5 + 163840*a^2*b*(e^(d*x + c) + e^(-d*x - c))^
5 + 184320*a*b^2*(e^(d*x + c) + e^(-d*x - c))^5 - 151680*a^3*(e^(d*x + c) + e^(-d*x - c))^3 - 542720*a^2*b*(e^
(d*x + c) + e^(-d*x - c))^3 - 491520*a*b^2*(e^(d*x + c) + e^(-d*x - c))^3 + 247040*a^3*(e^(d*x + c) + e^(-d*x
- c)) + 675840*a^2*b*(e^(d*x + c) + e^(-d*x - c)) + 491520*a*b^2*(e^(d*x + c) + e^(-d*x - c)))/(((e^(d*x + c)
+ e^(-d*x - c))^2 - 4)^5*d)