\(\int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [367]

Optimal result

Integrand size = 36, antiderivative size = 897 \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {3 a e f^2 x}{4 b^2 d^2}-\frac {3 a f^3 x^2}{8 b^2 d^2}-\frac {a^3 (e+f x)^4}{4 b^4 f}-\frac {a (e+f x)^4}{8 b^2 f}+\frac {6 a^2 f^2 (e+f x) \cosh (c+d x)}{b^3 d^3}+\frac {4 f^2 (e+f x) \cosh (c+d x)}{3 b d^3}+\frac {a^2 (e+f x)^3 \cosh (c+d x)}{b^3 d}+\frac {3 a f^3 \cosh ^2(c+d x)}{8 b^2 d^4}+\frac {3 a f (e+f x)^2 \cosh ^2(c+d x)}{4 b^2 d^2}+\frac {2 f^2 (e+f x) \cosh ^3(c+d x)}{9 b d^3}+\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 b d}+\frac {a^2 \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {a^2 \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d}+\frac {3 a^2 \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {3 a^2 \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {6 a^2 \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^3}+\frac {6 a^2 \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^3}+\frac {6 a^2 \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^4}-\frac {6 a^2 \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^4}-\frac {6 a^2 f^3 \sinh (c+d x)}{b^3 d^4}-\frac {14 f^3 \sinh (c+d x)}{9 b d^4}-\frac {3 a^2 f (e+f x)^2 \sinh (c+d x)}{b^3 d^2}-\frac {2 f (e+f x)^2 \sinh (c+d x)}{3 b d^2}-\frac {3 a f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^3}-\frac {a (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}-\frac {f (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b d^2}-\frac {2 f^3 \sinh ^3(c+d x)}{27 b d^4} \]

[Out]

-3/4*a*f^2*(f*x+e)*cosh(d*x+c)*sinh(d*x+c)/b^2/d^3+3*a^2*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^4/d^2-3*a^2*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^4
/d^2-6*a^2*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^4/d^3+6*a^2*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^4/d^3+4/3*f^2*(f*x+e)*cosh(d*x+c)/b/d^3-2/3*f*(
f*x+e)^2*sinh(d*x+c)/b/d^2+6*a^2*f^2*(f*x+e)*cosh(d*x+c)/b^3/d^3+3/4*a*f*(f*x+e)^2*cosh(d*x+c)^2/b^2/d^2-3*a^2
*f*(f*x+e)^2*sinh(d*x+c)/b^3/d^2+a^2*(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^4/d-1/
2*a*(f*x+e)^3*cosh(d*x+c)*sinh(d*x+c)/b^2/d-a^2*(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^4/d+a^2*(f*x+e)^3*cosh(d*x+c)/b^3/d-1/3*f*(f*x+e)^2*cosh(d*x+c)^2*sinh(d*x+c)/b/d^2+6*a^2*f^3*polylog(4,-
b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/b^4/d^4-6*a^2*f^3*polylog(4,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)
))*(a^2+b^2)^(1/2)/b^4/d^4-3/4*a*e*f^2*x/b^2/d^2-1/4*a^3*(f*x+e)^4/b^4/f+1/3*(f*x+e)^3*cosh(d*x+c)^3/b/d-2/27*
f^3*sinh(d*x+c)^3/b/d^4-1/8*a*(f*x+e)^4/b^2/f-14/9*f^3*sinh(d*x+c)/b/d^4-3/8*a*f^3*x^2/b^2/d^2+3/8*a*f^3*cosh(
d*x+c)^2/b^2/d^4+2/9*f^2*(f*x+e)*cosh(d*x+c)^3/b/d^3-6*a^2*f^3*sinh(d*x+c)/b^3/d^4

Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 897, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {5698, 5555, 3392, 3377, 2717, 2713, 32, 3391, 5684, 3403, 2296, 2221, 2611, 6744, 2320, 6724} \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a (e+f x)^4}{8 b^2 f}-\frac {a^3 (e+f x)^4}{4 b^4 f}+\frac {\cosh ^3(c+d x) (e+f x)^3}{3 b d}+\frac {a^2 \cosh (c+d x) (e+f x)^3}{b^3 d}+\frac {a^2 \sqrt {a^2+b^2} \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{b^4 d}-\frac {a^2 \sqrt {a^2+b^2} \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{b^4 d}-\frac {a \cosh (c+d x) \sinh (c+d x) (e+f x)^3}{2 b^2 d}+\frac {3 a f \cosh ^2(c+d x) (e+f x)^2}{4 b^2 d^2}+\frac {3 a^2 \sqrt {a^2+b^2} f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) (e+f x)^2}{b^4 d^2}-\frac {3 a^2 \sqrt {a^2+b^2} f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) (e+f x)^2}{b^4 d^2}-\frac {f \cosh ^2(c+d x) \sinh (c+d x) (e+f x)^2}{3 b d^2}-\frac {2 f \sinh (c+d x) (e+f x)^2}{3 b d^2}-\frac {3 a^2 f \sinh (c+d x) (e+f x)^2}{b^3 d^2}+\frac {2 f^2 \cosh ^3(c+d x) (e+f x)}{9 b d^3}+\frac {4 f^2 \cosh (c+d x) (e+f x)}{3 b d^3}+\frac {6 a^2 f^2 \cosh (c+d x) (e+f x)}{b^3 d^3}-\frac {6 a^2 \sqrt {a^2+b^2} f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) (e+f x)}{b^4 d^3}+\frac {6 a^2 \sqrt {a^2+b^2} f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) (e+f x)}{b^4 d^3}-\frac {3 a f^2 \cosh (c+d x) \sinh (c+d x) (e+f x)}{4 b^2 d^3}-\frac {2 f^3 \sinh ^3(c+d x)}{27 b d^4}-\frac {3 a f^3 x^2}{8 b^2 d^2}+\frac {3 a f^3 \cosh ^2(c+d x)}{8 b^2 d^4}-\frac {3 a e f^2 x}{4 b^2 d^2}+\frac {6 a^2 \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^4}-\frac {6 a^2 \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^4}-\frac {14 f^3 \sinh (c+d x)}{9 b d^4}-\frac {6 a^2 f^3 \sinh (c+d x)}{b^3 d^4} \]

[In]

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

[Out]

(-3*a*e*f^2*x)/(4*b^2*d^2) - (3*a*f^3*x^2)/(8*b^2*d^2) - (a^3*(e + f*x)^4)/(4*b^4*f) - (a*(e + f*x)^4)/(8*b^2*
f) + (6*a^2*f^2*(e + f*x)*Cosh[c + d*x])/(b^3*d^3) + (4*f^2*(e + f*x)*Cosh[c + d*x])/(3*b*d^3) + (a^2*(e + f*x
)^3*Cosh[c + d*x])/(b^3*d) + (3*a*f^3*Cosh[c + d*x]^2)/(8*b^2*d^4) + (3*a*f*(e + f*x)^2*Cosh[c + d*x]^2)/(4*b^
2*d^2) + (2*f^2*(e + f*x)*Cosh[c + d*x]^3)/(9*b*d^3) + ((e + f*x)^3*Cosh[c + d*x]^3)/(3*b*d) + (a^2*Sqrt[a^2 +
b^2]*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^4*d) - (a^2*Sqrt[a^2 + b^2]*(e + f*x)^3*L
og[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^4*d) + (3*a^2*Sqrt[a^2 + b^2]*f*(e + f*x)^2*PolyLog[2, -((b*
E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^4*d^2) - (3*a^2*Sqrt[a^2 + b^2]*f*(e + f*x)^2*PolyLog[2, -((b*E^(c +
d*x))/(a + Sqrt[a^2 + b^2]))])/(b^4*d^2) - (6*a^2*Sqrt[a^2 + b^2]*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(
a - Sqrt[a^2 + b^2]))])/(b^4*d^3) + (6*a^2*Sqrt[a^2 + b^2]*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqr
t[a^2 + b^2]))])/(b^4*d^3) + (6*a^2*Sqrt[a^2 + b^2]*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/
(b^4*d^4) - (6*a^2*Sqrt[a^2 + b^2]*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^4*d^4) - (6*a^
2*f^3*Sinh[c + d*x])/(b^3*d^4) - (14*f^3*Sinh[c + d*x])/(9*b*d^4) - (3*a^2*f*(e + f*x)^2*Sinh[c + d*x])/(b^3*d
^2) - (2*f*(e + f*x)^2*Sinh[c + d*x])/(3*b*d^2) - (3*a*f^2*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(4*b^2*d^3)
- (a*(e + f*x)^3*Cosh[c + d*x]*Sinh[c + d*x])/(2*b^2*d) - (f*(e + f*x)^2*Cosh[c + d*x]^2*Sinh[c + d*x])/(3*b*d
^2) - (2*f^3*Sinh[c + d*x]^3)/(27*b*d^4)

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 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 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 (c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b} \\ & = \frac {(e+f x)^3 \cosh ^3(c+d x)}{3 b d}-\frac {a \int (e+f x)^3 \cosh ^2(c+d x) \, dx}{b^2}+\frac {a^2 \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}-\frac {f \int (e+f x)^2 \cosh ^3(c+d x) \, dx}{b d} \\ & = \frac {3 a f (e+f x)^2 \cosh ^2(c+d x)}{4 b^2 d^2}+\frac {2 f^2 (e+f x) \cosh ^3(c+d x)}{9 b d^3}+\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 b d}-\frac {a (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}-\frac {f (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b d^2}-\frac {a^3 \int (e+f x)^3 \, dx}{b^4}+\frac {a^2 \int (e+f x)^3 \sinh (c+d x) \, dx}{b^3}-\frac {a \int (e+f x)^3 \, dx}{2 b^2}+\frac {\left (a^2 \left (a^2+b^2\right )\right ) \int \frac {(e+f x)^3}{a+b \sinh (c+d x)} \, dx}{b^4}-\frac {(2 f) \int (e+f x)^2 \cosh (c+d x) \, dx}{3 b d}-\frac {\left (3 a f^2\right ) \int (e+f x) \cosh ^2(c+d x) \, dx}{2 b^2 d^2}-\frac {\left (2 f^3\right ) \int \cosh ^3(c+d x) \, dx}{9 b d^3} \\ & = -\frac {a^3 (e+f x)^4}{4 b^4 f}-\frac {a (e+f x)^4}{8 b^2 f}+\frac {a^2 (e+f x)^3 \cosh (c+d x)}{b^3 d}+\frac {3 a f^3 \cosh ^2(c+d x)}{8 b^2 d^4}+\frac {3 a f (e+f x)^2 \cosh ^2(c+d x)}{4 b^2 d^2}+\frac {2 f^2 (e+f x) \cosh ^3(c+d x)}{9 b d^3}+\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 b d}-\frac {2 f (e+f x)^2 \sinh (c+d x)}{3 b d^2}-\frac {3 a f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^3}-\frac {a (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}-\frac {f (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b d^2}+\frac {\left (2 a^2 \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^4}-\frac {\left (3 a^2 f\right ) \int (e+f x)^2 \cosh (c+d x) \, dx}{b^3 d}-\frac {\left (3 a f^2\right ) \int (e+f x) \, dx}{4 b^2 d^2}+\frac {\left (4 f^2\right ) \int (e+f x) \sinh (c+d x) \, dx}{3 b d^2}-\frac {\left (2 i f^3\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-i \sinh (c+d x)\right )}{9 b d^4} \\ & = -\frac {3 a e f^2 x}{4 b^2 d^2}-\frac {3 a f^3 x^2}{8 b^2 d^2}-\frac {a^3 (e+f x)^4}{4 b^4 f}-\frac {a (e+f x)^4}{8 b^2 f}+\frac {4 f^2 (e+f x) \cosh (c+d x)}{3 b d^3}+\frac {a^2 (e+f x)^3 \cosh (c+d x)}{b^3 d}+\frac {3 a f^3 \cosh ^2(c+d x)}{8 b^2 d^4}+\frac {3 a f (e+f x)^2 \cosh ^2(c+d x)}{4 b^2 d^2}+\frac {2 f^2 (e+f x) \cosh ^3(c+d x)}{9 b d^3}+\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 b d}-\frac {2 f^3 \sinh (c+d x)}{9 b d^4}-\frac {3 a^2 f (e+f x)^2 \sinh (c+d x)}{b^3 d^2}-\frac {2 f (e+f x)^2 \sinh (c+d x)}{3 b d^2}-\frac {3 a f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^3}-\frac {a (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}-\frac {f (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b d^2}-\frac {2 f^3 \sinh ^3(c+d x)}{27 b d^4}+\frac {\left (2 a^2 \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^3}-\frac {\left (2 a^2 \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^3}+\frac {\left (6 a^2 f^2\right ) \int (e+f x) \sinh (c+d x) \, dx}{b^3 d^2}-\frac {\left (4 f^3\right ) \int \cosh (c+d x) \, dx}{3 b d^3} \\ & = -\frac {3 a e f^2 x}{4 b^2 d^2}-\frac {3 a f^3 x^2}{8 b^2 d^2}-\frac {a^3 (e+f x)^4}{4 b^4 f}-\frac {a (e+f x)^4}{8 b^2 f}+\frac {6 a^2 f^2 (e+f x) \cosh (c+d x)}{b^3 d^3}+\frac {4 f^2 (e+f x) \cosh (c+d x)}{3 b d^3}+\frac {a^2 (e+f x)^3 \cosh (c+d x)}{b^3 d}+\frac {3 a f^3 \cosh ^2(c+d x)}{8 b^2 d^4}+\frac {3 a f (e+f x)^2 \cosh ^2(c+d x)}{4 b^2 d^2}+\frac {2 f^2 (e+f x) \cosh ^3(c+d x)}{9 b d^3}+\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 b d}+\frac {a^2 \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^4 d}-\frac {a^2 \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^4 d}-\frac {14 f^3 \sinh (c+d x)}{9 b d^4}-\frac {3 a^2 f (e+f x)^2 \sinh (c+d x)}{b^3 d^2}-\frac {2 f (e+f x)^2 \sinh (c+d x)}{3 b d^2}-\frac {3 a f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^3}-\frac {a (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}-\frac {f (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b d^2}-\frac {2 f^3 \sinh ^3(c+d x)}{27 b d^4}-\frac {\left (3 a^2 \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^4 d}+\frac {\left (3 a^2 \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^4 d}-\frac {\left (6 a^2 f^3\right ) \int \cosh (c+d x) \, dx}{b^3 d^3} \\ & = -\frac {3 a e f^2 x}{4 b^2 d^2}-\frac {3 a f^3 x^2}{8 b^2 d^2}-\frac {a^3 (e+f x)^4}{4 b^4 f}-\frac {a (e+f x)^4}{8 b^2 f}+\frac {6 a^2 f^2 (e+f x) \cosh (c+d x)}{b^3 d^3}+\frac {4 f^2 (e+f x) \cosh (c+d x)}{3 b d^3}+\frac {a^2 (e+f x)^3 \cosh (c+d x)}{b^3 d}+\frac {3 a f^3 \cosh ^2(c+d x)}{8 b^2 d^4}+\frac {3 a f (e+f x)^2 \cosh ^2(c+d x)}{4 b^2 d^2}+\frac {2 f^2 (e+f x) \cosh ^3(c+d x)}{9 b d^3}+\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 b d}+\frac {a^2 \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^4 d}-\frac {a^2 \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^4 d}+\frac {3 a^2 \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^4 d^2}-\frac {3 a^2 \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^4 d^2}-\frac {6 a^2 f^3 \sinh (c+d x)}{b^3 d^4}-\frac {14 f^3 \sinh (c+d x)}{9 b d^4}-\frac {3 a^2 f (e+f x)^2 \sinh (c+d x)}{b^3 d^2}-\frac {2 f (e+f x)^2 \sinh (c+d x)}{3 b d^2}-\frac {3 a f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^3}-\frac {a (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}-\frac {f (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b d^2}-\frac {2 f^3 \sinh ^3(c+d x)}{27 b d^4}-\frac {\left (6 a^2 \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^4 d^2}+\frac {\left (6 a^2 \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^4 d^2} \\ & = -\frac {3 a e f^2 x}{4 b^2 d^2}-\frac {3 a f^3 x^2}{8 b^2 d^2}-\frac {a^3 (e+f x)^4}{4 b^4 f}-\frac {a (e+f x)^4}{8 b^2 f}+\frac {6 a^2 f^2 (e+f x) \cosh (c+d x)}{b^3 d^3}+\frac {4 f^2 (e+f x) \cosh (c+d x)}{3 b d^3}+\frac {a^2 (e+f x)^3 \cosh (c+d x)}{b^3 d}+\frac {3 a f^3 \cosh ^2(c+d x)}{8 b^2 d^4}+\frac {3 a f (e+f x)^2 \cosh ^2(c+d x)}{4 b^2 d^2}+\frac {2 f^2 (e+f x) \cosh ^3(c+d x)}{9 b d^3}+\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 b d}+\frac {a^2 \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^4 d}-\frac {a^2 \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^4 d}+\frac {3 a^2 \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^4 d^2}-\frac {3 a^2 \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^4 d^2}-\frac {6 a^2 \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^4 d^3}+\frac {6 a^2 \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^4 d^3}-\frac {6 a^2 f^3 \sinh (c+d x)}{b^3 d^4}-\frac {14 f^3 \sinh (c+d x)}{9 b d^4}-\frac {3 a^2 f (e+f x)^2 \sinh (c+d x)}{b^3 d^2}-\frac {2 f (e+f x)^2 \sinh (c+d x)}{3 b d^2}-\frac {3 a f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^3}-\frac {a (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}-\frac {f (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b d^2}-\frac {2 f^3 \sinh ^3(c+d x)}{27 b d^4}+\frac {\left (6 a^2 \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^4 d^3}-\frac {\left (6 a^2 \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^4 d^3} \\ & = -\frac {3 a e f^2 x}{4 b^2 d^2}-\frac {3 a f^3 x^2}{8 b^2 d^2}-\frac {a^3 (e+f x)^4}{4 b^4 f}-\frac {a (e+f x)^4}{8 b^2 f}+\frac {6 a^2 f^2 (e+f x) \cosh (c+d x)}{b^3 d^3}+\frac {4 f^2 (e+f x) \cosh (c+d x)}{3 b d^3}+\frac {a^2 (e+f x)^3 \cosh (c+d x)}{b^3 d}+\frac {3 a f^3 \cosh ^2(c+d x)}{8 b^2 d^4}+\frac {3 a f (e+f x)^2 \cosh ^2(c+d x)}{4 b^2 d^2}+\frac {2 f^2 (e+f x) \cosh ^3(c+d x)}{9 b d^3}+\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 b d}+\frac {a^2 \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^4 d}-\frac {a^2 \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^4 d}+\frac {3 a^2 \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^4 d^2}-\frac {3 a^2 \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^4 d^2}-\frac {6 a^2 \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^4 d^3}+\frac {6 a^2 \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^4 d^3}-\frac {6 a^2 f^3 \sinh (c+d x)}{b^3 d^4}-\frac {14 f^3 \sinh (c+d x)}{9 b d^4}-\frac {3 a^2 f (e+f x)^2 \sinh (c+d x)}{b^3 d^2}-\frac {2 f (e+f x)^2 \sinh (c+d x)}{3 b d^2}-\frac {3 a f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^3}-\frac {a (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}-\frac {f (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b d^2}-\frac {2 f^3 \sinh ^3(c+d x)}{27 b d^4}+\frac {\left (6 a^2 \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^4 d^4}-\frac {\left (6 a^2 \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^4 d^4} \\ & = -\frac {3 a e f^2 x}{4 b^2 d^2}-\frac {3 a f^3 x^2}{8 b^2 d^2}-\frac {a^3 (e+f x)^4}{4 b^4 f}-\frac {a (e+f x)^4}{8 b^2 f}+\frac {6 a^2 f^2 (e+f x) \cosh (c+d x)}{b^3 d^3}+\frac {4 f^2 (e+f x) \cosh (c+d x)}{3 b d^3}+\frac {a^2 (e+f x)^3 \cosh (c+d x)}{b^3 d}+\frac {3 a f^3 \cosh ^2(c+d x)}{8 b^2 d^4}+\frac {3 a f (e+f x)^2 \cosh ^2(c+d x)}{4 b^2 d^2}+\frac {2 f^2 (e+f x) \cosh ^3(c+d x)}{9 b d^3}+\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 b d}+\frac {a^2 \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^4 d}-\frac {a^2 \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^4 d}+\frac {3 a^2 \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^4 d^2}-\frac {3 a^2 \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^4 d^2}-\frac {6 a^2 \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^4 d^3}+\frac {6 a^2 \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^4 d^3}+\frac {6 a^2 \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^4 d^4}-\frac {6 a^2 \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^4 d^4}-\frac {6 a^2 f^3 \sinh (c+d x)}{b^3 d^4}-\frac {14 f^3 \sinh (c+d x)}{9 b d^4}-\frac {3 a^2 f (e+f x)^2 \sinh (c+d x)}{b^3 d^2}-\frac {2 f (e+f x)^2 \sinh (c+d x)}{3 b d^2}-\frac {3 a f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^3}-\frac {a (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}-\frac {f (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b d^2}-\frac {2 f^3 \sinh ^3(c+d x)}{27 b d^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.64 (sec) , antiderivative size = 1667, normalized size of antiderivative = 1.86 \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {432 a^3 d^4 e^3 x+216 a b^2 d^4 e^3 x+648 a^3 d^4 e^2 f x^2+324 a b^2 d^4 e^2 f x^2+432 a^3 d^4 e f^2 x^3+216 a b^2 d^4 e f^2 x^3+108 a^3 d^4 f^3 x^4+54 a b^2 d^4 f^3 x^4+864 a^2 \sqrt {a^2+b^2} d^3 e^3 \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )-432 a^2 b d^3 e^3 \cosh (c+d x)-108 b^3 d^3 e^3 \cosh (c+d x)-2592 a^2 b d e f^2 \cosh (c+d x)-648 b^3 d e f^2 \cosh (c+d x)-1296 a^2 b d^3 e^2 f x \cosh (c+d x)-324 b^3 d^3 e^2 f x \cosh (c+d x)-2592 a^2 b d f^3 x \cosh (c+d x)-648 b^3 d f^3 x \cosh (c+d x)-1296 a^2 b d^3 e f^2 x^2 \cosh (c+d x)-324 b^3 d^3 e f^2 x^2 \cosh (c+d x)-432 a^2 b d^3 f^3 x^3 \cosh (c+d x)-108 b^3 d^3 f^3 x^3 \cosh (c+d x)-162 a b^2 d^2 e^2 f \cosh (2 (c+d x))-81 a b^2 f^3 \cosh (2 (c+d x))-324 a b^2 d^2 e f^2 x \cosh (2 (c+d x))-162 a b^2 d^2 f^3 x^2 \cosh (2 (c+d x))-36 b^3 d^3 e^3 \cosh (3 (c+d x))-24 b^3 d e f^2 \cosh (3 (c+d x))-108 b^3 d^3 e^2 f x \cosh (3 (c+d x))-24 b^3 d f^3 x \cosh (3 (c+d x))-108 b^3 d^3 e f^2 x^2 \cosh (3 (c+d x))-36 b^3 d^3 f^3 x^3 \cosh (3 (c+d x))-1296 a^2 \sqrt {a^2+b^2} d^3 e^2 f x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-1296 a^2 \sqrt {a^2+b^2} d^3 e f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-432 a^2 \sqrt {a^2+b^2} d^3 f^3 x^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+1296 a^2 \sqrt {a^2+b^2} d^3 e^2 f x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+1296 a^2 \sqrt {a^2+b^2} d^3 e f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+432 a^2 \sqrt {a^2+b^2} d^3 f^3 x^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-1296 a^2 \sqrt {a^2+b^2} d^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+1296 a^2 \sqrt {a^2+b^2} d^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+2592 a^2 \sqrt {a^2+b^2} d e f^2 \operatorname {PolyLog}\left (3,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2592 a^2 \sqrt {a^2+b^2} d f^3 x \operatorname {PolyLog}\left (3,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-2592 a^2 \sqrt {a^2+b^2} d e f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2592 a^2 \sqrt {a^2+b^2} d f^3 x \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2592 a^2 \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2592 a^2 \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+1296 a^2 b d^2 e^2 f \sinh (c+d x)+324 b^3 d^2 e^2 f \sinh (c+d x)+2592 a^2 b f^3 \sinh (c+d x)+648 b^3 f^3 \sinh (c+d x)+2592 a^2 b d^2 e f^2 x \sinh (c+d x)+648 b^3 d^2 e f^2 x \sinh (c+d x)+1296 a^2 b d^2 f^3 x^2 \sinh (c+d x)+324 b^3 d^2 f^3 x^2 \sinh (c+d x)+108 a b^2 d^3 e^3 \sinh (2 (c+d x))+162 a b^2 d e f^2 \sinh (2 (c+d x))+324 a b^2 d^3 e^2 f x \sinh (2 (c+d x))+162 a b^2 d f^3 x \sinh (2 (c+d x))+324 a b^2 d^3 e f^2 x^2 \sinh (2 (c+d x))+108 a b^2 d^3 f^3 x^3 \sinh (2 (c+d x))+36 b^3 d^2 e^2 f \sinh (3 (c+d x))+8 b^3 f^3 \sinh (3 (c+d x))+72 b^3 d^2 e f^2 x \sinh (3 (c+d x))+36 b^3 d^2 f^3 x^2 \sinh (3 (c+d x))}{432 b^4 d^4} \]

[In]

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

[Out]

-1/432*(432*a^3*d^4*e^3*x + 216*a*b^2*d^4*e^3*x + 648*a^3*d^4*e^2*f*x^2 + 324*a*b^2*d^4*e^2*f*x^2 + 432*a^3*d^
4*e*f^2*x^3 + 216*a*b^2*d^4*e*f^2*x^3 + 108*a^3*d^4*f^3*x^4 + 54*a*b^2*d^4*f^3*x^4 + 864*a^2*Sqrt[a^2 + b^2]*d
^3*e^3*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] - 432*a^2*b*d^3*e^3*Cosh[c + d*x] - 108*b^3*d^3*e^3*Cosh[c
+ d*x] - 2592*a^2*b*d*e*f^2*Cosh[c + d*x] - 648*b^3*d*e*f^2*Cosh[c + d*x] - 1296*a^2*b*d^3*e^2*f*x*Cosh[c + d
*x] - 324*b^3*d^3*e^2*f*x*Cosh[c + d*x] - 2592*a^2*b*d*f^3*x*Cosh[c + d*x] - 648*b^3*d*f^3*x*Cosh[c + d*x] - 1
296*a^2*b*d^3*e*f^2*x^2*Cosh[c + d*x] - 324*b^3*d^3*e*f^2*x^2*Cosh[c + d*x] - 432*a^2*b*d^3*f^3*x^3*Cosh[c + d
*x] - 108*b^3*d^3*f^3*x^3*Cosh[c + d*x] - 162*a*b^2*d^2*e^2*f*Cosh[2*(c + d*x)] - 81*a*b^2*f^3*Cosh[2*(c + d*x
)] - 324*a*b^2*d^2*e*f^2*x*Cosh[2*(c + d*x)] - 162*a*b^2*d^2*f^3*x^2*Cosh[2*(c + d*x)] - 36*b^3*d^3*e^3*Cosh[3
*(c + d*x)] - 24*b^3*d*e*f^2*Cosh[3*(c + d*x)] - 108*b^3*d^3*e^2*f*x*Cosh[3*(c + d*x)] - 24*b^3*d*f^3*x*Cosh[3
*(c + d*x)] - 108*b^3*d^3*e*f^2*x^2*Cosh[3*(c + d*x)] - 36*b^3*d^3*f^3*x^3*Cosh[3*(c + d*x)] - 1296*a^2*Sqrt[a
^2 + b^2]*d^3*e^2*f*x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - 1296*a^2*Sqrt[a^2 + b^2]*d^3*e*f^2*x^2*
Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - 432*a^2*Sqrt[a^2 + b^2]*d^3*f^3*x^3*Log[1 + (b*E^(c + d*x))/(
a - Sqrt[a^2 + b^2])] + 1296*a^2*Sqrt[a^2 + b^2]*d^3*e^2*f*x*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] +
1296*a^2*Sqrt[a^2 + b^2]*d^3*e*f^2*x^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 432*a^2*Sqrt[a^2 + b^2
]*d^3*f^3*x^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - 1296*a^2*Sqrt[a^2 + b^2]*d^2*f*(e + f*x)^2*Poly
Log[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 1296*a^2*Sqrt[a^2 + b^2]*d^2*f*(e + f*x)^2*PolyLog[2, -((b*E^
(c + d*x))/(a + Sqrt[a^2 + b^2]))] + 2592*a^2*Sqrt[a^2 + b^2]*d*e*f^2*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^
2 + b^2])] + 2592*a^2*Sqrt[a^2 + b^2]*d*f^3*x*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 2592*a^2*Sq
rt[a^2 + b^2]*d*e*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] - 2592*a^2*Sqrt[a^2 + b^2]*d*f^3*x*
PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] - 2592*a^2*Sqrt[a^2 + b^2]*f^3*PolyLog[4, (b*E^(c + d*x))
/(-a + Sqrt[a^2 + b^2])] + 2592*a^2*Sqrt[a^2 + b^2]*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] +
1296*a^2*b*d^2*e^2*f*Sinh[c + d*x] + 324*b^3*d^2*e^2*f*Sinh[c + d*x] + 2592*a^2*b*f^3*Sinh[c + d*x] + 648*b^3
*f^3*Sinh[c + d*x] + 2592*a^2*b*d^2*e*f^2*x*Sinh[c + d*x] + 648*b^3*d^2*e*f^2*x*Sinh[c + d*x] + 1296*a^2*b*d^2
*f^3*x^2*Sinh[c + d*x] + 324*b^3*d^2*f^3*x^2*Sinh[c + d*x] + 108*a*b^2*d^3*e^3*Sinh[2*(c + d*x)] + 162*a*b^2*d
*e*f^2*Sinh[2*(c + d*x)] + 324*a*b^2*d^3*e^2*f*x*Sinh[2*(c + d*x)] + 162*a*b^2*d*f^3*x*Sinh[2*(c + d*x)] + 324
*a*b^2*d^3*e*f^2*x^2*Sinh[2*(c + d*x)] + 108*a*b^2*d^3*f^3*x^3*Sinh[2*(c + d*x)] + 36*b^3*d^2*e^2*f*Sinh[3*(c
+ d*x)] + 8*b^3*f^3*Sinh[3*(c + d*x)] + 72*b^3*d^2*e*f^2*x*Sinh[3*(c + d*x)] + 36*b^3*d^2*f^3*x^2*Sinh[3*(c +
d*x)])/(b^4*d^4)

Maple [F]

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

[In]

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

[Out]

int((f*x+e)^3*cosh(d*x+c)^2*sinh(d*x+c)^2/(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. 7042 vs. \(2 (825) = 1650\).

Time = 0.36 (sec) , antiderivative size = 7042, normalized size of antiderivative = 7.85 \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

1/24*e^3*(24*sqrt(a^2 + b^2)*a^2*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt(a^2 + b
^2)))/(b^4*d) - (3*a*b*e^(-d*x - c) - b^2 - 3*(4*a^2 + b^2)*e^(-2*d*x - 2*c))*e^(3*d*x + 3*c)/(b^3*d) - 12*(2*
a^3 + a*b^2)*(d*x + c)/(b^4*d) + (3*a*b*e^(-2*d*x - 2*c) + b^2*e^(-3*d*x - 3*c) + 3*(4*a^2 + b^2)*e^(-d*x - c)
)/(b^3*d)) - 1/864*(108*(2*a^3*d^4*f^3*e^(3*c) + a*b^2*d^4*f^3*e^(3*c))*x^4 + 432*(2*a^3*d^4*e*f^2*e^(3*c) + a
*b^2*d^4*e*f^2*e^(3*c))*x^3 + 648*(2*a^3*d^4*e^2*f*e^(3*c) + a*b^2*d^4*e^2*f*e^(3*c))*x^2 - 4*(9*b^3*d^3*f^3*x
^3*e^(6*c) + 9*(3*d^3*e*f^2 - d^2*f^3)*b^3*x^2*e^(6*c) + 3*(9*d^3*e^2*f - 6*d^2*e*f^2 + 2*d*f^3)*b^3*x*e^(6*c)
- (9*d^2*e^2*f - 6*d*e*f^2 + 2*f^3)*b^3*e^(6*c))*e^(3*d*x) + 27*(4*a*b^2*d^3*f^3*x^3*e^(5*c) + 6*(2*d^3*e*f^2
- d^2*f^3)*a*b^2*x^2*e^(5*c) + 6*(2*d^3*e^2*f - 2*d^2*e*f^2 + d*f^3)*a*b^2*x*e^(5*c) - 3*(2*d^2*e^2*f - 2*d*e
*f^2 + f^3)*a*b^2*e^(5*c))*e^(2*d*x) + 108*(12*(d^2*e^2*f - 2*d*e*f^2 + 2*f^3)*a^2*b*e^(4*c) + 3*(d^2*e^2*f -
2*d*e*f^2 + 2*f^3)*b^3*e^(4*c) - (4*a^2*b*d^3*f^3*e^(4*c) + b^3*d^3*f^3*e^(4*c))*x^3 - 3*(4*(d^3*e*f^2 - d^2*f
^3)*a^2*b*e^(4*c) + (d^3*e*f^2 - d^2*f^3)*b^3*e^(4*c))*x^2 - 3*(4*(d^3*e^2*f - 2*d^2*e*f^2 + 2*d*f^3)*a^2*b*e^
(4*c) + (d^3*e^2*f - 2*d^2*e*f^2 + 2*d*f^3)*b^3*e^(4*c))*x)*e^(d*x) - 108*(12*(d^2*e^2*f + 2*d*e*f^2 + 2*f^3)*
a^2*b*e^(2*c) + 3*(d^2*e^2*f + 2*d*e*f^2 + 2*f^3)*b^3*e^(2*c) + (4*a^2*b*d^3*f^3*e^(2*c) + b^3*d^3*f^3*e^(2*c)
)*x^3 + 3*(4*(d^3*e*f^2 + d^2*f^3)*a^2*b*e^(2*c) + (d^3*e*f^2 + d^2*f^3)*b^3*e^(2*c))*x^2 + 3*(4*(d^3*e^2*f +
2*d^2*e*f^2 + 2*d*f^3)*a^2*b*e^(2*c) + (d^3*e^2*f + 2*d^2*e*f^2 + 2*d*f^3)*b^3*e^(2*c))*x)*e^(-d*x) - 27*(4*a*
b^2*d^3*f^3*x^3*e^c + 6*(2*d^3*e*f^2 + d^2*f^3)*a*b^2*x^2*e^c + 6*(2*d^3*e^2*f + 2*d^2*e*f^2 + d*f^3)*a*b^2*x*
e^c + 3*(2*d^2*e^2*f + 2*d*e*f^2 + f^3)*a*b^2*e^c)*e^(-2*d*x) - 4*(9*b^3*d^3*f^3*x^3 + 9*(3*d^3*e*f^2 + d^2*f^
3)*b^3*x^2 + 3*(9*d^3*e^2*f + 6*d^2*e*f^2 + 2*d*f^3)*b^3*x + (9*d^2*e^2*f + 6*d*e*f^2 + 2*f^3)*b^3)*e^(-3*d*x)
)*e^(-3*c)/(b^4*d^4) + integrate(2*((a^4*f^3*e^c + a^2*b^2*f^3*e^c)*x^3 + 3*(a^4*e*f^2*e^c + a^2*b^2*e*f^2*e^c
)*x^2 + 3*(a^4*e^2*f*e^c + a^2*b^2*e^2*f*e^c)*x)*e^(d*x)/(b^5*e^(2*d*x + 2*c) + 2*a*b^4*e^(d*x + c) - b^5), x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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