\(\int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [373]

Optimal result

Integrand size = 36, antiderivative size = 819 \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {a^2 e f x}{2 b^3 d}-\frac {3 e f x}{16 b d}+\frac {a^2 f^2 x^2}{4 b^3 d}-\frac {3 f^2 x^2}{32 b d}-\frac {a^2 \left (a^2+b^2\right ) (e+f x)^3}{3 b^5 f}+\frac {2 a^3 f (e+f x) \cosh (c+d x)}{b^4 d^2}+\frac {4 a f (e+f x) \cosh (c+d x)}{3 b^2 d^2}+\frac {3 f^2 \cosh ^2(c+d x)}{32 b d^3}+\frac {2 a f (e+f x) \cosh ^3(c+d x)}{9 b^2 d^2}+\frac {f^2 \cosh ^4(c+d x)}{32 b d^3}+\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 b d}+\frac {a^2 \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d}+\frac {a^2 \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d}+\frac {2 a^2 \left (a^2+b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d^2}+\frac {2 a^2 \left (a^2+b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d^2}-\frac {2 a^2 \left (a^2+b^2\right ) f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d^3}-\frac {2 a^2 \left (a^2+b^2\right ) f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d^3}-\frac {2 a^3 f^2 \sinh (c+d x)}{b^4 d^3}-\frac {14 a f^2 \sinh (c+d x)}{9 b^2 d^3}-\frac {a^3 (e+f x)^2 \sinh (c+d x)}{b^4 d}-\frac {2 a (e+f x)^2 \sinh (c+d x)}{3 b^2 d}-\frac {a^2 f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^3 d^2}-\frac {3 f (e+f x) \cosh (c+d x) \sinh (c+d x)}{16 b d^2}-\frac {a (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^2 d}-\frac {f (e+f x) \cosh ^3(c+d x) \sinh (c+d x)}{8 b d^2}+\frac {a^2 f^2 \sinh ^2(c+d x)}{4 b^3 d^3}+\frac {a^2 (e+f x)^2 \sinh ^2(c+d x)}{2 b^3 d}-\frac {2 a f^2 \sinh ^3(c+d x)}{27 b^2 d^3} \]

[Out]

-3/16*e*f*x/b/d-14/9*a*f^2*sinh(d*x+c)/b^2/d^3-1/2*a^2*f*(f*x+e)*cosh(d*x+c)*sinh(d*x+c)/b^3/d^2+2*a^2*(a^2+b^
2)*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^5/d^2+2*a^2*(a^2+b^2)*f*(f*x+e)*polylog(2,-b*exp(d
*x+c)/(a+(a^2+b^2)^(1/2)))/b^5/d^2+1/2*a^2*e*f*x/b^3/d+2*a^3*f*(f*x+e)*cosh(d*x+c)/b^4/d^2+2/9*a*f*(f*x+e)*cos
h(d*x+c)^3/b^2/d^2-1/3*a*(f*x+e)^2*cosh(d*x+c)^2*sinh(d*x+c)/b^2/d-1/8*f*(f*x+e)*cosh(d*x+c)^3*sinh(d*x+c)/b/d
^2-2*a^2*(a^2+b^2)*f^2*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^5/d^3+4/3*a*f*(f*x+e)*cosh(d*x+c)/b^2/d^
2-3/16*f*(f*x+e)*cosh(d*x+c)*sinh(d*x+c)/b/d^2+a^2*(a^2+b^2)*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/
b^5/d+a^2*(a^2+b^2)*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^5/d-a^3*(f*x+e)^2*sinh(d*x+c)/b^4/d-2/3
*a*(f*x+e)^2*sinh(d*x+c)/b^2/d-2*a^2*(a^2+b^2)*f^2*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^5/d^3+3/32*f
^2*cosh(d*x+c)^2/b/d^3+1/32*f^2*cosh(d*x+c)^4/b/d^3+1/4*(f*x+e)^2*cosh(d*x+c)^4/b/d-3/32*f^2*x^2/b/d+1/4*a^2*f
^2*x^2/b^3/d-1/3*a^2*(a^2+b^2)*(f*x+e)^3/b^5/f-2*a^3*f^2*sinh(d*x+c)/b^4/d^3+1/4*a^2*f^2*sinh(d*x+c)^2/b^3/d^3
+1/2*a^2*(f*x+e)^2*sinh(d*x+c)^2/b^3/d-2/27*a*f^2*sinh(d*x+c)^3/b^2/d^3

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 819, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {5698, 5555, 3391, 3392, 3377, 2717, 2713, 5684, 5554, 5680, 2221, 2611, 2320, 6724} \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {f^2 \cosh ^4(c+d x)}{32 b d^3}+\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 b d}+\frac {2 a f (e+f x) \cosh ^3(c+d x)}{9 b^2 d^2}-\frac {f (e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{8 b d^2}+\frac {3 f^2 \cosh ^2(c+d x)}{32 b d^3}-\frac {a (e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 b^2 d}+\frac {4 a f (e+f x) \cosh (c+d x)}{3 b^2 d^2}+\frac {2 a^3 f (e+f x) \cosh (c+d x)}{b^4 d^2}-\frac {3 f (e+f x) \sinh (c+d x) \cosh (c+d x)}{16 b d^2}-\frac {a^2 f (e+f x) \sinh (c+d x) \cosh (c+d x)}{2 b^3 d^2}-\frac {a^2 \left (a^2+b^2\right ) (e+f x)^3}{3 b^5 f}-\frac {2 a f^2 \sinh ^3(c+d x)}{27 b^2 d^3}-\frac {3 f^2 x^2}{32 b d}+\frac {a^2 f^2 x^2}{4 b^3 d}+\frac {a^2 f^2 \sinh ^2(c+d x)}{4 b^3 d^3}+\frac {a^2 (e+f x)^2 \sinh ^2(c+d x)}{2 b^3 d}-\frac {3 e f x}{16 b d}+\frac {a^2 e f x}{2 b^3 d}+\frac {a^2 \left (a^2+b^2\right ) (e+f x)^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )}{b^5 d}+\frac {a^2 \left (a^2+b^2\right ) (e+f x)^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )}{b^5 d}+\frac {2 a^2 \left (a^2+b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d^2}+\frac {2 a^2 \left (a^2+b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d^2}-\frac {2 a^2 \left (a^2+b^2\right ) f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d^3}-\frac {2 a^2 \left (a^2+b^2\right ) f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d^3}-\frac {14 a f^2 \sinh (c+d x)}{9 b^2 d^3}-\frac {2 a^3 f^2 \sinh (c+d x)}{b^4 d^3}-\frac {2 a (e+f x)^2 \sinh (c+d x)}{3 b^2 d}-\frac {a^3 (e+f x)^2 \sinh (c+d x)}{b^4 d} \]

[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 2221

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

Rule 2320

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

Rule 2611

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

Rule 2713

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 2717

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

Rule 3377

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

Rule 3391

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

Rule 3392

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

Rule 5554

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

Rule 5555

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

Rule 5680

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

Rule 5684

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

Rule 5698

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

Rule 6724

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

Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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) \operatorname {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) \operatorname {PolyLog}\left (2,-\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 \operatorname {PolyLog}\left (2,-\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 \operatorname {PolyLog}\left (2,-\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) \operatorname {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) \operatorname {PolyLog}\left (2,-\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 ) \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^5 d^3}-\frac {\left (2 a^2 \left (a^2+b^2\right ) 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^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) \operatorname {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) \operatorname {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 \operatorname {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 \operatorname {PolyLog}\left (3,-\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] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(5113\) vs. \(2(819)=1638\).

Time = 15.71 (sec) , antiderivative size = 5113, normalized size of antiderivative = 6.24 \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \]

[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

Maple [F]

\[\int \frac {\left (f x +e \right )^{2} \cosh \left (d x +c \right )^{3} \sinh \left (d x +c \right )^{2}}{a +b \sinh \left (d x +c \right )}d x\]

[In]

int((f*x+e)^2*cosh(d*x+c)^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)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 7645 vs. \(2 (761) = 1522\).

Time = 0.38 (sec) , antiderivative size = 7645, normalized size of antiderivative = 9.33 \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

integrate((f*x+e)^2*cosh(d*x+c)^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)

Giac [F]

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

[In]

integrate((f*x+e)^2*cosh(d*x+c)^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)

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^3\,{\mathrm {sinh}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^2}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]

[In]

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

[Out]

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