\(\int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [368]

Optimal result

Integrand size = 36, antiderivative size = 649 \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a f^2 x}{4 b^2 d^2}-\frac {a^3 (e+f x)^3}{3 b^4 f}-\frac {a (e+f x)^3}{6 b^2 f}+\frac {2 a^2 f^2 \cosh (c+d x)}{b^3 d^3}+\frac {4 f^2 \cosh (c+d x)}{9 b d^3}+\frac {a^2 (e+f x)^2 \cosh (c+d x)}{b^3 d}+\frac {a f (e+f x) \cosh ^2(c+d x)}{2 b^2 d^2}+\frac {2 f^2 \cosh ^3(c+d x)}{27 b d^3}+\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 b d}+\frac {a^2 \sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {a^2 \sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d}+\frac {2 a^2 \sqrt {a^2+b^2} f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {2 a^2 \sqrt {a^2+b^2} f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {2 a^2 \sqrt {a^2+b^2} f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^3}+\frac {2 a^2 \sqrt {a^2+b^2} f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^3}-\frac {2 a^2 f (e+f x) \sinh (c+d x)}{b^3 d^2}-\frac {4 f (e+f x) \sinh (c+d x)}{9 b d^2}-\frac {a f^2 \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^3}-\frac {a (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}-\frac {2 f (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 649, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {5698, 5555, 3391, 3377, 2718, 3392, 32, 2715, 8, 5684, 3403, 2296, 2221, 2611, 2320, 6724} \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a^3 (e+f x)^3}{3 b^4 f}+\frac {2 a^2 f^2 \cosh (c+d x)}{b^3 d^3}-\frac {2 a^2 f (e+f x) \sinh (c+d x)}{b^3 d^2}+\frac {a^2 (e+f x)^2 \cosh (c+d x)}{b^3 d}-\frac {2 a^2 f^2 \sqrt {a^2+b^2} \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^3}+\frac {2 a^2 f^2 \sqrt {a^2+b^2} \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^3}+\frac {2 a^2 f \sqrt {a^2+b^2} (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {2 a^2 f \sqrt {a^2+b^2} (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^2}+\frac {a^2 \sqrt {a^2+b^2} (e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b^4 d}-\frac {a^2 \sqrt {a^2+b^2} (e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b^4 d}-\frac {a f^2 \sinh (c+d x) \cosh (c+d x)}{4 b^2 d^3}+\frac {a f (e+f x) \cosh ^2(c+d x)}{2 b^2 d^2}-\frac {a (e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 b^2 d}-\frac {a f^2 x}{4 b^2 d^2}-\frac {a (e+f x)^3}{6 b^2 f}+\frac {2 f^2 \cosh ^3(c+d x)}{27 b d^3}+\frac {4 f^2 \cosh (c+d x)}{9 b d^3}-\frac {4 f (e+f x) \sinh (c+d x)}{9 b d^2}-\frac {2 f (e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{9 b d^2}+\frac {(e+f x)^2 \cosh ^3(c+d x)}{3 b d} \]

[In]

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

[Out]

-1/4*(a*f^2*x)/(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*Sqr
t[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 - Sqr
t[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*Cos
h[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)
*Cosh[c + d*x]^2*Sinh[c + d*x])/(9*b*d^2)

Rule 8

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

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2296

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 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2718

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

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3391

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*((b*Sin[e + f*x])^n/(f^2*n^
2)), x] + (Dist[b^2*((n - 1)/n), Int[(c + d*x)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[b*(c + d*x)*Cos[e + f*x
]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]

Rule 3392

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*m*(c + d*x)^(m - 1)*((
b*Sin[e + f*x])^n/(f^2*n^2)), x] + (Dist[b^2*((n - 1)/n), Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[d^2*m*((m - 1)/(f^2*n^2)), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[b*(c + d*x)^m*Cos[e + f
*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]

Rule 3403

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 5555

Int[Cosh[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[(c +
d*x)^m*(Cosh[a + b*x]^(n + 1)/(b*(n + 1))), x] - Dist[d*(m/(b*(n + 1))), Int[(c + d*x)^(m - 1)*Cosh[a + b*x]^
(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]

Rule 5684

Int[(Cosh[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symb
ol] :> Dist[-a/b^2, Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2), x], x] + (Dist[1/b, Int[(e + f*x)^m*Cosh[c + d*x]^(
n - 2)*Sinh[c + d*x], x], x] + Dist[(a^2 + b^2)/b^2, Int[(e + f*x)^m*(Cosh[c + d*x]^(n - 2)/(a + b*Sinh[c + d*
x])), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0]

Rule 5698

Int[(Cosh[(c_.) + (d_.)*(x_)]^(p_.)*((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/b, Int[(e + f*x)^m*Cosh[c + d*x]^p*Sinh[c + d*x]^(n - 1), x], x]
- Dist[a/b, Int[(e + f*x)^m*Cosh[c + d*x]^p*(Sinh[c + d*x]^(n - 1)/(a + b*Sinh[c + d*x])), x], x] /; FreeQ[{a
, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps \begin{align*} \text {integral}& = \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) \operatorname {PolyLog}\left (2,-\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) \operatorname {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 (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 \operatorname {PolyLog}\left (2,-\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 \operatorname {PolyLog}\left (2,-\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) \operatorname {PolyLog}\left (2,-\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) \operatorname {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 (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 ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\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 ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\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) \operatorname {PolyLog}\left (2,-\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) \operatorname {PolyLog}\left (2,-\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 \operatorname {PolyLog}\left (3,-\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 \operatorname {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 (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] (verified)

Time = 2.97 (sec) , antiderivative size = 966, normalized size of antiderivative = 1.49 \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {216 a^3 d^3 e^2 x+108 a b^2 d^3 e^2 x+216 a^3 d^3 e f x^2+108 a b^2 d^3 e f x^2+72 a^3 d^3 f^2 x^3+36 a b^2 d^3 f^2 x^3+432 a^2 \sqrt {a^2+b^2} d^2 e^2 \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )-216 a^2 b d^2 e^2 \cosh (c+d x)-54 b^3 d^2 e^2 \cosh (c+d x)-432 a^2 b f^2 \cosh (c+d x)-108 b^3 f^2 \cosh (c+d x)-432 a^2 b d^2 e f x \cosh (c+d x)-108 b^3 d^2 e f x \cosh (c+d x)-216 a^2 b d^2 f^2 x^2 \cosh (c+d x)-54 b^3 d^2 f^2 x^2 \cosh (c+d x)-54 a b^2 d e f \cosh (2 (c+d x))-54 a b^2 d f^2 x \cosh (2 (c+d x))-18 b^3 d^2 e^2 \cosh (3 (c+d x))-4 b^3 f^2 \cosh (3 (c+d x))-36 b^3 d^2 e f x \cosh (3 (c+d x))-18 b^3 d^2 f^2 x^2 \cosh (3 (c+d x))-432 a^2 \sqrt {a^2+b^2} d^2 e f x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-216 a^2 \sqrt {a^2+b^2} d^2 f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+432 a^2 \sqrt {a^2+b^2} d^2 e f x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+216 a^2 \sqrt {a^2+b^2} d^2 f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-432 a^2 \sqrt {a^2+b^2} d f (e+f x) \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+432 a^2 \sqrt {a^2+b^2} d f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+432 a^2 \sqrt {a^2+b^2} f^2 \operatorname {PolyLog}\left (3,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-432 a^2 \sqrt {a^2+b^2} f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+432 a^2 b d e f \sinh (c+d x)+108 b^3 d e f \sinh (c+d x)+432 a^2 b d f^2 x \sinh (c+d x)+108 b^3 d f^2 x \sinh (c+d x)+54 a b^2 d^2 e^2 \sinh (2 (c+d x))+27 a b^2 f^2 \sinh (2 (c+d x))+108 a b^2 d^2 e f x \sinh (2 (c+d x))+54 a b^2 d^2 f^2 x^2 \sinh (2 (c+d x))+12 b^3 d e f \sinh (3 (c+d x))+12 b^3 d f^2 x \sinh (3 (c+d x))}{216 b^4 d^3} \]

[In]

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

[Out]

-1/216*(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]] - 2
16*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)] - 1
8*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 + 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 + 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*Si
nh[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)])/(b^4*d^3)

Maple [F]

\[\int \frac {\left (f x +e \right )^{2} \cosh \left (d x +c \right )^{2} \sinh \left (d x +c \right )^{2}}{a +b \sinh \left (d x +c \right )}d x\]

[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)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4311 vs. \(2 (595) = 1190\).

Time = 0.33 (sec) , antiderivative size = 4311, normalized size of antiderivative = 6.64 \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]

[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

Maxima [F]

\[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \]

[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]

1/24*e^2*(24*sqrt(a^2 + b^2)*a^2*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt(a^2 + b
^2)))/(b^4*d) - (3*a*b*e^(-d*x - c) - b^2 - 3*(4*a^2 + b^2)*e^(-2*d*x - 2*c))*e^(3*d*x + 3*c)/(b^3*d) - 12*(2*
a^3 + a*b^2)*(d*x + c)/(b^4*d) + (3*a*b*e^(-2*d*x - 2*c) + b^2*e^(-3*d*x - 3*c) + 3*(4*a^2 + b^2)*e^(-d*x - c)
)/(b^3*d)) - 1/432*(72*(2*a^3*d^3*f^2*e^(3*c) + a*b^2*d^3*f^2*e^(3*c))*x^3 + 216*(2*a^3*d^3*e*f*e^(3*c) + a*b^
2*d^3*e*f*e^(3*c))*x^2 - 2*(9*b^3*d^2*f^2*x^2*e^(6*c) + 6*(3*d^2*e*f - d*f^2)*b^3*x*e^(6*c) - 2*(3*d*e*f - f^2
)*b^3*e^(6*c))*e^(3*d*x) + 27*(2*a*b^2*d^2*f^2*x^2*e^(5*c) + 2*(2*d^2*e*f - d*f^2)*a*b^2*x*e^(5*c) - (2*d*e*f
- f^2)*a*b^2*e^(5*c))*e^(2*d*x) + 54*(8*(d*e*f - f^2)*a^2*b*e^(4*c) + 2*(d*e*f - f^2)*b^3*e^(4*c) - (4*a^2*b*d
^2*f^2*e^(4*c) + b^3*d^2*f^2*e^(4*c))*x^2 - 2*(4*(d^2*e*f - d*f^2)*a^2*b*e^(4*c) + (d^2*e*f - d*f^2)*b^3*e^(4*
c))*x)*e^(d*x) - 54*(8*(d*e*f + f^2)*a^2*b*e^(2*c) + 2*(d*e*f + f^2)*b^3*e^(2*c) + (4*a^2*b*d^2*f^2*e^(2*c) +
b^3*d^2*f^2*e^(2*c))*x^2 + 2*(4*(d^2*e*f + d*f^2)*a^2*b*e^(2*c) + (d^2*e*f + d*f^2)*b^3*e^(2*c))*x)*e^(-d*x) -
27*(2*a*b^2*d^2*f^2*x^2*e^c + 2*(2*d^2*e*f + d*f^2)*a*b^2*x*e^c + (2*d*e*f + f^2)*a*b^2*e^c)*e^(-2*d*x) - 2*(
9*b^3*d^2*f^2*x^2 + 6*(3*d^2*e*f + d*f^2)*b^3*x + 2*(3*d*e*f + f^2)*b^3)*e^(-3*d*x))*e^(-3*c)/(b^4*d^3) + inte
grate(2*((a^4*f^2*e^c + a^2*b^2*f^2*e^c)*x^2 + 2*(a^4*e*f*e^c + a^2*b^2*e*f*e^c)*x)*e^(d*x)/(b^5*e^(2*d*x + 2*
c) + 2*a*b^4*e^(d*x + c) - b^5), x)

Giac [F]

\[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^2\,{\mathrm {sinh}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^2}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]

[In]

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

[Out]

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