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

Optimal. Leaf size=819 \[ \frac{f^2 \cosh ^4(c+d x)}{32 b d^3}+\frac{(e+f x)^2 \cosh ^4(c+d x)}{4 b d}+\frac{2 a f (e+f x) \cosh ^3(c+d x)}{9 b^2 d^2}-\frac{f (e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{8 b d^2}+\frac{3 f^2 \cosh ^2(c+d x)}{32 b d^3}-\frac{a (e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 b^2 d}+\frac{4 a f (e+f x) \cosh (c+d x)}{3 b^2 d^2}+\frac{2 a^3 f (e+f x) \cosh (c+d x)}{b^4 d^2}-\frac{3 f (e+f x) \sinh (c+d x) \cosh (c+d x)}{16 b d^2}-\frac{a^2 f (e+f x) \sinh (c+d x) \cosh (c+d x)}{2 b^3 d^2}-\frac{a^2 \left (a^2+b^2\right ) (e+f x)^3}{3 b^5 f}-\frac{2 a f^2 \sinh ^3(c+d x)}{27 b^2 d^3}-\frac{3 f^2 x^2}{32 b d}+\frac{a^2 f^2 x^2}{4 b^3 d}+\frac{a^2 f^2 \sinh ^2(c+d x)}{4 b^3 d^3}+\frac{a^2 (e+f x)^2 \sinh ^2(c+d x)}{2 b^3 d}-\frac{3 e f x}{16 b d}+\frac{a^2 e f x}{2 b^3 d}+\frac{a^2 \left (a^2+b^2\right ) (e+f x)^2 \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right )}{b^5 d}+\frac{a^2 \left (a^2+b^2\right ) (e+f x)^2 \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right )}{b^5 d}+\frac{2 a^2 \left (a^2+b^2\right ) f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^5 d^2}+\frac{2 a^2 \left (a^2+b^2\right ) f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^5 d^2}-\frac{2 a^2 \left (a^2+b^2\right ) f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^5 d^3}-\frac{2 a^2 \left (a^2+b^2\right ) f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^5 d^3}-\frac{14 a f^2 \sinh (c+d x)}{9 b^2 d^3}-\frac{2 a^3 f^2 \sinh (c+d x)}{b^4 d^3}-\frac{2 a (e+f x)^2 \sinh (c+d x)}{3 b^2 d}-\frac{a^3 (e+f x)^2 \sinh (c+d x)}{b^4 d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.17102, antiderivative size = 819, normalized size of antiderivative = 1., number of steps used = 28, number of rules used = 14, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {5579, 5447, 3310, 3311, 3296, 2637, 2633, 5565, 5446, 5561, 2190, 2531, 2282, 6589} \[ \frac{f^2 \cosh ^4(c+d x)}{32 b d^3}+\frac{(e+f x)^2 \cosh ^4(c+d x)}{4 b d}+\frac{2 a f (e+f x) \cosh ^3(c+d x)}{9 b^2 d^2}-\frac{f (e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{8 b d^2}+\frac{3 f^2 \cosh ^2(c+d x)}{32 b d^3}-\frac{a (e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 b^2 d}+\frac{4 a f (e+f x) \cosh (c+d x)}{3 b^2 d^2}+\frac{2 a^3 f (e+f x) \cosh (c+d x)}{b^4 d^2}-\frac{3 f (e+f x) \sinh (c+d x) \cosh (c+d x)}{16 b d^2}-\frac{a^2 f (e+f x) \sinh (c+d x) \cosh (c+d x)}{2 b^3 d^2}-\frac{a^2 \left (a^2+b^2\right ) (e+f x)^3}{3 b^5 f}-\frac{2 a f^2 \sinh ^3(c+d x)}{27 b^2 d^3}-\frac{3 f^2 x^2}{32 b d}+\frac{a^2 f^2 x^2}{4 b^3 d}+\frac{a^2 f^2 \sinh ^2(c+d x)}{4 b^3 d^3}+\frac{a^2 (e+f x)^2 \sinh ^2(c+d x)}{2 b^3 d}-\frac{3 e f x}{16 b d}+\frac{a^2 e f x}{2 b^3 d}+\frac{a^2 \left (a^2+b^2\right ) (e+f x)^2 \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right )}{b^5 d}+\frac{a^2 \left (a^2+b^2\right ) (e+f x)^2 \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right )}{b^5 d}+\frac{2 a^2 \left (a^2+b^2\right ) f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^5 d^2}+\frac{2 a^2 \left (a^2+b^2\right ) f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^5 d^2}-\frac{2 a^2 \left (a^2+b^2\right ) f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^5 d^3}-\frac{2 a^2 \left (a^2+b^2\right ) f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^5 d^3}-\frac{14 a f^2 \sinh (c+d x)}{9 b^2 d^3}-\frac{2 a^3 f^2 \sinh (c+d x)}{b^4 d^3}-\frac{2 a (e+f x)^2 \sinh (c+d x)}{3 b^2 d}-\frac{a^3 (e+f x)^2 \sinh (c+d x)}{b^4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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 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 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rule 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 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 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 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 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 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 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]

Rubi steps

\begin{align*} \int \frac{(e+f x)^2 \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac{\int (e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x) \, dx}{b}-\frac{a \int \frac{(e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=\frac{(e+f x)^2 \cosh ^4(c+d x)}{4 b d}-\frac{a \int (e+f x)^2 \cosh ^3(c+d x) \, dx}{b^2}+\frac{a^2 \int \frac{(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}-\frac{f \int (e+f x) \cosh ^4(c+d x) \, dx}{2 b d}\\ &=\frac{2 a f (e+f x) \cosh ^3(c+d x)}{9 b^2 d^2}+\frac{f^2 \cosh ^4(c+d x)}{32 b d^3}+\frac{(e+f x)^2 \cosh ^4(c+d x)}{4 b d}-\frac{a (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^2 d}-\frac{f (e+f x) \cosh ^3(c+d x) \sinh (c+d x)}{8 b d^2}-\frac{a^3 \int (e+f x)^2 \cosh (c+d x) \, dx}{b^4}+\frac{a^2 \int (e+f x)^2 \cosh (c+d x) \sinh (c+d x) \, dx}{b^3}-\frac{(2 a) \int (e+f x)^2 \cosh (c+d x) \, dx}{3 b^2}+\frac{\left (a^2 \left (a^2+b^2\right )\right ) \int \frac{(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^4}-\frac{(3 f) \int (e+f x) \cosh ^2(c+d x) \, dx}{8 b d}-\frac{\left (2 a f^2\right ) \int \cosh ^3(c+d x) \, dx}{9 b^2 d^2}\\ &=-\frac{a^2 \left (a^2+b^2\right ) (e+f x)^3}{3 b^5 f}+\frac{3 f^2 \cosh ^2(c+d x)}{32 b d^3}+\frac{2 a f (e+f x) \cosh ^3(c+d x)}{9 b^2 d^2}+\frac{f^2 \cosh ^4(c+d x)}{32 b d^3}+\frac{(e+f x)^2 \cosh ^4(c+d x)}{4 b d}-\frac{a^3 (e+f x)^2 \sinh (c+d x)}{b^4 d}-\frac{2 a (e+f x)^2 \sinh (c+d x)}{3 b^2 d}-\frac{3 f (e+f x) \cosh (c+d x) \sinh (c+d x)}{16 b d^2}-\frac{a (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^2 d}-\frac{f (e+f x) \cosh ^3(c+d x) \sinh (c+d x)}{8 b d^2}+\frac{a^2 (e+f x)^2 \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)^2}{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)^2}{a+\sqrt{a^2+b^2}+b e^{c+d x}} \, dx}{b^4}+\frac{\left (2 a^3 f\right ) \int (e+f x) \sinh (c+d x) \, dx}{b^4 d}-\frac{\left (a^2 f\right ) \int (e+f x) \sinh ^2(c+d x) \, dx}{b^3 d}+\frac{(4 a f) \int (e+f x) \sinh (c+d x) \, dx}{3 b^2 d}-\frac{(3 f) \int (e+f x) \, dx}{16 b d}-\frac{\left (2 i a f^2\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-i \sinh (c+d x)\right )}{9 b^2 d^3}\\ &=-\frac{3 e f x}{16 b d}-\frac{3 f^2 x^2}{32 b d}-\frac{a^2 \left (a^2+b^2\right ) (e+f x)^3}{3 b^5 f}+\frac{2 a^3 f (e+f x) \cosh (c+d x)}{b^4 d^2}+\frac{4 a f (e+f x) \cosh (c+d x)}{3 b^2 d^2}+\frac{3 f^2 \cosh ^2(c+d x)}{32 b d^3}+\frac{2 a f (e+f x) \cosh ^3(c+d x)}{9 b^2 d^2}+\frac{f^2 \cosh ^4(c+d x)}{32 b d^3}+\frac{(e+f x)^2 \cosh ^4(c+d x)}{4 b d}+\frac{a^2 \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^5 d}+\frac{a^2 \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^5 d}-\frac{2 a f^2 \sinh (c+d x)}{9 b^2 d^3}-\frac{a^3 (e+f x)^2 \sinh (c+d x)}{b^4 d}-\frac{2 a (e+f x)^2 \sinh (c+d x)}{3 b^2 d}-\frac{a^2 f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^3 d^2}-\frac{3 f (e+f x) \cosh (c+d x) \sinh (c+d x)}{16 b d^2}-\frac{a (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^2 d}-\frac{f (e+f x) \cosh ^3(c+d x) \sinh (c+d x)}{8 b d^2}+\frac{a^2 f^2 \sinh ^2(c+d x)}{4 b^3 d^3}+\frac{a^2 (e+f x)^2 \sinh ^2(c+d x)}{2 b^3 d}-\frac{2 a f^2 \sinh ^3(c+d x)}{27 b^2 d^3}+\frac{\left (a^2 f\right ) \int (e+f x) \, dx}{2 b^3 d}-\frac{\left (2 a^2 \left (a^2+b^2\right ) f\right ) \int (e+f x) \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) \, dx}{b^5 d}-\frac{\left (2 a^2 \left (a^2+b^2\right ) f\right ) \int (e+f x) \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) \, dx}{b^5 d}-\frac{\left (2 a^3 f^2\right ) \int \cosh (c+d x) \, dx}{b^4 d^2}-\frac{\left (4 a f^2\right ) \int \cosh (c+d x) \, dx}{3 b^2 d^2}\\ &=\frac{a^2 e f x}{2 b^3 d}-\frac{3 e f x}{16 b d}+\frac{a^2 f^2 x^2}{4 b^3 d}-\frac{3 f^2 x^2}{32 b d}-\frac{a^2 \left (a^2+b^2\right ) (e+f x)^3}{3 b^5 f}+\frac{2 a^3 f (e+f x) \cosh (c+d x)}{b^4 d^2}+\frac{4 a f (e+f x) \cosh (c+d x)}{3 b^2 d^2}+\frac{3 f^2 \cosh ^2(c+d x)}{32 b d^3}+\frac{2 a f (e+f x) \cosh ^3(c+d x)}{9 b^2 d^2}+\frac{f^2 \cosh ^4(c+d x)}{32 b d^3}+\frac{(e+f x)^2 \cosh ^4(c+d x)}{4 b d}+\frac{a^2 \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^5 d}+\frac{a^2 \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^5 d}+\frac{2 a^2 \left (a^2+b^2\right ) f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^5 d^2}+\frac{2 a^2 \left (a^2+b^2\right ) f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^5 d^2}-\frac{2 a^3 f^2 \sinh (c+d x)}{b^4 d^3}-\frac{14 a f^2 \sinh (c+d x)}{9 b^2 d^3}-\frac{a^3 (e+f x)^2 \sinh (c+d x)}{b^4 d}-\frac{2 a (e+f x)^2 \sinh (c+d x)}{3 b^2 d}-\frac{a^2 f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^3 d^2}-\frac{3 f (e+f x) \cosh (c+d x) \sinh (c+d x)}{16 b d^2}-\frac{a (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^2 d}-\frac{f (e+f x) \cosh ^3(c+d x) \sinh (c+d x)}{8 b d^2}+\frac{a^2 f^2 \sinh ^2(c+d x)}{4 b^3 d^3}+\frac{a^2 (e+f x)^2 \sinh ^2(c+d x)}{2 b^3 d}-\frac{2 a f^2 \sinh ^3(c+d x)}{27 b^2 d^3}-\frac{\left (2 a^2 \left (a^2+b^2\right ) f^2\right ) \int \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) \, dx}{b^5 d^2}-\frac{\left (2 a^2 \left (a^2+b^2\right ) f^2\right ) \int \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) \, dx}{b^5 d^2}\\ &=\frac{a^2 e f x}{2 b^3 d}-\frac{3 e f x}{16 b d}+\frac{a^2 f^2 x^2}{4 b^3 d}-\frac{3 f^2 x^2}{32 b d}-\frac{a^2 \left (a^2+b^2\right ) (e+f x)^3}{3 b^5 f}+\frac{2 a^3 f (e+f x) \cosh (c+d x)}{b^4 d^2}+\frac{4 a f (e+f x) \cosh (c+d x)}{3 b^2 d^2}+\frac{3 f^2 \cosh ^2(c+d x)}{32 b d^3}+\frac{2 a f (e+f x) \cosh ^3(c+d x)}{9 b^2 d^2}+\frac{f^2 \cosh ^4(c+d x)}{32 b d^3}+\frac{(e+f x)^2 \cosh ^4(c+d x)}{4 b d}+\frac{a^2 \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^5 d}+\frac{a^2 \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^5 d}+\frac{2 a^2 \left (a^2+b^2\right ) f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^5 d^2}+\frac{2 a^2 \left (a^2+b^2\right ) f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^5 d^2}-\frac{2 a^3 f^2 \sinh (c+d x)}{b^4 d^3}-\frac{14 a f^2 \sinh (c+d x)}{9 b^2 d^3}-\frac{a^3 (e+f x)^2 \sinh (c+d x)}{b^4 d}-\frac{2 a (e+f x)^2 \sinh (c+d x)}{3 b^2 d}-\frac{a^2 f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^3 d^2}-\frac{3 f (e+f x) \cosh (c+d x) \sinh (c+d x)}{16 b d^2}-\frac{a (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^2 d}-\frac{f (e+f x) \cosh ^3(c+d x) \sinh (c+d x)}{8 b d^2}+\frac{a^2 f^2 \sinh ^2(c+d x)}{4 b^3 d^3}+\frac{a^2 (e+f x)^2 \sinh ^2(c+d x)}{2 b^3 d}-\frac{2 a f^2 \sinh ^3(c+d x)}{27 b^2 d^3}-\frac{\left (2 a^2 \left (a^2+b^2\right ) f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{b x}{-a+\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^5 d^3}-\frac{\left (2 a^2 \left (a^2+b^2\right ) f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{b x}{a+\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^5 d^3}\\ &=\frac{a^2 e f x}{2 b^3 d}-\frac{3 e f x}{16 b d}+\frac{a^2 f^2 x^2}{4 b^3 d}-\frac{3 f^2 x^2}{32 b d}-\frac{a^2 \left (a^2+b^2\right ) (e+f x)^3}{3 b^5 f}+\frac{2 a^3 f (e+f x) \cosh (c+d x)}{b^4 d^2}+\frac{4 a f (e+f x) \cosh (c+d x)}{3 b^2 d^2}+\frac{3 f^2 \cosh ^2(c+d x)}{32 b d^3}+\frac{2 a f (e+f x) \cosh ^3(c+d x)}{9 b^2 d^2}+\frac{f^2 \cosh ^4(c+d x)}{32 b d^3}+\frac{(e+f x)^2 \cosh ^4(c+d x)}{4 b d}+\frac{a^2 \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^5 d}+\frac{a^2 \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^5 d}+\frac{2 a^2 \left (a^2+b^2\right ) f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^5 d^2}+\frac{2 a^2 \left (a^2+b^2\right ) f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^5 d^2}-\frac{2 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 )}{b^5 d^3}-\frac{2 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 )}{b^5 d^3}-\frac{2 a^3 f^2 \sinh (c+d x)}{b^4 d^3}-\frac{14 a f^2 \sinh (c+d x)}{9 b^2 d^3}-\frac{a^3 (e+f x)^2 \sinh (c+d x)}{b^4 d}-\frac{2 a (e+f x)^2 \sinh (c+d x)}{3 b^2 d}-\frac{a^2 f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^3 d^2}-\frac{3 f (e+f x) \cosh (c+d x) \sinh (c+d x)}{16 b d^2}-\frac{a (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^2 d}-\frac{f (e+f x) \cosh ^3(c+d x) \sinh (c+d x)}{8 b d^2}+\frac{a^2 f^2 \sinh ^2(c+d x)}{4 b^3 d^3}+\frac{a^2 (e+f x)^2 \sinh ^2(c+d x)}{2 b^3 d}-\frac{2 a f^2 \sinh ^3(c+d x)}{27 b^2 d^3}\\ \end{align*}

Mathematica [B]  time = 19.3254, size = 5198, normalized size = 6.35 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((e + f*x)^2*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 = 0.243, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{2} \left ( \cosh \left ( dx+c \right ) \right ) ^{3} \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{a+b\sinh \left ( dx+c \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((f*x+e)^2*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., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/192*e^2*((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)) + 1/13824*(4608*(a^4*d^3*f^2*e^(4*c) + a
^2*b^2*d^3*f^2*e^(4*c))*x^3 + 13824*(a^4*d^3*e*f*e^(4*c) + a^2*b^2*d^3*e*f*e^(4*c))*x^2 + 27*(8*b^4*d^2*f^2*x^
2*e^(8*c) + 4*(4*d^2*e*f - d*f^2)*b^4*x*e^(8*c) - (4*d*e*f - f^2)*b^4*e^(8*c))*e^(4*d*x) - 64*(9*a*b^3*d^2*f^2
*x^2*e^(7*c) + 6*(3*d^2*e*f - d*f^2)*a*b^3*x*e^(7*c) - 2*(3*d*e*f - f^2)*a*b^3*e^(7*c))*e^(3*d*x) - 432*(2*(2*
d*e*f - f^2)*a^2*b^2*e^(6*c) + (2*d*e*f - f^2)*b^4*e^(6*c) - 2*(2*a^2*b^2*d^2*f^2*e^(6*c) + b^4*d^2*f^2*e^(6*c
))*x^2 - 2*(2*(2*d^2*e*f - d*f^2)*a^2*b^2*e^(6*c) + (2*d^2*e*f - d*f^2)*b^4*e^(6*c))*x)*e^(2*d*x) + 1728*(8*(d
*e*f - f^2)*a^3*b*e^(5*c) + 6*(d*e*f - f^2)*a*b^3*e^(5*c) - (4*a^3*b*d^2*f^2*e^(5*c) + 3*a*b^3*d^2*f^2*e^(5*c)
)*x^2 - 2*(4*(d^2*e*f - d*f^2)*a^3*b*e^(5*c) + 3*(d^2*e*f - d*f^2)*a*b^3*e^(5*c))*x)*e^(d*x) + 1728*(8*(d*e*f
+ f^2)*a^3*b*e^(3*c) + 6*(d*e*f + f^2)*a*b^3*e^(3*c) + (4*a^3*b*d^2*f^2*e^(3*c) + 3*a*b^3*d^2*f^2*e^(3*c))*x^2
+ 2*(4*(d^2*e*f + d*f^2)*a^3*b*e^(3*c) + 3*(d^2*e*f + d*f^2)*a*b^3*e^(3*c))*x)*e^(-d*x) + 432*(2*(2*d*e*f + f
^2)*a^2*b^2*e^(2*c) + (2*d*e*f + f^2)*b^4*e^(2*c) + 2*(2*a^2*b^2*d^2*f^2*e^(2*c) + b^4*d^2*f^2*e^(2*c))*x^2 +
2*(2*(2*d^2*e*f + d*f^2)*a^2*b^2*e^(2*c) + (2*d^2*e*f + d*f^2)*b^4*e^(2*c))*x)*e^(-2*d*x) + 64*(9*a*b^3*d^2*f^
2*x^2*e^c + 6*(3*d^2*e*f + d*f^2)*a*b^3*x*e^c + 2*(3*d*e*f + f^2)*a*b^3*e^c)*e^(-3*d*x) + 27*(8*b^4*d^2*f^2*x^
2 + 4*(4*d^2*e*f + d*f^2)*b^4*x + (4*d*e*f + f^2)*b^4)*e^(-4*d*x))*e^(-4*c)/(b^5*d^3) - integrate(-2*((a^4*b*f
^2 + a^2*b^3*f^2)*x^2 + 2*(a^4*b*e*f + a^2*b^3*e*f)*x - ((a^5*f^2*e^c + a^3*b^2*f^2*e^c)*x^2 + 2*(a^5*e*f*e^c
+ a^3*b^2*e*f*e^c)*x)*e^(d*x))/(b^6*e^(2*d*x + 2*c) + 2*a*b^5*e^(d*x + c) - b^6), x)

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Fricas [C]  time = 3.44704, size = 16764, normalized size = 20.47 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13824*(216*b^4*d^2*f^2*x^2 + 27*(8*b^4*d^2*f^2*x^2 + 8*b^4*d^2*e^2 - 4*b^4*d*e*f + b^4*f^2 + 4*(4*b^4*d^2*e*
f - b^4*d*f^2)*x)*cosh(d*x + c)^8 + 27*(8*b^4*d^2*f^2*x^2 + 8*b^4*d^2*e^2 - 4*b^4*d*e*f + b^4*f^2 + 4*(4*b^4*d
^2*e*f - b^4*d*f^2)*x)*sinh(d*x + c)^8 + 216*b^4*d^2*e^2 - 64*(9*a*b^3*d^2*f^2*x^2 + 9*a*b^3*d^2*e^2 - 6*a*b^3
*d*e*f + 2*a*b^3*f^2 + 6*(3*a*b^3*d^2*e*f - a*b^3*d*f^2)*x)*cosh(d*x + c)^7 - 8*(72*a*b^3*d^2*f^2*x^2 + 72*a*b
^3*d^2*e^2 - 48*a*b^3*d*e*f + 16*a*b^3*f^2 + 48*(3*a*b^3*d^2*e*f - a*b^3*d*f^2)*x - 27*(8*b^4*d^2*f^2*x^2 + 8*
b^4*d^2*e^2 - 4*b^4*d*e*f + b^4*f^2 + 4*(4*b^4*d^2*e*f - b^4*d*f^2)*x)*cosh(d*x + c))*sinh(d*x + c)^7 + 108*b^
4*d*e*f + 432*(2*(2*a^2*b^2 + b^4)*d^2*f^2*x^2 + 2*(2*a^2*b^2 + b^4)*d^2*e^2 - 2*(2*a^2*b^2 + b^4)*d*e*f + (2*
a^2*b^2 + b^4)*f^2 + 2*(2*(2*a^2*b^2 + b^4)*d^2*e*f - (2*a^2*b^2 + b^4)*d*f^2)*x)*cosh(d*x + c)^6 + 4*(216*(2*
a^2*b^2 + b^4)*d^2*f^2*x^2 + 216*(2*a^2*b^2 + b^4)*d^2*e^2 - 216*(2*a^2*b^2 + b^4)*d*e*f + 108*(2*a^2*b^2 + b^
4)*f^2 + 189*(8*b^4*d^2*f^2*x^2 + 8*b^4*d^2*e^2 - 4*b^4*d*e*f + b^4*f^2 + 4*(4*b^4*d^2*e*f - b^4*d*f^2)*x)*cos
h(d*x + c)^2 + 216*(2*(2*a^2*b^2 + b^4)*d^2*e*f - (2*a^2*b^2 + b^4)*d*f^2)*x - 112*(9*a*b^3*d^2*f^2*x^2 + 9*a*
b^3*d^2*e^2 - 6*a*b^3*d*e*f + 2*a*b^3*f^2 + 6*(3*a*b^3*d^2*e*f - a*b^3*d*f^2)*x)*cosh(d*x + c))*sinh(d*x + c)^
6 + 27*b^4*f^2 - 1728*((4*a^3*b + 3*a*b^3)*d^2*f^2*x^2 + (4*a^3*b + 3*a*b^3)*d^2*e^2 - 2*(4*a^3*b + 3*a*b^3)*d
*e*f + 2*(4*a^3*b + 3*a*b^3)*f^2 + 2*((4*a^3*b + 3*a*b^3)*d^2*e*f - (4*a^3*b + 3*a*b^3)*d*f^2)*x)*cosh(d*x + c
)^5 - 24*(72*(4*a^3*b + 3*a*b^3)*d^2*f^2*x^2 + 72*(4*a^3*b + 3*a*b^3)*d^2*e^2 - 144*(4*a^3*b + 3*a*b^3)*d*e*f
- 63*(8*b^4*d^2*f^2*x^2 + 8*b^4*d^2*e^2 - 4*b^4*d*e*f + b^4*f^2 + 4*(4*b^4*d^2*e*f - b^4*d*f^2)*x)*cosh(d*x +
c)^3 + 144*(4*a^3*b + 3*a*b^3)*f^2 + 56*(9*a*b^3*d^2*f^2*x^2 + 9*a*b^3*d^2*e^2 - 6*a*b^3*d*e*f + 2*a*b^3*f^2 +
6*(3*a*b^3*d^2*e*f - a*b^3*d*f^2)*x)*cosh(d*x + c)^2 + 144*((4*a^3*b + 3*a*b^3)*d^2*e*f - (4*a^3*b + 3*a*b^3)
*d*f^2)*x - 108*(2*(2*a^2*b^2 + b^4)*d^2*f^2*x^2 + 2*(2*a^2*b^2 + b^4)*d^2*e^2 - 2*(2*a^2*b^2 + b^4)*d*e*f + (
2*a^2*b^2 + b^4)*f^2 + 2*(2*(2*a^2*b^2 + b^4)*d^2*e*f - (2*a^2*b^2 + b^4)*d*f^2)*x)*cosh(d*x + c))*sinh(d*x +
c)^5 - 4608*((a^4 + a^2*b^2)*d^3*f^2*x^3 + 3*(a^4 + a^2*b^2)*d^3*e*f*x^2 + 3*(a^4 + a^2*b^2)*d^3*e^2*x + 6*(a^
4 + a^2*b^2)*c*d^2*e^2 - 6*(a^4 + a^2*b^2)*c^2*d*e*f + 2*(a^4 + a^2*b^2)*c^3*f^2)*cosh(d*x + c)^4 - 2*(2304*(a
^4 + a^2*b^2)*d^3*f^2*x^3 + 6912*(a^4 + a^2*b^2)*d^3*e*f*x^2 + 6912*(a^4 + a^2*b^2)*d^3*e^2*x + 13824*(a^4 + a
^2*b^2)*c*d^2*e^2 - 13824*(a^4 + a^2*b^2)*c^2*d*e*f + 4608*(a^4 + a^2*b^2)*c^3*f^2 - 945*(8*b^4*d^2*f^2*x^2 +
8*b^4*d^2*e^2 - 4*b^4*d*e*f + b^4*f^2 + 4*(4*b^4*d^2*e*f - b^4*d*f^2)*x)*cosh(d*x + c)^4 + 1120*(9*a*b^3*d^2*f
^2*x^2 + 9*a*b^3*d^2*e^2 - 6*a*b^3*d*e*f + 2*a*b^3*f^2 + 6*(3*a*b^3*d^2*e*f - a*b^3*d*f^2)*x)*cosh(d*x + c)^3
- 3240*(2*(2*a^2*b^2 + b^4)*d^2*f^2*x^2 + 2*(2*a^2*b^2 + b^4)*d^2*e^2 - 2*(2*a^2*b^2 + b^4)*d*e*f + (2*a^2*b^2
+ b^4)*f^2 + 2*(2*(2*a^2*b^2 + b^4)*d^2*e*f - (2*a^2*b^2 + b^4)*d*f^2)*x)*cosh(d*x + c)^2 + 4320*((4*a^3*b +
3*a*b^3)*d^2*f^2*x^2 + (4*a^3*b + 3*a*b^3)*d^2*e^2 - 2*(4*a^3*b + 3*a*b^3)*d*e*f + 2*(4*a^3*b + 3*a*b^3)*f^2 +
2*((4*a^3*b + 3*a*b^3)*d^2*e*f - (4*a^3*b + 3*a*b^3)*d*f^2)*x)*cosh(d*x + c))*sinh(d*x + c)^4 + 1728*((4*a^3*
b + 3*a*b^3)*d^2*f^2*x^2 + (4*a^3*b + 3*a*b^3)*d^2*e^2 + 2*(4*a^3*b + 3*a*b^3)*d*e*f + 2*(4*a^3*b + 3*a*b^3)*f
^2 + 2*((4*a^3*b + 3*a*b^3)*d^2*e*f + (4*a^3*b + 3*a*b^3)*d*f^2)*x)*cosh(d*x + c)^3 + 8*(216*(4*a^3*b + 3*a*b^
3)*d^2*f^2*x^2 + 189*(8*b^4*d^2*f^2*x^2 + 8*b^4*d^2*e^2 - 4*b^4*d*e*f + b^4*f^2 + 4*(4*b^4*d^2*e*f - b^4*d*f^2
)*x)*cosh(d*x + c)^5 + 216*(4*a^3*b + 3*a*b^3)*d^2*e^2 - 280*(9*a*b^3*d^2*f^2*x^2 + 9*a*b^3*d^2*e^2 - 6*a*b^3*
d*e*f + 2*a*b^3*f^2 + 6*(3*a*b^3*d^2*e*f - a*b^3*d*f^2)*x)*cosh(d*x + c)^4 + 432*(4*a^3*b + 3*a*b^3)*d*e*f + 1
080*(2*(2*a^2*b^2 + b^4)*d^2*f^2*x^2 + 2*(2*a^2*b^2 + b^4)*d^2*e^2 - 2*(2*a^2*b^2 + b^4)*d*e*f + (2*a^2*b^2 +
b^4)*f^2 + 2*(2*(2*a^2*b^2 + b^4)*d^2*e*f - (2*a^2*b^2 + b^4)*d*f^2)*x)*cosh(d*x + c)^3 + 432*(4*a^3*b + 3*a*b
^3)*f^2 - 2160*((4*a^3*b + 3*a*b^3)*d^2*f^2*x^2 + (4*a^3*b + 3*a*b^3)*d^2*e^2 - 2*(4*a^3*b + 3*a*b^3)*d*e*f +
2*(4*a^3*b + 3*a*b^3)*f^2 + 2*((4*a^3*b + 3*a*b^3)*d^2*e*f - (4*a^3*b + 3*a*b^3)*d*f^2)*x)*cosh(d*x + c)^2 + 4
32*((4*a^3*b + 3*a*b^3)*d^2*e*f + (4*a^3*b + 3*a*b^3)*d*f^2)*x - 2304*((a^4 + a^2*b^2)*d^3*f^2*x^3 + 3*(a^4 +
a^2*b^2)*d^3*e*f*x^2 + 3*(a^4 + a^2*b^2)*d^3*e^2*x + 6*(a^4 + a^2*b^2)*c*d^2*e^2 - 6*(a^4 + a^2*b^2)*c^2*d*e*f
+ 2*(a^4 + a^2*b^2)*c^3*f^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 432*(2*(2*a^2*b^2 + b^4)*d^2*f^2*x^2 + 2*(2*a^2
*b^2 + b^4)*d^2*e^2 + 2*(2*a^2*b^2 + b^4)*d*e*f + (2*a^2*b^2 + b^4)*f^2 + 2*(2*(2*a^2*b^2 + b^4)*d^2*e*f + (2*
a^2*b^2 + b^4)*d*f^2)*x)*cosh(d*x + c)^2 + 12*(72*(2*a^2*b^2 + b^4)*d^2*f^2*x^2 + 63*(8*b^4*d^2*f^2*x^2 + 8*b^
4*d^2*e^2 - 4*b^4*d*e*f + b^4*f^2 + 4*(4*b^4*d^2*e*f - b^4*d*f^2)*x)*cosh(d*x + c)^6 - 112*(9*a*b^3*d^2*f^2*x^
2 + 9*a*b^3*d^2*e^2 - 6*a*b^3*d*e*f + 2*a*b^3*f^2 + 6*(3*a*b^3*d^2*e*f - a*b^3*d*f^2)*x)*cosh(d*x + c)^5 + 72*
(2*a^2*b^2 + b^4)*d^2*e^2 + 540*(2*(2*a^2*b^2 + b^4)*d^2*f^2*x^2 + 2*(2*a^2*b^2 + b^4)*d^2*e^2 - 2*(2*a^2*b^2
+ b^4)*d*e*f + (2*a^2*b^2 + b^4)*f^2 + 2*(2*(2*a^2*b^2 + b^4)*d^2*e*f - (2*a^2*b^2 + b^4)*d*f^2)*x)*cosh(d*x +
c)^4 + 72*(2*a^2*b^2 + b^4)*d*e*f - 1440*((4*a^3*b + 3*a*b^3)*d^2*f^2*x^2 + (4*a^3*b + 3*a*b^3)*d^2*e^2 - 2*(
4*a^3*b + 3*a*b^3)*d*e*f + 2*(4*a^3*b + 3*a*b^3)*f^2 + 2*((4*a^3*b + 3*a*b^3)*d^2*e*f - (4*a^3*b + 3*a*b^3)*d*
f^2)*x)*cosh(d*x + c)^3 + 36*(2*a^2*b^2 + b^4)*f^2 - 2304*((a^4 + a^2*b^2)*d^3*f^2*x^3 + 3*(a^4 + a^2*b^2)*d^3
*e*f*x^2 + 3*(a^4 + a^2*b^2)*d^3*e^2*x + 6*(a^4 + a^2*b^2)*c*d^2*e^2 - 6*(a^4 + a^2*b^2)*c^2*d*e*f + 2*(a^4 +
a^2*b^2)*c^3*f^2)*cosh(d*x + c)^2 + 72*(2*(2*a^2*b^2 + b^4)*d^2*e*f + (2*a^2*b^2 + b^4)*d*f^2)*x + 432*((4*a^3
*b + 3*a*b^3)*d^2*f^2*x^2 + (4*a^3*b + 3*a*b^3)*d^2*e^2 + 2*(4*a^3*b + 3*a*b^3)*d*e*f + 2*(4*a^3*b + 3*a*b^3)*
f^2 + 2*((4*a^3*b + 3*a*b^3)*d^2*e*f + (4*a^3*b + 3*a*b^3)*d*f^2)*x)*cosh(d*x + c))*sinh(d*x + c)^2 + 108*(4*b
^4*d^2*e*f + b^4*d*f^2)*x + 64*(9*a*b^3*d^2*f^2*x^2 + 9*a*b^3*d^2*e^2 + 6*a*b^3*d*e*f + 2*a*b^3*f^2 + 6*(3*a*b
^3*d^2*e*f + a*b^3*d*f^2)*x)*cosh(d*x + c) + 27648*(((a^4 + a^2*b^2)*d*f^2*x + (a^4 + a^2*b^2)*d*e*f)*cosh(d*x
+ c)^4 + 4*((a^4 + a^2*b^2)*d*f^2*x + (a^4 + a^2*b^2)*d*e*f)*cosh(d*x + c)^3*sinh(d*x + c) + 6*((a^4 + a^2*b^
2)*d*f^2*x + (a^4 + a^2*b^2)*d*e*f)*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*((a^4 + a^2*b^2)*d*f^2*x + (a^4 + a^2*
b^2)*d*e*f)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^4 + a^2*b^2)*d*f^2*x + (a^4 + a^2*b^2)*d*e*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) + 27648*(((a^4 + a^2*b^2)*d*f^2*x + (a^4 + a^2*b^2)*d*e*f)*cosh(d*x + c)^4 + 4*((a^4 + a^2*b^2)*d*f^2*x +
(a^4 + a^2*b^2)*d*e*f)*cosh(d*x + c)^3*sinh(d*x + c) + 6*((a^4 + a^2*b^2)*d*f^2*x + (a^4 + a^2*b^2)*d*e*f)*cos
h(d*x + c)^2*sinh(d*x + c)^2 + 4*((a^4 + a^2*b^2)*d*f^2*x + (a^4 + a^2*b^2)*d*e*f)*cosh(d*x + c)*sinh(d*x + c)
^3 + ((a^4 + a^2*b^2)*d*f^2*x + (a^4 + a^2*b^2)*d*e*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) + 13824*(((a^4 + a^2*b^2)*d^2*e^2 -
2*(a^4 + a^2*b^2)*c*d*e*f + (a^4 + a^2*b^2)*c^2*f^2)*cosh(d*x + c)^4 + 4*((a^4 + a^2*b^2)*d^2*e^2 - 2*(a^4 +
a^2*b^2)*c*d*e*f + (a^4 + a^2*b^2)*c^2*f^2)*cosh(d*x + c)^3*sinh(d*x + c) + 6*((a^4 + a^2*b^2)*d^2*e^2 - 2*(a^
4 + a^2*b^2)*c*d*e*f + (a^4 + a^2*b^2)*c^2*f^2)*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*((a^4 + a^2*b^2)*d^2*e^2 -
2*(a^4 + a^2*b^2)*c*d*e*f + (a^4 + a^2*b^2)*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^4 + a^2*b^2)*d^2*e^2
- 2*(a^4 + a^2*b^2)*c*d*e*f + (a^4 + a^2*b^2)*c^2*f^2)*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) + 13824*(((a^4 + a^2*b^2)*d^2*e^2 - 2*(a^4 + a^2*b^2)*c*d*e*f + (a^4 +
a^2*b^2)*c^2*f^2)*cosh(d*x + c)^4 + 4*((a^4 + a^2*b^2)*d^2*e^2 - 2*(a^4 + a^2*b^2)*c*d*e*f + (a^4 + a^2*b^2)*
c^2*f^2)*cosh(d*x + c)^3*sinh(d*x + c) + 6*((a^4 + a^2*b^2)*d^2*e^2 - 2*(a^4 + a^2*b^2)*c*d*e*f + (a^4 + a^2*b
^2)*c^2*f^2)*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*((a^4 + a^2*b^2)*d^2*e^2 - 2*(a^4 + a^2*b^2)*c*d*e*f + (a^4 +
a^2*b^2)*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^4 + a^2*b^2)*d^2*e^2 - 2*(a^4 + a^2*b^2)*c*d*e*f + (a^4
+ a^2*b^2)*c^2*f^2)*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) + 13824*(((a^4 + a^2*b^2)*d^2*f^2*x^2 + 2*(a^4 + a^2*b^2)*d^2*e*f*x + 2*(a^4 + a^2*b^2)*c*d*e*f - (a^4 +
a^2*b^2)*c^2*f^2)*cosh(d*x + c)^4 + 4*((a^4 + a^2*b^2)*d^2*f^2*x^2 + 2*(a^4 + a^2*b^2)*d^2*e*f*x + 2*(a^4 + a^
2*b^2)*c*d*e*f - (a^4 + a^2*b^2)*c^2*f^2)*cosh(d*x + c)^3*sinh(d*x + c) + 6*((a^4 + a^2*b^2)*d^2*f^2*x^2 + 2*(
a^4 + a^2*b^2)*d^2*e*f*x + 2*(a^4 + a^2*b^2)*c*d*e*f - (a^4 + a^2*b^2)*c^2*f^2)*cosh(d*x + c)^2*sinh(d*x + c)^
2 + 4*((a^4 + a^2*b^2)*d^2*f^2*x^2 + 2*(a^4 + a^2*b^2)*d^2*e*f*x + 2*(a^4 + a^2*b^2)*c*d*e*f - (a^4 + a^2*b^2)
*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^4 + a^2*b^2)*d^2*f^2*x^2 + 2*(a^4 + a^2*b^2)*d^2*e*f*x + 2*(a^4
+ a^2*b^2)*c*d*e*f - (a^4 + a^2*b^2)*c^2*f^2)*sinh(d*x + c)^4)*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) + (b*co
sh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) + 13824*(((a^4 + a^2*b^2)*d^2*f^2*x^2 + 2*(a^4 +
a^2*b^2)*d^2*e*f*x + 2*(a^4 + a^2*b^2)*c*d*e*f - (a^4 + a^2*b^2)*c^2*f^2)*cosh(d*x + c)^4 + 4*((a^4 + a^2*b^2)
*d^2*f^2*x^2 + 2*(a^4 + a^2*b^2)*d^2*e*f*x + 2*(a^4 + a^2*b^2)*c*d*e*f - (a^4 + a^2*b^2)*c^2*f^2)*cosh(d*x + c
)^3*sinh(d*x + c) + 6*((a^4 + a^2*b^2)*d^2*f^2*x^2 + 2*(a^4 + a^2*b^2)*d^2*e*f*x + 2*(a^4 + a^2*b^2)*c*d*e*f -
(a^4 + a^2*b^2)*c^2*f^2)*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*((a^4 + a^2*b^2)*d^2*f^2*x^2 + 2*(a^4 + a^2*b^2)
*d^2*e*f*x + 2*(a^4 + a^2*b^2)*c*d*e*f - (a^4 + a^2*b^2)*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^4 + a^2*
b^2)*d^2*f^2*x^2 + 2*(a^4 + a^2*b^2)*d^2*e*f*x + 2*(a^4 + a^2*b^2)*c*d*e*f - (a^4 + a^2*b^2)*c^2*f^2)*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) - 27648*((a^4 + a^2*b^2)*f^2*cosh(d*x + c)^4 + 4*(a^4 + a^2*b^2)*f^2*cosh(d*x + c)^3*sinh(d*x + c) + 6*
(a^4 + a^2*b^2)*f^2*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*(a^4 + a^2*b^2)*f^2*cosh(d*x + c)*sinh(d*x + c)^3 + (a
^4 + a^2*b^2)*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) - 27648*((a^4 + a^2*b^2)*f^2*cosh(d*x + c)^4 + 4*(a^4 + a^2*b^2)*f^2*cosh(d
*x + c)^3*sinh(d*x + c) + 6*(a^4 + a^2*b^2)*f^2*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*(a^4 + a^2*b^2)*f^2*cosh(d
*x + c)*sinh(d*x + c)^3 + (a^4 + a^2*b^2)*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*(72*a*b^3*d^2*f^2*x^2 + 72*a*b^3*d^2*e^2 +
27*(8*b^4*d^2*f^2*x^2 + 8*b^4*d^2*e^2 - 4*b^4*d*e*f + b^4*f^2 + 4*(4*b^4*d^2*e*f - b^4*d*f^2)*x)*cosh(d*x + c)
^7 + 48*a*b^3*d*e*f - 56*(9*a*b^3*d^2*f^2*x^2 + 9*a*b^3*d^2*e^2 - 6*a*b^3*d*e*f + 2*a*b^3*f^2 + 6*(3*a*b^3*d^2
*e*f - a*b^3*d*f^2)*x)*cosh(d*x + c)^6 + 16*a*b^3*f^2 + 324*(2*(2*a^2*b^2 + b^4)*d^2*f^2*x^2 + 2*(2*a^2*b^2 +
b^4)*d^2*e^2 - 2*(2*a^2*b^2 + b^4)*d*e*f + (2*a^2*b^2 + b^4)*f^2 + 2*(2*(2*a^2*b^2 + b^4)*d^2*e*f - (2*a^2*b^2
+ b^4)*d*f^2)*x)*cosh(d*x + c)^5 - 1080*((4*a^3*b + 3*a*b^3)*d^2*f^2*x^2 + (4*a^3*b + 3*a*b^3)*d^2*e^2 - 2*(4
*a^3*b + 3*a*b^3)*d*e*f + 2*(4*a^3*b + 3*a*b^3)*f^2 + 2*((4*a^3*b + 3*a*b^3)*d^2*e*f - (4*a^3*b + 3*a*b^3)*d*f
^2)*x)*cosh(d*x + c)^4 - 2304*((a^4 + a^2*b^2)*d^3*f^2*x^3 + 3*(a^4 + a^2*b^2)*d^3*e*f*x^2 + 3*(a^4 + a^2*b^2)
*d^3*e^2*x + 6*(a^4 + a^2*b^2)*c*d^2*e^2 - 6*(a^4 + a^2*b^2)*c^2*d*e*f + 2*(a^4 + a^2*b^2)*c^3*f^2)*cosh(d*x +
c)^3 + 648*((4*a^3*b + 3*a*b^3)*d^2*f^2*x^2 + (4*a^3*b + 3*a*b^3)*d^2*e^2 + 2*(4*a^3*b + 3*a*b^3)*d*e*f + 2*(
4*a^3*b + 3*a*b^3)*f^2 + 2*((4*a^3*b + 3*a*b^3)*d^2*e*f + (4*a^3*b + 3*a*b^3)*d*f^2)*x)*cosh(d*x + c)^2 + 48*(
3*a*b^3*d^2*e*f + a*b^3*d*f^2)*x + 108*(2*(2*a^2*b^2 + b^4)*d^2*f^2*x^2 + 2*(2*a^2*b^2 + b^4)*d^2*e^2 + 2*(2*a
^2*b^2 + b^4)*d*e*f + (2*a^2*b^2 + b^4)*f^2 + 2*(2*(2*a^2*b^2 + b^4)*d^2*e*f + (2*a^2*b^2 + b^4)*d*f^2)*x)*cos
h(d*x + c))*sinh(d*x + c))/(b^5*d^3*cosh(d*x + c)^4 + 4*b^5*d^3*cosh(d*x + c)^3*sinh(d*x + c) + 6*b^5*d^3*cosh
(d*x + c)^2*sinh(d*x + c)^2 + 4*b^5*d^3*cosh(d*x + c)*sinh(d*x + c)^3 + b^5*d^3*sinh(d*x + c)^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (f x + e\right )}^{2} \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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