3.402 \(\int \frac{(e+f x)^2 \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\)

Optimal. Leaf size=1049 \[ \text{result too large to display} \]

[Out]

-(a^3*e*f*x)/(2*b^4*d) + (3*a*e*f*x)/(16*b^2*d) - (a^3*f^2*x^2)/(4*b^4*d) + (3*a*f^2*x^2)/(32*b^2*d) + (a^3*(a
^2 + b^2)*(e + f*x)^3)/(3*b^6*f) - (2*a^4*f*(e + f*x)*Cosh[c + d*x])/(b^5*d^2) - (4*a^2*f*(e + f*x)*Cosh[c + d
*x])/(3*b^3*d^2) + (f*(e + f*x)*Cosh[c + d*x])/(4*b*d^2) - (3*a*f^2*Cosh[c + d*x]^2)/(32*b^2*d^3) - (2*a^2*f*(
e + f*x)*Cosh[c + d*x]^3)/(9*b^3*d^2) - (a*f^2*Cosh[c + d*x]^4)/(32*b^2*d^3) - (a*(e + f*x)^2*Cosh[c + d*x]^4)
/(4*b^2*d) - (f*(e + f*x)*Cosh[3*c + 3*d*x])/(72*b*d^2) - (f*(e + f*x)*Cosh[5*c + 5*d*x])/(200*b*d^2) - (a^3*(
a^2 + b^2)*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^6*d) - (a^3*(a^2 + b^2)*(e + f*x)^2*
Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^6*d) - (2*a^3*(a^2 + b^2)*f*(e + f*x)*PolyLog[2, -((b*E^(c
+ d*x))/(a - Sqrt[a^2 + b^2]))])/(b^6*d^2) - (2*a^3*(a^2 + b^2)*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a +
Sqrt[a^2 + b^2]))])/(b^6*d^2) + (2*a^3*(a^2 + b^2)*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(
b^6*d^3) + (2*a^3*(a^2 + b^2)*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^6*d^3) + (2*a^4*f^2
*Sinh[c + d*x])/(b^5*d^3) + (14*a^2*f^2*Sinh[c + d*x])/(9*b^3*d^3) - (f^2*Sinh[c + d*x])/(4*b*d^3) + (a^4*(e +
f*x)^2*Sinh[c + d*x])/(b^5*d) + (2*a^2*(e + f*x)^2*Sinh[c + d*x])/(3*b^3*d) - ((e + f*x)^2*Sinh[c + d*x])/(8*
b*d) + (a^3*f*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(2*b^4*d^2) + (3*a*f*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x
])/(16*b^2*d^2) + (a^2*(e + f*x)^2*Cosh[c + d*x]^2*Sinh[c + d*x])/(3*b^3*d) + (a*f*(e + f*x)*Cosh[c + d*x]^3*S
inh[c + d*x])/(8*b^2*d^2) - (a^3*f^2*Sinh[c + d*x]^2)/(4*b^4*d^3) - (a^3*(e + f*x)^2*Sinh[c + d*x]^2)/(2*b^4*d
) + (2*a^2*f^2*Sinh[c + d*x]^3)/(27*b^3*d^3) + (f^2*Sinh[3*c + 3*d*x])/(216*b*d^3) + ((e + f*x)^2*Sinh[3*c + 3
*d*x])/(48*b*d) + (f^2*Sinh[5*c + 5*d*x])/(1000*b*d^3) + ((e + f*x)^2*Sinh[5*c + 5*d*x])/(80*b*d)

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Rubi [A]  time = 1.62347, antiderivative size = 1049, normalized size of antiderivative = 1., number of steps used = 40, number of rules used = 15, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {5579, 5448, 3296, 2637, 5447, 3310, 3311, 2633, 5565, 5446, 5561, 2190, 2531, 2282, 6589} \[ -\frac{2 f (e+f x) \cosh (c+d x) a^4}{b^5 d^2}+\frac{2 f^2 \sinh (c+d x) a^4}{b^5 d^3}+\frac{(e+f x)^2 \sinh (c+d x) a^4}{b^5 d}+\frac{\left (a^2+b^2\right ) (e+f x)^3 a^3}{3 b^6 f}-\frac{f^2 x^2 a^3}{4 b^4 d}-\frac{f^2 \sinh ^2(c+d x) a^3}{4 b^4 d^3}-\frac{(e+f x)^2 \sinh ^2(c+d x) a^3}{2 b^4 d}-\frac{e f x a^3}{2 b^4 d}-\frac{\left (a^2+b^2\right ) (e+f x)^2 \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) a^3}{b^6 d}-\frac{\left (a^2+b^2\right ) (e+f x)^2 \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) a^3}{b^6 d}-\frac{2 \left (a^2+b^2\right ) f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) a^3}{b^6 d^2}-\frac{2 \left (a^2+b^2\right ) f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) a^3}{b^6 d^2}+\frac{2 \left (a^2+b^2\right ) f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) a^3}{b^6 d^3}+\frac{2 \left (a^2+b^2\right ) f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) a^3}{b^6 d^3}+\frac{f (e+f x) \cosh (c+d x) \sinh (c+d x) a^3}{2 b^4 d^2}-\frac{2 f (e+f x) \cosh ^3(c+d x) a^2}{9 b^3 d^2}+\frac{2 f^2 \sinh ^3(c+d x) a^2}{27 b^3 d^3}-\frac{4 f (e+f x) \cosh (c+d x) a^2}{3 b^3 d^2}+\frac{14 f^2 \sinh (c+d x) a^2}{9 b^3 d^3}+\frac{2 (e+f x)^2 \sinh (c+d x) a^2}{3 b^3 d}+\frac{(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x) a^2}{3 b^3 d}-\frac{f^2 \cosh ^4(c+d x) a}{32 b^2 d^3}-\frac{(e+f x)^2 \cosh ^4(c+d x) a}{4 b^2 d}+\frac{3 f^2 x^2 a}{32 b^2 d}-\frac{3 f^2 \cosh ^2(c+d x) a}{32 b^2 d^3}+\frac{3 e f x a}{16 b^2 d}+\frac{f (e+f x) \cosh ^3(c+d x) \sinh (c+d x) a}{8 b^2 d^2}+\frac{3 f (e+f x) \cosh (c+d x) \sinh (c+d x) a}{16 b^2 d^2}+\frac{f (e+f x) \cosh (c+d x)}{4 b d^2}-\frac{f (e+f x) \cosh (3 c+3 d x)}{72 b d^2}-\frac{f (e+f x) \cosh (5 c+5 d x)}{200 b d^2}-\frac{f^2 \sinh (c+d x)}{4 b d^3}-\frac{(e+f x)^2 \sinh (c+d x)}{8 b d}+\frac{f^2 \sinh (3 c+3 d x)}{216 b d^3}+\frac{(e+f x)^2 \sinh (3 c+3 d x)}{48 b d}+\frac{f^2 \sinh (5 c+5 d x)}{1000 b d^3}+\frac{(e+f x)^2 \sinh (5 c+5 d x)}{80 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^3*e*f*x)/(2*b^4*d) + (3*a*e*f*x)/(16*b^2*d) - (a^3*f^2*x^2)/(4*b^4*d) + (3*a*f^2*x^2)/(32*b^2*d) + (a^3*(a
^2 + b^2)*(e + f*x)^3)/(3*b^6*f) - (2*a^4*f*(e + f*x)*Cosh[c + d*x])/(b^5*d^2) - (4*a^2*f*(e + f*x)*Cosh[c + d
*x])/(3*b^3*d^2) + (f*(e + f*x)*Cosh[c + d*x])/(4*b*d^2) - (3*a*f^2*Cosh[c + d*x]^2)/(32*b^2*d^3) - (2*a^2*f*(
e + f*x)*Cosh[c + d*x]^3)/(9*b^3*d^2) - (a*f^2*Cosh[c + d*x]^4)/(32*b^2*d^3) - (a*(e + f*x)^2*Cosh[c + d*x]^4)
/(4*b^2*d) - (f*(e + f*x)*Cosh[3*c + 3*d*x])/(72*b*d^2) - (f*(e + f*x)*Cosh[5*c + 5*d*x])/(200*b*d^2) - (a^3*(
a^2 + b^2)*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^6*d) - (a^3*(a^2 + b^2)*(e + f*x)^2*
Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^6*d) - (2*a^3*(a^2 + b^2)*f*(e + f*x)*PolyLog[2, -((b*E^(c
+ d*x))/(a - Sqrt[a^2 + b^2]))])/(b^6*d^2) - (2*a^3*(a^2 + b^2)*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a +
Sqrt[a^2 + b^2]))])/(b^6*d^2) + (2*a^3*(a^2 + b^2)*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(
b^6*d^3) + (2*a^3*(a^2 + b^2)*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^6*d^3) + (2*a^4*f^2
*Sinh[c + d*x])/(b^5*d^3) + (14*a^2*f^2*Sinh[c + d*x])/(9*b^3*d^3) - (f^2*Sinh[c + d*x])/(4*b*d^3) + (a^4*(e +
f*x)^2*Sinh[c + d*x])/(b^5*d) + (2*a^2*(e + f*x)^2*Sinh[c + d*x])/(3*b^3*d) - ((e + f*x)^2*Sinh[c + d*x])/(8*
b*d) + (a^3*f*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(2*b^4*d^2) + (3*a*f*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x
])/(16*b^2*d^2) + (a^2*(e + f*x)^2*Cosh[c + d*x]^2*Sinh[c + d*x])/(3*b^3*d) + (a*f*(e + f*x)*Cosh[c + d*x]^3*S
inh[c + d*x])/(8*b^2*d^2) - (a^3*f^2*Sinh[c + d*x]^2)/(4*b^4*d^3) - (a^3*(e + f*x)^2*Sinh[c + d*x]^2)/(2*b^4*d
) + (2*a^2*f^2*Sinh[c + d*x]^3)/(27*b^3*d^3) + (f^2*Sinh[3*c + 3*d*x])/(216*b*d^3) + ((e + f*x)^2*Sinh[3*c + 3
*d*x])/(48*b*d) + (f^2*Sinh[5*c + 5*d*x])/(1000*b*d^3) + ((e + f*x)^2*Sinh[5*c + 5*d*x])/(80*b*d)

Rule 5579

Int[(Cosh[(c_.) + (d_.)*(x_)]^(p_.)*((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/b, Int[(e + f*x)^m*Cosh[c + d*x]^p*Sinh[c + d*x]^(n - 1), x], x]
- Dist[a/b, Int[((e + f*x)^m*Cosh[c + d*x]^p*Sinh[c + d*x]^(n - 1))/(a + b*Sinh[c + d*x]), x], x] /; FreeQ[{a
, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]

Rule 5448

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 5447

Int[Cosh[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[((c
+ d*x)^m*Cosh[a + b*x]^(n + 1))/(b*(n + 1)), x] - Dist[(d*m)/(b*(n + 1)), Int[(c + d*x)^(m - 1)*Cosh[a + b*x]^
(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]

Rule 3310

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

Rule 3311

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*m*(c + d*x)^(m - 1)*(
b*Sin[e + f*x])^n)/(f^2*n^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[(d^2*m*(m - 1))/(f^2*n^2), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[(b*(c + d*x)^m*Cos[e +
f*x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rule 5565

Int[(Cosh[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symb
ol] :> -Dist[a/b^2, Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2), x], x] + (Dist[1/b, Int[(e + f*x)^m*Cosh[c + d*x]^(
n - 2)*Sinh[c + d*x], x], x] + Dist[(a^2 + b^2)/b^2, Int[((e + f*x)^m*Cosh[c + d*x]^(n - 2))/(a + b*Sinh[c + d
*x]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0]

Rule 5446

Int[Cosh[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Simp[((c
+ d*x)^m*Sinh[a + b*x]^(n + 1))/(b*(n + 1)), x] - Dist[(d*m)/(b*(n + 1)), Int[(c + d*x)^(m - 1)*Sinh[a + b*x]^
(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]

Rule 5561

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :
> -Simp[(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[((e + f*x)^m*E^(c + d*x))/(a - Rt[a^2 + b^2, 2] + b*E^(c +
d*x)), x] + Int[((e + f*x)^m*E^(c + d*x))/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x)), x]) /; FreeQ[{a, b, c, d, e,
f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin{align*} \int \frac{(e+f x)^2 \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac{\int (e+f x)^2 \cosh ^3(c+d x) \sinh ^2(c+d x) \, dx}{b}-\frac{a \int \frac{(e+f x)^2 \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=-\frac{a \int (e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x) \, dx}{b^2}+\frac{a^2 \int \frac{(e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}+\frac{\int \left (-\frac{1}{8} (e+f x)^2 \cosh (c+d x)+\frac{1}{16} (e+f x)^2 \cosh (3 c+3 d x)+\frac{1}{16} (e+f x)^2 \cosh (5 c+5 d x)\right ) \, dx}{b}\\ &=-\frac{a (e+f x)^2 \cosh ^4(c+d x)}{4 b^2 d}+\frac{a^2 \int (e+f x)^2 \cosh ^3(c+d x) \, dx}{b^3}-\frac{a^3 \int \frac{(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx}{b^3}+\frac{\int (e+f x)^2 \cosh (3 c+3 d x) \, dx}{16 b}+\frac{\int (e+f x)^2 \cosh (5 c+5 d x) \, dx}{16 b}-\frac{\int (e+f x)^2 \cosh (c+d x) \, dx}{8 b}+\frac{(a f) \int (e+f x) \cosh ^4(c+d x) \, dx}{2 b^2 d}\\ &=-\frac{2 a^2 f (e+f x) \cosh ^3(c+d x)}{9 b^3 d^2}-\frac{a f^2 \cosh ^4(c+d x)}{32 b^2 d^3}-\frac{a (e+f x)^2 \cosh ^4(c+d x)}{4 b^2 d}-\frac{(e+f x)^2 \sinh (c+d x)}{8 b d}+\frac{a^2 (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^3 d}+\frac{a f (e+f x) \cosh ^3(c+d x) \sinh (c+d x)}{8 b^2 d^2}+\frac{(e+f x)^2 \sinh (3 c+3 d x)}{48 b d}+\frac{(e+f x)^2 \sinh (5 c+5 d x)}{80 b d}+\frac{a^4 \int (e+f x)^2 \cosh (c+d x) \, dx}{b^5}-\frac{a^3 \int (e+f x)^2 \cosh (c+d x) \sinh (c+d x) \, dx}{b^4}+\frac{\left (2 a^2\right ) \int (e+f x)^2 \cosh (c+d x) \, dx}{3 b^3}-\frac{\left (a^3 \left (a^2+b^2\right )\right ) \int \frac{(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^5}+\frac{(3 a f) \int (e+f x) \cosh ^2(c+d x) \, dx}{8 b^2 d}-\frac{f \int (e+f x) \sinh (5 c+5 d x) \, dx}{40 b d}-\frac{f \int (e+f x) \sinh (3 c+3 d x) \, dx}{24 b d}+\frac{f \int (e+f x) \sinh (c+d x) \, dx}{4 b d}+\frac{\left (2 a^2 f^2\right ) \int \cosh ^3(c+d x) \, dx}{9 b^3 d^2}\\ &=\frac{a^3 \left (a^2+b^2\right ) (e+f x)^3}{3 b^6 f}+\frac{f (e+f x) \cosh (c+d x)}{4 b d^2}-\frac{3 a f^2 \cosh ^2(c+d x)}{32 b^2 d^3}-\frac{2 a^2 f (e+f x) \cosh ^3(c+d x)}{9 b^3 d^2}-\frac{a f^2 \cosh ^4(c+d x)}{32 b^2 d^3}-\frac{a (e+f x)^2 \cosh ^4(c+d x)}{4 b^2 d}-\frac{f (e+f x) \cosh (3 c+3 d x)}{72 b d^2}-\frac{f (e+f x) \cosh (5 c+5 d x)}{200 b d^2}+\frac{a^4 (e+f x)^2 \sinh (c+d x)}{b^5 d}+\frac{2 a^2 (e+f x)^2 \sinh (c+d x)}{3 b^3 d}-\frac{(e+f x)^2 \sinh (c+d x)}{8 b d}+\frac{3 a f (e+f x) \cosh (c+d x) \sinh (c+d x)}{16 b^2 d^2}+\frac{a^2 (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^3 d}+\frac{a f (e+f x) \cosh ^3(c+d x) \sinh (c+d x)}{8 b^2 d^2}-\frac{a^3 (e+f x)^2 \sinh ^2(c+d x)}{2 b^4 d}+\frac{(e+f x)^2 \sinh (3 c+3 d x)}{48 b d}+\frac{(e+f x)^2 \sinh (5 c+5 d x)}{80 b d}-\frac{\left (a^3 \left (a^2+b^2\right )\right ) \int \frac{e^{c+d x} (e+f x)^2}{a-\sqrt{a^2+b^2}+b e^{c+d x}} \, dx}{b^5}-\frac{\left (a^3 \left (a^2+b^2\right )\right ) \int \frac{e^{c+d x} (e+f x)^2}{a+\sqrt{a^2+b^2}+b e^{c+d x}} \, dx}{b^5}-\frac{\left (2 a^4 f\right ) \int (e+f x) \sinh (c+d x) \, dx}{b^5 d}+\frac{\left (a^3 f\right ) \int (e+f x) \sinh ^2(c+d x) \, dx}{b^4 d}-\frac{\left (4 a^2 f\right ) \int (e+f x) \sinh (c+d x) \, dx}{3 b^3 d}+\frac{(3 a f) \int (e+f x) \, dx}{16 b^2 d}+\frac{\left (2 i a^2 f^2\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-i \sinh (c+d x)\right )}{9 b^3 d^3}+\frac{f^2 \int \cosh (5 c+5 d x) \, dx}{200 b d^2}+\frac{f^2 \int \cosh (3 c+3 d x) \, dx}{72 b d^2}-\frac{f^2 \int \cosh (c+d x) \, dx}{4 b d^2}\\ &=\frac{3 a e f x}{16 b^2 d}+\frac{3 a f^2 x^2}{32 b^2 d}+\frac{a^3 \left (a^2+b^2\right ) (e+f x)^3}{3 b^6 f}-\frac{2 a^4 f (e+f x) \cosh (c+d x)}{b^5 d^2}-\frac{4 a^2 f (e+f x) \cosh (c+d x)}{3 b^3 d^2}+\frac{f (e+f x) \cosh (c+d x)}{4 b d^2}-\frac{3 a f^2 \cosh ^2(c+d x)}{32 b^2 d^3}-\frac{2 a^2 f (e+f x) \cosh ^3(c+d x)}{9 b^3 d^2}-\frac{a f^2 \cosh ^4(c+d x)}{32 b^2 d^3}-\frac{a (e+f x)^2 \cosh ^4(c+d x)}{4 b^2 d}-\frac{f (e+f x) \cosh (3 c+3 d x)}{72 b d^2}-\frac{f (e+f x) \cosh (5 c+5 d x)}{200 b d^2}-\frac{a^3 \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^6 d}-\frac{a^3 \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^6 d}+\frac{2 a^2 f^2 \sinh (c+d x)}{9 b^3 d^3}-\frac{f^2 \sinh (c+d x)}{4 b d^3}+\frac{a^4 (e+f x)^2 \sinh (c+d x)}{b^5 d}+\frac{2 a^2 (e+f x)^2 \sinh (c+d x)}{3 b^3 d}-\frac{(e+f x)^2 \sinh (c+d x)}{8 b d}+\frac{a^3 f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^4 d^2}+\frac{3 a f (e+f x) \cosh (c+d x) \sinh (c+d x)}{16 b^2 d^2}+\frac{a^2 (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^3 d}+\frac{a f (e+f x) \cosh ^3(c+d x) \sinh (c+d x)}{8 b^2 d^2}-\frac{a^3 f^2 \sinh ^2(c+d x)}{4 b^4 d^3}-\frac{a^3 (e+f x)^2 \sinh ^2(c+d x)}{2 b^4 d}+\frac{2 a^2 f^2 \sinh ^3(c+d x)}{27 b^3 d^3}+\frac{f^2 \sinh (3 c+3 d x)}{216 b d^3}+\frac{(e+f x)^2 \sinh (3 c+3 d x)}{48 b d}+\frac{f^2 \sinh (5 c+5 d x)}{1000 b d^3}+\frac{(e+f x)^2 \sinh (5 c+5 d x)}{80 b d}-\frac{\left (a^3 f\right ) \int (e+f x) \, dx}{2 b^4 d}+\frac{\left (2 a^3 \left (a^2+b^2\right ) f\right ) \int (e+f x) \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) \, dx}{b^6 d}+\frac{\left (2 a^3 \left (a^2+b^2\right ) f\right ) \int (e+f x) \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) \, dx}{b^6 d}+\frac{\left (2 a^4 f^2\right ) \int \cosh (c+d x) \, dx}{b^5 d^2}+\frac{\left (4 a^2 f^2\right ) \int \cosh (c+d x) \, dx}{3 b^3 d^2}\\ &=-\frac{a^3 e f x}{2 b^4 d}+\frac{3 a e f x}{16 b^2 d}-\frac{a^3 f^2 x^2}{4 b^4 d}+\frac{3 a f^2 x^2}{32 b^2 d}+\frac{a^3 \left (a^2+b^2\right ) (e+f x)^3}{3 b^6 f}-\frac{2 a^4 f (e+f x) \cosh (c+d x)}{b^5 d^2}-\frac{4 a^2 f (e+f x) \cosh (c+d x)}{3 b^3 d^2}+\frac{f (e+f x) \cosh (c+d x)}{4 b d^2}-\frac{3 a f^2 \cosh ^2(c+d x)}{32 b^2 d^3}-\frac{2 a^2 f (e+f x) \cosh ^3(c+d x)}{9 b^3 d^2}-\frac{a f^2 \cosh ^4(c+d x)}{32 b^2 d^3}-\frac{a (e+f x)^2 \cosh ^4(c+d x)}{4 b^2 d}-\frac{f (e+f x) \cosh (3 c+3 d x)}{72 b d^2}-\frac{f (e+f x) \cosh (5 c+5 d x)}{200 b d^2}-\frac{a^3 \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^6 d}-\frac{a^3 \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^6 d}-\frac{2 a^3 \left (a^2+b^2\right ) f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^6 d^2}-\frac{2 a^3 \left (a^2+b^2\right ) f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^6 d^2}+\frac{2 a^4 f^2 \sinh (c+d x)}{b^5 d^3}+\frac{14 a^2 f^2 \sinh (c+d x)}{9 b^3 d^3}-\frac{f^2 \sinh (c+d x)}{4 b d^3}+\frac{a^4 (e+f x)^2 \sinh (c+d x)}{b^5 d}+\frac{2 a^2 (e+f x)^2 \sinh (c+d x)}{3 b^3 d}-\frac{(e+f x)^2 \sinh (c+d x)}{8 b d}+\frac{a^3 f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^4 d^2}+\frac{3 a f (e+f x) \cosh (c+d x) \sinh (c+d x)}{16 b^2 d^2}+\frac{a^2 (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^3 d}+\frac{a f (e+f x) \cosh ^3(c+d x) \sinh (c+d x)}{8 b^2 d^2}-\frac{a^3 f^2 \sinh ^2(c+d x)}{4 b^4 d^3}-\frac{a^3 (e+f x)^2 \sinh ^2(c+d x)}{2 b^4 d}+\frac{2 a^2 f^2 \sinh ^3(c+d x)}{27 b^3 d^3}+\frac{f^2 \sinh (3 c+3 d x)}{216 b d^3}+\frac{(e+f x)^2 \sinh (3 c+3 d x)}{48 b d}+\frac{f^2 \sinh (5 c+5 d x)}{1000 b d^3}+\frac{(e+f x)^2 \sinh (5 c+5 d x)}{80 b d}+\frac{\left (2 a^3 \left (a^2+b^2\right ) f^2\right ) \int \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) \, dx}{b^6 d^2}+\frac{\left (2 a^3 \left (a^2+b^2\right ) f^2\right ) \int \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) \, dx}{b^6 d^2}\\ &=-\frac{a^3 e f x}{2 b^4 d}+\frac{3 a e f x}{16 b^2 d}-\frac{a^3 f^2 x^2}{4 b^4 d}+\frac{3 a f^2 x^2}{32 b^2 d}+\frac{a^3 \left (a^2+b^2\right ) (e+f x)^3}{3 b^6 f}-\frac{2 a^4 f (e+f x) \cosh (c+d x)}{b^5 d^2}-\frac{4 a^2 f (e+f x) \cosh (c+d x)}{3 b^3 d^2}+\frac{f (e+f x) \cosh (c+d x)}{4 b d^2}-\frac{3 a f^2 \cosh ^2(c+d x)}{32 b^2 d^3}-\frac{2 a^2 f (e+f x) \cosh ^3(c+d x)}{9 b^3 d^2}-\frac{a f^2 \cosh ^4(c+d x)}{32 b^2 d^3}-\frac{a (e+f x)^2 \cosh ^4(c+d x)}{4 b^2 d}-\frac{f (e+f x) \cosh (3 c+3 d x)}{72 b d^2}-\frac{f (e+f x) \cosh (5 c+5 d x)}{200 b d^2}-\frac{a^3 \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^6 d}-\frac{a^3 \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^6 d}-\frac{2 a^3 \left (a^2+b^2\right ) f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^6 d^2}-\frac{2 a^3 \left (a^2+b^2\right ) f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^6 d^2}+\frac{2 a^4 f^2 \sinh (c+d x)}{b^5 d^3}+\frac{14 a^2 f^2 \sinh (c+d x)}{9 b^3 d^3}-\frac{f^2 \sinh (c+d x)}{4 b d^3}+\frac{a^4 (e+f x)^2 \sinh (c+d x)}{b^5 d}+\frac{2 a^2 (e+f x)^2 \sinh (c+d x)}{3 b^3 d}-\frac{(e+f x)^2 \sinh (c+d x)}{8 b d}+\frac{a^3 f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^4 d^2}+\frac{3 a f (e+f x) \cosh (c+d x) \sinh (c+d x)}{16 b^2 d^2}+\frac{a^2 (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^3 d}+\frac{a f (e+f x) \cosh ^3(c+d x) \sinh (c+d x)}{8 b^2 d^2}-\frac{a^3 f^2 \sinh ^2(c+d x)}{4 b^4 d^3}-\frac{a^3 (e+f x)^2 \sinh ^2(c+d x)}{2 b^4 d}+\frac{2 a^2 f^2 \sinh ^3(c+d x)}{27 b^3 d^3}+\frac{f^2 \sinh (3 c+3 d x)}{216 b d^3}+\frac{(e+f x)^2 \sinh (3 c+3 d x)}{48 b d}+\frac{f^2 \sinh (5 c+5 d x)}{1000 b d^3}+\frac{(e+f x)^2 \sinh (5 c+5 d x)}{80 b d}+\frac{\left (2 a^3 \left (a^2+b^2\right ) f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{b x}{-a+\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^6 d^3}+\frac{\left (2 a^3 \left (a^2+b^2\right ) f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{b x}{a+\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^6 d^3}\\ &=-\frac{a^3 e f x}{2 b^4 d}+\frac{3 a e f x}{16 b^2 d}-\frac{a^3 f^2 x^2}{4 b^4 d}+\frac{3 a f^2 x^2}{32 b^2 d}+\frac{a^3 \left (a^2+b^2\right ) (e+f x)^3}{3 b^6 f}-\frac{2 a^4 f (e+f x) \cosh (c+d x)}{b^5 d^2}-\frac{4 a^2 f (e+f x) \cosh (c+d x)}{3 b^3 d^2}+\frac{f (e+f x) \cosh (c+d x)}{4 b d^2}-\frac{3 a f^2 \cosh ^2(c+d x)}{32 b^2 d^3}-\frac{2 a^2 f (e+f x) \cosh ^3(c+d x)}{9 b^3 d^2}-\frac{a f^2 \cosh ^4(c+d x)}{32 b^2 d^3}-\frac{a (e+f x)^2 \cosh ^4(c+d x)}{4 b^2 d}-\frac{f (e+f x) \cosh (3 c+3 d x)}{72 b d^2}-\frac{f (e+f x) \cosh (5 c+5 d x)}{200 b d^2}-\frac{a^3 \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^6 d}-\frac{a^3 \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^6 d}-\frac{2 a^3 \left (a^2+b^2\right ) f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^6 d^2}-\frac{2 a^3 \left (a^2+b^2\right ) f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^6 d^2}+\frac{2 a^3 \left (a^2+b^2\right ) f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^6 d^3}+\frac{2 a^3 \left (a^2+b^2\right ) f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^6 d^3}+\frac{2 a^4 f^2 \sinh (c+d x)}{b^5 d^3}+\frac{14 a^2 f^2 \sinh (c+d x)}{9 b^3 d^3}-\frac{f^2 \sinh (c+d x)}{4 b d^3}+\frac{a^4 (e+f x)^2 \sinh (c+d x)}{b^5 d}+\frac{2 a^2 (e+f x)^2 \sinh (c+d x)}{3 b^3 d}-\frac{(e+f x)^2 \sinh (c+d x)}{8 b d}+\frac{a^3 f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^4 d^2}+\frac{3 a f (e+f x) \cosh (c+d x) \sinh (c+d x)}{16 b^2 d^2}+\frac{a^2 (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^3 d}+\frac{a f (e+f x) \cosh ^3(c+d x) \sinh (c+d x)}{8 b^2 d^2}-\frac{a^3 f^2 \sinh ^2(c+d x)}{4 b^4 d^3}-\frac{a^3 (e+f x)^2 \sinh ^2(c+d x)}{2 b^4 d}+\frac{2 a^2 f^2 \sinh ^3(c+d x)}{27 b^3 d^3}+\frac{f^2 \sinh (3 c+3 d x)}{216 b d^3}+\frac{(e+f x)^2 \sinh (3 c+3 d x)}{48 b d}+\frac{f^2 \sinh (5 c+5 d x)}{1000 b d^3}+\frac{(e+f x)^2 \sinh (5 c+5 d x)}{80 b d}\\ \end{align*}

Mathematica [B]  time = 14.9595, size = 3179, normalized size = 3.03 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((8*a^3*(a^2 + b^2)*(6*e^2*E^(2*c)*x + 6*e*E^(2*c)*f*x^2 + 2*E^(2*c)*f^2*x^3 + (6*a*Sqrt[a^2 + b^2]*e^2*ArcTan
[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]])/(Sqrt[-(a^2 + b^2)^2]*d) + (6*a*Sqrt[-(a^2 + b^2)^2]*e^2*E^(2*c)*ArcTa
n[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]])/((a^2 + b^2)^(3/2)*d) - (6*a*Sqrt[-(a^2 + b^2)^2]*e^2*ArcTanh[(a + b*
E^(c + d*x))/Sqrt[a^2 + b^2]])/((-a^2 - b^2)^(3/2)*d) + (6*a*Sqrt[-(a^2 + b^2)^2]*e^2*E^(2*c)*ArcTanh[(a + b*E
^(c + d*x))/Sqrt[a^2 + b^2]])/((-a^2 - b^2)^(3/2)*d) + (3*e^2*Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))])
/d - (3*e^2*E^(2*c)*Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))])/d + (6*e*f*x*Log[1 + (b*E^(2*c + d*x))/(a
*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*e*E^(2*c)*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E
^(2*c)])])/d + (3*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (3*E^(2*c)*f^2*x
^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (6*e*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E
^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*e*E^(2*c)*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(
2*c)])])/d + (3*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (3*E^(2*c)*f^2*x^2
*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*(-1 + E^(2*c))*f*(e + f*x)*PolyLog[2,
-((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^2 - (6*(-1 + E^(2*c))*f*(e + f*x)*PolyLog[2, -((b
*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^2 - (6*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sq
rt[(a^2 + b^2)*E^(2*c)]))])/d^3 + (6*E^(2*c)*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2
*c)]))])/d^3 - (6*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 + (6*E^(2*c)*f
^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3))/(3*b^6*(-1 + E^(2*c))) - (8*a^3
*(a^2 + b^2)*e^2*x*(1 + Cosh[2*c] + Sinh[2*c]))/(b^6*(-1 + Cosh[2*c] + Sinh[2*c])) - (8*a^3*(a^2 + b^2)*e*f*x^
2*(1 + Cosh[2*c] + Sinh[2*c]))/(b^6*(-1 + Cosh[2*c] + Sinh[2*c])) - (8*a^3*(a^2 + b^2)*f^2*x^3*(1 + Cosh[2*c]
+ Sinh[2*c]))/(3*b^6*(-1 + Cosh[2*c] + Sinh[2*c])) + ((-8*a^4 - 6*a^2*b^2 + b^4)*(d^2*e^2 + 2*d*e*f + 2*f^2)*(
Cosh[c]/(2*b^5*d^3) - Sinh[c]/(2*b^5*d^3)) + (8*a^4*d*e*f + 6*a^2*b^2*d*e*f - b^4*d*e*f + 8*a^4*f^2 + 6*a^2*b^
2*f^2 - b^4*f^2)*(-((x*Cosh[c])/(b^5*d^2)) + (x*Sinh[c])/(b^5*d^2)) + (-8*a^4 - 6*a^2*b^2 + b^4)*((f^2*x^2*Cos
h[c])/(2*b^5*d) - (f^2*x^2*Sinh[c])/(2*b^5*d)))*(Cosh[d*x] - Sinh[d*x]) + ((-8*a^4 - 6*a^2*b^2 + b^4)*(d^2*e^2
- 2*d*e*f + 2*f^2)*(-Cosh[c]/(2*b^5*d^3) - Sinh[c]/(2*b^5*d^3)) + (x*(8*a^4*d*e*f*Cosh[c] + 6*a^2*b^2*d*e*f*C
osh[c] - b^4*d*e*f*Cosh[c] - 8*a^4*f^2*Cosh[c] - 6*a^2*b^2*f^2*Cosh[c] + b^4*f^2*Cosh[c] + 8*a^4*d*e*f*Sinh[c]
+ 6*a^2*b^2*d*e*f*Sinh[c] - b^4*d*e*f*Sinh[c] - 8*a^4*f^2*Sinh[c] - 6*a^2*b^2*f^2*Sinh[c] + b^4*f^2*Sinh[c]))
/(b^5*d^2) + (-8*a^4 - 6*a^2*b^2 + b^4)*(-(f^2*x^2*Cosh[c])/(2*b^5*d) - (f^2*x^2*Sinh[c])/(2*b^5*d)))*(Cosh[d*
x] + Sinh[d*x]) + ((2*a^2 + b^2)*(2*d^2*e^2 + 2*d*e*f + f^2)*(-(a*Cosh[2*c])/(4*b^4*d^3) + (a*Sinh[2*c])/(4*b^
4*d^3)) + (4*a^3*d*e*f + 2*a*b^2*d*e*f + 2*a^3*f^2 + a*b^2*f^2)*(-(x*Cosh[2*c])/(2*b^4*d^2) + (x*Sinh[2*c])/(2
*b^4*d^2)) + (2*a^2 + b^2)*(-(a*f^2*x^2*Cosh[2*c])/(2*b^4*d) + (a*f^2*x^2*Sinh[2*c])/(2*b^4*d)))*(Cosh[2*d*x]
- Sinh[2*d*x]) + ((2*a^2 + b^2)*(2*d^2*e^2 - 2*d*e*f + f^2)*(-(a*Cosh[2*c])/(4*b^4*d^3) - (a*Sinh[2*c])/(4*b^4
*d^3)) + (x*(-4*a^3*d*e*f*Cosh[2*c] - 2*a*b^2*d*e*f*Cosh[2*c] + 2*a^3*f^2*Cosh[2*c] + a*b^2*f^2*Cosh[2*c] - 4*
a^3*d*e*f*Sinh[2*c] - 2*a*b^2*d*e*f*Sinh[2*c] + 2*a^3*f^2*Sinh[2*c] + a*b^2*f^2*Sinh[2*c]))/(2*b^4*d^2) + (2*a
^2 + b^2)*(-(a*f^2*x^2*Cosh[2*c])/(2*b^4*d) - (a*f^2*x^2*Sinh[2*c])/(2*b^4*d)))*(Cosh[2*d*x] + Sinh[2*d*x]) +
((4*a^2 + b^2)*(9*d^2*e^2 + 6*d*e*f + 2*f^2)*(-Cosh[3*c]/(108*b^3*d^3) + Sinh[3*c]/(108*b^3*d^3)) + (12*a^2*d*
e*f + 3*b^2*d*e*f + 4*a^2*f^2 + b^2*f^2)*(-(x*Cosh[3*c])/(18*b^3*d^2) + (x*Sinh[3*c])/(18*b^3*d^2)) + (4*a^2 +
b^2)*(-(f^2*x^2*Cosh[3*c])/(12*b^3*d) + (f^2*x^2*Sinh[3*c])/(12*b^3*d)))*(Cosh[3*d*x] - Sinh[3*d*x]) + ((4*a^
2 + b^2)*(9*d^2*e^2 - 6*d*e*f + 2*f^2)*(Cosh[3*c]/(108*b^3*d^3) + Sinh[3*c]/(108*b^3*d^3)) + (x*(12*a^2*d*e*f*
Cosh[3*c] + 3*b^2*d*e*f*Cosh[3*c] - 4*a^2*f^2*Cosh[3*c] - b^2*f^2*Cosh[3*c] + 12*a^2*d*e*f*Sinh[3*c] + 3*b^2*d
*e*f*Sinh[3*c] - 4*a^2*f^2*Sinh[3*c] - b^2*f^2*Sinh[3*c]))/(18*b^3*d^2) + (4*a^2 + b^2)*((f^2*x^2*Cosh[3*c])/(
12*b^3*d) + (f^2*x^2*Sinh[3*c])/(12*b^3*d)))*(Cosh[3*d*x] + Sinh[3*d*x]) + (-(a*f^2*x^2*Cosh[4*c])/(8*b^2*d) +
(a*f^2*x^2*Sinh[4*c])/(8*b^2*d) + (8*d^2*e^2 + 4*d*e*f + f^2)*(-(a*Cosh[4*c])/(64*b^2*d^3) + (a*Sinh[4*c])/(6
4*b^2*d^3)) + (4*a*d*e*f + a*f^2)*(-(x*Cosh[4*c])/(16*b^2*d^2) + (x*Sinh[4*c])/(16*b^2*d^2)))*(Cosh[4*d*x] - S
inh[4*d*x]) + (-(a*f^2*x^2*Cosh[4*c])/(8*b^2*d) - (a*f^2*x^2*Sinh[4*c])/(8*b^2*d) + (8*d^2*e^2 - 4*d*e*f + f^2
)*(-(a*Cosh[4*c])/(64*b^2*d^3) - (a*Sinh[4*c])/(64*b^2*d^3)) + (x*(-4*a*d*e*f*Cosh[4*c] + a*f^2*Cosh[4*c] - 4*
a*d*e*f*Sinh[4*c] + a*f^2*Sinh[4*c]))/(16*b^2*d^2))*(Cosh[4*d*x] + Sinh[4*d*x]) + (-(f^2*x^2*Cosh[5*c])/(20*b*
d) + (f^2*x^2*Sinh[5*c])/(20*b*d) + (25*d^2*e^2 + 10*d*e*f + 2*f^2)*(-Cosh[5*c]/(500*b*d^3) + Sinh[5*c]/(500*b
*d^3)) + (5*d*e*f + f^2)*(-(x*Cosh[5*c])/(50*b*d^2) + (x*Sinh[5*c])/(50*b*d^2)))*(Cosh[5*d*x] - Sinh[5*d*x]) +
((f^2*x^2*Cosh[5*c])/(20*b*d) + (f^2*x^2*Sinh[5*c])/(20*b*d) + (25*d^2*e^2 - 10*d*e*f + 2*f^2)*(Cosh[5*c]/(50
0*b*d^3) + Sinh[5*c]/(500*b*d^3)) + (x*(5*d*e*f*Cosh[5*c] - f^2*Cosh[5*c] + 5*d*e*f*Sinh[5*c] - f^2*Sinh[5*c])
)/(50*b*d^2))*(Cosh[5*d*x] + Sinh[5*d*x]))/8

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Maple [F]  time = 0.211, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{2} \left ( \cosh \left ( dx+c \right ) \right ) ^{3} \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{a+b\sinh \left ( dx+c \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/960*e^2*((15*a*b^3*e^(-d*x - c) - 6*b^4 - 10*(4*a^2*b^2 + b^4)*e^(-2*d*x - 2*c) + 60*(2*a^3*b + a*b^3)*e^(-
3*d*x - 3*c) - 60*(8*a^4 + 6*a^2*b^2 - b^4)*e^(-4*d*x - 4*c))*e^(5*d*x + 5*c)/(b^5*d) + 960*(a^5 + a^3*b^2)*(d
*x + c)/(b^6*d) + (15*a*b^3*e^(-4*d*x - 4*c) + 6*b^4*e^(-5*d*x - 5*c) + 60*(8*a^4 + 6*a^2*b^2 - b^4)*e^(-d*x -
c) + 60*(2*a^3*b + a*b^3)*e^(-2*d*x - 2*c) + 10*(4*a^2*b^2 + b^4)*e^(-3*d*x - 3*c))/(b^5*d) + 960*(a^5 + a^3*
b^2)*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(b^6*d)) - 1/1728000*(576000*(a^5*d^3*f^2*e^(5*c) + a^3*b
^2*d^3*f^2*e^(5*c))*x^3 + 1728000*(a^5*d^3*e*f*e^(5*c) + a^3*b^2*d^3*e*f*e^(5*c))*x^2 - 432*(25*b^5*d^2*f^2*x^
2*e^(10*c) + 10*(5*d^2*e*f - d*f^2)*b^5*x*e^(10*c) - 2*(5*d*e*f - f^2)*b^5*e^(10*c))*e^(5*d*x) + 3375*(8*a*b^4
*d^2*f^2*x^2*e^(9*c) + 4*(4*d^2*e*f - d*f^2)*a*b^4*x*e^(9*c) - (4*d*e*f - f^2)*a*b^4*e^(9*c))*e^(4*d*x) + 2000
*(8*(3*d*e*f - f^2)*a^2*b^3*e^(8*c) + 2*(3*d*e*f - f^2)*b^5*e^(8*c) - 9*(4*a^2*b^3*d^2*f^2*e^(8*c) + b^5*d^2*f
^2*e^(8*c))*x^2 - 6*(4*(3*d^2*e*f - d*f^2)*a^2*b^3*e^(8*c) + (3*d^2*e*f - d*f^2)*b^5*e^(8*c))*x)*e^(3*d*x) - 5
4000*(2*(2*d*e*f - f^2)*a^3*b^2*e^(7*c) + (2*d*e*f - f^2)*a*b^4*e^(7*c) - 2*(2*a^3*b^2*d^2*f^2*e^(7*c) + a*b^4
*d^2*f^2*e^(7*c))*x^2 - 2*(2*(2*d^2*e*f - d*f^2)*a^3*b^2*e^(7*c) + (2*d^2*e*f - d*f^2)*a*b^4*e^(7*c))*x)*e^(2*
d*x) + 108000*(16*(d*e*f - f^2)*a^4*b*e^(6*c) + 12*(d*e*f - f^2)*a^2*b^3*e^(6*c) - 2*(d*e*f - f^2)*b^5*e^(6*c)
- (8*a^4*b*d^2*f^2*e^(6*c) + 6*a^2*b^3*d^2*f^2*e^(6*c) - b^5*d^2*f^2*e^(6*c))*x^2 - 2*(8*(d^2*e*f - d*f^2)*a^
4*b*e^(6*c) + 6*(d^2*e*f - d*f^2)*a^2*b^3*e^(6*c) - (d^2*e*f - d*f^2)*b^5*e^(6*c))*x)*e^(d*x) + 108000*(16*(d*
e*f + f^2)*a^4*b*e^(4*c) + 12*(d*e*f + f^2)*a^2*b^3*e^(4*c) - 2*(d*e*f + f^2)*b^5*e^(4*c) + (8*a^4*b*d^2*f^2*e
^(4*c) + 6*a^2*b^3*d^2*f^2*e^(4*c) - b^5*d^2*f^2*e^(4*c))*x^2 + 2*(8*(d^2*e*f + d*f^2)*a^4*b*e^(4*c) + 6*(d^2*
e*f + d*f^2)*a^2*b^3*e^(4*c) - (d^2*e*f + d*f^2)*b^5*e^(4*c))*x)*e^(-d*x) + 54000*(2*(2*d*e*f + f^2)*a^3*b^2*e
^(3*c) + (2*d*e*f + f^2)*a*b^4*e^(3*c) + 2*(2*a^3*b^2*d^2*f^2*e^(3*c) + a*b^4*d^2*f^2*e^(3*c))*x^2 + 2*(2*(2*d
^2*e*f + d*f^2)*a^3*b^2*e^(3*c) + (2*d^2*e*f + d*f^2)*a*b^4*e^(3*c))*x)*e^(-2*d*x) + 2000*(8*(3*d*e*f + f^2)*a
^2*b^3*e^(2*c) + 2*(3*d*e*f + f^2)*b^5*e^(2*c) + 9*(4*a^2*b^3*d^2*f^2*e^(2*c) + b^5*d^2*f^2*e^(2*c))*x^2 + 6*(
4*(3*d^2*e*f + d*f^2)*a^2*b^3*e^(2*c) + (3*d^2*e*f + d*f^2)*b^5*e^(2*c))*x)*e^(-3*d*x) + 3375*(8*a*b^4*d^2*f^2
*x^2*e^c + 4*(4*d^2*e*f + d*f^2)*a*b^4*x*e^c + (4*d*e*f + f^2)*a*b^4*e^c)*e^(-4*d*x) + 432*(25*b^5*d^2*f^2*x^2
+ 10*(5*d^2*e*f + d*f^2)*b^5*x + 2*(5*d*e*f + f^2)*b^5)*e^(-5*d*x))*e^(-5*c)/(b^6*d^3) + integrate(-2*((a^5*b
*f^2 + a^3*b^3*f^2)*x^2 + 2*(a^5*b*e*f + a^3*b^3*e*f)*x - ((a^6*f^2*e^c + a^4*b^2*f^2*e^c)*x^2 + 2*(a^6*e*f*e^
c + a^4*b^2*e*f*e^c)*x)*e^(d*x))/(b^7*e^(2*d*x + 2*c) + 2*a*b^6*e^(d*x + c) - b^7), x)

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Fricas [C]  time = 4.60862, size = 24563, normalized size = 23.42 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1728000*(10800*b^5*d^2*f^2*x^2 - 432*(25*b^5*d^2*f^2*x^2 + 25*b^5*d^2*e^2 - 10*b^5*d*e*f + 2*b^5*f^2 + 10*(
5*b^5*d^2*e*f - b^5*d*f^2)*x)*cosh(d*x + c)^10 - 432*(25*b^5*d^2*f^2*x^2 + 25*b^5*d^2*e^2 - 10*b^5*d*e*f + 2*b
^5*f^2 + 10*(5*b^5*d^2*e*f - b^5*d*f^2)*x)*sinh(d*x + c)^10 + 3375*(8*a*b^4*d^2*f^2*x^2 + 8*a*b^4*d^2*e^2 - 4*
a*b^4*d*e*f + a*b^4*f^2 + 4*(4*a*b^4*d^2*e*f - a*b^4*d*f^2)*x)*cosh(d*x + c)^9 + 135*(200*a*b^4*d^2*f^2*x^2 +
200*a*b^4*d^2*e^2 - 100*a*b^4*d*e*f + 25*a*b^4*f^2 + 100*(4*a*b^4*d^2*e*f - a*b^4*d*f^2)*x - 32*(25*b^5*d^2*f^
2*x^2 + 25*b^5*d^2*e^2 - 10*b^5*d*e*f + 2*b^5*f^2 + 10*(5*b^5*d^2*e*f - b^5*d*f^2)*x)*cosh(d*x + c))*sinh(d*x
+ c)^9 + 10800*b^5*d^2*e^2 - 2000*(9*(4*a^2*b^3 + b^5)*d^2*f^2*x^2 + 9*(4*a^2*b^3 + b^5)*d^2*e^2 - 6*(4*a^2*b^
3 + b^5)*d*e*f + 2*(4*a^2*b^3 + b^5)*f^2 + 6*(3*(4*a^2*b^3 + b^5)*d^2*e*f - (4*a^2*b^3 + b^5)*d*f^2)*x)*cosh(d
*x + c)^8 - 5*(3600*(4*a^2*b^3 + b^5)*d^2*f^2*x^2 + 3600*(4*a^2*b^3 + b^5)*d^2*e^2 - 2400*(4*a^2*b^3 + b^5)*d*
e*f + 800*(4*a^2*b^3 + b^5)*f^2 + 3888*(25*b^5*d^2*f^2*x^2 + 25*b^5*d^2*e^2 - 10*b^5*d*e*f + 2*b^5*f^2 + 10*(5
*b^5*d^2*e*f - b^5*d*f^2)*x)*cosh(d*x + c)^2 + 2400*(3*(4*a^2*b^3 + b^5)*d^2*e*f - (4*a^2*b^3 + b^5)*d*f^2)*x
- 6075*(8*a*b^4*d^2*f^2*x^2 + 8*a*b^4*d^2*e^2 - 4*a*b^4*d*e*f + a*b^4*f^2 + 4*(4*a*b^4*d^2*e*f - a*b^4*d*f^2)*
x)*cosh(d*x + c))*sinh(d*x + c)^8 + 4320*b^5*d*e*f + 54000*(2*(2*a^3*b^2 + a*b^4)*d^2*f^2*x^2 + 2*(2*a^3*b^2 +
a*b^4)*d^2*e^2 - 2*(2*a^3*b^2 + a*b^4)*d*e*f + (2*a^3*b^2 + a*b^4)*f^2 + 2*(2*(2*a^3*b^2 + a*b^4)*d^2*e*f - (
2*a^3*b^2 + a*b^4)*d*f^2)*x)*cosh(d*x + c)^7 + 20*(5400*(2*a^3*b^2 + a*b^4)*d^2*f^2*x^2 + 5400*(2*a^3*b^2 + a*
b^4)*d^2*e^2 - 5400*(2*a^3*b^2 + a*b^4)*d*e*f - 2592*(25*b^5*d^2*f^2*x^2 + 25*b^5*d^2*e^2 - 10*b^5*d*e*f + 2*b
^5*f^2 + 10*(5*b^5*d^2*e*f - b^5*d*f^2)*x)*cosh(d*x + c)^3 + 2700*(2*a^3*b^2 + a*b^4)*f^2 + 6075*(8*a*b^4*d^2*
f^2*x^2 + 8*a*b^4*d^2*e^2 - 4*a*b^4*d*e*f + a*b^4*f^2 + 4*(4*a*b^4*d^2*e*f - a*b^4*d*f^2)*x)*cosh(d*x + c)^2 +
5400*(2*(2*a^3*b^2 + a*b^4)*d^2*e*f - (2*a^3*b^2 + a*b^4)*d*f^2)*x - 800*(9*(4*a^2*b^3 + b^5)*d^2*f^2*x^2 + 9
*(4*a^2*b^3 + b^5)*d^2*e^2 - 6*(4*a^2*b^3 + b^5)*d*e*f + 2*(4*a^2*b^3 + b^5)*f^2 + 6*(3*(4*a^2*b^3 + b^5)*d^2*
e*f - (4*a^2*b^3 + b^5)*d*f^2)*x)*cosh(d*x + c))*sinh(d*x + c)^7 + 864*b^5*f^2 - 108000*((8*a^4*b + 6*a^2*b^3
- b^5)*d^2*f^2*x^2 + (8*a^4*b + 6*a^2*b^3 - b^5)*d^2*e^2 - 2*(8*a^4*b + 6*a^2*b^3 - b^5)*d*e*f + 2*(8*a^4*b +
6*a^2*b^3 - b^5)*f^2 + 2*((8*a^4*b + 6*a^2*b^3 - b^5)*d^2*e*f - (8*a^4*b + 6*a^2*b^3 - b^5)*d*f^2)*x)*cosh(d*x
+ c)^6 - 20*(5400*(8*a^4*b + 6*a^2*b^3 - b^5)*d^2*f^2*x^2 + 5400*(8*a^4*b + 6*a^2*b^3 - b^5)*d^2*e^2 + 4536*(
25*b^5*d^2*f^2*x^2 + 25*b^5*d^2*e^2 - 10*b^5*d*e*f + 2*b^5*f^2 + 10*(5*b^5*d^2*e*f - b^5*d*f^2)*x)*cosh(d*x +
c)^4 - 10800*(8*a^4*b + 6*a^2*b^3 - b^5)*d*e*f - 14175*(8*a*b^4*d^2*f^2*x^2 + 8*a*b^4*d^2*e^2 - 4*a*b^4*d*e*f
+ a*b^4*f^2 + 4*(4*a*b^4*d^2*e*f - a*b^4*d*f^2)*x)*cosh(d*x + c)^3 + 10800*(8*a^4*b + 6*a^2*b^3 - b^5)*f^2 + 2
800*(9*(4*a^2*b^3 + b^5)*d^2*f^2*x^2 + 9*(4*a^2*b^3 + b^5)*d^2*e^2 - 6*(4*a^2*b^3 + b^5)*d*e*f + 2*(4*a^2*b^3
+ b^5)*f^2 + 6*(3*(4*a^2*b^3 + b^5)*d^2*e*f - (4*a^2*b^3 + b^5)*d*f^2)*x)*cosh(d*x + c)^2 + 10800*((8*a^4*b +
6*a^2*b^3 - b^5)*d^2*e*f - (8*a^4*b + 6*a^2*b^3 - b^5)*d*f^2)*x - 18900*(2*(2*a^3*b^2 + a*b^4)*d^2*f^2*x^2 + 2
*(2*a^3*b^2 + a*b^4)*d^2*e^2 - 2*(2*a^3*b^2 + a*b^4)*d*e*f + (2*a^3*b^2 + a*b^4)*f^2 + 2*(2*(2*a^3*b^2 + a*b^4
)*d^2*e*f - (2*a^3*b^2 + a*b^4)*d*f^2)*x)*cosh(d*x + c))*sinh(d*x + c)^6 - 576000*((a^5 + a^3*b^2)*d^3*f^2*x^3
+ 3*(a^5 + a^3*b^2)*d^3*e*f*x^2 + 3*(a^5 + a^3*b^2)*d^3*e^2*x + 6*(a^5 + a^3*b^2)*c*d^2*e^2 - 6*(a^5 + a^3*b^
2)*c^2*d*e*f + 2*(a^5 + a^3*b^2)*c^3*f^2)*cosh(d*x + c)^5 - 2*(288000*(a^5 + a^3*b^2)*d^3*f^2*x^3 + 864000*(a^
5 + a^3*b^2)*d^3*e*f*x^2 + 864000*(a^5 + a^3*b^2)*d^3*e^2*x + 1728000*(a^5 + a^3*b^2)*c*d^2*e^2 - 1728000*(a^5
+ a^3*b^2)*c^2*d*e*f + 576000*(a^5 + a^3*b^2)*c^3*f^2 + 54432*(25*b^5*d^2*f^2*x^2 + 25*b^5*d^2*e^2 - 10*b^5*d
*e*f + 2*b^5*f^2 + 10*(5*b^5*d^2*e*f - b^5*d*f^2)*x)*cosh(d*x + c)^5 - 212625*(8*a*b^4*d^2*f^2*x^2 + 8*a*b^4*d
^2*e^2 - 4*a*b^4*d*e*f + a*b^4*f^2 + 4*(4*a*b^4*d^2*e*f - a*b^4*d*f^2)*x)*cosh(d*x + c)^4 + 56000*(9*(4*a^2*b^
3 + b^5)*d^2*f^2*x^2 + 9*(4*a^2*b^3 + b^5)*d^2*e^2 - 6*(4*a^2*b^3 + b^5)*d*e*f + 2*(4*a^2*b^3 + b^5)*f^2 + 6*(
3*(4*a^2*b^3 + b^5)*d^2*e*f - (4*a^2*b^3 + b^5)*d*f^2)*x)*cosh(d*x + c)^3 - 567000*(2*(2*a^3*b^2 + a*b^4)*d^2*
f^2*x^2 + 2*(2*a^3*b^2 + a*b^4)*d^2*e^2 - 2*(2*a^3*b^2 + a*b^4)*d*e*f + (2*a^3*b^2 + a*b^4)*f^2 + 2*(2*(2*a^3*
b^2 + a*b^4)*d^2*e*f - (2*a^3*b^2 + a*b^4)*d*f^2)*x)*cosh(d*x + c)^2 + 324000*((8*a^4*b + 6*a^2*b^3 - b^5)*d^2
*f^2*x^2 + (8*a^4*b + 6*a^2*b^3 - b^5)*d^2*e^2 - 2*(8*a^4*b + 6*a^2*b^3 - b^5)*d*e*f + 2*(8*a^4*b + 6*a^2*b^3
- b^5)*f^2 + 2*((8*a^4*b + 6*a^2*b^3 - b^5)*d^2*e*f - (8*a^4*b + 6*a^2*b^3 - b^5)*d*f^2)*x)*cosh(d*x + c))*sin
h(d*x + c)^5 + 108000*((8*a^4*b + 6*a^2*b^3 - b^5)*d^2*f^2*x^2 + (8*a^4*b + 6*a^2*b^3 - b^5)*d^2*e^2 + 2*(8*a^
4*b + 6*a^2*b^3 - b^5)*d*e*f + 2*(8*a^4*b + 6*a^2*b^3 - b^5)*f^2 + 2*((8*a^4*b + 6*a^2*b^3 - b^5)*d^2*e*f + (8
*a^4*b + 6*a^2*b^3 - b^5)*d*f^2)*x)*cosh(d*x + c)^4 + 10*(10800*(8*a^4*b + 6*a^2*b^3 - b^5)*d^2*f^2*x^2 - 9072
*(25*b^5*d^2*f^2*x^2 + 25*b^5*d^2*e^2 - 10*b^5*d*e*f + 2*b^5*f^2 + 10*(5*b^5*d^2*e*f - b^5*d*f^2)*x)*cosh(d*x
+ c)^6 + 42525*(8*a*b^4*d^2*f^2*x^2 + 8*a*b^4*d^2*e^2 - 4*a*b^4*d*e*f + a*b^4*f^2 + 4*(4*a*b^4*d^2*e*f - a*b^4
*d*f^2)*x)*cosh(d*x + c)^5 + 10800*(8*a^4*b + 6*a^2*b^3 - b^5)*d^2*e^2 - 14000*(9*(4*a^2*b^3 + b^5)*d^2*f^2*x^
2 + 9*(4*a^2*b^3 + b^5)*d^2*e^2 - 6*(4*a^2*b^3 + b^5)*d*e*f + 2*(4*a^2*b^3 + b^5)*f^2 + 6*(3*(4*a^2*b^3 + b^5)
*d^2*e*f - (4*a^2*b^3 + b^5)*d*f^2)*x)*cosh(d*x + c)^4 + 21600*(8*a^4*b + 6*a^2*b^3 - b^5)*d*e*f + 189000*(2*(
2*a^3*b^2 + a*b^4)*d^2*f^2*x^2 + 2*(2*a^3*b^2 + a*b^4)*d^2*e^2 - 2*(2*a^3*b^2 + a*b^4)*d*e*f + (2*a^3*b^2 + a*
b^4)*f^2 + 2*(2*(2*a^3*b^2 + a*b^4)*d^2*e*f - (2*a^3*b^2 + a*b^4)*d*f^2)*x)*cosh(d*x + c)^3 + 21600*(8*a^4*b +
6*a^2*b^3 - b^5)*f^2 - 162000*((8*a^4*b + 6*a^2*b^3 - b^5)*d^2*f^2*x^2 + (8*a^4*b + 6*a^2*b^3 - b^5)*d^2*e^2
- 2*(8*a^4*b + 6*a^2*b^3 - b^5)*d*e*f + 2*(8*a^4*b + 6*a^2*b^3 - b^5)*f^2 + 2*((8*a^4*b + 6*a^2*b^3 - b^5)*d^2
*e*f - (8*a^4*b + 6*a^2*b^3 - b^5)*d*f^2)*x)*cosh(d*x + c)^2 + 21600*((8*a^4*b + 6*a^2*b^3 - b^5)*d^2*e*f + (8
*a^4*b + 6*a^2*b^3 - b^5)*d*f^2)*x - 288000*((a^5 + a^3*b^2)*d^3*f^2*x^3 + 3*(a^5 + a^3*b^2)*d^3*e*f*x^2 + 3*(
a^5 + a^3*b^2)*d^3*e^2*x + 6*(a^5 + a^3*b^2)*c*d^2*e^2 - 6*(a^5 + a^3*b^2)*c^2*d*e*f + 2*(a^5 + a^3*b^2)*c^3*f
^2)*cosh(d*x + c))*sinh(d*x + c)^4 + 54000*(2*(2*a^3*b^2 + a*b^4)*d^2*f^2*x^2 + 2*(2*a^3*b^2 + a*b^4)*d^2*e^2
+ 2*(2*a^3*b^2 + a*b^4)*d*e*f + (2*a^3*b^2 + a*b^4)*f^2 + 2*(2*(2*a^3*b^2 + a*b^4)*d^2*e*f + (2*a^3*b^2 + a*b^
4)*d*f^2)*x)*cosh(d*x + c)^3 - 20*(2592*(25*b^5*d^2*f^2*x^2 + 25*b^5*d^2*e^2 - 10*b^5*d*e*f + 2*b^5*f^2 + 10*(
5*b^5*d^2*e*f - b^5*d*f^2)*x)*cosh(d*x + c)^7 - 5400*(2*a^3*b^2 + a*b^4)*d^2*f^2*x^2 - 14175*(8*a*b^4*d^2*f^2*
x^2 + 8*a*b^4*d^2*e^2 - 4*a*b^4*d*e*f + a*b^4*f^2 + 4*(4*a*b^4*d^2*e*f - a*b^4*d*f^2)*x)*cosh(d*x + c)^6 + 560
0*(9*(4*a^2*b^3 + b^5)*d^2*f^2*x^2 + 9*(4*a^2*b^3 + b^5)*d^2*e^2 - 6*(4*a^2*b^3 + b^5)*d*e*f + 2*(4*a^2*b^3 +
b^5)*f^2 + 6*(3*(4*a^2*b^3 + b^5)*d^2*e*f - (4*a^2*b^3 + b^5)*d*f^2)*x)*cosh(d*x + c)^5 - 5400*(2*a^3*b^2 + a*
b^4)*d^2*e^2 - 94500*(2*(2*a^3*b^2 + a*b^4)*d^2*f^2*x^2 + 2*(2*a^3*b^2 + a*b^4)*d^2*e^2 - 2*(2*a^3*b^2 + a*b^4
)*d*e*f + (2*a^3*b^2 + a*b^4)*f^2 + 2*(2*(2*a^3*b^2 + a*b^4)*d^2*e*f - (2*a^3*b^2 + a*b^4)*d*f^2)*x)*cosh(d*x
+ c)^4 - 5400*(2*a^3*b^2 + a*b^4)*d*e*f + 108000*((8*a^4*b + 6*a^2*b^3 - b^5)*d^2*f^2*x^2 + (8*a^4*b + 6*a^2*b
^3 - b^5)*d^2*e^2 - 2*(8*a^4*b + 6*a^2*b^3 - b^5)*d*e*f + 2*(8*a^4*b + 6*a^2*b^3 - b^5)*f^2 + 2*((8*a^4*b + 6*
a^2*b^3 - b^5)*d^2*e*f - (8*a^4*b + 6*a^2*b^3 - b^5)*d*f^2)*x)*cosh(d*x + c)^3 - 2700*(2*a^3*b^2 + a*b^4)*f^2
+ 288000*((a^5 + a^3*b^2)*d^3*f^2*x^3 + 3*(a^5 + a^3*b^2)*d^3*e*f*x^2 + 3*(a^5 + a^3*b^2)*d^3*e^2*x + 6*(a^5 +
a^3*b^2)*c*d^2*e^2 - 6*(a^5 + a^3*b^2)*c^2*d*e*f + 2*(a^5 + a^3*b^2)*c^3*f^2)*cosh(d*x + c)^2 - 5400*(2*(2*a^
3*b^2 + a*b^4)*d^2*e*f + (2*a^3*b^2 + a*b^4)*d*f^2)*x - 21600*((8*a^4*b + 6*a^2*b^3 - b^5)*d^2*f^2*x^2 + (8*a^
4*b + 6*a^2*b^3 - b^5)*d^2*e^2 + 2*(8*a^4*b + 6*a^2*b^3 - b^5)*d*e*f + 2*(8*a^4*b + 6*a^2*b^3 - b^5)*f^2 + 2*(
(8*a^4*b + 6*a^2*b^3 - b^5)*d^2*e*f + (8*a^4*b + 6*a^2*b^3 - b^5)*d*f^2)*x)*cosh(d*x + c))*sinh(d*x + c)^3 + 2
000*(9*(4*a^2*b^3 + b^5)*d^2*f^2*x^2 + 9*(4*a^2*b^3 + b^5)*d^2*e^2 + 6*(4*a^2*b^3 + b^5)*d*e*f + 2*(4*a^2*b^3
+ b^5)*f^2 + 6*(3*(4*a^2*b^3 + b^5)*d^2*e*f + (4*a^2*b^3 + b^5)*d*f^2)*x)*cosh(d*x + c)^2 - 20*(972*(25*b^5*d^
2*f^2*x^2 + 25*b^5*d^2*e^2 - 10*b^5*d*e*f + 2*b^5*f^2 + 10*(5*b^5*d^2*e*f - b^5*d*f^2)*x)*cosh(d*x + c)^8 - 60
75*(8*a*b^4*d^2*f^2*x^2 + 8*a*b^4*d^2*e^2 - 4*a*b^4*d*e*f + a*b^4*f^2 + 4*(4*a*b^4*d^2*e*f - a*b^4*d*f^2)*x)*c
osh(d*x + c)^7 - 900*(4*a^2*b^3 + b^5)*d^2*f^2*x^2 + 2800*(9*(4*a^2*b^3 + b^5)*d^2*f^2*x^2 + 9*(4*a^2*b^3 + b^
5)*d^2*e^2 - 6*(4*a^2*b^3 + b^5)*d*e*f + 2*(4*a^2*b^3 + b^5)*f^2 + 6*(3*(4*a^2*b^3 + b^5)*d^2*e*f - (4*a^2*b^3
+ b^5)*d*f^2)*x)*cosh(d*x + c)^6 - 56700*(2*(2*a^3*b^2 + a*b^4)*d^2*f^2*x^2 + 2*(2*a^3*b^2 + a*b^4)*d^2*e^2 -
2*(2*a^3*b^2 + a*b^4)*d*e*f + (2*a^3*b^2 + a*b^4)*f^2 + 2*(2*(2*a^3*b^2 + a*b^4)*d^2*e*f - (2*a^3*b^2 + a*b^4
)*d*f^2)*x)*cosh(d*x + c)^5 - 900*(4*a^2*b^3 + b^5)*d^2*e^2 + 81000*((8*a^4*b + 6*a^2*b^3 - b^5)*d^2*f^2*x^2 +
(8*a^4*b + 6*a^2*b^3 - b^5)*d^2*e^2 - 2*(8*a^4*b + 6*a^2*b^3 - b^5)*d*e*f + 2*(8*a^4*b + 6*a^2*b^3 - b^5)*f^2
+ 2*((8*a^4*b + 6*a^2*b^3 - b^5)*d^2*e*f - (8*a^4*b + 6*a^2*b^3 - b^5)*d*f^2)*x)*cosh(d*x + c)^4 - 600*(4*a^2
*b^3 + b^5)*d*e*f + 288000*((a^5 + a^3*b^2)*d^3*f^2*x^3 + 3*(a^5 + a^3*b^2)*d^3*e*f*x^2 + 3*(a^5 + a^3*b^2)*d^
3*e^2*x + 6*(a^5 + a^3*b^2)*c*d^2*e^2 - 6*(a^5 + a^3*b^2)*c^2*d*e*f + 2*(a^5 + a^3*b^2)*c^3*f^2)*cosh(d*x + c)
^3 - 200*(4*a^2*b^3 + b^5)*f^2 - 32400*((8*a^4*b + 6*a^2*b^3 - b^5)*d^2*f^2*x^2 + (8*a^4*b + 6*a^2*b^3 - b^5)*
d^2*e^2 + 2*(8*a^4*b + 6*a^2*b^3 - b^5)*d*e*f + 2*(8*a^4*b + 6*a^2*b^3 - b^5)*f^2 + 2*((8*a^4*b + 6*a^2*b^3 -
b^5)*d^2*e*f + (8*a^4*b + 6*a^2*b^3 - b^5)*d*f^2)*x)*cosh(d*x + c)^2 - 600*(3*(4*a^2*b^3 + b^5)*d^2*e*f + (4*a
^2*b^3 + b^5)*d*f^2)*x - 8100*(2*(2*a^3*b^2 + a*b^4)*d^2*f^2*x^2 + 2*(2*a^3*b^2 + a*b^4)*d^2*e^2 + 2*(2*a^3*b^
2 + a*b^4)*d*e*f + (2*a^3*b^2 + a*b^4)*f^2 + 2*(2*(2*a^3*b^2 + a*b^4)*d^2*e*f + (2*a^3*b^2 + a*b^4)*d*f^2)*x)*
cosh(d*x + c))*sinh(d*x + c)^2 + 4320*(5*b^5*d^2*e*f + b^5*d*f^2)*x + 3375*(8*a*b^4*d^2*f^2*x^2 + 8*a*b^4*d^2*
e^2 + 4*a*b^4*d*e*f + a*b^4*f^2 + 4*(4*a*b^4*d^2*e*f + a*b^4*d*f^2)*x)*cosh(d*x + c) + 3456000*(((a^5 + a^3*b^
2)*d*f^2*x + (a^5 + a^3*b^2)*d*e*f)*cosh(d*x + c)^5 + 5*((a^5 + a^3*b^2)*d*f^2*x + (a^5 + a^3*b^2)*d*e*f)*cosh
(d*x + c)^4*sinh(d*x + c) + 10*((a^5 + a^3*b^2)*d*f^2*x + (a^5 + a^3*b^2)*d*e*f)*cosh(d*x + c)^3*sinh(d*x + c)
^2 + 10*((a^5 + a^3*b^2)*d*f^2*x + (a^5 + a^3*b^2)*d*e*f)*cosh(d*x + c)^2*sinh(d*x + c)^3 + 5*((a^5 + a^3*b^2)
*d*f^2*x + (a^5 + a^3*b^2)*d*e*f)*cosh(d*x + c)*sinh(d*x + c)^4 + ((a^5 + a^3*b^2)*d*f^2*x + (a^5 + a^3*b^2)*d
*e*f)*sinh(d*x + c)^5)*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^
2 + b^2)/b^2) - b)/b + 1) + 3456000*(((a^5 + a^3*b^2)*d*f^2*x + (a^5 + a^3*b^2)*d*e*f)*cosh(d*x + c)^5 + 5*((a
^5 + a^3*b^2)*d*f^2*x + (a^5 + a^3*b^2)*d*e*f)*cosh(d*x + c)^4*sinh(d*x + c) + 10*((a^5 + a^3*b^2)*d*f^2*x + (
a^5 + a^3*b^2)*d*e*f)*cosh(d*x + c)^3*sinh(d*x + c)^2 + 10*((a^5 + a^3*b^2)*d*f^2*x + (a^5 + a^3*b^2)*d*e*f)*c
osh(d*x + c)^2*sinh(d*x + c)^3 + 5*((a^5 + a^3*b^2)*d*f^2*x + (a^5 + a^3*b^2)*d*e*f)*cosh(d*x + c)*sinh(d*x +
c)^4 + ((a^5 + a^3*b^2)*d*f^2*x + (a^5 + a^3*b^2)*d*e*f)*sinh(d*x + c)^5)*dilog((a*cosh(d*x + c) + a*sinh(d*x
+ c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + 1728000*(((a^5 + a^3*b^2)*d^2*e
^2 - 2*(a^5 + a^3*b^2)*c*d*e*f + (a^5 + a^3*b^2)*c^2*f^2)*cosh(d*x + c)^5 + 5*((a^5 + a^3*b^2)*d^2*e^2 - 2*(a^
5 + a^3*b^2)*c*d*e*f + (a^5 + a^3*b^2)*c^2*f^2)*cosh(d*x + c)^4*sinh(d*x + c) + 10*((a^5 + a^3*b^2)*d^2*e^2 -
2*(a^5 + a^3*b^2)*c*d*e*f + (a^5 + a^3*b^2)*c^2*f^2)*cosh(d*x + c)^3*sinh(d*x + c)^2 + 10*((a^5 + a^3*b^2)*d^2
*e^2 - 2*(a^5 + a^3*b^2)*c*d*e*f + (a^5 + a^3*b^2)*c^2*f^2)*cosh(d*x + c)^2*sinh(d*x + c)^3 + 5*((a^5 + a^3*b^
2)*d^2*e^2 - 2*(a^5 + a^3*b^2)*c*d*e*f + (a^5 + a^3*b^2)*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^4 + ((a^5 + a^3*
b^2)*d^2*e^2 - 2*(a^5 + a^3*b^2)*c*d*e*f + (a^5 + a^3*b^2)*c^2*f^2)*sinh(d*x + c)^5)*log(2*b*cosh(d*x + c) + 2
*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + 1728000*(((a^5 + a^3*b^2)*d^2*e^2 - 2*(a^5 + a^3*b^2)*c*
d*e*f + (a^5 + a^3*b^2)*c^2*f^2)*cosh(d*x + c)^5 + 5*((a^5 + a^3*b^2)*d^2*e^2 - 2*(a^5 + a^3*b^2)*c*d*e*f + (a
^5 + a^3*b^2)*c^2*f^2)*cosh(d*x + c)^4*sinh(d*x + c) + 10*((a^5 + a^3*b^2)*d^2*e^2 - 2*(a^5 + a^3*b^2)*c*d*e*f
+ (a^5 + a^3*b^2)*c^2*f^2)*cosh(d*x + c)^3*sinh(d*x + c)^2 + 10*((a^5 + a^3*b^2)*d^2*e^2 - 2*(a^5 + a^3*b^2)*
c*d*e*f + (a^5 + a^3*b^2)*c^2*f^2)*cosh(d*x + c)^2*sinh(d*x + c)^3 + 5*((a^5 + a^3*b^2)*d^2*e^2 - 2*(a^5 + a^3
*b^2)*c*d*e*f + (a^5 + a^3*b^2)*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^4 + ((a^5 + a^3*b^2)*d^2*e^2 - 2*(a^5 + a
^3*b^2)*c*d*e*f + (a^5 + a^3*b^2)*c^2*f^2)*sinh(d*x + c)^5)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sq
rt((a^2 + b^2)/b^2) + 2*a) + 1728000*(((a^5 + a^3*b^2)*d^2*f^2*x^2 + 2*(a^5 + a^3*b^2)*d^2*e*f*x + 2*(a^5 + a^
3*b^2)*c*d*e*f - (a^5 + a^3*b^2)*c^2*f^2)*cosh(d*x + c)^5 + 5*((a^5 + a^3*b^2)*d^2*f^2*x^2 + 2*(a^5 + a^3*b^2)
*d^2*e*f*x + 2*(a^5 + a^3*b^2)*c*d*e*f - (a^5 + a^3*b^2)*c^2*f^2)*cosh(d*x + c)^4*sinh(d*x + c) + 10*((a^5 + a
^3*b^2)*d^2*f^2*x^2 + 2*(a^5 + a^3*b^2)*d^2*e*f*x + 2*(a^5 + a^3*b^2)*c*d*e*f - (a^5 + a^3*b^2)*c^2*f^2)*cosh(
d*x + c)^3*sinh(d*x + c)^2 + 10*((a^5 + a^3*b^2)*d^2*f^2*x^2 + 2*(a^5 + a^3*b^2)*d^2*e*f*x + 2*(a^5 + a^3*b^2)
*c*d*e*f - (a^5 + a^3*b^2)*c^2*f^2)*cosh(d*x + c)^2*sinh(d*x + c)^3 + 5*((a^5 + a^3*b^2)*d^2*f^2*x^2 + 2*(a^5
+ a^3*b^2)*d^2*e*f*x + 2*(a^5 + a^3*b^2)*c*d*e*f - (a^5 + a^3*b^2)*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^4 + ((
a^5 + a^3*b^2)*d^2*f^2*x^2 + 2*(a^5 + a^3*b^2)*d^2*e*f*x + 2*(a^5 + a^3*b^2)*c*d*e*f - (a^5 + a^3*b^2)*c^2*f^2
)*sinh(d*x + c)^5)*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b
^2)/b^2) - b)/b) + 1728000*(((a^5 + a^3*b^2)*d^2*f^2*x^2 + 2*(a^5 + a^3*b^2)*d^2*e*f*x + 2*(a^5 + a^3*b^2)*c*d
*e*f - (a^5 + a^3*b^2)*c^2*f^2)*cosh(d*x + c)^5 + 5*((a^5 + a^3*b^2)*d^2*f^2*x^2 + 2*(a^5 + a^3*b^2)*d^2*e*f*x
+ 2*(a^5 + a^3*b^2)*c*d*e*f - (a^5 + a^3*b^2)*c^2*f^2)*cosh(d*x + c)^4*sinh(d*x + c) + 10*((a^5 + a^3*b^2)*d^
2*f^2*x^2 + 2*(a^5 + a^3*b^2)*d^2*e*f*x + 2*(a^5 + a^3*b^2)*c*d*e*f - (a^5 + a^3*b^2)*c^2*f^2)*cosh(d*x + c)^3
*sinh(d*x + c)^2 + 10*((a^5 + a^3*b^2)*d^2*f^2*x^2 + 2*(a^5 + a^3*b^2)*d^2*e*f*x + 2*(a^5 + a^3*b^2)*c*d*e*f -
(a^5 + a^3*b^2)*c^2*f^2)*cosh(d*x + c)^2*sinh(d*x + c)^3 + 5*((a^5 + a^3*b^2)*d^2*f^2*x^2 + 2*(a^5 + a^3*b^2)
*d^2*e*f*x + 2*(a^5 + a^3*b^2)*c*d*e*f - (a^5 + a^3*b^2)*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^4 + ((a^5 + a^3*
b^2)*d^2*f^2*x^2 + 2*(a^5 + a^3*b^2)*d^2*e*f*x + 2*(a^5 + a^3*b^2)*c*d*e*f - (a^5 + a^3*b^2)*c^2*f^2)*sinh(d*x
+ c)^5)*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) -
b)/b) - 3456000*((a^5 + a^3*b^2)*f^2*cosh(d*x + c)^5 + 5*(a^5 + a^3*b^2)*f^2*cosh(d*x + c)^4*sinh(d*x + c) +
10*(a^5 + a^3*b^2)*f^2*cosh(d*x + c)^3*sinh(d*x + c)^2 + 10*(a^5 + a^3*b^2)*f^2*cosh(d*x + c)^2*sinh(d*x + c)^
3 + 5*(a^5 + a^3*b^2)*f^2*cosh(d*x + c)*sinh(d*x + c)^4 + (a^5 + a^3*b^2)*f^2*sinh(d*x + c)^5)*polylog(3, (a*c
osh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) - 3456000*((a^5
+ a^3*b^2)*f^2*cosh(d*x + c)^5 + 5*(a^5 + a^3*b^2)*f^2*cosh(d*x + c)^4*sinh(d*x + c) + 10*(a^5 + a^3*b^2)*f^2
*cosh(d*x + c)^3*sinh(d*x + c)^2 + 10*(a^5 + a^3*b^2)*f^2*cosh(d*x + c)^2*sinh(d*x + c)^3 + 5*(a^5 + a^3*b^2)*
f^2*cosh(d*x + c)*sinh(d*x + c)^4 + (a^5 + a^3*b^2)*f^2*sinh(d*x + c)^5)*polylog(3, (a*cosh(d*x + c) + a*sinh(
d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) + 5*(5400*a*b^4*d^2*f^2*x^2 - 864*(25
*b^5*d^2*f^2*x^2 + 25*b^5*d^2*e^2 - 10*b^5*d*e*f + 2*b^5*f^2 + 10*(5*b^5*d^2*e*f - b^5*d*f^2)*x)*cosh(d*x + c)
^9 + 5400*a*b^4*d^2*e^2 + 6075*(8*a*b^4*d^2*f^2*x^2 + 8*a*b^4*d^2*e^2 - 4*a*b^4*d*e*f + a*b^4*f^2 + 4*(4*a*b^4
*d^2*e*f - a*b^4*d*f^2)*x)*cosh(d*x + c)^8 + 2700*a*b^4*d*e*f - 3200*(9*(4*a^2*b^3 + b^5)*d^2*f^2*x^2 + 9*(4*a
^2*b^3 + b^5)*d^2*e^2 - 6*(4*a^2*b^3 + b^5)*d*e*f + 2*(4*a^2*b^3 + b^5)*f^2 + 6*(3*(4*a^2*b^3 + b^5)*d^2*e*f -
(4*a^2*b^3 + b^5)*d*f^2)*x)*cosh(d*x + c)^7 + 675*a*b^4*f^2 + 75600*(2*(2*a^3*b^2 + a*b^4)*d^2*f^2*x^2 + 2*(2
*a^3*b^2 + a*b^4)*d^2*e^2 - 2*(2*a^3*b^2 + a*b^4)*d*e*f + (2*a^3*b^2 + a*b^4)*f^2 + 2*(2*(2*a^3*b^2 + a*b^4)*d
^2*e*f - (2*a^3*b^2 + a*b^4)*d*f^2)*x)*cosh(d*x + c)^6 - 129600*((8*a^4*b + 6*a^2*b^3 - b^5)*d^2*f^2*x^2 + (8*
a^4*b + 6*a^2*b^3 - b^5)*d^2*e^2 - 2*(8*a^4*b + 6*a^2*b^3 - b^5)*d*e*f + 2*(8*a^4*b + 6*a^2*b^3 - b^5)*f^2 + 2
*((8*a^4*b + 6*a^2*b^3 - b^5)*d^2*e*f - (8*a^4*b + 6*a^2*b^3 - b^5)*d*f^2)*x)*cosh(d*x + c)^5 - 576000*((a^5 +
a^3*b^2)*d^3*f^2*x^3 + 3*(a^5 + a^3*b^2)*d^3*e*f*x^2 + 3*(a^5 + a^3*b^2)*d^3*e^2*x + 6*(a^5 + a^3*b^2)*c*d^2*
e^2 - 6*(a^5 + a^3*b^2)*c^2*d*e*f + 2*(a^5 + a^3*b^2)*c^3*f^2)*cosh(d*x + c)^4 + 86400*((8*a^4*b + 6*a^2*b^3 -
b^5)*d^2*f^2*x^2 + (8*a^4*b + 6*a^2*b^3 - b^5)*d^2*e^2 + 2*(8*a^4*b + 6*a^2*b^3 - b^5)*d*e*f + 2*(8*a^4*b + 6
*a^2*b^3 - b^5)*f^2 + 2*((8*a^4*b + 6*a^2*b^3 - b^5)*d^2*e*f + (8*a^4*b + 6*a^2*b^3 - b^5)*d*f^2)*x)*cosh(d*x
+ c)^3 + 32400*(2*(2*a^3*b^2 + a*b^4)*d^2*f^2*x^2 + 2*(2*a^3*b^2 + a*b^4)*d^2*e^2 + 2*(2*a^3*b^2 + a*b^4)*d*e*
f + (2*a^3*b^2 + a*b^4)*f^2 + 2*(2*(2*a^3*b^2 + a*b^4)*d^2*e*f + (2*a^3*b^2 + a*b^4)*d*f^2)*x)*cosh(d*x + c)^2
+ 2700*(4*a*b^4*d^2*e*f + a*b^4*d*f^2)*x + 800*(9*(4*a^2*b^3 + b^5)*d^2*f^2*x^2 + 9*(4*a^2*b^3 + b^5)*d^2*e^2
+ 6*(4*a^2*b^3 + b^5)*d*e*f + 2*(4*a^2*b^3 + b^5)*f^2 + 6*(3*(4*a^2*b^3 + b^5)*d^2*e*f + (4*a^2*b^3 + b^5)*d*
f^2)*x)*cosh(d*x + c))*sinh(d*x + c))/(b^6*d^3*cosh(d*x + c)^5 + 5*b^6*d^3*cosh(d*x + c)^4*sinh(d*x + c) + 10*
b^6*d^3*cosh(d*x + c)^3*sinh(d*x + c)^2 + 10*b^6*d^3*cosh(d*x + c)^2*sinh(d*x + c)^3 + 5*b^6*d^3*cosh(d*x + c)
*sinh(d*x + c)^4 + b^6*d^3*sinh(d*x + c)^5)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)**2*cosh(d*x+c)**3*sinh(d*x+c)**3/(a+b*sinh(d*x+c)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (f x + e\right )}^{2} \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^2*cosh(d*x+c)^3*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

integrate((f*x + e)^2*cosh(d*x + c)^3*sinh(d*x + c)^3/(b*sinh(d*x + c) + a), x)