3.1.66 \(\int \cosh ^3(c+d x) (a+b \text {sech}^2(c+d x))^3 \, dx\) [66]

Optimal. Leaf size=81 \[ \frac {b^2 (6 a+b) \text {ArcTan}(\sinh (c+d x))}{2 d}+\frac {a^2 (a+3 b) \sinh (c+d x)}{d}+\frac {a^3 \sinh ^3(c+d x)}{3 d}+\frac {b^3 \text {sech}(c+d x) \tanh (c+d x)}{2 d} \]

[Out]

1/2*b^2*(6*a+b)*arctan(sinh(d*x+c))/d+a^2*(a+3*b)*sinh(d*x+c)/d+1/3*a^3*sinh(d*x+c)^3/d+1/2*b^3*sech(d*x+c)*ta
nh(d*x+c)/d

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Rubi [A]
time = 0.06, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {4232, 398, 393, 209} \begin {gather*} \frac {a^3 \sinh ^3(c+d x)}{3 d}+\frac {a^2 (a+3 b) \sinh (c+d x)}{d}+\frac {b^2 (6 a+b) \text {ArcTan}(\sinh (c+d x))}{2 d}+\frac {b^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]^3*(a + b*Sech[c + d*x]^2)^3,x]

[Out]

(b^2*(6*a + b)*ArcTan[Sinh[c + d*x]])/(2*d) + (a^2*(a + 3*b)*Sinh[c + d*x])/d + (a^3*Sinh[c + d*x]^3)/(3*d) +
(b^3*Sech[c + d*x]*Tanh[c + d*x])/(2*d)

Rule 209

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

Rule 393

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

Rule 398

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rule 4232

Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = Fr
eeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[ExpandToSum[b + a*(1 - ff^2*x^2)^(n/2), x]^p/(1 - ff^2*x^2)^
((m + n*p + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n
/2] && IntegerQ[p]

Rubi steps

\begin {align*} \int \cosh ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b+a x^2\right )^3}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (a^2 (a+3 b)+a^3 x^2+\frac {b^2 (3 a+b)+3 a b^2 x^2}{\left (1+x^2\right )^2}\right ) \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {a^2 (a+3 b) \sinh (c+d x)}{d}+\frac {a^3 \sinh ^3(c+d x)}{3 d}+\frac {\text {Subst}\left (\int \frac {b^2 (3 a+b)+3 a b^2 x^2}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {a^2 (a+3 b) \sinh (c+d x)}{d}+\frac {a^3 \sinh ^3(c+d x)}{3 d}+\frac {b^3 \text {sech}(c+d x) \tanh (c+d x)}{2 d}+\frac {\left (b^2 (6 a+b)\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{2 d}\\ &=\frac {b^2 (6 a+b) \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac {a^2 (a+3 b) \sinh (c+d x)}{d}+\frac {a^3 \sinh ^3(c+d x)}{3 d}+\frac {b^3 \text {sech}(c+d x) \tanh (c+d x)}{2 d}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 6.67, size = 483, normalized size = 5.96 \begin {gather*} \frac {\coth ^3(c+d x) \text {csch}^2(c+d x) (a \cosh (c+d x)+b \text {sech}(c+d x))^3 \left (-256 \, _5F_4\left (\frac {3}{2},2,2,2,2;1,1,1,\frac {11}{2};-\sinh ^2(c+d x)\right ) \sinh ^8(c+d x) \left (a+b+a \sinh ^2(c+d x)\right )^3-\frac {315 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \left (b^3 \left (2401+1875 \sinh ^2(c+d x)+243 \sinh ^4(c+d x)-47 \sinh ^6(c+d x)\right )+3 a^2 b \cosh ^4(c+d x) \left (2401+1875 \sinh ^2(c+d x)+243 \sinh ^4(c+d x)+\sinh ^6(c+d x)\right )+a^3 \cosh ^6(c+d x) \left (2401+1875 \sinh ^2(c+d x)+243 \sinh ^4(c+d x)+\sinh ^6(c+d x)\right )+3 a b^2 \left (2401+4276 \sinh ^2(c+d x)+2118 \sinh ^4(c+d x)+148 \sinh ^6(c+d x)+\sinh ^8(c+d x)\right )\right )}{\sqrt {-\sinh ^2(c+d x)}}+21 \left (b^3 \left (36015+16120 \sinh ^2(c+d x)+1473 \sinh ^4(c+d x)\right )+3 a b^2 \left (36015+52135 \sinh ^2(c+d x)+17593 \sinh ^4(c+d x)+753 \sinh ^6(c+d x)\right )+3 a^2 b \left (36015+88150 \sinh ^2(c+d x)+69728 \sinh ^4(c+d x)+19786 \sinh ^6(c+d x)+753 \sinh ^8(c+d x)\right )+a^3 \left (36015+124165 \sinh ^2(c+d x)+157878 \sinh ^4(c+d x)+89514 \sinh ^6(c+d x)+19579 \sinh ^8(c+d x)+753 \sinh ^{10}(c+d x)\right )\right )\right )}{3780 d (a+2 b+a \cosh (2 c+2 d x))^3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Cosh[c + d*x]^3*(a + b*Sech[c + d*x]^2)^3,x]

[Out]

(Coth[c + d*x]^3*Csch[c + d*x]^2*(a*Cosh[c + d*x] + b*Sech[c + d*x])^3*(-256*HypergeometricPFQ[{3/2, 2, 2, 2,
2}, {1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^8*(a + b + a*Sinh[c + d*x]^2)^3 - (315*ArcTanh[Sqrt[-Sinh
[c + d*x]^2]]*(b^3*(2401 + 1875*Sinh[c + d*x]^2 + 243*Sinh[c + d*x]^4 - 47*Sinh[c + d*x]^6) + 3*a^2*b*Cosh[c +
d*x]^4*(2401 + 1875*Sinh[c + d*x]^2 + 243*Sinh[c + d*x]^4 + Sinh[c + d*x]^6) + a^3*Cosh[c + d*x]^6*(2401 + 18
75*Sinh[c + d*x]^2 + 243*Sinh[c + d*x]^4 + Sinh[c + d*x]^6) + 3*a*b^2*(2401 + 4276*Sinh[c + d*x]^2 + 2118*Sinh
[c + d*x]^4 + 148*Sinh[c + d*x]^6 + Sinh[c + d*x]^8)))/Sqrt[-Sinh[c + d*x]^2] + 21*(b^3*(36015 + 16120*Sinh[c
+ d*x]^2 + 1473*Sinh[c + d*x]^4) + 3*a*b^2*(36015 + 52135*Sinh[c + d*x]^2 + 17593*Sinh[c + d*x]^4 + 753*Sinh[c
+ d*x]^6) + 3*a^2*b*(36015 + 88150*Sinh[c + d*x]^2 + 69728*Sinh[c + d*x]^4 + 19786*Sinh[c + d*x]^6 + 753*Sinh
[c + d*x]^8) + a^3*(36015 + 124165*Sinh[c + d*x]^2 + 157878*Sinh[c + d*x]^4 + 89514*Sinh[c + d*x]^6 + 19579*Si
nh[c + d*x]^8 + 753*Sinh[c + d*x]^10))))/(3780*d*(a + 2*b + a*Cosh[2*c + 2*d*x])^3)

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Maple [C] Result contains complex when optimal does not.
time = 1.91, size = 215, normalized size = 2.65

method	result	size
risch	\(\frac {{\mathrm e}^{3 d x +3 c} a^{3}}{24 d}+\frac {3 \,{\mathrm e}^{d x +c} a^{3}}{8 d}+\frac {3 \,{\mathrm e}^{d x +c} a^{2} b}{2 d}-\frac {3 \,{\mathrm e}^{-d x -c} a^{3}}{8 d}-\frac {3 \,{\mathrm e}^{-d x -c} a^{2} b}{2 d}-\frac {{\mathrm e}^{-3 d x -3 c} a^{3}}{24 d}+\frac {b^{3} {\mathrm e}^{d x +c} \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}+\frac {3 i b^{2} \ln \left ({\mathrm e}^{d x +c}+i\right ) a}{d}+\frac {i b^{3} \ln \left ({\mathrm e}^{d x +c}+i\right )}{2 d}-\frac {3 i b^{2} \ln \left ({\mathrm e}^{d x +c}-i\right ) a}{d}-\frac {i b^{3} \ln \left ({\mathrm e}^{d x +c}-i\right )}{2 d}\)	\(215\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(d*x+c)^3*(a+b*sech(d*x+c)^2)^3,x,method=_RETURNVERBOSE)

[Out]

1/24/d*exp(3*d*x+3*c)*a^3+3/8/d*exp(d*x+c)*a^3+3/2/d*exp(d*x+c)*a^2*b-3/8/d*exp(-d*x-c)*a^3-3/2/d*exp(-d*x-c)*
a^2*b-1/24/d*exp(-3*d*x-3*c)*a^3+b^3*exp(d*x+c)*(exp(2*d*x+2*c)-1)/d/(1+exp(2*d*x+2*c))^2+3*I*b^2/d*ln(exp(d*x
+c)+I)*a+1/2*I*b^3/d*ln(exp(d*x+c)+I)-3*I*b^2/d*ln(exp(d*x+c)-I)*a-1/2*I*b^3/d*ln(exp(d*x+c)-I)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 179 vs. \(2 (75) = 150\).
time = 0.52, size = 179, normalized size = 2.21 \begin {gather*} -b^{3} {\left (\frac {\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} + \frac {1}{24} \, a^{3} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} - \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + \frac {3}{2} \, a^{2} b {\left (\frac {e^{\left (d x + c\right )}}{d} - \frac {e^{\left (-d x - c\right )}}{d}\right )} - \frac {6 \, a b^{2} \arctan \left (e^{\left (-d x - c\right )}\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^3*(a+b*sech(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

-b^3*(arctan(e^(-d*x - c))/d - (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
))) + 1/24*a^3*(e^(3*d*x + 3*c)/d + 9*e^(d*x + c)/d - 9*e^(-d*x - c)/d - e^(-3*d*x - 3*c)/d) + 3/2*a^2*b*(e^(d
*x + c)/d - e^(-d*x - c)/d) - 6*a*b^2*arctan(e^(-d*x - c))/d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1409 vs. \(2 (75) = 150\).
time = 0.42, size = 1409, normalized size = 17.40 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^3*(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

1/24*(a^3*cosh(d*x + c)^10 + 10*a^3*cosh(d*x + c)*sinh(d*x + c)^9 + a^3*sinh(d*x + c)^10 + (11*a^3 + 36*a^2*b)
*cosh(d*x + c)^8 + (45*a^3*cosh(d*x + c)^2 + 11*a^3 + 36*a^2*b)*sinh(d*x + c)^8 + 8*(15*a^3*cosh(d*x + c)^3 +
(11*a^3 + 36*a^2*b)*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(5*a^3 + 18*a^2*b + 12*b^3)*cosh(d*x + c)^6 + 2*(105*a^
3*cosh(d*x + c)^4 + 5*a^3 + 18*a^2*b + 12*b^3 + 14*(11*a^3 + 36*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(6
3*a^3*cosh(d*x + c)^5 + 14*(11*a^3 + 36*a^2*b)*cosh(d*x + c)^3 + 3*(5*a^3 + 18*a^2*b + 12*b^3)*cosh(d*x + c))*
sinh(d*x + c)^5 - 2*(5*a^3 + 18*a^2*b + 12*b^3)*cosh(d*x + c)^4 + 2*(105*a^3*cosh(d*x + c)^6 + 35*(11*a^3 + 36
*a^2*b)*cosh(d*x + c)^4 - 5*a^3 - 18*a^2*b - 12*b^3 + 15*(5*a^3 + 18*a^2*b + 12*b^3)*cosh(d*x + c)^2)*sinh(d*x
+ c)^4 + 8*(15*a^3*cosh(d*x + c)^7 + 7*(11*a^3 + 36*a^2*b)*cosh(d*x + c)^5 + 5*(5*a^3 + 18*a^2*b + 12*b^3)*co
sh(d*x + c)^3 - (5*a^3 + 18*a^2*b + 12*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 - a^3 - (11*a^3 + 36*a^2*b)*cosh(d*
x + c)^2 + (45*a^3*cosh(d*x + c)^8 + 28*(11*a^3 + 36*a^2*b)*cosh(d*x + c)^6 + 30*(5*a^3 + 18*a^2*b + 12*b^3)*c
osh(d*x + c)^4 - 11*a^3 - 36*a^2*b - 12*(5*a^3 + 18*a^2*b + 12*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 24*((6*
a*b^2 + b^3)*cosh(d*x + c)^7 + 7*(6*a*b^2 + b^3)*cosh(d*x + c)*sinh(d*x + c)^6 + (6*a*b^2 + b^3)*sinh(d*x + c)
^7 + 2*(6*a*b^2 + b^3)*cosh(d*x + c)^5 + (12*a*b^2 + 2*b^3 + 21*(6*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)
^5 + 5*(7*(6*a*b^2 + b^3)*cosh(d*x + c)^3 + 2*(6*a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c)^4 + (6*a*b^2 + b^3)
*cosh(d*x + c)^3 + (35*(6*a*b^2 + b^3)*cosh(d*x + c)^4 + 6*a*b^2 + b^3 + 20*(6*a*b^2 + b^3)*cosh(d*x + c)^2)*s
inh(d*x + c)^3 + (21*(6*a*b^2 + b^3)*cosh(d*x + c)^5 + 20*(6*a*b^2 + b^3)*cosh(d*x + c)^3 + 3*(6*a*b^2 + b^3)*
cosh(d*x + c))*sinh(d*x + c)^2 + (7*(6*a*b^2 + b^3)*cosh(d*x + c)^6 + 10*(6*a*b^2 + b^3)*cosh(d*x + c)^4 + 3*(
6*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c))*arctan(cosh(d*x + c) + sinh(d*x + c)) + 2*(5*a^3*cosh(d*x + c)^
9 + 4*(11*a^3 + 36*a^2*b)*cosh(d*x + c)^7 + 6*(5*a^3 + 18*a^2*b + 12*b^3)*cosh(d*x + c)^5 - 4*(5*a^3 + 18*a^2*
b + 12*b^3)*cosh(d*x + c)^3 - (11*a^3 + 36*a^2*b)*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^7 + 7*d*cosh(
d*x + c)*sinh(d*x + c)^6 + d*sinh(d*x + c)^7 + 2*d*cosh(d*x + c)^5 + (21*d*cosh(d*x + c)^2 + 2*d)*sinh(d*x + c
)^5 + 5*(7*d*cosh(d*x + c)^3 + 2*d*cosh(d*x + c))*sinh(d*x + c)^4 + d*cosh(d*x + c)^3 + (35*d*cosh(d*x + c)^4
+ 20*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^3 + (21*d*cosh(d*x + c)^5 + 20*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c)
)*sinh(d*x + c)^2 + (7*d*cosh(d*x + c)^6 + 10*d*cosh(d*x + c)^4 + 3*d*cosh(d*x + c)^2)*sinh(d*x + c))

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)**3*(a+b*sech(d*x+c)**2)**3,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3005 deep

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (75) = 150\).
time = 0.42, size = 163, normalized size = 2.01 \begin {gather*} \frac {a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 12 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 36 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + \frac {24 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}}{{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4} + 6 \, {\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (6 \, a b^{2} + b^{3}\right )}}{24 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^3*(a+b*sech(d*x+c)^2)^3,x, algorithm="giac")

[Out]

1/24*(a^3*(e^(d*x + c) - e^(-d*x - c))^3 + 12*a^3*(e^(d*x + c) - e^(-d*x - c)) + 36*a^2*b*(e^(d*x + c) - e^(-d
*x - c)) + 24*b^3*(e^(d*x + c) - e^(-d*x - c))/((e^(d*x + c) - e^(-d*x - c))^2 + 4) + 6*(pi + 2*arctan(1/2*(e^
(2*d*x + 2*c) - 1)*e^(-d*x - c)))*(6*a*b^2 + b^3))/d

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Mupad [B]
time = 0.22, size = 218, normalized size = 2.69 \begin {gather*} \frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (b^3\,\sqrt {d^2}+6\,a\,b^2\,\sqrt {d^2}\right )}{d\,\sqrt {36\,a^2\,b^4+12\,a\,b^5+b^6}}\right )\,\sqrt {36\,a^2\,b^4+12\,a\,b^5+b^6}}{\sqrt {d^2}}-\frac {a^3\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,d}+\frac {a^3\,{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,d}-\frac {3\,a^2\,{\mathrm {e}}^{-c-d\,x}\,\left (a+4\,b\right )}{8\,d}+\frac {3\,a^2\,{\mathrm {e}}^{c+d\,x}\,\left (a+4\,b\right )}{8\,d}+\frac {b^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {2\,b^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(c + d*x)^3*(a + b/cosh(c + d*x)^2)^3,x)

[Out]

(atan((exp(d*x)*exp(c)*(b^3*(d^2)^(1/2) + 6*a*b^2*(d^2)^(1/2)))/(d*(12*a*b^5 + b^6 + 36*a^2*b^4)^(1/2)))*(12*a
*b^5 + b^6 + 36*a^2*b^4)^(1/2))/(d^2)^(1/2) - (a^3*exp(- 3*c - 3*d*x))/(24*d) + (a^3*exp(3*c + 3*d*x))/(24*d)
- (3*a^2*exp(- c - d*x)*(a + 4*b))/(8*d) + (3*a^2*exp(c + d*x)*(a + 4*b))/(8*d) + (b^3*exp(c + d*x))/(d*(exp(2
*c + 2*d*x) + 1)) - (2*b^3*exp(c + d*x))/(d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1))

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