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

Optimal. Leaf size=649 \[ \frac{2 a^2 f \sqrt{a^2+b^2} (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^4 d^2}-\frac{2 a^2 f \sqrt{a^2+b^2} (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^4 d^2}-\frac{2 a^2 f^2 \sqrt{a^2+b^2} \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^4 d^3}+\frac{2 a^2 f^2 \sqrt{a^2+b^2} \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^4 d^3}-\frac{2 a^2 f (e+f x) \sinh (c+d x)}{b^3 d^2}+\frac{2 a^2 f^2 \cosh (c+d x)}{b^3 d^3}+\frac{a^2 \sqrt{a^2+b^2} (e+f x)^2 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b^4 d}-\frac{a^2 \sqrt{a^2+b^2} (e+f x)^2 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b^4 d}+\frac{a^2 (e+f x)^2 \cosh (c+d x)}{b^3 d}-\frac{a^3 (e+f x)^3}{3 b^4 f}+\frac{a f (e+f x) \cosh ^2(c+d x)}{2 b^2 d^2}-\frac{a f^2 \sinh (c+d x) \cosh (c+d x)}{4 b^2 d^3}-\frac{a (e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 b^2 d}-\frac{a f^2 x}{4 b^2 d^2}-\frac{a (e+f x)^3}{6 b^2 f}-\frac{4 f (e+f x) \sinh (c+d x)}{9 b d^2}-\frac{2 f (e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{9 b d^2}+\frac{2 f^2 \cosh ^3(c+d x)}{27 b d^3}+\frac{4 f^2 \cosh (c+d x)}{9 b d^3}+\frac{(e+f x)^2 \cosh ^3(c+d x)}{3 b d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.20186, antiderivative size = 649, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 16, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {5579, 5447, 3310, 3296, 2638, 3311, 32, 2635, 8, 5565, 3322, 2264, 2190, 2531, 2282, 6589} \[ \frac{2 a^2 f \sqrt{a^2+b^2} (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^4 d^2}-\frac{2 a^2 f \sqrt{a^2+b^2} (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^4 d^2}-\frac{2 a^2 f^2 \sqrt{a^2+b^2} \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^4 d^3}+\frac{2 a^2 f^2 \sqrt{a^2+b^2} \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^4 d^3}-\frac{2 a^2 f (e+f x) \sinh (c+d x)}{b^3 d^2}+\frac{2 a^2 f^2 \cosh (c+d x)}{b^3 d^3}+\frac{a^2 \sqrt{a^2+b^2} (e+f x)^2 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b^4 d}-\frac{a^2 \sqrt{a^2+b^2} (e+f x)^2 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b^4 d}+\frac{a^2 (e+f x)^2 \cosh (c+d x)}{b^3 d}-\frac{a^3 (e+f x)^3}{3 b^4 f}+\frac{a f (e+f x) \cosh ^2(c+d x)}{2 b^2 d^2}-\frac{a f^2 \sinh (c+d x) \cosh (c+d x)}{4 b^2 d^3}-\frac{a (e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 b^2 d}-\frac{a f^2 x}{4 b^2 d^2}-\frac{a (e+f x)^3}{6 b^2 f}-\frac{4 f (e+f x) \sinh (c+d x)}{9 b d^2}-\frac{2 f (e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{9 b d^2}+\frac{2 f^2 \cosh ^3(c+d x)}{27 b d^3}+\frac{4 f^2 \cosh (c+d x)}{9 b d^3}+\frac{(e+f x)^2 \cosh ^3(c+d x)}{3 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 2638

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

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 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 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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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 3322

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Dist[2,
Int[((c + d*x)^m*E^(-(I*e) + f*fz*x))/(-(I*b) + 2*a*E^(-(I*e) + f*fz*x) + I*b*E^(2*(-(I*e) + f*fz*x))), x], x]
/; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 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 ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac{\int (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x) \, dx}{b}-\frac{a \int \frac{(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=\frac{(e+f x)^2 \cosh ^3(c+d x)}{3 b d}-\frac{a \int (e+f x)^2 \cosh ^2(c+d x) \, dx}{b^2}+\frac{a^2 \int \frac{(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}-\frac{(2 f) \int (e+f x) \cosh ^3(c+d x) \, dx}{3 b d}\\ &=\frac{a f (e+f x) \cosh ^2(c+d x)}{2 b^2 d^2}+\frac{2 f^2 \cosh ^3(c+d x)}{27 b d^3}+\frac{(e+f x)^2 \cosh ^3(c+d x)}{3 b d}-\frac{a (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}-\frac{2 f (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b d^2}-\frac{a^3 \int (e+f x)^2 \, dx}{b^4}+\frac{a^2 \int (e+f x)^2 \sinh (c+d x) \, dx}{b^3}-\frac{a \int (e+f x)^2 \, dx}{2 b^2}+\frac{\left (a^2 \left (a^2+b^2\right )\right ) \int \frac{(e+f x)^2}{a+b \sinh (c+d x)} \, dx}{b^4}-\frac{(4 f) \int (e+f x) \cosh (c+d x) \, dx}{9 b d}-\frac{\left (a f^2\right ) \int \cosh ^2(c+d x) \, dx}{2 b^2 d^2}\\ &=-\frac{a^3 (e+f x)^3}{3 b^4 f}-\frac{a (e+f x)^3}{6 b^2 f}+\frac{a^2 (e+f x)^2 \cosh (c+d x)}{b^3 d}+\frac{a f (e+f x) \cosh ^2(c+d x)}{2 b^2 d^2}+\frac{2 f^2 \cosh ^3(c+d x)}{27 b d^3}+\frac{(e+f x)^2 \cosh ^3(c+d x)}{3 b d}-\frac{4 f (e+f x) \sinh (c+d x)}{9 b d^2}-\frac{a f^2 \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^3}-\frac{a (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}-\frac{2 f (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b d^2}+\frac{\left (2 a^2 \left (a^2+b^2\right )\right ) \int \frac{e^{c+d x} (e+f x)^2}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{b^4}-\frac{\left (2 a^2 f\right ) \int (e+f x) \cosh (c+d x) \, dx}{b^3 d}-\frac{\left (a f^2\right ) \int 1 \, dx}{4 b^2 d^2}+\frac{\left (4 f^2\right ) \int \sinh (c+d x) \, dx}{9 b d^2}\\ &=-\frac{a f^2 x}{4 b^2 d^2}-\frac{a^3 (e+f x)^3}{3 b^4 f}-\frac{a (e+f x)^3}{6 b^2 f}+\frac{4 f^2 \cosh (c+d x)}{9 b d^3}+\frac{a^2 (e+f x)^2 \cosh (c+d x)}{b^3 d}+\frac{a f (e+f x) \cosh ^2(c+d x)}{2 b^2 d^2}+\frac{2 f^2 \cosh ^3(c+d x)}{27 b d^3}+\frac{(e+f x)^2 \cosh ^3(c+d x)}{3 b d}-\frac{2 a^2 f (e+f x) \sinh (c+d x)}{b^3 d^2}-\frac{4 f (e+f x) \sinh (c+d x)}{9 b d^2}-\frac{a f^2 \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^3}-\frac{a (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}-\frac{2 f (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b d^2}+\frac{\left (2 a^2 \sqrt{a^2+b^2}\right ) \int \frac{e^{c+d x} (e+f x)^2}{2 a-2 \sqrt{a^2+b^2}+2 b e^{c+d x}} \, dx}{b^3}-\frac{\left (2 a^2 \sqrt{a^2+b^2}\right ) \int \frac{e^{c+d x} (e+f x)^2}{2 a+2 \sqrt{a^2+b^2}+2 b e^{c+d x}} \, dx}{b^3}+\frac{\left (2 a^2 f^2\right ) \int \sinh (c+d x) \, dx}{b^3 d^2}\\ &=-\frac{a f^2 x}{4 b^2 d^2}-\frac{a^3 (e+f x)^3}{3 b^4 f}-\frac{a (e+f x)^3}{6 b^2 f}+\frac{2 a^2 f^2 \cosh (c+d x)}{b^3 d^3}+\frac{4 f^2 \cosh (c+d x)}{9 b d^3}+\frac{a^2 (e+f x)^2 \cosh (c+d x)}{b^3 d}+\frac{a f (e+f x) \cosh ^2(c+d x)}{2 b^2 d^2}+\frac{2 f^2 \cosh ^3(c+d x)}{27 b d^3}+\frac{(e+f x)^2 \cosh ^3(c+d x)}{3 b d}+\frac{a^2 \sqrt{a^2+b^2} (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^4 d}-\frac{a^2 \sqrt{a^2+b^2} (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^4 d}-\frac{2 a^2 f (e+f x) \sinh (c+d x)}{b^3 d^2}-\frac{4 f (e+f x) \sinh (c+d x)}{9 b d^2}-\frac{a f^2 \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^3}-\frac{a (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}-\frac{2 f (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b d^2}-\frac{\left (2 a^2 \sqrt{a^2+b^2} f\right ) \int (e+f x) \log \left (1+\frac{2 b e^{c+d x}}{2 a-2 \sqrt{a^2+b^2}}\right ) \, dx}{b^4 d}+\frac{\left (2 a^2 \sqrt{a^2+b^2} f\right ) \int (e+f x) \log \left (1+\frac{2 b e^{c+d x}}{2 a+2 \sqrt{a^2+b^2}}\right ) \, dx}{b^4 d}\\ &=-\frac{a f^2 x}{4 b^2 d^2}-\frac{a^3 (e+f x)^3}{3 b^4 f}-\frac{a (e+f x)^3}{6 b^2 f}+\frac{2 a^2 f^2 \cosh (c+d x)}{b^3 d^3}+\frac{4 f^2 \cosh (c+d x)}{9 b d^3}+\frac{a^2 (e+f x)^2 \cosh (c+d x)}{b^3 d}+\frac{a f (e+f x) \cosh ^2(c+d x)}{2 b^2 d^2}+\frac{2 f^2 \cosh ^3(c+d x)}{27 b d^3}+\frac{(e+f x)^2 \cosh ^3(c+d x)}{3 b d}+\frac{a^2 \sqrt{a^2+b^2} (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^4 d}-\frac{a^2 \sqrt{a^2+b^2} (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^4 d}+\frac{2 a^2 \sqrt{a^2+b^2} f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^4 d^2}-\frac{2 a^2 \sqrt{a^2+b^2} f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^4 d^2}-\frac{2 a^2 f (e+f x) \sinh (c+d x)}{b^3 d^2}-\frac{4 f (e+f x) \sinh (c+d x)}{9 b d^2}-\frac{a f^2 \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^3}-\frac{a (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}-\frac{2 f (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b d^2}-\frac{\left (2 a^2 \sqrt{a^2+b^2} f^2\right ) \int \text{Li}_2\left (-\frac{2 b e^{c+d x}}{2 a-2 \sqrt{a^2+b^2}}\right ) \, dx}{b^4 d^2}+\frac{\left (2 a^2 \sqrt{a^2+b^2} f^2\right ) \int \text{Li}_2\left (-\frac{2 b e^{c+d x}}{2 a+2 \sqrt{a^2+b^2}}\right ) \, dx}{b^4 d^2}\\ &=-\frac{a f^2 x}{4 b^2 d^2}-\frac{a^3 (e+f x)^3}{3 b^4 f}-\frac{a (e+f x)^3}{6 b^2 f}+\frac{2 a^2 f^2 \cosh (c+d x)}{b^3 d^3}+\frac{4 f^2 \cosh (c+d x)}{9 b d^3}+\frac{a^2 (e+f x)^2 \cosh (c+d x)}{b^3 d}+\frac{a f (e+f x) \cosh ^2(c+d x)}{2 b^2 d^2}+\frac{2 f^2 \cosh ^3(c+d x)}{27 b d^3}+\frac{(e+f x)^2 \cosh ^3(c+d x)}{3 b d}+\frac{a^2 \sqrt{a^2+b^2} (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^4 d}-\frac{a^2 \sqrt{a^2+b^2} (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^4 d}+\frac{2 a^2 \sqrt{a^2+b^2} f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^4 d^2}-\frac{2 a^2 \sqrt{a^2+b^2} f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^4 d^2}-\frac{2 a^2 f (e+f x) \sinh (c+d x)}{b^3 d^2}-\frac{4 f (e+f x) \sinh (c+d x)}{9 b d^2}-\frac{a f^2 \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^3}-\frac{a (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}-\frac{2 f (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b d^2}-\frac{\left (2 a^2 \sqrt{a^2+b^2} 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^4 d^3}+\frac{\left (2 a^2 \sqrt{a^2+b^2} 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^4 d^3}\\ &=-\frac{a f^2 x}{4 b^2 d^2}-\frac{a^3 (e+f x)^3}{3 b^4 f}-\frac{a (e+f x)^3}{6 b^2 f}+\frac{2 a^2 f^2 \cosh (c+d x)}{b^3 d^3}+\frac{4 f^2 \cosh (c+d x)}{9 b d^3}+\frac{a^2 (e+f x)^2 \cosh (c+d x)}{b^3 d}+\frac{a f (e+f x) \cosh ^2(c+d x)}{2 b^2 d^2}+\frac{2 f^2 \cosh ^3(c+d x)}{27 b d^3}+\frac{(e+f x)^2 \cosh ^3(c+d x)}{3 b d}+\frac{a^2 \sqrt{a^2+b^2} (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^4 d}-\frac{a^2 \sqrt{a^2+b^2} (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^4 d}+\frac{2 a^2 \sqrt{a^2+b^2} f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^4 d^2}-\frac{2 a^2 \sqrt{a^2+b^2} f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^4 d^2}-\frac{2 a^2 \sqrt{a^2+b^2} f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^4 d^3}+\frac{2 a^2 \sqrt{a^2+b^2} f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^4 d^3}-\frac{2 a^2 f (e+f x) \sinh (c+d x)}{b^3 d^2}-\frac{4 f (e+f x) \sinh (c+d x)}{9 b d^2}-\frac{a f^2 \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^3}-\frac{a (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}-\frac{2 f (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b d^2}\\ \end{align*}

Mathematica [A]  time = 4.98927, size = 966, normalized size = 1.49 \[ -\frac{-54 d^2 e^2 \cosh (c+d x) b^3-108 f^2 \cosh (c+d x) b^3-54 d^2 f^2 x^2 \cosh (c+d x) b^3-108 d^2 e f x \cosh (c+d x) b^3-18 d^2 e^2 \cosh (3 (c+d x)) b^3-4 f^2 \cosh (3 (c+d x)) b^3-18 d^2 f^2 x^2 \cosh (3 (c+d x)) b^3-36 d^2 e f x \cosh (3 (c+d x)) b^3+108 d e f \sinh (c+d x) b^3+108 d f^2 x \sinh (c+d x) b^3+12 d e f \sinh (3 (c+d x)) b^3+12 d f^2 x \sinh (3 (c+d x)) b^3+36 a d^3 f^2 x^3 b^2+108 a d^3 e f x^2 b^2+108 a d^3 e^2 x b^2-54 a d e f \cosh (2 (c+d x)) b^2-54 a d f^2 x \cosh (2 (c+d x)) b^2+54 a d^2 e^2 \sinh (2 (c+d x)) b^2+27 a f^2 \sinh (2 (c+d x)) b^2+54 a d^2 f^2 x^2 \sinh (2 (c+d x)) b^2+108 a d^2 e f x \sinh (2 (c+d x)) b^2-216 a^2 d^2 e^2 \cosh (c+d x) b-432 a^2 f^2 \cosh (c+d x) b-216 a^2 d^2 f^2 x^2 \cosh (c+d x) b-432 a^2 d^2 e f x \cosh (c+d x) b+432 a^2 d e f \sinh (c+d x) b+432 a^2 d f^2 x \sinh (c+d x) b+72 a^3 d^3 f^2 x^3+216 a^3 d^3 e f x^2+216 a^3 d^3 e^2 x+432 a^2 \sqrt{a^2+b^2} d^2 e^2 \tanh ^{-1}\left (\frac{a+b e^{c+d x}}{\sqrt{a^2+b^2}}\right )-216 a^2 \sqrt{a^2+b^2} d^2 f^2 x^2 \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right )-432 a^2 \sqrt{a^2+b^2} d^2 e f x \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right )+216 a^2 \sqrt{a^2+b^2} d^2 f^2 x^2 \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right )+432 a^2 \sqrt{a^2+b^2} d^2 e f x \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right )-432 a^2 \sqrt{a^2+b^2} d f (e+f x) \text{PolyLog}\left (2,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )+432 a^2 \sqrt{a^2+b^2} d f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )+432 a^2 \sqrt{a^2+b^2} f^2 \text{PolyLog}\left (3,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )-432 a^2 \sqrt{a^2+b^2} f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{216 b^4 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(216*a^3*d^3*e^2*x + 108*a*b^2*d^3*e^2*x + 216*a^3*d^3*e*f*x^2 + 108*a*b^2*d^3*e*f*x^2 + 72*a^3*d^3*f^2*x^3 +
36*a*b^2*d^3*f^2*x^3 + 432*a^2*Sqrt[a^2 + b^2]*d^2*e^2*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] - 216*a^2
*b*d^2*e^2*Cosh[c + d*x] - 54*b^3*d^2*e^2*Cosh[c + d*x] - 432*a^2*b*f^2*Cosh[c + d*x] - 108*b^3*f^2*Cosh[c + d
*x] - 432*a^2*b*d^2*e*f*x*Cosh[c + d*x] - 108*b^3*d^2*e*f*x*Cosh[c + d*x] - 216*a^2*b*d^2*f^2*x^2*Cosh[c + d*x
] - 54*b^3*d^2*f^2*x^2*Cosh[c + d*x] - 54*a*b^2*d*e*f*Cosh[2*(c + d*x)] - 54*a*b^2*d*f^2*x*Cosh[2*(c + d*x)] -
18*b^3*d^2*e^2*Cosh[3*(c + d*x)] - 4*b^3*f^2*Cosh[3*(c + d*x)] - 36*b^3*d^2*e*f*x*Cosh[3*(c + d*x)] - 18*b^3*
d^2*f^2*x^2*Cosh[3*(c + d*x)] - 432*a^2*Sqrt[a^2 + b^2]*d^2*e*f*x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]
)] - 216*a^2*Sqrt[a^2 + b^2]*d^2*f^2*x^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 432*a^2*Sqrt[a^2 + b
^2]*d^2*e*f*x*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 216*a^2*Sqrt[a^2 + b^2]*d^2*f^2*x^2*Log[1 + (b*
E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - 432*a^2*Sqrt[a^2 + b^2]*d*f*(e + f*x)*PolyLog[2, (b*E^(c + d*x))/(-a + S
qrt[a^2 + b^2])] + 432*a^2*Sqrt[a^2 + b^2]*d*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]
+ 432*a^2*Sqrt[a^2 + b^2]*f^2*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 432*a^2*Sqrt[a^2 + b^2]*f^2
*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] + 432*a^2*b*d*e*f*Sinh[c + d*x] + 108*b^3*d*e*f*Sinh[c +
d*x] + 432*a^2*b*d*f^2*x*Sinh[c + d*x] + 108*b^3*d*f^2*x*Sinh[c + d*x] + 54*a*b^2*d^2*e^2*Sinh[2*(c + d*x)] +
27*a*b^2*f^2*Sinh[2*(c + d*x)] + 108*a*b^2*d^2*e*f*x*Sinh[2*(c + d*x)] + 54*a*b^2*d^2*f^2*x^2*Sinh[2*(c + d*x
)] + 12*b^3*d*e*f*Sinh[3*(c + d*x)] + 12*b^3*d*f^2*x*Sinh[3*(c + d*x)])/(216*b^4*d^3)

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Maple [F]  time = 0.228, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{2} \left ( \cosh \left ( dx+c \right ) \right ) ^{2} \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)^2*sinh(d*x+c)^2/(a+b*sinh(d*x+c)),x)

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [C]  time = 3.49251, size = 9708, normalized size = 14.96 \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)^2*sinh(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

1/432*(18*b^3*d^2*f^2*x^2 + 18*b^3*d^2*e^2 + 2*(9*b^3*d^2*f^2*x^2 + 9*b^3*d^2*e^2 - 6*b^3*d*e*f + 2*b^3*f^2 +
6*(3*b^3*d^2*e*f - b^3*d*f^2)*x)*cosh(d*x + c)^6 + 2*(9*b^3*d^2*f^2*x^2 + 9*b^3*d^2*e^2 - 6*b^3*d*e*f + 2*b^3*
f^2 + 6*(3*b^3*d^2*e*f - b^3*d*f^2)*x)*sinh(d*x + c)^6 + 12*b^3*d*e*f - 27*(2*a*b^2*d^2*f^2*x^2 + 2*a*b^2*d^2*
e^2 - 2*a*b^2*d*e*f + a*b^2*f^2 + 2*(2*a*b^2*d^2*e*f - a*b^2*d*f^2)*x)*cosh(d*x + c)^5 - 3*(18*a*b^2*d^2*f^2*x
^2 + 18*a*b^2*d^2*e^2 - 18*a*b^2*d*e*f + 9*a*b^2*f^2 + 18*(2*a*b^2*d^2*e*f - a*b^2*d*f^2)*x - 4*(9*b^3*d^2*f^2
*x^2 + 9*b^3*d^2*e^2 - 6*b^3*d*e*f + 2*b^3*f^2 + 6*(3*b^3*d^2*e*f - b^3*d*f^2)*x)*cosh(d*x + c))*sinh(d*x + c)
^5 + 4*b^3*f^2 + 54*((4*a^2*b + b^3)*d^2*f^2*x^2 + (4*a^2*b + b^3)*d^2*e^2 - 2*(4*a^2*b + b^3)*d*e*f + 2*(4*a^
2*b + b^3)*f^2 + 2*((4*a^2*b + b^3)*d^2*e*f - (4*a^2*b + b^3)*d*f^2)*x)*cosh(d*x + c)^4 + 3*(18*(4*a^2*b + b^3
)*d^2*f^2*x^2 + 18*(4*a^2*b + b^3)*d^2*e^2 - 36*(4*a^2*b + b^3)*d*e*f + 36*(4*a^2*b + b^3)*f^2 + 10*(9*b^3*d^2
*f^2*x^2 + 9*b^3*d^2*e^2 - 6*b^3*d*e*f + 2*b^3*f^2 + 6*(3*b^3*d^2*e*f - b^3*d*f^2)*x)*cosh(d*x + c)^2 + 36*((4
*a^2*b + b^3)*d^2*e*f - (4*a^2*b + b^3)*d*f^2)*x - 45*(2*a*b^2*d^2*f^2*x^2 + 2*a*b^2*d^2*e^2 - 2*a*b^2*d*e*f +
a*b^2*f^2 + 2*(2*a*b^2*d^2*e*f - a*b^2*d*f^2)*x)*cosh(d*x + c))*sinh(d*x + c)^4 - 72*((2*a^3 + a*b^2)*d^3*f^2
*x^3 + 3*(2*a^3 + a*b^2)*d^3*e*f*x^2 + 3*(2*a^3 + a*b^2)*d^3*e^2*x)*cosh(d*x + c)^3 - 2*(36*(2*a^3 + a*b^2)*d^
3*f^2*x^3 + 108*(2*a^3 + a*b^2)*d^3*e*f*x^2 + 108*(2*a^3 + a*b^2)*d^3*e^2*x - 20*(9*b^3*d^2*f^2*x^2 + 9*b^3*d^
2*e^2 - 6*b^3*d*e*f + 2*b^3*f^2 + 6*(3*b^3*d^2*e*f - b^3*d*f^2)*x)*cosh(d*x + c)^3 + 135*(2*a*b^2*d^2*f^2*x^2
+ 2*a*b^2*d^2*e^2 - 2*a*b^2*d*e*f + a*b^2*f^2 + 2*(2*a*b^2*d^2*e*f - a*b^2*d*f^2)*x)*cosh(d*x + c)^2 - 108*((4
*a^2*b + b^3)*d^2*f^2*x^2 + (4*a^2*b + b^3)*d^2*e^2 - 2*(4*a^2*b + b^3)*d*e*f + 2*(4*a^2*b + b^3)*f^2 + 2*((4*
a^2*b + b^3)*d^2*e*f - (4*a^2*b + b^3)*d*f^2)*x)*cosh(d*x + c))*sinh(d*x + c)^3 + 54*((4*a^2*b + b^3)*d^2*f^2*
x^2 + (4*a^2*b + b^3)*d^2*e^2 + 2*(4*a^2*b + b^3)*d*e*f + 2*(4*a^2*b + b^3)*f^2 + 2*((4*a^2*b + b^3)*d^2*e*f +
(4*a^2*b + b^3)*d*f^2)*x)*cosh(d*x + c)^2 + 6*(9*(4*a^2*b + b^3)*d^2*f^2*x^2 + 9*(4*a^2*b + b^3)*d^2*e^2 + 5*
(9*b^3*d^2*f^2*x^2 + 9*b^3*d^2*e^2 - 6*b^3*d*e*f + 2*b^3*f^2 + 6*(3*b^3*d^2*e*f - b^3*d*f^2)*x)*cosh(d*x + c)^
4 + 18*(4*a^2*b + b^3)*d*e*f - 45*(2*a*b^2*d^2*f^2*x^2 + 2*a*b^2*d^2*e^2 - 2*a*b^2*d*e*f + a*b^2*f^2 + 2*(2*a*
b^2*d^2*e*f - a*b^2*d*f^2)*x)*cosh(d*x + c)^3 + 18*(4*a^2*b + b^3)*f^2 + 54*((4*a^2*b + b^3)*d^2*f^2*x^2 + (4*
a^2*b + b^3)*d^2*e^2 - 2*(4*a^2*b + b^3)*d*e*f + 2*(4*a^2*b + b^3)*f^2 + 2*((4*a^2*b + b^3)*d^2*e*f - (4*a^2*b
+ b^3)*d*f^2)*x)*cosh(d*x + c)^2 + 18*((4*a^2*b + b^3)*d^2*e*f + (4*a^2*b + b^3)*d*f^2)*x - 36*((2*a^3 + a*b^
2)*d^3*f^2*x^3 + 3*(2*a^3 + a*b^2)*d^3*e*f*x^2 + 3*(2*a^3 + a*b^2)*d^3*e^2*x)*cosh(d*x + c))*sinh(d*x + c)^2 +
864*((a^2*b*d*f^2*x + a^2*b*d*e*f)*cosh(d*x + c)^3 + 3*(a^2*b*d*f^2*x + a^2*b*d*e*f)*cosh(d*x + c)^2*sinh(d*x
+ c) + 3*(a^2*b*d*f^2*x + a^2*b*d*e*f)*cosh(d*x + c)*sinh(d*x + c)^2 + (a^2*b*d*f^2*x + a^2*b*d*e*f)*sinh(d*x
+ c)^3)*sqrt((a^2 + b^2)/b^2)*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) - 864*((a^2*b*d*f^2*x + a^2*b*d*e*f)*cosh(d*x + c)^3 + 3*(a^2*b*d*f^2*x + a^
2*b*d*e*f)*cosh(d*x + c)^2*sinh(d*x + c) + 3*(a^2*b*d*f^2*x + a^2*b*d*e*f)*cosh(d*x + c)*sinh(d*x + c)^2 + (a^
2*b*d*f^2*x + a^2*b*d*e*f)*sinh(d*x + c)^3)*sqrt((a^2 + b^2)/b^2)*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) - 432*((a^2*b*d^2*e^2 - 2*a^2*b*c*d*e*f +
a^2*b*c^2*f^2)*cosh(d*x + c)^3 + 3*(a^2*b*d^2*e^2 - 2*a^2*b*c*d*e*f + a^2*b*c^2*f^2)*cosh(d*x + c)^2*sinh(d*x
+ c) + 3*(a^2*b*d^2*e^2 - 2*a^2*b*c*d*e*f + a^2*b*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^2 + (a^2*b*d^2*e^2 - 2
*a^2*b*c*d*e*f + a^2*b*c^2*f^2)*sinh(d*x + c)^3)*sqrt((a^2 + b^2)/b^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x +
c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + 432*((a^2*b*d^2*e^2 - 2*a^2*b*c*d*e*f + a^2*b*c^2*f^2)*cosh(d*x + c)^3
+ 3*(a^2*b*d^2*e^2 - 2*a^2*b*c*d*e*f + a^2*b*c^2*f^2)*cosh(d*x + c)^2*sinh(d*x + c) + 3*(a^2*b*d^2*e^2 - 2*a^
2*b*c*d*e*f + a^2*b*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^2 + (a^2*b*d^2*e^2 - 2*a^2*b*c*d*e*f + a^2*b*c^2*f^2)
*sinh(d*x + c)^3)*sqrt((a^2 + b^2)/b^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2)
+ 2*a) + 432*((a^2*b*d^2*f^2*x^2 + 2*a^2*b*d^2*e*f*x + 2*a^2*b*c*d*e*f - a^2*b*c^2*f^2)*cosh(d*x + c)^3 + 3*(a
^2*b*d^2*f^2*x^2 + 2*a^2*b*d^2*e*f*x + 2*a^2*b*c*d*e*f - a^2*b*c^2*f^2)*cosh(d*x + c)^2*sinh(d*x + c) + 3*(a^2
*b*d^2*f^2*x^2 + 2*a^2*b*d^2*e*f*x + 2*a^2*b*c*d*e*f - a^2*b*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^2 + (a^2*b*d
^2*f^2*x^2 + 2*a^2*b*d^2*e*f*x + 2*a^2*b*c*d*e*f - a^2*b*c^2*f^2)*sinh(d*x + c)^3)*sqrt((a^2 + b^2)/b^2)*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) - 432*(
(a^2*b*d^2*f^2*x^2 + 2*a^2*b*d^2*e*f*x + 2*a^2*b*c*d*e*f - a^2*b*c^2*f^2)*cosh(d*x + c)^3 + 3*(a^2*b*d^2*f^2*x
^2 + 2*a^2*b*d^2*e*f*x + 2*a^2*b*c*d*e*f - a^2*b*c^2*f^2)*cosh(d*x + c)^2*sinh(d*x + c) + 3*(a^2*b*d^2*f^2*x^2
+ 2*a^2*b*d^2*e*f*x + 2*a^2*b*c*d*e*f - a^2*b*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^2 + (a^2*b*d^2*f^2*x^2 + 2
*a^2*b*d^2*e*f*x + 2*a^2*b*c*d*e*f - a^2*b*c^2*f^2)*sinh(d*x + c)^3)*sqrt((a^2 + b^2)/b^2)*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) - 864*(a^2*b*f^2*cosh
(d*x + c)^3 + 3*a^2*b*f^2*cosh(d*x + c)^2*sinh(d*x + c) + 3*a^2*b*f^2*cosh(d*x + c)*sinh(d*x + c)^2 + a^2*b*f^
2*sinh(d*x + c)^3)*sqrt((a^2 + b^2)/b^2)*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) + 864*(a^2*b*f^2*cosh(d*x + c)^3 + 3*a^2*b*f^2*cosh(d*x + c)^2*sinh(d
*x + c) + 3*a^2*b*f^2*cosh(d*x + c)*sinh(d*x + c)^2 + a^2*b*f^2*sinh(d*x + c)^3)*sqrt((a^2 + b^2)/b^2)*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) + 12*(3
*b^3*d^2*e*f + b^3*d*f^2)*x + 27*(2*a*b^2*d^2*f^2*x^2 + 2*a*b^2*d^2*e^2 + 2*a*b^2*d*e*f + a*b^2*f^2 + 2*(2*a*b
^2*d^2*e*f + a*b^2*d*f^2)*x)*cosh(d*x + c) + 3*(18*a*b^2*d^2*f^2*x^2 + 18*a*b^2*d^2*e^2 + 18*a*b^2*d*e*f + 4*(
9*b^3*d^2*f^2*x^2 + 9*b^3*d^2*e^2 - 6*b^3*d*e*f + 2*b^3*f^2 + 6*(3*b^3*d^2*e*f - b^3*d*f^2)*x)*cosh(d*x + c)^5
+ 9*a*b^2*f^2 - 45*(2*a*b^2*d^2*f^2*x^2 + 2*a*b^2*d^2*e^2 - 2*a*b^2*d*e*f + a*b^2*f^2 + 2*(2*a*b^2*d^2*e*f -
a*b^2*d*f^2)*x)*cosh(d*x + c)^4 + 72*((4*a^2*b + b^3)*d^2*f^2*x^2 + (4*a^2*b + b^3)*d^2*e^2 - 2*(4*a^2*b + b^3
)*d*e*f + 2*(4*a^2*b + b^3)*f^2 + 2*((4*a^2*b + b^3)*d^2*e*f - (4*a^2*b + b^3)*d*f^2)*x)*cosh(d*x + c)^3 - 72*
((2*a^3 + a*b^2)*d^3*f^2*x^3 + 3*(2*a^3 + a*b^2)*d^3*e*f*x^2 + 3*(2*a^3 + a*b^2)*d^3*e^2*x)*cosh(d*x + c)^2 +
18*(2*a*b^2*d^2*e*f + a*b^2*d*f^2)*x + 36*((4*a^2*b + b^3)*d^2*f^2*x^2 + (4*a^2*b + b^3)*d^2*e^2 + 2*(4*a^2*b
+ b^3)*d*e*f + 2*(4*a^2*b + b^3)*f^2 + 2*((4*a^2*b + b^3)*d^2*e*f + (4*a^2*b + b^3)*d*f^2)*x)*cosh(d*x + c))*s
inh(d*x + c))/(b^4*d^3*cosh(d*x + c)^3 + 3*b^4*d^3*cosh(d*x + c)^2*sinh(d*x + c) + 3*b^4*d^3*cosh(d*x + c)*sin
h(d*x + c)^2 + b^4*d^3*sinh(d*x + c)^3)

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

[Out]

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