### 3.69 $$\int \text{csch}(c+d x) (a+b \tanh ^3(c+d x))^3 \, dx$$

Optimal. Leaf size=219 $\frac{3 a^2 b \tan ^{-1}(\sinh (c+d x))}{2 d}-\frac{3 a^2 b \tanh (c+d x) \text{sech}(c+d x)}{2 d}-\frac{a^3 \tanh ^{-1}(\cosh (c+d x))}{d}-\frac{3 a b^2 \text{sech}^5(c+d x)}{5 d}+\frac{2 a b^2 \text{sech}^3(c+d x)}{d}-\frac{3 a b^2 \text{sech}(c+d x)}{d}+\frac{35 b^3 \tan ^{-1}(\sinh (c+d x))}{128 d}-\frac{b^3 \tanh ^7(c+d x) \text{sech}(c+d x)}{8 d}-\frac{7 b^3 \tanh ^5(c+d x) \text{sech}(c+d x)}{48 d}-\frac{35 b^3 \tanh ^3(c+d x) \text{sech}(c+d x)}{192 d}-\frac{35 b^3 \tanh (c+d x) \text{sech}(c+d x)}{128 d}$

[Out]

(3*a^2*b*ArcTan[Sinh[c + d*x]])/(2*d) + (35*b^3*ArcTan[Sinh[c + d*x]])/(128*d) - (a^3*ArcTanh[Cosh[c + d*x]])/
d - (3*a*b^2*Sech[c + d*x])/d + (2*a*b^2*Sech[c + d*x]^3)/d - (3*a*b^2*Sech[c + d*x]^5)/(5*d) - (3*a^2*b*Sech[
c + d*x]*Tanh[c + d*x])/(2*d) - (35*b^3*Sech[c + d*x]*Tanh[c + d*x])/(128*d) - (35*b^3*Sech[c + d*x]*Tanh[c +
d*x]^3)/(192*d) - (7*b^3*Sech[c + d*x]*Tanh[c + d*x]^5)/(48*d) - (b^3*Sech[c + d*x]*Tanh[c + d*x]^7)/(8*d)

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Rubi [A]  time = 0.2635, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 5, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.238, Rules used = {3666, 3770, 2611, 2606, 194} $\frac{3 a^2 b \tan ^{-1}(\sinh (c+d x))}{2 d}-\frac{3 a^2 b \tanh (c+d x) \text{sech}(c+d x)}{2 d}-\frac{a^3 \tanh ^{-1}(\cosh (c+d x))}{d}-\frac{3 a b^2 \text{sech}^5(c+d x)}{5 d}+\frac{2 a b^2 \text{sech}^3(c+d x)}{d}-\frac{3 a b^2 \text{sech}(c+d x)}{d}+\frac{35 b^3 \tan ^{-1}(\sinh (c+d x))}{128 d}-\frac{b^3 \tanh ^7(c+d x) \text{sech}(c+d x)}{8 d}-\frac{7 b^3 \tanh ^5(c+d x) \text{sech}(c+d x)}{48 d}-\frac{35 b^3 \tanh ^3(c+d x) \text{sech}(c+d x)}{192 d}-\frac{35 b^3 \tanh (c+d x) \text{sech}(c+d x)}{128 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(3*a^2*b*ArcTan[Sinh[c + d*x]])/(2*d) + (35*b^3*ArcTan[Sinh[c + d*x]])/(128*d) - (a^3*ArcTanh[Cosh[c + d*x]])/
d - (3*a*b^2*Sech[c + d*x])/d + (2*a*b^2*Sech[c + d*x]^3)/d - (3*a*b^2*Sech[c + d*x]^5)/(5*d) - (3*a^2*b*Sech[
c + d*x]*Tanh[c + d*x])/(2*d) - (35*b^3*Sech[c + d*x]*Tanh[c + d*x])/(128*d) - (35*b^3*Sech[c + d*x]*Tanh[c +
d*x]^3)/(192*d) - (7*b^3*Sech[c + d*x]*Tanh[c + d*x]^5)/(48*d) - (b^3*Sech[c + d*x]*Tanh[c + d*x]^7)/(8*d)

Rule 3666

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]
:> Int[ExpandTrig[(d*sin[e + f*x])^m*(a + b*(c*tan[e + f*x])^n)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n},
x] && IGtQ[p, 0]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 2611

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a*Sec[e
+ f*x])^m*(b*Tan[e + f*x])^(n - 1))/(f*(m + n - 1)), x] - Dist[(b^2*(n - 1))/(m + n - 1), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \text{csch}(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx &=i \int \left (-i a^3 \text{csch}(c+d x)-3 i a^2 b \text{sech}(c+d x) \tanh ^2(c+d x)-3 i a b^2 \text{sech}(c+d x) \tanh ^5(c+d x)-i b^3 \text{sech}(c+d x) \tanh ^8(c+d x)\right ) \, dx\\ &=a^3 \int \text{csch}(c+d x) \, dx+\left (3 a^2 b\right ) \int \text{sech}(c+d x) \tanh ^2(c+d x) \, dx+\left (3 a b^2\right ) \int \text{sech}(c+d x) \tanh ^5(c+d x) \, dx+b^3 \int \text{sech}(c+d x) \tanh ^8(c+d x) \, dx\\ &=-\frac{a^3 \tanh ^{-1}(\cosh (c+d x))}{d}-\frac{3 a^2 b \text{sech}(c+d x) \tanh (c+d x)}{2 d}-\frac{b^3 \text{sech}(c+d x) \tanh ^7(c+d x)}{8 d}+\frac{1}{2} \left (3 a^2 b\right ) \int \text{sech}(c+d x) \, dx+\frac{1}{8} \left (7 b^3\right ) \int \text{sech}(c+d x) \tanh ^6(c+d x) \, dx-\frac{\left (3 a b^2\right ) \operatorname{Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\text{sech}(c+d x)\right )}{d}\\ &=\frac{3 a^2 b \tan ^{-1}(\sinh (c+d x))}{2 d}-\frac{a^3 \tanh ^{-1}(\cosh (c+d x))}{d}-\frac{3 a^2 b \text{sech}(c+d x) \tanh (c+d x)}{2 d}-\frac{7 b^3 \text{sech}(c+d x) \tanh ^5(c+d x)}{48 d}-\frac{b^3 \text{sech}(c+d x) \tanh ^7(c+d x)}{8 d}+\frac{1}{48} \left (35 b^3\right ) \int \text{sech}(c+d x) \tanh ^4(c+d x) \, dx-\frac{\left (3 a b^2\right ) \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\text{sech}(c+d x)\right )}{d}\\ &=\frac{3 a^2 b \tan ^{-1}(\sinh (c+d x))}{2 d}-\frac{a^3 \tanh ^{-1}(\cosh (c+d x))}{d}-\frac{3 a b^2 \text{sech}(c+d x)}{d}+\frac{2 a b^2 \text{sech}^3(c+d x)}{d}-\frac{3 a b^2 \text{sech}^5(c+d x)}{5 d}-\frac{3 a^2 b \text{sech}(c+d x) \tanh (c+d x)}{2 d}-\frac{35 b^3 \text{sech}(c+d x) \tanh ^3(c+d x)}{192 d}-\frac{7 b^3 \text{sech}(c+d x) \tanh ^5(c+d x)}{48 d}-\frac{b^3 \text{sech}(c+d x) \tanh ^7(c+d x)}{8 d}+\frac{1}{64} \left (35 b^3\right ) \int \text{sech}(c+d x) \tanh ^2(c+d x) \, dx\\ &=\frac{3 a^2 b \tan ^{-1}(\sinh (c+d x))}{2 d}-\frac{a^3 \tanh ^{-1}(\cosh (c+d x))}{d}-\frac{3 a b^2 \text{sech}(c+d x)}{d}+\frac{2 a b^2 \text{sech}^3(c+d x)}{d}-\frac{3 a b^2 \text{sech}^5(c+d x)}{5 d}-\frac{3 a^2 b \text{sech}(c+d x) \tanh (c+d x)}{2 d}-\frac{35 b^3 \text{sech}(c+d x) \tanh (c+d x)}{128 d}-\frac{35 b^3 \text{sech}(c+d x) \tanh ^3(c+d x)}{192 d}-\frac{7 b^3 \text{sech}(c+d x) \tanh ^5(c+d x)}{48 d}-\frac{b^3 \text{sech}(c+d x) \tanh ^7(c+d x)}{8 d}+\frac{1}{128} \left (35 b^3\right ) \int \text{sech}(c+d x) \, dx\\ &=\frac{3 a^2 b \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac{35 b^3 \tan ^{-1}(\sinh (c+d x))}{128 d}-\frac{a^3 \tanh ^{-1}(\cosh (c+d x))}{d}-\frac{3 a b^2 \text{sech}(c+d x)}{d}+\frac{2 a b^2 \text{sech}^3(c+d x)}{d}-\frac{3 a b^2 \text{sech}^5(c+d x)}{5 d}-\frac{3 a^2 b \text{sech}(c+d x) \tanh (c+d x)}{2 d}-\frac{35 b^3 \text{sech}(c+d x) \tanh (c+d x)}{128 d}-\frac{35 b^3 \text{sech}(c+d x) \tanh ^3(c+d x)}{192 d}-\frac{7 b^3 \text{sech}(c+d x) \tanh ^5(c+d x)}{48 d}-\frac{b^3 \text{sech}(c+d x) \tanh ^7(c+d x)}{8 d}\\ \end{align*}

Mathematica [A]  time = 2.88559, size = 154, normalized size = 0.7 $\frac{-45 b \text{sech}(c+d x) \left (\left (64 a^2+31 b^2\right ) \tanh (c+d x)+128 a b\right )+30 \left (b \left (192 a^2+35 b^2\right ) \tan ^{-1}\left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )+64 a^3 \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )\right )-8 b^2 \text{sech}^5(c+d x) (144 a+125 b \tanh (c+d x))+10 b^2 \text{sech}^3(c+d x) (384 a+163 b \tanh (c+d x))+240 b^3 \tanh (c+d x) \text{sech}^7(c+d x)}{1920 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(30*(b*(192*a^2 + 35*b^2)*ArcTan[Tanh[(c + d*x)/2]] + 64*a^3*Log[Tanh[(c + d*x)/2]]) + 240*b^3*Sech[c + d*x]^7
*Tanh[c + d*x] - 8*b^2*Sech[c + d*x]^5*(144*a + 125*b*Tanh[c + d*x]) + 10*b^2*Sech[c + d*x]^3*(384*a + 163*b*T
anh[c + d*x]) - 45*b*Sech[c + d*x]*(128*a*b + (64*a^2 + 31*b^2)*Tanh[c + d*x]))/(1920*d)

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Maple [A]  time = 0.096, size = 387, normalized size = 1.8 \begin{align*} -2\,{\frac{{a}^{3}{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) }{d}}-3\,{\frac{{a}^{2}b\sinh \left ( dx+c \right ) }{d \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}}+{\frac{3\,{a}^{2}b{\rm sech} \left (dx+c\right )\tanh \left ( dx+c \right ) }{2\,d}}+3\,{\frac{{a}^{2}b\arctan \left ({{\rm e}^{dx+c}} \right ) }{d}}-3\,{\frac{a{b}^{2} \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}{d \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}}-{\frac{12\,a{b}^{2} \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{5\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}}+{\frac{8\,a{b}^{2} \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{5\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}}+{\frac{8\,a{b}^{2} \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{5\,d\cosh \left ( dx+c \right ) }}-{\frac{8\,a{b}^{2}\cosh \left ( dx+c \right ) }{5\,d}}-{\frac{{b}^{3} \left ( \sinh \left ( dx+c \right ) \right ) ^{7}}{d \left ( \cosh \left ( dx+c \right ) \right ) ^{8}}}-{\frac{7\,{b}^{3} \left ( \sinh \left ( dx+c \right ) \right ) ^{5}}{3\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{8}}}-{\frac{7\,{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 ) }{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}}{8\,d}}+{\frac{7\,{b}^{3}\tanh \left ( dx+c \right ) \left ({\rm sech} \left (dx+c\right ) \right ) ^{5}}{48\,d}}+{\frac{35\,{b}^{3} \left ({\rm sech} \left (dx+c\right ) \right ) ^{3}\tanh \left ( dx+c \right ) }{192\,d}}+{\frac{35\,{b}^{3}{\rm sech} \left (dx+c\right )\tanh \left ( dx+c \right ) }{128\,d}}+{\frac{35\,{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(csch(d*x+c)*(a+b*tanh(d*x+c)^3)^3,x)

[Out]

-2/d*a^3*arctanh(exp(d*x+c))-3/d*a^2*b*sinh(d*x+c)/cosh(d*x+c)^2+3/2*a^2*b*sech(d*x+c)*tanh(d*x+c)/d+3/d*a^2*b
*arctan(exp(d*x+c))-3/d*a*b^2*sinh(d*x+c)^4/cosh(d*x+c)^5-12/5/d*a*b^2*sinh(d*x+c)^2/cosh(d*x+c)^5+8/5/d*a*b^2
*sinh(d*x+c)^2/cosh(d*x+c)^3+8/5/d*a*b^2*sinh(d*x+c)^2/cosh(d*x+c)-8/5*a*b^2*cosh(d*x+c)/d-1/d*b^3*sinh(d*x+c)
^7/cosh(d*x+c)^8-7/3/d*b^3*sinh(d*x+c)^5/cosh(d*x+c)^8-7/3/d*b^3*sinh(d*x+c)^3/cosh(d*x+c)^8-1/d*b^3*sinh(d*x+
c)/cosh(d*x+c)^8+1/8/d*b^3*tanh(d*x+c)*sech(d*x+c)^7+7/48/d*b^3*tanh(d*x+c)*sech(d*x+c)^5+35/192*b^3*sech(d*x+
c)^3*tanh(d*x+c)/d+35/128*b^3*sech(d*x+c)*tanh(d*x+c)/d+35/64/d*b^3*arctan(exp(d*x+c))

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Maxima [B]  time = 1.72007, size = 883, normalized size = 4.03 \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)*(a+b*tanh(d*x+c)^3)^3,x, algorithm="maxima")

[Out]

-1/192*b^3*(105*arctan(e^(-d*x - c))/d + (279*e^(-d*x - c) + 91*e^(-3*d*x - 3*c) + 1799*e^(-5*d*x - 5*c) - 108
5*e^(-7*d*x - 7*c) + 1085*e^(-9*d*x - 9*c) - 1799*e^(-11*d*x - 11*c) - 91*e^(-13*d*x - 13*c) - 279*e^(-15*d*x
- 15*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*a^2*b*(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))) - 2/5*a*b^2*(1
5*e^(-d*x - 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^(-1
0*d*x - 10*c) + 1)) + 20*e^(-3*d*x - 3*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)) + 58*e^(-5*d*x - 5*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)) + 20*e^(-7*d*x - 7*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^(-
9*d*x - 9*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))) + a^3*log(tanh(1/2*d*x + 1/2*c))/d

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Fricas [B]  time = 3.60023, size = 19499, normalized size = 89.04 \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)*(a+b*tanh(d*x+c)^3)^3,x, algorithm="fricas")

[Out]

-1/960*(45*(64*a^2*b + 128*a*b^2 + 31*b^3)*cosh(d*x + c)^15 + 675*(64*a^2*b + 128*a*b^2 + 31*b^3)*cosh(d*x + c
)*sinh(d*x + c)^14 + 45*(64*a^2*b + 128*a*b^2 + 31*b^3)*sinh(d*x + c)^15 + 5*(2880*a^2*b + 4992*a*b^2 + 91*b^3
)*cosh(d*x + c)^13 + 5*(2880*a^2*b + 4992*a*b^2 + 91*b^3 + 945*(64*a^2*b + 128*a*b^2 + 31*b^3)*cosh(d*x + c)^2
)*sinh(d*x + c)^13 + 65*(315*(64*a^2*b + 128*a*b^2 + 31*b^3)*cosh(d*x + c)^3 + (2880*a^2*b + 4992*a*b^2 + 91*b
^3)*cosh(d*x + c))*sinh(d*x + c)^12 + (25920*a^2*b + 62592*a*b^2 + 8995*b^3)*cosh(d*x + c)^11 + (61425*(64*a^2
*b + 128*a*b^2 + 31*b^3)*cosh(d*x + c)^4 + 25920*a^2*b + 62592*a*b^2 + 8995*b^3 + 390*(2880*a^2*b + 4992*a*b^2
+ 91*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^11 + 11*(12285*(64*a^2*b + 128*a*b^2 + 31*b^3)*cosh(d*x + c)^5 + 130
*(2880*a^2*b + 4992*a*b^2 + 91*b^3)*cosh(d*x + c)^3 + (25920*a^2*b + 62592*a*b^2 + 8995*b^3)*cosh(d*x + c))*si
nh(d*x + c)^10 + (14400*a^2*b + 103296*a*b^2 - 5425*b^3)*cosh(d*x + c)^9 + (225225*(64*a^2*b + 128*a*b^2 + 31*
b^3)*cosh(d*x + c)^6 + 3575*(2880*a^2*b + 4992*a*b^2 + 91*b^3)*cosh(d*x + c)^4 + 14400*a^2*b + 103296*a*b^2 -
5425*b^3 + 55*(25920*a^2*b + 62592*a*b^2 + 8995*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^9 + 3*(96525*(64*a^2*b + 1
28*a*b^2 + 31*b^3)*cosh(d*x + c)^7 + 2145*(2880*a^2*b + 4992*a*b^2 + 91*b^3)*cosh(d*x + c)^5 + 55*(25920*a^2*b
+ 62592*a*b^2 + 8995*b^3)*cosh(d*x + c)^3 + 3*(14400*a^2*b + 103296*a*b^2 - 5425*b^3)*cosh(d*x + c))*sinh(d*x
+ c)^8 - (14400*a^2*b - 103296*a*b^2 - 5425*b^3)*cosh(d*x + c)^7 + (289575*(64*a^2*b + 128*a*b^2 + 31*b^3)*co
sh(d*x + c)^8 + 8580*(2880*a^2*b + 4992*a*b^2 + 91*b^3)*cosh(d*x + c)^6 + 330*(25920*a^2*b + 62592*a*b^2 + 899
5*b^3)*cosh(d*x + c)^4 - 14400*a^2*b + 103296*a*b^2 + 5425*b^3 + 36*(14400*a^2*b + 103296*a*b^2 - 5425*b^3)*co
sh(d*x + c)^2)*sinh(d*x + c)^7 + (225225*(64*a^2*b + 128*a*b^2 + 31*b^3)*cosh(d*x + c)^9 + 8580*(2880*a^2*b +
4992*a*b^2 + 91*b^3)*cosh(d*x + c)^7 + 462*(25920*a^2*b + 62592*a*b^2 + 8995*b^3)*cosh(d*x + c)^5 + 84*(14400*
a^2*b + 103296*a*b^2 - 5425*b^3)*cosh(d*x + c)^3 - 7*(14400*a^2*b - 103296*a*b^2 - 5425*b^3)*cosh(d*x + c))*si
nh(d*x + c)^6 - (25920*a^2*b - 62592*a*b^2 + 8995*b^3)*cosh(d*x + c)^5 + (135135*(64*a^2*b + 128*a*b^2 + 31*b^
3)*cosh(d*x + c)^10 + 6435*(2880*a^2*b + 4992*a*b^2 + 91*b^3)*cosh(d*x + c)^8 + 462*(25920*a^2*b + 62592*a*b^2
+ 8995*b^3)*cosh(d*x + c)^6 + 126*(14400*a^2*b + 103296*a*b^2 - 5425*b^3)*cosh(d*x + c)^4 - 25920*a^2*b + 625
92*a*b^2 - 8995*b^3 - 21*(14400*a^2*b - 103296*a*b^2 - 5425*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + (61425*(64
*a^2*b + 128*a*b^2 + 31*b^3)*cosh(d*x + c)^11 + 3575*(2880*a^2*b + 4992*a*b^2 + 91*b^3)*cosh(d*x + c)^9 + 330*
(25920*a^2*b + 62592*a*b^2 + 8995*b^3)*cosh(d*x + c)^7 + 126*(14400*a^2*b + 103296*a*b^2 - 5425*b^3)*cosh(d*x
+ c)^5 - 35*(14400*a^2*b - 103296*a*b^2 - 5425*b^3)*cosh(d*x + c)^3 - 5*(25920*a^2*b - 62592*a*b^2 + 8995*b^3)
*cosh(d*x + c))*sinh(d*x + c)^4 - 5*(2880*a^2*b - 4992*a*b^2 + 91*b^3)*cosh(d*x + c)^3 + (20475*(64*a^2*b + 12
8*a*b^2 + 31*b^3)*cosh(d*x + c)^12 + 1430*(2880*a^2*b + 4992*a*b^2 + 91*b^3)*cosh(d*x + c)^10 + 165*(25920*a^2
*b + 62592*a*b^2 + 8995*b^3)*cosh(d*x + c)^8 + 84*(14400*a^2*b + 103296*a*b^2 - 5425*b^3)*cosh(d*x + c)^6 - 35
*(14400*a^2*b - 103296*a*b^2 - 5425*b^3)*cosh(d*x + c)^4 - 14400*a^2*b + 24960*a*b^2 - 455*b^3 - 10*(25920*a^2
*b - 62592*a*b^2 + 8995*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + (4725*(64*a^2*b + 128*a*b^2 + 31*b^3)*cosh(d*x
+ c)^13 + 390*(2880*a^2*b + 4992*a*b^2 + 91*b^3)*cosh(d*x + c)^11 + 55*(25920*a^2*b + 62592*a*b^2 + 8995*b^3)
*cosh(d*x + c)^9 + 36*(14400*a^2*b + 103296*a*b^2 - 5425*b^3)*cosh(d*x + c)^7 - 21*(14400*a^2*b - 103296*a*b^2
- 5425*b^3)*cosh(d*x + c)^5 - 10*(25920*a^2*b - 62592*a*b^2 + 8995*b^3)*cosh(d*x + c)^3 - 15*(2880*a^2*b - 49
92*a*b^2 + 91*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 - 15*((192*a^2*b + 35*b^3)*cosh(d*x + c)^16 + 16*(192*a^2*b
+ 35*b^3)*cosh(d*x + c)*sinh(d*x + c)^15 + (192*a^2*b + 35*b^3)*sinh(d*x + c)^16 + 8*(192*a^2*b + 35*b^3)*cosh
(d*x + c)^14 + 8*(192*a^2*b + 35*b^3 + 15*(192*a^2*b + 35*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^14 + 112*(5*(192
*a^2*b + 35*b^3)*cosh(d*x + c)^3 + (192*a^2*b + 35*b^3)*cosh(d*x + c))*sinh(d*x + c)^13 + 28*(192*a^2*b + 35*b
^3)*cosh(d*x + c)^12 + 28*(65*(192*a^2*b + 35*b^3)*cosh(d*x + c)^4 + 192*a^2*b + 35*b^3 + 26*(192*a^2*b + 35*b
^3)*cosh(d*x + c)^2)*sinh(d*x + c)^12 + 112*(39*(192*a^2*b + 35*b^3)*cosh(d*x + c)^5 + 26*(192*a^2*b + 35*b^3)
*cosh(d*x + c)^3 + 3*(192*a^2*b + 35*b^3)*cosh(d*x + c))*sinh(d*x + c)^11 + 56*(192*a^2*b + 35*b^3)*cosh(d*x +
c)^10 + 56*(143*(192*a^2*b + 35*b^3)*cosh(d*x + c)^6 + 143*(192*a^2*b + 35*b^3)*cosh(d*x + c)^4 + 192*a^2*b +
35*b^3 + 33*(192*a^2*b + 35*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^10 + 16*(715*(192*a^2*b + 35*b^3)*cosh(d*x +
c)^7 + 1001*(192*a^2*b + 35*b^3)*cosh(d*x + c)^5 + 385*(192*a^2*b + 35*b^3)*cosh(d*x + c)^3 + 35*(192*a^2*b +
35*b^3)*cosh(d*x + c))*sinh(d*x + c)^9 + 70*(192*a^2*b + 35*b^3)*cosh(d*x + c)^8 + 2*(6435*(192*a^2*b + 35*b^3
)*cosh(d*x + c)^8 + 12012*(192*a^2*b + 35*b^3)*cosh(d*x + c)^6 + 6930*(192*a^2*b + 35*b^3)*cosh(d*x + c)^4 + 6
720*a^2*b + 1225*b^3 + 1260*(192*a^2*b + 35*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 16*(715*(192*a^2*b + 35*b^
3)*cosh(d*x + c)^9 + 1716*(192*a^2*b + 35*b^3)*cosh(d*x + c)^7 + 1386*(192*a^2*b + 35*b^3)*cosh(d*x + c)^5 + 4
20*(192*a^2*b + 35*b^3)*cosh(d*x + c)^3 + 35*(192*a^2*b + 35*b^3)*cosh(d*x + c))*sinh(d*x + c)^7 + 56*(192*a^2
*b + 35*b^3)*cosh(d*x + c)^6 + 56*(143*(192*a^2*b + 35*b^3)*cosh(d*x + c)^10 + 429*(192*a^2*b + 35*b^3)*cosh(d
*x + c)^8 + 462*(192*a^2*b + 35*b^3)*cosh(d*x + c)^6 + 210*(192*a^2*b + 35*b^3)*cosh(d*x + c)^4 + 192*a^2*b +
35*b^3 + 35*(192*a^2*b + 35*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 112*(39*(192*a^2*b + 35*b^3)*cosh(d*x + c)
^11 + 143*(192*a^2*b + 35*b^3)*cosh(d*x + c)^9 + 198*(192*a^2*b + 35*b^3)*cosh(d*x + c)^7 + 126*(192*a^2*b + 3
5*b^3)*cosh(d*x + c)^5 + 35*(192*a^2*b + 35*b^3)*cosh(d*x + c)^3 + 3*(192*a^2*b + 35*b^3)*cosh(d*x + c))*sinh(
d*x + c)^5 + 28*(192*a^2*b + 35*b^3)*cosh(d*x + c)^4 + 28*(65*(192*a^2*b + 35*b^3)*cosh(d*x + c)^12 + 286*(192
*a^2*b + 35*b^3)*cosh(d*x + c)^10 + 495*(192*a^2*b + 35*b^3)*cosh(d*x + c)^8 + 420*(192*a^2*b + 35*b^3)*cosh(d
*x + c)^6 + 175*(192*a^2*b + 35*b^3)*cosh(d*x + c)^4 + 192*a^2*b + 35*b^3 + 30*(192*a^2*b + 35*b^3)*cosh(d*x +
c)^2)*sinh(d*x + c)^4 + 112*(5*(192*a^2*b + 35*b^3)*cosh(d*x + c)^13 + 26*(192*a^2*b + 35*b^3)*cosh(d*x + c)^
11 + 55*(192*a^2*b + 35*b^3)*cosh(d*x + c)^9 + 60*(192*a^2*b + 35*b^3)*cosh(d*x + c)^7 + 35*(192*a^2*b + 35*b^
3)*cosh(d*x + c)^5 + 10*(192*a^2*b + 35*b^3)*cosh(d*x + c)^3 + (192*a^2*b + 35*b^3)*cosh(d*x + c))*sinh(d*x +
c)^3 + 192*a^2*b + 35*b^3 + 8*(192*a^2*b + 35*b^3)*cosh(d*x + c)^2 + 8*(15*(192*a^2*b + 35*b^3)*cosh(d*x + c)^
14 + 91*(192*a^2*b + 35*b^3)*cosh(d*x + c)^12 + 231*(192*a^2*b + 35*b^3)*cosh(d*x + c)^10 + 315*(192*a^2*b + 3
5*b^3)*cosh(d*x + c)^8 + 245*(192*a^2*b + 35*b^3)*cosh(d*x + c)^6 + 105*(192*a^2*b + 35*b^3)*cosh(d*x + c)^4 +
192*a^2*b + 35*b^3 + 21*(192*a^2*b + 35*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 16*((192*a^2*b + 35*b^3)*cosh
(d*x + c)^15 + 7*(192*a^2*b + 35*b^3)*cosh(d*x + c)^13 + 21*(192*a^2*b + 35*b^3)*cosh(d*x + c)^11 + 35*(192*a^
2*b + 35*b^3)*cosh(d*x + c)^9 + 35*(192*a^2*b + 35*b^3)*cosh(d*x + c)^7 + 21*(192*a^2*b + 35*b^3)*cosh(d*x + c
)^5 + 7*(192*a^2*b + 35*b^3)*cosh(d*x + c)^3 + (192*a^2*b + 35*b^3)*cosh(d*x + c))*sinh(d*x + c))*arctan(cosh(
d*x + c) + sinh(d*x + c)) - 45*(64*a^2*b - 128*a*b^2 + 31*b^3)*cosh(d*x + c) + 960*(a^3*cosh(d*x + c)^16 + 16*
a^3*cosh(d*x + c)*sinh(d*x + c)^15 + a^3*sinh(d*x + c)^16 + 8*a^3*cosh(d*x + c)^14 + 28*a^3*cosh(d*x + c)^12 +
8*(15*a^3*cosh(d*x + c)^2 + a^3)*sinh(d*x + c)^14 + 112*(5*a^3*cosh(d*x + c)^3 + a^3*cosh(d*x + c))*sinh(d*x
+ c)^13 + 56*a^3*cosh(d*x + c)^10 + 28*(65*a^3*cosh(d*x + c)^4 + 26*a^3*cosh(d*x + c)^2 + a^3)*sinh(d*x + c)^1
2 + 112*(39*a^3*cosh(d*x + c)^5 + 26*a^3*cosh(d*x + c)^3 + 3*a^3*cosh(d*x + c))*sinh(d*x + c)^11 + 70*a^3*cosh
(d*x + c)^8 + 56*(143*a^3*cosh(d*x + c)^6 + 143*a^3*cosh(d*x + c)^4 + 33*a^3*cosh(d*x + c)^2 + a^3)*sinh(d*x +
c)^10 + 16*(715*a^3*cosh(d*x + c)^7 + 1001*a^3*cosh(d*x + c)^5 + 385*a^3*cosh(d*x + c)^3 + 35*a^3*cosh(d*x +
c))*sinh(d*x + c)^9 + 56*a^3*cosh(d*x + c)^6 + 2*(6435*a^3*cosh(d*x + c)^8 + 12012*a^3*cosh(d*x + c)^6 + 6930*
a^3*cosh(d*x + c)^4 + 1260*a^3*cosh(d*x + c)^2 + 35*a^3)*sinh(d*x + c)^8 + 16*(715*a^3*cosh(d*x + c)^9 + 1716*
a^3*cosh(d*x + c)^7 + 1386*a^3*cosh(d*x + c)^5 + 420*a^3*cosh(d*x + c)^3 + 35*a^3*cosh(d*x + c))*sinh(d*x + c)
^7 + 28*a^3*cosh(d*x + c)^4 + 56*(143*a^3*cosh(d*x + c)^10 + 429*a^3*cosh(d*x + c)^8 + 462*a^3*cosh(d*x + c)^6
+ 210*a^3*cosh(d*x + c)^4 + 35*a^3*cosh(d*x + c)^2 + a^3)*sinh(d*x + c)^6 + 112*(39*a^3*cosh(d*x + c)^11 + 14
3*a^3*cosh(d*x + c)^9 + 198*a^3*cosh(d*x + c)^7 + 126*a^3*cosh(d*x + c)^5 + 35*a^3*cosh(d*x + c)^3 + 3*a^3*cos
h(d*x + c))*sinh(d*x + c)^5 + 8*a^3*cosh(d*x + c)^2 + 28*(65*a^3*cosh(d*x + c)^12 + 286*a^3*cosh(d*x + c)^10 +
495*a^3*cosh(d*x + c)^8 + 420*a^3*cosh(d*x + c)^6 + 175*a^3*cosh(d*x + c)^4 + 30*a^3*cosh(d*x + c)^2 + a^3)*s
inh(d*x + c)^4 + 112*(5*a^3*cosh(d*x + c)^13 + 26*a^3*cosh(d*x + c)^11 + 55*a^3*cosh(d*x + c)^9 + 60*a^3*cosh(
d*x + c)^7 + 35*a^3*cosh(d*x + c)^5 + 10*a^3*cosh(d*x + c)^3 + a^3*cosh(d*x + c))*sinh(d*x + c)^3 + a^3 + 8*(1
5*a^3*cosh(d*x + c)^14 + 91*a^3*cosh(d*x + c)^12 + 231*a^3*cosh(d*x + c)^10 + 315*a^3*cosh(d*x + c)^8 + 245*a^
3*cosh(d*x + c)^6 + 105*a^3*cosh(d*x + c)^4 + 21*a^3*cosh(d*x + c)^2 + a^3)*sinh(d*x + c)^2 + 16*(a^3*cosh(d*x
+ c)^15 + 7*a^3*cosh(d*x + c)^13 + 21*a^3*cosh(d*x + c)^11 + 35*a^3*cosh(d*x + c)^9 + 35*a^3*cosh(d*x + c)^7
+ 21*a^3*cosh(d*x + c)^5 + 7*a^3*cosh(d*x + c)^3 + a^3*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(
d*x + c) + 1) - 960*(a^3*cosh(d*x + c)^16 + 16*a^3*cosh(d*x + c)*sinh(d*x + c)^15 + a^3*sinh(d*x + c)^16 + 8*a
^3*cosh(d*x + c)^14 + 28*a^3*cosh(d*x + c)^12 + 8*(15*a^3*cosh(d*x + c)^2 + a^3)*sinh(d*x + c)^14 + 112*(5*a^3
*cosh(d*x + c)^3 + a^3*cosh(d*x + c))*sinh(d*x + c)^13 + 56*a^3*cosh(d*x + c)^10 + 28*(65*a^3*cosh(d*x + c)^4
+ 26*a^3*cosh(d*x + c)^2 + a^3)*sinh(d*x + c)^12 + 112*(39*a^3*cosh(d*x + c)^5 + 26*a^3*cosh(d*x + c)^3 + 3*a^
3*cosh(d*x + c))*sinh(d*x + c)^11 + 70*a^3*cosh(d*x + c)^8 + 56*(143*a^3*cosh(d*x + c)^6 + 143*a^3*cosh(d*x +
c)^4 + 33*a^3*cosh(d*x + c)^2 + a^3)*sinh(d*x + c)^10 + 16*(715*a^3*cosh(d*x + c)^7 + 1001*a^3*cosh(d*x + c)^5
+ 385*a^3*cosh(d*x + c)^3 + 35*a^3*cosh(d*x + c))*sinh(d*x + c)^9 + 56*a^3*cosh(d*x + c)^6 + 2*(6435*a^3*cosh
(d*x + c)^8 + 12012*a^3*cosh(d*x + c)^6 + 6930*a^3*cosh(d*x + c)^4 + 1260*a^3*cosh(d*x + c)^2 + 35*a^3)*sinh(d
*x + c)^8 + 16*(715*a^3*cosh(d*x + c)^9 + 1716*a^3*cosh(d*x + c)^7 + 1386*a^3*cosh(d*x + c)^5 + 420*a^3*cosh(d
*x + c)^3 + 35*a^3*cosh(d*x + c))*sinh(d*x + c)^7 + 28*a^3*cosh(d*x + c)^4 + 56*(143*a^3*cosh(d*x + c)^10 + 42
9*a^3*cosh(d*x + c)^8 + 462*a^3*cosh(d*x + c)^6 + 210*a^3*cosh(d*x + c)^4 + 35*a^3*cosh(d*x + c)^2 + a^3)*sinh
(d*x + c)^6 + 112*(39*a^3*cosh(d*x + c)^11 + 143*a^3*cosh(d*x + c)^9 + 198*a^3*cosh(d*x + c)^7 + 126*a^3*cosh(
d*x + c)^5 + 35*a^3*cosh(d*x + c)^3 + 3*a^3*cosh(d*x + c))*sinh(d*x + c)^5 + 8*a^3*cosh(d*x + c)^2 + 28*(65*a^
3*cosh(d*x + c)^12 + 286*a^3*cosh(d*x + c)^10 + 495*a^3*cosh(d*x + c)^8 + 420*a^3*cosh(d*x + c)^6 + 175*a^3*co
sh(d*x + c)^4 + 30*a^3*cosh(d*x + c)^2 + a^3)*sinh(d*x + c)^4 + 112*(5*a^3*cosh(d*x + c)^13 + 26*a^3*cosh(d*x
+ c)^11 + 55*a^3*cosh(d*x + c)^9 + 60*a^3*cosh(d*x + c)^7 + 35*a^3*cosh(d*x + c)^5 + 10*a^3*cosh(d*x + c)^3 +
a^3*cosh(d*x + c))*sinh(d*x + c)^3 + a^3 + 8*(15*a^3*cosh(d*x + c)^14 + 91*a^3*cosh(d*x + c)^12 + 231*a^3*cosh
(d*x + c)^10 + 315*a^3*cosh(d*x + c)^8 + 245*a^3*cosh(d*x + c)^6 + 105*a^3*cosh(d*x + c)^4 + 21*a^3*cosh(d*x +
c)^2 + a^3)*sinh(d*x + c)^2 + 16*(a^3*cosh(d*x + c)^15 + 7*a^3*cosh(d*x + c)^13 + 21*a^3*cosh(d*x + c)^11 + 3
5*a^3*cosh(d*x + c)^9 + 35*a^3*cosh(d*x + c)^7 + 21*a^3*cosh(d*x + c)^5 + 7*a^3*cosh(d*x + c)^3 + a^3*cosh(d*x
+ c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - 1) + (675*(64*a^2*b + 128*a*b^2 + 31*b^3)*cosh(d*x +
c)^14 + 65*(2880*a^2*b + 4992*a*b^2 + 91*b^3)*cosh(d*x + c)^12 + 11*(25920*a^2*b + 62592*a*b^2 + 8995*b^3)*co
sh(d*x + c)^10 + 9*(14400*a^2*b + 103296*a*b^2 - 5425*b^3)*cosh(d*x + c)^8 - 7*(14400*a^2*b - 103296*a*b^2 - 5
425*b^3)*cosh(d*x + c)^6 - 5*(25920*a^2*b - 62592*a*b^2 + 8995*b^3)*cosh(d*x + c)^4 - 2880*a^2*b + 5760*a*b^2
- 1395*b^3 - 15*(2880*a^2*b - 4992*a*b^2 + 91*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 + 1716*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*cosh(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 + 9
1*d*cosh(d*x + c)^12 + 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*cos
h(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*c
osh(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}{\left (c + d x \right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 2.28561, size = 570, normalized size = 2.6 \begin{align*} -\frac{960 \, a^{3} \log \left (e^{\left (d x + c\right )} + 1\right ) - 960 \, a^{3} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) - 15 \,{\left (192 \, a^{2} b e^{c} + 35 \, b^{3} e^{c}\right )} \arctan \left (e^{\left (d x + c\right )}\right ) e^{\left (-c\right )} + \frac{2880 \, a^{2} b e^{\left (15 \, d x + 15 \, c\right )} + 5760 \, a b^{2} e^{\left (15 \, d x + 15 \, c\right )} + 1395 \, b^{3} e^{\left (15 \, d x + 15 \, c\right )} + 14400 \, a^{2} b e^{\left (13 \, d x + 13 \, c\right )} + 24960 \, a b^{2} e^{\left (13 \, d x + 13 \, c\right )} + 455 \, b^{3} e^{\left (13 \, d x + 13 \, c\right )} + 25920 \, a^{2} b e^{\left (11 \, d x + 11 \, c\right )} + 62592 \, a b^{2} e^{\left (11 \, d x + 11 \, c\right )} + 8995 \, b^{3} e^{\left (11 \, d x + 11 \, c\right )} + 14400 \, a^{2} b e^{\left (9 \, d x + 9 \, c\right )} + 103296 \, a b^{2} e^{\left (9 \, d x + 9 \, c\right )} - 5425 \, b^{3} e^{\left (9 \, d x + 9 \, c\right )} - 14400 \, a^{2} b e^{\left (7 \, d x + 7 \, c\right )} + 103296 \, a b^{2} e^{\left (7 \, d x + 7 \, c\right )} + 5425 \, b^{3} e^{\left (7 \, d x + 7 \, c\right )} - 25920 \, a^{2} b e^{\left (5 \, d x + 5 \, c\right )} + 62592 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} - 8995 \, b^{3} e^{\left (5 \, d x + 5 \, c\right )} - 14400 \, a^{2} b e^{\left (3 \, d x + 3 \, c\right )} + 24960 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 455 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} - 2880 \, a^{2} b e^{\left (d x + c\right )} + 5760 \, a b^{2} e^{\left (d x + c\right )} - 1395 \, b^{3} e^{\left (d x + c\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{8}}}{960 \, d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/960*(960*a^3*log(e^(d*x + c) + 1) - 960*a^3*log(abs(e^(d*x + c) - 1)) - 15*(192*a^2*b*e^c + 35*b^3*e^c)*arc
tan(e^(d*x + c))*e^(-c) + (2880*a^2*b*e^(15*d*x + 15*c) + 5760*a*b^2*e^(15*d*x + 15*c) + 1395*b^3*e^(15*d*x +
15*c) + 14400*a^2*b*e^(13*d*x + 13*c) + 24960*a*b^2*e^(13*d*x + 13*c) + 455*b^3*e^(13*d*x + 13*c) + 25920*a^2*
b*e^(11*d*x + 11*c) + 62592*a*b^2*e^(11*d*x + 11*c) + 8995*b^3*e^(11*d*x + 11*c) + 14400*a^2*b*e^(9*d*x + 9*c)
+ 103296*a*b^2*e^(9*d*x + 9*c) - 5425*b^3*e^(9*d*x + 9*c) - 14400*a^2*b*e^(7*d*x + 7*c) + 103296*a*b^2*e^(7*d
*x + 7*c) + 5425*b^3*e^(7*d*x + 7*c) - 25920*a^2*b*e^(5*d*x + 5*c) + 62592*a*b^2*e^(5*d*x + 5*c) - 8995*b^3*e^
(5*d*x + 5*c) - 14400*a^2*b*e^(3*d*x + 3*c) + 24960*a*b^2*e^(3*d*x + 3*c) - 455*b^3*e^(3*d*x + 3*c) - 2880*a^2
*b*e^(d*x + c) + 5760*a*b^2*e^(d*x + c) - 1395*b^3*e^(d*x + c))/(e^(2*d*x + 2*c) + 1)^8)/d