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

Optimal. Leaf size=1038 \[ -\frac{(e+f x)^4}{32 b f}+\frac{a^2 (e+f x)^4}{8 b^3 f}+\frac{a^4 (e+f x)^4}{4 b^5 f}-\frac{a \cosh ^3(c+d x) (e+f x)^3}{3 b^2 d}-\frac{a^3 \cosh (c+d x) (e+f x)^3}{b^4 d}-\frac{a^3 \sqrt{a^2+b^2} \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) (e+f x)^3}{b^5 d}+\frac{a^3 \sqrt{a^2+b^2} \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) (e+f x)^3}{b^5 d}+\frac{a^2 \cosh (c+d x) \sinh (c+d x) (e+f x)^3}{2 b^3 d}+\frac{\sinh (4 c+4 d x) (e+f x)^3}{32 b d}-\frac{3 a^2 f \cosh ^2(c+d x) (e+f x)^2}{4 b^3 d^2}-\frac{3 f \cosh (4 c+4 d x) (e+f x)^2}{128 b d^2}-\frac{3 a^3 \sqrt{a^2+b^2} f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) (e+f x)^2}{b^5 d^2}+\frac{3 a^3 \sqrt{a^2+b^2} f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) (e+f x)^2}{b^5 d^2}+\frac{a f \cosh ^2(c+d x) \sinh (c+d x) (e+f x)^2}{3 b^2 d^2}+\frac{2 a f \sinh (c+d x) (e+f x)^2}{3 b^2 d^2}+\frac{3 a^3 f \sinh (c+d x) (e+f x)^2}{b^4 d^2}-\frac{2 a f^2 \cosh ^3(c+d x) (e+f x)}{9 b^2 d^3}-\frac{4 a f^2 \cosh (c+d x) (e+f x)}{3 b^2 d^3}-\frac{6 a^3 f^2 \cosh (c+d x) (e+f x)}{b^4 d^3}+\frac{6 a^3 \sqrt{a^2+b^2} f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) (e+f x)}{b^5 d^3}-\frac{6 a^3 \sqrt{a^2+b^2} f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) (e+f x)}{b^5 d^3}+\frac{3 a^2 f^2 \cosh (c+d x) \sinh (c+d x) (e+f x)}{4 b^3 d^3}+\frac{3 f^2 \sinh (4 c+4 d x) (e+f x)}{256 b d^3}+\frac{2 a f^3 \sinh ^3(c+d x)}{27 b^2 d^4}+\frac{3 a^2 f^3 x^2}{8 b^3 d^2}-\frac{3 a^2 f^3 \cosh ^2(c+d x)}{8 b^3 d^4}+\frac{3 a^2 e f^2 x}{4 b^3 d^2}-\frac{3 f^3 \cosh (4 c+4 d x)}{1024 b d^4}-\frac{6 a^3 \sqrt{a^2+b^2} f^3 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^5 d^4}+\frac{6 a^3 \sqrt{a^2+b^2} f^3 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^5 d^4}+\frac{14 a f^3 \sinh (c+d x)}{9 b^2 d^4}+\frac{6 a^3 f^3 \sinh (c+d x)}{b^4 d^4} \]

[Out]

(3*a^2*e*f^2*x)/(4*b^3*d^2) + (3*a^2*f^3*x^2)/(8*b^3*d^2) + (a^4*(e + f*x)^4)/(4*b^5*f) + (a^2*(e + f*x)^4)/(8
*b^3*f) - (e + f*x)^4/(32*b*f) - (6*a^3*f^2*(e + f*x)*Cosh[c + d*x])/(b^4*d^3) - (4*a*f^2*(e + f*x)*Cosh[c + d
*x])/(3*b^2*d^3) - (a^3*(e + f*x)^3*Cosh[c + d*x])/(b^4*d) - (3*a^2*f^3*Cosh[c + d*x]^2)/(8*b^3*d^4) - (3*a^2*
f*(e + f*x)^2*Cosh[c + d*x]^2)/(4*b^3*d^2) - (2*a*f^2*(e + f*x)*Cosh[c + d*x]^3)/(9*b^2*d^3) - (a*(e + f*x)^3*
Cosh[c + d*x]^3)/(3*b^2*d) - (3*f^3*Cosh[4*c + 4*d*x])/(1024*b*d^4) - (3*f*(e + f*x)^2*Cosh[4*c + 4*d*x])/(128
*b*d^2) - (a^3*Sqrt[a^2 + b^2]*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^5*d) + (a^3*Sqrt
[a^2 + b^2]*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^5*d) - (3*a^3*Sqrt[a^2 + b^2]*f*(e
+ f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^5*d^2) + (3*a^3*Sqrt[a^2 + b^2]*f*(e + f*x)^
2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^5*d^2) + (6*a^3*Sqrt[a^2 + b^2]*f^2*(e + f*x)*PolyL
og[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^5*d^3) - (6*a^3*Sqrt[a^2 + b^2]*f^2*(e + f*x)*PolyLog[3, -
((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^5*d^3) - (6*a^3*Sqrt[a^2 + b^2]*f^3*PolyLog[4, -((b*E^(c + d*x))/
(a - Sqrt[a^2 + b^2]))])/(b^5*d^4) + (6*a^3*Sqrt[a^2 + b^2]*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b
^2]))])/(b^5*d^4) + (6*a^3*f^3*Sinh[c + d*x])/(b^4*d^4) + (14*a*f^3*Sinh[c + d*x])/(9*b^2*d^4) + (3*a^3*f*(e +
f*x)^2*Sinh[c + d*x])/(b^4*d^2) + (2*a*f*(e + f*x)^2*Sinh[c + d*x])/(3*b^2*d^2) + (3*a^2*f^2*(e + f*x)*Cosh[c
+ d*x]*Sinh[c + d*x])/(4*b^3*d^3) + (a^2*(e + f*x)^3*Cosh[c + d*x]*Sinh[c + d*x])/(2*b^3*d) + (a*f*(e + f*x)^
2*Cosh[c + d*x]^2*Sinh[c + d*x])/(3*b^2*d^2) + (2*a*f^3*Sinh[c + d*x]^3)/(27*b^2*d^4) + (3*f^2*(e + f*x)*Sinh[
4*c + 4*d*x])/(256*b*d^3) + ((e + f*x)^3*Sinh[4*c + 4*d*x])/(32*b*d)

________________________________________________________________________________________

Rubi [A]  time = 1.82668, antiderivative size = 1038, normalized size of antiderivative = 1., number of steps used = 38, number of rules used = 18, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5579, 5448, 3296, 2638, 5447, 3311, 2637, 2633, 32, 3310, 5565, 3322, 2264, 2190, 2531, 6609, 2282, 6589} \[ -\frac{(e+f x)^4}{32 b f}+\frac{a^2 (e+f x)^4}{8 b^3 f}+\frac{a^4 (e+f x)^4}{4 b^5 f}-\frac{a \cosh ^3(c+d x) (e+f x)^3}{3 b^2 d}-\frac{a^3 \cosh (c+d x) (e+f x)^3}{b^4 d}-\frac{a^3 \sqrt{a^2+b^2} \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) (e+f x)^3}{b^5 d}+\frac{a^3 \sqrt{a^2+b^2} \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) (e+f x)^3}{b^5 d}+\frac{a^2 \cosh (c+d x) \sinh (c+d x) (e+f x)^3}{2 b^3 d}+\frac{\sinh (4 c+4 d x) (e+f x)^3}{32 b d}-\frac{3 a^2 f \cosh ^2(c+d x) (e+f x)^2}{4 b^3 d^2}-\frac{3 f \cosh (4 c+4 d x) (e+f x)^2}{128 b d^2}-\frac{3 a^3 \sqrt{a^2+b^2} f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) (e+f x)^2}{b^5 d^2}+\frac{3 a^3 \sqrt{a^2+b^2} f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) (e+f x)^2}{b^5 d^2}+\frac{a f \cosh ^2(c+d x) \sinh (c+d x) (e+f x)^2}{3 b^2 d^2}+\frac{2 a f \sinh (c+d x) (e+f x)^2}{3 b^2 d^2}+\frac{3 a^3 f \sinh (c+d x) (e+f x)^2}{b^4 d^2}-\frac{2 a f^2 \cosh ^3(c+d x) (e+f x)}{9 b^2 d^3}-\frac{4 a f^2 \cosh (c+d x) (e+f x)}{3 b^2 d^3}-\frac{6 a^3 f^2 \cosh (c+d x) (e+f x)}{b^4 d^3}+\frac{6 a^3 \sqrt{a^2+b^2} f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) (e+f x)}{b^5 d^3}-\frac{6 a^3 \sqrt{a^2+b^2} f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) (e+f x)}{b^5 d^3}+\frac{3 a^2 f^2 \cosh (c+d x) \sinh (c+d x) (e+f x)}{4 b^3 d^3}+\frac{3 f^2 \sinh (4 c+4 d x) (e+f x)}{256 b d^3}+\frac{2 a f^3 \sinh ^3(c+d x)}{27 b^2 d^4}+\frac{3 a^2 f^3 x^2}{8 b^3 d^2}-\frac{3 a^2 f^3 \cosh ^2(c+d x)}{8 b^3 d^4}+\frac{3 a^2 e f^2 x}{4 b^3 d^2}-\frac{3 f^3 \cosh (4 c+4 d x)}{1024 b d^4}-\frac{6 a^3 \sqrt{a^2+b^2} f^3 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^5 d^4}+\frac{6 a^3 \sqrt{a^2+b^2} f^3 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^5 d^4}+\frac{14 a f^3 \sinh (c+d x)}{9 b^2 d^4}+\frac{6 a^3 f^3 \sinh (c+d x)}{b^4 d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a^2*e*f^2*x)/(4*b^3*d^2) + (3*a^2*f^3*x^2)/(8*b^3*d^2) + (a^4*(e + f*x)^4)/(4*b^5*f) + (a^2*(e + f*x)^4)/(8
*b^3*f) - (e + f*x)^4/(32*b*f) - (6*a^3*f^2*(e + f*x)*Cosh[c + d*x])/(b^4*d^3) - (4*a*f^2*(e + f*x)*Cosh[c + d
*x])/(3*b^2*d^3) - (a^3*(e + f*x)^3*Cosh[c + d*x])/(b^4*d) - (3*a^2*f^3*Cosh[c + d*x]^2)/(8*b^3*d^4) - (3*a^2*
f*(e + f*x)^2*Cosh[c + d*x]^2)/(4*b^3*d^2) - (2*a*f^2*(e + f*x)*Cosh[c + d*x]^3)/(9*b^2*d^3) - (a*(e + f*x)^3*
Cosh[c + d*x]^3)/(3*b^2*d) - (3*f^3*Cosh[4*c + 4*d*x])/(1024*b*d^4) - (3*f*(e + f*x)^2*Cosh[4*c + 4*d*x])/(128
*b*d^2) - (a^3*Sqrt[a^2 + b^2]*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^5*d) + (a^3*Sqrt
[a^2 + b^2]*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^5*d) - (3*a^3*Sqrt[a^2 + b^2]*f*(e
+ f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^5*d^2) + (3*a^3*Sqrt[a^2 + b^2]*f*(e + f*x)^
2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^5*d^2) + (6*a^3*Sqrt[a^2 + b^2]*f^2*(e + f*x)*PolyL
og[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^5*d^3) - (6*a^3*Sqrt[a^2 + b^2]*f^2*(e + f*x)*PolyLog[3, -
((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^5*d^3) - (6*a^3*Sqrt[a^2 + b^2]*f^3*PolyLog[4, -((b*E^(c + d*x))/
(a - Sqrt[a^2 + b^2]))])/(b^5*d^4) + (6*a^3*Sqrt[a^2 + b^2]*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b
^2]))])/(b^5*d^4) + (6*a^3*f^3*Sinh[c + d*x])/(b^4*d^4) + (14*a*f^3*Sinh[c + d*x])/(9*b^2*d^4) + (3*a^3*f*(e +
f*x)^2*Sinh[c + d*x])/(b^4*d^2) + (2*a*f*(e + f*x)^2*Sinh[c + d*x])/(3*b^2*d^2) + (3*a^2*f^2*(e + f*x)*Cosh[c
+ d*x]*Sinh[c + d*x])/(4*b^3*d^3) + (a^2*(e + f*x)^3*Cosh[c + d*x]*Sinh[c + d*x])/(2*b^3*d) + (a*f*(e + f*x)^
2*Cosh[c + d*x]^2*Sinh[c + d*x])/(3*b^2*d^2) + (2*a*f^3*Sinh[c + d*x]^3)/(27*b^2*d^4) + (3*f^2*(e + f*x)*Sinh[
4*c + 4*d*x])/(256*b*d^3) + ((e + f*x)^3*Sinh[4*c + 4*d*x])/(32*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 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[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 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 2637

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

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 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -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 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 3322

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Dist[2,
Int[((c + d*x)^m*E^(-(I*e) + f*fz*x))/(-(I*b) + 2*a*E^(-(I*e) + f*fz*x) + I*b*E^(2*(-(I*e) + f*fz*x))), x], x]
/; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 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 6609

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[((e + f*x)^m*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p])/(b*c*p*Log[F]), x] - Dist[(f*m)/(b*c*p*Log[F]), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, 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)^3 \cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac{\int (e+f x)^3 \cosh ^2(c+d x) \sinh ^2(c+d x) \, dx}{b}-\frac{a \int \frac{(e+f x)^3 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=-\frac{a \int (e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x) \, dx}{b^2}+\frac{a^2 \int \frac{(e+f x)^3 \cosh ^2(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)^3+\frac{1}{8} (e+f x)^3 \cosh (4 c+4 d x)\right ) \, dx}{b}\\ &=-\frac{(e+f x)^4}{32 b f}-\frac{a (e+f x)^3 \cosh ^3(c+d x)}{3 b^2 d}+\frac{a^2 \int (e+f x)^3 \cosh ^2(c+d x) \, dx}{b^3}-\frac{a^3 \int \frac{(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b^3}+\frac{\int (e+f x)^3 \cosh (4 c+4 d x) \, dx}{8 b}+\frac{(a f) \int (e+f x)^2 \cosh ^3(c+d x) \, dx}{b^2 d}\\ &=-\frac{(e+f x)^4}{32 b f}-\frac{3 a^2 f (e+f x)^2 \cosh ^2(c+d x)}{4 b^3 d^2}-\frac{2 a f^2 (e+f x) \cosh ^3(c+d x)}{9 b^2 d^3}-\frac{a (e+f x)^3 \cosh ^3(c+d x)}{3 b^2 d}+\frac{a^2 (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b^3 d}+\frac{a f (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^2 d^2}+\frac{(e+f x)^3 \sinh (4 c+4 d x)}{32 b d}+\frac{a^4 \int (e+f x)^3 \, dx}{b^5}-\frac{a^3 \int (e+f x)^3 \sinh (c+d x) \, dx}{b^4}+\frac{a^2 \int (e+f x)^3 \, dx}{2 b^3}-\frac{\left (a^3 \left (a^2+b^2\right )\right ) \int \frac{(e+f x)^3}{a+b \sinh (c+d x)} \, dx}{b^5}+\frac{(2 a f) \int (e+f x)^2 \cosh (c+d x) \, dx}{3 b^2 d}-\frac{(3 f) \int (e+f x)^2 \sinh (4 c+4 d x) \, dx}{32 b d}+\frac{\left (3 a^2 f^2\right ) \int (e+f x) \cosh ^2(c+d x) \, dx}{2 b^3 d^2}+\frac{\left (2 a f^3\right ) \int \cosh ^3(c+d x) \, dx}{9 b^2 d^3}\\ &=\frac{a^4 (e+f x)^4}{4 b^5 f}+\frac{a^2 (e+f x)^4}{8 b^3 f}-\frac{(e+f x)^4}{32 b f}-\frac{a^3 (e+f x)^3 \cosh (c+d x)}{b^4 d}-\frac{3 a^2 f^3 \cosh ^2(c+d x)}{8 b^3 d^4}-\frac{3 a^2 f (e+f x)^2 \cosh ^2(c+d x)}{4 b^3 d^2}-\frac{2 a f^2 (e+f x) \cosh ^3(c+d x)}{9 b^2 d^3}-\frac{a (e+f x)^3 \cosh ^3(c+d x)}{3 b^2 d}-\frac{3 f (e+f x)^2 \cosh (4 c+4 d x)}{128 b d^2}+\frac{2 a f (e+f x)^2 \sinh (c+d x)}{3 b^2 d^2}+\frac{3 a^2 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b^3 d^3}+\frac{a^2 (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b^3 d}+\frac{a f (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^2 d^2}+\frac{(e+f x)^3 \sinh (4 c+4 d x)}{32 b d}-\frac{\left (2 a^3 \left (a^2+b^2\right )\right ) \int \frac{e^{c+d x} (e+f x)^3}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{b^5}+\frac{\left (3 a^3 f\right ) \int (e+f x)^2 \cosh (c+d x) \, dx}{b^4 d}+\frac{\left (3 a^2 f^2\right ) \int (e+f x) \, dx}{4 b^3 d^2}-\frac{\left (4 a f^2\right ) \int (e+f x) \sinh (c+d x) \, dx}{3 b^2 d^2}+\frac{\left (3 f^2\right ) \int (e+f x) \cosh (4 c+4 d x) \, dx}{64 b d^2}+\frac{\left (2 i a f^3\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-i \sinh (c+d x)\right )}{9 b^2 d^4}\\ &=\frac{3 a^2 e f^2 x}{4 b^3 d^2}+\frac{3 a^2 f^3 x^2}{8 b^3 d^2}+\frac{a^4 (e+f x)^4}{4 b^5 f}+\frac{a^2 (e+f x)^4}{8 b^3 f}-\frac{(e+f x)^4}{32 b f}-\frac{4 a f^2 (e+f x) \cosh (c+d x)}{3 b^2 d^3}-\frac{a^3 (e+f x)^3 \cosh (c+d x)}{b^4 d}-\frac{3 a^2 f^3 \cosh ^2(c+d x)}{8 b^3 d^4}-\frac{3 a^2 f (e+f x)^2 \cosh ^2(c+d x)}{4 b^3 d^2}-\frac{2 a f^2 (e+f x) \cosh ^3(c+d x)}{9 b^2 d^3}-\frac{a (e+f x)^3 \cosh ^3(c+d x)}{3 b^2 d}-\frac{3 f (e+f x)^2 \cosh (4 c+4 d x)}{128 b d^2}+\frac{2 a f^3 \sinh (c+d x)}{9 b^2 d^4}+\frac{3 a^3 f (e+f x)^2 \sinh (c+d x)}{b^4 d^2}+\frac{2 a f (e+f x)^2 \sinh (c+d x)}{3 b^2 d^2}+\frac{3 a^2 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b^3 d^3}+\frac{a^2 (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b^3 d}+\frac{a f (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^2 d^2}+\frac{2 a f^3 \sinh ^3(c+d x)}{27 b^2 d^4}+\frac{3 f^2 (e+f x) \sinh (4 c+4 d x)}{256 b d^3}+\frac{(e+f x)^3 \sinh (4 c+4 d x)}{32 b d}-\frac{\left (2 a^3 \sqrt{a^2+b^2}\right ) \int \frac{e^{c+d x} (e+f x)^3}{2 a-2 \sqrt{a^2+b^2}+2 b e^{c+d x}} \, dx}{b^4}+\frac{\left (2 a^3 \sqrt{a^2+b^2}\right ) \int \frac{e^{c+d x} (e+f x)^3}{2 a+2 \sqrt{a^2+b^2}+2 b e^{c+d x}} \, dx}{b^4}-\frac{\left (6 a^3 f^2\right ) \int (e+f x) \sinh (c+d x) \, dx}{b^4 d^2}+\frac{\left (4 a f^3\right ) \int \cosh (c+d x) \, dx}{3 b^2 d^3}-\frac{\left (3 f^3\right ) \int \sinh (4 c+4 d x) \, dx}{256 b d^3}\\ &=\frac{3 a^2 e f^2 x}{4 b^3 d^2}+\frac{3 a^2 f^3 x^2}{8 b^3 d^2}+\frac{a^4 (e+f x)^4}{4 b^5 f}+\frac{a^2 (e+f x)^4}{8 b^3 f}-\frac{(e+f x)^4}{32 b f}-\frac{6 a^3 f^2 (e+f x) \cosh (c+d x)}{b^4 d^3}-\frac{4 a f^2 (e+f x) \cosh (c+d x)}{3 b^2 d^3}-\frac{a^3 (e+f x)^3 \cosh (c+d x)}{b^4 d}-\frac{3 a^2 f^3 \cosh ^2(c+d x)}{8 b^3 d^4}-\frac{3 a^2 f (e+f x)^2 \cosh ^2(c+d x)}{4 b^3 d^2}-\frac{2 a f^2 (e+f x) \cosh ^3(c+d x)}{9 b^2 d^3}-\frac{a (e+f x)^3 \cosh ^3(c+d x)}{3 b^2 d}-\frac{3 f^3 \cosh (4 c+4 d x)}{1024 b d^4}-\frac{3 f (e+f x)^2 \cosh (4 c+4 d x)}{128 b d^2}-\frac{a^3 \sqrt{a^2+b^2} (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^5 d}+\frac{a^3 \sqrt{a^2+b^2} (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^5 d}+\frac{14 a f^3 \sinh (c+d x)}{9 b^2 d^4}+\frac{3 a^3 f (e+f x)^2 \sinh (c+d x)}{b^4 d^2}+\frac{2 a f (e+f x)^2 \sinh (c+d x)}{3 b^2 d^2}+\frac{3 a^2 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b^3 d^3}+\frac{a^2 (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b^3 d}+\frac{a f (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^2 d^2}+\frac{2 a f^3 \sinh ^3(c+d x)}{27 b^2 d^4}+\frac{3 f^2 (e+f x) \sinh (4 c+4 d x)}{256 b d^3}+\frac{(e+f x)^3 \sinh (4 c+4 d x)}{32 b d}+\frac{\left (3 a^3 \sqrt{a^2+b^2} f\right ) \int (e+f x)^2 \log \left (1+\frac{2 b e^{c+d x}}{2 a-2 \sqrt{a^2+b^2}}\right ) \, dx}{b^5 d}-\frac{\left (3 a^3 \sqrt{a^2+b^2} f\right ) \int (e+f x)^2 \log \left (1+\frac{2 b e^{c+d x}}{2 a+2 \sqrt{a^2+b^2}}\right ) \, dx}{b^5 d}+\frac{\left (6 a^3 f^3\right ) \int \cosh (c+d x) \, dx}{b^4 d^3}\\ &=\frac{3 a^2 e f^2 x}{4 b^3 d^2}+\frac{3 a^2 f^3 x^2}{8 b^3 d^2}+\frac{a^4 (e+f x)^4}{4 b^5 f}+\frac{a^2 (e+f x)^4}{8 b^3 f}-\frac{(e+f x)^4}{32 b f}-\frac{6 a^3 f^2 (e+f x) \cosh (c+d x)}{b^4 d^3}-\frac{4 a f^2 (e+f x) \cosh (c+d x)}{3 b^2 d^3}-\frac{a^3 (e+f x)^3 \cosh (c+d x)}{b^4 d}-\frac{3 a^2 f^3 \cosh ^2(c+d x)}{8 b^3 d^4}-\frac{3 a^2 f (e+f x)^2 \cosh ^2(c+d x)}{4 b^3 d^2}-\frac{2 a f^2 (e+f x) \cosh ^3(c+d x)}{9 b^2 d^3}-\frac{a (e+f x)^3 \cosh ^3(c+d x)}{3 b^2 d}-\frac{3 f^3 \cosh (4 c+4 d x)}{1024 b d^4}-\frac{3 f (e+f x)^2 \cosh (4 c+4 d x)}{128 b d^2}-\frac{a^3 \sqrt{a^2+b^2} (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^5 d}+\frac{a^3 \sqrt{a^2+b^2} (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^5 d}-\frac{3 a^3 \sqrt{a^2+b^2} f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^5 d^2}+\frac{3 a^3 \sqrt{a^2+b^2} f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^5 d^2}+\frac{6 a^3 f^3 \sinh (c+d x)}{b^4 d^4}+\frac{14 a f^3 \sinh (c+d x)}{9 b^2 d^4}+\frac{3 a^3 f (e+f x)^2 \sinh (c+d x)}{b^4 d^2}+\frac{2 a f (e+f x)^2 \sinh (c+d x)}{3 b^2 d^2}+\frac{3 a^2 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b^3 d^3}+\frac{a^2 (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b^3 d}+\frac{a f (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^2 d^2}+\frac{2 a f^3 \sinh ^3(c+d x)}{27 b^2 d^4}+\frac{3 f^2 (e+f x) \sinh (4 c+4 d x)}{256 b d^3}+\frac{(e+f x)^3 \sinh (4 c+4 d x)}{32 b d}+\frac{\left (6 a^3 \sqrt{a^2+b^2} f^2\right ) \int (e+f x) \text{Li}_2\left (-\frac{2 b e^{c+d x}}{2 a-2 \sqrt{a^2+b^2}}\right ) \, dx}{b^5 d^2}-\frac{\left (6 a^3 \sqrt{a^2+b^2} f^2\right ) \int (e+f x) \text{Li}_2\left (-\frac{2 b e^{c+d x}}{2 a+2 \sqrt{a^2+b^2}}\right ) \, dx}{b^5 d^2}\\ &=\frac{3 a^2 e f^2 x}{4 b^3 d^2}+\frac{3 a^2 f^3 x^2}{8 b^3 d^2}+\frac{a^4 (e+f x)^4}{4 b^5 f}+\frac{a^2 (e+f x)^4}{8 b^3 f}-\frac{(e+f x)^4}{32 b f}-\frac{6 a^3 f^2 (e+f x) \cosh (c+d x)}{b^4 d^3}-\frac{4 a f^2 (e+f x) \cosh (c+d x)}{3 b^2 d^3}-\frac{a^3 (e+f x)^3 \cosh (c+d x)}{b^4 d}-\frac{3 a^2 f^3 \cosh ^2(c+d x)}{8 b^3 d^4}-\frac{3 a^2 f (e+f x)^2 \cosh ^2(c+d x)}{4 b^3 d^2}-\frac{2 a f^2 (e+f x) \cosh ^3(c+d x)}{9 b^2 d^3}-\frac{a (e+f x)^3 \cosh ^3(c+d x)}{3 b^2 d}-\frac{3 f^3 \cosh (4 c+4 d x)}{1024 b d^4}-\frac{3 f (e+f x)^2 \cosh (4 c+4 d x)}{128 b d^2}-\frac{a^3 \sqrt{a^2+b^2} (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^5 d}+\frac{a^3 \sqrt{a^2+b^2} (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^5 d}-\frac{3 a^3 \sqrt{a^2+b^2} f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^5 d^2}+\frac{3 a^3 \sqrt{a^2+b^2} f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^5 d^2}+\frac{6 a^3 \sqrt{a^2+b^2} f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^5 d^3}-\frac{6 a^3 \sqrt{a^2+b^2} f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^5 d^3}+\frac{6 a^3 f^3 \sinh (c+d x)}{b^4 d^4}+\frac{14 a f^3 \sinh (c+d x)}{9 b^2 d^4}+\frac{3 a^3 f (e+f x)^2 \sinh (c+d x)}{b^4 d^2}+\frac{2 a f (e+f x)^2 \sinh (c+d x)}{3 b^2 d^2}+\frac{3 a^2 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b^3 d^3}+\frac{a^2 (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b^3 d}+\frac{a f (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^2 d^2}+\frac{2 a f^3 \sinh ^3(c+d x)}{27 b^2 d^4}+\frac{3 f^2 (e+f x) \sinh (4 c+4 d x)}{256 b d^3}+\frac{(e+f x)^3 \sinh (4 c+4 d x)}{32 b d}-\frac{\left (6 a^3 \sqrt{a^2+b^2} f^3\right ) \int \text{Li}_3\left (-\frac{2 b e^{c+d x}}{2 a-2 \sqrt{a^2+b^2}}\right ) \, dx}{b^5 d^3}+\frac{\left (6 a^3 \sqrt{a^2+b^2} f^3\right ) \int \text{Li}_3\left (-\frac{2 b e^{c+d x}}{2 a+2 \sqrt{a^2+b^2}}\right ) \, dx}{b^5 d^3}\\ &=\frac{3 a^2 e f^2 x}{4 b^3 d^2}+\frac{3 a^2 f^3 x^2}{8 b^3 d^2}+\frac{a^4 (e+f x)^4}{4 b^5 f}+\frac{a^2 (e+f x)^4}{8 b^3 f}-\frac{(e+f x)^4}{32 b f}-\frac{6 a^3 f^2 (e+f x) \cosh (c+d x)}{b^4 d^3}-\frac{4 a f^2 (e+f x) \cosh (c+d x)}{3 b^2 d^3}-\frac{a^3 (e+f x)^3 \cosh (c+d x)}{b^4 d}-\frac{3 a^2 f^3 \cosh ^2(c+d x)}{8 b^3 d^4}-\frac{3 a^2 f (e+f x)^2 \cosh ^2(c+d x)}{4 b^3 d^2}-\frac{2 a f^2 (e+f x) \cosh ^3(c+d x)}{9 b^2 d^3}-\frac{a (e+f x)^3 \cosh ^3(c+d x)}{3 b^2 d}-\frac{3 f^3 \cosh (4 c+4 d x)}{1024 b d^4}-\frac{3 f (e+f x)^2 \cosh (4 c+4 d x)}{128 b d^2}-\frac{a^3 \sqrt{a^2+b^2} (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^5 d}+\frac{a^3 \sqrt{a^2+b^2} (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^5 d}-\frac{3 a^3 \sqrt{a^2+b^2} f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^5 d^2}+\frac{3 a^3 \sqrt{a^2+b^2} f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^5 d^2}+\frac{6 a^3 \sqrt{a^2+b^2} f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^5 d^3}-\frac{6 a^3 \sqrt{a^2+b^2} f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^5 d^3}+\frac{6 a^3 f^3 \sinh (c+d x)}{b^4 d^4}+\frac{14 a f^3 \sinh (c+d x)}{9 b^2 d^4}+\frac{3 a^3 f (e+f x)^2 \sinh (c+d x)}{b^4 d^2}+\frac{2 a f (e+f x)^2 \sinh (c+d x)}{3 b^2 d^2}+\frac{3 a^2 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b^3 d^3}+\frac{a^2 (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b^3 d}+\frac{a f (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^2 d^2}+\frac{2 a f^3 \sinh ^3(c+d x)}{27 b^2 d^4}+\frac{3 f^2 (e+f x) \sinh (4 c+4 d x)}{256 b d^3}+\frac{(e+f x)^3 \sinh (4 c+4 d x)}{32 b d}-\frac{\left (6 a^3 \sqrt{a^2+b^2} f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (\frac{b x}{-a+\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^5 d^4}+\frac{\left (6 a^3 \sqrt{a^2+b^2} f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (-\frac{b x}{a+\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^5 d^4}\\ &=\frac{3 a^2 e f^2 x}{4 b^3 d^2}+\frac{3 a^2 f^3 x^2}{8 b^3 d^2}+\frac{a^4 (e+f x)^4}{4 b^5 f}+\frac{a^2 (e+f x)^4}{8 b^3 f}-\frac{(e+f x)^4}{32 b f}-\frac{6 a^3 f^2 (e+f x) \cosh (c+d x)}{b^4 d^3}-\frac{4 a f^2 (e+f x) \cosh (c+d x)}{3 b^2 d^3}-\frac{a^3 (e+f x)^3 \cosh (c+d x)}{b^4 d}-\frac{3 a^2 f^3 \cosh ^2(c+d x)}{8 b^3 d^4}-\frac{3 a^2 f (e+f x)^2 \cosh ^2(c+d x)}{4 b^3 d^2}-\frac{2 a f^2 (e+f x) \cosh ^3(c+d x)}{9 b^2 d^3}-\frac{a (e+f x)^3 \cosh ^3(c+d x)}{3 b^2 d}-\frac{3 f^3 \cosh (4 c+4 d x)}{1024 b d^4}-\frac{3 f (e+f x)^2 \cosh (4 c+4 d x)}{128 b d^2}-\frac{a^3 \sqrt{a^2+b^2} (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^5 d}+\frac{a^3 \sqrt{a^2+b^2} (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^5 d}-\frac{3 a^3 \sqrt{a^2+b^2} f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^5 d^2}+\frac{3 a^3 \sqrt{a^2+b^2} f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^5 d^2}+\frac{6 a^3 \sqrt{a^2+b^2} f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^5 d^3}-\frac{6 a^3 \sqrt{a^2+b^2} f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^5 d^3}-\frac{6 a^3 \sqrt{a^2+b^2} f^3 \text{Li}_4\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^5 d^4}+\frac{6 a^3 \sqrt{a^2+b^2} f^3 \text{Li}_4\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^5 d^4}+\frac{6 a^3 f^3 \sinh (c+d x)}{b^4 d^4}+\frac{14 a f^3 \sinh (c+d x)}{9 b^2 d^4}+\frac{3 a^3 f (e+f x)^2 \sinh (c+d x)}{b^4 d^2}+\frac{2 a f (e+f x)^2 \sinh (c+d x)}{3 b^2 d^2}+\frac{3 a^2 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b^3 d^3}+\frac{a^2 (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b^3 d}+\frac{a f (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^2 d^2}+\frac{2 a f^3 \sinh ^3(c+d x)}{27 b^2 d^4}+\frac{3 f^2 (e+f x) \sinh (4 c+4 d x)}{256 b d^3}+\frac{(e+f x)^3 \sinh (4 c+4 d x)}{32 b d}\\ \end{align*}

Mathematica [C]  time = 29.1008, size = 7520, normalized size = 7.24 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [F]  time = 0.238, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{3} \left ( \cosh \left ( dx+c \right ) \right ) ^{2} \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)^3*cosh(d*x+c)^2*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [C]  time = 4.20574, size = 22876, normalized size = 22.04 \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)^3*cosh(d*x+c)^2*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-1/55296*(864*b^4*d^3*f^3*x^3 + 864*b^4*d^3*e^3 + 648*b^4*d^2*e^2*f - 27*(32*b^4*d^3*f^3*x^3 + 32*b^4*d^3*e^3
- 24*b^4*d^2*e^2*f + 12*b^4*d*e*f^2 - 3*b^4*f^3 + 24*(4*b^4*d^3*e*f^2 - b^4*d^2*f^3)*x^2 + 12*(8*b^4*d^3*e^2*f
- 4*b^4*d^2*e*f^2 + b^4*d*f^3)*x)*cosh(d*x + c)^8 - 27*(32*b^4*d^3*f^3*x^3 + 32*b^4*d^3*e^3 - 24*b^4*d^2*e^2*
f + 12*b^4*d*e*f^2 - 3*b^4*f^3 + 24*(4*b^4*d^3*e*f^2 - b^4*d^2*f^3)*x^2 + 12*(8*b^4*d^3*e^2*f - 4*b^4*d^2*e*f^
2 + b^4*d*f^3)*x)*sinh(d*x + c)^8 + 324*b^4*d*e*f^2 + 256*(9*a*b^3*d^3*f^3*x^3 + 9*a*b^3*d^3*e^3 - 9*a*b^3*d^2
*e^2*f + 6*a*b^3*d*e*f^2 - 2*a*b^3*f^3 + 9*(3*a*b^3*d^3*e*f^2 - a*b^3*d^2*f^3)*x^2 + 3*(9*a*b^3*d^3*e^2*f - 6*
a*b^3*d^2*e*f^2 + 2*a*b^3*d*f^3)*x)*cosh(d*x + c)^7 + 8*(288*a*b^3*d^3*f^3*x^3 + 288*a*b^3*d^3*e^3 - 288*a*b^3
*d^2*e^2*f + 192*a*b^3*d*e*f^2 - 64*a*b^3*f^3 + 288*(3*a*b^3*d^3*e*f^2 - a*b^3*d^2*f^3)*x^2 + 96*(9*a*b^3*d^3*
e^2*f - 6*a*b^3*d^2*e*f^2 + 2*a*b^3*d*f^3)*x - 27*(32*b^4*d^3*f^3*x^3 + 32*b^4*d^3*e^3 - 24*b^4*d^2*e^2*f + 12
*b^4*d*e*f^2 - 3*b^4*f^3 + 24*(4*b^4*d^3*e*f^2 - b^4*d^2*f^3)*x^2 + 12*(8*b^4*d^3*e^2*f - 4*b^4*d^2*e*f^2 + b^
4*d*f^3)*x)*cosh(d*x + c))*sinh(d*x + c)^7 + 81*b^4*f^3 - 1728*(4*a^2*b^2*d^3*f^3*x^3 + 4*a^2*b^2*d^3*e^3 - 6*
a^2*b^2*d^2*e^2*f + 6*a^2*b^2*d*e*f^2 - 3*a^2*b^2*f^3 + 6*(2*a^2*b^2*d^3*e*f^2 - a^2*b^2*d^2*f^3)*x^2 + 6*(2*a
^2*b^2*d^3*e^2*f - 2*a^2*b^2*d^2*e*f^2 + a^2*b^2*d*f^3)*x)*cosh(d*x + c)^6 - 4*(1728*a^2*b^2*d^3*f^3*x^3 + 172
8*a^2*b^2*d^3*e^3 - 2592*a^2*b^2*d^2*e^2*f + 2592*a^2*b^2*d*e*f^2 - 1296*a^2*b^2*f^3 + 2592*(2*a^2*b^2*d^3*e*f
^2 - a^2*b^2*d^2*f^3)*x^2 + 189*(32*b^4*d^3*f^3*x^3 + 32*b^4*d^3*e^3 - 24*b^4*d^2*e^2*f + 12*b^4*d*e*f^2 - 3*b
^4*f^3 + 24*(4*b^4*d^3*e*f^2 - b^4*d^2*f^3)*x^2 + 12*(8*b^4*d^3*e^2*f - 4*b^4*d^2*e*f^2 + b^4*d*f^3)*x)*cosh(d
*x + c)^2 + 2592*(2*a^2*b^2*d^3*e^2*f - 2*a^2*b^2*d^2*e*f^2 + a^2*b^2*d*f^3)*x - 448*(9*a*b^3*d^3*f^3*x^3 + 9*
a*b^3*d^3*e^3 - 9*a*b^3*d^2*e^2*f + 6*a*b^3*d*e*f^2 - 2*a*b^3*f^3 + 9*(3*a*b^3*d^3*e*f^2 - a*b^3*d^2*f^3)*x^2
+ 3*(9*a*b^3*d^3*e^2*f - 6*a*b^3*d^2*e*f^2 + 2*a*b^3*d*f^3)*x)*cosh(d*x + c))*sinh(d*x + c)^6 + 6912*((4*a^3*b
+ a*b^3)*d^3*f^3*x^3 + (4*a^3*b + a*b^3)*d^3*e^3 - 3*(4*a^3*b + a*b^3)*d^2*e^2*f + 6*(4*a^3*b + a*b^3)*d*e*f^
2 - 6*(4*a^3*b + a*b^3)*f^3 + 3*((4*a^3*b + a*b^3)*d^3*e*f^2 - (4*a^3*b + a*b^3)*d^2*f^3)*x^2 + 3*((4*a^3*b +
a*b^3)*d^3*e^2*f - 2*(4*a^3*b + a*b^3)*d^2*e*f^2 + 2*(4*a^3*b + a*b^3)*d*f^3)*x)*cosh(d*x + c)^5 + 24*(288*(4*
a^3*b + a*b^3)*d^3*f^3*x^3 + 288*(4*a^3*b + a*b^3)*d^3*e^3 - 864*(4*a^3*b + a*b^3)*d^2*e^2*f + 1728*(4*a^3*b +
a*b^3)*d*e*f^2 - 1728*(4*a^3*b + a*b^3)*f^3 - 63*(32*b^4*d^3*f^3*x^3 + 32*b^4*d^3*e^3 - 24*b^4*d^2*e^2*f + 12
*b^4*d*e*f^2 - 3*b^4*f^3 + 24*(4*b^4*d^3*e*f^2 - b^4*d^2*f^3)*x^2 + 12*(8*b^4*d^3*e^2*f - 4*b^4*d^2*e*f^2 + b^
4*d*f^3)*x)*cosh(d*x + c)^3 + 864*((4*a^3*b + a*b^3)*d^3*e*f^2 - (4*a^3*b + a*b^3)*d^2*f^3)*x^2 + 224*(9*a*b^3
*d^3*f^3*x^3 + 9*a*b^3*d^3*e^3 - 9*a*b^3*d^2*e^2*f + 6*a*b^3*d*e*f^2 - 2*a*b^3*f^3 + 9*(3*a*b^3*d^3*e*f^2 - a*
b^3*d^2*f^3)*x^2 + 3*(9*a*b^3*d^3*e^2*f - 6*a*b^3*d^2*e*f^2 + 2*a*b^3*d*f^3)*x)*cosh(d*x + c)^2 + 864*((4*a^3*
b + a*b^3)*d^3*e^2*f - 2*(4*a^3*b + a*b^3)*d^2*e*f^2 + 2*(4*a^3*b + a*b^3)*d*f^3)*x - 432*(4*a^2*b^2*d^3*f^3*x
^3 + 4*a^2*b^2*d^3*e^3 - 6*a^2*b^2*d^2*e^2*f + 6*a^2*b^2*d*e*f^2 - 3*a^2*b^2*f^3 + 6*(2*a^2*b^2*d^3*e*f^2 - a^
2*b^2*d^2*f^3)*x^2 + 6*(2*a^2*b^2*d^3*e^2*f - 2*a^2*b^2*d^2*e*f^2 + a^2*b^2*d*f^3)*x)*cosh(d*x + c))*sinh(d*x
+ c)^5 - 1728*((8*a^4 + 4*a^2*b^2 - b^4)*d^4*f^3*x^4 + 4*(8*a^4 + 4*a^2*b^2 - b^4)*d^4*e*f^2*x^3 + 6*(8*a^4 +
4*a^2*b^2 - b^4)*d^4*e^2*f*x^2 + 4*(8*a^4 + 4*a^2*b^2 - b^4)*d^4*e^3*x)*cosh(d*x + c)^4 - 2*(864*(8*a^4 + 4*a^
2*b^2 - b^4)*d^4*f^3*x^4 + 3456*(8*a^4 + 4*a^2*b^2 - b^4)*d^4*e*f^2*x^3 + 5184*(8*a^4 + 4*a^2*b^2 - b^4)*d^4*e
^2*f*x^2 + 3456*(8*a^4 + 4*a^2*b^2 - b^4)*d^4*e^3*x + 945*(32*b^4*d^3*f^3*x^3 + 32*b^4*d^3*e^3 - 24*b^4*d^2*e^
2*f + 12*b^4*d*e*f^2 - 3*b^4*f^3 + 24*(4*b^4*d^3*e*f^2 - b^4*d^2*f^3)*x^2 + 12*(8*b^4*d^3*e^2*f - 4*b^4*d^2*e*
f^2 + b^4*d*f^3)*x)*cosh(d*x + c)^4 - 4480*(9*a*b^3*d^3*f^3*x^3 + 9*a*b^3*d^3*e^3 - 9*a*b^3*d^2*e^2*f + 6*a*b^
3*d*e*f^2 - 2*a*b^3*f^3 + 9*(3*a*b^3*d^3*e*f^2 - a*b^3*d^2*f^3)*x^2 + 3*(9*a*b^3*d^3*e^2*f - 6*a*b^3*d^2*e*f^2
+ 2*a*b^3*d*f^3)*x)*cosh(d*x + c)^3 + 12960*(4*a^2*b^2*d^3*f^3*x^3 + 4*a^2*b^2*d^3*e^3 - 6*a^2*b^2*d^2*e^2*f
+ 6*a^2*b^2*d*e*f^2 - 3*a^2*b^2*f^3 + 6*(2*a^2*b^2*d^3*e*f^2 - a^2*b^2*d^2*f^3)*x^2 + 6*(2*a^2*b^2*d^3*e^2*f -
2*a^2*b^2*d^2*e*f^2 + a^2*b^2*d*f^3)*x)*cosh(d*x + c)^2 - 17280*((4*a^3*b + a*b^3)*d^3*f^3*x^3 + (4*a^3*b + a
*b^3)*d^3*e^3 - 3*(4*a^3*b + a*b^3)*d^2*e^2*f + 6*(4*a^3*b + a*b^3)*d*e*f^2 - 6*(4*a^3*b + a*b^3)*f^3 + 3*((4*
a^3*b + a*b^3)*d^3*e*f^2 - (4*a^3*b + a*b^3)*d^2*f^3)*x^2 + 3*((4*a^3*b + a*b^3)*d^3*e^2*f - 2*(4*a^3*b + a*b^
3)*d^2*e*f^2 + 2*(4*a^3*b + a*b^3)*d*f^3)*x)*cosh(d*x + c))*sinh(d*x + c)^4 + 6912*((4*a^3*b + a*b^3)*d^3*f^3*
x^3 + (4*a^3*b + a*b^3)*d^3*e^3 + 3*(4*a^3*b + a*b^3)*d^2*e^2*f + 6*(4*a^3*b + a*b^3)*d*e*f^2 + 6*(4*a^3*b + a
*b^3)*f^3 + 3*((4*a^3*b + a*b^3)*d^3*e*f^2 + (4*a^3*b + a*b^3)*d^2*f^3)*x^2 + 3*((4*a^3*b + a*b^3)*d^3*e^2*f +
2*(4*a^3*b + a*b^3)*d^2*e*f^2 + 2*(4*a^3*b + a*b^3)*d*f^3)*x)*cosh(d*x + c)^3 + 8*(864*(4*a^3*b + a*b^3)*d^3*
f^3*x^3 + 864*(4*a^3*b + a*b^3)*d^3*e^3 + 2592*(4*a^3*b + a*b^3)*d^2*e^2*f - 189*(32*b^4*d^3*f^3*x^3 + 32*b^4*
d^3*e^3 - 24*b^4*d^2*e^2*f + 12*b^4*d*e*f^2 - 3*b^4*f^3 + 24*(4*b^4*d^3*e*f^2 - b^4*d^2*f^3)*x^2 + 12*(8*b^4*d
^3*e^2*f - 4*b^4*d^2*e*f^2 + b^4*d*f^3)*x)*cosh(d*x + c)^5 + 5184*(4*a^3*b + a*b^3)*d*e*f^2 + 1120*(9*a*b^3*d^
3*f^3*x^3 + 9*a*b^3*d^3*e^3 - 9*a*b^3*d^2*e^2*f + 6*a*b^3*d*e*f^2 - 2*a*b^3*f^3 + 9*(3*a*b^3*d^3*e*f^2 - a*b^3
*d^2*f^3)*x^2 + 3*(9*a*b^3*d^3*e^2*f - 6*a*b^3*d^2*e*f^2 + 2*a*b^3*d*f^3)*x)*cosh(d*x + c)^4 + 5184*(4*a^3*b +
a*b^3)*f^3 - 4320*(4*a^2*b^2*d^3*f^3*x^3 + 4*a^2*b^2*d^3*e^3 - 6*a^2*b^2*d^2*e^2*f + 6*a^2*b^2*d*e*f^2 - 3*a^
2*b^2*f^3 + 6*(2*a^2*b^2*d^3*e*f^2 - a^2*b^2*d^2*f^3)*x^2 + 6*(2*a^2*b^2*d^3*e^2*f - 2*a^2*b^2*d^2*e*f^2 + a^2
*b^2*d*f^3)*x)*cosh(d*x + c)^3 + 2592*((4*a^3*b + a*b^3)*d^3*e*f^2 + (4*a^3*b + a*b^3)*d^2*f^3)*x^2 + 8640*((4
*a^3*b + a*b^3)*d^3*f^3*x^3 + (4*a^3*b + a*b^3)*d^3*e^3 - 3*(4*a^3*b + a*b^3)*d^2*e^2*f + 6*(4*a^3*b + a*b^3)*
d*e*f^2 - 6*(4*a^3*b + a*b^3)*f^3 + 3*((4*a^3*b + a*b^3)*d^3*e*f^2 - (4*a^3*b + a*b^3)*d^2*f^3)*x^2 + 3*((4*a^
3*b + a*b^3)*d^3*e^2*f - 2*(4*a^3*b + a*b^3)*d^2*e*f^2 + 2*(4*a^3*b + a*b^3)*d*f^3)*x)*cosh(d*x + c)^2 + 2592*
((4*a^3*b + a*b^3)*d^3*e^2*f + 2*(4*a^3*b + a*b^3)*d^2*e*f^2 + 2*(4*a^3*b + a*b^3)*d*f^3)*x - 864*((8*a^4 + 4*
a^2*b^2 - b^4)*d^4*f^3*x^4 + 4*(8*a^4 + 4*a^2*b^2 - b^4)*d^4*e*f^2*x^3 + 6*(8*a^4 + 4*a^2*b^2 - b^4)*d^4*e^2*f
*x^2 + 4*(8*a^4 + 4*a^2*b^2 - b^4)*d^4*e^3*x)*cosh(d*x + c))*sinh(d*x + c)^3 + 648*(4*b^4*d^3*e*f^2 + b^4*d^2*
f^3)*x^2 + 1728*(4*a^2*b^2*d^3*f^3*x^3 + 4*a^2*b^2*d^3*e^3 + 6*a^2*b^2*d^2*e^2*f + 6*a^2*b^2*d*e*f^2 + 3*a^2*b
^2*f^3 + 6*(2*a^2*b^2*d^3*e*f^2 + a^2*b^2*d^2*f^3)*x^2 + 6*(2*a^2*b^2*d^3*e^2*f + 2*a^2*b^2*d^2*e*f^2 + a^2*b^
2*d*f^3)*x)*cosh(d*x + c)^2 + 12*(576*a^2*b^2*d^3*f^3*x^3 + 576*a^2*b^2*d^3*e^3 + 864*a^2*b^2*d^2*e^2*f + 864*
a^2*b^2*d*e*f^2 + 432*a^2*b^2*f^3 - 63*(32*b^4*d^3*f^3*x^3 + 32*b^4*d^3*e^3 - 24*b^4*d^2*e^2*f + 12*b^4*d*e*f^
2 - 3*b^4*f^3 + 24*(4*b^4*d^3*e*f^2 - b^4*d^2*f^3)*x^2 + 12*(8*b^4*d^3*e^2*f - 4*b^4*d^2*e*f^2 + b^4*d*f^3)*x)
*cosh(d*x + c)^6 + 448*(9*a*b^3*d^3*f^3*x^3 + 9*a*b^3*d^3*e^3 - 9*a*b^3*d^2*e^2*f + 6*a*b^3*d*e*f^2 - 2*a*b^3*
f^3 + 9*(3*a*b^3*d^3*e*f^2 - a*b^3*d^2*f^3)*x^2 + 3*(9*a*b^3*d^3*e^2*f - 6*a*b^3*d^2*e*f^2 + 2*a*b^3*d*f^3)*x)
*cosh(d*x + c)^5 - 2160*(4*a^2*b^2*d^3*f^3*x^3 + 4*a^2*b^2*d^3*e^3 - 6*a^2*b^2*d^2*e^2*f + 6*a^2*b^2*d*e*f^2 -
3*a^2*b^2*f^3 + 6*(2*a^2*b^2*d^3*e*f^2 - a^2*b^2*d^2*f^3)*x^2 + 6*(2*a^2*b^2*d^3*e^2*f - 2*a^2*b^2*d^2*e*f^2
+ a^2*b^2*d*f^3)*x)*cosh(d*x + c)^4 + 5760*((4*a^3*b + a*b^3)*d^3*f^3*x^3 + (4*a^3*b + a*b^3)*d^3*e^3 - 3*(4*a
^3*b + a*b^3)*d^2*e^2*f + 6*(4*a^3*b + a*b^3)*d*e*f^2 - 6*(4*a^3*b + a*b^3)*f^3 + 3*((4*a^3*b + a*b^3)*d^3*e*f
^2 - (4*a^3*b + a*b^3)*d^2*f^3)*x^2 + 3*((4*a^3*b + a*b^3)*d^3*e^2*f - 2*(4*a^3*b + a*b^3)*d^2*e*f^2 + 2*(4*a^
3*b + a*b^3)*d*f^3)*x)*cosh(d*x + c)^3 + 864*(2*a^2*b^2*d^3*e*f^2 + a^2*b^2*d^2*f^3)*x^2 - 864*((8*a^4 + 4*a^2
*b^2 - b^4)*d^4*f^3*x^4 + 4*(8*a^4 + 4*a^2*b^2 - b^4)*d^4*e*f^2*x^3 + 6*(8*a^4 + 4*a^2*b^2 - b^4)*d^4*e^2*f*x^
2 + 4*(8*a^4 + 4*a^2*b^2 - b^4)*d^4*e^3*x)*cosh(d*x + c)^2 + 864*(2*a^2*b^2*d^3*e^2*f + 2*a^2*b^2*d^2*e*f^2 +
a^2*b^2*d*f^3)*x + 1728*((4*a^3*b + a*b^3)*d^3*f^3*x^3 + (4*a^3*b + a*b^3)*d^3*e^3 + 3*(4*a^3*b + a*b^3)*d^2*e
^2*f + 6*(4*a^3*b + a*b^3)*d*e*f^2 + 6*(4*a^3*b + a*b^3)*f^3 + 3*((4*a^3*b + a*b^3)*d^3*e*f^2 + (4*a^3*b + a*b
^3)*d^2*f^3)*x^2 + 3*((4*a^3*b + a*b^3)*d^3*e^2*f + 2*(4*a^3*b + a*b^3)*d^2*e*f^2 + 2*(4*a^3*b + a*b^3)*d*f^3)
*x)*cosh(d*x + c))*sinh(d*x + c)^2 + 165888*((a^3*b*d^2*f^3*x^2 + 2*a^3*b*d^2*e*f^2*x + a^3*b*d^2*e^2*f)*cosh(
d*x + c)^4 + 4*(a^3*b*d^2*f^3*x^2 + 2*a^3*b*d^2*e*f^2*x + a^3*b*d^2*e^2*f)*cosh(d*x + c)^3*sinh(d*x + c) + 6*(
a^3*b*d^2*f^3*x^2 + 2*a^3*b*d^2*e*f^2*x + a^3*b*d^2*e^2*f)*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*(a^3*b*d^2*f^3*
x^2 + 2*a^3*b*d^2*e*f^2*x + a^3*b*d^2*e^2*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^3*b*d^2*f^3*x^2 + 2*a^3*b*d^2*
e*f^2*x + a^3*b*d^2*e^2*f)*sinh(d*x + c)^4)*sqrt((a^2 + b^2)/b^2)*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) - 165888*((a^3*b*d^2*f^3*x^2 + 2*a^3*b*d^
2*e*f^2*x + a^3*b*d^2*e^2*f)*cosh(d*x + c)^4 + 4*(a^3*b*d^2*f^3*x^2 + 2*a^3*b*d^2*e*f^2*x + a^3*b*d^2*e^2*f)*c
osh(d*x + c)^3*sinh(d*x + c) + 6*(a^3*b*d^2*f^3*x^2 + 2*a^3*b*d^2*e*f^2*x + a^3*b*d^2*e^2*f)*cosh(d*x + c)^2*s
inh(d*x + c)^2 + 4*(a^3*b*d^2*f^3*x^2 + 2*a^3*b*d^2*e*f^2*x + a^3*b*d^2*e^2*f)*cosh(d*x + c)*sinh(d*x + c)^3 +
(a^3*b*d^2*f^3*x^2 + 2*a^3*b*d^2*e*f^2*x + a^3*b*d^2*e^2*f)*sinh(d*x + c)^4)*sqrt((a^2 + b^2)/b^2)*dilog((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)/b + 1) - 55296
*((a^3*b*d^3*e^3 - 3*a^3*b*c*d^2*e^2*f + 3*a^3*b*c^2*d*e*f^2 - a^3*b*c^3*f^3)*cosh(d*x + c)^4 + 4*(a^3*b*d^3*e
^3 - 3*a^3*b*c*d^2*e^2*f + 3*a^3*b*c^2*d*e*f^2 - a^3*b*c^3*f^3)*cosh(d*x + c)^3*sinh(d*x + c) + 6*(a^3*b*d^3*e
^3 - 3*a^3*b*c*d^2*e^2*f + 3*a^3*b*c^2*d*e*f^2 - a^3*b*c^3*f^3)*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*(a^3*b*d^3
*e^3 - 3*a^3*b*c*d^2*e^2*f + 3*a^3*b*c^2*d*e*f^2 - a^3*b*c^3*f^3)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^3*b*d^3*e
^3 - 3*a^3*b*c*d^2*e^2*f + 3*a^3*b*c^2*d*e*f^2 - a^3*b*c^3*f^3)*sinh(d*x + c)^4)*sqrt((a^2 + b^2)/b^2)*log(2*b
*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + 55296*((a^3*b*d^3*e^3 - 3*a^3*b*c*d^2*
e^2*f + 3*a^3*b*c^2*d*e*f^2 - a^3*b*c^3*f^3)*cosh(d*x + c)^4 + 4*(a^3*b*d^3*e^3 - 3*a^3*b*c*d^2*e^2*f + 3*a^3*
b*c^2*d*e*f^2 - a^3*b*c^3*f^3)*cosh(d*x + c)^3*sinh(d*x + c) + 6*(a^3*b*d^3*e^3 - 3*a^3*b*c*d^2*e^2*f + 3*a^3*
b*c^2*d*e*f^2 - a^3*b*c^3*f^3)*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*(a^3*b*d^3*e^3 - 3*a^3*b*c*d^2*e^2*f + 3*a^
3*b*c^2*d*e*f^2 - a^3*b*c^3*f^3)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^3*b*d^3*e^3 - 3*a^3*b*c*d^2*e^2*f + 3*a^3*
b*c^2*d*e*f^2 - a^3*b*c^3*f^3)*sinh(d*x + c)^4)*sqrt((a^2 + b^2)/b^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c
) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + 55296*((a^3*b*d^3*f^3*x^3 + 3*a^3*b*d^3*e*f^2*x^2 + 3*a^3*b*d^3*e^2*f*x
+ 3*a^3*b*c*d^2*e^2*f - 3*a^3*b*c^2*d*e*f^2 + a^3*b*c^3*f^3)*cosh(d*x + c)^4 + 4*(a^3*b*d^3*f^3*x^3 + 3*a^3*b
*d^3*e*f^2*x^2 + 3*a^3*b*d^3*e^2*f*x + 3*a^3*b*c*d^2*e^2*f - 3*a^3*b*c^2*d*e*f^2 + a^3*b*c^3*f^3)*cosh(d*x + c
)^3*sinh(d*x + c) + 6*(a^3*b*d^3*f^3*x^3 + 3*a^3*b*d^3*e*f^2*x^2 + 3*a^3*b*d^3*e^2*f*x + 3*a^3*b*c*d^2*e^2*f -
3*a^3*b*c^2*d*e*f^2 + a^3*b*c^3*f^3)*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*(a^3*b*d^3*f^3*x^3 + 3*a^3*b*d^3*e*f
^2*x^2 + 3*a^3*b*d^3*e^2*f*x + 3*a^3*b*c*d^2*e^2*f - 3*a^3*b*c^2*d*e*f^2 + a^3*b*c^3*f^3)*cosh(d*x + c)*sinh(d
*x + c)^3 + (a^3*b*d^3*f^3*x^3 + 3*a^3*b*d^3*e*f^2*x^2 + 3*a^3*b*d^3*e^2*f*x + 3*a^3*b*c*d^2*e^2*f - 3*a^3*b*c
^2*d*e*f^2 + a^3*b*c^3*f^3)*sinh(d*x + c)^4)*sqrt((a^2 + b^2)/b^2)*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) - 55296*((a^3*b*d^3*f^3*x^3 + 3*a^3*b*d^3*e*f
^2*x^2 + 3*a^3*b*d^3*e^2*f*x + 3*a^3*b*c*d^2*e^2*f - 3*a^3*b*c^2*d*e*f^2 + a^3*b*c^3*f^3)*cosh(d*x + c)^4 + 4*
(a^3*b*d^3*f^3*x^3 + 3*a^3*b*d^3*e*f^2*x^2 + 3*a^3*b*d^3*e^2*f*x + 3*a^3*b*c*d^2*e^2*f - 3*a^3*b*c^2*d*e*f^2 +
a^3*b*c^3*f^3)*cosh(d*x + c)^3*sinh(d*x + c) + 6*(a^3*b*d^3*f^3*x^3 + 3*a^3*b*d^3*e*f^2*x^2 + 3*a^3*b*d^3*e^2
*f*x + 3*a^3*b*c*d^2*e^2*f - 3*a^3*b*c^2*d*e*f^2 + a^3*b*c^3*f^3)*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*(a^3*b*d
^3*f^3*x^3 + 3*a^3*b*d^3*e*f^2*x^2 + 3*a^3*b*d^3*e^2*f*x + 3*a^3*b*c*d^2*e^2*f - 3*a^3*b*c^2*d*e*f^2 + a^3*b*c
^3*f^3)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^3*b*d^3*f^3*x^3 + 3*a^3*b*d^3*e*f^2*x^2 + 3*a^3*b*d^3*e^2*f*x + 3*a
^3*b*c*d^2*e^2*f - 3*a^3*b*c^2*d*e*f^2 + a^3*b*c^3*f^3)*sinh(d*x + c)^4)*sqrt((a^2 + b^2)/b^2)*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) + 331776*(a^3*b*f
^3*cosh(d*x + c)^4 + 4*a^3*b*f^3*cosh(d*x + c)^3*sinh(d*x + c) + 6*a^3*b*f^3*cosh(d*x + c)^2*sinh(d*x + c)^2 +
4*a^3*b*f^3*cosh(d*x + c)*sinh(d*x + c)^3 + a^3*b*f^3*sinh(d*x + c)^4)*sqrt((a^2 + b^2)/b^2)*polylog(4, (a*co
sh(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) - 331776*(a^3*b*
f^3*cosh(d*x + c)^4 + 4*a^3*b*f^3*cosh(d*x + c)^3*sinh(d*x + c) + 6*a^3*b*f^3*cosh(d*x + c)^2*sinh(d*x + c)^2
+ 4*a^3*b*f^3*cosh(d*x + c)*sinh(d*x + c)^3 + a^3*b*f^3*sinh(d*x + c)^4)*sqrt((a^2 + b^2)/b^2)*polylog(4, (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) - 331776*((a^3*
b*d*f^3*x + a^3*b*d*e*f^2)*cosh(d*x + c)^4 + 4*(a^3*b*d*f^3*x + a^3*b*d*e*f^2)*cosh(d*x + c)^3*sinh(d*x + c) +
6*(a^3*b*d*f^3*x + a^3*b*d*e*f^2)*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*(a^3*b*d*f^3*x + a^3*b*d*e*f^2)*cosh(d*
x + c)*sinh(d*x + c)^3 + (a^3*b*d*f^3*x + a^3*b*d*e*f^2)*sinh(d*x + c)^4)*sqrt((a^2 + b^2)/b^2)*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) + 331776*((a^3
*b*d*f^3*x + a^3*b*d*e*f^2)*cosh(d*x + c)^4 + 4*(a^3*b*d*f^3*x + a^3*b*d*e*f^2)*cosh(d*x + c)^3*sinh(d*x + c)
+ 6*(a^3*b*d*f^3*x + a^3*b*d*e*f^2)*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*(a^3*b*d*f^3*x + a^3*b*d*e*f^2)*cosh(d
*x + c)*sinh(d*x + c)^3 + (a^3*b*d*f^3*x + a^3*b*d*e*f^2)*sinh(d*x + c)^4)*sqrt((a^2 + b^2)/b^2)*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) + 324*(8*b^4*
d^3*e^2*f + 4*b^4*d^2*e*f^2 + b^4*d*f^3)*x + 256*(9*a*b^3*d^3*f^3*x^3 + 9*a*b^3*d^3*e^3 + 9*a*b^3*d^2*e^2*f +
6*a*b^3*d*e*f^2 + 2*a*b^3*f^3 + 9*(3*a*b^3*d^3*e*f^2 + a*b^3*d^2*f^3)*x^2 + 3*(9*a*b^3*d^3*e^2*f + 6*a*b^3*d^2
*e*f^2 + 2*a*b^3*d*f^3)*x)*cosh(d*x + c) + 8*(288*a*b^3*d^3*f^3*x^3 + 288*a*b^3*d^3*e^3 + 288*a*b^3*d^2*e^2*f
+ 192*a*b^3*d*e*f^2 - 27*(32*b^4*d^3*f^3*x^3 + 32*b^4*d^3*e^3 - 24*b^4*d^2*e^2*f + 12*b^4*d*e*f^2 - 3*b^4*f^3
+ 24*(4*b^4*d^3*e*f^2 - b^4*d^2*f^3)*x^2 + 12*(8*b^4*d^3*e^2*f - 4*b^4*d^2*e*f^2 + b^4*d*f^3)*x)*cosh(d*x + c)
^7 + 64*a*b^3*f^3 + 224*(9*a*b^3*d^3*f^3*x^3 + 9*a*b^3*d^3*e^3 - 9*a*b^3*d^2*e^2*f + 6*a*b^3*d*e*f^2 - 2*a*b^3
*f^3 + 9*(3*a*b^3*d^3*e*f^2 - a*b^3*d^2*f^3)*x^2 + 3*(9*a*b^3*d^3*e^2*f - 6*a*b^3*d^2*e*f^2 + 2*a*b^3*d*f^3)*x
)*cosh(d*x + c)^6 - 1296*(4*a^2*b^2*d^3*f^3*x^3 + 4*a^2*b^2*d^3*e^3 - 6*a^2*b^2*d^2*e^2*f + 6*a^2*b^2*d*e*f^2
- 3*a^2*b^2*f^3 + 6*(2*a^2*b^2*d^3*e*f^2 - a^2*b^2*d^2*f^3)*x^2 + 6*(2*a^2*b^2*d^3*e^2*f - 2*a^2*b^2*d^2*e*f^2
+ a^2*b^2*d*f^3)*x)*cosh(d*x + c)^5 + 4320*((4*a^3*b + a*b^3)*d^3*f^3*x^3 + (4*a^3*b + a*b^3)*d^3*e^3 - 3*(4*
a^3*b + a*b^3)*d^2*e^2*f + 6*(4*a^3*b + a*b^3)*d*e*f^2 - 6*(4*a^3*b + a*b^3)*f^3 + 3*((4*a^3*b + a*b^3)*d^3*e*
f^2 - (4*a^3*b + a*b^3)*d^2*f^3)*x^2 + 3*((4*a^3*b + a*b^3)*d^3*e^2*f - 2*(4*a^3*b + a*b^3)*d^2*e*f^2 + 2*(4*a
^3*b + a*b^3)*d*f^3)*x)*cosh(d*x + c)^4 - 864*((8*a^4 + 4*a^2*b^2 - b^4)*d^4*f^3*x^4 + 4*(8*a^4 + 4*a^2*b^2 -
b^4)*d^4*e*f^2*x^3 + 6*(8*a^4 + 4*a^2*b^2 - b^4)*d^4*e^2*f*x^2 + 4*(8*a^4 + 4*a^2*b^2 - b^4)*d^4*e^3*x)*cosh(d
*x + c)^3 + 288*(3*a*b^3*d^3*e*f^2 + a*b^3*d^2*f^3)*x^2 + 2592*((4*a^3*b + a*b^3)*d^3*f^3*x^3 + (4*a^3*b + a*b
^3)*d^3*e^3 + 3*(4*a^3*b + a*b^3)*d^2*e^2*f + 6*(4*a^3*b + a*b^3)*d*e*f^2 + 6*(4*a^3*b + a*b^3)*f^3 + 3*((4*a^
3*b + a*b^3)*d^3*e*f^2 + (4*a^3*b + a*b^3)*d^2*f^3)*x^2 + 3*((4*a^3*b + a*b^3)*d^3*e^2*f + 2*(4*a^3*b + a*b^3)
*d^2*e*f^2 + 2*(4*a^3*b + a*b^3)*d*f^3)*x)*cosh(d*x + c)^2 + 96*(9*a*b^3*d^3*e^2*f + 6*a*b^3*d^2*e*f^2 + 2*a*b
^3*d*f^3)*x + 432*(4*a^2*b^2*d^3*f^3*x^3 + 4*a^2*b^2*d^3*e^3 + 6*a^2*b^2*d^2*e^2*f + 6*a^2*b^2*d*e*f^2 + 3*a^2
*b^2*f^3 + 6*(2*a^2*b^2*d^3*e*f^2 + a^2*b^2*d^2*f^3)*x^2 + 6*(2*a^2*b^2*d^3*e^2*f + 2*a^2*b^2*d^2*e*f^2 + a^2*
b^2*d*f^3)*x)*cosh(d*x + c))*sinh(d*x + c))/(b^5*d^4*cosh(d*x + c)^4 + 4*b^5*d^4*cosh(d*x + c)^3*sinh(d*x + c)
+ 6*b^5*d^4*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*b^5*d^4*cosh(d*x + c)*sinh(d*x + c)^3 + b^5*d^4*sinh(d*x + c)
^4)

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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)**3*cosh(d*x+c)**2*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 )}^{3} \cosh \left (d x + c\right )^{2} \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)^3*cosh(d*x+c)^2*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

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