3.5.2 \(\int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [402]

Optimal. Leaf size=1049 \[ -\frac {a^3 e f x}{2 b^4 d}+\frac {3 a e f x}{16 b^2 d}-\frac {a^3 f^2 x^2}{4 b^4 d}+\frac {3 a f^2 x^2}{32 b^2 d}+\frac {a^3 \left (a^2+b^2\right ) (e+f x)^3}{3 b^6 f}-\frac {2 a^4 f (e+f x) \cosh (c+d x)}{b^5 d^2}-\frac {4 a^2 f (e+f x) \cosh (c+d x)}{3 b^3 d^2}+\frac {f (e+f x) \cosh (c+d x)}{4 b d^2}-\frac {3 a f^2 \cosh ^2(c+d x)}{32 b^2 d^3}-\frac {2 a^2 f (e+f x) \cosh ^3(c+d x)}{9 b^3 d^2}-\frac {a f^2 \cosh ^4(c+d x)}{32 b^2 d^3}-\frac {a (e+f x)^2 \cosh ^4(c+d x)}{4 b^2 d}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 b d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 b d^2}-\frac {a^3 \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^6 d}-\frac {a^3 \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^6 d}-\frac {2 a^3 \left (a^2+b^2\right ) f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^6 d^2}-\frac {2 a^3 \left (a^2+b^2\right ) f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^6 d^2}+\frac {2 a^3 \left (a^2+b^2\right ) f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^6 d^3}+\frac {2 a^3 \left (a^2+b^2\right ) f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^6 d^3}+\frac {2 a^4 f^2 \sinh (c+d x)}{b^5 d^3}+\frac {14 a^2 f^2 \sinh (c+d x)}{9 b^3 d^3}-\frac {f^2 \sinh (c+d x)}{4 b d^3}+\frac {a^4 (e+f x)^2 \sinh (c+d x)}{b^5 d}+\frac {2 a^2 (e+f x)^2 \sinh (c+d x)}{3 b^3 d}-\frac {(e+f x)^2 \sinh (c+d x)}{8 b d}+\frac {a^3 f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^4 d^2}+\frac {3 a f (e+f x) \cosh (c+d x) \sinh (c+d x)}{16 b^2 d^2}+\frac {a^2 (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^3 d}+\frac {a f (e+f x) \cosh ^3(c+d x) \sinh (c+d x)}{8 b^2 d^2}-\frac {a^3 f^2 \sinh ^2(c+d x)}{4 b^4 d^3}-\frac {a^3 (e+f x)^2 \sinh ^2(c+d x)}{2 b^4 d}+\frac {2 a^2 f^2 \sinh ^3(c+d x)}{27 b^3 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 b d^3}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 b d}+\frac {f^2 \sinh (5 c+5 d x)}{1000 b d^3}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 b d} \]

[Out]

-2*a^3*(a^2+b^2)*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^6/d^2-2*a^3*(a^2+b^2)*f*(f*x+e)*poly
log(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^6/d^2+1/2*a^3*f*(f*x+e)*cosh(d*x+c)*sinh(d*x+c)/b^4/d^2+1/8*a*f*(f*
x+e)*cosh(d*x+c)^3*sinh(d*x+c)/b^2/d^2+1/4*f*(f*x+e)*cosh(d*x+c)/b/d^2-1/8*(f*x+e)^2*sinh(d*x+c)/b/d-1/4*f^2*s
inh(d*x+c)/b/d^3-1/2*a^3*e*f*x/b^4/d-2*a^4*f*(f*x+e)*cosh(d*x+c)/b^5/d^2-2/9*a^2*f*(f*x+e)*cosh(d*x+c)^3/b^3/d
^2+3/16*a*e*f*x/b^2/d-4/3*a^2*f*(f*x+e)*cosh(d*x+c)/b^3/d^2+a^4*(f*x+e)^2*sinh(d*x+c)/b^5/d+1/216*f^2*sinh(3*d
*x+3*c)/b/d^3+1/48*(f*x+e)^2*sinh(3*d*x+3*c)/b/d+1/1000*f^2*sinh(5*d*x+5*c)/b/d^3+1/80*(f*x+e)^2*sinh(5*d*x+5*
c)/b/d-a^3*(a^2+b^2)*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^6/d-a^3*(a^2+b^2)*(f*x+e)^2*ln(1+b*exp
(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^6/d+2*a^3*(a^2+b^2)*f^2*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^6/d^3+2*
a^3*(a^2+b^2)*f^2*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^6/d^3+3/16*a*f*(f*x+e)*cosh(d*x+c)*sinh(d*x+c
)/b^2/d^2+1/3*a^2*(f*x+e)^2*cosh(d*x+c)^2*sinh(d*x+c)/b^3/d+3/32*a*f^2*x^2/b^2/d+14/9*a^2*f^2*sinh(d*x+c)/b^3/
d^3-1/4*a^3*f^2*x^2/b^4/d+1/3*a^3*(a^2+b^2)*(f*x+e)^3/b^6/f-3/32*a*f^2*cosh(d*x+c)^2/b^2/d^3-1/32*a*f^2*cosh(d
*x+c)^4/b^2/d^3-1/4*a*(f*x+e)^2*cosh(d*x+c)^4/b^2/d-1/72*f*(f*x+e)*cosh(3*d*x+3*c)/b/d^2-1/200*f*(f*x+e)*cosh(
5*d*x+5*c)/b/d^2+2*a^4*f^2*sinh(d*x+c)/b^5/d^3+2/3*a^2*(f*x+e)^2*sinh(d*x+c)/b^3/d-1/4*a^3*f^2*sinh(d*x+c)^2/b
^4/d^3-1/2*a^3*(f*x+e)^2*sinh(d*x+c)^2/b^4/d+2/27*a^2*f^2*sinh(d*x+c)^3/b^3/d^3

________________________________________________________________________________________

Rubi [A]
time = 1.11, antiderivative size = 1049, normalized size of antiderivative = 1.00, number of steps used = 40, number of rules used = 15, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5698, 5556, 3377, 2717, 5555, 3391, 3392, 2713, 5684, 5554, 5680, 2221, 2611, 2320, 6724} \begin {gather*} -\frac {2 f (e+f x) \cosh (c+d x) a^4}{b^5 d^2}+\frac {2 f^2 \sinh (c+d x) a^4}{b^5 d^3}+\frac {(e+f x)^2 \sinh (c+d x) a^4}{b^5 d}+\frac {\left (a^2+b^2\right ) (e+f x)^3 a^3}{3 b^6 f}-\frac {f^2 x^2 a^3}{4 b^4 d}-\frac {f^2 \sinh ^2(c+d x) a^3}{4 b^4 d^3}-\frac {(e+f x)^2 \sinh ^2(c+d x) a^3}{2 b^4 d}-\frac {e f x a^3}{2 b^4 d}-\frac {\left (a^2+b^2\right ) (e+f x)^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) a^3}{b^6 d}-\frac {\left (a^2+b^2\right ) (e+f x)^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) a^3}{b^6 d}-\frac {2 \left (a^2+b^2\right ) f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) a^3}{b^6 d^2}-\frac {2 \left (a^2+b^2\right ) f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) a^3}{b^6 d^2}+\frac {2 \left (a^2+b^2\right ) f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) a^3}{b^6 d^3}+\frac {2 \left (a^2+b^2\right ) f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) a^3}{b^6 d^3}+\frac {f (e+f x) \cosh (c+d x) \sinh (c+d x) a^3}{2 b^4 d^2}-\frac {2 f (e+f x) \cosh ^3(c+d x) a^2}{9 b^3 d^2}+\frac {2 f^2 \sinh ^3(c+d x) a^2}{27 b^3 d^3}-\frac {4 f (e+f x) \cosh (c+d x) a^2}{3 b^3 d^2}+\frac {14 f^2 \sinh (c+d x) a^2}{9 b^3 d^3}+\frac {2 (e+f x)^2 \sinh (c+d x) a^2}{3 b^3 d}+\frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x) a^2}{3 b^3 d}-\frac {f^2 \cosh ^4(c+d x) a}{32 b^2 d^3}-\frac {(e+f x)^2 \cosh ^4(c+d x) a}{4 b^2 d}+\frac {3 f^2 x^2 a}{32 b^2 d}-\frac {3 f^2 \cosh ^2(c+d x) a}{32 b^2 d^3}+\frac {3 e f x a}{16 b^2 d}+\frac {f (e+f x) \cosh ^3(c+d x) \sinh (c+d x) a}{8 b^2 d^2}+\frac {3 f (e+f x) \cosh (c+d x) \sinh (c+d x) a}{16 b^2 d^2}+\frac {f (e+f x) \cosh (c+d x)}{4 b d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 b d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 b d^2}-\frac {f^2 \sinh (c+d x)}{4 b d^3}-\frac {(e+f x)^2 \sinh (c+d x)}{8 b d}+\frac {f^2 \sinh (3 c+3 d x)}{216 b d^3}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 b d}+\frac {f^2 \sinh (5 c+5 d x)}{1000 b d^3}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 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 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 5554

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

Rule 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 5680

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

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

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Mathematica [A]
time = 7.54, size = 1170, normalized size = 1.12 \begin {gather*} \frac {1}{8} \left (-\frac {8 a^3 \left (a^2+b^2\right ) e^2 x \coth (c)}{b^6}-\frac {8 a^3 \left (a^2+b^2\right ) e f x^2 \coth (c)}{b^6}-\frac {8 a^3 \left (a^2+b^2\right ) f^2 x^3 \coth (c)}{3 b^6}-\frac {8 a^3 \left (a^2+b^2\right ) \left (2 d^3 e^{2 c} x \left (3 e^2+3 e f x+f^2 x^2\right )+3 \left (1-e^{2 c}\right ) \left (d^2 e^2 \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )+2 d^2 e f x \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+d^2 f^2 x^2 \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+2 d^2 e f x \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+d^2 f^2 x^2 \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+2 d f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+2 d f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-2 f^2 \text {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-2 f^2 \text {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )\right )\right )}{3 b^6 d^3 \left (1-e^{2 c}\right )}+\frac {\left (-8 a^4-6 a^2 b^2+b^4\right ) \left (2 f^2+2 d f (e+f x)+d^2 (e+f x)^2\right ) (\cosh (c+d x)-\sinh (c+d x))}{2 b^5 d^3}+\frac {\left (8 a^4+6 a^2 b^2-b^4\right ) \left (2 f^2-2 d f (e+f x)+d^2 (e+f x)^2\right ) (\cosh (c+d x)+\sinh (c+d x))}{2 b^5 d^3}+\frac {a \left (2 a^2+b^2\right ) \left (f^2+2 d f (e+f x)+2 d^2 (e+f x)^2\right ) (-\cosh (2 (c+d x))+\sinh (2 (c+d x)))}{4 b^4 d^3}-\frac {a \left (2 a^2+b^2\right ) \left (f^2-2 d f (e+f x)+2 d^2 (e+f x)^2\right ) (\cosh (2 (c+d x))+\sinh (2 (c+d x)))}{4 b^4 d^3}+\frac {\left (4 a^2+b^2\right ) \left (2 f^2+6 d f (e+f x)+9 d^2 (e+f x)^2\right ) (-\cosh (3 (c+d x))+\sinh (3 (c+d x)))}{108 b^3 d^3}+\frac {\left (4 a^2+b^2\right ) \left (2 f^2-6 d f (e+f x)+9 d^2 (e+f x)^2\right ) (\cosh (3 (c+d x))+\sinh (3 (c+d x)))}{108 b^3 d^3}+\frac {a \left (f^2+4 d f (e+f x)+8 d^2 (e+f x)^2\right ) (-\cosh (4 (c+d x))+\sinh (4 (c+d x)))}{64 b^2 d^3}-\frac {a \left (f^2-4 d f (e+f x)+8 d^2 (e+f x)^2\right ) (\cosh (4 (c+d x))+\sinh (4 (c+d x)))}{64 b^2 d^3}+\frac {\left (2 f^2+10 d f (e+f x)+25 d^2 (e+f x)^2\right ) (-\cosh (5 (c+d x))+\sinh (5 (c+d x)))}{500 b d^3}+\frac {\left (2 f^2-10 d f (e+f x)+25 d^2 (e+f x)^2\right ) (\cosh (5 (c+d x))+\sinh (5 (c+d x)))}{500 b d^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-8*a^3*(a^2 + b^2)*e^2*x*Coth[c])/b^6 - (8*a^3*(a^2 + b^2)*e*f*x^2*Coth[c])/b^6 - (8*a^3*(a^2 + b^2)*f^2*x^3
*Coth[c])/(3*b^6) - (8*a^3*(a^2 + b^2)*(2*d^3*E^(2*c)*x*(3*e^2 + 3*e*f*x + f^2*x^2) + 3*(1 - E^(2*c))*(d^2*e^2
*Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))] + 2*d^2*e*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 +
b^2)*E^(2*c)])] + d^2*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] + 2*d^2*e*f*x*Log
[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] + d^2*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c + S
qrt[(a^2 + b^2)*E^(2*c)])] + 2*d*f*(e + f*x)*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]
))] + 2*d*f*(e + f*x)*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))] - 2*f^2*PolyLog[3,
-((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] - 2*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt
[(a^2 + b^2)*E^(2*c)]))])))/(3*b^6*d^3*(1 - E^(2*c))) + ((-8*a^4 - 6*a^2*b^2 + b^4)*(2*f^2 + 2*d*f*(e + f*x) +
d^2*(e + f*x)^2)*(Cosh[c + d*x] - Sinh[c + d*x]))/(2*b^5*d^3) + ((8*a^4 + 6*a^2*b^2 - b^4)*(2*f^2 - 2*d*f*(e
+ f*x) + d^2*(e + f*x)^2)*(Cosh[c + d*x] + Sinh[c + d*x]))/(2*b^5*d^3) + (a*(2*a^2 + b^2)*(f^2 + 2*d*f*(e + f*
x) + 2*d^2*(e + f*x)^2)*(-Cosh[2*(c + d*x)] + Sinh[2*(c + d*x)]))/(4*b^4*d^3) - (a*(2*a^2 + b^2)*(f^2 - 2*d*f*
(e + f*x) + 2*d^2*(e + f*x)^2)*(Cosh[2*(c + d*x)] + Sinh[2*(c + d*x)]))/(4*b^4*d^3) + ((4*a^2 + b^2)*(2*f^2 +
6*d*f*(e + f*x) + 9*d^2*(e + f*x)^2)*(-Cosh[3*(c + d*x)] + Sinh[3*(c + d*x)]))/(108*b^3*d^3) + ((4*a^2 + b^2)*
(2*f^2 - 6*d*f*(e + f*x) + 9*d^2*(e + f*x)^2)*(Cosh[3*(c + d*x)] + Sinh[3*(c + d*x)]))/(108*b^3*d^3) + (a*(f^2
+ 4*d*f*(e + f*x) + 8*d^2*(e + f*x)^2)*(-Cosh[4*(c + d*x)] + Sinh[4*(c + d*x)]))/(64*b^2*d^3) - (a*(f^2 - 4*d
*f*(e + f*x) + 8*d^2*(e + f*x)^2)*(Cosh[4*(c + d*x)] + Sinh[4*(c + d*x)]))/(64*b^2*d^3) + ((2*f^2 + 10*d*f*(e
+ f*x) + 25*d^2*(e + f*x)^2)*(-Cosh[5*(c + d*x)] + Sinh[5*(c + d*x)]))/(500*b*d^3) + ((2*f^2 - 10*d*f*(e + f*x
) + 25*d^2*(e + f*x)^2)*(Cosh[5*(c + d*x)] + Sinh[5*(c + d*x)]))/(500*b*d^3))/8

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Maple [F]
time = 2.60, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{2} \left (\cosh ^{3}\left (d x +c \right )\right ) \left (\sinh ^{3}\left (d x +c \right )\right )}{a +b \sinh \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 17953 vs. \(2 (997) = 1994\).
time = 0.63, size = 17953, normalized size = 17.11 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1728000*(10800*b^5*d^2*f^2*x^2 - 432*(25*b^5*d^2*f^2*x^2 - 10*b^5*d*f^2*x + 25*b^5*d^2*cosh(1)^2 + 25*b^5*d
^2*sinh(1)^2 + 2*b^5*f^2 + 10*(5*b^5*d^2*f*x - b^5*d*f)*cosh(1) + 10*(5*b^5*d^2*f*x + 5*b^5*d^2*cosh(1) - b^5*
d*f)*sinh(1))*cosh(d*x + c)^10 - 432*(25*b^5*d^2*f^2*x^2 - 10*b^5*d*f^2*x + 25*b^5*d^2*cosh(1)^2 + 25*b^5*d^2*
sinh(1)^2 + 2*b^5*f^2 + 10*(5*b^5*d^2*f*x - b^5*d*f)*cosh(1) + 10*(5*b^5*d^2*f*x + 5*b^5*d^2*cosh(1) - b^5*d*f
)*sinh(1))*sinh(d*x + c)^10 + 3375*(8*a*b^4*d^2*f^2*x^2 - 4*a*b^4*d*f^2*x + 8*a*b^4*d^2*cosh(1)^2 + 8*a*b^4*d^
2*sinh(1)^2 + a*b^4*f^2 + 4*(4*a*b^4*d^2*f*x - a*b^4*d*f)*cosh(1) + 4*(4*a*b^4*d^2*f*x + 4*a*b^4*d^2*cosh(1) -
a*b^4*d*f)*sinh(1))*cosh(d*x + c)^9 + 135*(200*a*b^4*d^2*f^2*x^2 - 100*a*b^4*d*f^2*x + 200*a*b^4*d^2*cosh(1)^
2 + 200*a*b^4*d^2*sinh(1)^2 + 25*a*b^4*f^2 + 100*(4*a*b^4*d^2*f*x - a*b^4*d*f)*cosh(1) - 32*(25*b^5*d^2*f^2*x^
2 - 10*b^5*d*f^2*x + 25*b^5*d^2*cosh(1)^2 + 25*b^5*d^2*sinh(1)^2 + 2*b^5*f^2 + 10*(5*b^5*d^2*f*x - b^5*d*f)*co
sh(1) + 10*(5*b^5*d^2*f*x + 5*b^5*d^2*cosh(1) - b^5*d*f)*sinh(1))*cosh(d*x + c) + 100*(4*a*b^4*d^2*f*x + 4*a*b
^4*d^2*cosh(1) - a*b^4*d*f)*sinh(1))*sinh(d*x + c)^9 + 4320*b^5*d*f^2*x + 10800*b^5*d^2*cosh(1)^2 - 2000*(9*(4
*a^2*b^3 + b^5)*d^2*f^2*x^2 - 6*(4*a^2*b^3 + b^5)*d*f^2*x + 9*(4*a^2*b^3 + b^5)*d^2*cosh(1)^2 + 9*(4*a^2*b^3 +
b^5)*d^2*sinh(1)^2 + 2*(4*a^2*b^3 + b^5)*f^2 + 6*(3*(4*a^2*b^3 + b^5)*d^2*f*x - (4*a^2*b^3 + b^5)*d*f)*cosh(1
) + 6*(3*(4*a^2*b^3 + b^5)*d^2*f*x + 3*(4*a^2*b^3 + b^5)*d^2*cosh(1) - (4*a^2*b^3 + b^5)*d*f)*sinh(1))*cosh(d*
x + c)^8 + 10800*b^5*d^2*sinh(1)^2 - 5*(3600*(4*a^2*b^3 + b^5)*d^2*f^2*x^2 - 2400*(4*a^2*b^3 + b^5)*d*f^2*x +
3600*(4*a^2*b^3 + b^5)*d^2*cosh(1)^2 + 3600*(4*a^2*b^3 + b^5)*d^2*sinh(1)^2 + 800*(4*a^2*b^3 + b^5)*f^2 + 3888
*(25*b^5*d^2*f^2*x^2 - 10*b^5*d*f^2*x + 25*b^5*d^2*cosh(1)^2 + 25*b^5*d^2*sinh(1)^2 + 2*b^5*f^2 + 10*(5*b^5*d^
2*f*x - b^5*d*f)*cosh(1) + 10*(5*b^5*d^2*f*x + 5*b^5*d^2*cosh(1) - b^5*d*f)*sinh(1))*cosh(d*x + c)^2 + 2400*(3
*(4*a^2*b^3 + b^5)*d^2*f*x - (4*a^2*b^3 + b^5)*d*f)*cosh(1) - 6075*(8*a*b^4*d^2*f^2*x^2 - 4*a*b^4*d*f^2*x + 8*
a*b^4*d^2*cosh(1)^2 + 8*a*b^4*d^2*sinh(1)^2 + a*b^4*f^2 + 4*(4*a*b^4*d^2*f*x - a*b^4*d*f)*cosh(1) + 4*(4*a*b^4
*d^2*f*x + 4*a*b^4*d^2*cosh(1) - a*b^4*d*f)*sinh(1))*cosh(d*x + c) + 2400*(3*(4*a^2*b^3 + b^5)*d^2*f*x + 3*(4*
a^2*b^3 + b^5)*d^2*cosh(1) - (4*a^2*b^3 + b^5)*d*f)*sinh(1))*sinh(d*x + c)^8 + 54000*(2*(2*a^3*b^2 + a*b^4)*d^
2*f^2*x^2 - 2*(2*a^3*b^2 + a*b^4)*d*f^2*x + 2*(2*a^3*b^2 + a*b^4)*d^2*cosh(1)^2 + 2*(2*a^3*b^2 + a*b^4)*d^2*si
nh(1)^2 + (2*a^3*b^2 + a*b^4)*f^2 + 2*(2*(2*a^3*b^2 + a*b^4)*d^2*f*x - (2*a^3*b^2 + a*b^4)*d*f)*cosh(1) + 2*(2
*(2*a^3*b^2 + a*b^4)*d^2*f*x + 2*(2*a^3*b^2 + a*b^4)*d^2*cosh(1) - (2*a^3*b^2 + a*b^4)*d*f)*sinh(1))*cosh(d*x
+ c)^7 + 20*(5400*(2*a^3*b^2 + a*b^4)*d^2*f^2*x^2 - 5400*(2*a^3*b^2 + a*b^4)*d*f^2*x + 5400*(2*a^3*b^2 + a*b^4
)*d^2*cosh(1)^2 + 5400*(2*a^3*b^2 + a*b^4)*d^2*sinh(1)^2 - 2592*(25*b^5*d^2*f^2*x^2 - 10*b^5*d*f^2*x + 25*b^5*
d^2*cosh(1)^2 + 25*b^5*d^2*sinh(1)^2 + 2*b^5*f^2 + 10*(5*b^5*d^2*f*x - b^5*d*f)*cosh(1) + 10*(5*b^5*d^2*f*x +
5*b^5*d^2*cosh(1) - b^5*d*f)*sinh(1))*cosh(d*x + c)^3 + 2700*(2*a^3*b^2 + a*b^4)*f^2 + 6075*(8*a*b^4*d^2*f^2*x
^2 - 4*a*b^4*d*f^2*x + 8*a*b^4*d^2*cosh(1)^2 + 8*a*b^4*d^2*sinh(1)^2 + a*b^4*f^2 + 4*(4*a*b^4*d^2*f*x - a*b^4*
d*f)*cosh(1) + 4*(4*a*b^4*d^2*f*x + 4*a*b^4*d^2*cosh(1) - a*b^4*d*f)*sinh(1))*cosh(d*x + c)^2 + 5400*(2*(2*a^3
*b^2 + a*b^4)*d^2*f*x - (2*a^3*b^2 + a*b^4)*d*f)*cosh(1) - 800*(9*(4*a^2*b^3 + b^5)*d^2*f^2*x^2 - 6*(4*a^2*b^3
+ b^5)*d*f^2*x + 9*(4*a^2*b^3 + b^5)*d^2*cosh(1)^2 + 9*(4*a^2*b^3 + b^5)*d^2*sinh(1)^2 + 2*(4*a^2*b^3 + b^5)*
f^2 + 6*(3*(4*a^2*b^3 + b^5)*d^2*f*x - (4*a^2*b^3 + b^5)*d*f)*cosh(1) + 6*(3*(4*a^2*b^3 + b^5)*d^2*f*x + 3*(4*
a^2*b^3 + b^5)*d^2*cosh(1) - (4*a^2*b^3 + b^5)*d*f)*sinh(1))*cosh(d*x + c) + 5400*(2*(2*a^3*b^2 + a*b^4)*d^2*f
*x + 2*(2*a^3*b^2 + a*b^4)*d^2*cosh(1) - (2*a^3*b^2 + a*b^4)*d*f)*sinh(1))*sinh(d*x + c)^7 + 864*b^5*f^2 - 108
000*((8*a^4*b + 6*a^2*b^3 - b^5)*d^2*f^2*x^2 - 2*(8*a^4*b + 6*a^2*b^3 - b^5)*d*f^2*x + (8*a^4*b + 6*a^2*b^3 -
b^5)*d^2*cosh(1)^2 + (8*a^4*b + 6*a^2*b^3 - b^5)*d^2*sinh(1)^2 + 2*(8*a^4*b + 6*a^2*b^3 - b^5)*f^2 + 2*((8*a^4
*b + 6*a^2*b^3 - b^5)*d^2*f*x - (8*a^4*b + 6*a^2*b^3 - b^5)*d*f)*cosh(1) + 2*((8*a^4*b + 6*a^2*b^3 - b^5)*d^2*
f*x + (8*a^4*b + 6*a^2*b^3 - b^5)*d^2*cosh(1) - (8*a^4*b + 6*a^2*b^3 - b^5)*d*f)*sinh(1))*cosh(d*x + c)^6 - 20
*(5400*(8*a^4*b + 6*a^2*b^3 - b^5)*d^2*f^2*x^2 - 10800*(8*a^4*b + 6*a^2*b^3 - b^5)*d*f^2*x + 5400*(8*a^4*b + 6
*a^2*b^3 - b^5)*d^2*cosh(1)^2 + 4536*(25*b^5*d^2*f^2*x^2 - 10*b^5*d*f^2*x + 25*b^5*d^2*cosh(1)^2 + 25*b^5*d^2*
sinh(1)^2 + 2*b^5*f^2 + 10*(5*b^5*d^2*f*x - b^5*d*f)*cosh(1) + 10*(5*b^5*d^2*f*x + 5*b^5*d^2*cosh(1) - b^5*d*f
)*sinh(1))*cosh(d*x + c)^4 + 5400*(8*a^4*b + 6*a^2*b^3 - b^5)*d^2*sinh(1)^2 - 14175*(8*a*b^4*d^2*f^2*x^2 - 4*a
*b^4*d*f^2*x + 8*a*b^4*d^2*cosh(1)^2 + 8*a*b^4*d^2*sinh(1)^2 + a*b^4*f^2 + 4*(4*a*b^4*d^2*f*x - a*b^4*d*f)*cos
h(1) + 4*(4*a*b^4*d^2*f*x + 4*a*b^4*d^2*cosh(1)...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^3\,{\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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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