\(\int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [397]

Optimal result

Integrand size = 36, antiderivative size = 755 \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {a^2 f^2 x}{4 b^3 d^2}+\frac {a^4 (e+f x)^3}{3 b^5 f}+\frac {a^2 (e+f x)^3}{6 b^3 f}-\frac {(e+f x)^3}{24 b f}-\frac {2 a^3 f^2 \cosh (c+d x)}{b^4 d^3}-\frac {4 a f^2 \cosh (c+d x)}{9 b^2 d^3}-\frac {a^3 (e+f x)^2 \cosh (c+d x)}{b^4 d}-\frac {a^2 f (e+f x) \cosh ^2(c+d x)}{2 b^3 d^2}-\frac {2 a f^2 \cosh ^3(c+d x)}{27 b^2 d^3}-\frac {a (e+f x)^2 \cosh ^3(c+d x)}{3 b^2 d}-\frac {f (e+f x) \cosh (4 c+4 d x)}{64 b d^2}-\frac {a^3 \sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d}+\frac {a^3 \sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d}-\frac {2 a^3 \sqrt {a^2+b^2} f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d^2}+\frac {2 a^3 \sqrt {a^2+b^2} f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d^2}+\frac {2 a^3 \sqrt {a^2+b^2} f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d^3}-\frac {2 a^3 \sqrt {a^2+b^2} f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d^3}+\frac {2 a^3 f (e+f x) \sinh (c+d x)}{b^4 d^2}+\frac {4 a f (e+f x) \sinh (c+d x)}{9 b^2 d^2}+\frac {a^2 f^2 \cosh (c+d x) \sinh (c+d x)}{4 b^3 d^3}+\frac {a^2 (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b^3 d}+\frac {2 a f (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b^2 d^2}+\frac {f^2 \sinh (4 c+4 d x)}{256 b d^3}+\frac {(e+f x)^2 \sinh (4 c+4 d x)}{32 b d} \]

[Out]

1/4*a^2*f^2*x/b^3/d^2+1/3*a^4*(f*x+e)^3/b^5/f+1/6*a^2*(f*x+e)^3/b^3/f-1/24*(f*x+e)^3/b/f-2*a^3*f^2*cosh(d*x+c)
/b^4/d^3-4/9*a*f^2*cosh(d*x+c)/b^2/d^3-a^3*(f*x+e)^2*cosh(d*x+c)/b^4/d-1/2*a^2*f*(f*x+e)*cosh(d*x+c)^2/b^3/d^2
-2/27*a*f^2*cosh(d*x+c)^3/b^2/d^3-1/3*a*(f*x+e)^2*cosh(d*x+c)^3/b^2/d-1/64*f*(f*x+e)*cosh(4*d*x+4*c)/b/d^2+2*a
^3*f*(f*x+e)*sinh(d*x+c)/b^4/d^2+4/9*a*f*(f*x+e)*sinh(d*x+c)/b^2/d^2+1/4*a^2*f^2*cosh(d*x+c)*sinh(d*x+c)/b^3/d
^3+1/2*a^2*(f*x+e)^2*cosh(d*x+c)*sinh(d*x+c)/b^3/d+2/9*a*f*(f*x+e)*cosh(d*x+c)^2*sinh(d*x+c)/b^2/d^2+1/256*f^2
*sinh(4*d*x+4*c)/b/d^3+1/32*(f*x+e)^2*sinh(4*d*x+4*c)/b/d-a^3*(f*x+e)^2*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)^2*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/b^5/d-2*a^3*f*(f*x
+e)*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/b^5/d^2+2*a^3*f*(f*x+e)*polylog(2,-b*exp(d*x+
c)/(a+(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/b^5/d^2+2*a^3*f^2*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))*(a^2+b^
2)^(1/2)/b^5/d^3-2*a^3*f^2*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/b^5/d^3

Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 755, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5698, 5556, 3377, 2717, 5555, 3391, 2718, 3392, 32, 2715, 8, 5684, 3403, 2296, 2221, 2611, 2320, 6724} \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {a^4 (e+f x)^3}{3 b^5 f}-\frac {2 a^3 f^2 \cosh (c+d x)}{b^4 d^3}+\frac {2 a^3 f (e+f x) \sinh (c+d x)}{b^4 d^2}-\frac {a^3 (e+f x)^2 \cosh (c+d x)}{b^4 d}+\frac {a^2 f^2 \sinh (c+d x) \cosh (c+d x)}{4 b^3 d^3}-\frac {a^2 f (e+f x) \cosh ^2(c+d x)}{2 b^3 d^2}+\frac {a^2 (e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 b^3 d}+\frac {a^2 f^2 x}{4 b^3 d^2}+\frac {a^2 (e+f x)^3}{6 b^3 f}+\frac {2 a^3 f^2 \sqrt {a^2+b^2} \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d^3}-\frac {2 a^3 f^2 \sqrt {a^2+b^2} \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d^3}-\frac {2 a^3 f \sqrt {a^2+b^2} (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d^2}+\frac {2 a^3 f \sqrt {a^2+b^2} (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d^2}-\frac {a^3 \sqrt {a^2+b^2} (e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b^5 d}+\frac {a^3 \sqrt {a^2+b^2} (e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b^5 d}-\frac {2 a f^2 \cosh ^3(c+d x)}{27 b^2 d^3}-\frac {4 a f^2 \cosh (c+d x)}{9 b^2 d^3}+\frac {4 a f (e+f x) \sinh (c+d x)}{9 b^2 d^2}+\frac {2 a f (e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{9 b^2 d^2}-\frac {a (e+f x)^2 \cosh ^3(c+d x)}{3 b^2 d}+\frac {f^2 \sinh (4 c+4 d x)}{256 b d^3}-\frac {f (e+f x) \cosh (4 c+4 d x)}{64 b d^2}+\frac {(e+f x)^2 \sinh (4 c+4 d x)}{32 b d}-\frac {(e+f x)^3}{24 b f} \]

[In]

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

[Out]

(a^2*f^2*x)/(4*b^3*d^2) + (a^4*(e + f*x)^3)/(3*b^5*f) + (a^2*(e + f*x)^3)/(6*b^3*f) - (e + f*x)^3/(24*b*f) - (
2*a^3*f^2*Cosh[c + d*x])/(b^4*d^3) - (4*a*f^2*Cosh[c + d*x])/(9*b^2*d^3) - (a^3*(e + f*x)^2*Cosh[c + d*x])/(b^
4*d) - (a^2*f*(e + f*x)*Cosh[c + d*x]^2)/(2*b^3*d^2) - (2*a*f^2*Cosh[c + d*x]^3)/(27*b^2*d^3) - (a*(e + f*x)^2
*Cosh[c + d*x]^3)/(3*b^2*d) - (f*(e + f*x)*Cosh[4*c + 4*d*x])/(64*b*d^2) - (a^3*Sqrt[a^2 + b^2]*(e + f*x)^2*Lo
g[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)^2*Log[1 + (b*E^(c + d*x
))/(a + Sqrt[a^2 + b^2])])/(b^5*d) - (2*a^3*Sqrt[a^2 + b^2]*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt
[a^2 + b^2]))])/(b^5*d^2) + (2*a^3*Sqrt[a^2 + b^2]*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^
2]))])/(b^5*d^2) + (2*a^3*Sqrt[a^2 + b^2]*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^5*d^3)
- (2*a^3*Sqrt[a^2 + b^2]*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^5*d^3) + (2*a^3*f*(e + f
*x)*Sinh[c + d*x])/(b^4*d^2) + (4*a*f*(e + f*x)*Sinh[c + d*x])/(9*b^2*d^2) + (a^2*f^2*Cosh[c + d*x]*Sinh[c + d
*x])/(4*b^3*d^3) + (a^2*(e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x])/(2*b^3*d) + (2*a*f*(e + f*x)*Cosh[c + d*x]^2*
Sinh[c + d*x])/(9*b^2*d^2) + (f^2*Sinh[4*c + 4*d*x])/(256*b*d^3) + ((e + f*x)^2*Sinh[4*c + 4*d*x])/(32*b*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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 2715

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

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x)^2 \cosh ^2(c+d x) \sinh ^2(c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b} \\ & = -\frac {a \int (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x) \, dx}{b^2}+\frac {a^2 \int \frac {(e+f x)^2 \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)^2+\frac {1}{8} (e+f x)^2 \cosh (4 c+4 d x)\right ) \, dx}{b} \\ & = -\frac {(e+f x)^3}{24 b f}-\frac {a (e+f x)^2 \cosh ^3(c+d x)}{3 b^2 d}+\frac {a^2 \int (e+f x)^2 \cosh ^2(c+d x) \, dx}{b^3}-\frac {a^3 \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b^3}+\frac {\int (e+f x)^2 \cosh (4 c+4 d x) \, dx}{8 b}+\frac {(2 a f) \int (e+f x) \cosh ^3(c+d x) \, dx}{3 b^2 d} \\ & = -\frac {(e+f x)^3}{24 b f}-\frac {a^2 f (e+f x) \cosh ^2(c+d x)}{2 b^3 d^2}-\frac {2 a f^2 \cosh ^3(c+d x)}{27 b^2 d^3}-\frac {a (e+f x)^2 \cosh ^3(c+d x)}{3 b^2 d}+\frac {a^2 (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b^3 d}+\frac {2 a f (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b^2 d^2}+\frac {(e+f x)^2 \sinh (4 c+4 d x)}{32 b d}+\frac {a^4 \int (e+f x)^2 \, dx}{b^5}-\frac {a^3 \int (e+f x)^2 \sinh (c+d x) \, dx}{b^4}+\frac {a^2 \int (e+f x)^2 \, dx}{2 b^3}-\frac {\left (a^3 \left (a^2+b^2\right )\right ) \int \frac {(e+f x)^2}{a+b \sinh (c+d x)} \, dx}{b^5}+\frac {(4 a f) \int (e+f x) \cosh (c+d x) \, dx}{9 b^2 d}-\frac {f \int (e+f x) \sinh (4 c+4 d x) \, dx}{16 b d}+\frac {\left (a^2 f^2\right ) \int \cosh ^2(c+d x) \, dx}{2 b^3 d^2} \\ & = \frac {a^4 (e+f x)^3}{3 b^5 f}+\frac {a^2 (e+f x)^3}{6 b^3 f}-\frac {(e+f x)^3}{24 b f}-\frac {a^3 (e+f x)^2 \cosh (c+d x)}{b^4 d}-\frac {a^2 f (e+f x) \cosh ^2(c+d x)}{2 b^3 d^2}-\frac {2 a f^2 \cosh ^3(c+d x)}{27 b^2 d^3}-\frac {a (e+f x)^2 \cosh ^3(c+d x)}{3 b^2 d}-\frac {f (e+f x) \cosh (4 c+4 d x)}{64 b d^2}+\frac {4 a f (e+f x) \sinh (c+d x)}{9 b^2 d^2}+\frac {a^2 f^2 \cosh (c+d x) \sinh (c+d x)}{4 b^3 d^3}+\frac {a^2 (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b^3 d}+\frac {2 a f (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b^2 d^2}+\frac {(e+f x)^2 \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)^2}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{b^5}+\frac {\left (2 a^3 f\right ) \int (e+f x) \cosh (c+d x) \, dx}{b^4 d}+\frac {\left (a^2 f^2\right ) \int 1 \, dx}{4 b^3 d^2}-\frac {\left (4 a f^2\right ) \int \sinh (c+d x) \, dx}{9 b^2 d^2}+\frac {f^2 \int \cosh (4 c+4 d x) \, dx}{64 b d^2} \\ & = \frac {a^2 f^2 x}{4 b^3 d^2}+\frac {a^4 (e+f x)^3}{3 b^5 f}+\frac {a^2 (e+f x)^3}{6 b^3 f}-\frac {(e+f x)^3}{24 b f}-\frac {4 a f^2 \cosh (c+d x)}{9 b^2 d^3}-\frac {a^3 (e+f x)^2 \cosh (c+d x)}{b^4 d}-\frac {a^2 f (e+f x) \cosh ^2(c+d x)}{2 b^3 d^2}-\frac {2 a f^2 \cosh ^3(c+d x)}{27 b^2 d^3}-\frac {a (e+f x)^2 \cosh ^3(c+d x)}{3 b^2 d}-\frac {f (e+f x) \cosh (4 c+4 d x)}{64 b d^2}+\frac {2 a^3 f (e+f x) \sinh (c+d x)}{b^4 d^2}+\frac {4 a f (e+f x) \sinh (c+d x)}{9 b^2 d^2}+\frac {a^2 f^2 \cosh (c+d x) \sinh (c+d x)}{4 b^3 d^3}+\frac {a^2 (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b^3 d}+\frac {2 a f (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b^2 d^2}+\frac {f^2 \sinh (4 c+4 d x)}{256 b d^3}+\frac {(e+f x)^2 \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)^2}{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)^2}{2 a+2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{b^4}-\frac {\left (2 a^3 f^2\right ) \int \sinh (c+d x) \, dx}{b^4 d^2} \\ & = \frac {a^2 f^2 x}{4 b^3 d^2}+\frac {a^4 (e+f x)^3}{3 b^5 f}+\frac {a^2 (e+f x)^3}{6 b^3 f}-\frac {(e+f x)^3}{24 b f}-\frac {2 a^3 f^2 \cosh (c+d x)}{b^4 d^3}-\frac {4 a f^2 \cosh (c+d x)}{9 b^2 d^3}-\frac {a^3 (e+f x)^2 \cosh (c+d x)}{b^4 d}-\frac {a^2 f (e+f x) \cosh ^2(c+d x)}{2 b^3 d^2}-\frac {2 a f^2 \cosh ^3(c+d x)}{27 b^2 d^3}-\frac {a (e+f x)^2 \cosh ^3(c+d x)}{3 b^2 d}-\frac {f (e+f x) \cosh (4 c+4 d x)}{64 b d^2}-\frac {a^3 \sqrt {a^2+b^2} (e+f x)^2 \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)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d}+\frac {2 a^3 f (e+f x) \sinh (c+d x)}{b^4 d^2}+\frac {4 a f (e+f x) \sinh (c+d x)}{9 b^2 d^2}+\frac {a^2 f^2 \cosh (c+d x) \sinh (c+d x)}{4 b^3 d^3}+\frac {a^2 (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b^3 d}+\frac {2 a f (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b^2 d^2}+\frac {f^2 \sinh (4 c+4 d x)}{256 b d^3}+\frac {(e+f x)^2 \sinh (4 c+4 d x)}{32 b d}+\frac {\left (2 a^3 \sqrt {a^2+b^2} f\right ) \int (e+f x) \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 (2 a^3 \sqrt {a^2+b^2} f\right ) \int (e+f x) \log \left (1+\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{b^5 d} \\ & = \frac {a^2 f^2 x}{4 b^3 d^2}+\frac {a^4 (e+f x)^3}{3 b^5 f}+\frac {a^2 (e+f x)^3}{6 b^3 f}-\frac {(e+f x)^3}{24 b f}-\frac {2 a^3 f^2 \cosh (c+d x)}{b^4 d^3}-\frac {4 a f^2 \cosh (c+d x)}{9 b^2 d^3}-\frac {a^3 (e+f x)^2 \cosh (c+d x)}{b^4 d}-\frac {a^2 f (e+f x) \cosh ^2(c+d x)}{2 b^3 d^2}-\frac {2 a f^2 \cosh ^3(c+d x)}{27 b^2 d^3}-\frac {a (e+f x)^2 \cosh ^3(c+d x)}{3 b^2 d}-\frac {f (e+f x) \cosh (4 c+4 d x)}{64 b d^2}-\frac {a^3 \sqrt {a^2+b^2} (e+f x)^2 \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)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d}-\frac {2 a^3 \sqrt {a^2+b^2} f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d^2}+\frac {2 a^3 \sqrt {a^2+b^2} f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d^2}+\frac {2 a^3 f (e+f x) \sinh (c+d x)}{b^4 d^2}+\frac {4 a f (e+f x) \sinh (c+d x)}{9 b^2 d^2}+\frac {a^2 f^2 \cosh (c+d x) \sinh (c+d x)}{4 b^3 d^3}+\frac {a^2 (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b^3 d}+\frac {2 a f (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b^2 d^2}+\frac {f^2 \sinh (4 c+4 d x)}{256 b d^3}+\frac {(e+f x)^2 \sinh (4 c+4 d x)}{32 b d}+\frac {\left (2 a^3 \sqrt {a^2+b^2} f^2\right ) \int \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 (2 a^3 \sqrt {a^2+b^2} f^2\right ) \int \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 {a^2 f^2 x}{4 b^3 d^2}+\frac {a^4 (e+f x)^3}{3 b^5 f}+\frac {a^2 (e+f x)^3}{6 b^3 f}-\frac {(e+f x)^3}{24 b f}-\frac {2 a^3 f^2 \cosh (c+d x)}{b^4 d^3}-\frac {4 a f^2 \cosh (c+d x)}{9 b^2 d^3}-\frac {a^3 (e+f x)^2 \cosh (c+d x)}{b^4 d}-\frac {a^2 f (e+f x) \cosh ^2(c+d x)}{2 b^3 d^2}-\frac {2 a f^2 \cosh ^3(c+d x)}{27 b^2 d^3}-\frac {a (e+f x)^2 \cosh ^3(c+d x)}{3 b^2 d}-\frac {f (e+f x) \cosh (4 c+4 d x)}{64 b d^2}-\frac {a^3 \sqrt {a^2+b^2} (e+f x)^2 \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)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d}-\frac {2 a^3 \sqrt {a^2+b^2} f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d^2}+\frac {2 a^3 \sqrt {a^2+b^2} f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d^2}+\frac {2 a^3 f (e+f x) \sinh (c+d x)}{b^4 d^2}+\frac {4 a f (e+f x) \sinh (c+d x)}{9 b^2 d^2}+\frac {a^2 f^2 \cosh (c+d x) \sinh (c+d x)}{4 b^3 d^3}+\frac {a^2 (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b^3 d}+\frac {2 a f (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b^2 d^2}+\frac {f^2 \sinh (4 c+4 d x)}{256 b d^3}+\frac {(e+f x)^2 \sinh (4 c+4 d x)}{32 b d}+\frac {\left (2 a^3 \sqrt {a^2+b^2} f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {b x}{-a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^5 d^3}-\frac {\left (2 a^3 \sqrt {a^2+b^2} f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^5 d^3} \\ & = \frac {a^2 f^2 x}{4 b^3 d^2}+\frac {a^4 (e+f x)^3}{3 b^5 f}+\frac {a^2 (e+f x)^3}{6 b^3 f}-\frac {(e+f x)^3}{24 b f}-\frac {2 a^3 f^2 \cosh (c+d x)}{b^4 d^3}-\frac {4 a f^2 \cosh (c+d x)}{9 b^2 d^3}-\frac {a^3 (e+f x)^2 \cosh (c+d x)}{b^4 d}-\frac {a^2 f (e+f x) \cosh ^2(c+d x)}{2 b^3 d^2}-\frac {2 a f^2 \cosh ^3(c+d x)}{27 b^2 d^3}-\frac {a (e+f x)^2 \cosh ^3(c+d x)}{3 b^2 d}-\frac {f (e+f x) \cosh (4 c+4 d x)}{64 b d^2}-\frac {a^3 \sqrt {a^2+b^2} (e+f x)^2 \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)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d}-\frac {2 a^3 \sqrt {a^2+b^2} f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d^2}+\frac {2 a^3 \sqrt {a^2+b^2} f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d^2}+\frac {2 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 )}{b^5 d^3}-\frac {2 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 )}{b^5 d^3}+\frac {2 a^3 f (e+f x) \sinh (c+d x)}{b^4 d^2}+\frac {4 a f (e+f x) \sinh (c+d x)}{9 b^2 d^2}+\frac {a^2 f^2 \cosh (c+d x) \sinh (c+d x)}{4 b^3 d^3}+\frac {a^2 (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b^3 d}+\frac {2 a f (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b^2 d^2}+\frac {f^2 \sinh (4 c+4 d x)}{256 b d^3}+\frac {(e+f x)^2 \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. \(3579\) vs. \(2(755)=1510\).

Time = 13.14 (sec) , antiderivative size = 3579, normalized size of antiderivative = 4.74 \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

-1/8*(e^2*(c/d + x - (2*a*ArcTan[(b - a*Tanh[(c + d*x)/2])/Sqrt[-a^2 - b^2]])/(Sqrt[-a^2 - b^2]*d)))/b - (e*f*
(x^2 - (2*a*(d*x*(Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2
])]) + PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])
)]))/(Sqrt[a^2 + b^2]*d^2)))/(8*b) - (f^2*(x^3 - (3*a*(d^2*x^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])]
- d^2*x^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 2*d*x*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b
^2])] - 2*d*x*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] - 2*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a
^2 + b^2])] + 2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(Sqrt[a^2 + b^2]*d^3)))/(24*b) - (f^2*(
2*(4*a^2 + b^2)*x^3 - (6*a*(4*a^2 + 3*b^2)*(d^2*x^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - d^2*x^2*L
og[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 2*d*x*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 2*d
*x*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] - 2*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])]
+ 2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(Sqrt[a^2 + b^2]*d^3) - (24*a*b*Cosh[d*x]*((2 + d^
2*x^2)*Cosh[c] - 2*d*x*Sinh[c]))/d^3 + (3*b^2*Cosh[2*d*x]*(-2*d*x*Cosh[2*c] + (1 + 2*d^2*x^2)*Sinh[2*c]))/d^3
- (24*a*b*(-2*d*x*Cosh[c] + (2 + d^2*x^2)*Sinh[c])*Sinh[d*x])/d^3 + (3*b^2*((1 + 2*d^2*x^2)*Cosh[2*c] - 2*d*x*
Sinh[2*c])*Sinh[2*d*x])/d^3))/(96*b^3) - (e^2*((4*a^2 + b^2)*(c + d*x) - (2*a*(4*a^2 + 3*b^2)*ArcTan[(b - a*Ta
nh[(c + d*x)/2])/Sqrt[-a^2 - b^2]])/Sqrt[-a^2 - b^2] - 4*a*b*Cosh[c + d*x] + b^2*Sinh[2*(c + d*x)]))/(16*b^3*d
) - (e*f*((4*a^2 + b^2)*(-c + d*x)*(c + d*x) - 8*a*b*d*x*Cosh[c + d*x] - b^2*Cosh[2*(c + d*x)] - (2*a*(4*a^2 +
3*b^2)*(2*c*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] + (c + d*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 +
b^2])] - (c + d*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2
+ b^2])] - PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/Sqrt[a^2 + b^2] + 8*a*b*Sinh[c + d*x] + 2*b^
2*d*x*Sinh[2*(c + d*x)]))/(16*b^3*d^2) + (e^2*(6*(16*a^4 + 12*a^2*b^2 + b^4)*(c + d*x) - (12*a*(16*a^4 + 20*a^
2*b^2 + 5*b^4)*ArcTan[(b - a*Tanh[(c + d*x)/2])/Sqrt[-a^2 - b^2]])/Sqrt[-a^2 - b^2] - 48*a*b*(2*a^2 + b^2)*Cos
h[c + d*x] - 8*a*b^3*Cosh[3*(c + d*x)] + 6*b^2*(4*a^2 + b^2)*Sinh[2*(c + d*x)] + 3*b^4*Sinh[4*(c + d*x)]))/(96
*b^5*d) + (e*f*(-576*a^4*Sqrt[a^2 + b^2]*c^2 - 432*a^2*b^2*Sqrt[a^2 + b^2]*c^2 - 36*b^4*Sqrt[a^2 + b^2]*c^2 +
576*a^4*Sqrt[a^2 + b^2]*d^2*x^2 + 432*a^2*b^2*Sqrt[a^2 + b^2]*d^2*x^2 + 36*b^4*Sqrt[a^2 + b^2]*d^2*x^2 - 2304*
a^5*c*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] - 2880*a^3*b^2*c*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2
]] - 720*a*b^4*c*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] - 1152*a^3*b*Sqrt[a^2 + b^2]*d*x*Cosh[c + d*x] -
576*a*b^3*Sqrt[a^2 + b^2]*d*x*Cosh[c + d*x] - 144*a^2*b^2*Sqrt[a^2 + b^2]*Cosh[2*(c + d*x)] - 36*b^4*Sqrt[a^2
+ b^2]*Cosh[2*(c + d*x)] - 96*a*b^3*Sqrt[a^2 + b^2]*d*x*Cosh[3*(c + d*x)] - 9*b^4*Sqrt[a^2 + b^2]*Cosh[4*(c +
d*x)] - 1152*a^5*c*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - 1440*a^3*b^2*c*Log[1 + (b*E^(c + d*x))/(a
- Sqrt[a^2 + b^2])] - 360*a*b^4*c*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - 1152*a^5*d*x*Log[1 + (b*E^
(c + d*x))/(a - Sqrt[a^2 + b^2])] - 1440*a^3*b^2*d*x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - 360*a*b^
4*d*x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 1152*a^5*c*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]
)] + 1440*a^3*b^2*c*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 360*a*b^4*c*Log[1 + (b*E^(c + d*x))/(a +
Sqrt[a^2 + b^2])] + 1152*a^5*d*x*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 1440*a^3*b^2*d*x*Log[1 + (b*
E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 360*a*b^4*d*x*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - 72*(16*a^
5 + 20*a^3*b^2 + 5*a*b^4)*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 72*(16*a^5 + 20*a^3*b^2 + 5*a*b
^4)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] + 1152*a^3*b*Sqrt[a^2 + b^2]*Sinh[c + d*x] + 576*a*b^
3*Sqrt[a^2 + b^2]*Sinh[c + d*x] + 288*a^2*b^2*Sqrt[a^2 + b^2]*d*x*Sinh[2*(c + d*x)] + 72*b^4*Sqrt[a^2 + b^2]*d
*x*Sinh[2*(c + d*x)] + 32*a*b^3*Sqrt[a^2 + b^2]*Sinh[3*(c + d*x)] + 36*b^4*Sqrt[a^2 + b^2]*d*x*Sinh[4*(c + d*x
)]))/(576*b^5*Sqrt[a^2 + b^2]*d^2) + (f^2*((16*a^4*x^3)/(3*b^5) + (4*a^2*x^3)/b^3 + x^3/(3*b) - (32*a^3*Cosh[c
+ d*x])/(b^4*d^3) - (16*a*Cosh[c + d*x])/(b^2*d^3) - (16*a^3*x^2*Cosh[c + d*x])/(b^4*d) - (8*a*x^2*Cosh[c + d
*x])/(b^2*d) - (4*a^2*x*Cosh[2*(c + d*x)])/(b^3*d^2) - (x*Cosh[2*(c + d*x)])/(b*d^2) - (8*a*Cosh[3*(c + d*x)])
/(27*b^2*d^3) - (4*a*x^2*Cosh[3*(c + d*x)])/(3*b^2*d) - (x*Cosh[4*(c + d*x)])/(4*b*d^2) - (16*a^5*x^2*Log[1 +
(b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^5*Sqrt[a^2 + b^2]*d) - (20*a^3*x^2*Log[1 + (b*E^(c + d*x))/(a - Sqr
t[a^2 + b^2])])/(b^3*Sqrt[a^2 + b^2]*d) - (5*a*x^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*Sqrt[a^2
+ b^2]*d) + (16*a^5*x^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^5*Sqrt[a^2 + b^2]*d) + (20*a^3*x^2
*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^3*Sqrt[a^2 + b^2]*d) + (5*a*x^2*Log[1 + (b*E^(c + d*x))/(a
+ Sqrt[a^2 + b^2])])/(b*Sqrt[a^2 + b^2]*d) - (2*(16*a^5 + 20*a^3*b^2 + 5*a*b^4)*x*PolyLog[2, (b*E^(c + d*x))/
(-a + Sqrt[a^2 + b^2])])/(b^5*Sqrt[a^2 + b^2]*d^2) + (2*(16*a^5 + 20*a^3*b^2 + 5*a*b^4)*x*PolyLog[2, -((b*E^(c
+ d*x))/(a + Sqrt[a^2 + b^2]))])/(b^5*Sqrt[a^2 + b^2]*d^2) + (32*a^5*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^
2 + b^2])])/(b^5*Sqrt[a^2 + b^2]*d^3) + (40*a^3*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])])/(b^3*Sqrt[
a^2 + b^2]*d^3) + (10*a*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])])/(b*Sqrt[a^2 + b^2]*d^3) - (32*a^5*
PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^5*Sqrt[a^2 + b^2]*d^3) - (40*a^3*PolyLog[3, -((b*E^(c
+ d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*Sqrt[a^2 + b^2]*d^3) - (10*a*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2
+ b^2]))])/(b*Sqrt[a^2 + b^2]*d^3) + (32*a^3*x*Sinh[c + d*x])/(b^4*d^2) + (16*a*x*Sinh[c + d*x])/(b^2*d^2) +
(2*a^2*Sinh[2*(c + d*x)])/(b^3*d^3) + Sinh[2*(c + d*x)]/(2*b*d^3) + (4*a^2*x^2*Sinh[2*(c + d*x)])/(b^3*d) + (x
^2*Sinh[2*(c + d*x)])/(b*d) + (8*a*x*Sinh[3*(c + d*x)])/(9*b^2*d^2) + Sinh[4*(c + d*x)]/(16*b*d^3) + (x^2*Sinh
[4*(c + d*x)])/(2*b*d)))/16

Maple [F]

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

[In]

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

[Out]

int((f*x+e)^2*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. 6459 vs. \(2 (693) = 1386\).

Time = 0.37 (sec) , antiderivative size = 6459, normalized size of antiderivative = 8.55 \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

-1/192*e^2*(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)) + 1/13824*(576*(
8*a^4*d^3*f^2*e^(4*c) + 4*a^2*b^2*d^3*f^2*e^(4*c) - b^4*d^3*f^2*e^(4*c))*x^3 + 1728*(8*a^4*d^3*e*f*e^(4*c) + 4
*a^2*b^2*d^3*e*f*e^(4*c) - b^4*d^3*e*f*e^(4*c))*x^2 + 27*(8*b^4*d^2*f^2*x^2*e^(8*c) + 4*(4*d^2*e*f - d*f^2)*b^
4*x*e^(8*c) - (4*d*e*f - f^2)*b^4*e^(8*c))*e^(4*d*x) - 64*(9*a*b^3*d^2*f^2*x^2*e^(7*c) + 6*(3*d^2*e*f - d*f^2)
*a*b^3*x*e^(7*c) - 2*(3*d*e*f - f^2)*a*b^3*e^(7*c))*e^(3*d*x) + 864*(2*a^2*b^2*d^2*f^2*x^2*e^(6*c) + 2*(2*d^2*
e*f - d*f^2)*a^2*b^2*x*e^(6*c) - (2*d*e*f - f^2)*a^2*b^2*e^(6*c))*e^(2*d*x) + 1728*(8*(d*e*f - f^2)*a^3*b*e^(5
*c) + 2*(d*e*f - f^2)*a*b^3*e^(5*c) - (4*a^3*b*d^2*f^2*e^(5*c) + a*b^3*d^2*f^2*e^(5*c))*x^2 - 2*(4*(d^2*e*f -
d*f^2)*a^3*b*e^(5*c) + (d^2*e*f - d*f^2)*a*b^3*e^(5*c))*x)*e^(d*x) - 1728*(8*(d*e*f + f^2)*a^3*b*e^(3*c) + 2*(
d*e*f + f^2)*a*b^3*e^(3*c) + (4*a^3*b*d^2*f^2*e^(3*c) + a*b^3*d^2*f^2*e^(3*c))*x^2 + 2*(4*(d^2*e*f + d*f^2)*a^
3*b*e^(3*c) + (d^2*e*f + d*f^2)*a*b^3*e^(3*c))*x)*e^(-d*x) - 864*(2*a^2*b^2*d^2*f^2*x^2*e^(2*c) + 2*(2*d^2*e*f
+ d*f^2)*a^2*b^2*x*e^(2*c) + (2*d*e*f + f^2)*a^2*b^2*e^(2*c))*e^(-2*d*x) - 64*(9*a*b^3*d^2*f^2*x^2*e^c + 6*(3
*d^2*e*f + d*f^2)*a*b^3*x*e^c + 2*(3*d*e*f + f^2)*a*b^3*e^c)*e^(-3*d*x) - 27*(8*b^4*d^2*f^2*x^2 + 4*(4*d^2*e*f
+ d*f^2)*b^4*x + (4*d*e*f + f^2)*b^4)*e^(-4*d*x))*e^(-4*c)/(b^5*d^3) - integrate(2*((a^5*f^2*e^c + a^3*b^2*f^
2*e^c)*x^2 + 2*(a^5*e*f*e^c + a^3*b^2*e*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]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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