### 3.72 $$\int \text{csch}^4(c+d x) (a+b \tanh ^3(c+d x))^3 \, dx$$

Optimal. Leaf size=138 $-\frac{3 a^2 b \tanh ^2(c+d x)}{2 d}+\frac{3 a^2 b \log (\tanh (c+d x))}{d}-\frac{a^3 \coth ^3(c+d x)}{3 d}+\frac{a^3 \coth (c+d x)}{d}-\frac{3 a b^2 \tanh ^5(c+d x)}{5 d}+\frac{a b^2 \tanh ^3(c+d x)}{d}-\frac{b^3 \tanh ^8(c+d x)}{8 d}+\frac{b^3 \tanh ^6(c+d x)}{6 d}$

[Out]

(a^3*Coth[c + d*x])/d - (a^3*Coth[c + d*x]^3)/(3*d) + (3*a^2*b*Log[Tanh[c + d*x]])/d - (3*a^2*b*Tanh[c + d*x]^
2)/(2*d) + (a*b^2*Tanh[c + d*x]^3)/d - (3*a*b^2*Tanh[c + d*x]^5)/(5*d) + (b^3*Tanh[c + d*x]^6)/(6*d) - (b^3*Ta
nh[c + d*x]^8)/(8*d)

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Rubi [A]  time = 0.115098, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.087, Rules used = {3663, 1802} $-\frac{3 a^2 b \tanh ^2(c+d x)}{2 d}+\frac{3 a^2 b \log (\tanh (c+d x))}{d}-\frac{a^3 \coth ^3(c+d x)}{3 d}+\frac{a^3 \coth (c+d x)}{d}-\frac{3 a b^2 \tanh ^5(c+d x)}{5 d}+\frac{a b^2 \tanh ^3(c+d x)}{d}-\frac{b^3 \tanh ^8(c+d x)}{8 d}+\frac{b^3 \tanh ^6(c+d x)}{6 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(a^3*Coth[c + d*x])/d - (a^3*Coth[c + d*x]^3)/(3*d) + (3*a^2*b*Log[Tanh[c + d*x]])/d - (3*a^2*b*Tanh[c + d*x]^
2)/(2*d) + (a*b^2*Tanh[c + d*x]^3)/d - (3*a*b^2*Tanh[c + d*x]^5)/(5*d) + (b^3*Tanh[c + d*x]^6)/(6*d) - (b^3*Ta
nh[c + d*x]^8)/(8*d)

Rule 3663

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(c*ff^(m + 1))/f, Subst[Int[(x^m*(a + b*(ff*x)^n)^p)/(c^2 + ff^2*x^2
)^(m/2 + 1), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[m/2]

Rule 1802

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin{align*} \int \text{csch}^4(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right ) \left (a+b x^3\right )^3}{x^4} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^3}{x^4}-\frac{a^3}{x^2}+\frac{3 a^2 b}{x}-3 a^2 b x+3 a b^2 x^2-3 a b^2 x^4+b^3 x^5-b^3 x^7\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{a^3 \coth (c+d x)}{d}-\frac{a^3 \coth ^3(c+d x)}{3 d}+\frac{3 a^2 b \log (\tanh (c+d x))}{d}-\frac{3 a^2 b \tanh ^2(c+d x)}{2 d}+\frac{a b^2 \tanh ^3(c+d x)}{d}-\frac{3 a b^2 \tanh ^5(c+d x)}{5 d}+\frac{b^3 \tanh ^6(c+d x)}{6 d}-\frac{b^3 \tanh ^8(c+d x)}{8 d}\\ \end{align*}

Mathematica [A]  time = 0.168904, size = 213, normalized size = 1.54 $\frac{3 a^2 b \text{sech}^2(c+d x)}{2 d}+\frac{3 a^2 b \log (\sinh (c+d x))}{d}-\frac{3 a^2 b \log (\cosh (c+d x))}{d}+\frac{2 a^3 \coth (c+d x)}{3 d}-\frac{a^3 \coth (c+d x) \text{csch}^2(c+d x)}{3 d}+\frac{2 a b^2 \tanh (c+d x)}{5 d}-\frac{3 a b^2 \tanh (c+d x) \text{sech}^4(c+d x)}{5 d}+\frac{a b^2 \tanh (c+d x) \text{sech}^2(c+d x)}{5 d}-\frac{b^3 \text{sech}^8(c+d x)}{8 d}+\frac{b^3 \text{sech}^6(c+d x)}{3 d}-\frac{b^3 \text{sech}^4(c+d x)}{4 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*a^3*Coth[c + d*x])/(3*d) - (a^3*Coth[c + d*x]*Csch[c + d*x]^2)/(3*d) - (3*a^2*b*Log[Cosh[c + d*x]])/d + (3*
a^2*b*Log[Sinh[c + d*x]])/d + (3*a^2*b*Sech[c + d*x]^2)/(2*d) - (b^3*Sech[c + d*x]^4)/(4*d) + (b^3*Sech[c + d*
x]^6)/(3*d) - (b^3*Sech[c + d*x]^8)/(8*d) + (2*a*b^2*Tanh[c + d*x])/(5*d) + (a*b^2*Sech[c + d*x]^2*Tanh[c + d*
x])/(5*d) - (3*a*b^2*Sech[c + d*x]^4*Tanh[c + d*x])/(5*d)

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Maple [B]  time = 0.106, size = 275, normalized size = 2. \begin{align*}{\frac{2\,{a}^{3}{\rm coth} \left (dx+c\right )}{3\,d}}-{\frac{{a}^{3}{\rm coth} \left (dx+c\right ) \left ({\rm csch} \left (dx+c\right ) \right ) ^{2}}{3\,d}}+{\frac{3\,{a}^{2}b}{2\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}}+3\,{\frac{{a}^{2}b\ln \left ( \tanh \left ( dx+c \right ) \right ) }{d}}-{\frac{3\,a{b}^{2}\sinh \left ( dx+c \right ) }{4\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}}+{\frac{2\,a{b}^{2}\tanh \left ( dx+c \right ) }{5\,d}}+{\frac{3\,a{b}^{2}\tanh \left ( dx+c \right ) \left ({\rm sech} \left (dx+c\right ) \right ) ^{4}}{20\,d}}+{\frac{a{b}^{2}\tanh \left ( dx+c \right ) \left ({\rm sech} \left (dx+c\right ) \right ) ^{2}}{5\,d}}-{\frac{{b}^{3} \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}{4\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{8}}}-{\frac{{b}^{3} \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{8\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{8}}}+{\frac{{b}^{3} \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{24\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{6}}}+{\frac{{b}^{3} \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{24\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{4}}}+{\frac{{b}^{3} \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{24\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*a^3*coth(d*x+c)/d-1/3/d*a^3*coth(d*x+c)*csch(d*x+c)^2+3/2/d*a^2*b/cosh(d*x+c)^2+3*a^2*b*ln(tanh(d*x+c))/d-
3/4/d*a*b^2*sinh(d*x+c)/cosh(d*x+c)^5+2/5*a*b^2*tanh(d*x+c)/d+3/20/d*a*b^2*tanh(d*x+c)*sech(d*x+c)^4+1/5/d*a*b
^2*tanh(d*x+c)*sech(d*x+c)^2-1/4/d*b^3*sinh(d*x+c)^4/cosh(d*x+c)^8-1/8/d*b^3*sinh(d*x+c)^2/cosh(d*x+c)^8+1/24/
d*b^3*sinh(d*x+c)^2/cosh(d*x+c)^6+1/24/d*b^3*sinh(d*x+c)^2/cosh(d*x+c)^4+1/24/d*b^3*sinh(d*x+c)^2/cosh(d*x+c)^
2

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Maxima [B]  time = 1.80962, size = 1346, normalized size = 9.75 \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)^4*(a+b*tanh(d*x+c)^3)^3,x, algorithm="maxima")

[Out]

3*a^2*b*(log(e^(-d*x - c) + 1)/d + log(e^(-d*x - c) - 1)/d - log(e^(-2*d*x - 2*c) + 1)/d + 2*e^(-2*d*x - 2*c)/
(d*(2*e^(-2*d*x - 2*c) + e^(-4*d*x - 4*c) + 1))) + 4/5*a*b^2*(5*e^(-2*d*x - 2*c)/(d*(5*e^(-2*d*x - 2*c) + 10*e
^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) + 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) + 1)) - 5*e^(-4*d*x - 4*c)/(d*
(5*e^(-2*d*x - 2*c) + 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) + 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) + 1)
) + 15*e^(-6*d*x - 6*c)/(d*(5*e^(-2*d*x - 2*c) + 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) + 5*e^(-8*d*x - 8*c
) + e^(-10*d*x - 10*c) + 1)) + 1/(d*(5*e^(-2*d*x - 2*c) + 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) + 5*e^(-8*
d*x - 8*c) + e^(-10*d*x - 10*c) + 1))) + 4/3*a^3*(3*e^(-2*d*x - 2*c)/(d*(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*
c) + e^(-6*d*x - 6*c) - 1)) - 1/(d*(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) - 1))) - 4/3*b^
3*(3*e^(-4*d*x - 4*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)) - 4*e^(-6*d
*x - 6*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)) + 10*e^(-8*d*x - 8*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)) - 4*e^(-10*d*x - 10*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)) + 3*e^(-12*d*x - 12*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)))

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Fricas [B]  time = 3.63018, size = 25978, normalized size = 188.25 \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)^4*(a+b*tanh(d*x+c)^3)^3,x, algorithm="fricas")

[Out]

1/15*(90*a^2*b*cosh(d*x + c)^20 + 1800*a^2*b*cosh(d*x + c)*sinh(d*x + c)^19 + 90*a^2*b*sinh(d*x + c)^20 - 30*(
2*a^3 - 9*a^2*b + 6*a*b^2 + 2*b^3)*cosh(d*x + c)^18 + 30*(570*a^2*b*cosh(d*x + c)^2 - 2*a^3 + 9*a^2*b - 6*a*b^
2 - 2*b^3)*sinh(d*x + c)^18 + 540*(190*a^2*b*cosh(d*x + c)^3 - (2*a^3 - 9*a^2*b + 6*a*b^2 + 2*b^3)*cosh(d*x +
c))*sinh(d*x + c)^17 - 20*(23*a^3 - 3*a*b^2 - 13*b^3)*cosh(d*x + c)^16 + 10*(43605*a^2*b*cosh(d*x + c)^4 - 46*
a^3 + 6*a*b^2 + 26*b^3 - 459*(2*a^3 - 9*a^2*b + 6*a*b^2 + 2*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^16 + 160*(8721
*a^2*b*cosh(d*x + c)^5 - 153*(2*a^3 - 9*a^2*b + 6*a*b^2 + 2*b^3)*cosh(d*x + c)^3 - 2*(23*a^3 - 3*a*b^2 - 13*b^
3)*cosh(d*x + c))*sinh(d*x + c)^15 - 20*(76*a^3 + 36*a^2*b - 24*a*b^2 + 31*b^3)*cosh(d*x + c)^14 + 20*(174420*
a^2*b*cosh(d*x + c)^6 - 4590*(2*a^3 - 9*a^2*b + 6*a*b^2 + 2*b^3)*cosh(d*x + c)^4 - 76*a^3 - 36*a^2*b + 24*a*b^
2 - 31*b^3 - 120*(23*a^3 - 3*a*b^2 - 13*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^14 + 40*(174420*a^2*b*cosh(d*x + c
)^7 - 6426*(2*a^3 - 9*a^2*b + 6*a*b^2 + 2*b^3)*cosh(d*x + c)^5 - 280*(23*a^3 - 3*a*b^2 - 13*b^3)*cosh(d*x + c)
^3 - 7*(76*a^3 + 36*a^2*b - 24*a*b^2 + 31*b^3)*cosh(d*x + c))*sinh(d*x + c)^13 - 4*(700*a^3 + 135*a^2*b + 48*a
*b^2 - 245*b^3)*cosh(d*x + c)^12 + 4*(2834325*a^2*b*cosh(d*x + c)^8 - 139230*(2*a^3 - 9*a^2*b + 6*a*b^2 + 2*b^
3)*cosh(d*x + c)^6 - 9100*(23*a^3 - 3*a*b^2 - 13*b^3)*cosh(d*x + c)^4 - 700*a^3 - 135*a^2*b - 48*a*b^2 + 245*b
^3 - 455*(76*a^3 + 36*a^2*b - 24*a*b^2 + 31*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^12 + 16*(944775*a^2*b*cosh(d*x
+ c)^9 - 59670*(2*a^3 - 9*a^2*b + 6*a*b^2 + 2*b^3)*cosh(d*x + c)^7 - 5460*(23*a^3 - 3*a*b^2 - 13*b^3)*cosh(d*
x + c)^5 - 455*(76*a^3 + 36*a^2*b - 24*a*b^2 + 31*b^3)*cosh(d*x + c)^3 - 3*(700*a^3 + 135*a^2*b + 48*a*b^2 - 2
45*b^3)*cosh(d*x + c))*sinh(d*x + c)^11 - 20*(154*a^3 - 27*a^2*b + 18*a*b^2 + 49*b^3)*cosh(d*x + c)^10 + 4*(41
57010*a^2*b*cosh(d*x + c)^10 - 328185*(2*a^3 - 9*a^2*b + 6*a*b^2 + 2*b^3)*cosh(d*x + c)^8 - 40040*(23*a^3 - 3*
a*b^2 - 13*b^3)*cosh(d*x + c)^6 - 5005*(76*a^3 + 36*a^2*b - 24*a*b^2 + 31*b^3)*cosh(d*x + c)^4 - 770*a^3 + 135
*a^2*b - 90*a*b^2 - 245*b^3 - 66*(700*a^3 + 135*a^2*b + 48*a*b^2 - 245*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^10
+ 40*(377910*a^2*b*cosh(d*x + c)^11 - 36465*(2*a^3 - 9*a^2*b + 6*a*b^2 + 2*b^3)*cosh(d*x + c)^9 - 5720*(23*a^3
- 3*a*b^2 - 13*b^3)*cosh(d*x + c)^7 - 1001*(76*a^3 + 36*a^2*b - 24*a*b^2 + 31*b^3)*cosh(d*x + c)^5 - 22*(700*
a^3 + 135*a^2*b + 48*a*b^2 - 245*b^3)*cosh(d*x + c)^3 - 5*(154*a^3 - 27*a^2*b + 18*a*b^2 + 49*b^3)*cosh(d*x +
c))*sinh(d*x + c)^9 - 4*(490*a^3 - 180*a^2*b - 54*a*b^2 - 155*b^3)*cosh(d*x + c)^8 + 4*(2834325*a^2*b*cosh(d*x
+ c)^12 - 328185*(2*a^3 - 9*a^2*b + 6*a*b^2 + 2*b^3)*cosh(d*x + c)^10 - 64350*(23*a^3 - 3*a*b^2 - 13*b^3)*cos
h(d*x + c)^8 - 15015*(76*a^3 + 36*a^2*b - 24*a*b^2 + 31*b^3)*cosh(d*x + c)^6 - 495*(700*a^3 + 135*a^2*b + 48*a
*b^2 - 245*b^3)*cosh(d*x + c)^4 - 490*a^3 + 180*a^2*b + 54*a*b^2 + 155*b^3 - 225*(154*a^3 - 27*a^2*b + 18*a*b^
2 + 49*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 32*(218025*a^2*b*cosh(d*x + c)^13 - 29835*(2*a^3 - 9*a^2*b + 6*
a*b^2 + 2*b^3)*cosh(d*x + c)^11 - 7150*(23*a^3 - 3*a*b^2 - 13*b^3)*cosh(d*x + c)^9 - 2145*(76*a^3 + 36*a^2*b -
24*a*b^2 + 31*b^3)*cosh(d*x + c)^7 - 99*(700*a^3 + 135*a^2*b + 48*a*b^2 - 245*b^3)*cosh(d*x + c)^5 - 75*(154*
a^3 - 27*a^2*b + 18*a*b^2 + 49*b^3)*cosh(d*x + c)^3 - (490*a^3 - 180*a^2*b - 54*a*b^2 - 155*b^3)*cosh(d*x + c)
)*sinh(d*x + c)^7 - 20*(28*a^3 + 13*b^3)*cosh(d*x + c)^6 + 4*(872100*a^2*b*cosh(d*x + c)^14 - 139230*(2*a^3 -
9*a^2*b + 6*a*b^2 + 2*b^3)*cosh(d*x + c)^12 - 40040*(23*a^3 - 3*a*b^2 - 13*b^3)*cosh(d*x + c)^10 - 15015*(76*a
^3 + 36*a^2*b - 24*a*b^2 + 31*b^3)*cosh(d*x + c)^8 - 924*(700*a^3 + 135*a^2*b + 48*a*b^2 - 245*b^3)*cosh(d*x +
c)^6 - 1050*(154*a^3 - 27*a^2*b + 18*a*b^2 + 49*b^3)*cosh(d*x + c)^4 - 140*a^3 - 65*b^3 - 28*(490*a^3 - 180*a
^2*b - 54*a*b^2 - 155*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(174420*a^2*b*cosh(d*x + c)^15 - 32130*(2*a^3
- 9*a^2*b + 6*a*b^2 + 2*b^3)*cosh(d*x + c)^13 - 10920*(23*a^3 - 3*a*b^2 - 13*b^3)*cosh(d*x + c)^11 - 5005*(76*
a^3 + 36*a^2*b - 24*a*b^2 + 31*b^3)*cosh(d*x + c)^9 - 396*(700*a^3 + 135*a^2*b + 48*a*b^2 - 245*b^3)*cosh(d*x
+ c)^7 - 630*(154*a^3 - 27*a^2*b + 18*a*b^2 + 49*b^3)*cosh(d*x + c)^5 - 28*(490*a^3 - 180*a^2*b - 54*a*b^2 - 1
55*b^3)*cosh(d*x + c)^3 - 15*(28*a^3 + 13*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(40*a^3 - 135*a^2*b - 48*a*b
^2 + 30*b^3)*cosh(d*x + c)^4 + 2*(218025*a^2*b*cosh(d*x + c)^16 - 45900*(2*a^3 - 9*a^2*b + 6*a*b^2 + 2*b^3)*co
sh(d*x + c)^14 - 18200*(23*a^3 - 3*a*b^2 - 13*b^3)*cosh(d*x + c)^12 - 10010*(76*a^3 + 36*a^2*b - 24*a*b^2 + 31
*b^3)*cosh(d*x + c)^10 - 990*(700*a^3 + 135*a^2*b + 48*a*b^2 - 245*b^3)*cosh(d*x + c)^8 - 2100*(154*a^3 - 27*a
^2*b + 18*a*b^2 + 49*b^3)*cosh(d*x + c)^6 - 140*(490*a^3 - 180*a^2*b - 54*a*b^2 - 155*b^3)*cosh(d*x + c)^4 + 4
0*a^3 - 135*a^2*b - 48*a*b^2 + 30*b^3 - 150*(28*a^3 + 13*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(12825*a^2*
b*cosh(d*x + c)^17 - 3060*(2*a^3 - 9*a^2*b + 6*a*b^2 + 2*b^3)*cosh(d*x + c)^15 - 1400*(23*a^3 - 3*a*b^2 - 13*b
^3)*cosh(d*x + c)^13 - 910*(76*a^3 + 36*a^2*b - 24*a*b^2 + 31*b^3)*cosh(d*x + c)^11 - 110*(700*a^3 + 135*a^2*b
+ 48*a*b^2 - 245*b^3)*cosh(d*x + c)^9 - 300*(154*a^3 - 27*a^2*b + 18*a*b^2 + 49*b^3)*cosh(d*x + c)^7 - 28*(49
0*a^3 - 180*a^2*b - 54*a*b^2 - 155*b^3)*cosh(d*x + c)^5 - 50*(28*a^3 + 13*b^3)*cosh(d*x + c)^3 + (40*a^3 - 135
*a^2*b - 48*a*b^2 + 30*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 20*a^3 + 12*a*b^2 + 10*(10*a^3 - 9*a^2*b + 6*a*b^
2)*cosh(d*x + c)^2 + 2*(8550*a^2*b*cosh(d*x + c)^18 - 2295*(2*a^3 - 9*a^2*b + 6*a*b^2 + 2*b^3)*cosh(d*x + c)^1
6 - 1200*(23*a^3 - 3*a*b^2 - 13*b^3)*cosh(d*x + c)^14 - 910*(76*a^3 + 36*a^2*b - 24*a*b^2 + 31*b^3)*cosh(d*x +
c)^12 - 132*(700*a^3 + 135*a^2*b + 48*a*b^2 - 245*b^3)*cosh(d*x + c)^10 - 450*(154*a^3 - 27*a^2*b + 18*a*b^2
+ 49*b^3)*cosh(d*x + c)^8 - 56*(490*a^3 - 180*a^2*b - 54*a*b^2 - 155*b^3)*cosh(d*x + c)^6 - 150*(28*a^3 + 13*b
^3)*cosh(d*x + c)^4 + 50*a^3 - 45*a^2*b + 30*a*b^2 + 6*(40*a^3 - 135*a^2*b - 48*a*b^2 + 30*b^3)*cosh(d*x + c)^
2)*sinh(d*x + c)^2 - 45*(a^2*b*cosh(d*x + c)^22 + 22*a^2*b*cosh(d*x + c)*sinh(d*x + c)^21 + a^2*b*sinh(d*x + c
)^22 + 5*a^2*b*cosh(d*x + c)^20 + 7*a^2*b*cosh(d*x + c)^18 + (231*a^2*b*cosh(d*x + c)^2 + 5*a^2*b)*sinh(d*x +
c)^20 + 20*(77*a^2*b*cosh(d*x + c)^3 + 5*a^2*b*cosh(d*x + c))*sinh(d*x + c)^19 - 5*a^2*b*cosh(d*x + c)^16 + (7
315*a^2*b*cosh(d*x + c)^4 + 950*a^2*b*cosh(d*x + c)^2 + 7*a^2*b)*sinh(d*x + c)^18 + 6*(4389*a^2*b*cosh(d*x + c
)^5 + 950*a^2*b*cosh(d*x + c)^3 + 21*a^2*b*cosh(d*x + c))*sinh(d*x + c)^17 - 22*a^2*b*cosh(d*x + c)^14 + (7461
3*a^2*b*cosh(d*x + c)^6 + 24225*a^2*b*cosh(d*x + c)^4 + 1071*a^2*b*cosh(d*x + c)^2 - 5*a^2*b)*sinh(d*x + c)^16
+ 16*(10659*a^2*b*cosh(d*x + c)^7 + 4845*a^2*b*cosh(d*x + c)^5 + 357*a^2*b*cosh(d*x + c)^3 - 5*a^2*b*cosh(d*x
+ c))*sinh(d*x + c)^15 - 14*a^2*b*cosh(d*x + c)^12 + 2*(159885*a^2*b*cosh(d*x + c)^8 + 96900*a^2*b*cosh(d*x +
c)^6 + 10710*a^2*b*cosh(d*x + c)^4 - 300*a^2*b*cosh(d*x + c)^2 - 11*a^2*b)*sinh(d*x + c)^14 + 4*(124355*a^2*b
*cosh(d*x + c)^9 + 96900*a^2*b*cosh(d*x + c)^7 + 14994*a^2*b*cosh(d*x + c)^5 - 700*a^2*b*cosh(d*x + c)^3 - 77*
a^2*b*cosh(d*x + c))*sinh(d*x + c)^13 + 14*a^2*b*cosh(d*x + c)^10 + 2*(323323*a^2*b*cosh(d*x + c)^10 + 314925*
a^2*b*cosh(d*x + c)^8 + 64974*a^2*b*cosh(d*x + c)^6 - 4550*a^2*b*cosh(d*x + c)^4 - 1001*a^2*b*cosh(d*x + c)^2
- 7*a^2*b)*sinh(d*x + c)^12 + 8*(88179*a^2*b*cosh(d*x + c)^11 + 104975*a^2*b*cosh(d*x + c)^9 + 27846*a^2*b*cos
h(d*x + c)^7 - 2730*a^2*b*cosh(d*x + c)^5 - 1001*a^2*b*cosh(d*x + c)^3 - 21*a^2*b*cosh(d*x + c))*sinh(d*x + c)
^11 + 22*a^2*b*cosh(d*x + c)^8 + 2*(323323*a^2*b*cosh(d*x + c)^12 + 461890*a^2*b*cosh(d*x + c)^10 + 153153*a^2
*b*cosh(d*x + c)^8 - 20020*a^2*b*cosh(d*x + c)^6 - 11011*a^2*b*cosh(d*x + c)^4 - 462*a^2*b*cosh(d*x + c)^2 + 7
*a^2*b)*sinh(d*x + c)^10 + 4*(124355*a^2*b*cosh(d*x + c)^13 + 209950*a^2*b*cosh(d*x + c)^11 + 85085*a^2*b*cosh
(d*x + c)^9 - 14300*a^2*b*cosh(d*x + c)^7 - 11011*a^2*b*cosh(d*x + c)^5 - 770*a^2*b*cosh(d*x + c)^3 + 35*a^2*b
*cosh(d*x + c))*sinh(d*x + c)^9 + 5*a^2*b*cosh(d*x + c)^6 + 2*(159885*a^2*b*cosh(d*x + c)^14 + 314925*a^2*b*co
sh(d*x + c)^12 + 153153*a^2*b*cosh(d*x + c)^10 - 32175*a^2*b*cosh(d*x + c)^8 - 33033*a^2*b*cosh(d*x + c)^6 - 3
465*a^2*b*cosh(d*x + c)^4 + 315*a^2*b*cosh(d*x + c)^2 + 11*a^2*b)*sinh(d*x + c)^8 + 16*(10659*a^2*b*cosh(d*x +
c)^15 + 24225*a^2*b*cosh(d*x + c)^13 + 13923*a^2*b*cosh(d*x + c)^11 - 3575*a^2*b*cosh(d*x + c)^9 - 4719*a^2*b
*cosh(d*x + c)^7 - 693*a^2*b*cosh(d*x + c)^5 + 105*a^2*b*cosh(d*x + c)^3 + 11*a^2*b*cosh(d*x + c))*sinh(d*x +
c)^7 - 7*a^2*b*cosh(d*x + c)^4 + (74613*a^2*b*cosh(d*x + c)^16 + 193800*a^2*b*cosh(d*x + c)^14 + 129948*a^2*b*
cosh(d*x + c)^12 - 40040*a^2*b*cosh(d*x + c)^10 - 66066*a^2*b*cosh(d*x + c)^8 - 12936*a^2*b*cosh(d*x + c)^6 +
2940*a^2*b*cosh(d*x + c)^4 + 616*a^2*b*cosh(d*x + c)^2 + 5*a^2*b)*sinh(d*x + c)^6 + 2*(13167*a^2*b*cosh(d*x +
c)^17 + 38760*a^2*b*cosh(d*x + c)^15 + 29988*a^2*b*cosh(d*x + c)^13 - 10920*a^2*b*cosh(d*x + c)^11 - 22022*a^2
*b*cosh(d*x + c)^9 - 5544*a^2*b*cosh(d*x + c)^7 + 1764*a^2*b*cosh(d*x + c)^5 + 616*a^2*b*cosh(d*x + c)^3 + 15*
a^2*b*cosh(d*x + c))*sinh(d*x + c)^5 - 5*a^2*b*cosh(d*x + c)^2 + (7315*a^2*b*cosh(d*x + c)^18 + 24225*a^2*b*co
sh(d*x + c)^16 + 21420*a^2*b*cosh(d*x + c)^14 - 9100*a^2*b*cosh(d*x + c)^12 - 22022*a^2*b*cosh(d*x + c)^10 - 6
930*a^2*b*cosh(d*x + c)^8 + 2940*a^2*b*cosh(d*x + c)^6 + 1540*a^2*b*cosh(d*x + c)^4 + 75*a^2*b*cosh(d*x + c)^2
- 7*a^2*b)*sinh(d*x + c)^4 + 4*(385*a^2*b*cosh(d*x + c)^19 + 1425*a^2*b*cosh(d*x + c)^17 + 1428*a^2*b*cosh(d*
x + c)^15 - 700*a^2*b*cosh(d*x + c)^13 - 2002*a^2*b*cosh(d*x + c)^11 - 770*a^2*b*cosh(d*x + c)^9 + 420*a^2*b*c
osh(d*x + c)^7 + 308*a^2*b*cosh(d*x + c)^5 + 25*a^2*b*cosh(d*x + c)^3 - 7*a^2*b*cosh(d*x + c))*sinh(d*x + c)^3
- a^2*b + (231*a^2*b*cosh(d*x + c)^20 + 950*a^2*b*cosh(d*x + c)^18 + 1071*a^2*b*cosh(d*x + c)^16 - 600*a^2*b*
cosh(d*x + c)^14 - 2002*a^2*b*cosh(d*x + c)^12 - 924*a^2*b*cosh(d*x + c)^10 + 630*a^2*b*cosh(d*x + c)^8 + 616*
a^2*b*cosh(d*x + c)^6 + 75*a^2*b*cosh(d*x + c)^4 - 42*a^2*b*cosh(d*x + c)^2 - 5*a^2*b)*sinh(d*x + c)^2 + 2*(11
*a^2*b*cosh(d*x + c)^21 + 50*a^2*b*cosh(d*x + c)^19 + 63*a^2*b*cosh(d*x + c)^17 - 40*a^2*b*cosh(d*x + c)^15 -
154*a^2*b*cosh(d*x + c)^13 - 84*a^2*b*cosh(d*x + c)^11 + 70*a^2*b*cosh(d*x + c)^9 + 88*a^2*b*cosh(d*x + c)^7 +
15*a^2*b*cosh(d*x + c)^5 - 14*a^2*b*cosh(d*x + c)^3 - 5*a^2*b*cosh(d*x + c))*sinh(d*x + c))*log(2*cosh(d*x +
c)/(cosh(d*x + c) - sinh(d*x + c))) + 45*(a^2*b*cosh(d*x + c)^22 + 22*a^2*b*cosh(d*x + c)*sinh(d*x + c)^21 + a
^2*b*sinh(d*x + c)^22 + 5*a^2*b*cosh(d*x + c)^20 + 7*a^2*b*cosh(d*x + c)^18 + (231*a^2*b*cosh(d*x + c)^2 + 5*a
^2*b)*sinh(d*x + c)^20 + 20*(77*a^2*b*cosh(d*x + c)^3 + 5*a^2*b*cosh(d*x + c))*sinh(d*x + c)^19 - 5*a^2*b*cosh
(d*x + c)^16 + (7315*a^2*b*cosh(d*x + c)^4 + 950*a^2*b*cosh(d*x + c)^2 + 7*a^2*b)*sinh(d*x + c)^18 + 6*(4389*a
^2*b*cosh(d*x + c)^5 + 950*a^2*b*cosh(d*x + c)^3 + 21*a^2*b*cosh(d*x + c))*sinh(d*x + c)^17 - 22*a^2*b*cosh(d*
x + c)^14 + (74613*a^2*b*cosh(d*x + c)^6 + 24225*a^2*b*cosh(d*x + c)^4 + 1071*a^2*b*cosh(d*x + c)^2 - 5*a^2*b)
*sinh(d*x + c)^16 + 16*(10659*a^2*b*cosh(d*x + c)^7 + 4845*a^2*b*cosh(d*x + c)^5 + 357*a^2*b*cosh(d*x + c)^3 -
5*a^2*b*cosh(d*x + c))*sinh(d*x + c)^15 - 14*a^2*b*cosh(d*x + c)^12 + 2*(159885*a^2*b*cosh(d*x + c)^8 + 96900
*a^2*b*cosh(d*x + c)^6 + 10710*a^2*b*cosh(d*x + c)^4 - 300*a^2*b*cosh(d*x + c)^2 - 11*a^2*b)*sinh(d*x + c)^14
+ 4*(124355*a^2*b*cosh(d*x + c)^9 + 96900*a^2*b*cosh(d*x + c)^7 + 14994*a^2*b*cosh(d*x + c)^5 - 700*a^2*b*cosh
(d*x + c)^3 - 77*a^2*b*cosh(d*x + c))*sinh(d*x + c)^13 + 14*a^2*b*cosh(d*x + c)^10 + 2*(323323*a^2*b*cosh(d*x
+ c)^10 + 314925*a^2*b*cosh(d*x + c)^8 + 64974*a^2*b*cosh(d*x + c)^6 - 4550*a^2*b*cosh(d*x + c)^4 - 1001*a^2*b
*cosh(d*x + c)^2 - 7*a^2*b)*sinh(d*x + c)^12 + 8*(88179*a^2*b*cosh(d*x + c)^11 + 104975*a^2*b*cosh(d*x + c)^9
+ 27846*a^2*b*cosh(d*x + c)^7 - 2730*a^2*b*cosh(d*x + c)^5 - 1001*a^2*b*cosh(d*x + c)^3 - 21*a^2*b*cosh(d*x +
c))*sinh(d*x + c)^11 + 22*a^2*b*cosh(d*x + c)^8 + 2*(323323*a^2*b*cosh(d*x + c)^12 + 461890*a^2*b*cosh(d*x + c
)^10 + 153153*a^2*b*cosh(d*x + c)^8 - 20020*a^2*b*cosh(d*x + c)^6 - 11011*a^2*b*cosh(d*x + c)^4 - 462*a^2*b*co
sh(d*x + c)^2 + 7*a^2*b)*sinh(d*x + c)^10 + 4*(124355*a^2*b*cosh(d*x + c)^13 + 209950*a^2*b*cosh(d*x + c)^11 +
85085*a^2*b*cosh(d*x + c)^9 - 14300*a^2*b*cosh(d*x + c)^7 - 11011*a^2*b*cosh(d*x + c)^5 - 770*a^2*b*cosh(d*x
+ c)^3 + 35*a^2*b*cosh(d*x + c))*sinh(d*x + c)^9 + 5*a^2*b*cosh(d*x + c)^6 + 2*(159885*a^2*b*cosh(d*x + c)^14
+ 314925*a^2*b*cosh(d*x + c)^12 + 153153*a^2*b*cosh(d*x + c)^10 - 32175*a^2*b*cosh(d*x + c)^8 - 33033*a^2*b*co
sh(d*x + c)^6 - 3465*a^2*b*cosh(d*x + c)^4 + 315*a^2*b*cosh(d*x + c)^2 + 11*a^2*b)*sinh(d*x + c)^8 + 16*(10659
*a^2*b*cosh(d*x + c)^15 + 24225*a^2*b*cosh(d*x + c)^13 + 13923*a^2*b*cosh(d*x + c)^11 - 3575*a^2*b*cosh(d*x +
c)^9 - 4719*a^2*b*cosh(d*x + c)^7 - 693*a^2*b*cosh(d*x + c)^5 + 105*a^2*b*cosh(d*x + c)^3 + 11*a^2*b*cosh(d*x
+ c))*sinh(d*x + c)^7 - 7*a^2*b*cosh(d*x + c)^4 + (74613*a^2*b*cosh(d*x + c)^16 + 193800*a^2*b*cosh(d*x + c)^1
4 + 129948*a^2*b*cosh(d*x + c)^12 - 40040*a^2*b*cosh(d*x + c)^10 - 66066*a^2*b*cosh(d*x + c)^8 - 12936*a^2*b*c
osh(d*x + c)^6 + 2940*a^2*b*cosh(d*x + c)^4 + 616*a^2*b*cosh(d*x + c)^2 + 5*a^2*b)*sinh(d*x + c)^6 + 2*(13167*
a^2*b*cosh(d*x + c)^17 + 38760*a^2*b*cosh(d*x + c)^15 + 29988*a^2*b*cosh(d*x + c)^13 - 10920*a^2*b*cosh(d*x +
c)^11 - 22022*a^2*b*cosh(d*x + c)^9 - 5544*a^2*b*cosh(d*x + c)^7 + 1764*a^2*b*cosh(d*x + c)^5 + 616*a^2*b*cosh
(d*x + c)^3 + 15*a^2*b*cosh(d*x + c))*sinh(d*x + c)^5 - 5*a^2*b*cosh(d*x + c)^2 + (7315*a^2*b*cosh(d*x + c)^18
+ 24225*a^2*b*cosh(d*x + c)^16 + 21420*a^2*b*cosh(d*x + c)^14 - 9100*a^2*b*cosh(d*x + c)^12 - 22022*a^2*b*cos
h(d*x + c)^10 - 6930*a^2*b*cosh(d*x + c)^8 + 2940*a^2*b*cosh(d*x + c)^6 + 1540*a^2*b*cosh(d*x + c)^4 + 75*a^2*
b*cosh(d*x + c)^2 - 7*a^2*b)*sinh(d*x + c)^4 + 4*(385*a^2*b*cosh(d*x + c)^19 + 1425*a^2*b*cosh(d*x + c)^17 + 1
428*a^2*b*cosh(d*x + c)^15 - 700*a^2*b*cosh(d*x + c)^13 - 2002*a^2*b*cosh(d*x + c)^11 - 770*a^2*b*cosh(d*x + c
)^9 + 420*a^2*b*cosh(d*x + c)^7 + 308*a^2*b*cosh(d*x + c)^5 + 25*a^2*b*cosh(d*x + c)^3 - 7*a^2*b*cosh(d*x + c)
)*sinh(d*x + c)^3 - a^2*b + (231*a^2*b*cosh(d*x + c)^20 + 950*a^2*b*cosh(d*x + c)^18 + 1071*a^2*b*cosh(d*x + c
)^16 - 600*a^2*b*cosh(d*x + c)^14 - 2002*a^2*b*cosh(d*x + c)^12 - 924*a^2*b*cosh(d*x + c)^10 + 630*a^2*b*cosh(
d*x + c)^8 + 616*a^2*b*cosh(d*x + c)^6 + 75*a^2*b*cosh(d*x + c)^4 - 42*a^2*b*cosh(d*x + c)^2 - 5*a^2*b)*sinh(d
*x + c)^2 + 2*(11*a^2*b*cosh(d*x + c)^21 + 50*a^2*b*cosh(d*x + c)^19 + 63*a^2*b*cosh(d*x + c)^17 - 40*a^2*b*co
sh(d*x + c)^15 - 154*a^2*b*cosh(d*x + c)^13 - 84*a^2*b*cosh(d*x + c)^11 + 70*a^2*b*cosh(d*x + c)^9 + 88*a^2*b*
cosh(d*x + c)^7 + 15*a^2*b*cosh(d*x + c)^5 - 14*a^2*b*cosh(d*x + c)^3 - 5*a^2*b*cosh(d*x + c))*sinh(d*x + c))*
log(2*sinh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) + 4*(450*a^2*b*cosh(d*x + c)^19 - 135*(2*a^3 - 9*a^2*b +
6*a*b^2 + 2*b^3)*cosh(d*x + c)^17 - 80*(23*a^3 - 3*a*b^2 - 13*b^3)*cosh(d*x + c)^15 - 70*(76*a^3 + 36*a^2*b -
24*a*b^2 + 31*b^3)*cosh(d*x + c)^13 - 12*(700*a^3 + 135*a^2*b + 48*a*b^2 - 245*b^3)*cosh(d*x + c)^11 - 50*(154
*a^3 - 27*a^2*b + 18*a*b^2 + 49*b^3)*cosh(d*x + c)^9 - 8*(490*a^3 - 180*a^2*b - 54*a*b^2 - 155*b^3)*cosh(d*x +
c)^7 - 30*(28*a^3 + 13*b^3)*cosh(d*x + c)^5 + 2*(40*a^3 - 135*a^2*b - 48*a*b^2 + 30*b^3)*cosh(d*x + c)^3 + 5*
(10*a^3 - 9*a^2*b + 6*a*b^2)*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^22 + 22*d*cosh(d*x + c)*sinh(d*x +
c)^21 + d*sinh(d*x + c)^22 + 5*d*cosh(d*x + c)^20 + (231*d*cosh(d*x + c)^2 + 5*d)*sinh(d*x + c)^20 + 20*(77*d
*cosh(d*x + c)^3 + 5*d*cosh(d*x + c))*sinh(d*x + c)^19 + 7*d*cosh(d*x + c)^18 + (7315*d*cosh(d*x + c)^4 + 950*
d*cosh(d*x + c)^2 + 7*d)*sinh(d*x + c)^18 + 6*(4389*d*cosh(d*x + c)^5 + 950*d*cosh(d*x + c)^3 + 21*d*cosh(d*x
+ c))*sinh(d*x + c)^17 - 5*d*cosh(d*x + c)^16 + (74613*d*cosh(d*x + c)^6 + 24225*d*cosh(d*x + c)^4 + 1071*d*co
sh(d*x + c)^2 - 5*d)*sinh(d*x + c)^16 + 16*(10659*d*cosh(d*x + c)^7 + 4845*d*cosh(d*x + c)^5 + 357*d*cosh(d*x
+ c)^3 - 5*d*cosh(d*x + c))*sinh(d*x + c)^15 - 22*d*cosh(d*x + c)^14 + 2*(159885*d*cosh(d*x + c)^8 + 96900*d*c
osh(d*x + c)^6 + 10710*d*cosh(d*x + c)^4 - 300*d*cosh(d*x + c)^2 - 11*d)*sinh(d*x + c)^14 + 4*(124355*d*cosh(d
*x + c)^9 + 96900*d*cosh(d*x + c)^7 + 14994*d*cosh(d*x + c)^5 - 700*d*cosh(d*x + c)^3 - 77*d*cosh(d*x + c))*si
nh(d*x + c)^13 - 14*d*cosh(d*x + c)^12 + 2*(323323*d*cosh(d*x + c)^10 + 314925*d*cosh(d*x + c)^8 + 64974*d*cos
h(d*x + c)^6 - 4550*d*cosh(d*x + c)^4 - 1001*d*cosh(d*x + c)^2 - 7*d)*sinh(d*x + c)^12 + 8*(88179*d*cosh(d*x +
c)^11 + 104975*d*cosh(d*x + c)^9 + 27846*d*cosh(d*x + c)^7 - 2730*d*cosh(d*x + c)^5 - 1001*d*cosh(d*x + c)^3
- 21*d*cosh(d*x + c))*sinh(d*x + c)^11 + 14*d*cosh(d*x + c)^10 + 2*(323323*d*cosh(d*x + c)^12 + 461890*d*cosh(
d*x + c)^10 + 153153*d*cosh(d*x + c)^8 - 20020*d*cosh(d*x + c)^6 - 11011*d*cosh(d*x + c)^4 - 462*d*cosh(d*x +
c)^2 + 7*d)*sinh(d*x + c)^10 + 4*(124355*d*cosh(d*x + c)^13 + 209950*d*cosh(d*x + c)^11 + 85085*d*cosh(d*x + c
)^9 - 14300*d*cosh(d*x + c)^7 - 11011*d*cosh(d*x + c)^5 - 770*d*cosh(d*x + c)^3 + 35*d*cosh(d*x + c))*sinh(d*x
+ c)^9 + 22*d*cosh(d*x + c)^8 + 2*(159885*d*cosh(d*x + c)^14 + 314925*d*cosh(d*x + c)^12 + 153153*d*cosh(d*x
+ c)^10 - 32175*d*cosh(d*x + c)^8 - 33033*d*cosh(d*x + c)^6 - 3465*d*cosh(d*x + c)^4 + 315*d*cosh(d*x + c)^2 +
11*d)*sinh(d*x + c)^8 + 16*(10659*d*cosh(d*x + c)^15 + 24225*d*cosh(d*x + c)^13 + 13923*d*cosh(d*x + c)^11 -
3575*d*cosh(d*x + c)^9 - 4719*d*cosh(d*x + c)^7 - 693*d*cosh(d*x + c)^5 + 105*d*cosh(d*x + c)^3 + 11*d*cosh(d*
x + c))*sinh(d*x + c)^7 + 5*d*cosh(d*x + c)^6 + (74613*d*cosh(d*x + c)^16 + 193800*d*cosh(d*x + c)^14 + 129948
*d*cosh(d*x + c)^12 - 40040*d*cosh(d*x + c)^10 - 66066*d*cosh(d*x + c)^8 - 12936*d*cosh(d*x + c)^6 + 2940*d*co
sh(d*x + c)^4 + 616*d*cosh(d*x + c)^2 + 5*d)*sinh(d*x + c)^6 + 2*(13167*d*cosh(d*x + c)^17 + 38760*d*cosh(d*x
+ c)^15 + 29988*d*cosh(d*x + c)^13 - 10920*d*cosh(d*x + c)^11 - 22022*d*cosh(d*x + c)^9 - 5544*d*cosh(d*x + c)
^7 + 1764*d*cosh(d*x + c)^5 + 616*d*cosh(d*x + c)^3 + 15*d*cosh(d*x + c))*sinh(d*x + c)^5 - 7*d*cosh(d*x + c)^
4 + (7315*d*cosh(d*x + c)^18 + 24225*d*cosh(d*x + c)^16 + 21420*d*cosh(d*x + c)^14 - 9100*d*cosh(d*x + c)^12 -
22022*d*cosh(d*x + c)^10 - 6930*d*cosh(d*x + c)^8 + 2940*d*cosh(d*x + c)^6 + 1540*d*cosh(d*x + c)^4 + 75*d*co
sh(d*x + c)^2 - 7*d)*sinh(d*x + c)^4 + 4*(385*d*cosh(d*x + c)^19 + 1425*d*cosh(d*x + c)^17 + 1428*d*cosh(d*x +
c)^15 - 700*d*cosh(d*x + c)^13 - 2002*d*cosh(d*x + c)^11 - 770*d*cosh(d*x + c)^9 + 420*d*cosh(d*x + c)^7 + 30
8*d*cosh(d*x + c)^5 + 25*d*cosh(d*x + c)^3 - 7*d*cosh(d*x + c))*sinh(d*x + c)^3 - 5*d*cosh(d*x + c)^2 + (231*d
*cosh(d*x + c)^20 + 950*d*cosh(d*x + c)^18 + 1071*d*cosh(d*x + c)^16 - 600*d*cosh(d*x + c)^14 - 2002*d*cosh(d*
x + c)^12 - 924*d*cosh(d*x + c)^10 + 630*d*cosh(d*x + c)^8 + 616*d*cosh(d*x + c)^6 + 75*d*cosh(d*x + c)^4 - 42
*d*cosh(d*x + c)^2 - 5*d)*sinh(d*x + c)^2 + 2*(11*d*cosh(d*x + c)^21 + 50*d*cosh(d*x + c)^19 + 63*d*cosh(d*x +
c)^17 - 40*d*cosh(d*x + c)^15 - 154*d*cosh(d*x + c)^13 - 84*d*cosh(d*x + c)^11 + 70*d*cosh(d*x + c)^9 + 88*d*
cosh(d*x + c)^7 + 15*d*cosh(d*x + c)^5 - 14*d*cosh(d*x + c)^3 - 5*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 ^{3}{\left (c + d x \right )}\right )^{3} \operatorname{csch}^{4}{\left (c + d x \right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)**4*(a+b*tanh(d*x+c)**3)**3,x)

[Out]

Integral((a + b*tanh(c + d*x)**3)**3*csch(c + d*x)**4, x)

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Giac [B]  time = 2.87652, size = 590, normalized size = 4.28 \begin{align*} -\frac{2520 \, a^{2} b \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) - 2520 \, a^{2} b \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right ) + \frac{140 \,{\left (33 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 99 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 24 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 99 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 8 \, a^{3} - 33 \, a^{2} b\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}} - \frac{6849 \, a^{2} b e^{\left (16 \, d x + 16 \, c\right )} + 59832 \, a^{2} b e^{\left (14 \, d x + 14 \, c\right )} + 222012 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} - 10080 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} - 3360 \, b^{3} e^{\left (12 \, d x + 12 \, c\right )} + 459144 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} - 26880 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 4480 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 580230 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} - 23520 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 11200 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 459144 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 10752 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 4480 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 222012 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 8736 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 3360 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 59832 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 5376 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 6849 \, a^{2} b - 672 \, a b^{2}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{8}}}{840 \, d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/840*(2520*a^2*b*log(e^(2*d*x + 2*c) + 1) - 2520*a^2*b*log(abs(e^(2*d*x + 2*c) - 1)) + 140*(33*a^2*b*e^(6*d*
x + 6*c) - 99*a^2*b*e^(4*d*x + 4*c) + 24*a^3*e^(2*d*x + 2*c) + 99*a^2*b*e^(2*d*x + 2*c) - 8*a^3 - 33*a^2*b)/(e
^(2*d*x + 2*c) - 1)^3 - (6849*a^2*b*e^(16*d*x + 16*c) + 59832*a^2*b*e^(14*d*x + 14*c) + 222012*a^2*b*e^(12*d*x
+ 12*c) - 10080*a*b^2*e^(12*d*x + 12*c) - 3360*b^3*e^(12*d*x + 12*c) + 459144*a^2*b*e^(10*d*x + 10*c) - 26880
*a*b^2*e^(10*d*x + 10*c) + 4480*b^3*e^(10*d*x + 10*c) + 580230*a^2*b*e^(8*d*x + 8*c) - 23520*a*b^2*e^(8*d*x +
8*c) - 11200*b^3*e^(8*d*x + 8*c) + 459144*a^2*b*e^(6*d*x + 6*c) - 10752*a*b^2*e^(6*d*x + 6*c) + 4480*b^3*e^(6*
d*x + 6*c) + 222012*a^2*b*e^(4*d*x + 4*c) - 8736*a*b^2*e^(4*d*x + 4*c) - 3360*b^3*e^(4*d*x + 4*c) + 59832*a^2*
b*e^(2*d*x + 2*c) - 5376*a*b^2*e^(2*d*x + 2*c) + 6849*a^2*b - 672*a*b^2)/(e^(2*d*x + 2*c) + 1)^8)/d