3.4.72 \(\int \frac {(e+f x)^3 \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [372]

Optimal. Leaf size=1123 \[ \frac {3 a^2 f^3 x}{8 b^3 d^3}-\frac {45 f^3 x}{256 b d^3}+\frac {a^2 (e+f x)^3}{4 b^3 d}-\frac {3 (e+f x)^3}{32 b d}-\frac {a^2 \left (a^2+b^2\right ) (e+f x)^4}{4 b^5 f}+\frac {6 a^3 f^3 \cosh (c+d x)}{b^4 d^4}+\frac {40 a f^3 \cosh (c+d x)}{9 b^2 d^4}+\frac {3 a^3 f (e+f x)^2 \cosh (c+d x)}{b^4 d^2}+\frac {2 a f (e+f x)^2 \cosh (c+d x)}{b^2 d^2}+\frac {9 f^2 (e+f x) \cosh ^2(c+d x)}{32 b d^3}+\frac {2 a f^3 \cosh ^3(c+d x)}{27 b^2 d^4}+\frac {a f (e+f x)^2 \cosh ^3(c+d x)}{3 b^2 d^2}+\frac {3 f^2 (e+f x) \cosh ^4(c+d x)}{32 b d^3}+\frac {(e+f x)^3 \cosh ^4(c+d x)}{4 b d}+\frac {a^2 \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d}+\frac {a^2 \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d}+\frac {3 a^2 \left (a^2+b^2\right ) f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d^2}+\frac {3 a^2 \left (a^2+b^2\right ) f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d^2}-\frac {6 a^2 \left (a^2+b^2\right ) f^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d^3}-\frac {6 a^2 \left (a^2+b^2\right ) f^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d^3}+\frac {6 a^2 \left (a^2+b^2\right ) f^3 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d^4}+\frac {6 a^2 \left (a^2+b^2\right ) f^3 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d^4}-\frac {6 a^3 f^2 (e+f x) \sinh (c+d x)}{b^4 d^3}-\frac {40 a f^2 (e+f x) \sinh (c+d x)}{9 b^2 d^3}-\frac {a^3 (e+f x)^3 \sinh (c+d x)}{b^4 d}-\frac {2 a (e+f x)^3 \sinh (c+d x)}{3 b^2 d}-\frac {3 a^2 f^3 \cosh (c+d x) \sinh (c+d x)}{8 b^3 d^4}-\frac {45 f^3 \cosh (c+d x) \sinh (c+d x)}{256 b d^4}-\frac {3 a^2 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b^3 d^2}-\frac {9 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{32 b d^2}-\frac {2 a f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b^2 d^3}-\frac {a (e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^2 d}-\frac {3 f^3 \cosh ^3(c+d x) \sinh (c+d x)}{128 b d^4}-\frac {3 f (e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{16 b d^2}+\frac {3 a^2 f^2 (e+f x) \sinh ^2(c+d x)}{4 b^3 d^3}+\frac {a^2 (e+f x)^3 \sinh ^2(c+d x)}{2 b^3 d} \]

[Out]

-a^3*(f*x+e)^3*sinh(d*x+c)/b^4/d-3/32*(f*x+e)^3/b/d-45/256*f^3*x/b/d^3+a^2*(a^2+b^2)*(f*x+e)^3*ln(1+b*exp(d*x+
c)/(a-(a^2+b^2)^(1/2)))/b^5/d+a^2*(a^2+b^2)*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^5/d+6*a^2*(a^2+
b^2)*f^3*polylog(4,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^5/d^4+6*a^2*(a^2+b^2)*f^3*polylog(4,-b*exp(d*x+c)/(a+(
a^2+b^2)^(1/2)))/b^5/d^4+2*a*f*(f*x+e)^2*cosh(d*x+c)/b^2/d^2-40/9*a*f^2*(f*x+e)*sinh(d*x+c)/b^2/d^3-9/32*f*(f*
x+e)^2*cosh(d*x+c)*sinh(d*x+c)/b/d^2+3*a^3*f*(f*x+e)^2*cosh(d*x+c)/b^4/d^2+1/3*a*f*(f*x+e)^2*cosh(d*x+c)^3/b^2
/d^2-6*a^3*f^2*(f*x+e)*sinh(d*x+c)/b^4/d^3-3/8*a^2*f^3*cosh(d*x+c)*sinh(d*x+c)/b^3/d^4-1/3*a*(f*x+e)^3*cosh(d*
x+c)^2*sinh(d*x+c)/b^2/d-3/16*f*(f*x+e)^2*cosh(d*x+c)^3*sinh(d*x+c)/b/d^2+3/4*a^2*f^2*(f*x+e)*sinh(d*x+c)^2/b^
3/d^3+1/4*a^2*(f*x+e)^3/b^3/d+1/4*(f*x+e)^3*cosh(d*x+c)^4/b/d+3*a^2*(a^2+b^2)*f*(f*x+e)^2*polylog(2,-b*exp(d*x
+c)/(a-(a^2+b^2)^(1/2)))/b^5/d^2+3*a^2*(a^2+b^2)*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^5/
d^2-6*a^2*(a^2+b^2)*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^5/d^3-6*a^2*(a^2+b^2)*f^2*(f*x+
e)*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^5/d^3-45/256*f^3*cosh(d*x+c)*sinh(d*x+c)/b/d^4+40/9*a*f^3*co
sh(d*x+c)/b^2/d^4+3/8*a^2*f^3*x/b^3/d^3-3/4*a^2*f*(f*x+e)^2*cosh(d*x+c)*sinh(d*x+c)/b^3/d^2-2/9*a*f^2*(f*x+e)*
cosh(d*x+c)^2*sinh(d*x+c)/b^2/d^3-1/4*a^2*(a^2+b^2)*(f*x+e)^4/b^5/f+6*a^3*f^3*cosh(d*x+c)/b^4/d^4+9/32*f^2*(f*
x+e)*cosh(d*x+c)^2/b/d^3+2/27*a*f^3*cosh(d*x+c)^3/b^2/d^4+3/32*f^2*(f*x+e)*cosh(d*x+c)^4/b/d^3-2/3*a*(f*x+e)^3
*sinh(d*x+c)/b^2/d-3/128*f^3*cosh(d*x+c)^3*sinh(d*x+c)/b/d^4+1/2*a^2*(f*x+e)^3*sinh(d*x+c)^2/b^3/d

________________________________________________________________________________________

Rubi [A]
time = 1.08, antiderivative size = 1123, normalized size of antiderivative = 1.00, number of steps used = 40, number of rules used = 17, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.472, Rules used = {5698, 5555, 3392, 32, 2715, 8, 3377, 2718, 3391, 5684, 5554, 5680, 2221, 2611, 6744, 2320, 6724} \begin {gather*} -\frac {a^2 \left (a^2+b^2\right ) (e+f x)^4}{4 b^5 f}+\frac {\cosh ^4(c+d x) (e+f x)^3}{4 b d}+\frac {a^2 \sinh ^2(c+d x) (e+f x)^3}{2 b^3 d}+\frac {a^2 \left (a^2+b^2\right ) \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{b^5 d}+\frac {a^2 \left (a^2+b^2\right ) \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{b^5 d}-\frac {a \cosh ^2(c+d x) \sinh (c+d x) (e+f x)^3}{3 b^2 d}-\frac {2 a \sinh (c+d x) (e+f x)^3}{3 b^2 d}-\frac {a^3 \sinh (c+d x) (e+f x)^3}{b^4 d}-\frac {3 (e+f x)^3}{32 b d}+\frac {a^2 (e+f x)^3}{4 b^3 d}+\frac {a f \cosh ^3(c+d x) (e+f x)^2}{3 b^2 d^2}+\frac {2 a f \cosh (c+d x) (e+f x)^2}{b^2 d^2}+\frac {3 a^3 f \cosh (c+d x) (e+f x)^2}{b^4 d^2}+\frac {3 a^2 \left (a^2+b^2\right ) f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) (e+f x)^2}{b^5 d^2}+\frac {3 a^2 \left (a^2+b^2\right ) f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) (e+f x)^2}{b^5 d^2}-\frac {3 f \cosh ^3(c+d x) \sinh (c+d x) (e+f x)^2}{16 b d^2}-\frac {9 f \cosh (c+d x) \sinh (c+d x) (e+f x)^2}{32 b d^2}-\frac {3 a^2 f \cosh (c+d x) \sinh (c+d x) (e+f x)^2}{4 b^3 d^2}+\frac {3 f^2 \cosh ^4(c+d x) (e+f x)}{32 b d^3}+\frac {9 f^2 \cosh ^2(c+d x) (e+f x)}{32 b d^3}+\frac {3 a^2 f^2 \sinh ^2(c+d x) (e+f x)}{4 b^3 d^3}-\frac {6 a^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 ) (e+f x)}{b^5 d^3}-\frac {6 a^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 ) (e+f x)}{b^5 d^3}-\frac {40 a f^2 \sinh (c+d x) (e+f x)}{9 b^2 d^3}-\frac {6 a^3 f^2 \sinh (c+d x) (e+f x)}{b^4 d^3}-\frac {2 a f^2 \cosh ^2(c+d x) \sinh (c+d x) (e+f x)}{9 b^2 d^3}+\frac {2 a f^3 \cosh ^3(c+d x)}{27 b^2 d^4}-\frac {45 f^3 x}{256 b d^3}+\frac {3 a^2 f^3 x}{8 b^3 d^3}+\frac {40 a f^3 \cosh (c+d x)}{9 b^2 d^4}+\frac {6 a^3 f^3 \cosh (c+d x)}{b^4 d^4}+\frac {6 a^2 \left (a^2+b^2\right ) f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d^4}+\frac {6 a^2 \left (a^2+b^2\right ) f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d^4}-\frac {3 f^3 \cosh ^3(c+d x) \sinh (c+d x)}{128 b d^4}-\frac {45 f^3 \cosh (c+d x) \sinh (c+d x)}{256 b d^4}-\frac {3 a^2 f^3 \cosh (c+d x) \sinh (c+d x)}{8 b^3 d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a^2*f^3*x)/(8*b^3*d^3) - (45*f^3*x)/(256*b*d^3) + (a^2*(e + f*x)^3)/(4*b^3*d) - (3*(e + f*x)^3)/(32*b*d) -
(a^2*(a^2 + b^2)*(e + f*x)^4)/(4*b^5*f) + (6*a^3*f^3*Cosh[c + d*x])/(b^4*d^4) + (40*a*f^3*Cosh[c + d*x])/(9*b^
2*d^4) + (3*a^3*f*(e + f*x)^2*Cosh[c + d*x])/(b^4*d^2) + (2*a*f*(e + f*x)^2*Cosh[c + d*x])/(b^2*d^2) + (9*f^2*
(e + f*x)*Cosh[c + d*x]^2)/(32*b*d^3) + (2*a*f^3*Cosh[c + d*x]^3)/(27*b^2*d^4) + (a*f*(e + f*x)^2*Cosh[c + d*x
]^3)/(3*b^2*d^2) + (3*f^2*(e + f*x)*Cosh[c + d*x]^4)/(32*b*d^3) + ((e + f*x)^3*Cosh[c + d*x]^4)/(4*b*d) + (a^2
*(a^2 + b^2)*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^5*d) + (a^2*(a^2 + b^2)*(e + f*x)^
3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^5*d) + (3*a^2*(a^2 + b^2)*f*(e + f*x)^2*PolyLog[2, -((b*E
^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^5*d^2) + (3*a^2*(a^2 + b^2)*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))
/(a + Sqrt[a^2 + b^2]))])/(b^5*d^2) - (6*a^2*(a^2 + b^2)*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[
a^2 + b^2]))])/(b^5*d^3) - (6*a^2*(a^2 + b^2)*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])
)])/(b^5*d^3) + (6*a^2*(a^2 + b^2)*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^5*d^4) + (6*a^
2*(a^2 + b^2)*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^5*d^4) - (6*a^3*f^2*(e + f*x)*Sinh[
c + d*x])/(b^4*d^3) - (40*a*f^2*(e + f*x)*Sinh[c + d*x])/(9*b^2*d^3) - (a^3*(e + f*x)^3*Sinh[c + d*x])/(b^4*d)
- (2*a*(e + f*x)^3*Sinh[c + d*x])/(3*b^2*d) - (3*a^2*f^3*Cosh[c + d*x]*Sinh[c + d*x])/(8*b^3*d^4) - (45*f^3*C
osh[c + d*x]*Sinh[c + d*x])/(256*b*d^4) - (3*a^2*f*(e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x])/(4*b^3*d^2) - (9*f
*(e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x])/(32*b*d^2) - (2*a*f^2*(e + f*x)*Cosh[c + d*x]^2*Sinh[c + d*x])/(9*b^
2*d^3) - (a*(e + f*x)^3*Cosh[c + d*x]^2*Sinh[c + d*x])/(3*b^2*d) - (3*f^3*Cosh[c + d*x]^3*Sinh[c + d*x])/(128*
b*d^4) - (3*f*(e + f*x)^2*Cosh[c + d*x]^3*Sinh[c + d*x])/(16*b*d^2) + (3*a^2*f^2*(e + f*x)*Sinh[c + d*x]^2)/(4
*b^3*d^3) + (a^2*(e + f*x)^3*Sinh[c + d*x]^2)/(2*b^3*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 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 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 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]

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 ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^3 \cosh ^3(c+d x) \sinh (c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=\frac {(e+f x)^3 \cosh ^4(c+d x)}{4 b d}-\frac {a \int (e+f x)^3 \cosh ^3(c+d x) \, dx}{b^2}+\frac {a^2 \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}-\frac {(3 f) \int (e+f x)^2 \cosh ^4(c+d x) \, dx}{4 b d}\\ &=\frac {a f (e+f x)^2 \cosh ^3(c+d x)}{3 b^2 d^2}+\frac {3 f^2 (e+f x) \cosh ^4(c+d x)}{32 b d^3}+\frac {(e+f x)^3 \cosh ^4(c+d x)}{4 b d}-\frac {a (e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^2 d}-\frac {3 f (e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{16 b d^2}-\frac {a^3 \int (e+f x)^3 \cosh (c+d x) \, dx}{b^4}+\frac {a^2 \int (e+f x)^3 \cosh (c+d x) \sinh (c+d x) \, dx}{b^3}-\frac {(2 a) \int (e+f x)^3 \cosh (c+d x) \, dx}{3 b^2}+\frac {\left (a^2 \left (a^2+b^2\right )\right ) \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^4}-\frac {(9 f) \int (e+f x)^2 \cosh ^2(c+d x) \, dx}{16 b d}-\frac {\left (2 a f^2\right ) \int (e+f x) \cosh ^3(c+d x) \, dx}{3 b^2 d^2}-\frac {\left (3 f^3\right ) \int \cosh ^4(c+d x) \, dx}{32 b d^3}\\ &=-\frac {a^2 \left (a^2+b^2\right ) (e+f x)^4}{4 b^5 f}+\frac {9 f^2 (e+f x) \cosh ^2(c+d x)}{32 b d^3}+\frac {2 a f^3 \cosh ^3(c+d x)}{27 b^2 d^4}+\frac {a f (e+f x)^2 \cosh ^3(c+d x)}{3 b^2 d^2}+\frac {3 f^2 (e+f x) \cosh ^4(c+d x)}{32 b d^3}+\frac {(e+f x)^3 \cosh ^4(c+d x)}{4 b d}-\frac {a^3 (e+f x)^3 \sinh (c+d x)}{b^4 d}-\frac {2 a (e+f x)^3 \sinh (c+d x)}{3 b^2 d}-\frac {9 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{32 b d^2}-\frac {2 a f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b^2 d^3}-\frac {a (e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^2 d}-\frac {3 f^3 \cosh ^3(c+d x) \sinh (c+d x)}{128 b d^4}-\frac {3 f (e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{16 b d^2}+\frac {a^2 (e+f x)^3 \sinh ^2(c+d x)}{2 b^3 d}+\frac {\left (a^2 \left (a^2+b^2\right )\right ) \int \frac {e^{c+d x} (e+f x)^3}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{b^4}+\frac {\left (a^2 \left (a^2+b^2\right )\right ) \int \frac {e^{c+d x} (e+f x)^3}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{b^4}+\frac {\left (3 a^3 f\right ) \int (e+f x)^2 \sinh (c+d x) \, dx}{b^4 d}-\frac {\left (3 a^2 f\right ) \int (e+f x)^2 \sinh ^2(c+d x) \, dx}{2 b^3 d}+\frac {(2 a f) \int (e+f x)^2 \sinh (c+d x) \, dx}{b^2 d}-\frac {(9 f) \int (e+f x)^2 \, dx}{32 b d}-\frac {\left (4 a f^2\right ) \int (e+f x) \cosh (c+d x) \, dx}{9 b^2 d^2}-\frac {\left (9 f^3\right ) \int \cosh ^2(c+d x) \, dx}{128 b d^3}-\frac {\left (9 f^3\right ) \int \cosh ^2(c+d x) \, dx}{32 b d^3}\\ &=-\frac {3 (e+f x)^3}{32 b d}-\frac {a^2 \left (a^2+b^2\right ) (e+f x)^4}{4 b^5 f}+\frac {3 a^3 f (e+f x)^2 \cosh (c+d x)}{b^4 d^2}+\frac {2 a f (e+f x)^2 \cosh (c+d x)}{b^2 d^2}+\frac {9 f^2 (e+f x) \cosh ^2(c+d x)}{32 b d^3}+\frac {2 a f^3 \cosh ^3(c+d x)}{27 b^2 d^4}+\frac {a f (e+f x)^2 \cosh ^3(c+d x)}{3 b^2 d^2}+\frac {3 f^2 (e+f x) \cosh ^4(c+d x)}{32 b d^3}+\frac {(e+f x)^3 \cosh ^4(c+d x)}{4 b d}+\frac {a^2 \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d}+\frac {a^2 \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d}-\frac {4 a f^2 (e+f x) \sinh (c+d x)}{9 b^2 d^3}-\frac {a^3 (e+f x)^3 \sinh (c+d x)}{b^4 d}-\frac {2 a (e+f x)^3 \sinh (c+d x)}{3 b^2 d}-\frac {45 f^3 \cosh (c+d x) \sinh (c+d x)}{256 b d^4}-\frac {3 a^2 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b^3 d^2}-\frac {9 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{32 b d^2}-\frac {2 a f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b^2 d^3}-\frac {a (e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^2 d}-\frac {3 f^3 \cosh ^3(c+d x) \sinh (c+d x)}{128 b d^4}-\frac {3 f (e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{16 b d^2}+\frac {3 a^2 f^2 (e+f x) \sinh ^2(c+d x)}{4 b^3 d^3}+\frac {a^2 (e+f x)^3 \sinh ^2(c+d x)}{2 b^3 d}+\frac {\left (3 a^2 f\right ) \int (e+f x)^2 \, dx}{4 b^3 d}-\frac {\left (3 a^2 \left (a^2+b^2\right ) 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^5 d}-\frac {\left (3 a^2 \left (a^2+b^2\right ) 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^5 d}-\frac {\left (6 a^3 f^2\right ) \int (e+f x) \cosh (c+d x) \, dx}{b^4 d^2}-\frac {\left (4 a f^2\right ) \int (e+f x) \cosh (c+d x) \, dx}{b^2 d^2}-\frac {\left (3 a^2 f^3\right ) \int \sinh ^2(c+d x) \, dx}{4 b^3 d^3}+\frac {\left (4 a f^3\right ) \int \sinh (c+d x) \, dx}{9 b^2 d^3}-\frac {\left (9 f^3\right ) \int 1 \, dx}{256 b d^3}-\frac {\left (9 f^3\right ) \int 1 \, dx}{64 b d^3}\\ &=-\frac {45 f^3 x}{256 b d^3}+\frac {a^2 (e+f x)^3}{4 b^3 d}-\frac {3 (e+f x)^3}{32 b d}-\frac {a^2 \left (a^2+b^2\right ) (e+f x)^4}{4 b^5 f}+\frac {4 a f^3 \cosh (c+d x)}{9 b^2 d^4}+\frac {3 a^3 f (e+f x)^2 \cosh (c+d x)}{b^4 d^2}+\frac {2 a f (e+f x)^2 \cosh (c+d x)}{b^2 d^2}+\frac {9 f^2 (e+f x) \cosh ^2(c+d x)}{32 b d^3}+\frac {2 a f^3 \cosh ^3(c+d x)}{27 b^2 d^4}+\frac {a f (e+f x)^2 \cosh ^3(c+d x)}{3 b^2 d^2}+\frac {3 f^2 (e+f x) \cosh ^4(c+d x)}{32 b d^3}+\frac {(e+f x)^3 \cosh ^4(c+d x)}{4 b d}+\frac {a^2 \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d}+\frac {a^2 \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d}+\frac {3 a^2 \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d^2}+\frac {3 a^2 \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d^2}-\frac {6 a^3 f^2 (e+f x) \sinh (c+d x)}{b^4 d^3}-\frac {40 a f^2 (e+f x) \sinh (c+d x)}{9 b^2 d^3}-\frac {a^3 (e+f x)^3 \sinh (c+d x)}{b^4 d}-\frac {2 a (e+f x)^3 \sinh (c+d x)}{3 b^2 d}-\frac {3 a^2 f^3 \cosh (c+d x) \sinh (c+d x)}{8 b^3 d^4}-\frac {45 f^3 \cosh (c+d x) \sinh (c+d x)}{256 b d^4}-\frac {3 a^2 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b^3 d^2}-\frac {9 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{32 b d^2}-\frac {2 a f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b^2 d^3}-\frac {a (e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^2 d}-\frac {3 f^3 \cosh ^3(c+d x) \sinh (c+d x)}{128 b d^4}-\frac {3 f (e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{16 b d^2}+\frac {3 a^2 f^2 (e+f x) \sinh ^2(c+d x)}{4 b^3 d^3}+\frac {a^2 (e+f x)^3 \sinh ^2(c+d x)}{2 b^3 d}-\frac {\left (6 a^2 \left (a^2+b^2\right ) 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^5 d^2}-\frac {\left (6 a^2 \left (a^2+b^2\right ) 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^5 d^2}+\frac {\left (6 a^3 f^3\right ) \int \sinh (c+d x) \, dx}{b^4 d^3}+\frac {\left (3 a^2 f^3\right ) \int 1 \, dx}{8 b^3 d^3}+\frac {\left (4 a f^3\right ) \int \sinh (c+d x) \, dx}{b^2 d^3}\\ &=\frac {3 a^2 f^3 x}{8 b^3 d^3}-\frac {45 f^3 x}{256 b d^3}+\frac {a^2 (e+f x)^3}{4 b^3 d}-\frac {3 (e+f x)^3}{32 b d}-\frac {a^2 \left (a^2+b^2\right ) (e+f x)^4}{4 b^5 f}+\frac {6 a^3 f^3 \cosh (c+d x)}{b^4 d^4}+\frac {40 a f^3 \cosh (c+d x)}{9 b^2 d^4}+\frac {3 a^3 f (e+f x)^2 \cosh (c+d x)}{b^4 d^2}+\frac {2 a f (e+f x)^2 \cosh (c+d x)}{b^2 d^2}+\frac {9 f^2 (e+f x) \cosh ^2(c+d x)}{32 b d^3}+\frac {2 a f^3 \cosh ^3(c+d x)}{27 b^2 d^4}+\frac {a f (e+f x)^2 \cosh ^3(c+d x)}{3 b^2 d^2}+\frac {3 f^2 (e+f x) \cosh ^4(c+d x)}{32 b d^3}+\frac {(e+f x)^3 \cosh ^4(c+d x)}{4 b d}+\frac {a^2 \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d}+\frac {a^2 \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d}+\frac {3 a^2 \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d^2}+\frac {3 a^2 \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d^2}-\frac {6 a^2 \left (a^2+b^2\right ) f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d^3}-\frac {6 a^2 \left (a^2+b^2\right ) f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d^3}-\frac {6 a^3 f^2 (e+f x) \sinh (c+d x)}{b^4 d^3}-\frac {40 a f^2 (e+f x) \sinh (c+d x)}{9 b^2 d^3}-\frac {a^3 (e+f x)^3 \sinh (c+d x)}{b^4 d}-\frac {2 a (e+f x)^3 \sinh (c+d x)}{3 b^2 d}-\frac {3 a^2 f^3 \cosh (c+d x) \sinh (c+d x)}{8 b^3 d^4}-\frac {45 f^3 \cosh (c+d x) \sinh (c+d x)}{256 b d^4}-\frac {3 a^2 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b^3 d^2}-\frac {9 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{32 b d^2}-\frac {2 a f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b^2 d^3}-\frac {a (e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^2 d}-\frac {3 f^3 \cosh ^3(c+d x) \sinh (c+d x)}{128 b d^4}-\frac {3 f (e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{16 b d^2}+\frac {3 a^2 f^2 (e+f x) \sinh ^2(c+d x)}{4 b^3 d^3}+\frac {a^2 (e+f x)^3 \sinh ^2(c+d x)}{2 b^3 d}+\frac {\left (6 a^2 \left (a^2+b^2\right ) f^3\right ) \int \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{b^5 d^3}+\frac {\left (6 a^2 \left (a^2+b^2\right ) f^3\right ) \int \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{b^5 d^3}\\ &=\frac {3 a^2 f^3 x}{8 b^3 d^3}-\frac {45 f^3 x}{256 b d^3}+\frac {a^2 (e+f x)^3}{4 b^3 d}-\frac {3 (e+f x)^3}{32 b d}-\frac {a^2 \left (a^2+b^2\right ) (e+f x)^4}{4 b^5 f}+\frac {6 a^3 f^3 \cosh (c+d x)}{b^4 d^4}+\frac {40 a f^3 \cosh (c+d x)}{9 b^2 d^4}+\frac {3 a^3 f (e+f x)^2 \cosh (c+d x)}{b^4 d^2}+\frac {2 a f (e+f x)^2 \cosh (c+d x)}{b^2 d^2}+\frac {9 f^2 (e+f x) \cosh ^2(c+d x)}{32 b d^3}+\frac {2 a f^3 \cosh ^3(c+d x)}{27 b^2 d^4}+\frac {a f (e+f x)^2 \cosh ^3(c+d x)}{3 b^2 d^2}+\frac {3 f^2 (e+f x) \cosh ^4(c+d x)}{32 b d^3}+\frac {(e+f x)^3 \cosh ^4(c+d x)}{4 b d}+\frac {a^2 \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d}+\frac {a^2 \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d}+\frac {3 a^2 \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d^2}+\frac {3 a^2 \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d^2}-\frac {6 a^2 \left (a^2+b^2\right ) f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d^3}-\frac {6 a^2 \left (a^2+b^2\right ) f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d^3}-\frac {6 a^3 f^2 (e+f x) \sinh (c+d x)}{b^4 d^3}-\frac {40 a f^2 (e+f x) \sinh (c+d x)}{9 b^2 d^3}-\frac {a^3 (e+f x)^3 \sinh (c+d x)}{b^4 d}-\frac {2 a (e+f x)^3 \sinh (c+d x)}{3 b^2 d}-\frac {3 a^2 f^3 \cosh (c+d x) \sinh (c+d x)}{8 b^3 d^4}-\frac {45 f^3 \cosh (c+d x) \sinh (c+d x)}{256 b d^4}-\frac {3 a^2 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b^3 d^2}-\frac {9 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{32 b d^2}-\frac {2 a f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b^2 d^3}-\frac {a (e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^2 d}-\frac {3 f^3 \cosh ^3(c+d x) \sinh (c+d x)}{128 b d^4}-\frac {3 f (e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{16 b d^2}+\frac {3 a^2 f^2 (e+f x) \sinh ^2(c+d x)}{4 b^3 d^3}+\frac {a^2 (e+f x)^3 \sinh ^2(c+d x)}{2 b^3 d}+\frac {\left (6 a^2 \left (a^2+b^2\right ) 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^5 d^4}+\frac {\left (6 a^2 \left (a^2+b^2\right ) 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^5 d^4}\\ &=\frac {3 a^2 f^3 x}{8 b^3 d^3}-\frac {45 f^3 x}{256 b d^3}+\frac {a^2 (e+f x)^3}{4 b^3 d}-\frac {3 (e+f x)^3}{32 b d}-\frac {a^2 \left (a^2+b^2\right ) (e+f x)^4}{4 b^5 f}+\frac {6 a^3 f^3 \cosh (c+d x)}{b^4 d^4}+\frac {40 a f^3 \cosh (c+d x)}{9 b^2 d^4}+\frac {3 a^3 f (e+f x)^2 \cosh (c+d x)}{b^4 d^2}+\frac {2 a f (e+f x)^2 \cosh (c+d x)}{b^2 d^2}+\frac {9 f^2 (e+f x) \cosh ^2(c+d x)}{32 b d^3}+\frac {2 a f^3 \cosh ^3(c+d x)}{27 b^2 d^4}+\frac {a f (e+f x)^2 \cosh ^3(c+d x)}{3 b^2 d^2}+\frac {3 f^2 (e+f x) \cosh ^4(c+d x)}{32 b d^3}+\frac {(e+f x)^3 \cosh ^4(c+d x)}{4 b d}+\frac {a^2 \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d}+\frac {a^2 \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d}+\frac {3 a^2 \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d^2}+\frac {3 a^2 \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d^2}-\frac {6 a^2 \left (a^2+b^2\right ) f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d^3}-\frac {6 a^2 \left (a^2+b^2\right ) f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d^3}+\frac {6 a^2 \left (a^2+b^2\right ) f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d^4}+\frac {6 a^2 \left (a^2+b^2\right ) f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d^4}-\frac {6 a^3 f^2 (e+f x) \sinh (c+d x)}{b^4 d^3}-\frac {40 a f^2 (e+f x) \sinh (c+d x)}{9 b^2 d^3}-\frac {a^3 (e+f x)^3 \sinh (c+d x)}{b^4 d}-\frac {2 a (e+f x)^3 \sinh (c+d x)}{3 b^2 d}-\frac {3 a^2 f^3 \cosh (c+d x) \sinh (c+d x)}{8 b^3 d^4}-\frac {45 f^3 \cosh (c+d x) \sinh (c+d x)}{256 b d^4}-\frac {3 a^2 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b^3 d^2}-\frac {9 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{32 b d^2}-\frac {2 a f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b^2 d^3}-\frac {a (e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^2 d}-\frac {3 f^3 \cosh ^3(c+d x) \sinh (c+d x)}{128 b d^4}-\frac {3 f (e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{16 b d^2}+\frac {3 a^2 f^2 (e+f x) \sinh ^2(c+d x)}{4 b^3 d^3}+\frac {a^2 (e+f x)^3 \sinh ^2(c+d x)}{2 b^3 d}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(6348\) vs. \(2(1123)=2246\).
time = 16.98, size = 6348, normalized size = 5.65 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Result too large to show

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Maple [F]
time = 2.24, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{3} \left (\cosh ^{3}\left (d x +c \right )\right ) \left (\sinh ^{2}\left (d x +c \right )\right )}{a +b \sinh \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

-1/192*((8*a*b^2*e^(-d*x - c) - 3*b^3 - 12*(2*a^2*b + b^3)*e^(-2*d*x - 2*c) + 24*(4*a^3 + 3*a*b^2)*e^(-3*d*x -
3*c))*e^(4*d*x + 4*c)/(b^4*d) - 192*(a^4 + a^2*b^2)*(d*x + c)/(b^5*d) - (8*a*b^2*e^(-3*d*x - 3*c) + 3*b^3*e^(
-4*d*x - 4*c) + 24*(4*a^3 + 3*a*b^2)*e^(-d*x - c) + 12*(2*a^2*b + b^3)*e^(-2*d*x - 2*c))/(b^4*d) - 192*(a^4 +
a^2*b^2)*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(b^5*d))*e^3 + 1/55296*(13824*(a^4*d^4*f^3*e^(4*c) +
a^2*b^2*d^4*f^3*e^(4*c))*x^4 + 55296*(a^4*d^4*f^2*e^(4*c) + a^2*b^2*d^4*f^2*e^(4*c))*x^3*e + 82944*(a^4*d^4*f*
e^(4*c) + a^2*b^2*d^4*f*e^(4*c))*x^2*e^2 + 27*(32*b^4*d^3*f^3*x^3*e^(8*c) - 3*b^4*f^3*e^(8*c) - 24*b^4*d^2*f*e
^(8*c + 2) + 12*b^4*d*f^2*e^(8*c + 1) - 24*(b^4*d^2*f^3*e^(8*c) - 4*b^4*d^3*f^2*e^(8*c + 1))*x^2 + 12*(b^4*d*f
^3*e^(8*c) + 8*b^4*d^3*f*e^(8*c + 2) - 4*b^4*d^2*f^2*e^(8*c + 1))*x)*e^(4*d*x) - 256*(9*a*b^3*d^3*f^3*x^3*e^(7
*c) - 2*a*b^3*f^3*e^(7*c) - 9*a*b^3*d^2*f*e^(7*c + 2) + 6*a*b^3*d*f^2*e^(7*c + 1) - 9*(a*b^3*d^2*f^3*e^(7*c) -
3*a*b^3*d^3*f^2*e^(7*c + 1))*x^2 + 3*(2*a*b^3*d*f^3*e^(7*c) + 9*a*b^3*d^3*f*e^(7*c + 2) - 6*a*b^3*d^2*f^2*e^(
7*c + 1))*x)*e^(3*d*x) - 864*(6*a^2*b^2*f^3*e^(6*c) + 3*b^4*f^3*e^(6*c) - 4*(2*a^2*b^2*d^3*f^3*e^(6*c) + b^4*d
^3*f^3*e^(6*c))*x^3 + 6*(2*a^2*b^2*d^2*f^3*e^(6*c) + b^4*d^2*f^3*e^(6*c) - 2*(2*a^2*b^2*d^3*f^2*e^(6*c) + b^4*
d^3*f^2*e^(6*c))*e)*x^2 - 6*(2*a^2*b^2*d*f^3*e^(6*c) + b^4*d*f^3*e^(6*c) + 2*(2*a^2*b^2*d^3*f*e^(6*c) + b^4*d^
3*f*e^(6*c))*e^2 - 2*(2*a^2*b^2*d^2*f^2*e^(6*c) + b^4*d^2*f^2*e^(6*c))*e)*x + 6*(2*a^2*b^2*d^2*f*e^(6*c) + b^4
*d^2*f*e^(6*c))*e^2 - 6*(2*a^2*b^2*d*f^2*e^(6*c) + b^4*d*f^2*e^(6*c))*e)*e^(2*d*x) + 6912*(24*a^3*b*f^3*e^(5*c
) + 18*a*b^3*f^3*e^(5*c) - (4*a^3*b*d^3*f^3*e^(5*c) + 3*a*b^3*d^3*f^3*e^(5*c))*x^3 + 3*(4*a^3*b*d^2*f^3*e^(5*c
) + 3*a*b^3*d^2*f^3*e^(5*c) - (4*a^3*b*d^3*f^2*e^(5*c) + 3*a*b^3*d^3*f^2*e^(5*c))*e)*x^2 - 3*(8*a^3*b*d*f^3*e^
(5*c) + 6*a*b^3*d*f^3*e^(5*c) + (4*a^3*b*d^3*f*e^(5*c) + 3*a*b^3*d^3*f*e^(5*c))*e^2 - 2*(4*a^3*b*d^2*f^2*e^(5*
c) + 3*a*b^3*d^2*f^2*e^(5*c))*e)*x + 3*(4*a^3*b*d^2*f*e^(5*c) + 3*a*b^3*d^2*f*e^(5*c))*e^2 - 6*(4*a^3*b*d*f^2*
e^(5*c) + 3*a*b^3*d*f^2*e^(5*c))*e)*e^(d*x) + 6912*(24*a^3*b*f^3*e^(3*c) + 18*a*b^3*f^3*e^(3*c) + (4*a^3*b*d^3
*f^3*e^(3*c) + 3*a*b^3*d^3*f^3*e^(3*c))*x^3 + 3*(4*a^3*b*d^2*f^3*e^(3*c) + 3*a*b^3*d^2*f^3*e^(3*c) + (4*a^3*b*
d^3*f^2*e^(3*c) + 3*a*b^3*d^3*f^2*e^(3*c))*e)*x^2 + 3*(8*a^3*b*d*f^3*e^(3*c) + 6*a*b^3*d*f^3*e^(3*c) + (4*a^3*
b*d^3*f*e^(3*c) + 3*a*b^3*d^3*f*e^(3*c))*e^2 + 2*(4*a^3*b*d^2*f^2*e^(3*c) + 3*a*b^3*d^2*f^2*e^(3*c))*e)*x + 3*
(4*a^3*b*d^2*f*e^(3*c) + 3*a*b^3*d^2*f*e^(3*c))*e^2 + 6*(4*a^3*b*d*f^2*e^(3*c) + 3*a*b^3*d*f^2*e^(3*c))*e)*e^(
-d*x) + 864*(6*a^2*b^2*f^3*e^(2*c) + 3*b^4*f^3*e^(2*c) + 4*(2*a^2*b^2*d^3*f^3*e^(2*c) + b^4*d^3*f^3*e^(2*c))*x
^3 + 6*(2*a^2*b^2*d^2*f^3*e^(2*c) + b^4*d^2*f^3*e^(2*c) + 2*(2*a^2*b^2*d^3*f^2*e^(2*c) + b^4*d^3*f^2*e^(2*c))*
e)*x^2 + 6*(2*a^2*b^2*d*f^3*e^(2*c) + b^4*d*f^3*e^(2*c) + 2*(2*a^2*b^2*d^3*f*e^(2*c) + b^4*d^3*f*e^(2*c))*e^2
+ 2*(2*a^2*b^2*d^2*f^2*e^(2*c) + b^4*d^2*f^2*e^(2*c))*e)*x + 6*(2*a^2*b^2*d^2*f*e^(2*c) + b^4*d^2*f*e^(2*c))*e
^2 + 6*(2*a^2*b^2*d*f^2*e^(2*c) + b^4*d*f^2*e^(2*c))*e)*e^(-2*d*x) + 256*(9*a*b^3*d^3*f^3*x^3*e^c + 9*a*b^3*d^
2*f*e^(c + 2) + 6*a*b^3*d*f^2*e^(c + 1) + 2*a*b^3*f^3*e^c + 9*(3*a*b^3*d^3*f^2*e^(c + 1) + a*b^3*d^2*f^3*e^c)*
x^2 + 3*(9*a*b^3*d^3*f*e^(c + 2) + 6*a*b^3*d^2*f^2*e^(c + 1) + 2*a*b^3*d*f^3*e^c)*x)*e^(-3*d*x) + 27*(32*b^4*d
^3*f^3*x^3 + 24*b^4*d^2*f*e^2 + 12*b^4*d*f^2*e + 3*b^4*f^3 + 24*(4*b^4*d^3*f^2*e + b^4*d^2*f^3)*x^2 + 12*(8*b^
4*d^3*f*e^2 + 4*b^4*d^2*f^2*e + b^4*d*f^3)*x)*e^(-4*d*x))*e^(-4*c)/(b^5*d^4) - integrate(-2*((a^4*b*f^3 + a^2*
b^3*f^3)*x^3 + 3*(a^4*b*f^2 + a^2*b^3*f^2)*x^2*e + 3*(a^4*b*f + a^2*b^3*f)*x*e^2 - ((a^5*f^3*e^c + a^3*b^2*f^3
*e^c)*x^3 + 3*(a^5*f^2*e^c + a^3*b^2*f^2*e^c)*x^2*e + 3*(a^5*f*e^c + a^3*b^2*f*e^c)*x*e^2)*e^(d*x))/(b^6*e^(2*
d*x + 2*c) + 2*a*b^5*e^(d*x + c) - b^6), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 24245 vs. \(2 (1077) = 2154\).
time = 0.62, size = 24245, normalized size = 21.59 \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)^3*sinh(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

1/55296*(864*b^4*d^3*f^3*x^3 + 648*b^4*d^2*f^3*x^2 + 864*b^4*d^3*cosh(1)^3 + 864*b^4*d^3*sinh(1)^3 + 324*b^4*d
*f^3*x + 27*(32*b^4*d^3*f^3*x^3 - 24*b^4*d^2*f^3*x^2 + 32*b^4*d^3*cosh(1)^3 + 32*b^4*d^3*sinh(1)^3 + 12*b^4*d*
f^3*x - 3*b^4*f^3 + 24*(4*b^4*d^3*f*x - b^4*d^2*f)*cosh(1)^2 + 24*(4*b^4*d^3*f*x + 4*b^4*d^3*cosh(1) - b^4*d^2
*f)*sinh(1)^2 + 12*(8*b^4*d^3*f^2*x^2 - 4*b^4*d^2*f^2*x + b^4*d*f^2)*cosh(1) + 12*(8*b^4*d^3*f^2*x^2 - 4*b^4*d
^2*f^2*x + 8*b^4*d^3*cosh(1)^2 + b^4*d*f^2 + 4*(4*b^4*d^3*f*x - b^4*d^2*f)*cosh(1))*sinh(1))*cosh(d*x + c)^8 +
27*(32*b^4*d^3*f^3*x^3 - 24*b^4*d^2*f^3*x^2 + 32*b^4*d^3*cosh(1)^3 + 32*b^4*d^3*sinh(1)^3 + 12*b^4*d*f^3*x -
3*b^4*f^3 + 24*(4*b^4*d^3*f*x - b^4*d^2*f)*cosh(1)^2 + 24*(4*b^4*d^3*f*x + 4*b^4*d^3*cosh(1) - b^4*d^2*f)*sinh
(1)^2 + 12*(8*b^4*d^3*f^2*x^2 - 4*b^4*d^2*f^2*x + b^4*d*f^2)*cosh(1) + 12*(8*b^4*d^3*f^2*x^2 - 4*b^4*d^2*f^2*x
+ 8*b^4*d^3*cosh(1)^2 + b^4*d*f^2 + 4*(4*b^4*d^3*f*x - b^4*d^2*f)*cosh(1))*sinh(1))*sinh(d*x + c)^8 - 256*(9*
a*b^3*d^3*f^3*x^3 - 9*a*b^3*d^2*f^3*x^2 + 9*a*b^3*d^3*cosh(1)^3 + 9*a*b^3*d^3*sinh(1)^3 + 6*a*b^3*d*f^3*x - 2*
a*b^3*f^3 + 9*(3*a*b^3*d^3*f*x - a*b^3*d^2*f)*cosh(1)^2 + 9*(3*a*b^3*d^3*f*x + 3*a*b^3*d^3*cosh(1) - a*b^3*d^2
*f)*sinh(1)^2 + 3*(9*a*b^3*d^3*f^2*x^2 - 6*a*b^3*d^2*f^2*x + 2*a*b^3*d*f^2)*cosh(1) + 3*(9*a*b^3*d^3*f^2*x^2 -
6*a*b^3*d^2*f^2*x + 9*a*b^3*d^3*cosh(1)^2 + 2*a*b^3*d*f^2 + 6*(3*a*b^3*d^3*f*x - a*b^3*d^2*f)*cosh(1))*sinh(1
))*cosh(d*x + c)^7 - 8*(288*a*b^3*d^3*f^3*x^3 - 288*a*b^3*d^2*f^3*x^2 + 288*a*b^3*d^3*cosh(1)^3 + 288*a*b^3*d^
3*sinh(1)^3 + 192*a*b^3*d*f^3*x - 64*a*b^3*f^3 + 288*(3*a*b^3*d^3*f*x - a*b^3*d^2*f)*cosh(1)^2 + 288*(3*a*b^3*
d^3*f*x + 3*a*b^3*d^3*cosh(1) - a*b^3*d^2*f)*sinh(1)^2 + 96*(9*a*b^3*d^3*f^2*x^2 - 6*a*b^3*d^2*f^2*x + 2*a*b^3
*d*f^2)*cosh(1) - 27*(32*b^4*d^3*f^3*x^3 - 24*b^4*d^2*f^3*x^2 + 32*b^4*d^3*cosh(1)^3 + 32*b^4*d^3*sinh(1)^3 +
12*b^4*d*f^3*x - 3*b^4*f^3 + 24*(4*b^4*d^3*f*x - b^4*d^2*f)*cosh(1)^2 + 24*(4*b^4*d^3*f*x + 4*b^4*d^3*cosh(1)
- b^4*d^2*f)*sinh(1)^2 + 12*(8*b^4*d^3*f^2*x^2 - 4*b^4*d^2*f^2*x + b^4*d*f^2)*cosh(1) + 12*(8*b^4*d^3*f^2*x^2
- 4*b^4*d^2*f^2*x + 8*b^4*d^3*cosh(1)^2 + b^4*d*f^2 + 4*(4*b^4*d^3*f*x - b^4*d^2*f)*cosh(1))*sinh(1))*cosh(d*x
+ c) + 96*(9*a*b^3*d^3*f^2*x^2 - 6*a*b^3*d^2*f^2*x + 9*a*b^3*d^3*cosh(1)^2 + 2*a*b^3*d*f^2 + 6*(3*a*b^3*d^3*f
*x - a*b^3*d^2*f)*cosh(1))*sinh(1))*sinh(d*x + c)^7 + 81*b^4*f^3 + 864*(4*(2*a^2*b^2 + b^4)*d^3*f^3*x^3 - 6*(2
*a^2*b^2 + b^4)*d^2*f^3*x^2 + 4*(2*a^2*b^2 + b^4)*d^3*cosh(1)^3 + 4*(2*a^2*b^2 + b^4)*d^3*sinh(1)^3 + 6*(2*a^2
*b^2 + b^4)*d*f^3*x - 3*(2*a^2*b^2 + b^4)*f^3 + 6*(2*(2*a^2*b^2 + b^4)*d^3*f*x - (2*a^2*b^2 + b^4)*d^2*f)*cosh
(1)^2 + 6*(2*(2*a^2*b^2 + b^4)*d^3*f*x + 2*(2*a^2*b^2 + b^4)*d^3*cosh(1) - (2*a^2*b^2 + b^4)*d^2*f)*sinh(1)^2
+ 6*(2*(2*a^2*b^2 + b^4)*d^3*f^2*x^2 - 2*(2*a^2*b^2 + b^4)*d^2*f^2*x + (2*a^2*b^2 + b^4)*d*f^2)*cosh(1) + 6*(2
*(2*a^2*b^2 + b^4)*d^3*f^2*x^2 - 2*(2*a^2*b^2 + b^4)*d^2*f^2*x + 2*(2*a^2*b^2 + b^4)*d^3*cosh(1)^2 + (2*a^2*b^
2 + b^4)*d*f^2 + 2*(2*(2*a^2*b^2 + b^4)*d^3*f*x - (2*a^2*b^2 + b^4)*d^2*f)*cosh(1))*sinh(1))*cosh(d*x + c)^6 +
4*(864*(2*a^2*b^2 + b^4)*d^3*f^3*x^3 - 1296*(2*a^2*b^2 + b^4)*d^2*f^3*x^2 + 864*(2*a^2*b^2 + b^4)*d^3*cosh(1)
^3 + 864*(2*a^2*b^2 + b^4)*d^3*sinh(1)^3 + 1296*(2*a^2*b^2 + b^4)*d*f^3*x - 648*(2*a^2*b^2 + b^4)*f^3 + 1296*(
2*(2*a^2*b^2 + b^4)*d^3*f*x - (2*a^2*b^2 + b^4)*d^2*f)*cosh(1)^2 + 189*(32*b^4*d^3*f^3*x^3 - 24*b^4*d^2*f^3*x^
2 + 32*b^4*d^3*cosh(1)^3 + 32*b^4*d^3*sinh(1)^3 + 12*b^4*d*f^3*x - 3*b^4*f^3 + 24*(4*b^4*d^3*f*x - b^4*d^2*f)*
cosh(1)^2 + 24*(4*b^4*d^3*f*x + 4*b^4*d^3*cosh(1) - b^4*d^2*f)*sinh(1)^2 + 12*(8*b^4*d^3*f^2*x^2 - 4*b^4*d^2*f
^2*x + b^4*d*f^2)*cosh(1) + 12*(8*b^4*d^3*f^2*x^2 - 4*b^4*d^2*f^2*x + 8*b^4*d^3*cosh(1)^2 + b^4*d*f^2 + 4*(4*b
^4*d^3*f*x - b^4*d^2*f)*cosh(1))*sinh(1))*cosh(d*x + c)^2 + 1296*(2*(2*a^2*b^2 + b^4)*d^3*f*x + 2*(2*a^2*b^2 +
b^4)*d^3*cosh(1) - (2*a^2*b^2 + b^4)*d^2*f)*sinh(1)^2 + 1296*(2*(2*a^2*b^2 + b^4)*d^3*f^2*x^2 - 2*(2*a^2*b^2
+ b^4)*d^2*f^2*x + (2*a^2*b^2 + b^4)*d*f^2)*cosh(1) - 448*(9*a*b^3*d^3*f^3*x^3 - 9*a*b^3*d^2*f^3*x^2 + 9*a*b^3
*d^3*cosh(1)^3 + 9*a*b^3*d^3*sinh(1)^3 + 6*a*b^3*d*f^3*x - 2*a*b^3*f^3 + 9*(3*a*b^3*d^3*f*x - a*b^3*d^2*f)*cos
h(1)^2 + 9*(3*a*b^3*d^3*f*x + 3*a*b^3*d^3*cosh(1) - a*b^3*d^2*f)*sinh(1)^2 + 3*(9*a*b^3*d^3*f^2*x^2 - 6*a*b^3*
d^2*f^2*x + 2*a*b^3*d*f^2)*cosh(1) + 3*(9*a*b^3*d^3*f^2*x^2 - 6*a*b^3*d^2*f^2*x + 9*a*b^3*d^3*cosh(1)^2 + 2*a*
b^3*d*f^2 + 6*(3*a*b^3*d^3*f*x - a*b^3*d^2*f)*cosh(1))*sinh(1))*cosh(d*x + c) + 1296*(2*(2*a^2*b^2 + b^4)*d^3*
f^2*x^2 - 2*(2*a^2*b^2 + b^4)*d^2*f^2*x + 2*(2*a^2*b^2 + b^4)*d^3*cosh(1)^2 + (2*a^2*b^2 + b^4)*d*f^2 + 2*(2*(
2*a^2*b^2 + b^4)*d^3*f*x - (2*a^2*b^2 + b^4)*d^2*f)*cosh(1))*sinh(1))*sinh(d*x + c)^6 - 6912*((4*a^3*b + 3*a*b
^3)*d^3*f^3*x^3 - 3*(4*a^3*b + 3*a*b^3)*d^2*f^3*x^2 + (4*a^3*b + 3*a*b^3)*d^3*cosh(1)^3 + (4*a^3*b + 3*a*b^3)*
d^3*sinh(1)^3 + 6*(4*a^3*b + 3*a*b^3)*d*f^3*x - 6*(4*a^3*b + 3*a*b^3)*f^3 + 3*((4*a^3*b + 3*a*b^3)*d^3*f*x - (
4*a^3*b + 3*a*b^3)*d^2*f)*cosh(1)^2 + 3*((4*a^3...

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

[Out]

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

[Out]

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

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