∫ csch3(c + dx)(a + b tanh3(c + dx))3 dx

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

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

[Out]

3/2*a^2*b*arctan(sinh(d*x+c))/d+5/128*b^3*arctan(sinh(d*x+c))/d+1/2*a^3*arctanh(cosh(d*x+c))/d-1/2*a^3*coth(d*

x+c)*csch(d*x+c)/d-a*b^2*sech(d*x+c)^3/d+3/5*a*b^2*sech(d*x+c)^5/d+3/2*a^2*b*sech(d*x+c)*tanh(d*x+c)/d+5/128*b

^3*sech(d*x+c)*tanh(d*x+c)/d-5/64*b^3*sech(d*x+c)^3*tanh(d*x+c)/d-5/48*b^3*sech(d*x+c)^3*tanh(d*x+c)^3/d-1/8*b

^3*sech(d*x+c)^3*tanh(d*x+c)^5/d

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Rubi [A] time = 0.32, antiderivative size = 232, normalized size of antiderivative = 1.00, 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 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 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 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 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]

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*}

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Mathematica [A] time = 6.37, size = 243, normalized size = 1.05 \[ -\frac {a^3 \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {a^3 \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {a^3 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {\text {sech}^2(c+d x) \left (192 a^2 b \sinh (c+d x)+5 b^3 \sinh (c+d x)\right )}{128 d}+\frac {b \left (192 a^2+5 b^2\right ) \tan ^{-1}\left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{64 d}+\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}-\frac {a b^2 \text {sech}^3(c+d x)}{d}-\frac {b^3 \tanh (c+d x) \text {sech}^7(c+d x)}{8 d}+\frac {17 b^3 \tanh (c+d x) \text {sech}^5(c+d x)}{48 d}-\frac {59 b^3 \tanh (c+d x) \text {sech}^3(c+d x)}{192 d} \]

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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fricas [B] time = 0.61, size = 10985, normalized size = 47.35 \[ \text {result too large to display} \]

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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giac [B] time = 0.84, size = 434, normalized size = 1.87 \[ \frac {480 \, a^{3} \log \left (e^{\left (d x + c\right )} + 1\right ) - 480 \, a^{3} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) + 15 \, {\left (192 \, a^{2} b e^{c} + 5 \, b^{3} e^{c}\right )} \arctan \left (e^{\left (d x + c\right )}\right ) e^{\left (-c\right )} - \frac {960 \, {\left (a^{3} e^{\left (3 \, d x + 3 \, c\right )} + a^{3} e^{\left (d x + c\right )}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}} + \frac {2880 \, a^{2} b e^{\left (15 \, d x + 15 \, c\right )} + 75 \, b^{3} e^{\left (15 \, d x + 15 \, c\right )} + 14400 \, a^{2} b e^{\left (13 \, d x + 13 \, c\right )} - 7680 \, a b^{2} e^{\left (13 \, d x + 13 \, c\right )} - 1985 \, b^{3} e^{\left (13 \, d x + 13 \, c\right )} + 25920 \, a^{2} b e^{\left (11 \, d x + 11 \, c\right )} - 19968 \, a b^{2} e^{\left (11 \, d x + 11 \, c\right )} + 4475 \, b^{3} e^{\left (11 \, d x + 11 \, c\right )} + 14400 \, a^{2} b e^{\left (9 \, d x + 9 \, c\right )} - 21504 \, a b^{2} e^{\left (9 \, d x + 9 \, c\right )} - 8825 \, b^{3} e^{\left (9 \, d x + 9 \, c\right )} - 14400 \, a^{2} b e^{\left (7 \, d x + 7 \, c\right )} - 21504 \, a b^{2} e^{\left (7 \, d x + 7 \, c\right )} + 8825 \, b^{3} e^{\left (7 \, d x + 7 \, c\right )} - 25920 \, a^{2} b e^{\left (5 \, d x + 5 \, c\right )} - 19968 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} - 4475 \, b^{3} e^{\left (5 \, d x + 5 \, c\right )} - 14400 \, a^{2} b e^{\left (3 \, d x + 3 \, c\right )} - 7680 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 1985 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} - 2880 \, a^{2} b e^{\left (d x + c\right )} - 75 \, b^{3} e^{\left (d x + c\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{8}}}{960 \, d} \]

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

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maple [A] time = 0.61, size = 286, normalized size = 1.23 \[ -\frac {a^{3} \coth \left (d x +c \right ) \mathrm {csch}\left (d x +c \right )}{2 d}+\frac {a^{3} \arctanh \left ({\mathrm e}^{d x +c}\right )}{d}+\frac {3 a^{2} b \,\mathrm {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2 d}+\frac {3 a^{2} b \arctan \left ({\mathrm e}^{d x +c}\right )}{d}-\frac {a \,b^{2} \left (\sinh ^{2}\left (d x +c \right )\right )}{d \cosh \left (d x +c \right )^{5}}-\frac {2 a \,b^{2}}{5 d \cosh \left (d x +c \right )^{5}}-\frac {b^{3} \left (\sinh ^{5}\left (d x +c \right )\right )}{3 d \cosh \left (d x +c \right )^{8}}-\frac {b^{3} \left (\sinh ^{3}\left (d x +c \right )\right )}{3 d \cosh \left (d x +c \right )^{8}}-\frac {b^{3} \sinh \left (d x +c \right )}{7 d \cosh \left (d x +c \right )^{8}}+\frac {b^{3} \tanh \left (d x +c \right ) \mathrm {sech}\left (d x +c \right )^{7}}{56 d}+\frac {b^{3} \tanh \left (d x +c \right ) \mathrm {sech}\left (d x +c \right )^{5}}{48 d}+\frac {5 b^{3} \tanh \left (d x +c \right ) \mathrm {sech}\left (d x +c \right )^{3}}{192 d}+\frac {5 b^{3} \mathrm {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{128 d}+\frac {5 b^{3} \arctan \left ({\mathrm e}^{d x +c}\right )}{64 d} \]

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

rctan(exp(d*x+c))-1/d*a*b^2*sinh(d*x+c)^2/cosh(d*x+c)^5-2/5/d*a*b^2/cosh(d*x+c)^5-1/3/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)*sec

h(d*x+c)^7+1/48/d*b^3*tanh(d*x+c)*sech(d*x+c)^5+5/192/d*b^3*tanh(d*x+c)*sech(d*x+c)^3+5/128/d*b^3*sech(d*x+c)*

tanh(d*x+c)+5/64/d*b^3*arctan(exp(d*x+c))

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maxima [B] time = 0.43, size = 586, normalized size = 2.53 \[ -\frac {1}{192} \, b^{3} {\left (\frac {15 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {15 \, e^{\left (-d x - c\right )} - 397 \, e^{\left (-3 \, d x - 3 \, c\right )} + 895 \, e^{\left (-5 \, d x - 5 \, c\right )} - 1765 \, e^{\left (-7 \, d x - 7 \, c\right )} + 1765 \, e^{\left (-9 \, d x - 9 \, c\right )} - 895 \, e^{\left (-11 \, d x - 11 \, c\right )} + 397 \, e^{\left (-13 \, d x - 13 \, c\right )} - 15 \, e^{\left (-15 \, d x - 15 \, c\right )}}{d {\left (8 \, e^{\left (-2 \, d x - 2 \, c\right )} + 28 \, e^{\left (-4 \, d x - 4 \, c\right )} + 56 \, e^{\left (-6 \, d x - 6 \, c\right )} + 70 \, e^{\left (-8 \, d x - 8 \, c\right )} + 56 \, e^{\left (-10 \, d x - 10 \, c\right )} + 28 \, e^{\left (-12 \, d x - 12 \, c\right )} + 8 \, e^{\left (-14 \, d x - 14 \, c\right )} + e^{\left (-16 \, d x - 16 \, c\right )} + 1\right )}}\right )} - 3 \, a^{2} b {\left (\frac {\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} + \frac {1}{2} \, a^{3} {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} - \frac {8}{5} \, a b^{2} {\left (\frac {5 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} - \frac {2 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} + \frac {5 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}}\right )} \]

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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mupad [B] time = 8.05, size = 731, normalized size = 3.15 \[ \frac {a^3\,\ln \left ({\mathrm {e}}^{c+d\,x}+1\right )}{2\,d}-\frac {a^3\,\ln \left ({\mathrm {e}}^{c+d\,x}-1\right )}{2\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (192\,a^2\,b+5\,b^3\right )}{64\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (2245\,b^3+3264\,a\,b^2\right )}{120\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (1325\,b^3+768\,a\,b^2\right )}{20\,d\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}-\frac {500\,b^3\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (6\,{\mathrm {e}}^{2\,c+2\,d\,x}+15\,{\mathrm {e}}^{4\,c+4\,d\,x}+20\,{\mathrm {e}}^{6\,c+6\,d\,x}+15\,{\mathrm {e}}^{8\,c+8\,d\,x}+6\,{\mathrm {e}}^{10\,c+10\,d\,x}+{\mathrm {e}}^{12\,c+12\,d\,x}+1\right )}+\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (1025\,b^3+144\,a\,b^2\right )}{15\,d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1\right )}+\frac {112\,b^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left (7\,{\mathrm {e}}^{2\,c+2\,d\,x}+21\,{\mathrm {e}}^{4\,c+4\,d\,x}+35\,{\mathrm {e}}^{6\,c+6\,d\,x}+35\,{\mathrm {e}}^{8\,c+8\,d\,x}+21\,{\mathrm {e}}^{10\,c+10\,d\,x}+7\,{\mathrm {e}}^{12\,c+12\,d\,x}+{\mathrm {e}}^{14\,c+14\,d\,x}+1\right )}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (576\,a^2\,b+768\,a\,b^2+251\,b^3\right )}{96\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {32\,b^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left (8\,{\mathrm {e}}^{2\,c+2\,d\,x}+28\,{\mathrm {e}}^{4\,c+4\,d\,x}+56\,{\mathrm {e}}^{6\,c+6\,d\,x}+70\,{\mathrm {e}}^{8\,c+8\,d\,x}+56\,{\mathrm {e}}^{10\,c+10\,d\,x}+28\,{\mathrm {e}}^{12\,c+12\,d\,x}+8\,{\mathrm {e}}^{14\,c+14\,d\,x}+{\mathrm {e}}^{16\,c+16\,d\,x}+1\right )}-\frac {a^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {2\,a^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {b\,\ln \left ({\mathrm {e}}^{c+d\,x}-\mathrm {i}\right )\,\left (192\,a^2+5\,b^2\right )\,1{}\mathrm {i}}{128\,d}+\frac {b\,\ln \left ({\mathrm {e}}^{c+d\,x}+1{}\mathrm {i}\right )\,\left (192\,a^2+5\,b^2\right )\,1{}\mathrm {i}}{128\,d} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*tanh(c + d*x)^3)^3/sinh(c + d*x)^3,x)

[Out]

(a^3*log(exp(c + d*x) + 1))/(2*d) - (a^3*log(exp(c + d*x) - 1))/(2*d) + (exp(c + d*x)*(192*a^2*b + 5*b^3))/(64

*d*(exp(2*c + 2*d*x) + 1)) + (exp(c + d*x)*(3264*a*b^2 + 2245*b^3))/(120*d*(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4

*d*x) + exp(6*c + 6*d*x) + 1)) - (b*log(exp(c + d*x) - 1i)*(192*a^2 + 5*b^2)*1i)/(128*d) + (b*log(exp(c + d*x)

+ 1i)*(192*a^2 + 5*b^2)*1i)/(128*d) - (exp(c + d*x)*(768*a*b^2 + 1325*b^3))/(20*d*(4*exp(2*c + 2*d*x) + 6*exp

(4*c + 4*d*x) + 4*exp(6*c + 6*d*x) + exp(8*c + 8*d*x) + 1)) - (500*b^3*exp(c + d*x))/(3*d*(6*exp(2*c + 2*d*x)

+ 15*exp(4*c + 4*d*x) + 20*exp(6*c + 6*d*x) + 15*exp(8*c + 8*d*x) + 6*exp(10*c + 10*d*x) + exp(12*c + 12*d*x)

+ 1)) + (2*exp(c + d*x)*(144*a*b^2 + 1025*b^3))/(15*d*(5*exp(2*c + 2*d*x) + 10*exp(4*c + 4*d*x) + 10*exp(6*c +

6*d*x) + 5*exp(8*c + 8*d*x) + exp(10*c + 10*d*x) + 1)) + (112*b^3*exp(c + d*x))/(d*(7*exp(2*c + 2*d*x) + 21*e

xp(4*c + 4*d*x) + 35*exp(6*c + 6*d*x) + 35*exp(8*c + 8*d*x) + 21*exp(10*c + 10*d*x) + 7*exp(12*c + 12*d*x) + e

xp(14*c + 14*d*x) + 1)) - (exp(c + d*x)*(768*a*b^2 + 576*a^2*b + 251*b^3))/(96*d*(2*exp(2*c + 2*d*x) + exp(4*c

+ 4*d*x) + 1)) - (32*b^3*exp(c + d*x))/(d*(8*exp(2*c + 2*d*x) + 28*exp(4*c + 4*d*x) + 56*exp(6*c + 6*d*x) + 7

0*exp(8*c + 8*d*x) + 56*exp(10*c + 10*d*x) + 28*exp(12*c + 12*d*x) + 8*exp(14*c + 14*d*x) + exp(16*c + 16*d*x)

+ 1)) - (a^3*exp(c + d*x))/(d*(exp(2*c + 2*d*x) - 1)) - (2*a^3*exp(c + d*x))/(d*(exp(4*c + 4*d*x) - 2*exp(2*c

+ 2*d*x) + 1))

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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \tanh ^{3}{\left (c + d x \right )}\right )^{3} \operatorname {csch}^{3}{\left (c + d x \right )}\, dx \]

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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