∫ sinh(c+dx)___ (a+bsech2(c+dx))3 dx

3.44 \(\int \frac{\sinh (c+d x)}{(a+b \text{sech}^2(c+d x))^3} \, dx\)

Optimal. Leaf size=116 \[ -\frac{5 \cosh ^3(c+d x)}{8 a^2 d \left (a \cosh ^2(c+d x)+b\right )}-\frac{15 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \cosh (c+d x)}{\sqrt{b}}\right )}{8 a^{7/2} d}+\frac{15 \cosh (c+d x)}{8 a^3 d}-\frac{\cosh ^5(c+d x)}{4 a d \left (a \cosh ^2(c+d x)+b\right )^2} \]

[Out]

(-15*Sqrt[b]*ArcTan[(Sqrt[a]*Cosh[c + d*x])/Sqrt[b]])/(8*a^(7/2)*d) + (15*Cosh[c + d*x])/(8*a^3*d) - Cosh[c +

d*x]^5/(4*a*d*(b + a*Cosh[c + d*x]^2)^2) - (5*Cosh[c + d*x]^3)/(8*a^2*d*(b + a*Cosh[c + d*x]^2))

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Rubi [A] time = 0.0841521, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {4133, 288, 321, 205} \[ -\frac{5 \cosh ^3(c+d x)}{8 a^2 d \left (a \cosh ^2(c+d x)+b\right )}-\frac{15 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \cosh (c+d x)}{\sqrt{b}}\right )}{8 a^{7/2} d}+\frac{15 \cosh (c+d x)}{8 a^3 d}-\frac{\cosh ^5(c+d x)}{4 a d \left (a \cosh ^2(c+d x)+b\right )^2} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-15*Sqrt[b]*ArcTan[(Sqrt[a]*Cosh[c + d*x])/Sqrt[b]])/(8*a^(7/2)*d) + (15*Cosh[c + d*x])/(8*a^3*d) - Cosh[c +

d*x]^5/(4*a*d*(b + a*Cosh[c + d*x]^2)^2) - (5*Cosh[c + d*x]^3)/(8*a^2*d*(b + a*Cosh[c + d*x]^2))

Rule 4133

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = F

reeFactors[Cos[e + f*x], x]}, -Dist[ff/f, Subst[Int[((1 - ff^2*x^2)^((m - 1)/2)*(b + a*(ff*x)^n)^p)/(ff*x)^(n*

p), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n] && IntegerQ[p

]

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^

n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x

], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && !ILtQ[(m + n*(p + 1) + 1)/n, 0]

&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n

)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p

, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,

c, n, m, p, x]

Rule 205

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

&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{\sinh (c+d x)}{\left (a+b \text{sech}^2(c+d x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^6}{\left (b+a x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{\cosh ^5(c+d x)}{4 a d \left (b+a \cosh ^2(c+d x)\right )^2}+\frac{5 \operatorname{Subst}\left (\int \frac{x^4}{\left (b+a x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{4 a d}\\ &=-\frac{\cosh ^5(c+d x)}{4 a d \left (b+a \cosh ^2(c+d x)\right )^2}-\frac{5 \cosh ^3(c+d x)}{8 a^2 d \left (b+a \cosh ^2(c+d x)\right )}+\frac{15 \operatorname{Subst}\left (\int \frac{x^2}{b+a x^2} \, dx,x,\cosh (c+d x)\right )}{8 a^2 d}\\ &=\frac{15 \cosh (c+d x)}{8 a^3 d}-\frac{\cosh ^5(c+d x)}{4 a d \left (b+a \cosh ^2(c+d x)\right )^2}-\frac{5 \cosh ^3(c+d x)}{8 a^2 d \left (b+a \cosh ^2(c+d x)\right )}-\frac{(15 b) \operatorname{Subst}\left (\int \frac{1}{b+a x^2} \, dx,x,\cosh (c+d x)\right )}{8 a^3 d}\\ &=-\frac{15 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \cosh (c+d x)}{\sqrt{b}}\right )}{8 a^{7/2} d}+\frac{15 \cosh (c+d x)}{8 a^3 d}-\frac{\cosh ^5(c+d x)}{4 a d \left (b+a \cosh ^2(c+d x)\right )^2}-\frac{5 \cosh ^3(c+d x)}{8 a^2 d \left (b+a \cosh ^2(c+d x)\right )}\\ \end{align*}

Mathematica [C] time = 9.16809, size = 453, normalized size = 3.91 \[ \frac{\text{sech}^6(c+d x) (a \cosh (2 (c+d x))+a+2 b)^3 \left (\frac{8 \cosh (c+d x) \left (3 \left (a^4+48 a b^3\right ) \cosh (2 (c+d x))+16 b^3 (9 a+14 b)\right )}{a^3 b^2 (a \cosh (2 (c+d x))+a+2 b)^2}-\frac{15 \left (\left (a^3+64 b^3\right ) \tan ^{-1}\left (\frac{\sinh (c) \tanh \left (\frac{d x}{2}\right ) \left (\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right )+\cosh (c) \left (\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left (\frac{d x}{2}\right )\right )}{\sqrt{b}}\right )+\left (a^3+64 b^3\right ) \tan ^{-1}\left (\frac{\sinh (c) \tanh \left (\frac{d x}{2}\right ) \left (\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right )+\cosh (c) \left (\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left (\frac{d x}{2}\right )\right )}{\sqrt{b}}\right )+a^3 \left (-\left (\tan ^{-1}\left (\frac{\sqrt{a}-i \sqrt{a+b} \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b}}\right )+\tan ^{-1}\left (\frac{\sqrt{a}+i \sqrt{a+b} \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b}}\right )\right )\right )\right )}{a^{7/2} b^{5/2}}+\frac{512 \sinh (c) \sinh (d x)}{a^3}+\frac{512 \cosh (c) \cosh (d x)}{a^3}-\frac{6 a \sinh (4 (c+d x)) \text{csch}(c+d x)}{b^2 (a \cosh (2 (c+d x))+a+2 b)^2}\right )}{4096 d \left (a+b \text{sech}^2(c+d x)\right )^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((a + 2*b + a*Cosh[2*(c + d*x)])^3*Sech[c + d*x]^6*((-15*((a^3 + 64*b^3)*ArcTan[((Sqrt[a] - I*Sqrt[a + b]*Sqrt

[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*

Tanh[(d*x)/2]))/Sqrt[b]] + (a^3 + 64*b^3)*ArcTan[((Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c

]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]] - a^3*

(ArcTan[(Sqrt[a] - I*Sqrt[a + b]*Tanh[(c + d*x)/2])/Sqrt[b]] + ArcTan[(Sqrt[a] + I*Sqrt[a + b]*Tanh[(c + d*x)/

2])/Sqrt[b]])))/(a^(7/2)*b^(5/2)) + (512*Cosh[c]*Cosh[d*x])/a^3 + (8*Cosh[c + d*x]*(16*b^3*(9*a + 14*b) + 3*(a

^4 + 48*a*b^3)*Cosh[2*(c + d*x)]))/(a^3*b^2*(a + 2*b + a*Cosh[2*(c + d*x)])^2) + (512*Sinh[c]*Sinh[d*x])/a^3 -

(6*a*Csch[c + d*x]*Sinh[4*(c + d*x)])/(b^2*(a + 2*b + a*Cosh[2*(c + d*x)])^2)))/(4096*d*(a + b*Sech[c + d*x]^

2)^3)

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Maple [A] time = 0.035, size = 107, normalized size = 0.9 \begin{align*}{\frac{7\,{b}^{2} \left ({\rm sech} \left (dx+c\right ) \right ) ^{3}}{8\,d{a}^{3} \left ( a+b \left ({\rm sech} \left (dx+c\right ) \right ) ^{2} \right ) ^{2}}}+{\frac{9\,b{\rm sech} \left (dx+c\right )}{8\,d{a}^{2} \left ( a+b \left ({\rm sech} \left (dx+c\right ) \right ) ^{2} \right ) ^{2}}}+{\frac{15\,b}{8\,d{a}^{3}}\arctan \left ({b{\rm sech} \left (dx+c\right ){\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{1}{d{a}^{3}{\rm sech} \left (dx+c\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7/8/d/a^3*b^2/(a+b*sech(d*x+c)^2)^2*sech(d*x+c)^3+9/8/d/a^2*b/(a+b*sech(d*x+c)^2)^2*sech(d*x+c)+15/8/d/a^3*b/(

a*b)^(1/2)*arctan(sech(d*x+c)*b/(a*b)^(1/2))+1/d/a^3/sech(d*x+c)

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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \, a^{2} e^{\left (10 \, d x + 10 \, c\right )} + 2 \, a^{2} + 5 \,{\left (2 \, a^{2} e^{\left (8 \, c\right )} + 5 \, a b e^{\left (8 \, c\right )}\right )} e^{\left (8 \, d x\right )} + 5 \,{\left (4 \, a^{2} e^{\left (6 \, c\right )} + 15 \, a b e^{\left (6 \, c\right )} + 12 \, b^{2} e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} + 5 \,{\left (4 \, a^{2} e^{\left (4 \, c\right )} + 15 \, a b e^{\left (4 \, c\right )} + 12 \, b^{2} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 5 \,{\left (2 \, a^{2} e^{\left (2 \, c\right )} + 5 \, a b e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}{4 \,{\left (a^{5} d e^{\left (9 \, d x + 9 \, c\right )} + a^{5} d e^{\left (d x + c\right )} + 4 \,{\left (a^{5} d e^{\left (7 \, c\right )} + 2 \, a^{4} b d e^{\left (7 \, c\right )}\right )} e^{\left (7 \, d x\right )} + 2 \,{\left (3 \, a^{5} d e^{\left (5 \, c\right )} + 8 \, a^{4} b d e^{\left (5 \, c\right )} + 8 \, a^{3} b^{2} d e^{\left (5 \, c\right )}\right )} e^{\left (5 \, d x\right )} + 4 \,{\left (a^{5} d e^{\left (3 \, c\right )} + 2 \, a^{4} b d e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )}\right )}} - \frac{1}{2} \, \int \frac{15 \,{\left (b e^{\left (3 \, d x + 3 \, c\right )} - b e^{\left (d x + c\right )}\right )}}{2 \,{\left (a^{4} e^{\left (4 \, d x + 4 \, c\right )} + a^{4} + 2 \,{\left (a^{4} e^{\left (2 \, c\right )} + 2 \, a^{3} b e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

*e^(6*c) + 12*b^2*e^(6*c))*e^(6*d*x) + 5*(4*a^2*e^(4*c) + 15*a*b*e^(4*c) + 12*b^2*e^(4*c))*e^(4*d*x) + 5*(2*a^

2*e^(2*c) + 5*a*b*e^(2*c))*e^(2*d*x))/(a^5*d*e^(9*d*x + 9*c) + a^5*d*e^(d*x + c) + 4*(a^5*d*e^(7*c) + 2*a^4*b*

d*e^(7*c))*e^(7*d*x) + 2*(3*a^5*d*e^(5*c) + 8*a^4*b*d*e^(5*c) + 8*a^3*b^2*d*e^(5*c))*e^(5*d*x) + 4*(a^5*d*e^(3

*c) + 2*a^4*b*d*e^(3*c))*e^(3*d*x)) - 1/2*integrate(15/2*(b*e^(3*d*x + 3*c) - b*e^(d*x + c))/(a^4*e^(4*d*x + 4

*c) + a^4 + 2*(a^4*e^(2*c) + 2*a^3*b*e^(2*c))*e^(2*d*x)), x)

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Fricas [B] time = 2.96016, size = 12176, normalized size = 104.97 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/16*(8*a^2*cosh(d*x + c)^10 + 80*a^2*cosh(d*x + c)*sinh(d*x + c)^9 + 8*a^2*sinh(d*x + c)^10 + 20*(2*a^2 + 5*

a*b)*cosh(d*x + c)^8 + 20*(18*a^2*cosh(d*x + c)^2 + 2*a^2 + 5*a*b)*sinh(d*x + c)^8 + 160*(6*a^2*cosh(d*x + c)^

3 + (2*a^2 + 5*a*b)*cosh(d*x + c))*sinh(d*x + c)^7 + 20*(4*a^2 + 15*a*b + 12*b^2)*cosh(d*x + c)^6 + 20*(84*a^2

*cosh(d*x + c)^4 + 28*(2*a^2 + 5*a*b)*cosh(d*x + c)^2 + 4*a^2 + 15*a*b + 12*b^2)*sinh(d*x + c)^6 + 8*(252*a^2*

cosh(d*x + c)^5 + 140*(2*a^2 + 5*a*b)*cosh(d*x + c)^3 + 15*(4*a^2 + 15*a*b + 12*b^2)*cosh(d*x + c))*sinh(d*x +

c)^5 + 20*(4*a^2 + 15*a*b + 12*b^2)*cosh(d*x + c)^4 + 20*(84*a^2*cosh(d*x + c)^6 + 70*(2*a^2 + 5*a*b)*cosh(d*

x + c)^4 + 15*(4*a^2 + 15*a*b + 12*b^2)*cosh(d*x + c)^2 + 4*a^2 + 15*a*b + 12*b^2)*sinh(d*x + c)^4 + 80*(12*a^

2*cosh(d*x + c)^7 + 14*(2*a^2 + 5*a*b)*cosh(d*x + c)^5 + 5*(4*a^2 + 15*a*b + 12*b^2)*cosh(d*x + c)^3 + (4*a^2

+ 15*a*b + 12*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 20*(2*a^2 + 5*a*b)*cosh(d*x + c)^2 + 20*(18*a^2*cosh(d*x +

c)^8 + 28*(2*a^2 + 5*a*b)*cosh(d*x + c)^6 + 15*(4*a^2 + 15*a*b + 12*b^2)*cosh(d*x + c)^4 + 6*(4*a^2 + 15*a*b

+ 12*b^2)*cosh(d*x + c)^2 + 2*a^2 + 5*a*b)*sinh(d*x + c)^2 + 15*(a^2*cosh(d*x + c)^9 + 9*a^2*cosh(d*x + c)*sin

h(d*x + c)^8 + a^2*sinh(d*x + c)^9 + 4*(a^2 + 2*a*b)*cosh(d*x + c)^7 + 4*(9*a^2*cosh(d*x + c)^2 + a^2 + 2*a*b)

*sinh(d*x + c)^7 + 28*(3*a^2*cosh(d*x + c)^3 + (a^2 + 2*a*b)*cosh(d*x + c))*sinh(d*x + c)^6 + 2*(3*a^2 + 8*a*b

+ 8*b^2)*cosh(d*x + c)^5 + 2*(63*a^2*cosh(d*x + c)^4 + 42*(a^2 + 2*a*b)*cosh(d*x + c)^2 + 3*a^2 + 8*a*b + 8*b

^2)*sinh(d*x + c)^5 + 2*(63*a^2*cosh(d*x + c)^5 + 70*(a^2 + 2*a*b)*cosh(d*x + c)^3 + 5*(3*a^2 + 8*a*b + 8*b^2)

*cosh(d*x + c))*sinh(d*x + c)^4 + 4*(a^2 + 2*a*b)*cosh(d*x + c)^3 + 4*(21*a^2*cosh(d*x + c)^6 + 35*(a^2 + 2*a*

b)*cosh(d*x + c)^4 + 5*(3*a^2 + 8*a*b + 8*b^2)*cosh(d*x + c)^2 + a^2 + 2*a*b)*sinh(d*x + c)^3 + a^2*cosh(d*x +

c) + 4*(9*a^2*cosh(d*x + c)^7 + 21*(a^2 + 2*a*b)*cosh(d*x + c)^5 + 5*(3*a^2 + 8*a*b + 8*b^2)*cosh(d*x + c)^3

+ 3*(a^2 + 2*a*b)*cosh(d*x + c))*sinh(d*x + c)^2 + (9*a^2*cosh(d*x + c)^8 + 28*(a^2 + 2*a*b)*cosh(d*x + c)^6 +

10*(3*a^2 + 8*a*b + 8*b^2)*cosh(d*x + c)^4 + 12*(a^2 + 2*a*b)*cosh(d*x + c)^2 + a^2)*sinh(d*x + c))*sqrt(-b/a

)*log((a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 + 2*(a - 2*b)*cosh(d*x + c)^2

+ 2*(3*a*cosh(d*x + c)^2 + a - 2*b)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 + (a - 2*b)*cosh(d*x + c))*sinh(d*

x + c) - 4*(a*cosh(d*x + c)^3 + 3*a*cosh(d*x + c)*sinh(d*x + c)^2 + a*sinh(d*x + c)^3 + a*cosh(d*x + c) + (3*a

*cosh(d*x + c)^2 + a)*sinh(d*x + c))*sqrt(-b/a) + a)/(a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 +

a*sinh(d*x + c)^4 + 2*(a + 2*b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 + a + 2*b)*sinh(d*x + c)^2 + 4*(a*cos

h(d*x + c)^3 + (a + 2*b)*cosh(d*x + c))*sinh(d*x + c) + a)) + 8*a^2 + 40*(2*a^2*cosh(d*x + c)^9 + 4*(2*a^2 + 5

*a*b)*cosh(d*x + c)^7 + 3*(4*a^2 + 15*a*b + 12*b^2)*cosh(d*x + c)^5 + 2*(4*a^2 + 15*a*b + 12*b^2)*cosh(d*x + c

)^3 + (2*a^2 + 5*a*b)*cosh(d*x + c))*sinh(d*x + c))/(a^5*d*cosh(d*x + c)^9 + 9*a^5*d*cosh(d*x + c)*sinh(d*x +

c)^8 + a^5*d*sinh(d*x + c)^9 + 4*(a^5 + 2*a^4*b)*d*cosh(d*x + c)^7 + 4*(9*a^5*d*cosh(d*x + c)^2 + (a^5 + 2*a^4

*b)*d)*sinh(d*x + c)^7 + a^5*d*cosh(d*x + c) + 2*(3*a^5 + 8*a^4*b + 8*a^3*b^2)*d*cosh(d*x + c)^5 + 28*(3*a^5*d

*cosh(d*x + c)^3 + (a^5 + 2*a^4*b)*d*cosh(d*x + c))*sinh(d*x + c)^6 + 2*(63*a^5*d*cosh(d*x + c)^4 + 42*(a^5 +

2*a^4*b)*d*cosh(d*x + c)^2 + (3*a^5 + 8*a^4*b + 8*a^3*b^2)*d)*sinh(d*x + c)^5 + 4*(a^5 + 2*a^4*b)*d*cosh(d*x +

c)^3 + 2*(63*a^5*d*cosh(d*x + c)^5 + 70*(a^5 + 2*a^4*b)*d*cosh(d*x + c)^3 + 5*(3*a^5 + 8*a^4*b + 8*a^3*b^2)*d

*cosh(d*x + c))*sinh(d*x + c)^4 + 4*(21*a^5*d*cosh(d*x + c)^6 + 35*(a^5 + 2*a^4*b)*d*cosh(d*x + c)^4 + 5*(3*a^

5 + 8*a^4*b + 8*a^3*b^2)*d*cosh(d*x + c)^2 + (a^5 + 2*a^4*b)*d)*sinh(d*x + c)^3 + 4*(9*a^5*d*cosh(d*x + c)^7 +

21*(a^5 + 2*a^4*b)*d*cosh(d*x + c)^5 + 5*(3*a^5 + 8*a^4*b + 8*a^3*b^2)*d*cosh(d*x + c)^3 + 3*(a^5 + 2*a^4*b)*

d*cosh(d*x + c))*sinh(d*x + c)^2 + (9*a^5*d*cosh(d*x + c)^8 + 28*(a^5 + 2*a^4*b)*d*cosh(d*x + c)^6 + a^5*d + 1

0*(3*a^5 + 8*a^4*b + 8*a^3*b^2)*d*cosh(d*x + c)^4 + 12*(a^5 + 2*a^4*b)*d*cosh(d*x + c)^2)*sinh(d*x + c)), 1/8*

(4*a^2*cosh(d*x + c)^10 + 40*a^2*cosh(d*x + c)*sinh(d*x + c)^9 + 4*a^2*sinh(d*x + c)^10 + 10*(2*a^2 + 5*a*b)*c

osh(d*x + c)^8 + 10*(18*a^2*cosh(d*x + c)^2 + 2*a^2 + 5*a*b)*sinh(d*x + c)^8 + 80*(6*a^2*cosh(d*x + c)^3 + (2*

a^2 + 5*a*b)*cosh(d*x + c))*sinh(d*x + c)^7 + 10*(4*a^2 + 15*a*b + 12*b^2)*cosh(d*x + c)^6 + 10*(84*a^2*cosh(d

*x + c)^4 + 28*(2*a^2 + 5*a*b)*cosh(d*x + c)^2 + 4*a^2 + 15*a*b + 12*b^2)*sinh(d*x + c)^6 + 4*(252*a^2*cosh(d*

x + c)^5 + 140*(2*a^2 + 5*a*b)*cosh(d*x + c)^3 + 15*(4*a^2 + 15*a*b + 12*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 +

10*(4*a^2 + 15*a*b + 12*b^2)*cosh(d*x + c)^4 + 10*(84*a^2*cosh(d*x + c)^6 + 70*(2*a^2 + 5*a*b)*cosh(d*x + c)^

4 + 15*(4*a^2 + 15*a*b + 12*b^2)*cosh(d*x + c)^2 + 4*a^2 + 15*a*b + 12*b^2)*sinh(d*x + c)^4 + 40*(12*a^2*cosh(

d*x + c)^7 + 14*(2*a^2 + 5*a*b)*cosh(d*x + c)^5 + 5*(4*a^2 + 15*a*b + 12*b^2)*cosh(d*x + c)^3 + (4*a^2 + 15*a*

b + 12*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 10*(2*a^2 + 5*a*b)*cosh(d*x + c)^2 + 10*(18*a^2*cosh(d*x + c)^8 +

28*(2*a^2 + 5*a*b)*cosh(d*x + c)^6 + 15*(4*a^2 + 15*a*b + 12*b^2)*cosh(d*x + c)^4 + 6*(4*a^2 + 15*a*b + 12*b^

2)*cosh(d*x + c)^2 + 2*a^2 + 5*a*b)*sinh(d*x + c)^2 + 15*(a^2*cosh(d*x + c)^9 + 9*a^2*cosh(d*x + c)*sinh(d*x +

c)^8 + a^2*sinh(d*x + c)^9 + 4*(a^2 + 2*a*b)*cosh(d*x + c)^7 + 4*(9*a^2*cosh(d*x + c)^2 + a^2 + 2*a*b)*sinh(d

*x + c)^7 + 28*(3*a^2*cosh(d*x + c)^3 + (a^2 + 2*a*b)*cosh(d*x + c))*sinh(d*x + c)^6 + 2*(3*a^2 + 8*a*b + 8*b^

2)*cosh(d*x + c)^5 + 2*(63*a^2*cosh(d*x + c)^4 + 42*(a^2 + 2*a*b)*cosh(d*x + c)^2 + 3*a^2 + 8*a*b + 8*b^2)*sin

h(d*x + c)^5 + 2*(63*a^2*cosh(d*x + c)^5 + 70*(a^2 + 2*a*b)*cosh(d*x + c)^3 + 5*(3*a^2 + 8*a*b + 8*b^2)*cosh(d

*x + c))*sinh(d*x + c)^4 + 4*(a^2 + 2*a*b)*cosh(d*x + c)^3 + 4*(21*a^2*cosh(d*x + c)^6 + 35*(a^2 + 2*a*b)*cosh

(d*x + c)^4 + 5*(3*a^2 + 8*a*b + 8*b^2)*cosh(d*x + c)^2 + a^2 + 2*a*b)*sinh(d*x + c)^3 + a^2*cosh(d*x + c) + 4

*(9*a^2*cosh(d*x + c)^7 + 21*(a^2 + 2*a*b)*cosh(d*x + c)^5 + 5*(3*a^2 + 8*a*b + 8*b^2)*cosh(d*x + c)^3 + 3*(a^

2 + 2*a*b)*cosh(d*x + c))*sinh(d*x + c)^2 + (9*a^2*cosh(d*x + c)^8 + 28*(a^2 + 2*a*b)*cosh(d*x + c)^6 + 10*(3*

a^2 + 8*a*b + 8*b^2)*cosh(d*x + c)^4 + 12*(a^2 + 2*a*b)*cosh(d*x + c)^2 + a^2)*sinh(d*x + c))*sqrt(b/a)*arctan

(1/2*(a*cosh(d*x + c)^3 + 3*a*cosh(d*x + c)*sinh(d*x + c)^2 + a*sinh(d*x + c)^3 + (a + 4*b)*cosh(d*x + c) + (3

*a*cosh(d*x + c)^2 + a + 4*b)*sinh(d*x + c))*sqrt(b/a)/b) - 15*(a^2*cosh(d*x + c)^9 + 9*a^2*cosh(d*x + c)*sinh

(d*x + c)^8 + a^2*sinh(d*x + c)^9 + 4*(a^2 + 2*a*b)*cosh(d*x + c)^7 + 4*(9*a^2*cosh(d*x + c)^2 + a^2 + 2*a*b)*

sinh(d*x + c)^7 + 28*(3*a^2*cosh(d*x + c)^3 + (a^2 + 2*a*b)*cosh(d*x + c))*sinh(d*x + c)^6 + 2*(3*a^2 + 8*a*b

+ 8*b^2)*cosh(d*x + c)^5 + 2*(63*a^2*cosh(d*x + c)^4 + 42*(a^2 + 2*a*b)*cosh(d*x + c)^2 + 3*a^2 + 8*a*b + 8*b^

2)*sinh(d*x + c)^5 + 2*(63*a^2*cosh(d*x + c)^5 + 70*(a^2 + 2*a*b)*cosh(d*x + c)^3 + 5*(3*a^2 + 8*a*b + 8*b^2)*

cosh(d*x + c))*sinh(d*x + c)^4 + 4*(a^2 + 2*a*b)*cosh(d*x + c)^3 + 4*(21*a^2*cosh(d*x + c)^6 + 35*(a^2 + 2*a*b

)*cosh(d*x + c)^4 + 5*(3*a^2 + 8*a*b + 8*b^2)*cosh(d*x + c)^2 + a^2 + 2*a*b)*sinh(d*x + c)^3 + a^2*cosh(d*x +

c) + 4*(9*a^2*cosh(d*x + c)^7 + 21*(a^2 + 2*a*b)*cosh(d*x + c)^5 + 5*(3*a^2 + 8*a*b + 8*b^2)*cosh(d*x + c)^3 +

3*(a^2 + 2*a*b)*cosh(d*x + c))*sinh(d*x + c)^2 + (9*a^2*cosh(d*x + c)^8 + 28*(a^2 + 2*a*b)*cosh(d*x + c)^6 +

10*(3*a^2 + 8*a*b + 8*b^2)*cosh(d*x + c)^4 + 12*(a^2 + 2*a*b)*cosh(d*x + c)^2 + a^2)*sinh(d*x + c))*sqrt(b/a)*

arctan(1/2*(a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt(b/a)/b) + 4*a^2 + 20*(2*a^2*cosh(d*x + c)^9 + 4*(2*a^2 + 5