3.372 \(\int \frac {(e+f x)^3 \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\)
Optimal. Leaf size=1123 \[ -\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} \]
[Out]
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+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-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+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-3/4*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*(f*x+e)^3/b^3/d+1/4*(f*x+e)^3*cosh(d*x+c)^4/b/d-45/256*f^3*x/b/d^3+3/8*a^2*
f^3*x/b^3/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+40/9*a*f^3*cosh(d*x+c)/b^2/d^4-
45/256*f^3*cosh(d*x+c)*sinh(d*x+c)/b/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^2-3/32*(f*x+e)^3/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-a^3*(f*x+e)^3*sinh(d*x+c)/b^4/d
________________________________________________________________________________________
Rubi [A] time = 1.52, 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
= {5579, 5447, 3311, 32, 2635, 8, 3296, 2638, 3310, 5565, 5446, 5561, 2190, 2531, 6609, 2282,
6589} \[ -\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 {PolyLog}\left (2,-\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 {PolyLog}\left (2,-\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 {PolyLog}\left (3,-\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 {PolyLog}\left (3,-\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 {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 {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} \]
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 2190
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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2282
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 2531
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 2635
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] && Integer
Q[2*n]
Rule 2638
Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Rule 3296
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 3310
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 3311
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 5446
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 5447
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 5561
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 5565
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 5579
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 6589
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 6609
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 ) \operatorname {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 ) \operatorname {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] time = 40.41, size = 8706, normalized size = 7.75 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
[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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fricas [C] time = 0.99, size = 12603, normalized size = 11.22 \[ \text {result too large to display} \]
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 + 864*b^4*d^3*e^3 + 648*b^4*d^2*e^2*f + 27*(32*b^4*d^3*f^3*x^3 + 32*b^4*d^3*e^3 -
24*b^4*d^2*e^2*f + 12*b^4*d*e*f^2 - 3*b^4*f^3 + 24*(4*b^4*d^3*e*f^2 - b^4*d^2*f^3)*x^2 + 12*(8*b^4*d^3*e^2*f
- 4*b^4*d^2*e*f^2 + b^4*d*f^3)*x)*cosh(d*x + c)^8 + 27*(32*b^4*d^3*f^3*x^3 + 32*b^4*d^3*e^3 - 24*b^4*d^2*e^2*f
+ 12*b^4*d*e*f^2 - 3*b^4*f^3 + 24*(4*b^4*d^3*e*f^2 - b^4*d^2*f^3)*x^2 + 12*(8*b^4*d^3*e^2*f - 4*b^4*d^2*e*f^2
+ b^4*d*f^3)*x)*sinh(d*x + c)^8 + 324*b^4*d*e*f^2 - 256*(9*a*b^3*d^3*f^3*x^3 + 9*a*b^3*d^3*e^3 - 9*a*b^3*d^2*
e^2*f + 6*a*b^3*d*e*f^2 - 2*a*b^3*f^3 + 9*(3*a*b^3*d^3*e*f^2 - a*b^3*d^2*f^3)*x^2 + 3*(9*a*b^3*d^3*e^2*f - 6*a
*b^3*d^2*e*f^2 + 2*a*b^3*d*f^3)*x)*cosh(d*x + c)^7 - 8*(288*a*b^3*d^3*f^3*x^3 + 288*a*b^3*d^3*e^3 - 288*a*b^3*
d^2*e^2*f + 192*a*b^3*d*e*f^2 - 64*a*b^3*f^3 + 288*(3*a*b^3*d^3*e*f^2 - a*b^3*d^2*f^3)*x^2 + 96*(9*a*b^3*d^3*e
^2*f - 6*a*b^3*d^2*e*f^2 + 2*a*b^3*d*f^3)*x - 27*(32*b^4*d^3*f^3*x^3 + 32*b^4*d^3*e^3 - 24*b^4*d^2*e^2*f + 12*
b^4*d*e*f^2 - 3*b^4*f^3 + 24*(4*b^4*d^3*e*f^2 - b^4*d^2*f^3)*x^2 + 12*(8*b^4*d^3*e^2*f - 4*b^4*d^2*e*f^2 + b^4
*d*f^3)*x)*cosh(d*x + c))*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 + 4*(2*a^2*b^2 +
b^4)*d^3*e^3 - 6*(2*a^2*b^2 + b^4)*d^2*e^2*f + 6*(2*a^2*b^2 + b^4)*d*e*f^2 - 3*(2*a^2*b^2 + b^4)*f^3 + 6*(2*(
2*a^2*b^2 + b^4)*d^3*e*f^2 - (2*a^2*b^2 + b^4)*d^2*f^3)*x^2 + 6*(2*(2*a^2*b^2 + b^4)*d^3*e^2*f - 2*(2*a^2*b^2
+ b^4)*d^2*e*f^2 + (2*a^2*b^2 + b^4)*d*f^3)*x)*cosh(d*x + c)^6 + 4*(864*(2*a^2*b^2 + b^4)*d^3*f^3*x^3 + 864*(2
*a^2*b^2 + b^4)*d^3*e^3 - 1296*(2*a^2*b^2 + b^4)*d^2*e^2*f + 1296*(2*a^2*b^2 + b^4)*d*e*f^2 - 648*(2*a^2*b^2 +
b^4)*f^3 + 1296*(2*(2*a^2*b^2 + b^4)*d^3*e*f^2 - (2*a^2*b^2 + b^4)*d^2*f^3)*x^2 + 189*(32*b^4*d^3*f^3*x^3 + 3
2*b^4*d^3*e^3 - 24*b^4*d^2*e^2*f + 12*b^4*d*e*f^2 - 3*b^4*f^3 + 24*(4*b^4*d^3*e*f^2 - b^4*d^2*f^3)*x^2 + 12*(8
*b^4*d^3*e^2*f - 4*b^4*d^2*e*f^2 + b^4*d*f^3)*x)*cosh(d*x + c)^2 + 1296*(2*(2*a^2*b^2 + b^4)*d^3*e^2*f - 2*(2*
a^2*b^2 + b^4)*d^2*e*f^2 + (2*a^2*b^2 + b^4)*d*f^3)*x - 448*(9*a*b^3*d^3*f^3*x^3 + 9*a*b^3*d^3*e^3 - 9*a*b^3*d
^2*e^2*f + 6*a*b^3*d*e*f^2 - 2*a*b^3*f^3 + 9*(3*a*b^3*d^3*e*f^2 - a*b^3*d^2*f^3)*x^2 + 3*(9*a*b^3*d^3*e^2*f -
6*a*b^3*d^2*e*f^2 + 2*a*b^3*d*f^3)*x)*cosh(d*x + c))*sinh(d*x + c)^6 - 6912*((4*a^3*b + 3*a*b^3)*d^3*f^3*x^3 +
(4*a^3*b + 3*a*b^3)*d^3*e^3 - 3*(4*a^3*b + 3*a*b^3)*d^2*e^2*f + 6*(4*a^3*b + 3*a*b^3)*d*e*f^2 - 6*(4*a^3*b +
3*a*b^3)*f^3 + 3*((4*a^3*b + 3*a*b^3)*d^3*e*f^2 - (4*a^3*b + 3*a*b^3)*d^2*f^3)*x^2 + 3*((4*a^3*b + 3*a*b^3)*d^
3*e^2*f - 2*(4*a^3*b + 3*a*b^3)*d^2*e*f^2 + 2*(4*a^3*b + 3*a*b^3)*d*f^3)*x)*cosh(d*x + c)^5 - 24*(288*(4*a^3*b
+ 3*a*b^3)*d^3*f^3*x^3 + 288*(4*a^3*b + 3*a*b^3)*d^3*e^3 - 864*(4*a^3*b + 3*a*b^3)*d^2*e^2*f + 1728*(4*a^3*b
+ 3*a*b^3)*d*e*f^2 - 1728*(4*a^3*b + 3*a*b^3)*f^3 - 63*(32*b^4*d^3*f^3*x^3 + 32*b^4*d^3*e^3 - 24*b^4*d^2*e^2*f
+ 12*b^4*d*e*f^2 - 3*b^4*f^3 + 24*(4*b^4*d^3*e*f^2 - b^4*d^2*f^3)*x^2 + 12*(8*b^4*d^3*e^2*f - 4*b^4*d^2*e*f^2
+ b^4*d*f^3)*x)*cosh(d*x + c)^3 + 864*((4*a^3*b + 3*a*b^3)*d^3*e*f^2 - (4*a^3*b + 3*a*b^3)*d^2*f^3)*x^2 + 224
*(9*a*b^3*d^3*f^3*x^3 + 9*a*b^3*d^3*e^3 - 9*a*b^3*d^2*e^2*f + 6*a*b^3*d*e*f^2 - 2*a*b^3*f^3 + 9*(3*a*b^3*d^3*e
*f^2 - a*b^3*d^2*f^3)*x^2 + 3*(9*a*b^3*d^3*e^2*f - 6*a*b^3*d^2*e*f^2 + 2*a*b^3*d*f^3)*x)*cosh(d*x + c)^2 + 864
*((4*a^3*b + 3*a*b^3)*d^3*e^2*f - 2*(4*a^3*b + 3*a*b^3)*d^2*e*f^2 + 2*(4*a^3*b + 3*a*b^3)*d*f^3)*x - 216*(4*(2
*a^2*b^2 + b^4)*d^3*f^3*x^3 + 4*(2*a^2*b^2 + b^4)*d^3*e^3 - 6*(2*a^2*b^2 + b^4)*d^2*e^2*f + 6*(2*a^2*b^2 + b^4
)*d*e*f^2 - 3*(2*a^2*b^2 + b^4)*f^3 + 6*(2*(2*a^2*b^2 + b^4)*d^3*e*f^2 - (2*a^2*b^2 + b^4)*d^2*f^3)*x^2 + 6*(2
*(2*a^2*b^2 + b^4)*d^3*e^2*f - 2*(2*a^2*b^2 + b^4)*d^2*e*f^2 + (2*a^2*b^2 + b^4)*d*f^3)*x)*cosh(d*x + c))*sinh
(d*x + c)^5 - 13824*((a^4 + a^2*b^2)*d^4*f^3*x^4 + 4*(a^4 + a^2*b^2)*d^4*e*f^2*x^3 + 6*(a^4 + a^2*b^2)*d^4*e^2
*f*x^2 + 4*(a^4 + a^2*b^2)*d^4*e^3*x + 8*(a^4 + a^2*b^2)*c*d^3*e^3 - 12*(a^4 + a^2*b^2)*c^2*d^2*e^2*f + 8*(a^4
+ a^2*b^2)*c^3*d*e*f^2 - 2*(a^4 + a^2*b^2)*c^4*f^3)*cosh(d*x + c)^4 - 2*(6912*(a^4 + a^2*b^2)*d^4*f^3*x^4 + 2
7648*(a^4 + a^2*b^2)*d^4*e*f^2*x^3 + 41472*(a^4 + a^2*b^2)*d^4*e^2*f*x^2 + 27648*(a^4 + a^2*b^2)*d^4*e^3*x + 5
5296*(a^4 + a^2*b^2)*c*d^3*e^3 - 82944*(a^4 + a^2*b^2)*c^2*d^2*e^2*f + 55296*(a^4 + a^2*b^2)*c^3*d*e*f^2 - 138
24*(a^4 + a^2*b^2)*c^4*f^3 - 945*(32*b^4*d^3*f^3*x^3 + 32*b^4*d^3*e^3 - 24*b^4*d^2*e^2*f + 12*b^4*d*e*f^2 - 3*
b^4*f^3 + 24*(4*b^4*d^3*e*f^2 - b^4*d^2*f^3)*x^2 + 12*(8*b^4*d^3*e^2*f - 4*b^4*d^2*e*f^2 + b^4*d*f^3)*x)*cosh(
d*x + c)^4 + 4480*(9*a*b^3*d^3*f^3*x^3 + 9*a*b^3*d^3*e^3 - 9*a*b^3*d^2*e^2*f + 6*a*b^3*d*e*f^2 - 2*a*b^3*f^3 +
9*(3*a*b^3*d^3*e*f^2 - a*b^3*d^2*f^3)*x^2 + 3*(9*a*b^3*d^3*e^2*f - 6*a*b^3*d^2*e*f^2 + 2*a*b^3*d*f^3)*x)*cosh
(d*x + c)^3 - 6480*(4*(2*a^2*b^2 + b^4)*d^3*f^3*x^3 + 4*(2*a^2*b^2 + b^4)*d^3*e^3 - 6*(2*a^2*b^2 + b^4)*d^2*e^
2*f + 6*(2*a^2*b^2 + b^4)*d*e*f^2 - 3*(2*a^2*b^2 + b^4)*f^3 + 6*(2*(2*a^2*b^2 + b^4)*d^3*e*f^2 - (2*a^2*b^2 +
b^4)*d^2*f^3)*x^2 + 6*(2*(2*a^2*b^2 + b^4)*d^3*e^2*f - 2*(2*a^2*b^2 + b^4)*d^2*e*f^2 + (2*a^2*b^2 + b^4)*d*f^3
)*x)*cosh(d*x + c)^2 + 17280*((4*a^3*b + 3*a*b^3)*d^3*f^3*x^3 + (4*a^3*b + 3*a*b^3)*d^3*e^3 - 3*(4*a^3*b + 3*a
*b^3)*d^2*e^2*f + 6*(4*a^3*b + 3*a*b^3)*d*e*f^2 - 6*(4*a^3*b + 3*a*b^3)*f^3 + 3*((4*a^3*b + 3*a*b^3)*d^3*e*f^2
- (4*a^3*b + 3*a*b^3)*d^2*f^3)*x^2 + 3*((4*a^3*b + 3*a*b^3)*d^3*e^2*f - 2*(4*a^3*b + 3*a*b^3)*d^2*e*f^2 + 2*(
4*a^3*b + 3*a*b^3)*d*f^3)*x)*cosh(d*x + c))*sinh(d*x + c)^4 + 6912*((4*a^3*b + 3*a*b^3)*d^3*f^3*x^3 + (4*a^3*b
+ 3*a*b^3)*d^3*e^3 + 3*(4*a^3*b + 3*a*b^3)*d^2*e^2*f + 6*(4*a^3*b + 3*a*b^3)*d*e*f^2 + 6*(4*a^3*b + 3*a*b^3)*
f^3 + 3*((4*a^3*b + 3*a*b^3)*d^3*e*f^2 + (4*a^3*b + 3*a*b^3)*d^2*f^3)*x^2 + 3*((4*a^3*b + 3*a*b^3)*d^3*e^2*f +
2*(4*a^3*b + 3*a*b^3)*d^2*e*f^2 + 2*(4*a^3*b + 3*a*b^3)*d*f^3)*x)*cosh(d*x + c)^3 + 8*(864*(4*a^3*b + 3*a*b^3
)*d^3*f^3*x^3 + 864*(4*a^3*b + 3*a*b^3)*d^3*e^3 + 2592*(4*a^3*b + 3*a*b^3)*d^2*e^2*f + 189*(32*b^4*d^3*f^3*x^3
+ 32*b^4*d^3*e^3 - 24*b^4*d^2*e^2*f + 12*b^4*d*e*f^2 - 3*b^4*f^3 + 24*(4*b^4*d^3*e*f^2 - b^4*d^2*f^3)*x^2 + 1
2*(8*b^4*d^3*e^2*f - 4*b^4*d^2*e*f^2 + b^4*d*f^3)*x)*cosh(d*x + c)^5 + 5184*(4*a^3*b + 3*a*b^3)*d*e*f^2 - 1120
*(9*a*b^3*d^3*f^3*x^3 + 9*a*b^3*d^3*e^3 - 9*a*b^3*d^2*e^2*f + 6*a*b^3*d*e*f^2 - 2*a*b^3*f^3 + 9*(3*a*b^3*d^3*e
*f^2 - a*b^3*d^2*f^3)*x^2 + 3*(9*a*b^3*d^3*e^2*f - 6*a*b^3*d^2*e*f^2 + 2*a*b^3*d*f^3)*x)*cosh(d*x + c)^4 + 518
4*(4*a^3*b + 3*a*b^3)*f^3 + 2160*(4*(2*a^2*b^2 + b^4)*d^3*f^3*x^3 + 4*(2*a^2*b^2 + b^4)*d^3*e^3 - 6*(2*a^2*b^2
+ b^4)*d^2*e^2*f + 6*(2*a^2*b^2 + b^4)*d*e*f^2 - 3*(2*a^2*b^2 + b^4)*f^3 + 6*(2*(2*a^2*b^2 + b^4)*d^3*e*f^2 -
(2*a^2*b^2 + b^4)*d^2*f^3)*x^2 + 6*(2*(2*a^2*b^2 + b^4)*d^3*e^2*f - 2*(2*a^2*b^2 + b^4)*d^2*e*f^2 + (2*a^2*b^
2 + b^4)*d*f^3)*x)*cosh(d*x + c)^3 + 2592*((4*a^3*b + 3*a*b^3)*d^3*e*f^2 + (4*a^3*b + 3*a*b^3)*d^2*f^3)*x^2 -
8640*((4*a^3*b + 3*a*b^3)*d^3*f^3*x^3 + (4*a^3*b + 3*a*b^3)*d^3*e^3 - 3*(4*a^3*b + 3*a*b^3)*d^2*e^2*f + 6*(4*a
^3*b + 3*a*b^3)*d*e*f^2 - 6*(4*a^3*b + 3*a*b^3)*f^3 + 3*((4*a^3*b + 3*a*b^3)*d^3*e*f^2 - (4*a^3*b + 3*a*b^3)*d
^2*f^3)*x^2 + 3*((4*a^3*b + 3*a*b^3)*d^3*e^2*f - 2*(4*a^3*b + 3*a*b^3)*d^2*e*f^2 + 2*(4*a^3*b + 3*a*b^3)*d*f^3
)*x)*cosh(d*x + c)^2 + 2592*((4*a^3*b + 3*a*b^3)*d^3*e^2*f + 2*(4*a^3*b + 3*a*b^3)*d^2*e*f^2 + 2*(4*a^3*b + 3*
a*b^3)*d*f^3)*x - 6912*((a^4 + a^2*b^2)*d^4*f^3*x^4 + 4*(a^4 + a^2*b^2)*d^4*e*f^2*x^3 + 6*(a^4 + a^2*b^2)*d^4*
e^2*f*x^2 + 4*(a^4 + a^2*b^2)*d^4*e^3*x + 8*(a^4 + a^2*b^2)*c*d^3*e^3 - 12*(a^4 + a^2*b^2)*c^2*d^2*e^2*f + 8*(
a^4 + a^2*b^2)*c^3*d*e*f^2 - 2*(a^4 + a^2*b^2)*c^4*f^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 648*(4*b^4*d^3*e*f^2
+ b^4*d^2*f^3)*x^2 + 864*(4*(2*a^2*b^2 + b^4)*d^3*f^3*x^3 + 4*(2*a^2*b^2 + b^4)*d^3*e^3 + 6*(2*a^2*b^2 + b^4)*
d^2*e^2*f + 6*(2*a^2*b^2 + b^4)*d*e*f^2 + 3*(2*a^2*b^2 + b^4)*f^3 + 6*(2*(2*a^2*b^2 + b^4)*d^3*e*f^2 + (2*a^2*
b^2 + b^4)*d^2*f^3)*x^2 + 6*(2*(2*a^2*b^2 + b^4)*d^3*e^2*f + 2*(2*a^2*b^2 + b^4)*d^2*e*f^2 + (2*a^2*b^2 + b^4)
*d*f^3)*x)*cosh(d*x + c)^2 + 12*(288*(2*a^2*b^2 + b^4)*d^3*f^3*x^3 + 288*(2*a^2*b^2 + b^4)*d^3*e^3 + 63*(32*b^
4*d^3*f^3*x^3 + 32*b^4*d^3*e^3 - 24*b^4*d^2*e^2*f + 12*b^4*d*e*f^2 - 3*b^4*f^3 + 24*(4*b^4*d^3*e*f^2 - b^4*d^2
*f^3)*x^2 + 12*(8*b^4*d^3*e^2*f - 4*b^4*d^2*e*f^2 + b^4*d*f^3)*x)*cosh(d*x + c)^6 + 432*(2*a^2*b^2 + b^4)*d^2*
e^2*f - 448*(9*a*b^3*d^3*f^3*x^3 + 9*a*b^3*d^3*e^3 - 9*a*b^3*d^2*e^2*f + 6*a*b^3*d*e*f^2 - 2*a*b^3*f^3 + 9*(3*
a*b^3*d^3*e*f^2 - a*b^3*d^2*f^3)*x^2 + 3*(9*a*b^3*d^3*e^2*f - 6*a*b^3*d^2*e*f^2 + 2*a*b^3*d*f^3)*x)*cosh(d*x +
c)^5 + 432*(2*a^2*b^2 + b^4)*d*e*f^2 + 1080*(4*(2*a^2*b^2 + b^4)*d^3*f^3*x^3 + 4*(2*a^2*b^2 + b^4)*d^3*e^3 -
6*(2*a^2*b^2 + b^4)*d^2*e^2*f + 6*(2*a^2*b^2 + b^4)*d*e*f^2 - 3*(2*a^2*b^2 + b^4)*f^3 + 6*(2*(2*a^2*b^2 + b^4)
*d^3*e*f^2 - (2*a^2*b^2 + b^4)*d^2*f^3)*x^2 + 6*(2*(2*a^2*b^2 + b^4)*d^3*e^2*f - 2*(2*a^2*b^2 + b^4)*d^2*e*f^2
+ (2*a^2*b^2 + b^4)*d*f^3)*x)*cosh(d*x + c)^4 + 216*(2*a^2*b^2 + b^4)*f^3 - 5760*((4*a^3*b + 3*a*b^3)*d^3*f^3
*x^3 + (4*a^3*b + 3*a*b^3)*d^3*e^3 - 3*(4*a^3*b + 3*a*b^3)*d^2*e^2*f + 6*(4*a^3*b + 3*a*b^3)*d*e*f^2 - 6*(4*a^
3*b + 3*a*b^3)*f^3 + 3*((4*a^3*b + 3*a*b^3)*d^3*e*f^2 - (4*a^3*b + 3*a*b^3)*d^2*f^3)*x^2 + 3*((4*a^3*b + 3*a*b
^3)*d^3*e^2*f - 2*(4*a^3*b + 3*a*b^3)*d^2*e*f^2 + 2*(4*a^3*b + 3*a*b^3)*d*f^3)*x)*cosh(d*x + c)^3 + 432*(2*(2*
a^2*b^2 + b^4)*d^3*e*f^2 + (2*a^2*b^2 + b^4)*d^2*f^3)*x^2 - 6912*((a^4 + a^2*b^2)*d^4*f^3*x^4 + 4*(a^4 + a^2*b
^2)*d^4*e*f^2*x^3 + 6*(a^4 + a^2*b^2)*d^4*e^2*f*x^2 + 4*(a^4 + a^2*b^2)*d^4*e^3*x + 8*(a^4 + a^2*b^2)*c*d^3*e^
3 - 12*(a^4 + a^2*b^2)*c^2*d^2*e^2*f + 8*(a^4 + a^2*b^2)*c^3*d*e*f^2 - 2*(a^4 + a^2*b^2)*c^4*f^3)*cosh(d*x + c
)^2 + 432*(2*(2*a^2*b^2 + b^4)*d^3*e^2*f + 2*(2*a^2*b^2 + b^4)*d^2*e*f^2 + (2*a^2*b^2 + b^4)*d*f^3)*x + 1728*(
(4*a^3*b + 3*a*b^3)*d^3*f^3*x^3 + (4*a^3*b + 3*a*b^3)*d^3*e^3 + 3*(4*a^3*b + 3*a*b^3)*d^2*e^2*f + 6*(4*a^3*b +
3*a*b^3)*d*e*f^2 + 6*(4*a^3*b + 3*a*b^3)*f^3 + 3*((4*a^3*b + 3*a*b^3)*d^3*e*f^2 + (4*a^3*b + 3*a*b^3)*d^2*f^3
)*x^2 + 3*((4*a^3*b + 3*a*b^3)*d^3*e^2*f + 2*(4*a^3*b + 3*a*b^3)*d^2*e*f^2 + 2*(4*a^3*b + 3*a*b^3)*d*f^3)*x)*c
osh(d*x + c))*sinh(d*x + c)^2 + 324*(8*b^4*d^3*e^2*f + 4*b^4*d^2*e*f^2 + b^4*d*f^3)*x + 256*(9*a*b^3*d^3*f^3*x
^3 + 9*a*b^3*d^3*e^3 + 9*a*b^3*d^2*e^2*f + 6*a*b^3*d*e*f^2 + 2*a*b^3*f^3 + 9*(3*a*b^3*d^3*e*f^2 + a*b^3*d^2*f^
3)*x^2 + 3*(9*a*b^3*d^3*e^2*f + 6*a*b^3*d^2*e*f^2 + 2*a*b^3*d*f^3)*x)*cosh(d*x + c) + 165888*(((a^4 + a^2*b^2)
*d^2*f^3*x^2 + 2*(a^4 + a^2*b^2)*d^2*e*f^2*x + (a^4 + a^2*b^2)*d^2*e^2*f)*cosh(d*x + c)^4 + 4*((a^4 + a^2*b^2)
*d^2*f^3*x^2 + 2*(a^4 + a^2*b^2)*d^2*e*f^2*x + (a^4 + a^2*b^2)*d^2*e^2*f)*cosh(d*x + c)^3*sinh(d*x + c) + 6*((
a^4 + a^2*b^2)*d^2*f^3*x^2 + 2*(a^4 + a^2*b^2)*d^2*e*f^2*x + (a^4 + a^2*b^2)*d^2*e^2*f)*cosh(d*x + c)^2*sinh(d
*x + c)^2 + 4*((a^4 + a^2*b^2)*d^2*f^3*x^2 + 2*(a^4 + a^2*b^2)*d^2*e*f^2*x + (a^4 + a^2*b^2)*d^2*e^2*f)*cosh(d
*x + c)*sinh(d*x + c)^3 + ((a^4 + a^2*b^2)*d^2*f^3*x^2 + 2*(a^4 + a^2*b^2)*d^2*e*f^2*x + (a^4 + a^2*b^2)*d^2*e
^2*f)*sinh(d*x + c)^4)*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^
2 + b^2)/b^2) - b)/b + 1) + 165888*(((a^4 + a^2*b^2)*d^2*f^3*x^2 + 2*(a^4 + a^2*b^2)*d^2*e*f^2*x + (a^4 + a^2*
b^2)*d^2*e^2*f)*cosh(d*x + c)^4 + 4*((a^4 + a^2*b^2)*d^2*f^3*x^2 + 2*(a^4 + a^2*b^2)*d^2*e*f^2*x + (a^4 + a^2*
b^2)*d^2*e^2*f)*cosh(d*x + c)^3*sinh(d*x + c) + 6*((a^4 + a^2*b^2)*d^2*f^3*x^2 + 2*(a^4 + a^2*b^2)*d^2*e*f^2*x
+ (a^4 + a^2*b^2)*d^2*e^2*f)*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*((a^4 + a^2*b^2)*d^2*f^3*x^2 + 2*(a^4 + a^2*
b^2)*d^2*e*f^2*x + (a^4 + a^2*b^2)*d^2*e^2*f)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^4 + a^2*b^2)*d^2*f^3*x^2 + 2
*(a^4 + a^2*b^2)*d^2*e*f^2*x + (a^4 + a^2*b^2)*d^2*e^2*f)*sinh(d*x + c)^4)*dilog((a*cosh(d*x + c) + a*sinh(d*x
+ c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + 55296*(((a^4 + a^2*b^2)*d^3*e^
3 - 3*(a^4 + a^2*b^2)*c*d^2*e^2*f + 3*(a^4 + a^2*b^2)*c^2*d*e*f^2 - (a^4 + a^2*b^2)*c^3*f^3)*cosh(d*x + c)^4 +
4*((a^4 + a^2*b^2)*d^3*e^3 - 3*(a^4 + a^2*b^2)*c*d^2*e^2*f + 3*(a^4 + a^2*b^2)*c^2*d*e*f^2 - (a^4 + a^2*b^2)*
c^3*f^3)*cosh(d*x + c)^3*sinh(d*x + c) + 6*((a^4 + a^2*b^2)*d^3*e^3 - 3*(a^4 + a^2*b^2)*c*d^2*e^2*f + 3*(a^4 +
a^2*b^2)*c^2*d*e*f^2 - (a^4 + a^2*b^2)*c^3*f^3)*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*((a^4 + a^2*b^2)*d^3*e^3
- 3*(a^4 + a^2*b^2)*c*d^2*e^2*f + 3*(a^4 + a^2*b^2)*c^2*d*e*f^2 - (a^4 + a^2*b^2)*c^3*f^3)*cosh(d*x + c)*sinh(
d*x + c)^3 + ((a^4 + a^2*b^2)*d^3*e^3 - 3*(a^4 + a^2*b^2)*c*d^2*e^2*f + 3*(a^4 + a^2*b^2)*c^2*d*e*f^2 - (a^4 +
a^2*b^2)*c^3*f^3)*sinh(d*x + c)^4)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*
a) + 55296*(((a^4 + a^2*b^2)*d^3*e^3 - 3*(a^4 + a^2*b^2)*c*d^2*e^2*f + 3*(a^4 + a^2*b^2)*c^2*d*e*f^2 - (a^4 +
a^2*b^2)*c^3*f^3)*cosh(d*x + c)^4 + 4*((a^4 + a^2*b^2)*d^3*e^3 - 3*(a^4 + a^2*b^2)*c*d^2*e^2*f + 3*(a^4 + a^2*
b^2)*c^2*d*e*f^2 - (a^4 + a^2*b^2)*c^3*f^3)*cosh(d*x + c)^3*sinh(d*x + c) + 6*((a^4 + a^2*b^2)*d^3*e^3 - 3*(a^
4 + a^2*b^2)*c*d^2*e^2*f + 3*(a^4 + a^2*b^2)*c^2*d*e*f^2 - (a^4 + a^2*b^2)*c^3*f^3)*cosh(d*x + c)^2*sinh(d*x +
c)^2 + 4*((a^4 + a^2*b^2)*d^3*e^3 - 3*(a^4 + a^2*b^2)*c*d^2*e^2*f + 3*(a^4 + a^2*b^2)*c^2*d*e*f^2 - (a^4 + a^
2*b^2)*c^3*f^3)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^4 + a^2*b^2)*d^3*e^3 - 3*(a^4 + a^2*b^2)*c*d^2*e^2*f + 3*(
a^4 + a^2*b^2)*c^2*d*e*f^2 - (a^4 + a^2*b^2)*c^3*f^3)*sinh(d*x + c)^4)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x +
c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + 55296*(((a^4 + a^2*b^2)*d^3*f^3*x^3 + 3*(a^4 + a^2*b^2)*d^3*e*f^2*x^2
+ 3*(a^4 + a^2*b^2)*d^3*e^2*f*x + 3*(a^4 + a^2*b^2)*c*d^2*e^2*f - 3*(a^4 + a^2*b^2)*c^2*d*e*f^2 + (a^4 + a^2*b
^2)*c^3*f^3)*cosh(d*x + c)^4 + 4*((a^4 + a^2*b^2)*d^3*f^3*x^3 + 3*(a^4 + a^2*b^2)*d^3*e*f^2*x^2 + 3*(a^4 + a^2
*b^2)*d^3*e^2*f*x + 3*(a^4 + a^2*b^2)*c*d^2*e^2*f - 3*(a^4 + a^2*b^2)*c^2*d*e*f^2 + (a^4 + a^2*b^2)*c^3*f^3)*c
osh(d*x + c)^3*sinh(d*x + c) + 6*((a^4 + a^2*b^2)*d^3*f^3*x^3 + 3*(a^4 + a^2*b^2)*d^3*e*f^2*x^2 + 3*(a^4 + a^2
*b^2)*d^3*e^2*f*x + 3*(a^4 + a^2*b^2)*c*d^2*e^2*f - 3*(a^4 + a^2*b^2)*c^2*d*e*f^2 + (a^4 + a^2*b^2)*c^3*f^3)*c
osh(d*x + c)^2*sinh(d*x + c)^2 + 4*((a^4 + a^2*b^2)*d^3*f^3*x^3 + 3*(a^4 + a^2*b^2)*d^3*e*f^2*x^2 + 3*(a^4 + a
^2*b^2)*d^3*e^2*f*x + 3*(a^4 + a^2*b^2)*c*d^2*e^2*f - 3*(a^4 + a^2*b^2)*c^2*d*e*f^2 + (a^4 + a^2*b^2)*c^3*f^3)
*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^4 + a^2*b^2)*d^3*f^3*x^3 + 3*(a^4 + a^2*b^2)*d^3*e*f^2*x^2 + 3*(a^4 + a^2
*b^2)*d^3*e^2*f*x + 3*(a^4 + a^2*b^2)*c*d^2*e^2*f - 3*(a^4 + a^2*b^2)*c^2*d*e*f^2 + (a^4 + a^2*b^2)*c^3*f^3)*s
inh(d*x + c)^4)*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)
/b^2) - b)/b) + 55296*(((a^4 + a^2*b^2)*d^3*f^3*x^3 + 3*(a^4 + a^2*b^2)*d^3*e*f^2*x^2 + 3*(a^4 + a^2*b^2)*d^3*
e^2*f*x + 3*(a^4 + a^2*b^2)*c*d^2*e^2*f - 3*(a^4 + a^2*b^2)*c^2*d*e*f^2 + (a^4 + a^2*b^2)*c^3*f^3)*cosh(d*x +
c)^4 + 4*((a^4 + a^2*b^2)*d^3*f^3*x^3 + 3*(a^4 + a^2*b^2)*d^3*e*f^2*x^2 + 3*(a^4 + a^2*b^2)*d^3*e^2*f*x + 3*(a
^4 + a^2*b^2)*c*d^2*e^2*f - 3*(a^4 + a^2*b^2)*c^2*d*e*f^2 + (a^4 + a^2*b^2)*c^3*f^3)*cosh(d*x + c)^3*sinh(d*x
+ c) + 6*((a^4 + a^2*b^2)*d^3*f^3*x^3 + 3*(a^4 + a^2*b^2)*d^3*e*f^2*x^2 + 3*(a^4 + a^2*b^2)*d^3*e^2*f*x + 3*(a
^4 + a^2*b^2)*c*d^2*e^2*f - 3*(a^4 + a^2*b^2)*c^2*d*e*f^2 + (a^4 + a^2*b^2)*c^3*f^3)*cosh(d*x + c)^2*sinh(d*x
+ c)^2 + 4*((a^4 + a^2*b^2)*d^3*f^3*x^3 + 3*(a^4 + a^2*b^2)*d^3*e*f^2*x^2 + 3*(a^4 + a^2*b^2)*d^3*e^2*f*x + 3*
(a^4 + a^2*b^2)*c*d^2*e^2*f - 3*(a^4 + a^2*b^2)*c^2*d*e*f^2 + (a^4 + a^2*b^2)*c^3*f^3)*cosh(d*x + c)*sinh(d*x
+ c)^3 + ((a^4 + a^2*b^2)*d^3*f^3*x^3 + 3*(a^4 + a^2*b^2)*d^3*e*f^2*x^2 + 3*(a^4 + a^2*b^2)*d^3*e^2*f*x + 3*(a
^4 + a^2*b^2)*c*d^2*e^2*f - 3*(a^4 + a^2*b^2)*c^2*d*e*f^2 + (a^4 + a^2*b^2)*c^3*f^3)*sinh(d*x + c)^4)*log(-(a*
cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) + 331776*(
(a^4 + a^2*b^2)*f^3*cosh(d*x + c)^4 + 4*(a^4 + a^2*b^2)*f^3*cosh(d*x + c)^3*sinh(d*x + c) + 6*(a^4 + a^2*b^2)*
f^3*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*(a^4 + a^2*b^2)*f^3*cosh(d*x + c)*sinh(d*x + c)^3 + (a^4 + a^2*b^2)*f^
3*sinh(d*x + c)^4)*polylog(4, (a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a
^2 + b^2)/b^2))/b) + 331776*((a^4 + a^2*b^2)*f^3*cosh(d*x + c)^4 + 4*(a^4 + a^2*b^2)*f^3*cosh(d*x + c)^3*sinh(
d*x + c) + 6*(a^4 + a^2*b^2)*f^3*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*(a^4 + a^2*b^2)*f^3*cosh(d*x + c)*sinh(d*
x + c)^3 + (a^4 + a^2*b^2)*f^3*sinh(d*x + c)^4)*polylog(4, (a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x +
c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) - 331776*(((a^4 + a^2*b^2)*d*f^3*x + (a^4 + a^2*b^2)*d*e*f^2)*
cosh(d*x + c)^4 + 4*((a^4 + a^2*b^2)*d*f^3*x + (a^4 + a^2*b^2)*d*e*f^2)*cosh(d*x + c)^3*sinh(d*x + c) + 6*((a^
4 + a^2*b^2)*d*f^3*x + (a^4 + a^2*b^2)*d*e*f^2)*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*((a^4 + a^2*b^2)*d*f^3*x +
(a^4 + a^2*b^2)*d*e*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^4 + a^2*b^2)*d*f^3*x + (a^4 + a^2*b^2)*d*e*f^2)*
sinh(d*x + c)^4)*polylog(3, (a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2
+ b^2)/b^2))/b) - 331776*(((a^4 + a^2*b^2)*d*f^3*x + (a^4 + a^2*b^2)*d*e*f^2)*cosh(d*x + c)^4 + 4*((a^4 + a^2
*b^2)*d*f^3*x + (a^4 + a^2*b^2)*d*e*f^2)*cosh(d*x + c)^3*sinh(d*x + c) + 6*((a^4 + a^2*b^2)*d*f^3*x + (a^4 + a
^2*b^2)*d*e*f^2)*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*((a^4 + a^2*b^2)*d*f^3*x + (a^4 + a^2*b^2)*d*e*f^2)*cosh(
d*x + c)*sinh(d*x + c)^3 + ((a^4 + a^2*b^2)*d*f^3*x + (a^4 + a^2*b^2)*d*e*f^2)*sinh(d*x + c)^4)*polylog(3, (a*
cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) + 8*(288*a*b^3
*d^3*f^3*x^3 + 288*a*b^3*d^3*e^3 + 288*a*b^3*d^2*e^2*f + 192*a*b^3*d*e*f^2 + 27*(32*b^4*d^3*f^3*x^3 + 32*b^4*d
^3*e^3 - 24*b^4*d^2*e^2*f + 12*b^4*d*e*f^2 - 3*b^4*f^3 + 24*(4*b^4*d^3*e*f^2 - b^4*d^2*f^3)*x^2 + 12*(8*b^4*d^
3*e^2*f - 4*b^4*d^2*e*f^2 + b^4*d*f^3)*x)*cosh(d*x + c)^7 + 64*a*b^3*f^3 - 224*(9*a*b^3*d^3*f^3*x^3 + 9*a*b^3*
d^3*e^3 - 9*a*b^3*d^2*e^2*f + 6*a*b^3*d*e*f^2 - 2*a*b^3*f^3 + 9*(3*a*b^3*d^3*e*f^2 - a*b^3*d^2*f^3)*x^2 + 3*(9
*a*b^3*d^3*e^2*f - 6*a*b^3*d^2*e*f^2 + 2*a*b^3*d*f^3)*x)*cosh(d*x + c)^6 + 648*(4*(2*a^2*b^2 + b^4)*d^3*f^3*x^
3 + 4*(2*a^2*b^2 + b^4)*d^3*e^3 - 6*(2*a^2*b^2 + b^4)*d^2*e^2*f + 6*(2*a^2*b^2 + b^4)*d*e*f^2 - 3*(2*a^2*b^2 +
b^4)*f^3 + 6*(2*(2*a^2*b^2 + b^4)*d^3*e*f^2 - (2*a^2*b^2 + b^4)*d^2*f^3)*x^2 + 6*(2*(2*a^2*b^2 + b^4)*d^3*e^2
*f - 2*(2*a^2*b^2 + b^4)*d^2*e*f^2 + (2*a^2*b^2 + b^4)*d*f^3)*x)*cosh(d*x + c)^5 - 4320*((4*a^3*b + 3*a*b^3)*d
^3*f^3*x^3 + (4*a^3*b + 3*a*b^3)*d^3*e^3 - 3*(4*a^3*b + 3*a*b^3)*d^2*e^2*f + 6*(4*a^3*b + 3*a*b^3)*d*e*f^2 - 6
*(4*a^3*b + 3*a*b^3)*f^3 + 3*((4*a^3*b + 3*a*b^3)*d^3*e*f^2 - (4*a^3*b + 3*a*b^3)*d^2*f^3)*x^2 + 3*((4*a^3*b +
3*a*b^3)*d^3*e^2*f - 2*(4*a^3*b + 3*a*b^3)*d^2*e*f^2 + 2*(4*a^3*b + 3*a*b^3)*d*f^3)*x)*cosh(d*x + c)^4 - 6912
*((a^4 + a^2*b^2)*d^4*f^3*x^4 + 4*(a^4 + a^2*b^2)*d^4*e*f^2*x^3 + 6*(a^4 + a^2*b^2)*d^4*e^2*f*x^2 + 4*(a^4 + a
^2*b^2)*d^4*e^3*x + 8*(a^4 + a^2*b^2)*c*d^3*e^3 - 12*(a^4 + a^2*b^2)*c^2*d^2*e^2*f + 8*(a^4 + a^2*b^2)*c^3*d*e
*f^2 - 2*(a^4 + a^2*b^2)*c^4*f^3)*cosh(d*x + c)^3 + 288*(3*a*b^3*d^3*e*f^2 + a*b^3*d^2*f^3)*x^2 + 2592*((4*a^3
*b + 3*a*b^3)*d^3*f^3*x^3 + (4*a^3*b + 3*a*b^3)*d^3*e^3 + 3*(4*a^3*b + 3*a*b^3)*d^2*e^2*f + 6*(4*a^3*b + 3*a*b
^3)*d*e*f^2 + 6*(4*a^3*b + 3*a*b^3)*f^3 + 3*((4*a^3*b + 3*a*b^3)*d^3*e*f^2 + (4*a^3*b + 3*a*b^3)*d^2*f^3)*x^2
+ 3*((4*a^3*b + 3*a*b^3)*d^3*e^2*f + 2*(4*a^3*b + 3*a*b^3)*d^2*e*f^2 + 2*(4*a^3*b + 3*a*b^3)*d*f^3)*x)*cosh(d*
x + c)^2 + 96*(9*a*b^3*d^3*e^2*f + 6*a*b^3*d^2*e*f^2 + 2*a*b^3*d*f^3)*x + 216*(4*(2*a^2*b^2 + b^4)*d^3*f^3*x^3
+ 4*(2*a^2*b^2 + b^4)*d^3*e^3 + 6*(2*a^2*b^2 + b^4)*d^2*e^2*f + 6*(2*a^2*b^2 + b^4)*d*e*f^2 + 3*(2*a^2*b^2 +
b^4)*f^3 + 6*(2*(2*a^2*b^2 + b^4)*d^3*e*f^2 + (2*a^2*b^2 + b^4)*d^2*f^3)*x^2 + 6*(2*(2*a^2*b^2 + b^4)*d^3*e^2*
f + 2*(2*a^2*b^2 + b^4)*d^2*e*f^2 + (2*a^2*b^2 + b^4)*d*f^3)*x)*cosh(d*x + c))*sinh(d*x + c))/(b^5*d^4*cosh(d*
x + c)^4 + 4*b^5*d^4*cosh(d*x + c)^3*sinh(d*x + c) + 6*b^5*d^4*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*b^5*d^4*cos
h(d*x + c)*sinh(d*x + c)^3 + b^5*d^4*sinh(d*x + c)^4)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (f x + e\right )}^{3} \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a}\,{d x} \]
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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maple [F] time = 0.72, 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 \[ \text {result too large to display} \]
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*e^3*((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)) + 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*e*f^2*e^(4*c) + a^2*b^2*d^4*e*f^2*e^(4*c))*x^3 + 82944*(a^4*d^4*
e^2*f*e^(4*c) + a^2*b^2*d^4*e^2*f*e^(4*c))*x^2 + 27*(32*b^4*d^3*f^3*x^3*e^(8*c) + 24*(4*d^3*e*f^2 - d^2*f^3)*b
^4*x^2*e^(8*c) + 12*(8*d^3*e^2*f - 4*d^2*e*f^2 + d*f^3)*b^4*x*e^(8*c) - 3*(8*d^2*e^2*f - 4*d*e*f^2 + f^3)*b^4*
e^(8*c))*e^(4*d*x) - 256*(9*a*b^3*d^3*f^3*x^3*e^(7*c) + 9*(3*d^3*e*f^2 - d^2*f^3)*a*b^3*x^2*e^(7*c) + 3*(9*d^3
*e^2*f - 6*d^2*e*f^2 + 2*d*f^3)*a*b^3*x*e^(7*c) - (9*d^2*e^2*f - 6*d*e*f^2 + 2*f^3)*a*b^3*e^(7*c))*e^(3*d*x) -
864*(6*(2*d^2*e^2*f - 2*d*e*f^2 + f^3)*a^2*b^2*e^(6*c) + 3*(2*d^2*e^2*f - 2*d*e*f^2 + f^3)*b^4*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*(2*d^3*e*f^2 - d^2*f^3)*a^2*b^2*e^(6*c) + (2*d^3*e*
f^2 - d^2*f^3)*b^4*e^(6*c))*x^2 - 6*(2*(2*d^3*e^2*f - 2*d^2*e*f^2 + d*f^3)*a^2*b^2*e^(6*c) + (2*d^3*e^2*f - 2*
d^2*e*f^2 + d*f^3)*b^4*e^(6*c))*x)*e^(2*d*x) + 6912*(12*(d^2*e^2*f - 2*d*e*f^2 + 2*f^3)*a^3*b*e^(5*c) + 9*(d^2
*e^2*f - 2*d*e*f^2 + 2*f^3)*a*b^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*(d^
3*e*f^2 - d^2*f^3)*a^3*b*e^(5*c) + 3*(d^3*e*f^2 - d^2*f^3)*a*b^3*e^(5*c))*x^2 - 3*(4*(d^3*e^2*f - 2*d^2*e*f^2
+ 2*d*f^3)*a^3*b*e^(5*c) + 3*(d^3*e^2*f - 2*d^2*e*f^2 + 2*d*f^3)*a*b^3*e^(5*c))*x)*e^(d*x) + 6912*(12*(d^2*e^2
*f + 2*d*e*f^2 + 2*f^3)*a^3*b*e^(3*c) + 9*(d^2*e^2*f + 2*d*e*f^2 + 2*f^3)*a*b^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*(d^3*e*f^2 + d^2*f^3)*a^3*b*e^(3*c) + 3*(d^3*e*f^2 + d^2*f^3)*a*b^3
*e^(3*c))*x^2 + 3*(4*(d^3*e^2*f + 2*d^2*e*f^2 + 2*d*f^3)*a^3*b*e^(3*c) + 3*(d^3*e^2*f + 2*d^2*e*f^2 + 2*d*f^3)
*a*b^3*e^(3*c))*x)*e^(-d*x) + 864*(6*(2*d^2*e^2*f + 2*d*e*f^2 + f^3)*a^2*b^2*e^(2*c) + 3*(2*d^2*e^2*f + 2*d*e*
f^2 + f^3)*b^4*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*(2*d^3*e*f^2 + d^2*f^3
)*a^2*b^2*e^(2*c) + (2*d^3*e*f^2 + d^2*f^3)*b^4*e^(2*c))*x^2 + 6*(2*(2*d^3*e^2*f + 2*d^2*e*f^2 + d*f^3)*a^2*b^
2*e^(2*c) + (2*d^3*e^2*f + 2*d^2*e*f^2 + d*f^3)*b^4*e^(2*c))*x)*e^(-2*d*x) + 256*(9*a*b^3*d^3*f^3*x^3*e^c + 9*
(3*d^3*e*f^2 + d^2*f^3)*a*b^3*x^2*e^c + 3*(9*d^3*e^2*f + 6*d^2*e*f^2 + 2*d*f^3)*a*b^3*x*e^c + (9*d^2*e^2*f + 6
*d*e*f^2 + 2*f^3)*a*b^3*e^c)*e^(-3*d*x) + 27*(32*b^4*d^3*f^3*x^3 + 24*(4*d^3*e*f^2 + d^2*f^3)*b^4*x^2 + 12*(8*
d^3*e^2*f + 4*d^2*e*f^2 + d*f^3)*b^4*x + 3*(8*d^2*e^2*f + 4*d*e*f^2 + f^3)*b^4)*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*e*f^2 + a^2*b^3*e*f^2)*x^2 + 3*(a^4*b*e^2*f + a^2*b^
3*e^2*f)*x - ((a^5*f^3*e^c + a^3*b^2*f^3*e^c)*x^3 + 3*(a^5*e*f^2*e^c + a^3*b^2*e*f^2*e^c)*x^2 + 3*(a^5*e^2*f*e
^c + a^3*b^2*e^2*f*e^c)*x)*e^(d*x))/(b^6*e^(2*d*x + 2*c) + 2*a*b^5*e^(d*x + c) - b^6), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^3\,{\mathrm {sinh}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^3}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]
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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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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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