\(\int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [396]

Optimal result

Integrand size = 36, antiderivative size = 1038 \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\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 \operatorname {PolyLog}\left (2,-\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 \operatorname {PolyLog}\left (2,-\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) \operatorname {PolyLog}\left (3,-\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) \operatorname {PolyLog}\left (3,-\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 \operatorname {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 \operatorname {PolyLog}\left (4,-\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} \]

[Out]

-a^3*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/b^5/d+a^3*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a
+(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/b^5/d-a^3*(f*x+e)^3*cosh(d*x+c)/b^4/d-4/3*a*f^2*(f*x+e)*cosh(d*x+c)/b^2/d^3
+2/3*a*f*(f*x+e)^2*sinh(d*x+c)/b^2/d^2+14/9*a*f^3*sinh(d*x+c)/b^2/d^4+3/8*a^2*f^3*x^2/b^3/d^2-3/8*a^2*f^3*cosh
(d*x+c)^2/b^3/d^4-1/3*a*(f*x+e)^3*cosh(d*x+c)^3/b^2/d-3/128*f*(f*x+e)^2*cosh(4*d*x+4*c)/b/d^2+6*a^3*f^3*sinh(d
*x+c)/b^4/d^4+2/27*a*f^3*sinh(d*x+c)^3/b^2/d^4+3/256*f^2*(f*x+e)*sinh(4*d*x+4*c)/b/d^3-1/32*(f*x+e)^4/b/f+3/4*
a^2*e*f^2*x/b^3/d^2-6*a^3*f^2*(f*x+e)*cosh(d*x+c)/b^4/d^3-3/4*a^2*f*(f*x+e)^2*cosh(d*x+c)^2/b^3/d^2-2/9*a*f^2*
(f*x+e)*cosh(d*x+c)^3/b^2/d^3+3*a^3*f*(f*x+e)^2*sinh(d*x+c)/b^4/d^2+1/2*a^2*(f*x+e)^3*cosh(d*x+c)*sinh(d*x+c)/
b^3/d-6*a^3*f^3*polylog(4,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/b^5/d^4+6*a^3*f^3*polylog(4,-b*ex
p(d*x+c)/(a+(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/b^5/d^4+3/4*a^2*f^2*(f*x+e)*cosh(d*x+c)*sinh(d*x+c)/b^3/d^3+1/3*
a*f*(f*x+e)^2*cosh(d*x+c)^2*sinh(d*x+c)/b^2/d^2-3*a^3*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))
*(a^2+b^2)^(1/2)/b^5/d^2+3*a^3*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/b^5/d^
2+6*a^3*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/b^5/d^3-6*a^3*f^2*(f*x+e)*pol
ylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/b^5/d^3+1/4*a^4*(f*x+e)^4/b^5/f-3/1024*f^3*cosh(4*d*
x+4*c)/b/d^4+1/32*(f*x+e)^3*sinh(4*d*x+4*c)/b/d+1/8*a^2*(f*x+e)^4/b^3/f

Rubi [A] (verified)

Time = 1.23 (sec) , antiderivative size = 1038, normalized size of antiderivative = 1.00, number of steps used = 38, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5698, 5556, 3377, 2718, 5555, 3392, 2717, 2713, 32, 3391, 5684, 3403, 2296, 2221, 2611, 6744, 2320, 6724} \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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} \]

[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 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 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2296

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 2320

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 2611

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 2713

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 2717

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

Rule 2718

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

Rule 3377

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 3391

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 3392

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 3403

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 5555

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 5556

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 5684

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 5698

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 6724

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]

Rule 6744

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]

Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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 \operatorname {PolyLog}\left (2,-\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 \operatorname {PolyLog}\left (2,-\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) \operatorname {PolyLog}\left (2,-\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) \operatorname {PolyLog}\left (2,-\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 \operatorname {PolyLog}\left (2,-\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 \operatorname {PolyLog}\left (2,-\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) \operatorname {PolyLog}\left (3,-\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) \operatorname {PolyLog}\left (3,-\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 \operatorname {PolyLog}\left (3,-\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 \operatorname {PolyLog}\left (3,-\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 \operatorname {PolyLog}\left (2,-\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 \operatorname {PolyLog}\left (2,-\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) \operatorname {PolyLog}\left (3,-\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) \operatorname {PolyLog}\left (3,-\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 ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,\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 ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,-\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 \operatorname {PolyLog}\left (2,-\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 \operatorname {PolyLog}\left (2,-\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) \operatorname {PolyLog}\left (3,-\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) \operatorname {PolyLog}\left (3,-\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 \operatorname {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 \operatorname {PolyLog}\left (4,-\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 [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(5984\) vs. \(2(1038)=2076\).

Time = 21.24 (sec) , antiderivative size = 5984, normalized size of antiderivative = 5.76 \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \]

[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

Maple [F]

\[\int \frac {\left (f x +e \right )^{3} \cosh \left (d x +c \right )^{2} \sinh \left (d x +c \right )^{3}}{a +b \sinh \left (d x +c \right )}d x\]

[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)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 10658 vs. \(2 (956) = 1912\).

Time = 0.45 (sec) , antiderivative size = 10658, normalized size of antiderivative = 10.27 \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

[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]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]

[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

Maxima [F]

\[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \]

[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]

-1/192*e^3*(192*sqrt(a^2 + b^2)*a^3*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt(a^2
+ b^2)))/(b^5*d) + (8*a*b^2*e^(-d*x - c) - 24*a^2*b*e^(-2*d*x - 2*c) - 3*b^3 + 24*(4*a^3 + a*b^2)*e^(-3*d*x -
3*c))*e^(4*d*x + 4*c)/(b^4*d) - 24*(8*a^4 + 4*a^2*b^2 - b^4)*(d*x + c)/(b^5*d) + (24*a^2*b*e^(-2*d*x - 2*c) +
8*a*b^2*e^(-3*d*x - 3*c) + 3*b^3*e^(-4*d*x - 4*c) + 24*(4*a^3 + a*b^2)*e^(-d*x - c))/(b^4*d)) + 1/55296*(1728*
(8*a^4*d^4*f^3*e^(4*c) + 4*a^2*b^2*d^4*f^3*e^(4*c) - b^4*d^4*f^3*e^(4*c))*x^4 + 6912*(8*a^4*d^4*e*f^2*e^(4*c)
+ 4*a^2*b^2*d^4*e*f^2*e^(4*c) - b^4*d^4*e*f^2*e^(4*c))*x^3 + 10368*(8*a^4*d^4*e^2*f*e^(4*c) + 4*a^2*b^2*d^4*e^
2*f*e^(4*c) - b^4*d^4*e^2*f*e^(4*c))*x^2 + 27*(32*b^4*d^3*f^3*x^3*e^(8*c) + 24*(4*d^3*e*f^2 - d^2*f^3)*b^4*x^2
*e^(8*c) + 12*(8*d^3*e^2*f - 4*d^2*e*f^2 + d*f^3)*b^4*x*e^(8*c) - 3*(8*d^2*e^2*f - 4*d*e*f^2 + f^3)*b^4*e^(8*c
))*e^(4*d*x) - 256*(9*a*b^3*d^3*f^3*x^3*e^(7*c) + 9*(3*d^3*e*f^2 - d^2*f^3)*a*b^3*x^2*e^(7*c) + 3*(9*d^3*e^2*f
- 6*d^2*e*f^2 + 2*d*f^3)*a*b^3*x*e^(7*c) - (9*d^2*e^2*f - 6*d*e*f^2 + 2*f^3)*a*b^3*e^(7*c))*e^(3*d*x) + 1728*
(4*a^2*b^2*d^3*f^3*x^3*e^(6*c) + 6*(2*d^3*e*f^2 - d^2*f^3)*a^2*b^2*x^2*e^(6*c) + 6*(2*d^3*e^2*f - 2*d^2*e*f^2
+ d*f^3)*a^2*b^2*x*e^(6*c) - 3*(2*d^2*e^2*f - 2*d*e*f^2 + f^3)*a^2*b^2*e^(6*c))*e^(2*d*x) + 6912*(12*(d^2*e^2*
f - 2*d*e*f^2 + 2*f^3)*a^3*b*e^(5*c) + 3*(d^2*e^2*f - 2*d*e*f^2 + 2*f^3)*a*b^3*e^(5*c) - (4*a^3*b*d^3*f^3*e^(5
*c) + a*b^3*d^3*f^3*e^(5*c))*x^3 - 3*(4*(d^3*e*f^2 - d^2*f^3)*a^3*b*e^(5*c) + (d^3*e*f^2 - d^2*f^3)*a*b^3*e^(5
*c))*x^2 - 3*(4*(d^3*e^2*f - 2*d^2*e*f^2 + 2*d*f^3)*a^3*b*e^(5*c) + (d^3*e^2*f - 2*d^2*e*f^2 + 2*d*f^3)*a*b^3*
e^(5*c))*x)*e^(d*x) - 6912*(12*(d^2*e^2*f + 2*d*e*f^2 + 2*f^3)*a^3*b*e^(3*c) + 3*(d^2*e^2*f + 2*d*e*f^2 + 2*f^
3)*a*b^3*e^(3*c) + (4*a^3*b*d^3*f^3*e^(3*c) + a*b^3*d^3*f^3*e^(3*c))*x^3 + 3*(4*(d^3*e*f^2 + d^2*f^3)*a^3*b*e^
(3*c) + (d^3*e*f^2 + d^2*f^3)*a*b^3*e^(3*c))*x^2 + 3*(4*(d^3*e^2*f + 2*d^2*e*f^2 + 2*d*f^3)*a^3*b*e^(3*c) + (d
^3*e^2*f + 2*d^2*e*f^2 + 2*d*f^3)*a*b^3*e^(3*c))*x)*e^(-d*x) - 1728*(4*a^2*b^2*d^3*f^3*x^3*e^(2*c) + 6*(2*d^3*
e*f^2 + d^2*f^3)*a^2*b^2*x^2*e^(2*c) + 6*(2*d^3*e^2*f + 2*d^2*e*f^2 + d*f^3)*a^2*b^2*x*e^(2*c) + 3*(2*d^2*e^2*
f + 2*d*e*f^2 + f^3)*a^2*b^2*e^(2*c))*e^(-2*d*x) - 256*(9*a*b^3*d^3*f^3*x^3*e^c + 9*(3*d^3*e*f^2 + d^2*f^3)*a*
b^3*x^2*e^c + 3*(9*d^3*e^2*f + 6*d^2*e*f^2 + 2*d*f^3)*a*b^3*x*e^c + (9*d^2*e^2*f + 6*d*e*f^2 + 2*f^3)*a*b^3*e^
c)*e^(-3*d*x) - 27*(32*b^4*d^3*f^3*x^3 + 24*(4*d^3*e*f^2 + d^2*f^3)*b^4*x^2 + 12*(8*d^3*e^2*f + 4*d^2*e*f^2 +
d*f^3)*b^4*x + 3*(8*d^2*e^2*f + 4*d*e*f^2 + f^3)*b^4)*e^(-4*d*x))*e^(-4*c)/(b^5*d^4) - integrate(2*((a^5*f^3*e
^c + a^3*b^2*f^3*e^c)*x^3 + 3*(a^5*e*f^2*e^c + a^3*b^2*e*f^2*e^c)*x^2 + 3*(a^5*e^2*f*e^c + a^3*b^2*e^2*f*e^c)*
x)*e^(d*x)/(b^6*e^(2*d*x + 2*c) + 2*a*b^5*e^(d*x + c) - b^6), x)

Giac [F]

\[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^2\,{\mathrm {sinh}\left (c+d\,x\right )}^3\,{\left (e+f\,x\right )}^3}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]

[In]

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

[Out]

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