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

Optimal. Leaf size=232 \[ \frac{3 a^2 b \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac{3 a^2 b \tanh (c+d x) \text{sech}(c+d x)}{2 d}+\frac{a^3 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{a^3 \coth (c+d x) \text{csch}(c+d x)}{2 d}+\frac{3 a b^2 \text{sech}^5(c+d x)}{5 d}-\frac{a b^2 \text{sech}^3(c+d x)}{d}+\frac{5 b^3 \tan ^{-1}(\sinh (c+d x))}{128 d}-\frac{b^3 \tanh ^5(c+d x) \text{sech}^3(c+d x)}{8 d}-\frac{5 b^3 \tanh ^3(c+d x) \text{sech}^3(c+d x)}{48 d}-\frac{5 b^3 \tanh (c+d x) \text{sech}^3(c+d x)}{64 d}+\frac{5 b^3 \tanh (c+d x) \text{sech}(c+d x)}{128 d} \]

[Out]

(3*a^2*b*ArcTan[Sinh[c + d*x]])/(2*d) + (5*b^3*ArcTan[Sinh[c + d*x]])/(128*d) + (a^3*ArcTanh[Cosh[c + d*x]])/(
2*d) - (a^3*Coth[c + d*x]*Csch[c + d*x])/(2*d) - (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) + (5*b^3*Sech[c + d*x]*Tanh[c + d*x])/(128*d) - (5*b^3*Sech[c + d
*x]^3*Tanh[c + d*x])/(64*d) - (5*b^3*Sech[c + d*x]^3*Tanh[c + d*x]^3)/(48*d) - (b^3*Sech[c + d*x]^3*Tanh[c + d
*x]^5)/(8*d)

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Rubi [A]  time = 0.316071, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3666, 3768, 3770, 2606, 14, 2611} \[ \frac{3 a^2 b \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac{3 a^2 b \tanh (c+d x) \text{sech}(c+d x)}{2 d}+\frac{a^3 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{a^3 \coth (c+d x) \text{csch}(c+d x)}{2 d}+\frac{3 a b^2 \text{sech}^5(c+d x)}{5 d}-\frac{a b^2 \text{sech}^3(c+d x)}{d}+\frac{5 b^3 \tan ^{-1}(\sinh (c+d x))}{128 d}-\frac{b^3 \tanh ^5(c+d x) \text{sech}^3(c+d x)}{8 d}-\frac{5 b^3 \tanh ^3(c+d x) \text{sech}^3(c+d x)}{48 d}-\frac{5 b^3 \tanh (c+d x) \text{sech}^3(c+d x)}{64 d}+\frac{5 b^3 \tanh (c+d x) \text{sech}(c+d x)}{128 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a^2*b*ArcTan[Sinh[c + d*x]])/(2*d) + (5*b^3*ArcTan[Sinh[c + d*x]])/(128*d) + (a^3*ArcTanh[Cosh[c + d*x]])/(
2*d) - (a^3*Coth[c + d*x]*Csch[c + d*x])/(2*d) - (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) + (5*b^3*Sech[c + d*x]*Tanh[c + d*x])/(128*d) - (5*b^3*Sech[c + d
*x]^3*Tanh[c + d*x])/(64*d) - (5*b^3*Sech[c + d*x]^3*Tanh[c + d*x]^3)/(48*d) - (b^3*Sech[c + d*x]^3*Tanh[c + d
*x]^5)/(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 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

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

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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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]

Rubi steps

\begin{align*} \int \text{csch}^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx &=-\left (i \int \left (i a^3 \text{csch}^3(c+d x)+3 i a^2 b \text{sech}^3(c+d x)+3 i a b^2 \text{sech}^3(c+d x) \tanh ^3(c+d x)+i b^3 \text{sech}^3(c+d x) \tanh ^6(c+d x)\right ) \, dx\right )\\ &=a^3 \int \text{csch}^3(c+d x) \, dx+\left (3 a^2 b\right ) \int \text{sech}^3(c+d x) \, dx+\left (3 a b^2\right ) \int \text{sech}^3(c+d x) \tanh ^3(c+d x) \, dx+b^3 \int \text{sech}^3(c+d x) \tanh ^6(c+d x) \, dx\\ &=-\frac{a^3 \coth (c+d x) \text{csch}(c+d x)}{2 d}+\frac{3 a^2 b \text{sech}(c+d x) \tanh (c+d x)}{2 d}-\frac{b^3 \text{sech}^3(c+d x) \tanh ^5(c+d x)}{8 d}-\frac{1}{2} a^3 \int \text{csch}(c+d x) \, dx+\frac{1}{2} \left (3 a^2 b\right ) \int \text{sech}(c+d x) \, dx+\frac{1}{8} \left (5 b^3\right ) \int \text{sech}^3(c+d x) \tanh ^4(c+d x) \, dx+\frac{\left (3 a b^2\right ) \operatorname{Subst}\left (\int x^2 \left (-1+x^2\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))}{2 d}-\frac{a^3 \coth (c+d x) \text{csch}(c+d x)}{2 d}+\frac{3 a^2 b \text{sech}(c+d x) \tanh (c+d x)}{2 d}-\frac{5 b^3 \text{sech}^3(c+d x) \tanh ^3(c+d x)}{48 d}-\frac{b^3 \text{sech}^3(c+d x) \tanh ^5(c+d x)}{8 d}+\frac{1}{16} \left (5 b^3\right ) \int \text{sech}^3(c+d x) \tanh ^2(c+d x) \, dx+\frac{\left (3 a b^2\right ) \operatorname{Subst}\left (\int \left (-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))}{2 d}-\frac{a^3 \coth (c+d x) \text{csch}(c+d x)}{2 d}-\frac{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{5 b^3 \text{sech}^3(c+d x) \tanh (c+d x)}{64 d}-\frac{5 b^3 \text{sech}^3(c+d x) \tanh ^3(c+d x)}{48 d}-\frac{b^3 \text{sech}^3(c+d x) \tanh ^5(c+d x)}{8 d}+\frac{1}{64} \left (5 b^3\right ) \int \text{sech}^3(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))}{2 d}-\frac{a^3 \coth (c+d x) \text{csch}(c+d x)}{2 d}-\frac{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{5 b^3 \text{sech}(c+d x) \tanh (c+d x)}{128 d}-\frac{5 b^3 \text{sech}^3(c+d x) \tanh (c+d x)}{64 d}-\frac{5 b^3 \text{sech}^3(c+d x) \tanh ^3(c+d x)}{48 d}-\frac{b^3 \text{sech}^3(c+d x) \tanh ^5(c+d x)}{8 d}+\frac{1}{128} \left (5 b^3\right ) \int \text{sech}(c+d x) \, dx\\ &=\frac{3 a^2 b \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac{5 b^3 \tan ^{-1}(\sinh (c+d x))}{128 d}+\frac{a^3 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{a^3 \coth (c+d x) \text{csch}(c+d x)}{2 d}-\frac{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{5 b^3 \text{sech}(c+d x) \tanh (c+d x)}{128 d}-\frac{5 b^3 \text{sech}^3(c+d x) \tanh (c+d x)}{64 d}-\frac{5 b^3 \text{sech}^3(c+d x) \tanh ^3(c+d x)}{48 d}-\frac{b^3 \text{sech}^3(c+d x) \tanh ^5(c+d x)}{8 d}\\ \end{align*}

Mathematica [A]  time = 6.31567, size = 243, normalized size = 1.05 \[ \frac{b \left (192 a^2+5 b^2\right ) \tan ^{-1}\left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{64 d}+\frac{\text{sech}^2(c+d x) \left (192 a^2 b \sinh (c+d x)+5 b^3 \sinh (c+d x)\right )}{128 d}-\frac{a^3 \text{csch}^2\left (\frac{1}{2} (c+d x)\right )}{8 d}-\frac{a^3 \text{sech}^2\left (\frac{1}{2} (c+d x)\right )}{8 d}-\frac{a^3 \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{2 d}+\frac{3 a b^2 \text{sech}^5(c+d x)}{5 d}-\frac{a b^2 \text{sech}^3(c+d x)}{d}-\frac{b^3 \tanh (c+d x) \text{sech}^7(c+d x)}{8 d}+\frac{17 b^3 \tanh (c+d x) \text{sech}^5(c+d x)}{48 d}-\frac{59 b^3 \tanh (c+d x) \text{sech}^3(c+d x)}{192 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(192*a^2 + 5*b^2)*ArcTan[Tanh[(c + d*x)/2]])/(64*d) - (a^3*Csch[(c + d*x)/2]^2)/(8*d) - (a^3*Log[Tanh[(c +
d*x)/2]])/(2*d) - (a^3*Sech[(c + d*x)/2]^2)/(8*d) - (a*b^2*Sech[c + d*x]^3)/d + (3*a*b^2*Sech[c + d*x]^5)/(5*d
) + (Sech[c + d*x]^2*(192*a^2*b*Sinh[c + d*x] + 5*b^3*Sinh[c + d*x]))/(128*d) - (59*b^3*Sech[c + d*x]^3*Tanh[c
+ d*x])/(192*d) + (17*b^3*Sech[c + d*x]^5*Tanh[c + d*x])/(48*d) - (b^3*Sech[c + d*x]^7*Tanh[c + d*x])/(8*d)

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Maple [A]  time = 0.099, size = 334, normalized size = 1.4 \begin{align*} -{\frac{{a}^{3}{\rm coth} \left (dx+c\right ){\rm csch} \left (dx+c\right )}{2\,d}}+{\frac{{a}^{3}{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) }{d}}+{\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}}-{\frac{3\,a{b}^{2} \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{5\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}}+{\frac{2\,a{b}^{2} \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{5\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}}+{\frac{2\,a{b}^{2} \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{5\,d\cosh \left ( dx+c \right ) }}-{\frac{2\,a{b}^{2}\cosh \left ( dx+c \right ) }{5\,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(csch(d*x+c)^3*(a+b*tanh(d*x+c)^3)^3,x)

[Out]

-1/2*a^3*coth(d*x+c)*csch(d*x+c)/d+1/d*a^3*arctanh(exp(d*x+c))+3/2*a^2*b*sech(d*x+c)*tanh(d*x+c)/d+3/d*a^2*b*a
rctan(exp(d*x+c))-3/5/d*a*b^2*sinh(d*x+c)^2/cosh(d*x+c)^5+2/5/d*a*b^2*sinh(d*x+c)^2/cosh(d*x+c)^3+2/5/d*a*b^2*
sinh(d*x+c)^2/cosh(d*x+c)-2/5*a*b^2*cosh(d*x+c)/d-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*sech(d*x+c)*tanh(d*x+c)/d+5/64/d*b^3*arctan(ex
p(d*x+c))

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Maxima [B]  time = 1.77763, size = 791, normalized size = 3.41 \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)^3*(a+b*tanh(d*x+c)^3)^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))) - 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))) + 1/2*a^3*(log(e^
(-d*x - c) + 1)/d - log(e^(-d*x - c) - 1)/d + 2*(e^(-d*x - c) + e^(-3*d*x - 3*c))/(d*(2*e^(-2*d*x - 2*c) - e^(
-4*d*x - 4*c) - 1))) - 8/5*a*b^2*(5*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)) - 2*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)) + 5*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)))

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Fricas [B]  time = 3.9052, size = 30318, normalized size = 130.68 \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)^3*(a+b*tanh(d*x+c)^3)^3,x, algorithm="fricas")

[Out]

-1/960*(15*(64*a^3 - 192*a^2*b - 5*b^3)*cosh(d*x + c)^19 + 285*(64*a^3 - 192*a^2*b - 5*b^3)*cosh(d*x + c)*sinh
(d*x + c)^18 + 15*(64*a^3 - 192*a^2*b - 5*b^3)*sinh(d*x + c)^19 + 5*(1728*a^3 - 1728*a^2*b + 1536*a*b^2 + 427*
b^3)*cosh(d*x + c)^17 + 5*(1728*a^3 - 1728*a^2*b + 1536*a*b^2 + 427*b^3 + 513*(64*a^3 - 192*a^2*b - 5*b^3)*cos
h(d*x + c)^2)*sinh(d*x + c)^17 + 85*(171*(64*a^3 - 192*a^2*b - 5*b^3)*cosh(d*x + c)^3 + (1728*a^3 - 1728*a^2*b
+ 1536*a*b^2 + 427*b^3)*cosh(d*x + c))*sinh(d*x + c)^16 + 24*(1440*a^3 + 192*a*b^2 - 355*b^3)*cosh(d*x + c)^1
5 + 4*(14535*(64*a^3 - 192*a^2*b - 5*b^3)*cosh(d*x + c)^4 + 8640*a^3 + 1152*a*b^2 - 2130*b^3 + 170*(1728*a^3 -
1728*a^2*b + 1536*a*b^2 + 427*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^15 + 20*(8721*(64*a^3 - 192*a^2*b - 5*b^3)*
cosh(d*x + c)^5 + 170*(1728*a^3 - 1728*a^2*b + 1536*a*b^2 + 427*b^3)*cosh(d*x + c)^3 + 18*(1440*a^3 + 192*a*b^
2 - 355*b^3)*cosh(d*x + c))*sinh(d*x + c)^14 + 16*(5040*a^3 + 1440*a^2*b - 672*a*b^2 + 1235*b^3)*cosh(d*x + c)
^13 + 4*(101745*(64*a^3 - 192*a^2*b - 5*b^3)*cosh(d*x + c)^6 + 2975*(1728*a^3 - 1728*a^2*b + 1536*a*b^2 + 427*
b^3)*cosh(d*x + c)^4 + 20160*a^3 + 5760*a^2*b - 2688*a*b^2 + 4940*b^3 + 630*(1440*a^3 + 192*a*b^2 - 355*b^3)*c
osh(d*x + c)^2)*sinh(d*x + c)^13 + 52*(14535*(64*a^3 - 192*a^2*b - 5*b^3)*cosh(d*x + c)^7 + 595*(1728*a^3 - 17
28*a^2*b + 1536*a*b^2 + 427*b^3)*cosh(d*x + c)^5 + 210*(1440*a^3 + 192*a*b^2 - 355*b^3)*cosh(d*x + c)^3 + 4*(5
040*a^3 + 1440*a^2*b - 672*a*b^2 + 1235*b^3)*cosh(d*x + c))*sinh(d*x + c)^12 + 2*(60480*a^3 + 8640*a^2*b - 768
*a*b^2 - 15475*b^3)*cosh(d*x + c)^11 + 2*(566865*(64*a^3 - 192*a^2*b - 5*b^3)*cosh(d*x + c)^8 + 30940*(1728*a^
3 - 1728*a^2*b + 1536*a*b^2 + 427*b^3)*cosh(d*x + c)^6 + 16380*(1440*a^3 + 192*a*b^2 - 355*b^3)*cosh(d*x + c)^
4 + 60480*a^3 + 8640*a^2*b - 768*a*b^2 - 15475*b^3 + 624*(5040*a^3 + 1440*a^2*b - 672*a*b^2 + 1235*b^3)*cosh(d
*x + c)^2)*sinh(d*x + c)^11 + 22*(62985*(64*a^3 - 192*a^2*b - 5*b^3)*cosh(d*x + c)^9 + 4420*(1728*a^3 - 1728*a
^2*b + 1536*a*b^2 + 427*b^3)*cosh(d*x + c)^7 + 3276*(1440*a^3 + 192*a*b^2 - 355*b^3)*cosh(d*x + c)^5 + 208*(50
40*a^3 + 1440*a^2*b - 672*a*b^2 + 1235*b^3)*cosh(d*x + c)^3 + (60480*a^3 + 8640*a^2*b - 768*a*b^2 - 15475*b^3)
*cosh(d*x + c))*sinh(d*x + c)^10 + 2*(60480*a^3 - 8640*a^2*b - 768*a*b^2 + 15475*b^3)*cosh(d*x + c)^9 + 2*(692
835*(64*a^3 - 192*a^2*b - 5*b^3)*cosh(d*x + c)^10 + 60775*(1728*a^3 - 1728*a^2*b + 1536*a*b^2 + 427*b^3)*cosh(
d*x + c)^8 + 60060*(1440*a^3 + 192*a*b^2 - 355*b^3)*cosh(d*x + c)^6 + 5720*(5040*a^3 + 1440*a^2*b - 672*a*b^2
+ 1235*b^3)*cosh(d*x + c)^4 + 60480*a^3 - 8640*a^2*b - 768*a*b^2 + 15475*b^3 + 55*(60480*a^3 + 8640*a^2*b - 76
8*a*b^2 - 15475*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^9 + 2*(566865*(64*a^3 - 192*a^2*b - 5*b^3)*cosh(d*x + c)^1
1 + 60775*(1728*a^3 - 1728*a^2*b + 1536*a*b^2 + 427*b^3)*cosh(d*x + c)^9 + 77220*(1440*a^3 + 192*a*b^2 - 355*b
^3)*cosh(d*x + c)^7 + 10296*(5040*a^3 + 1440*a^2*b - 672*a*b^2 + 1235*b^3)*cosh(d*x + c)^5 + 165*(60480*a^3 +
8640*a^2*b - 768*a*b^2 - 15475*b^3)*cosh(d*x + c)^3 + 9*(60480*a^3 - 8640*a^2*b - 768*a*b^2 + 15475*b^3)*cosh(
d*x + c))*sinh(d*x + c)^8 + 16*(5040*a^3 - 1440*a^2*b - 672*a*b^2 - 1235*b^3)*cosh(d*x + c)^7 + 4*(188955*(64*
a^3 - 192*a^2*b - 5*b^3)*cosh(d*x + c)^12 + 24310*(1728*a^3 - 1728*a^2*b + 1536*a*b^2 + 427*b^3)*cosh(d*x + c)
^10 + 38610*(1440*a^3 + 192*a*b^2 - 355*b^3)*cosh(d*x + c)^8 + 6864*(5040*a^3 + 1440*a^2*b - 672*a*b^2 + 1235*
b^3)*cosh(d*x + c)^6 + 165*(60480*a^3 + 8640*a^2*b - 768*a*b^2 - 15475*b^3)*cosh(d*x + c)^4 + 20160*a^3 - 5760
*a^2*b - 2688*a*b^2 - 4940*b^3 + 18*(60480*a^3 - 8640*a^2*b - 768*a*b^2 + 15475*b^3)*cosh(d*x + c)^2)*sinh(d*x
+ c)^7 + 4*(101745*(64*a^3 - 192*a^2*b - 5*b^3)*cosh(d*x + c)^13 + 15470*(1728*a^3 - 1728*a^2*b + 1536*a*b^2
+ 427*b^3)*cosh(d*x + c)^11 + 30030*(1440*a^3 + 192*a*b^2 - 355*b^3)*cosh(d*x + c)^9 + 6864*(5040*a^3 + 1440*a
^2*b - 672*a*b^2 + 1235*b^3)*cosh(d*x + c)^7 + 231*(60480*a^3 + 8640*a^2*b - 768*a*b^2 - 15475*b^3)*cosh(d*x +
c)^5 + 42*(60480*a^3 - 8640*a^2*b - 768*a*b^2 + 15475*b^3)*cosh(d*x + c)^3 + 28*(5040*a^3 - 1440*a^2*b - 672*
a*b^2 - 1235*b^3)*cosh(d*x + c))*sinh(d*x + c)^6 + 24*(1440*a^3 + 192*a*b^2 + 355*b^3)*cosh(d*x + c)^5 + 4*(43
605*(64*a^3 - 192*a^2*b - 5*b^3)*cosh(d*x + c)^14 + 7735*(1728*a^3 - 1728*a^2*b + 1536*a*b^2 + 427*b^3)*cosh(d
*x + c)^12 + 18018*(1440*a^3 + 192*a*b^2 - 355*b^3)*cosh(d*x + c)^10 + 5148*(5040*a^3 + 1440*a^2*b - 672*a*b^2
+ 1235*b^3)*cosh(d*x + c)^8 + 231*(60480*a^3 + 8640*a^2*b - 768*a*b^2 - 15475*b^3)*cosh(d*x + c)^6 + 63*(6048
0*a^3 - 8640*a^2*b - 768*a*b^2 + 15475*b^3)*cosh(d*x + c)^4 + 8640*a^3 + 1152*a*b^2 + 2130*b^3 + 84*(5040*a^3
- 1440*a^2*b - 672*a*b^2 - 1235*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 4*(14535*(64*a^3 - 192*a^2*b - 5*b^3)*
cosh(d*x + c)^15 + 2975*(1728*a^3 - 1728*a^2*b + 1536*a*b^2 + 427*b^3)*cosh(d*x + c)^13 + 8190*(1440*a^3 + 192
*a*b^2 - 355*b^3)*cosh(d*x + c)^11 + 2860*(5040*a^3 + 1440*a^2*b - 672*a*b^2 + 1235*b^3)*cosh(d*x + c)^9 + 165
*(60480*a^3 + 8640*a^2*b - 768*a*b^2 - 15475*b^3)*cosh(d*x + c)^7 + 63*(60480*a^3 - 8640*a^2*b - 768*a*b^2 + 1
5475*b^3)*cosh(d*x + c)^5 + 140*(5040*a^3 - 1440*a^2*b - 672*a*b^2 - 1235*b^3)*cosh(d*x + c)^3 + 30*(1440*a^3
+ 192*a*b^2 + 355*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 + 5*(1728*a^3 + 1728*a^2*b + 1536*a*b^2 - 427*b^3)*cosh(
d*x + c)^3 + (14535*(64*a^3 - 192*a^2*b - 5*b^3)*cosh(d*x + c)^16 + 3400*(1728*a^3 - 1728*a^2*b + 1536*a*b^2 +
427*b^3)*cosh(d*x + c)^14 + 10920*(1440*a^3 + 192*a*b^2 - 355*b^3)*cosh(d*x + c)^12 + 4576*(5040*a^3 + 1440*a
^2*b - 672*a*b^2 + 1235*b^3)*cosh(d*x + c)^10 + 330*(60480*a^3 + 8640*a^2*b - 768*a*b^2 - 15475*b^3)*cosh(d*x
+ c)^8 + 168*(60480*a^3 - 8640*a^2*b - 768*a*b^2 + 15475*b^3)*cosh(d*x + c)^6 + 560*(5040*a^3 - 1440*a^2*b - 6
72*a*b^2 - 1235*b^3)*cosh(d*x + c)^4 + 8640*a^3 + 8640*a^2*b + 7680*a*b^2 - 2135*b^3 + 240*(1440*a^3 + 192*a*b
^2 + 355*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + (2565*(64*a^3 - 192*a^2*b - 5*b^3)*cosh(d*x + c)^17 + 680*(17
28*a^3 - 1728*a^2*b + 1536*a*b^2 + 427*b^3)*cosh(d*x + c)^15 + 2520*(1440*a^3 + 192*a*b^2 - 355*b^3)*cosh(d*x
+ c)^13 + 1248*(5040*a^3 + 1440*a^2*b - 672*a*b^2 + 1235*b^3)*cosh(d*x + c)^11 + 110*(60480*a^3 + 8640*a^2*b -
768*a*b^2 - 15475*b^3)*cosh(d*x + c)^9 + 72*(60480*a^3 - 8640*a^2*b - 768*a*b^2 + 15475*b^3)*cosh(d*x + c)^7
+ 336*(5040*a^3 - 1440*a^2*b - 672*a*b^2 - 1235*b^3)*cosh(d*x + c)^5 + 240*(1440*a^3 + 192*a*b^2 + 355*b^3)*co
sh(d*x + c)^3 + 15*(1728*a^3 + 1728*a^2*b + 1536*a*b^2 - 427*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 - 15*((192*a^
2*b + 5*b^3)*cosh(d*x + c)^20 + 20*(192*a^2*b + 5*b^3)*cosh(d*x + c)*sinh(d*x + c)^19 + (192*a^2*b + 5*b^3)*si
nh(d*x + c)^20 + 6*(192*a^2*b + 5*b^3)*cosh(d*x + c)^18 + 2*(576*a^2*b + 15*b^3 + 95*(192*a^2*b + 5*b^3)*cosh(
d*x + c)^2)*sinh(d*x + c)^18 + 12*(95*(192*a^2*b + 5*b^3)*cosh(d*x + c)^3 + 9*(192*a^2*b + 5*b^3)*cosh(d*x + c
))*sinh(d*x + c)^17 + 13*(192*a^2*b + 5*b^3)*cosh(d*x + c)^16 + (4845*(192*a^2*b + 5*b^3)*cosh(d*x + c)^4 + 24
96*a^2*b + 65*b^3 + 918*(192*a^2*b + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^16 + 16*(969*(192*a^2*b + 5*b^3)*co
sh(d*x + c)^5 + 306*(192*a^2*b + 5*b^3)*cosh(d*x + c)^3 + 13*(192*a^2*b + 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^
15 + 8*(192*a^2*b + 5*b^3)*cosh(d*x + c)^14 + 8*(4845*(192*a^2*b + 5*b^3)*cosh(d*x + c)^6 + 2295*(192*a^2*b +
5*b^3)*cosh(d*x + c)^4 + 192*a^2*b + 5*b^3 + 195*(192*a^2*b + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^14 + 16*(4
845*(192*a^2*b + 5*b^3)*cosh(d*x + c)^7 + 3213*(192*a^2*b + 5*b^3)*cosh(d*x + c)^5 + 455*(192*a^2*b + 5*b^3)*c
osh(d*x + c)^3 + 7*(192*a^2*b + 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^13 - 14*(192*a^2*b + 5*b^3)*cosh(d*x + c)^
12 + 2*(62985*(192*a^2*b + 5*b^3)*cosh(d*x + c)^8 + 55692*(192*a^2*b + 5*b^3)*cosh(d*x + c)^6 + 11830*(192*a^2
*b + 5*b^3)*cosh(d*x + c)^4 - 1344*a^2*b - 35*b^3 + 364*(192*a^2*b + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^12
+ 8*(20995*(192*a^2*b + 5*b^3)*cosh(d*x + c)^9 + 23868*(192*a^2*b + 5*b^3)*cosh(d*x + c)^7 + 7098*(192*a^2*b +
5*b^3)*cosh(d*x + c)^5 + 364*(192*a^2*b + 5*b^3)*cosh(d*x + c)^3 - 21*(192*a^2*b + 5*b^3)*cosh(d*x + c))*sinh
(d*x + c)^11 - 28*(192*a^2*b + 5*b^3)*cosh(d*x + c)^10 + 4*(46189*(192*a^2*b + 5*b^3)*cosh(d*x + c)^10 + 65637
*(192*a^2*b + 5*b^3)*cosh(d*x + c)^8 + 26026*(192*a^2*b + 5*b^3)*cosh(d*x + c)^6 + 2002*(192*a^2*b + 5*b^3)*co
sh(d*x + c)^4 - 1344*a^2*b - 35*b^3 - 231*(192*a^2*b + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^10 + 8*(20995*(19
2*a^2*b + 5*b^3)*cosh(d*x + c)^11 + 36465*(192*a^2*b + 5*b^3)*cosh(d*x + c)^9 + 18590*(192*a^2*b + 5*b^3)*cosh
(d*x + c)^7 + 2002*(192*a^2*b + 5*b^3)*cosh(d*x + c)^5 - 385*(192*a^2*b + 5*b^3)*cosh(d*x + c)^3 - 35*(192*a^2
*b + 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^9 - 14*(192*a^2*b + 5*b^3)*cosh(d*x + c)^8 + 2*(62985*(192*a^2*b + 5*
b^3)*cosh(d*x + c)^12 + 131274*(192*a^2*b + 5*b^3)*cosh(d*x + c)^10 + 83655*(192*a^2*b + 5*b^3)*cosh(d*x + c)^
8 + 12012*(192*a^2*b + 5*b^3)*cosh(d*x + c)^6 - 3465*(192*a^2*b + 5*b^3)*cosh(d*x + c)^4 - 1344*a^2*b - 35*b^3
- 630*(192*a^2*b + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 16*(4845*(192*a^2*b + 5*b^3)*cosh(d*x + c)^13 +
11934*(192*a^2*b + 5*b^3)*cosh(d*x + c)^11 + 9295*(192*a^2*b + 5*b^3)*cosh(d*x + c)^9 + 1716*(192*a^2*b + 5*b^
3)*cosh(d*x + c)^7 - 693*(192*a^2*b + 5*b^3)*cosh(d*x + c)^5 - 210*(192*a^2*b + 5*b^3)*cosh(d*x + c)^3 - 7*(19
2*a^2*b + 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^7 + 8*(192*a^2*b + 5*b^3)*cosh(d*x + c)^6 + 8*(4845*(192*a^2*b +
5*b^3)*cosh(d*x + c)^14 + 13923*(192*a^2*b + 5*b^3)*cosh(d*x + c)^12 + 13013*(192*a^2*b + 5*b^3)*cosh(d*x + c
)^10 + 3003*(192*a^2*b + 5*b^3)*cosh(d*x + c)^8 - 1617*(192*a^2*b + 5*b^3)*cosh(d*x + c)^6 - 735*(192*a^2*b +
5*b^3)*cosh(d*x + c)^4 + 192*a^2*b + 5*b^3 - 49*(192*a^2*b + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 16*(969
*(192*a^2*b + 5*b^3)*cosh(d*x + c)^15 + 3213*(192*a^2*b + 5*b^3)*cosh(d*x + c)^13 + 3549*(192*a^2*b + 5*b^3)*c
osh(d*x + c)^11 + 1001*(192*a^2*b + 5*b^3)*cosh(d*x + c)^9 - 693*(192*a^2*b + 5*b^3)*cosh(d*x + c)^7 - 441*(19
2*a^2*b + 5*b^3)*cosh(d*x + c)^5 - 49*(192*a^2*b + 5*b^3)*cosh(d*x + c)^3 + 3*(192*a^2*b + 5*b^3)*cosh(d*x + c
))*sinh(d*x + c)^5 + 13*(192*a^2*b + 5*b^3)*cosh(d*x + c)^4 + (4845*(192*a^2*b + 5*b^3)*cosh(d*x + c)^16 + 183
60*(192*a^2*b + 5*b^3)*cosh(d*x + c)^14 + 23660*(192*a^2*b + 5*b^3)*cosh(d*x + c)^12 + 8008*(192*a^2*b + 5*b^3
)*cosh(d*x + c)^10 - 6930*(192*a^2*b + 5*b^3)*cosh(d*x + c)^8 - 5880*(192*a^2*b + 5*b^3)*cosh(d*x + c)^6 - 980
*(192*a^2*b + 5*b^3)*cosh(d*x + c)^4 + 2496*a^2*b + 65*b^3 + 120*(192*a^2*b + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x
+ c)^4 + 4*(285*(192*a^2*b + 5*b^3)*cosh(d*x + c)^17 + 1224*(192*a^2*b + 5*b^3)*cosh(d*x + c)^15 + 1820*(192*
a^2*b + 5*b^3)*cosh(d*x + c)^13 + 728*(192*a^2*b + 5*b^3)*cosh(d*x + c)^11 - 770*(192*a^2*b + 5*b^3)*cosh(d*x
+ c)^9 - 840*(192*a^2*b + 5*b^3)*cosh(d*x + c)^7 - 196*(192*a^2*b + 5*b^3)*cosh(d*x + c)^5 + 40*(192*a^2*b + 5
*b^3)*cosh(d*x + c)^3 + 13*(192*a^2*b + 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 192*a^2*b + 5*b^3 + 6*(192*a^2
*b + 5*b^3)*cosh(d*x + c)^2 + 2*(95*(192*a^2*b + 5*b^3)*cosh(d*x + c)^18 + 459*(192*a^2*b + 5*b^3)*cosh(d*x +
c)^16 + 780*(192*a^2*b + 5*b^3)*cosh(d*x + c)^14 + 364*(192*a^2*b + 5*b^3)*cosh(d*x + c)^12 - 462*(192*a^2*b +
5*b^3)*cosh(d*x + c)^10 - 630*(192*a^2*b + 5*b^3)*cosh(d*x + c)^8 - 196*(192*a^2*b + 5*b^3)*cosh(d*x + c)^6 +
60*(192*a^2*b + 5*b^3)*cosh(d*x + c)^4 + 576*a^2*b + 15*b^3 + 39*(192*a^2*b + 5*b^3)*cosh(d*x + c)^2)*sinh(d*
x + c)^2 + 4*(5*(192*a^2*b + 5*b^3)*cosh(d*x + c)^19 + 27*(192*a^2*b + 5*b^3)*cosh(d*x + c)^17 + 52*(192*a^2*b
+ 5*b^3)*cosh(d*x + c)^15 + 28*(192*a^2*b + 5*b^3)*cosh(d*x + c)^13 - 42*(192*a^2*b + 5*b^3)*cosh(d*x + c)^11
- 70*(192*a^2*b + 5*b^3)*cosh(d*x + c)^9 - 28*(192*a^2*b + 5*b^3)*cosh(d*x + c)^7 + 12*(192*a^2*b + 5*b^3)*co
sh(d*x + c)^5 + 13*(192*a^2*b + 5*b^3)*cosh(d*x + c)^3 + 3*(192*a^2*b + 5*b^3)*cosh(d*x + c))*sinh(d*x + c))*a
rctan(cosh(d*x + c) + sinh(d*x + c)) + 15*(64*a^3 + 192*a^2*b + 5*b^3)*cosh(d*x + c) - 480*(a^3*cosh(d*x + c)^
20 + 20*a^3*cosh(d*x + c)*sinh(d*x + c)^19 + a^3*sinh(d*x + c)^20 + 6*a^3*cosh(d*x + c)^18 + 13*a^3*cosh(d*x +
c)^16 + 2*(95*a^3*cosh(d*x + c)^2 + 3*a^3)*sinh(d*x + c)^18 + 12*(95*a^3*cosh(d*x + c)^3 + 9*a^3*cosh(d*x + c
))*sinh(d*x + c)^17 + 8*a^3*cosh(d*x + c)^14 + (4845*a^3*cosh(d*x + c)^4 + 918*a^3*cosh(d*x + c)^2 + 13*a^3)*s
inh(d*x + c)^16 + 16*(969*a^3*cosh(d*x + c)^5 + 306*a^3*cosh(d*x + c)^3 + 13*a^3*cosh(d*x + c))*sinh(d*x + c)^
15 - 14*a^3*cosh(d*x + c)^12 + 8*(4845*a^3*cosh(d*x + c)^6 + 2295*a^3*cosh(d*x + c)^4 + 195*a^3*cosh(d*x + c)^
2 + a^3)*sinh(d*x + c)^14 + 16*(4845*a^3*cosh(d*x + c)^7 + 3213*a^3*cosh(d*x + c)^5 + 455*a^3*cosh(d*x + c)^3
+ 7*a^3*cosh(d*x + c))*sinh(d*x + c)^13 - 28*a^3*cosh(d*x + c)^10 + 2*(62985*a^3*cosh(d*x + c)^8 + 55692*a^3*c
osh(d*x + c)^6 + 11830*a^3*cosh(d*x + c)^4 + 364*a^3*cosh(d*x + c)^2 - 7*a^3)*sinh(d*x + c)^12 + 8*(20995*a^3*
cosh(d*x + c)^9 + 23868*a^3*cosh(d*x + c)^7 + 7098*a^3*cosh(d*x + c)^5 + 364*a^3*cosh(d*x + c)^3 - 21*a^3*cosh
(d*x + c))*sinh(d*x + c)^11 - 14*a^3*cosh(d*x + c)^8 + 4*(46189*a^3*cosh(d*x + c)^10 + 65637*a^3*cosh(d*x + c)
^8 + 26026*a^3*cosh(d*x + c)^6 + 2002*a^3*cosh(d*x + c)^4 - 231*a^3*cosh(d*x + c)^2 - 7*a^3)*sinh(d*x + c)^10
+ 8*(20995*a^3*cosh(d*x + c)^11 + 36465*a^3*cosh(d*x + c)^9 + 18590*a^3*cosh(d*x + c)^7 + 2002*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 + 8*a^3*cosh(d*x + c)^6 + 2*(62985*a^3*
cosh(d*x + c)^12 + 131274*a^3*cosh(d*x + c)^10 + 83655*a^3*cosh(d*x + c)^8 + 12012*a^3*cosh(d*x + c)^6 - 3465*
a^3*cosh(d*x + c)^4 - 630*a^3*cosh(d*x + c)^2 - 7*a^3)*sinh(d*x + c)^8 + 16*(4845*a^3*cosh(d*x + c)^13 + 11934
*a^3*cosh(d*x + c)^11 + 9295*a^3*cosh(d*x + c)^9 + 1716*a^3*cosh(d*x + c)^7 - 693*a^3*cosh(d*x + c)^5 - 210*a^
3*cosh(d*x + c)^3 - 7*a^3*cosh(d*x + c))*sinh(d*x + c)^7 + 13*a^3*cosh(d*x + c)^4 + 8*(4845*a^3*cosh(d*x + c)^
14 + 13923*a^3*cosh(d*x + c)^12 + 13013*a^3*cosh(d*x + c)^10 + 3003*a^3*cosh(d*x + c)^8 - 1617*a^3*cosh(d*x +
c)^6 - 735*a^3*cosh(d*x + c)^4 - 49*a^3*cosh(d*x + c)^2 + a^3)*sinh(d*x + c)^6 + 16*(969*a^3*cosh(d*x + c)^15
+ 3213*a^3*cosh(d*x + c)^13 + 3549*a^3*cosh(d*x + c)^11 + 1001*a^3*cosh(d*x + c)^9 - 693*a^3*cosh(d*x + c)^7 -
441*a^3*cosh(d*x + c)^5 - 49*a^3*cosh(d*x + c)^3 + 3*a^3*cosh(d*x + c))*sinh(d*x + c)^5 + 6*a^3*cosh(d*x + c)
^2 + (4845*a^3*cosh(d*x + c)^16 + 18360*a^3*cosh(d*x + c)^14 + 23660*a^3*cosh(d*x + c)^12 + 8008*a^3*cosh(d*x
+ c)^10 - 6930*a^3*cosh(d*x + c)^8 - 5880*a^3*cosh(d*x + c)^6 - 980*a^3*cosh(d*x + c)^4 + 120*a^3*cosh(d*x + c
)^2 + 13*a^3)*sinh(d*x + c)^4 + 4*(285*a^3*cosh(d*x + c)^17 + 1224*a^3*cosh(d*x + c)^15 + 1820*a^3*cosh(d*x +
c)^13 + 728*a^3*cosh(d*x + c)^11 - 770*a^3*cosh(d*x + c)^9 - 840*a^3*cosh(d*x + c)^7 - 196*a^3*cosh(d*x + c)^5
+ 40*a^3*cosh(d*x + c)^3 + 13*a^3*cosh(d*x + c))*sinh(d*x + c)^3 + a^3 + 2*(95*a^3*cosh(d*x + c)^18 + 459*a^3
*cosh(d*x + c)^16 + 780*a^3*cosh(d*x + c)^14 + 364*a^3*cosh(d*x + c)^12 - 462*a^3*cosh(d*x + c)^10 - 630*a^3*c
osh(d*x + c)^8 - 196*a^3*cosh(d*x + c)^6 + 60*a^3*cosh(d*x + c)^4 + 39*a^3*cosh(d*x + c)^2 + 3*a^3)*sinh(d*x +
c)^2 + 4*(5*a^3*cosh(d*x + c)^19 + 27*a^3*cosh(d*x + c)^17 + 52*a^3*cosh(d*x + c)^15 + 28*a^3*cosh(d*x + c)^1
3 - 42*a^3*cosh(d*x + c)^11 - 70*a^3*cosh(d*x + c)^9 - 28*a^3*cosh(d*x + c)^7 + 12*a^3*cosh(d*x + c)^5 + 13*a^
3*cosh(d*x + c)^3 + 3*a^3*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) + 480*(a^3*cosh
(d*x + c)^20 + 20*a^3*cosh(d*x + c)*sinh(d*x + c)^19 + a^3*sinh(d*x + c)^20 + 6*a^3*cosh(d*x + c)^18 + 13*a^3*
cosh(d*x + c)^16 + 2*(95*a^3*cosh(d*x + c)^2 + 3*a^3)*sinh(d*x + c)^18 + 12*(95*a^3*cosh(d*x + c)^3 + 9*a^3*co
sh(d*x + c))*sinh(d*x + c)^17 + 8*a^3*cosh(d*x + c)^14 + (4845*a^3*cosh(d*x + c)^4 + 918*a^3*cosh(d*x + c)^2 +
13*a^3)*sinh(d*x + c)^16 + 16*(969*a^3*cosh(d*x + c)^5 + 306*a^3*cosh(d*x + c)^3 + 13*a^3*cosh(d*x + c))*sinh
(d*x + c)^15 - 14*a^3*cosh(d*x + c)^12 + 8*(4845*a^3*cosh(d*x + c)^6 + 2295*a^3*cosh(d*x + c)^4 + 195*a^3*cosh
(d*x + c)^2 + a^3)*sinh(d*x + c)^14 + 16*(4845*a^3*cosh(d*x + c)^7 + 3213*a^3*cosh(d*x + c)^5 + 455*a^3*cosh(d
*x + c)^3 + 7*a^3*cosh(d*x + c))*sinh(d*x + c)^13 - 28*a^3*cosh(d*x + c)^10 + 2*(62985*a^3*cosh(d*x + c)^8 + 5
5692*a^3*cosh(d*x + c)^6 + 11830*a^3*cosh(d*x + c)^4 + 364*a^3*cosh(d*x + c)^2 - 7*a^3)*sinh(d*x + c)^12 + 8*(
20995*a^3*cosh(d*x + c)^9 + 23868*a^3*cosh(d*x + c)^7 + 7098*a^3*cosh(d*x + c)^5 + 364*a^3*cosh(d*x + c)^3 - 2
1*a^3*cosh(d*x + c))*sinh(d*x + c)^11 - 14*a^3*cosh(d*x + c)^8 + 4*(46189*a^3*cosh(d*x + c)^10 + 65637*a^3*cos
h(d*x + c)^8 + 26026*a^3*cosh(d*x + c)^6 + 2002*a^3*cosh(d*x + c)^4 - 231*a^3*cosh(d*x + c)^2 - 7*a^3)*sinh(d*
x + c)^10 + 8*(20995*a^3*cosh(d*x + c)^11 + 36465*a^3*cosh(d*x + c)^9 + 18590*a^3*cosh(d*x + c)^7 + 2002*a^3*c
osh(d*x + c)^5 - 385*a^3*cosh(d*x + c)^3 - 35*a^3*cosh(d*x + c))*sinh(d*x + c)^9 + 8*a^3*cosh(d*x + c)^6 + 2*(
62985*a^3*cosh(d*x + c)^12 + 131274*a^3*cosh(d*x + c)^10 + 83655*a^3*cosh(d*x + c)^8 + 12012*a^3*cosh(d*x + c)
^6 - 3465*a^3*cosh(d*x + c)^4 - 630*a^3*cosh(d*x + c)^2 - 7*a^3)*sinh(d*x + c)^8 + 16*(4845*a^3*cosh(d*x + c)^
13 + 11934*a^3*cosh(d*x + c)^11 + 9295*a^3*cosh(d*x + c)^9 + 1716*a^3*cosh(d*x + c)^7 - 693*a^3*cosh(d*x + c)^
5 - 210*a^3*cosh(d*x + c)^3 - 7*a^3*cosh(d*x + c))*sinh(d*x + c)^7 + 13*a^3*cosh(d*x + c)^4 + 8*(4845*a^3*cosh
(d*x + c)^14 + 13923*a^3*cosh(d*x + c)^12 + 13013*a^3*cosh(d*x + c)^10 + 3003*a^3*cosh(d*x + c)^8 - 1617*a^3*c
osh(d*x + c)^6 - 735*a^3*cosh(d*x + c)^4 - 49*a^3*cosh(d*x + c)^2 + a^3)*sinh(d*x + c)^6 + 16*(969*a^3*cosh(d*
x + c)^15 + 3213*a^3*cosh(d*x + c)^13 + 3549*a^3*cosh(d*x + c)^11 + 1001*a^3*cosh(d*x + c)^9 - 693*a^3*cosh(d*
x + c)^7 - 441*a^3*cosh(d*x + c)^5 - 49*a^3*cosh(d*x + c)^3 + 3*a^3*cosh(d*x + c))*sinh(d*x + c)^5 + 6*a^3*cos
h(d*x + c)^2 + (4845*a^3*cosh(d*x + c)^16 + 18360*a^3*cosh(d*x + c)^14 + 23660*a^3*cosh(d*x + c)^12 + 8008*a^3
*cosh(d*x + c)^10 - 6930*a^3*cosh(d*x + c)^8 - 5880*a^3*cosh(d*x + c)^6 - 980*a^3*cosh(d*x + c)^4 + 120*a^3*co
sh(d*x + c)^2 + 13*a^3)*sinh(d*x + c)^4 + 4*(285*a^3*cosh(d*x + c)^17 + 1224*a^3*cosh(d*x + c)^15 + 1820*a^3*c
osh(d*x + c)^13 + 728*a^3*cosh(d*x + c)^11 - 770*a^3*cosh(d*x + c)^9 - 840*a^3*cosh(d*x + c)^7 - 196*a^3*cosh(
d*x + c)^5 + 40*a^3*cosh(d*x + c)^3 + 13*a^3*cosh(d*x + c))*sinh(d*x + c)^3 + a^3 + 2*(95*a^3*cosh(d*x + c)^18
+ 459*a^3*cosh(d*x + c)^16 + 780*a^3*cosh(d*x + c)^14 + 364*a^3*cosh(d*x + c)^12 - 462*a^3*cosh(d*x + c)^10 -
630*a^3*cosh(d*x + c)^8 - 196*a^3*cosh(d*x + c)^6 + 60*a^3*cosh(d*x + c)^4 + 39*a^3*cosh(d*x + c)^2 + 3*a^3)*
sinh(d*x + c)^2 + 4*(5*a^3*cosh(d*x + c)^19 + 27*a^3*cosh(d*x + c)^17 + 52*a^3*cosh(d*x + c)^15 + 28*a^3*cosh(
d*x + c)^13 - 42*a^3*cosh(d*x + c)^11 - 70*a^3*cosh(d*x + c)^9 - 28*a^3*cosh(d*x + c)^7 + 12*a^3*cosh(d*x + c)
^5 + 13*a^3*cosh(d*x + c)^3 + 3*a^3*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - 1) + (28
5*(64*a^3 - 192*a^2*b - 5*b^3)*cosh(d*x + c)^18 + 85*(1728*a^3 - 1728*a^2*b + 1536*a*b^2 + 427*b^3)*cosh(d*x +
c)^16 + 360*(1440*a^3 + 192*a*b^2 - 355*b^3)*cosh(d*x + c)^14 + 208*(5040*a^3 + 1440*a^2*b - 672*a*b^2 + 1235
*b^3)*cosh(d*x + c)^12 + 22*(60480*a^3 + 8640*a^2*b - 768*a*b^2 - 15475*b^3)*cosh(d*x + c)^10 + 18*(60480*a^3
- 8640*a^2*b - 768*a*b^2 + 15475*b^3)*cosh(d*x + c)^8 + 112*(5040*a^3 - 1440*a^2*b - 672*a*b^2 - 1235*b^3)*cos
h(d*x + c)^6 + 120*(1440*a^3 + 192*a*b^2 + 355*b^3)*cosh(d*x + c)^4 + 960*a^3 + 2880*a^2*b + 75*b^3 + 15*(1728
*a^3 + 1728*a^2*b + 1536*a*b^2 - 427*b^3)*cosh(d*x + c)^2)*sinh(d*x + c))/(d*cosh(d*x + c)^20 + 20*d*cosh(d*x
+ c)*sinh(d*x + c)^19 + d*sinh(d*x + c)^20 + 6*d*cosh(d*x + c)^18 + 2*(95*d*cosh(d*x + c)^2 + 3*d)*sinh(d*x +
c)^18 + 12*(95*d*cosh(d*x + c)^3 + 9*d*cosh(d*x + c))*sinh(d*x + c)^17 + 13*d*cosh(d*x + c)^16 + (4845*d*cosh(
d*x + c)^4 + 918*d*cosh(d*x + c)^2 + 13*d)*sinh(d*x + c)^16 + 16*(969*d*cosh(d*x + c)^5 + 306*d*cosh(d*x + c)^
3 + 13*d*cosh(d*x + c))*sinh(d*x + c)^15 + 8*d*cosh(d*x + c)^14 + 8*(4845*d*cosh(d*x + c)^6 + 2295*d*cosh(d*x
+ c)^4 + 195*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^14 + 16*(4845*d*cosh(d*x + c)^7 + 3213*d*cosh(d*x + c)^5 + 4
55*d*cosh(d*x + c)^3 + 7*d*cosh(d*x + c))*sinh(d*x + c)^13 - 14*d*cosh(d*x + c)^12 + 2*(62985*d*cosh(d*x + c)^
8 + 55692*d*cosh(d*x + c)^6 + 11830*d*cosh(d*x + c)^4 + 364*d*cosh(d*x + c)^2 - 7*d)*sinh(d*x + c)^12 + 8*(209
95*d*cosh(d*x + c)^9 + 23868*d*cosh(d*x + c)^7 + 7098*d*cosh(d*x + c)^5 + 364*d*cosh(d*x + c)^3 - 21*d*cosh(d*
x + c))*sinh(d*x + c)^11 - 28*d*cosh(d*x + c)^10 + 4*(46189*d*cosh(d*x + c)^10 + 65637*d*cosh(d*x + c)^8 + 260
26*d*cosh(d*x + c)^6 + 2002*d*cosh(d*x + c)^4 - 231*d*cosh(d*x + c)^2 - 7*d)*sinh(d*x + c)^10 + 8*(20995*d*cos
h(d*x + c)^11 + 36465*d*cosh(d*x + c)^9 + 18590*d*cosh(d*x + c)^7 + 2002*d*cosh(d*x + c)^5 - 385*d*cosh(d*x +
c)^3 - 35*d*cosh(d*x + c))*sinh(d*x + c)^9 - 14*d*cosh(d*x + c)^8 + 2*(62985*d*cosh(d*x + c)^12 + 131274*d*cos
h(d*x + c)^10 + 83655*d*cosh(d*x + c)^8 + 12012*d*cosh(d*x + c)^6 - 3465*d*cosh(d*x + c)^4 - 630*d*cosh(d*x +
c)^2 - 7*d)*sinh(d*x + c)^8 + 16*(4845*d*cosh(d*x + c)^13 + 11934*d*cosh(d*x + c)^11 + 9295*d*cosh(d*x + c)^9
+ 1716*d*cosh(d*x + c)^7 - 693*d*cosh(d*x + c)^5 - 210*d*cosh(d*x + c)^3 - 7*d*cosh(d*x + c))*sinh(d*x + c)^7
+ 8*d*cosh(d*x + c)^6 + 8*(4845*d*cosh(d*x + c)^14 + 13923*d*cosh(d*x + c)^12 + 13013*d*cosh(d*x + c)^10 + 300
3*d*cosh(d*x + c)^8 - 1617*d*cosh(d*x + c)^6 - 735*d*cosh(d*x + c)^4 - 49*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)
^6 + 16*(969*d*cosh(d*x + c)^15 + 3213*d*cosh(d*x + c)^13 + 3549*d*cosh(d*x + c)^11 + 1001*d*cosh(d*x + c)^9 -
693*d*cosh(d*x + c)^7 - 441*d*cosh(d*x + c)^5 - 49*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^5 + 1
3*d*cosh(d*x + c)^4 + (4845*d*cosh(d*x + c)^16 + 18360*d*cosh(d*x + c)^14 + 23660*d*cosh(d*x + c)^12 + 8008*d*
cosh(d*x + c)^10 - 6930*d*cosh(d*x + c)^8 - 5880*d*cosh(d*x + c)^6 - 980*d*cosh(d*x + c)^4 + 120*d*cosh(d*x +
c)^2 + 13*d)*sinh(d*x + c)^4 + 4*(285*d*cosh(d*x + c)^17 + 1224*d*cosh(d*x + c)^15 + 1820*d*cosh(d*x + c)^13 +
728*d*cosh(d*x + c)^11 - 770*d*cosh(d*x + c)^9 - 840*d*cosh(d*x + c)^7 - 196*d*cosh(d*x + c)^5 + 40*d*cosh(d*
x + c)^3 + 13*d*cosh(d*x + c))*sinh(d*x + c)^3 + 6*d*cosh(d*x + c)^2 + 2*(95*d*cosh(d*x + c)^18 + 459*d*cosh(d
*x + c)^16 + 780*d*cosh(d*x + c)^14 + 364*d*cosh(d*x + c)^12 - 462*d*cosh(d*x + c)^10 - 630*d*cosh(d*x + c)^8
- 196*d*cosh(d*x + c)^6 + 60*d*cosh(d*x + c)^4 + 39*d*cosh(d*x + c)^2 + 3*d)*sinh(d*x + c)^2 + 4*(5*d*cosh(d*x
+ c)^19 + 27*d*cosh(d*x + c)^17 + 52*d*cosh(d*x + c)^15 + 28*d*cosh(d*x + c)^13 - 42*d*cosh(d*x + c)^11 - 70*
d*cosh(d*x + c)^9 - 28*d*cosh(d*x + c)^7 + 12*d*cosh(d*x + c)^5 + 13*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*si
nh(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}^{3}{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 2.56392, size = 586, normalized size = 2.53 \begin{align*} \frac{480 \, a^{3} \log \left (e^{\left (d x + c\right )} + 1\right ) - 480 \, a^{3} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) + 15 \,{\left (192 \, a^{2} b e^{c} + 5 \, b^{3} e^{c}\right )} \arctan \left (e^{\left (d x + c\right )}\right ) e^{\left (-c\right )} - \frac{960 \,{\left (a^{3} e^{\left (3 \, d x + 3 \, c\right )} + a^{3} e^{\left (d x + c\right )}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}} + \frac{2880 \, a^{2} b e^{\left (15 \, d x + 15 \, c\right )} + 75 \, b^{3} e^{\left (15 \, d x + 15 \, c\right )} + 14400 \, a^{2} b e^{\left (13 \, d x + 13 \, c\right )} - 7680 \, a b^{2} e^{\left (13 \, d x + 13 \, c\right )} - 1985 \, b^{3} e^{\left (13 \, d x + 13 \, c\right )} + 25920 \, a^{2} b e^{\left (11 \, d x + 11 \, c\right )} - 19968 \, a b^{2} e^{\left (11 \, d x + 11 \, c\right )} + 4475 \, b^{3} e^{\left (11 \, d x + 11 \, c\right )} + 14400 \, a^{2} b e^{\left (9 \, d x + 9 \, c\right )} - 21504 \, a b^{2} e^{\left (9 \, d x + 9 \, c\right )} - 8825 \, b^{3} e^{\left (9 \, d x + 9 \, c\right )} - 14400 \, a^{2} b e^{\left (7 \, d x + 7 \, c\right )} - 21504 \, a b^{2} e^{\left (7 \, d x + 7 \, c\right )} + 8825 \, b^{3} e^{\left (7 \, d x + 7 \, c\right )} - 25920 \, a^{2} b e^{\left (5 \, d x + 5 \, c\right )} - 19968 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} - 4475 \, b^{3} e^{\left (5 \, d x + 5 \, c\right )} - 14400 \, a^{2} b e^{\left (3 \, d x + 3 \, c\right )} - 7680 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 1985 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} - 2880 \, a^{2} b e^{\left (d x + c\right )} - 75 \, 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)^3*(a+b*tanh(d*x+c)^3)^3,x, algorithm="giac")

[Out]

1/960*(480*a^3*log(e^(d*x + c) + 1) - 480*a^3*log(abs(e^(d*x + c) - 1)) + 15*(192*a^2*b*e^c + 5*b^3*e^c)*arcta
n(e^(d*x + c))*e^(-c) - 960*(a^3*e^(3*d*x + 3*c) + a^3*e^(d*x + c))/(e^(2*d*x + 2*c) - 1)^2 + (2880*a^2*b*e^(1
5*d*x + 15*c) + 75*b^3*e^(15*d*x + 15*c) + 14400*a^2*b*e^(13*d*x + 13*c) - 7680*a*b^2*e^(13*d*x + 13*c) - 1985
*b^3*e^(13*d*x + 13*c) + 25920*a^2*b*e^(11*d*x + 11*c) - 19968*a*b^2*e^(11*d*x + 11*c) + 4475*b^3*e^(11*d*x +
11*c) + 14400*a^2*b*e^(9*d*x + 9*c) - 21504*a*b^2*e^(9*d*x + 9*c) - 8825*b^3*e^(9*d*x + 9*c) - 14400*a^2*b*e^(
7*d*x + 7*c) - 21504*a*b^2*e^(7*d*x + 7*c) + 8825*b^3*e^(7*d*x + 7*c) - 25920*a^2*b*e^(5*d*x + 5*c) - 19968*a*
b^2*e^(5*d*x + 5*c) - 4475*b^3*e^(5*d*x + 5*c) - 14400*a^2*b*e^(3*d*x + 3*c) - 7680*a*b^2*e^(3*d*x + 3*c) + 19
85*b^3*e^(3*d*x + 3*c) - 2880*a^2*b*e^(d*x + c) - 75*b^3*e^(d*x + c))/(e^(2*d*x + 2*c) + 1)^8)/d