3.4.91 \(\int \frac {(e+f x)^3 \cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [391]

Optimal. Leaf size=792 \[ -\frac {3 a f^3 x}{8 b^2 d^3}-\frac {a (e+f x)^3}{4 b^2 d}+\frac {a^3 (e+f x)^4}{4 b^4 f}-\frac {6 a^2 f^3 \cosh (c+d x)}{b^3 d^4}+\frac {14 f^3 \cosh (c+d x)}{9 b d^4}-\frac {3 a^2 f (e+f x)^2 \cosh (c+d x)}{b^3 d^2}+\frac {2 f (e+f x)^2 \cosh (c+d x)}{3 b d^2}-\frac {2 f^3 \cosh ^3(c+d x)}{27 b d^4}-\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {3 a^3 f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {3 a^3 f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^2}+\frac {6 a^3 f^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^3}+\frac {6 a^3 f^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^3}-\frac {6 a^3 f^3 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^4}-\frac {6 a^3 f^3 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^4}+\frac {6 a^2 f^2 (e+f x) \sinh (c+d x)}{b^3 d^3}-\frac {4 f^2 (e+f x) \sinh (c+d x)}{3 b d^3}+\frac {a^2 (e+f x)^3 \sinh (c+d x)}{b^3 d}+\frac {3 a f^3 \cosh (c+d x) \sinh (c+d x)}{8 b^2 d^4}+\frac {3 a f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^2}-\frac {3 a f^2 (e+f x) \sinh ^2(c+d x)}{4 b^2 d^3}-\frac {a (e+f x)^3 \sinh ^2(c+d x)}{2 b^2 d}-\frac {f (e+f x)^2 \cosh (c+d x) \sinh ^2(c+d x)}{3 b d^2}+\frac {2 f^2 (e+f x) \sinh ^3(c+d x)}{9 b d^3}+\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 b d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.85, antiderivative size = 792, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 15, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.441, Rules used = {5698, 5554, 3392, 3377, 2718, 2713, 32, 2715, 8, 5680, 2221, 2611, 6744, 2320, 6724} \begin {gather*} \frac {a^3 (e+f x)^4}{4 b^4 f}-\frac {6 a^2 f^3 \cosh (c+d x)}{b^3 d^4}+\frac {6 a^2 f^2 (e+f x) \sinh (c+d x)}{b^3 d^3}-\frac {3 a^2 f (e+f x)^2 \cosh (c+d x)}{b^3 d^2}+\frac {a^2 (e+f x)^3 \sinh (c+d x)}{b^3 d}-\frac {6 a^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^4}-\frac {6 a^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^4}+\frac {6 a^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^3}+\frac {6 a^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^3}-\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {a^3 (e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b^4 d}-\frac {a^3 (e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b^4 d}+\frac {3 a f^3 \sinh (c+d x) \cosh (c+d x)}{8 b^2 d^4}-\frac {3 a f^2 (e+f x) \sinh ^2(c+d x)}{4 b^2 d^3}+\frac {3 a f (e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{4 b^2 d^2}-\frac {a (e+f x)^3 \sinh ^2(c+d x)}{2 b^2 d}-\frac {3 a f^3 x}{8 b^2 d^3}-\frac {a (e+f x)^3}{4 b^2 d}-\frac {2 f^3 \cosh ^3(c+d x)}{27 b d^4}+\frac {14 f^3 \cosh (c+d x)}{9 b d^4}+\frac {2 f^2 (e+f x) \sinh ^3(c+d x)}{9 b d^3}-\frac {4 f^2 (e+f x) \sinh (c+d x)}{3 b d^3}+\frac {2 f (e+f x)^2 \cosh (c+d x)}{3 b d^2}-\frac {f (e+f x)^2 \sinh ^2(c+d x) \cosh (c+d x)}{3 b d^2}+\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*a*f^3*x)/(8*b^2*d^3) - (a*(e + f*x)^3)/(4*b^2*d) + (a^3*(e + f*x)^4)/(4*b^4*f) - (6*a^2*f^3*Cosh[c + d*x])
/(b^3*d^4) + (14*f^3*Cosh[c + d*x])/(9*b*d^4) - (3*a^2*f*(e + f*x)^2*Cosh[c + d*x])/(b^3*d^2) + (2*f*(e + f*x)
^2*Cosh[c + d*x])/(3*b*d^2) - (2*f^3*Cosh[c + d*x]^3)/(27*b*d^4) - (a^3*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a
- Sqrt[a^2 + b^2])])/(b^4*d) - (a^3*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^4*d) - (3*
a^3*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^4*d^2) - (3*a^3*f*(e + f*x)^2*PolyL
og[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^4*d^2) + (6*a^3*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))
/(a - Sqrt[a^2 + b^2]))])/(b^4*d^3) + (6*a^3*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))
])/(b^4*d^3) - (6*a^3*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^4*d^4) - (6*a^3*f^3*PolyLog
[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^4*d^4) + (6*a^2*f^2*(e + f*x)*Sinh[c + d*x])/(b^3*d^3) - (4*
f^2*(e + f*x)*Sinh[c + d*x])/(3*b*d^3) + (a^2*(e + f*x)^3*Sinh[c + d*x])/(b^3*d) + (3*a*f^3*Cosh[c + d*x]*Sinh
[c + d*x])/(8*b^2*d^4) + (3*a*f*(e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x])/(4*b^2*d^2) - (3*a*f^2*(e + f*x)*Sinh
[c + d*x]^2)/(4*b^2*d^3) - (a*(e + f*x)^3*Sinh[c + d*x]^2)/(2*b^2*d) - (f*(e + f*x)^2*Cosh[c + d*x]*Sinh[c + d
*x]^2)/(3*b*d^2) + (2*f^2*(e + f*x)*Sinh[c + d*x]^3)/(9*b*d^3) + ((e + f*x)^3*Sinh[c + d*x]^3)/(3*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 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 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 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 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 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 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*} \int \frac {(e+f x)^3 \cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^3 \cosh (c+d x) \sinh ^2(c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x)^3 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 b d}-\frac {a \int (e+f x)^3 \cosh (c+d x) \sinh (c+d x) \, dx}{b^2}+\frac {a^2 \int \frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}-\frac {f \int (e+f x)^2 \sinh ^3(c+d x) \, dx}{b d}\\ &=-\frac {a (e+f x)^3 \sinh ^2(c+d x)}{2 b^2 d}-\frac {f (e+f x)^2 \cosh (c+d x) \sinh ^2(c+d x)}{3 b d^2}+\frac {2 f^2 (e+f x) \sinh ^3(c+d x)}{9 b d^3}+\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 b d}+\frac {a^2 \int (e+f x)^3 \cosh (c+d x) \, dx}{b^3}-\frac {a^3 \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^3}+\frac {(3 a f) \int (e+f x)^2 \sinh ^2(c+d x) \, dx}{2 b^2 d}+\frac {(2 f) \int (e+f x)^2 \sinh (c+d x) \, dx}{3 b d}-\frac {\left (2 f^3\right ) \int \sinh ^3(c+d x) \, dx}{9 b d^3}\\ &=\frac {a^3 (e+f x)^4}{4 b^4 f}+\frac {2 f (e+f x)^2 \cosh (c+d x)}{3 b d^2}+\frac {a^2 (e+f x)^3 \sinh (c+d x)}{b^3 d}+\frac {3 a f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^2}-\frac {3 a f^2 (e+f x) \sinh ^2(c+d x)}{4 b^2 d^3}-\frac {a (e+f x)^3 \sinh ^2(c+d x)}{2 b^2 d}-\frac {f (e+f x)^2 \cosh (c+d x) \sinh ^2(c+d x)}{3 b d^2}+\frac {2 f^2 (e+f x) \sinh ^3(c+d x)}{9 b d^3}+\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 b d}-\frac {a^3 \int \frac {e^{c+d x} (e+f x)^3}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{b^3}-\frac {a^3 \int \frac {e^{c+d x} (e+f x)^3}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{b^3}-\frac {\left (3 a^2 f\right ) \int (e+f x)^2 \sinh (c+d x) \, dx}{b^3 d}-\frac {(3 a f) \int (e+f x)^2 \, dx}{4 b^2 d}-\frac {\left (4 f^2\right ) \int (e+f x) \cosh (c+d x) \, dx}{3 b d^2}+\frac {\left (2 f^3\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (c+d x)\right )}{9 b d^4}+\frac {\left (3 a f^3\right ) \int \sinh ^2(c+d x) \, dx}{4 b^2 d^3}\\ &=-\frac {a (e+f x)^3}{4 b^2 d}+\frac {a^3 (e+f x)^4}{4 b^4 f}+\frac {2 f^3 \cosh (c+d x)}{9 b d^4}-\frac {3 a^2 f (e+f x)^2 \cosh (c+d x)}{b^3 d^2}+\frac {2 f (e+f x)^2 \cosh (c+d x)}{3 b d^2}-\frac {2 f^3 \cosh ^3(c+d x)}{27 b d^4}-\frac {a^3 (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^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {4 f^2 (e+f x) \sinh (c+d x)}{3 b d^3}+\frac {a^2 (e+f x)^3 \sinh (c+d x)}{b^3 d}+\frac {3 a f^3 \cosh (c+d x) \sinh (c+d x)}{8 b^2 d^4}+\frac {3 a f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^2}-\frac {3 a f^2 (e+f x) \sinh ^2(c+d x)}{4 b^2 d^3}-\frac {a (e+f x)^3 \sinh ^2(c+d x)}{2 b^2 d}-\frac {f (e+f x)^2 \cosh (c+d x) \sinh ^2(c+d x)}{3 b d^2}+\frac {2 f^2 (e+f x) \sinh ^3(c+d x)}{9 b d^3}+\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 b d}+\frac {\left (3 a^3 f\right ) \int (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{b^4 d}+\frac {\left (3 a^3 f\right ) \int (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{b^4 d}+\frac {\left (6 a^2 f^2\right ) \int (e+f x) \cosh (c+d x) \, dx}{b^3 d^2}-\frac {\left (3 a f^3\right ) \int 1 \, dx}{8 b^2 d^3}+\frac {\left (4 f^3\right ) \int \sinh (c+d x) \, dx}{3 b d^3}\\ &=-\frac {3 a f^3 x}{8 b^2 d^3}-\frac {a (e+f x)^3}{4 b^2 d}+\frac {a^3 (e+f x)^4}{4 b^4 f}+\frac {14 f^3 \cosh (c+d x)}{9 b d^4}-\frac {3 a^2 f (e+f x)^2 \cosh (c+d x)}{b^3 d^2}+\frac {2 f (e+f x)^2 \cosh (c+d x)}{3 b d^2}-\frac {2 f^3 \cosh ^3(c+d x)}{27 b d^4}-\frac {a^3 (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^3 (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^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^2}+\frac {6 a^2 f^2 (e+f x) \sinh (c+d x)}{b^3 d^3}-\frac {4 f^2 (e+f x) \sinh (c+d x)}{3 b d^3}+\frac {a^2 (e+f x)^3 \sinh (c+d x)}{b^3 d}+\frac {3 a f^3 \cosh (c+d x) \sinh (c+d x)}{8 b^2 d^4}+\frac {3 a f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^2}-\frac {3 a f^2 (e+f x) \sinh ^2(c+d x)}{4 b^2 d^3}-\frac {a (e+f x)^3 \sinh ^2(c+d x)}{2 b^2 d}-\frac {f (e+f x)^2 \cosh (c+d x) \sinh ^2(c+d x)}{3 b d^2}+\frac {2 f^2 (e+f x) \sinh ^3(c+d x)}{9 b d^3}+\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 b d}+\frac {\left (6 a^3 f^2\right ) \int (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{b^4 d^2}+\frac {\left (6 a^3 f^2\right ) \int (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{b^4 d^2}-\frac {\left (6 a^2 f^3\right ) \int \sinh (c+d x) \, dx}{b^3 d^3}\\ &=-\frac {3 a f^3 x}{8 b^2 d^3}-\frac {a (e+f x)^3}{4 b^2 d}+\frac {a^3 (e+f x)^4}{4 b^4 f}-\frac {6 a^2 f^3 \cosh (c+d x)}{b^3 d^4}+\frac {14 f^3 \cosh (c+d x)}{9 b d^4}-\frac {3 a^2 f (e+f x)^2 \cosh (c+d x)}{b^3 d^2}+\frac {2 f (e+f x)^2 \cosh (c+d x)}{3 b d^2}-\frac {2 f^3 \cosh ^3(c+d x)}{27 b d^4}-\frac {a^3 (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^3 (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^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^2}+\frac {6 a^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^3}+\frac {6 a^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^3}+\frac {6 a^2 f^2 (e+f x) \sinh (c+d x)}{b^3 d^3}-\frac {4 f^2 (e+f x) \sinh (c+d x)}{3 b d^3}+\frac {a^2 (e+f x)^3 \sinh (c+d x)}{b^3 d}+\frac {3 a f^3 \cosh (c+d x) \sinh (c+d x)}{8 b^2 d^4}+\frac {3 a f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^2}-\frac {3 a f^2 (e+f x) \sinh ^2(c+d x)}{4 b^2 d^3}-\frac {a (e+f x)^3 \sinh ^2(c+d x)}{2 b^2 d}-\frac {f (e+f x)^2 \cosh (c+d x) \sinh ^2(c+d x)}{3 b d^2}+\frac {2 f^2 (e+f x) \sinh ^3(c+d x)}{9 b d^3}+\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 b d}-\frac {\left (6 a^3 f^3\right ) \int \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{b^4 d^3}-\frac {\left (6 a^3 f^3\right ) \int \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{b^4 d^3}\\ &=-\frac {3 a f^3 x}{8 b^2 d^3}-\frac {a (e+f x)^3}{4 b^2 d}+\frac {a^3 (e+f x)^4}{4 b^4 f}-\frac {6 a^2 f^3 \cosh (c+d x)}{b^3 d^4}+\frac {14 f^3 \cosh (c+d x)}{9 b d^4}-\frac {3 a^2 f (e+f x)^2 \cosh (c+d x)}{b^3 d^2}+\frac {2 f (e+f x)^2 \cosh (c+d x)}{3 b d^2}-\frac {2 f^3 \cosh ^3(c+d x)}{27 b d^4}-\frac {a^3 (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^3 (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^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^2}+\frac {6 a^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^3}+\frac {6 a^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^3}+\frac {6 a^2 f^2 (e+f x) \sinh (c+d x)}{b^3 d^3}-\frac {4 f^2 (e+f x) \sinh (c+d x)}{3 b d^3}+\frac {a^2 (e+f x)^3 \sinh (c+d x)}{b^3 d}+\frac {3 a f^3 \cosh (c+d x) \sinh (c+d x)}{8 b^2 d^4}+\frac {3 a f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^2}-\frac {3 a f^2 (e+f x) \sinh ^2(c+d x)}{4 b^2 d^3}-\frac {a (e+f x)^3 \sinh ^2(c+d x)}{2 b^2 d}-\frac {f (e+f x)^2 \cosh (c+d x) \sinh ^2(c+d x)}{3 b d^2}+\frac {2 f^2 (e+f x) \sinh ^3(c+d x)}{9 b d^3}+\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 b d}-\frac {\left (6 a^3 f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (\frac {b x}{-a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^4 d^4}-\frac {\left (6 a^3 f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^4 d^4}\\ &=-\frac {3 a f^3 x}{8 b^2 d^3}-\frac {a (e+f x)^3}{4 b^2 d}+\frac {a^3 (e+f x)^4}{4 b^4 f}-\frac {6 a^2 f^3 \cosh (c+d x)}{b^3 d^4}+\frac {14 f^3 \cosh (c+d x)}{9 b d^4}-\frac {3 a^2 f (e+f x)^2 \cosh (c+d x)}{b^3 d^2}+\frac {2 f (e+f x)^2 \cosh (c+d x)}{3 b d^2}-\frac {2 f^3 \cosh ^3(c+d x)}{27 b d^4}-\frac {a^3 (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^3 (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^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^2}+\frac {6 a^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^3}+\frac {6 a^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^3}-\frac {6 a^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^4}-\frac {6 a^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^4}+\frac {6 a^2 f^2 (e+f x) \sinh (c+d x)}{b^3 d^3}-\frac {4 f^2 (e+f x) \sinh (c+d x)}{3 b d^3}+\frac {a^2 (e+f x)^3 \sinh (c+d x)}{b^3 d}+\frac {3 a f^3 \cosh (c+d x) \sinh (c+d x)}{8 b^2 d^4}+\frac {3 a f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^2}-\frac {3 a f^2 (e+f x) \sinh ^2(c+d x)}{4 b^2 d^3}-\frac {a (e+f x)^3 \sinh ^2(c+d x)}{2 b^2 d}-\frac {f (e+f x)^2 \cosh (c+d x) \sinh ^2(c+d x)}{3 b d^2}+\frac {2 f^2 (e+f x) \sinh ^3(c+d x)}{9 b d^3}+\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 b d}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(3901\) vs. \(2(792)=1584\).
time = 6.47, size = 3901, normalized size = 4.93 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1296*a^3*c^2*d^2*e^2*E^(3*c)*f + 2592*a^3*c*d^3*e^2*E^(3*c)*f*x + 1296*a^3*d^4*e^2*E^(3*c)*f*x^2 + 864*a^3*d^
4*e*E^(3*c)*f^2*x^3 + 216*a^3*d^4*E^(3*c)*f^3*x^4 - 2592*a^2*b*d*e*E^(2*c)*f^2*Cosh[d*x] + 648*b^3*d*e*E^(2*c)
*f^2*Cosh[d*x] + 2592*a^2*b*d*e*E^(4*c)*f^2*Cosh[d*x] - 648*b^3*d*e*E^(4*c)*f^2*Cosh[d*x] - 2592*a^2*b*E^(2*c)
*f^3*Cosh[d*x] + 648*b^3*E^(2*c)*f^3*Cosh[d*x] - 2592*a^2*b*E^(4*c)*f^3*Cosh[d*x] + 648*b^3*E^(4*c)*f^3*Cosh[d
*x] - 2592*a^2*b*d^2*e*E^(2*c)*f^2*x*Cosh[d*x] + 648*b^3*d^2*e*E^(2*c)*f^2*x*Cosh[d*x] - 2592*a^2*b*d^2*e*E^(4
*c)*f^2*x*Cosh[d*x] + 648*b^3*d^2*e*E^(4*c)*f^2*x*Cosh[d*x] - 2592*a^2*b*d*E^(2*c)*f^3*x*Cosh[d*x] + 648*b^3*d
*E^(2*c)*f^3*x*Cosh[d*x] + 2592*a^2*b*d*E^(4*c)*f^3*x*Cosh[d*x] - 648*b^3*d*E^(4*c)*f^3*x*Cosh[d*x] - 1296*a^2
*b*d^3*e*E^(2*c)*f^2*x^2*Cosh[d*x] + 324*b^3*d^3*e*E^(2*c)*f^2*x^2*Cosh[d*x] + 1296*a^2*b*d^3*e*E^(4*c)*f^2*x^
2*Cosh[d*x] - 324*b^3*d^3*e*E^(4*c)*f^2*x^2*Cosh[d*x] - 1296*a^2*b*d^2*E^(2*c)*f^3*x^2*Cosh[d*x] + 324*b^3*d^2
*E^(2*c)*f^3*x^2*Cosh[d*x] - 1296*a^2*b*d^2*E^(4*c)*f^3*x^2*Cosh[d*x] + 324*b^3*d^2*E^(4*c)*f^3*x^2*Cosh[d*x]
- 432*a^2*b*d^3*E^(2*c)*f^3*x^3*Cosh[d*x] + 108*b^3*d^3*E^(2*c)*f^3*x^3*Cosh[d*x] + 432*a^2*b*d^3*E^(4*c)*f^3*
x^3*Cosh[d*x] - 108*b^3*d^3*E^(4*c)*f^3*x^3*Cosh[d*x] - 162*a*b^2*d*e*E^c*f^2*Cosh[2*d*x] - 162*a*b^2*d*e*E^(5
*c)*f^2*Cosh[2*d*x] - 81*a*b^2*E^c*f^3*Cosh[2*d*x] + 81*a*b^2*E^(5*c)*f^3*Cosh[2*d*x] - 324*a*b^2*d^2*e*E^c*f^
2*x*Cosh[2*d*x] + 324*a*b^2*d^2*e*E^(5*c)*f^2*x*Cosh[2*d*x] - 162*a*b^2*d*E^c*f^3*x*Cosh[2*d*x] - 162*a*b^2*d*
E^(5*c)*f^3*x*Cosh[2*d*x] - 324*a*b^2*d^3*e*E^c*f^2*x^2*Cosh[2*d*x] - 324*a*b^2*d^3*e*E^(5*c)*f^2*x^2*Cosh[2*d
*x] - 162*a*b^2*d^2*E^c*f^3*x^2*Cosh[2*d*x] + 162*a*b^2*d^2*E^(5*c)*f^3*x^2*Cosh[2*d*x] - 108*a*b^2*d^3*E^c*f^
3*x^3*Cosh[2*d*x] - 108*a*b^2*d^3*E^(5*c)*f^3*x^3*Cosh[2*d*x] - 24*b^3*d*e*f^2*Cosh[3*d*x] + 24*b^3*d*e*E^(6*c
)*f^2*Cosh[3*d*x] - 8*b^3*f^3*Cosh[3*d*x] - 8*b^3*E^(6*c)*f^3*Cosh[3*d*x] - 72*b^3*d^2*e*f^2*x*Cosh[3*d*x] - 7
2*b^3*d^2*e*E^(6*c)*f^2*x*Cosh[3*d*x] - 24*b^3*d*f^3*x*Cosh[3*d*x] + 24*b^3*d*E^(6*c)*f^3*x*Cosh[3*d*x] - 108*
b^3*d^3*e*f^2*x^2*Cosh[3*d*x] + 108*b^3*d^3*e*E^(6*c)*f^2*x^2*Cosh[3*d*x] - 36*b^3*d^2*f^3*x^2*Cosh[3*d*x] - 3
6*b^3*d^2*E^(6*c)*f^3*x^2*Cosh[3*d*x] - 36*b^3*d^3*f^3*x^3*Cosh[3*d*x] + 36*b^3*d^3*E^(6*c)*f^3*x^3*Cosh[3*d*x
] - 2592*a^2*b*d^2*e^2*E^(3*c)*f*Cosh[c + d*x] + 648*b^3*d^2*e^2*E^(3*c)*f*Cosh[c + d*x] - 216*a*b^2*d^3*e^3*E
^(3*c)*Cosh[2*(c + d*x)] - 648*a*b^2*d^3*e^2*E^(3*c)*f*x*Cosh[2*(c + d*x)] - 72*b^3*d^2*e^2*E^(3*c)*f*Cosh[3*(
c + d*x)] - 2592*a^3*c*d^2*e^2*E^(3*c)*f*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - 2592*a^3*d^3*e^2*E^(
3*c)*f*x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - 2592*a^3*c*d^2*e^2*E^(3*c)*f*Log[1 + (b*E^(c + d*x))
/(a + Sqrt[a^2 + b^2])] - 2592*a^3*d^3*e^2*E^(3*c)*f*x*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - 2592*a
^3*d^3*e*E^(3*c)*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] - 864*a^3*d^3*E^(3*c)*
f^3*x^3*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] - 2592*a^3*d^3*e*E^(3*c)*f^2*x^2*Log[1
+ (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] - 864*a^3*d^3*E^(3*c)*f^3*x^3*Log[1 + (b*E^(2*c + d*x
))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] - 864*a^3*d^3*e^3*E^(3*c)*Log[a + b*Sinh[c + d*x]] + 2592*a^3*c*d^2*e^
2*E^(3*c)*f*Log[a + b*Sinh[c + d*x]] - 2592*a^3*d^2*e^2*E^(3*c)*f*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 +
b^2])] - 2592*a^3*d^2*e^2*E^(3*c)*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] - 5184*a^3*d^2*e*E^(3
*c)*f^2*x*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] - 2592*a^3*d^2*E^(3*c)*f^3*x^2*
PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] - 5184*a^3*d^2*e*E^(3*c)*f^2*x*PolyLog[2,
-((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))] - 2592*a^3*d^2*E^(3*c)*f^3*x^2*PolyLog[2, -((b*E^(2
*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))] + 5184*a^3*d*e*E^(3*c)*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*
E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] + 5184*a^3*d*E^(3*c)*f^3*x*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^
2 + b^2)*E^(2*c)]))] + 5184*a^3*d*e*E^(3*c)*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*
c)]))] + 5184*a^3*d*E^(3*c)*f^3*x*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))] - 5184*
a^3*E^(3*c)*f^3*PolyLog[4, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] - 5184*a^3*E^(3*c)*f^3*Po
lyLog[4, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))] + 2592*a^2*b*d*e*E^(2*c)*f^2*Sinh[d*x] - 64
8*b^3*d*e*E^(2*c)*f^2*Sinh[d*x] + 2592*a^2*b*d*e*E^(4*c)*f^2*Sinh[d*x] - 648*b^3*d*e*E^(4*c)*f^2*Sinh[d*x] + 2
592*a^2*b*E^(2*c)*f^3*Sinh[d*x] - 648*b^3*E^(2*c)*f^3*Sinh[d*x] - 2592*a^2*b*E^(4*c)*f^3*Sinh[d*x] + 648*b^3*E
^(4*c)*f^3*Sinh[d*x] + 2592*a^2*b*d^2*e*E^(2*c)*f^2*x*Sinh[d*x] - 648*b^3*d^2*e*E^(2*c)*f^2*x*Sinh[d*x] - 2592
*a^2*b*d^2*e*E^(4*c)*f^2*x*Sinh[d*x] + 648*b^3*d^2*e*E^(4*c)*f^2*x*Sinh[d*x] + 2592*a^2*b*d*E^(2*c)*f^3*x*Sinh
[d*x] - 648*b^3*d*E^(2*c)*f^3*x*Sinh[d*x] + 259...

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Maple [F]
time = 2.32, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{3} \cosh \left (d x +c \right ) \left (\sinh ^{3}\left (d x +c \right )\right )}{a +b \sinh \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

-1/24*(24*(d*x + c)*a^3/(b^4*d) + 24*a^3*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(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) + (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))*e^3 - 1/864*(216*a^3*d^4*f^3*x^4*e^(3*c) + 864*a^3*d^4*
f^2*x^3*e^(3*c + 1) + 1296*a^3*d^4*f*x^2*e^(3*c + 2) - 4*(9*b^3*d^3*f^3*x^3*e^(6*c) - 2*b^3*f^3*e^(6*c) - 9*b^
3*d^2*f*e^(6*c + 2) + 6*b^3*d*f^2*e^(6*c + 1) - 9*(b^3*d^2*f^3*e^(6*c) - 3*b^3*d^3*f^2*e^(6*c + 1))*x^2 + 3*(2
*b^3*d*f^3*e^(6*c) + 9*b^3*d^3*f*e^(6*c + 2) - 6*b^3*d^2*f^2*e^(6*c + 1))*x)*e^(3*d*x) + 27*(4*a*b^2*d^3*f^3*x
^3*e^(5*c) - 3*a*b^2*f^3*e^(5*c) - 6*a*b^2*d^2*f*e^(5*c + 2) + 6*a*b^2*d*f^2*e^(5*c + 1) - 6*(a*b^2*d^2*f^3*e^
(5*c) - 2*a*b^2*d^3*f^2*e^(5*c + 1))*x^2 + 6*(a*b^2*d*f^3*e^(5*c) + 2*a*b^2*d^3*f*e^(5*c + 2) - 2*a*b^2*d^2*f^
2*e^(5*c + 1))*x)*e^(2*d*x) + 108*(24*a^2*b*f^3*e^(4*c) - 6*b^3*f^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*a^2*b*d^2*f^3*e^(4*c) - b^3*d^2*f^3*e^(4*c) - (4*a^2*b*d^3*f^2*e^(4*c) - b^3*d^3*f^
2*e^(4*c))*e)*x^2 - 3*(8*a^2*b*d*f^3*e^(4*c) - 2*b^3*d*f^3*e^(4*c) + (4*a^2*b*d^3*f*e^(4*c) - b^3*d^3*f*e^(4*c
))*e^2 - 2*(4*a^2*b*d^2*f^2*e^(4*c) - b^3*d^2*f^2*e^(4*c))*e)*x + 3*(4*a^2*b*d^2*f*e^(4*c) - b^3*d^2*f*e^(4*c)
)*e^2 - 6*(4*a^2*b*d*f^2*e^(4*c) - b^3*d*f^2*e^(4*c))*e)*e^(d*x) + 108*(24*a^2*b*f^3*e^(2*c) - 6*b^3*f^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*a^2*b*d^2*f^3*e^(2*c) - b^3*d^2*f^3*e^(2*c) +
(4*a^2*b*d^3*f^2*e^(2*c) - b^3*d^3*f^2*e^(2*c))*e)*x^2 + 3*(8*a^2*b*d*f^3*e^(2*c) - 2*b^3*d*f^3*e^(2*c) + (4*a
^2*b*d^3*f*e^(2*c) - b^3*d^3*f*e^(2*c))*e^2 + 2*(4*a^2*b*d^2*f^2*e^(2*c) - b^3*d^2*f^2*e^(2*c))*e)*x + 3*(4*a^
2*b*d^2*f*e^(2*c) - b^3*d^2*f*e^(2*c))*e^2 + 6*(4*a^2*b*d*f^2*e^(2*c) - b^3*d*f^2*e^(2*c))*e)*e^(-d*x) + 27*(4
*a*b^2*d^3*f^3*x^3*e^c + 6*a*b^2*d^2*f*e^(c + 2) + 6*a*b^2*d*f^2*e^(c + 1) + 3*a*b^2*f^3*e^c + 6*(2*a*b^2*d^3*
f^2*e^(c + 1) + a*b^2*d^2*f^3*e^c)*x^2 + 6*(2*a*b^2*d^3*f*e^(c + 2) + 2*a*b^2*d^2*f^2*e^(c + 1) + a*b^2*d*f^3*
e^c)*x)*e^(-2*d*x) + 4*(9*b^3*d^3*f^3*x^3 + 9*b^3*d^2*f*e^2 + 6*b^3*d*f^2*e + 2*b^3*f^3 + 9*(3*b^3*d^3*f^2*e +
b^3*d^2*f^3)*x^2 + 3*(9*b^3*d^3*f*e^2 + 6*b^3*d^2*f^2*e + 2*b^3*d*f^3)*x)*e^(-3*d*x))*e^(-3*c)/(b^4*d^4) + in
tegrate(-2*(a^3*b*f^3*x^3 + 3*a^3*b*f^2*x^2*e + 3*a^3*b*f*x*e^2 - (a^4*f^3*x^3*e^c + 3*a^4*f^2*x^2*e^(c + 1) +
3*a^4*f*x*e^(c + 2))*e^(d*x))/(b^5*e^(2*d*x + 2*c) + 2*a*b^4*e^(d*x + c) - b^5), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 13122 vs. \(2 (757) = 1514\).
time = 0.54, size = 13122, normalized size = 16.57 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/864*(36*b^3*d^3*f^3*x^3 + 36*b^3*d^2*f^3*x^2 + 36*b^3*d^3*cosh(1)^3 + 36*b^3*d^3*sinh(1)^3 + 24*b^3*d*f^3*x
- 4*(9*b^3*d^3*f^3*x^3 - 9*b^3*d^2*f^3*x^2 + 9*b^3*d^3*cosh(1)^3 + 9*b^3*d^3*sinh(1)^3 + 6*b^3*d*f^3*x - 2*b^
3*f^3 + 9*(3*b^3*d^3*f*x - b^3*d^2*f)*cosh(1)^2 + 9*(3*b^3*d^3*f*x + 3*b^3*d^3*cosh(1) - b^3*d^2*f)*sinh(1)^2
+ 3*(9*b^3*d^3*f^2*x^2 - 6*b^3*d^2*f^2*x + 2*b^3*d*f^2)*cosh(1) + 3*(9*b^3*d^3*f^2*x^2 - 6*b^3*d^2*f^2*x + 9*b
^3*d^3*cosh(1)^2 + 2*b^3*d*f^2 + 6*(3*b^3*d^3*f*x - b^3*d^2*f)*cosh(1))*sinh(1))*cosh(d*x + c)^6 - 4*(9*b^3*d^
3*f^3*x^3 - 9*b^3*d^2*f^3*x^2 + 9*b^3*d^3*cosh(1)^3 + 9*b^3*d^3*sinh(1)^3 + 6*b^3*d*f^3*x - 2*b^3*f^3 + 9*(3*b
^3*d^3*f*x - b^3*d^2*f)*cosh(1)^2 + 9*(3*b^3*d^3*f*x + 3*b^3*d^3*cosh(1) - b^3*d^2*f)*sinh(1)^2 + 3*(9*b^3*d^3
*f^2*x^2 - 6*b^3*d^2*f^2*x + 2*b^3*d*f^2)*cosh(1) + 3*(9*b^3*d^3*f^2*x^2 - 6*b^3*d^2*f^2*x + 9*b^3*d^3*cosh(1)
^2 + 2*b^3*d*f^2 + 6*(3*b^3*d^3*f*x - b^3*d^2*f)*cosh(1))*sinh(1))*sinh(d*x + c)^6 + 8*b^3*f^3 + 27*(4*a*b^2*d
^3*f^3*x^3 - 6*a*b^2*d^2*f^3*x^2 + 4*a*b^2*d^3*cosh(1)^3 + 4*a*b^2*d^3*sinh(1)^3 + 6*a*b^2*d*f^3*x - 3*a*b^2*f
^3 + 6*(2*a*b^2*d^3*f*x - a*b^2*d^2*f)*cosh(1)^2 + 6*(2*a*b^2*d^3*f*x + 2*a*b^2*d^3*cosh(1) - a*b^2*d^2*f)*sin
h(1)^2 + 6*(2*a*b^2*d^3*f^2*x^2 - 2*a*b^2*d^2*f^2*x + a*b^2*d*f^2)*cosh(1) + 6*(2*a*b^2*d^3*f^2*x^2 - 2*a*b^2*
d^2*f^2*x + 2*a*b^2*d^3*cosh(1)^2 + a*b^2*d*f^2 + 2*(2*a*b^2*d^3*f*x - a*b^2*d^2*f)*cosh(1))*sinh(1))*cosh(d*x
+ c)^5 + 3*(36*a*b^2*d^3*f^3*x^3 - 54*a*b^2*d^2*f^3*x^2 + 36*a*b^2*d^3*cosh(1)^3 + 36*a*b^2*d^3*sinh(1)^3 + 5
4*a*b^2*d*f^3*x - 27*a*b^2*f^3 + 54*(2*a*b^2*d^3*f*x - a*b^2*d^2*f)*cosh(1)^2 + 54*(2*a*b^2*d^3*f*x + 2*a*b^2*
d^3*cosh(1) - a*b^2*d^2*f)*sinh(1)^2 + 54*(2*a*b^2*d^3*f^2*x^2 - 2*a*b^2*d^2*f^2*x + a*b^2*d*f^2)*cosh(1) - 8*
(9*b^3*d^3*f^3*x^3 - 9*b^3*d^2*f^3*x^2 + 9*b^3*d^3*cosh(1)^3 + 9*b^3*d^3*sinh(1)^3 + 6*b^3*d*f^3*x - 2*b^3*f^3
+ 9*(3*b^3*d^3*f*x - b^3*d^2*f)*cosh(1)^2 + 9*(3*b^3*d^3*f*x + 3*b^3*d^3*cosh(1) - b^3*d^2*f)*sinh(1)^2 + 3*(
9*b^3*d^3*f^2*x^2 - 6*b^3*d^2*f^2*x + 2*b^3*d*f^2)*cosh(1) + 3*(9*b^3*d^3*f^2*x^2 - 6*b^3*d^2*f^2*x + 9*b^3*d^
3*cosh(1)^2 + 2*b^3*d*f^2 + 6*(3*b^3*d^3*f*x - b^3*d^2*f)*cosh(1))*sinh(1))*cosh(d*x + c) + 54*(2*a*b^2*d^3*f^
2*x^2 - 2*a*b^2*d^2*f^2*x + 2*a*b^2*d^3*cosh(1)^2 + a*b^2*d*f^2 + 2*(2*a*b^2*d^3*f*x - a*b^2*d^2*f)*cosh(1))*s
inh(1))*sinh(d*x + c)^5 - 108*((4*a^2*b - b^3)*d^3*f^3*x^3 - 3*(4*a^2*b - b^3)*d^2*f^3*x^2 + (4*a^2*b - b^3)*d
^3*cosh(1)^3 + (4*a^2*b - b^3)*d^3*sinh(1)^3 + 6*(4*a^2*b - b^3)*d*f^3*x - 6*(4*a^2*b - b^3)*f^3 + 3*((4*a^2*b
- b^3)*d^3*f*x - (4*a^2*b - b^3)*d^2*f)*cosh(1)^2 + 3*((4*a^2*b - b^3)*d^3*f*x + (4*a^2*b - b^3)*d^3*cosh(1)
- (4*a^2*b - b^3)*d^2*f)*sinh(1)^2 + 3*((4*a^2*b - b^3)*d^3*f^2*x^2 - 2*(4*a^2*b - b^3)*d^2*f^2*x + 2*(4*a^2*b
- b^3)*d*f^2)*cosh(1) + 3*((4*a^2*b - b^3)*d^3*f^2*x^2 - 2*(4*a^2*b - b^3)*d^2*f^2*x + (4*a^2*b - b^3)*d^3*co
sh(1)^2 + 2*(4*a^2*b - b^3)*d*f^2 + 2*((4*a^2*b - b^3)*d^3*f*x - (4*a^2*b - b^3)*d^2*f)*cosh(1))*sinh(1))*cosh
(d*x + c)^4 - 3*(36*(4*a^2*b - b^3)*d^3*f^3*x^3 - 108*(4*a^2*b - b^3)*d^2*f^3*x^2 + 36*(4*a^2*b - b^3)*d^3*cos
h(1)^3 + 36*(4*a^2*b - b^3)*d^3*sinh(1)^3 + 216*(4*a^2*b - b^3)*d*f^3*x - 216*(4*a^2*b - b^3)*f^3 + 108*((4*a^
2*b - b^3)*d^3*f*x - (4*a^2*b - b^3)*d^2*f)*cosh(1)^2 + 20*(9*b^3*d^3*f^3*x^3 - 9*b^3*d^2*f^3*x^2 + 9*b^3*d^3*
cosh(1)^3 + 9*b^3*d^3*sinh(1)^3 + 6*b^3*d*f^3*x - 2*b^3*f^3 + 9*(3*b^3*d^3*f*x - b^3*d^2*f)*cosh(1)^2 + 9*(3*b
^3*d^3*f*x + 3*b^3*d^3*cosh(1) - b^3*d^2*f)*sinh(1)^2 + 3*(9*b^3*d^3*f^2*x^2 - 6*b^3*d^2*f^2*x + 2*b^3*d*f^2)*
cosh(1) + 3*(9*b^3*d^3*f^2*x^2 - 6*b^3*d^2*f^2*x + 9*b^3*d^3*cosh(1)^2 + 2*b^3*d*f^2 + 6*(3*b^3*d^3*f*x - b^3*
d^2*f)*cosh(1))*sinh(1))*cosh(d*x + c)^2 + 108*((4*a^2*b - b^3)*d^3*f*x + (4*a^2*b - b^3)*d^3*cosh(1) - (4*a^2
*b - b^3)*d^2*f)*sinh(1)^2 + 108*((4*a^2*b - b^3)*d^3*f^2*x^2 - 2*(4*a^2*b - b^3)*d^2*f^2*x + 2*(4*a^2*b - b^3
)*d*f^2)*cosh(1) - 45*(4*a*b^2*d^3*f^3*x^3 - 6*a*b^2*d^2*f^3*x^2 + 4*a*b^2*d^3*cosh(1)^3 + 4*a*b^2*d^3*sinh(1)
^3 + 6*a*b^2*d*f^3*x - 3*a*b^2*f^3 + 6*(2*a*b^2*d^3*f*x - a*b^2*d^2*f)*cosh(1)^2 + 6*(2*a*b^2*d^3*f*x + 2*a*b^
2*d^3*cosh(1) - a*b^2*d^2*f)*sinh(1)^2 + 6*(2*a*b^2*d^3*f^2*x^2 - 2*a*b^2*d^2*f^2*x + a*b^2*d*f^2)*cosh(1) + 6
*(2*a*b^2*d^3*f^2*x^2 - 2*a*b^2*d^2*f^2*x + 2*a*b^2*d^3*cosh(1)^2 + a*b^2*d*f^2 + 2*(2*a*b^2*d^3*f*x - a*b^2*d
^2*f)*cosh(1))*sinh(1))*cosh(d*x + c) + 108*((4*a^2*b - b^3)*d^3*f^2*x^2 - 2*(4*a^2*b - b^3)*d^2*f^2*x + (4*a^
2*b - b^3)*d^3*cosh(1)^2 + 2*(4*a^2*b - b^3)*d*f^2 + 2*((4*a^2*b - b^3)*d^3*f*x - (4*a^2*b - b^3)*d^2*f)*cosh(
1))*sinh(1))*sinh(d*x + c)^4 - 216*(a^3*d^4*f^3*x^4 - 2*a^3*c^4*f^3 + 4*(a^3*d^4*x + 2*a^3*c*d^3)*cosh(1)^3 +
4*(a^3*d^4*x + 2*a^3*c*d^3)*sinh(1)^3 + 6*(a^3*d^4*f*x^2 - 2*a^3*c^2*d^2*f)*cosh(1)^2 + 6*(a^3*d^4*f*x^2 - 2*a
^3*c^2*d^2*f + 2*(a^3*d^4*x + 2*a^3*c*d^3)*cosh(1))*sinh(1)^2 + 4*(a^3*d^4*f^2*x^3 + 2*a^3*c^3*d*f^2)*cosh(1)
+ 4*(a^3*d^4*f^2*x^3 + 2*a^3*c^3*d*f^2 + 3*(a^3*d^4*x + 2*a^3*c*d^3)*cosh(1)^2 + 3*(a^3*d^4*f*x^2 - 2*a^3*c^2*
d^2*f)*cosh(1))*sinh(1))*cosh(d*x + c)^3 - 2*(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)**3*cosh(d*x+c)*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)^3*cosh(d*x+c)*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

integrate((f*x + e)^3*cosh(d*x + c)*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 )\,{\mathrm {sinh}\left (c+d\,x\right )}^3\,{\left (e+f\,x\right )}^3}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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