### 3.103 $$\int \text{sech}^3(c+d x) (a+b \tanh ^2(c+d x))^3 \, dx$$

Optimal. Leaf size=198 $\frac{\left (48 a^2 b+64 a^3+24 a b^2+5 b^3\right ) \tan ^{-1}(\sinh (c+d x))}{128 d}-\frac{b \left (72 a^2+52 a b+15 b^2\right ) \tanh (c+d x) \text{sech}^3(c+d x)}{192 d}+\frac{\left (48 a^2 b+64 a^3+24 a b^2+5 b^3\right ) \tanh (c+d x) \text{sech}(c+d x)}{128 d}-\frac{b \tanh (c+d x) \text{sech}^7(c+d x) \left ((a+b) \sinh ^2(c+d x)+a\right )^2}{8 d}-\frac{b (12 a+5 b) \tanh (c+d x) \text{sech}^5(c+d x) \left ((a+b) \sinh ^2(c+d x)+a\right )}{48 d}$

[Out]

((64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*ArcTan[Sinh[c + d*x]])/(128*d) + ((64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^
3)*Sech[c + d*x]*Tanh[c + d*x])/(128*d) - (b*(72*a^2 + 52*a*b + 15*b^2)*Sech[c + d*x]^3*Tanh[c + d*x])/(192*d)
- (b*(12*a + 5*b)*Sech[c + d*x]^5*(a + (a + b)*Sinh[c + d*x]^2)*Tanh[c + d*x])/(48*d) - (b*Sech[c + d*x]^7*(a
+ (a + b)*Sinh[c + d*x]^2)^2*Tanh[c + d*x])/(8*d)

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Rubi [A]  time = 0.239466, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.261, Rules used = {3676, 413, 526, 385, 199, 203} $\frac{\left (48 a^2 b+64 a^3+24 a b^2+5 b^3\right ) \tan ^{-1}(\sinh (c+d x))}{128 d}-\frac{b \left (72 a^2+52 a b+15 b^2\right ) \tanh (c+d x) \text{sech}^3(c+d x)}{192 d}+\frac{\left (48 a^2 b+64 a^3+24 a b^2+5 b^3\right ) \tanh (c+d x) \text{sech}(c+d x)}{128 d}-\frac{b \tanh (c+d x) \text{sech}^7(c+d x) \left ((a+b) \sinh ^2(c+d x)+a\right )^2}{8 d}-\frac{b (12 a+5 b) \tanh (c+d x) \text{sech}^5(c+d x) \left ((a+b) \sinh ^2(c+d x)+a\right )}{48 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*ArcTan[Sinh[c + d*x]])/(128*d) + ((64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^
3)*Sech[c + d*x]*Tanh[c + d*x])/(128*d) - (b*(72*a^2 + 52*a*b + 15*b^2)*Sech[c + d*x]^3*Tanh[c + d*x])/(192*d)
- (b*(12*a + 5*b)*Sech[c + d*x]^5*(a + (a + b)*Sinh[c + d*x]^2)*Tanh[c + d*x])/(48*d) - (b*Sech[c + d*x]^7*(a
+ (a + b)*Sinh[c + d*x]^2)^2*Tanh[c + d*x])/(8*d)

Rule 3676

Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_.), x_Symbol] :> With[{ff = F
reeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[ExpandToSum[b*(ff*x)^n + 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]

Rule 413

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[((a*d - c*b)*x*(a + b*x^n)^
(p + 1)*(c + d*x^n)^(q - 1))/(a*b*n*(p + 1)), x] - Dist[1/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)
^(q - 2)*Simp[c*(a*d - c*b*(n*(p + 1) + 1)) + d*(a*d*(n*(q - 1) + 1) - b*c*(n*(p + q) + 1))*x^n, x], x], x] /;
FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, n, p, q
, x]

Rule 526

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

Rule 385

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 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 203

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

Rubi steps

\begin{align*} \int \text{sech}^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+(a+b) x^2\right )^3}{\left (1+x^2\right )^5} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=-\frac{b \text{sech}^7(c+d x) \left (a+(a+b) \sinh ^2(c+d x)\right )^2 \tanh (c+d x)}{8 d}+\frac{\operatorname{Subst}\left (\int \frac{\left (a+(a+b) x^2\right ) \left (a (8 a+b)+(a+b) (8 a+5 b) x^2\right )}{\left (1+x^2\right )^4} \, dx,x,\sinh (c+d x)\right )}{8 d}\\ &=-\frac{b (12 a+5 b) \text{sech}^5(c+d x) \left (a+(a+b) \sinh ^2(c+d x)\right ) \tanh (c+d x)}{48 d}-\frac{b \text{sech}^7(c+d x) \left (a+(a+b) \sinh ^2(c+d x)\right )^2 \tanh (c+d x)}{8 d}-\frac{\operatorname{Subst}\left (\int \frac{-a \left (48 a^2+18 a b+5 b^2\right )-3 (a+b) \left (16 a^2+14 a b+5 b^2\right ) x^2}{\left (1+x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{48 d}\\ &=-\frac{b \left (72 a^2+52 a b+15 b^2\right ) \text{sech}^3(c+d x) \tanh (c+d x)}{192 d}-\frac{b (12 a+5 b) \text{sech}^5(c+d x) \left (a+(a+b) \sinh ^2(c+d x)\right ) \tanh (c+d x)}{48 d}-\frac{b \text{sech}^7(c+d x) \left (a+(a+b) \sinh ^2(c+d x)\right )^2 \tanh (c+d x)}{8 d}+\frac{\left (64 a^3+48 a^2 b+24 a b^2+5 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{64 d}\\ &=\frac{\left (64 a^3+48 a^2 b+24 a b^2+5 b^3\right ) \text{sech}(c+d x) \tanh (c+d x)}{128 d}-\frac{b \left (72 a^2+52 a b+15 b^2\right ) \text{sech}^3(c+d x) \tanh (c+d x)}{192 d}-\frac{b (12 a+5 b) \text{sech}^5(c+d x) \left (a+(a+b) \sinh ^2(c+d x)\right ) \tanh (c+d x)}{48 d}-\frac{b \text{sech}^7(c+d x) \left (a+(a+b) \sinh ^2(c+d x)\right )^2 \tanh (c+d x)}{8 d}+\frac{\left (64 a^3+48 a^2 b+24 a b^2+5 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{128 d}\\ &=\frac{\left (64 a^3+48 a^2 b+24 a b^2+5 b^3\right ) \tan ^{-1}(\sinh (c+d x))}{128 d}+\frac{\left (64 a^3+48 a^2 b+24 a b^2+5 b^3\right ) \text{sech}(c+d x) \tanh (c+d x)}{128 d}-\frac{b \left (72 a^2+52 a b+15 b^2\right ) \text{sech}^3(c+d x) \tanh (c+d x)}{192 d}-\frac{b (12 a+5 b) \text{sech}^5(c+d x) \left (a+(a+b) \sinh ^2(c+d x)\right ) \tanh (c+d x)}{48 d}-\frac{b \text{sech}^7(c+d x) \left (a+(a+b) \sinh ^2(c+d x)\right )^2 \tanh (c+d x)}{8 d}\\ \end{align*}

Mathematica [A]  time = 11.6773, size = 158, normalized size = 0.8 $\frac{6 \left (48 a^2 b+64 a^3+24 a b^2+5 b^3\right ) \tan ^{-1}\left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )-2 b \left (144 a^2+168 a b+59 b^2\right ) \tanh (c+d x) \text{sech}^3(c+d x)+3 \left (48 a^2 b+64 a^3+24 a b^2+5 b^3\right ) \tanh (c+d x) \text{sech}(c+d x)+8 b^2 (24 a+17 b) \tanh (c+d x) \text{sech}^5(c+d x)-48 b^3 \tanh (c+d x) \text{sech}^7(c+d x)}{384 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(6*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*ArcTan[Tanh[(c + d*x)/2]] + 3*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)
*Sech[c + d*x]*Tanh[c + d*x] - 2*b*(144*a^2 + 168*a*b + 59*b^2)*Sech[c + d*x]^3*Tanh[c + d*x] + 8*b^2*(24*a +
17*b)*Sech[c + d*x]^5*Tanh[c + d*x] - 48*b^3*Sech[c + d*x]^7*Tanh[c + d*x])/(384*d)

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Maple [B]  time = 0.071, size = 421, normalized size = 2.1 \begin{align*}{\frac{{a}^{3}{\rm sech} \left (dx+c\right )\tanh \left ( dx+c \right ) }{2\,d}}+{\frac{{a}^{3}\arctan \left ({{\rm e}^{dx+c}} \right ) }{d}}-{\frac{{a}^{2}b\sinh \left ( dx+c \right ) }{d \left ( \cosh \left ( dx+c \right ) \right ) ^{4}}}+{\frac{{a}^{2}b\tanh \left ( dx+c \right ) \left ({\rm sech} \left (dx+c\right ) \right ) ^{3}}{4\,d}}+{\frac{3\,{a}^{2}b{\rm sech} \left (dx+c\right )\tanh \left ( dx+c \right ) }{8\,d}}+{\frac{3\,{a}^{2}b\arctan \left ({{\rm e}^{dx+c}} \right ) }{4\,d}}-{\frac{a{b}^{2} \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{d \left ( \cosh \left ( dx+c \right ) \right ) ^{6}}}-{\frac{3\,a{b}^{2}\sinh \left ( dx+c \right ) }{5\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{6}}}+{\frac{a{b}^{2}\tanh \left ( dx+c \right ) \left ({\rm sech} \left (dx+c\right ) \right ) ^{5}}{10\,d}}+{\frac{a{b}^{2}\tanh \left ( dx+c \right ) \left ({\rm sech} \left (dx+c\right ) \right ) ^{3}}{8\,d}}+{\frac{3\,a{b}^{2}{\rm sech} \left (dx+c\right )\tanh \left ( dx+c \right ) }{16\,d}}+{\frac{3\,a{b}^{2}\arctan \left ({{\rm e}^{dx+c}} \right ) }{8\,d}}-{\frac{{b}^{3} \left ( \sinh \left ( dx+c \right ) \right ) ^{5}}{3\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{8}}}-{\frac{{b}^{3} \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{3\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{8}}}-{\frac{{b}^{3}\sinh \left ( dx+c \right ) }{7\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{8}}}+{\frac{{b}^{3}\tanh \left ( dx+c \right ) \left ({\rm sech} \left (dx+c\right ) \right ) ^{7}}{56\,d}}+{\frac{{b}^{3}\tanh \left ( dx+c \right ) \left ({\rm sech} \left (dx+c\right ) \right ) ^{5}}{48\,d}}+{\frac{5\,{b}^{3} \left ({\rm sech} \left (dx+c\right ) \right ) ^{3}\tanh \left ( dx+c \right ) }{192\,d}}+{\frac{5\,{b}^{3}{\rm sech} \left (dx+c\right )\tanh \left ( dx+c \right ) }{128\,d}}+{\frac{5\,{b}^{3}\arctan \left ({{\rm e}^{dx+c}} \right ) }{64\,d}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(sech(d*x+c)^3*(a+b*tanh(d*x+c)^2)^3,x)

[Out]

1/2/d*a^3*sech(d*x+c)*tanh(d*x+c)+1/d*a^3*arctan(exp(d*x+c))-1/d*a^2*b*sinh(d*x+c)/cosh(d*x+c)^4+1/4/d*a^2*b*t
anh(d*x+c)*sech(d*x+c)^3+3/8*a^2*b*sech(d*x+c)*tanh(d*x+c)/d+3/4/d*a^2*b*arctan(exp(d*x+c))-1/d*a*b^2*sinh(d*x
+c)^3/cosh(d*x+c)^6-3/5/d*a*b^2*sinh(d*x+c)/cosh(d*x+c)^6+1/10/d*a*b^2*tanh(d*x+c)*sech(d*x+c)^5+1/8/d*a*b^2*t
anh(d*x+c)*sech(d*x+c)^3+3/16/d*a*b^2*sech(d*x+c)*tanh(d*x+c)+3/8/d*a*b^2*arctan(exp(d*x+c))-1/3/d*b^3*sinh(d*
x+c)^5/cosh(d*x+c)^8-1/3/d*b^3*sinh(d*x+c)^3/cosh(d*x+c)^8-1/7/d*b^3*sinh(d*x+c)/cosh(d*x+c)^8+1/56/d*b^3*tanh
(d*x+c)*sech(d*x+c)^7+1/48/d*b^3*tanh(d*x+c)*sech(d*x+c)^5+5/192*b^3*sech(d*x+c)^3*tanh(d*x+c)/d+5/128*b^3*sec
h(d*x+c)*tanh(d*x+c)/d+5/64/d*b^3*arctan(exp(d*x+c))

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Maxima [B]  time = 1.66783, size = 747, normalized size = 3.77 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/192*b^3*(15*arctan(e^(-d*x - c))/d - (15*e^(-d*x - c) - 397*e^(-3*d*x - 3*c) + 895*e^(-5*d*x - 5*c) - 1765*
e^(-7*d*x - 7*c) + 1765*e^(-9*d*x - 9*c) - 895*e^(-11*d*x - 11*c) + 397*e^(-13*d*x - 13*c) - 15*e^(-15*d*x - 1
5*c))/(d*(8*e^(-2*d*x - 2*c) + 28*e^(-4*d*x - 4*c) + 56*e^(-6*d*x - 6*c) + 70*e^(-8*d*x - 8*c) + 56*e^(-10*d*x
- 10*c) + 28*e^(-12*d*x - 12*c) + 8*e^(-14*d*x - 14*c) + e^(-16*d*x - 16*c) + 1))) - 1/8*a*b^2*(3*arctan(e^(-
d*x - c))/d - (3*e^(-d*x - c) - 47*e^(-3*d*x - 3*c) + 78*e^(-5*d*x - 5*c) - 78*e^(-7*d*x - 7*c) + 47*e^(-9*d*x
- 9*c) - 3*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))) - 3/4*a^2*b*(arctan(e^(-d*x - c))/d - (e^(-d*x -
c) - 7*e^(-3*d*x - 3*c) + 7*e^(-5*d*x - 5*c) - e^(-7*d*x - 7*c))/(d*(4*e^(-2*d*x - 2*c) + 6*e^(-4*d*x - 4*c)
+ 4*e^(-6*d*x - 6*c) + e^(-8*d*x - 8*c) + 1))) - a^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)))

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Fricas [B]  time = 2.56761, size = 16058, normalized size = 81.1 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*(3*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^15 + 45*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*c
osh(d*x + c)*sinh(d*x + c)^14 + 3*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*sinh(d*x + c)^15 + (960*a^3 - 432*a^2
*b - 984*a*b^2 - 397*b^3)*cosh(d*x + c)^13 + (960*a^3 - 432*a^2*b - 984*a*b^2 - 397*b^3 + 315*(64*a^3 + 48*a^2
*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^13 + 13*(105*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh
(d*x + c)^3 + (960*a^3 - 432*a^2*b - 984*a*b^2 - 397*b^3)*cosh(d*x + c))*sinh(d*x + c)^12 + (1728*a^3 - 2160*a
^2*b - 312*a*b^2 + 895*b^3)*cosh(d*x + c)^11 + (4095*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^4 +
1728*a^3 - 2160*a^2*b - 312*a*b^2 + 895*b^3 + 78*(960*a^3 - 432*a^2*b - 984*a*b^2 - 397*b^3)*cosh(d*x + c)^2)*
sinh(d*x + c)^11 + 11*(819*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^5 + 26*(960*a^3 - 432*a^2*b -
984*a*b^2 - 397*b^3)*cosh(d*x + c)^3 + (1728*a^3 - 2160*a^2*b - 312*a*b^2 + 895*b^3)*cosh(d*x + c))*sinh(d*x +
c)^10 + (960*a^3 - 1584*a^2*b + 744*a*b^2 - 1765*b^3)*cosh(d*x + c)^9 + (15015*(64*a^3 + 48*a^2*b + 24*a*b^2
+ 5*b^3)*cosh(d*x + c)^6 + 715*(960*a^3 - 432*a^2*b - 984*a*b^2 - 397*b^3)*cosh(d*x + c)^4 + 960*a^3 - 1584*a^
2*b + 744*a*b^2 - 1765*b^3 + 55*(1728*a^3 - 2160*a^2*b - 312*a*b^2 + 895*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^9
+ 3*(6435*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^7 + 429*(960*a^3 - 432*a^2*b - 984*a*b^2 - 397
*b^3)*cosh(d*x + c)^5 + 55*(1728*a^3 - 2160*a^2*b - 312*a*b^2 + 895*b^3)*cosh(d*x + c)^3 + 3*(960*a^3 - 1584*a
^2*b + 744*a*b^2 - 1765*b^3)*cosh(d*x + c))*sinh(d*x + c)^8 - (960*a^3 - 1584*a^2*b + 744*a*b^2 - 1765*b^3)*co
sh(d*x + c)^7 + (19305*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^8 + 1716*(960*a^3 - 432*a^2*b - 98
4*a*b^2 - 397*b^3)*cosh(d*x + c)^6 + 330*(1728*a^3 - 2160*a^2*b - 312*a*b^2 + 895*b^3)*cosh(d*x + c)^4 - 960*a
^3 + 1584*a^2*b - 744*a*b^2 + 1765*b^3 + 36*(960*a^3 - 1584*a^2*b + 744*a*b^2 - 1765*b^3)*cosh(d*x + c)^2)*sin
h(d*x + c)^7 + (15015*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^9 + 1716*(960*a^3 - 432*a^2*b - 984
*a*b^2 - 397*b^3)*cosh(d*x + c)^7 + 462*(1728*a^3 - 2160*a^2*b - 312*a*b^2 + 895*b^3)*cosh(d*x + c)^5 + 84*(96
0*a^3 - 1584*a^2*b + 744*a*b^2 - 1765*b^3)*cosh(d*x + c)^3 - 7*(960*a^3 - 1584*a^2*b + 744*a*b^2 - 1765*b^3)*c
osh(d*x + c))*sinh(d*x + c)^6 - (1728*a^3 - 2160*a^2*b - 312*a*b^2 + 895*b^3)*cosh(d*x + c)^5 + (9009*(64*a^3
+ 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^10 + 1287*(960*a^3 - 432*a^2*b - 984*a*b^2 - 397*b^3)*cosh(d*x +
c)^8 + 462*(1728*a^3 - 2160*a^2*b - 312*a*b^2 + 895*b^3)*cosh(d*x + c)^6 + 126*(960*a^3 - 1584*a^2*b + 744*a*b
^2 - 1765*b^3)*cosh(d*x + c)^4 - 1728*a^3 + 2160*a^2*b + 312*a*b^2 - 895*b^3 - 21*(960*a^3 - 1584*a^2*b + 744*
a*b^2 - 1765*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + (4095*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c
)^11 + 715*(960*a^3 - 432*a^2*b - 984*a*b^2 - 397*b^3)*cosh(d*x + c)^9 + 330*(1728*a^3 - 2160*a^2*b - 312*a*b^
2 + 895*b^3)*cosh(d*x + c)^7 + 126*(960*a^3 - 1584*a^2*b + 744*a*b^2 - 1765*b^3)*cosh(d*x + c)^5 - 35*(960*a^3
- 1584*a^2*b + 744*a*b^2 - 1765*b^3)*cosh(d*x + c)^3 - 5*(1728*a^3 - 2160*a^2*b - 312*a*b^2 + 895*b^3)*cosh(d
*x + c))*sinh(d*x + c)^4 - (960*a^3 - 432*a^2*b - 984*a*b^2 - 397*b^3)*cosh(d*x + c)^3 + (1365*(64*a^3 + 48*a^
2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^12 + 286*(960*a^3 - 432*a^2*b - 984*a*b^2 - 397*b^3)*cosh(d*x + c)^10 +
165*(1728*a^3 - 2160*a^2*b - 312*a*b^2 + 895*b^3)*cosh(d*x + c)^8 + 84*(960*a^3 - 1584*a^2*b + 744*a*b^2 - 176
5*b^3)*cosh(d*x + c)^6 - 35*(960*a^3 - 1584*a^2*b + 744*a*b^2 - 1765*b^3)*cosh(d*x + c)^4 - 960*a^3 + 432*a^2*
b + 984*a*b^2 + 397*b^3 - 10*(1728*a^3 - 2160*a^2*b - 312*a*b^2 + 895*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 +
(315*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^13 + 78*(960*a^3 - 432*a^2*b - 984*a*b^2 - 397*b^3)*
cosh(d*x + c)^11 + 55*(1728*a^3 - 2160*a^2*b - 312*a*b^2 + 895*b^3)*cosh(d*x + c)^9 + 36*(960*a^3 - 1584*a^2*b
+ 744*a*b^2 - 1765*b^3)*cosh(d*x + c)^7 - 21*(960*a^3 - 1584*a^2*b + 744*a*b^2 - 1765*b^3)*cosh(d*x + c)^5 -
10*(1728*a^3 - 2160*a^2*b - 312*a*b^2 + 895*b^3)*cosh(d*x + c)^3 - 3*(960*a^3 - 432*a^2*b - 984*a*b^2 - 397*b^
3)*cosh(d*x + c))*sinh(d*x + c)^2 + 3*((64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^16 + 16*(64*a^3 +
48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)*sinh(d*x + c)^15 + (64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*sinh(d*x
+ c)^16 + 8*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^14 + 8*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3
+ 15*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^14 + 112*(5*(64*a^3 + 48*a^2*b + 24
*a*b^2 + 5*b^3)*cosh(d*x + c)^3 + (64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^13 + 28*
(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^12 + 28*(65*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d
*x + c)^4 + 64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3 + 26*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^2)*
sinh(d*x + c)^12 + 112*(39*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^5 + 26*(64*a^3 + 48*a^2*b + 24
*a*b^2 + 5*b^3)*cosh(d*x + c)^3 + 3*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^11 + 5
6*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^10 + 56*(143*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cos
h(d*x + c)^6 + 143*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^4 + 64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b
^3 + 33*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^10 + 16*(715*(64*a^3 + 48*a^2*b
+ 24*a*b^2 + 5*b^3)*cosh(d*x + c)^7 + 1001*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^5 + 385*(64*a^
3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^3 + 35*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c))*si
nh(d*x + c)^9 + 70*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^8 + 2*(6435*(64*a^3 + 48*a^2*b + 24*a*
b^2 + 5*b^3)*cosh(d*x + c)^8 + 12012*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^6 + 6930*(64*a^3 + 4
8*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^4 + 2240*a^3 + 1680*a^2*b + 840*a*b^2 + 175*b^3 + 1260*(64*a^3 + 48*
a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 16*(715*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*co
sh(d*x + c)^9 + 1716*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^7 + 1386*(64*a^3 + 48*a^2*b + 24*a*b
^2 + 5*b^3)*cosh(d*x + c)^5 + 420*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^3 + 35*(64*a^3 + 48*a^2
*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^7 + 56*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c
)^6 + 56*(143*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^10 + 429*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*
b^3)*cosh(d*x + c)^8 + 462*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^6 + 210*(64*a^3 + 48*a^2*b + 2
4*a*b^2 + 5*b^3)*cosh(d*x + c)^4 + 64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3 + 35*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5
*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 112*(39*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^11 + 143
*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^9 + 198*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x
+ c)^7 + 126*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^5 + 35*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3
)*cosh(d*x + c)^3 + 3*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 28*(64*a^3 + 48*
a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^4 + 28*(65*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^12 + 2
86*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^10 + 495*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d
*x + c)^8 + 420*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^6 + 175*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5
*b^3)*cosh(d*x + c)^4 + 64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3 + 30*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(
d*x + c)^2)*sinh(d*x + c)^4 + 112*(5*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^13 + 26*(64*a^3 + 48
*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^11 + 55*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^9 + 60*(
64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^7 + 35*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c
)^5 + 10*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^3 + (64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(
d*x + c))*sinh(d*x + c)^3 + 64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3 + 8*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*co
sh(d*x + c)^2 + 8*(15*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^14 + 91*(64*a^3 + 48*a^2*b + 24*a*b
^2 + 5*b^3)*cosh(d*x + c)^12 + 231*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^10 + 315*(64*a^3 + 48*
a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^8 + 245*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^6 + 105*(
64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^4 + 64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3 + 21*(64*a^3 + 48
*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 16*((64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(
d*x + c)^15 + 7*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^13 + 21*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5
*b^3)*cosh(d*x + c)^11 + 35*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^9 + 35*(64*a^3 + 48*a^2*b + 2
4*a*b^2 + 5*b^3)*cosh(d*x + c)^7 + 21*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^5 + 7*(64*a^3 + 48*
a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^3 + (64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c))*sinh(d*x + c
))*arctan(cosh(d*x + c) + sinh(d*x + c)) - 3*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c) + (45*(64*a^
3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^14 + 13*(960*a^3 - 432*a^2*b - 984*a*b^2 - 397*b^3)*cosh(d*x +
c)^12 + 11*(1728*a^3 - 2160*a^2*b - 312*a*b^2 + 895*b^3)*cosh(d*x + c)^10 + 9*(960*a^3 - 1584*a^2*b + 744*a*b^
2 - 1765*b^3)*cosh(d*x + c)^8 - 7*(960*a^3 - 1584*a^2*b + 744*a*b^2 - 1765*b^3)*cosh(d*x + c)^6 - 5*(1728*a^3
- 2160*a^2*b - 312*a*b^2 + 895*b^3)*cosh(d*x + c)^4 - 192*a^3 - 144*a^2*b - 72*a*b^2 - 15*b^3 - 3*(960*a^3 - 4
32*a^2*b - 984*a*b^2 - 397*b^3)*cosh(d*x + c)^2)*sinh(d*x + c))/(d*cosh(d*x + c)^16 + 16*d*cosh(d*x + c)*sinh(
d*x + c)^15 + d*sinh(d*x + c)^16 + 8*d*cosh(d*x + c)^14 + 8*(15*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^14 + 112*
(5*d*cosh(d*x + c)^3 + d*cosh(d*x + c))*sinh(d*x + c)^13 + 28*d*cosh(d*x + c)^12 + 28*(65*d*cosh(d*x + c)^4 +
26*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^12 + 112*(39*d*cosh(d*x + c)^5 + 26*d*cosh(d*x + c)^3 + 3*d*cosh(d*x +
c))*sinh(d*x + c)^11 + 56*d*cosh(d*x + c)^10 + 56*(143*d*cosh(d*x + c)^6 + 143*d*cosh(d*x + c)^4 + 33*d*cosh(
d*x + c)^2 + d)*sinh(d*x + c)^10 + 16*(715*d*cosh(d*x + c)^7 + 1001*d*cosh(d*x + c)^5 + 385*d*cosh(d*x + c)^3
+ 35*d*cosh(d*x + c))*sinh(d*x + c)^9 + 70*d*cosh(d*x + c)^8 + 2*(6435*d*cosh(d*x + c)^8 + 12012*d*cosh(d*x +
c)^6 + 6930*d*cosh(d*x + c)^4 + 1260*d*cosh(d*x + c)^2 + 35*d)*sinh(d*x + c)^8 + 16*(715*d*cosh(d*x + c)^9 + 1
716*d*cosh(d*x + c)^7 + 1386*d*cosh(d*x + c)^5 + 420*d*cosh(d*x + c)^3 + 35*d*cosh(d*x + c))*sinh(d*x + c)^7 +
56*d*cosh(d*x + c)^6 + 56*(143*d*cosh(d*x + c)^10 + 429*d*cosh(d*x + c)^8 + 462*d*cosh(d*x + c)^6 + 210*d*cos
h(d*x + c)^4 + 35*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^6 + 112*(39*d*cosh(d*x + c)^11 + 143*d*cosh(d*x + c)^9
+ 198*d*cosh(d*x + c)^7 + 126*d*cosh(d*x + c)^5 + 35*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^5 +
28*d*cosh(d*x + c)^4 + 28*(65*d*cosh(d*x + c)^12 + 286*d*cosh(d*x + c)^10 + 495*d*cosh(d*x + c)^8 + 420*d*cosh
(d*x + c)^6 + 175*d*cosh(d*x + c)^4 + 30*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^4 + 112*(5*d*cosh(d*x + c)^13 +
26*d*cosh(d*x + c)^11 + 55*d*cosh(d*x + c)^9 + 60*d*cosh(d*x + c)^7 + 35*d*cosh(d*x + c)^5 + 10*d*cosh(d*x + c
)^3 + d*cosh(d*x + c))*sinh(d*x + c)^3 + 8*d*cosh(d*x + c)^2 + 8*(15*d*cosh(d*x + c)^14 + 91*d*cosh(d*x + c)^1
2 + 231*d*cosh(d*x + c)^10 + 315*d*cosh(d*x + c)^8 + 245*d*cosh(d*x + c)^6 + 105*d*cosh(d*x + c)^4 + 21*d*cosh
(d*x + c)^2 + d)*sinh(d*x + c)^2 + 16*(d*cosh(d*x + c)^15 + 7*d*cosh(d*x + c)^13 + 21*d*cosh(d*x + c)^11 + 35*
d*cosh(d*x + c)^9 + 35*d*cosh(d*x + c)^7 + 21*d*cosh(d*x + c)^5 + 7*d*cosh(d*x + c)^3 + d*cosh(d*x + c))*sinh(
d*x + c) + d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{3} \operatorname{sech}^{3}{\left (c + d x \right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*tanh(c + d*x)**2)**3*sech(c + d*x)**3, x)

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Giac [B]  time = 1.70428, size = 698, normalized size = 3.53 \begin{align*} \frac{3 \,{\left (64 \, a^{3} e^{c} + 48 \, a^{2} b e^{c} + 24 \, a b^{2} e^{c} + 5 \, b^{3} e^{c}\right )} \arctan \left (e^{\left (d x + c\right )}\right ) e^{\left (-c\right )} + \frac{192 \, a^{3} e^{\left (15 \, d x + 15 \, c\right )} + 144 \, a^{2} b e^{\left (15 \, d x + 15 \, c\right )} + 72 \, a b^{2} e^{\left (15 \, d x + 15 \, c\right )} + 15 \, b^{3} e^{\left (15 \, d x + 15 \, c\right )} + 960 \, a^{3} e^{\left (13 \, d x + 13 \, c\right )} - 432 \, a^{2} b e^{\left (13 \, d x + 13 \, c\right )} - 984 \, a b^{2} e^{\left (13 \, d x + 13 \, c\right )} - 397 \, b^{3} e^{\left (13 \, d x + 13 \, c\right )} + 1728 \, a^{3} e^{\left (11 \, d x + 11 \, c\right )} - 2160 \, a^{2} b e^{\left (11 \, d x + 11 \, c\right )} - 312 \, a b^{2} e^{\left (11 \, d x + 11 \, c\right )} + 895 \, b^{3} e^{\left (11 \, d x + 11 \, c\right )} + 960 \, a^{3} e^{\left (9 \, d x + 9 \, c\right )} - 1584 \, a^{2} b e^{\left (9 \, d x + 9 \, c\right )} + 744 \, a b^{2} e^{\left (9 \, d x + 9 \, c\right )} - 1765 \, b^{3} e^{\left (9 \, d x + 9 \, c\right )} - 960 \, a^{3} e^{\left (7 \, d x + 7 \, c\right )} + 1584 \, a^{2} b e^{\left (7 \, d x + 7 \, c\right )} - 744 \, a b^{2} e^{\left (7 \, d x + 7 \, c\right )} + 1765 \, b^{3} e^{\left (7 \, d x + 7 \, c\right )} - 1728 \, a^{3} e^{\left (5 \, d x + 5 \, c\right )} + 2160 \, a^{2} b e^{\left (5 \, d x + 5 \, c\right )} + 312 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} - 895 \, b^{3} e^{\left (5 \, d x + 5 \, c\right )} - 960 \, a^{3} e^{\left (3 \, d x + 3 \, c\right )} + 432 \, a^{2} b e^{\left (3 \, d x + 3 \, c\right )} + 984 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 397 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} - 192 \, a^{3} e^{\left (d x + c\right )} - 144 \, a^{2} b e^{\left (d x + c\right )} - 72 \, a b^{2} e^{\left (d x + c\right )} - 15 \, b^{3} e^{\left (d x + c\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{8}}}{192 \, d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*(3*(64*a^3*e^c + 48*a^2*b*e^c + 24*a*b^2*e^c + 5*b^3*e^c)*arctan(e^(d*x + c))*e^(-c) + (192*a^3*e^(15*d*
x + 15*c) + 144*a^2*b*e^(15*d*x + 15*c) + 72*a*b^2*e^(15*d*x + 15*c) + 15*b^3*e^(15*d*x + 15*c) + 960*a^3*e^(1
3*d*x + 13*c) - 432*a^2*b*e^(13*d*x + 13*c) - 984*a*b^2*e^(13*d*x + 13*c) - 397*b^3*e^(13*d*x + 13*c) + 1728*a
^3*e^(11*d*x + 11*c) - 2160*a^2*b*e^(11*d*x + 11*c) - 312*a*b^2*e^(11*d*x + 11*c) + 895*b^3*e^(11*d*x + 11*c)
+ 960*a^3*e^(9*d*x + 9*c) - 1584*a^2*b*e^(9*d*x + 9*c) + 744*a*b^2*e^(9*d*x + 9*c) - 1765*b^3*e^(9*d*x + 9*c)
- 960*a^3*e^(7*d*x + 7*c) + 1584*a^2*b*e^(7*d*x + 7*c) - 744*a*b^2*e^(7*d*x + 7*c) + 1765*b^3*e^(7*d*x + 7*c)
- 1728*a^3*e^(5*d*x + 5*c) + 2160*a^2*b*e^(5*d*x + 5*c) + 312*a*b^2*e^(5*d*x + 5*c) - 895*b^3*e^(5*d*x + 5*c)
- 960*a^3*e^(3*d*x + 3*c) + 432*a^2*b*e^(3*d*x + 3*c) + 984*a*b^2*e^(3*d*x + 3*c) + 397*b^3*e^(3*d*x + 3*c) -
192*a^3*e^(d*x + c) - 144*a^2*b*e^(d*x + c) - 72*a*b^2*e^(d*x + c) - 15*b^3*e^(d*x + c))/(e^(2*d*x + 2*c) + 1)
^8)/d