\(\int \frac {(e+f x)^3 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [362]

Optimal result

Integrand size = 34, antiderivative size = 606 \[ \int \frac {(e+f x)^3 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {3 f^3 x}{8 b d^3}+\frac {(e+f x)^3}{4 b d}-\frac {a^2 (e+f x)^4}{4 b^3 f}+\frac {6 a f^3 \cosh (c+d x)}{b^2 d^4}+\frac {3 a f (e+f x)^2 \cosh (c+d x)}{b^2 d^2}+\frac {a^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {a^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {3 a^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {3 a^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}-\frac {6 a^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {6 a^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^3}+\frac {6 a^2 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^4}+\frac {6 a^2 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^4}-\frac {6 a f^2 (e+f x) \sinh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^3 \sinh (c+d x)}{b^2 d}-\frac {3 f^3 \cosh (c+d x) \sinh (c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^2}+\frac {3 f^2 (e+f x) \sinh ^2(c+d x)}{4 b d^3}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 b d} \]

[Out]

3/8*f^3*x/b/d^3+1/4*(f*x+e)^3/b/d-1/4*a^2*(f*x+e)^4/b^3/f+6*a*f^3*cosh(d*x+c)/b^2/d^4+3*a*f*(f*x+e)^2*cosh(d*x
+c)/b^2/d^2+a^2*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^3/d+a^2*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a+(a^2
+b^2)^(1/2)))/b^3/d+3*a^2*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^3/d^2+3*a^2*f*(f*x+e)^2*p
olylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^3/d^2-6*a^2*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/
2)))/b^3/d^3-6*a^2*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^3/d^3+6*a^2*f^3*polylog(4,-b*exp
(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^3/d^4+6*a^2*f^3*polylog(4,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^3/d^4-6*a*f^2*(f
*x+e)*sinh(d*x+c)/b^2/d^3-a*(f*x+e)^3*sinh(d*x+c)/b^2/d-3/8*f^3*cosh(d*x+c)*sinh(d*x+c)/b/d^4-3/4*f*(f*x+e)^2*
cosh(d*x+c)*sinh(d*x+c)/b/d^2+3/4*f^2*(f*x+e)*sinh(d*x+c)^2/b/d^3+1/2*(f*x+e)^3*sinh(d*x+c)^2/b/d

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 606, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {5698, 5554, 3392, 32, 2715, 8, 3377, 2718, 5680, 2221, 2611, 6744, 2320, 6724} \[ \int \frac {(e+f x)^3 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a^2 (e+f x)^4}{4 b^3 f}+\frac {6 a^2 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^4}+\frac {6 a^2 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^4}-\frac {6 a^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {6 a^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^3}+\frac {3 a^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {3 a^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {a^2 (e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b^3 d}+\frac {a^2 (e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b^3 d}+\frac {6 a f^3 \cosh (c+d x)}{b^2 d^4}-\frac {6 a f^2 (e+f x) \sinh (c+d x)}{b^2 d^3}+\frac {3 a f (e+f x)^2 \cosh (c+d x)}{b^2 d^2}-\frac {a (e+f x)^3 \sinh (c+d x)}{b^2 d}-\frac {3 f^3 \sinh (c+d x) \cosh (c+d x)}{8 b d^4}+\frac {3 f^2 (e+f x) \sinh ^2(c+d x)}{4 b d^3}-\frac {3 f (e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{4 b d^2}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 b d}+\frac {3 f^3 x}{8 b d^3}+\frac {(e+f x)^3}{4 b d} \]

[In]

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

[Out]

(3*f^3*x)/(8*b*d^3) + (e + f*x)^3/(4*b*d) - (a^2*(e + f*x)^4)/(4*b^3*f) + (6*a*f^3*Cosh[c + d*x])/(b^2*d^4) +
(3*a*f*(e + f*x)^2*Cosh[c + d*x])/(b^2*d^2) + (a^2*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])
/(b^3*d) + (a^2*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^3*d) + (3*a^2*f*(e + f*x)^2*Pol
yLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*d^2) + (3*a^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x
))/(a + Sqrt[a^2 + b^2]))])/(b^3*d^2) - (6*a^2*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]
))])/(b^3*d^3) - (6*a^2*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*d^3) + (6*a^2
*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*d^4) + (6*a^2*f^3*PolyLog[4, -((b*E^(c + d*x))
/(a + Sqrt[a^2 + b^2]))])/(b^3*d^4) - (6*a*f^2*(e + f*x)*Sinh[c + d*x])/(b^2*d^3) - (a*(e + f*x)^3*Sinh[c + d*
x])/(b^2*d) - (3*f^3*Cosh[c + d*x]*Sinh[c + d*x])/(8*b*d^4) - (3*f*(e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x])/(4
*b*d^2) + (3*f^2*(e + f*x)*Sinh[c + d*x]^2)/(4*b*d^3) + ((e + f*x)^3*Sinh[c + d*x]^2)/(2*b*d)

Rule 8

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

Rule 32

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

Rule 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 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 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 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 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]

Rule 6744

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

Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x)^3 \cosh (c+d x) \sinh (c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b} \\ & = \frac {(e+f x)^3 \sinh ^2(c+d x)}{2 b d}-\frac {a \int (e+f x)^3 \cosh (c+d x) \, dx}{b^2}+\frac {a^2 \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}-\frac {(3 f) \int (e+f x)^2 \sinh ^2(c+d x) \, dx}{2 b d} \\ & = -\frac {a^2 (e+f x)^4}{4 b^3 f}-\frac {a (e+f x)^3 \sinh (c+d x)}{b^2 d}-\frac {3 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^2}+\frac {3 f^2 (e+f x) \sinh ^2(c+d x)}{4 b d^3}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 b d}+\frac {a^2 \int \frac {e^{c+d x} (e+f x)^3}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{b^2}+\frac {a^2 \int \frac {e^{c+d x} (e+f x)^3}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{b^2}+\frac {(3 a f) \int (e+f x)^2 \sinh (c+d x) \, dx}{b^2 d}+\frac {(3 f) \int (e+f x)^2 \, dx}{4 b d}-\frac {\left (3 f^3\right ) \int \sinh ^2(c+d x) \, dx}{4 b d^3} \\ & = \frac {(e+f x)^3}{4 b d}-\frac {a^2 (e+f x)^4}{4 b^3 f}+\frac {3 a f (e+f x)^2 \cosh (c+d x)}{b^2 d^2}+\frac {a^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {a^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}-\frac {a (e+f x)^3 \sinh (c+d x)}{b^2 d}-\frac {3 f^3 \cosh (c+d x) \sinh (c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^2}+\frac {3 f^2 (e+f x) \sinh ^2(c+d x)}{4 b d^3}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 b d}-\frac {\left (3 a^2 f\right ) \int (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{b^3 d}-\frac {\left (3 a^2 f\right ) \int (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{b^3 d}-\frac {\left (6 a f^2\right ) \int (e+f x) \cosh (c+d x) \, dx}{b^2 d^2}+\frac {\left (3 f^3\right ) \int 1 \, dx}{8 b d^3} \\ & = \frac {3 f^3 x}{8 b d^3}+\frac {(e+f x)^3}{4 b d}-\frac {a^2 (e+f x)^4}{4 b^3 f}+\frac {3 a f (e+f x)^2 \cosh (c+d x)}{b^2 d^2}+\frac {a^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {a^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {3 a^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {3 a^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}-\frac {6 a f^2 (e+f x) \sinh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^3 \sinh (c+d x)}{b^2 d}-\frac {3 f^3 \cosh (c+d x) \sinh (c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^2}+\frac {3 f^2 (e+f x) \sinh ^2(c+d x)}{4 b d^3}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 b d}-\frac {\left (6 a^2 f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{b^3 d^2}-\frac {\left (6 a^2 f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{b^3 d^2}+\frac {\left (6 a f^3\right ) \int \sinh (c+d x) \, dx}{b^2 d^3} \\ & = \frac {3 f^3 x}{8 b d^3}+\frac {(e+f x)^3}{4 b d}-\frac {a^2 (e+f x)^4}{4 b^3 f}+\frac {6 a f^3 \cosh (c+d x)}{b^2 d^4}+\frac {3 a f (e+f x)^2 \cosh (c+d x)}{b^2 d^2}+\frac {a^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {a^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {3 a^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {3 a^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}-\frac {6 a^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {6 a^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {6 a f^2 (e+f x) \sinh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^3 \sinh (c+d x)}{b^2 d}-\frac {3 f^3 \cosh (c+d x) \sinh (c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^2}+\frac {3 f^2 (e+f x) \sinh ^2(c+d x)}{4 b d^3}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 b d}+\frac {\left (6 a^2 f^3\right ) \int \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{b^3 d^3}+\frac {\left (6 a^2 f^3\right ) \int \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{b^3 d^3} \\ & = \frac {3 f^3 x}{8 b d^3}+\frac {(e+f x)^3}{4 b d}-\frac {a^2 (e+f x)^4}{4 b^3 f}+\frac {6 a f^3 \cosh (c+d x)}{b^2 d^4}+\frac {3 a f (e+f x)^2 \cosh (c+d x)}{b^2 d^2}+\frac {a^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {a^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {3 a^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {3 a^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}-\frac {6 a^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {6 a^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {6 a f^2 (e+f x) \sinh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^3 \sinh (c+d x)}{b^2 d}-\frac {3 f^3 \cosh (c+d x) \sinh (c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^2}+\frac {3 f^2 (e+f x) \sinh ^2(c+d x)}{4 b d^3}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 b d}+\frac {\left (6 a^2 f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,\frac {b x}{-a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^3 d^4}+\frac {\left (6 a^2 f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,-\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^3 d^4} \\ & = \frac {3 f^3 x}{8 b d^3}+\frac {(e+f x)^3}{4 b d}-\frac {a^2 (e+f x)^4}{4 b^3 f}+\frac {6 a f^3 \cosh (c+d x)}{b^2 d^4}+\frac {3 a f (e+f x)^2 \cosh (c+d x)}{b^2 d^2}+\frac {a^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {a^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {3 a^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {3 a^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}-\frac {6 a^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {6 a^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^3}+\frac {6 a^2 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^4}+\frac {6 a^2 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^4}-\frac {6 a f^2 (e+f x) \sinh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^3 \sinh (c+d x)}{b^2 d}-\frac {3 f^3 \cosh (c+d x) \sinh (c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^2}+\frac {3 f^2 (e+f x) \sinh ^2(c+d x)}{4 b d^3}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 b d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(2684\) vs. \(2(606)=1212\).

Time = 15.55 (sec) , antiderivative size = 2684, normalized size of antiderivative = 4.43 \[ \int \frac {(e+f x)^3 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

-1/4*(e^3*Log[a + b*Sinh[c + d*x]])/(b*d) - (3*e^2*f*(-1/2*x^2/b + (x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 +
b^2])])/(b*d) + (x*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*d) + PolyLog[2, (b*E^(c + d*x))/(-a + Sq
rt[a^2 + b^2])]/(b*d^2) + PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]/(b*d^2)))/4 - (3*e*f^2*(-1/3*x^
3/b + (x^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*d) + (x^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2
+ b^2])])/(b*d) + (2*x*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*d^2) + (2*x*PolyLog[2, -((b*E^
(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*d^2) - (2*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])])/(b*d^3) -
(2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*d^3)))/4 - (f^3*(-1/4*x^4/b + (x^3*Log[1 + (b*E^(
c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*d) + (x^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*d) + (3*x^2*
PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*d^2) + (3*x^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[
a^2 + b^2]))])/(b*d^2) - (6*x*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*d^3) - (6*x*PolyLog[3,
-((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*d^3) + (6*PolyLog[4, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])])/(b
*d^4) + (6*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*d^4)))/4 + (e*f^2*(2*(4*a^2 + b^2)*x^3*Cot
h[c] - (2*(4*a^2 + b^2)*(2*x^3 - (3*b^2*(-1 + E^(2*c))*(d^2*x^2*Log[1 + ((a - Sqrt[a^2 + b^2])*E^(-c - d*x))/b
] - 2*d*x*PolyLog[2, ((-a + Sqrt[a^2 + b^2])*E^(-c - d*x))/b] - 2*PolyLog[3, ((-a + Sqrt[a^2 + b^2])*E^(-c - d
*x))/b]))/(Sqrt[a^2 + b^2]*(-a + Sqrt[a^2 + b^2])*d^3) - (3*b^2*(-1 + E^(2*c))*(d^2*x^2*Log[1 + ((a + Sqrt[a^2
+ b^2])*E^(-c - d*x))/b] - 2*d*x*PolyLog[2, -(((a + Sqrt[a^2 + b^2])*E^(-c - d*x))/b)] - 2*PolyLog[3, -(((a +
Sqrt[a^2 + b^2])*E^(-c - d*x))/b)]))/(Sqrt[a^2 + b^2]*(a + Sqrt[a^2 + b^2])*d^3) + (3*a*(-1 + E^(2*c))*(d^2*x
^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 2*d*x*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] -
2*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])]))/(Sqrt[a^2 + b^2]*d^3) - (3*a*(-1 + E^(2*c))*(d^2*x^2*L
og[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 2*d*x*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] - 2
*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(Sqrt[a^2 + b^2]*d^3)))/(-1 + E^(2*c)) - (24*a*b*Cosh[
d*x]*(-2*d*x*Cosh[c] + (2 + d^2*x^2)*Sinh[c]))/d^3 + (3*b^2*Cosh[2*d*x]*((1 + 2*d^2*x^2)*Cosh[2*c] - 2*d*x*Sin
h[2*c]))/d^3 - (24*a*b*((2 + d^2*x^2)*Cosh[c] - 2*d*x*Sinh[c])*Sinh[d*x])/d^3 + (3*b^2*(-2*d*x*Cosh[2*c] + (1
+ 2*d^2*x^2)*Sinh[2*c])*Sinh[2*d*x])/d^3))/(8*b^3) + (f^3*((4*a^2 + b^2)*x^4*Coth[c] - (2*(4*a^2 + b^2)*(x^4 -
(2*b^2*(-1 + E^(2*c))*(d^3*x^3*Log[1 + ((a - Sqrt[a^2 + b^2])*E^(-c - d*x))/b] - 3*d^2*x^2*PolyLog[2, ((-a +
Sqrt[a^2 + b^2])*E^(-c - d*x))/b] - 6*d*x*PolyLog[3, ((-a + Sqrt[a^2 + b^2])*E^(-c - d*x))/b] - 6*PolyLog[4, (
(-a + Sqrt[a^2 + b^2])*E^(-c - d*x))/b]))/(Sqrt[a^2 + b^2]*(-a + Sqrt[a^2 + b^2])*d^4) - (2*b^2*(-1 + E^(2*c))
*(d^3*x^3*Log[1 + ((a + Sqrt[a^2 + b^2])*E^(-c - d*x))/b] - 3*d^2*x^2*PolyLog[2, -(((a + Sqrt[a^2 + b^2])*E^(-
c - d*x))/b)] - 6*d*x*PolyLog[3, -(((a + Sqrt[a^2 + b^2])*E^(-c - d*x))/b)] - 6*PolyLog[4, -(((a + Sqrt[a^2 +
b^2])*E^(-c - d*x))/b)]))/(Sqrt[a^2 + b^2]*(a + Sqrt[a^2 + b^2])*d^4) + (2*a*(-1 + E^(2*c))*(d^3*x^3*Log[1 + (
b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 3*d^2*x^2*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 6*d*x*P
olyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 6*PolyLog[4, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])]))/(Sq
rt[a^2 + b^2]*d^4) - (2*a*(-1 + E^(2*c))*(d^3*x^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 3*d^2*x^2*P
olyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] - 6*d*x*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])
)] + 6*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(Sqrt[a^2 + b^2]*d^4)))/(-1 + E^(2*c)) - (16*a*b
*Cosh[d*x]*(-3*(2 + d^2*x^2)*Cosh[c] + d*x*(6 + d^2*x^2)*Sinh[c]))/d^4 + (b^2*Cosh[2*d*x]*(2*d*x*(3 + 2*d^2*x^
2)*Cosh[2*c] - 3*(1 + 2*d^2*x^2)*Sinh[2*c]))/d^4 - (16*a*b*(d*x*(6 + d^2*x^2)*Cosh[c] - 3*(2 + d^2*x^2)*Sinh[c
])*Sinh[d*x])/d^4 + (b^2*(-3*(1 + 2*d^2*x^2)*Cosh[2*c] + 2*d*x*(3 + 2*d^2*x^2)*Sinh[2*c])*Sinh[2*d*x])/d^4))/(
16*b^3) + (e^3*((4*a^2 + b^2)*Log[a + b*Sinh[c + d*x]] - 4*a*b*Sinh[c + d*x] + 2*b^2*Sinh[c + d*x]^2))/(4*b^3*
d) + (3*e^2*f*(8*a*b*Cosh[c + d*x] + 2*b^2*d*x*Cosh[2*(c + d*x)] + (4*a^2 + b^2)*(2*c*(c + d*x) - (c + d*x)^2
+ 2*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 2*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a
^2 + b^2])] - 2*c*Log[b - 2*a*E^(c + d*x) - b*E^(2*(c + d*x))] + 2*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 +
b^2])] + 2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]) - 8*a*b*d*x*Sinh[c + d*x] - b^2*Sinh[2*(c +
d*x)]))/(8*b^3*d^2)

Maple [F]

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

[In]

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

[Out]

int((f*x+e)^3*cosh(d*x+c)*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. 3891 vs. \(2 (566) = 1132\).

Time = 0.33 (sec) , antiderivative size = 3891, normalized size of antiderivative = 6.42 \[ \int \frac {(e+f x)^3 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/32*(4*b^2*d^3*f^3*x^3 + 4*b^2*d^3*e^3 + 6*b^2*d^2*e^2*f + 6*b^2*d*e*f^2 + 3*b^2*f^3 + (4*b^2*d^3*f^3*x^3 + 4
*b^2*d^3*e^3 - 6*b^2*d^2*e^2*f + 6*b^2*d*e*f^2 - 3*b^2*f^3 + 6*(2*b^2*d^3*e*f^2 - b^2*d^2*f^3)*x^2 + 6*(2*b^2*
d^3*e^2*f - 2*b^2*d^2*e*f^2 + b^2*d*f^3)*x)*cosh(d*x + c)^4 + (4*b^2*d^3*f^3*x^3 + 4*b^2*d^3*e^3 - 6*b^2*d^2*e
^2*f + 6*b^2*d*e*f^2 - 3*b^2*f^3 + 6*(2*b^2*d^3*e*f^2 - b^2*d^2*f^3)*x^2 + 6*(2*b^2*d^3*e^2*f - 2*b^2*d^2*e*f^
2 + b^2*d*f^3)*x)*sinh(d*x + c)^4 - 16*(a*b*d^3*f^3*x^3 + a*b*d^3*e^3 - 3*a*b*d^2*e^2*f + 6*a*b*d*e*f^2 - 6*a*
b*f^3 + 3*(a*b*d^3*e*f^2 - a*b*d^2*f^3)*x^2 + 3*(a*b*d^3*e^2*f - 2*a*b*d^2*e*f^2 + 2*a*b*d*f^3)*x)*cosh(d*x +
c)^3 - 4*(4*a*b*d^3*f^3*x^3 + 4*a*b*d^3*e^3 - 12*a*b*d^2*e^2*f + 24*a*b*d*e*f^2 - 24*a*b*f^3 + 12*(a*b*d^3*e*f
^2 - a*b*d^2*f^3)*x^2 + 12*(a*b*d^3*e^2*f - 2*a*b*d^2*e*f^2 + 2*a*b*d*f^3)*x - (4*b^2*d^3*f^3*x^3 + 4*b^2*d^3*
e^3 - 6*b^2*d^2*e^2*f + 6*b^2*d*e*f^2 - 3*b^2*f^3 + 6*(2*b^2*d^3*e*f^2 - b^2*d^2*f^3)*x^2 + 6*(2*b^2*d^3*e^2*f
- 2*b^2*d^2*e*f^2 + b^2*d*f^3)*x)*cosh(d*x + c))*sinh(d*x + c)^3 + 6*(2*b^2*d^3*e*f^2 + b^2*d^2*f^3)*x^2 - 8*
(a^2*d^4*f^3*x^4 + 4*a^2*d^4*e*f^2*x^3 + 6*a^2*d^4*e^2*f*x^2 + 4*a^2*d^4*e^3*x + 8*a^2*c*d^3*e^3 - 12*a^2*c^2*
d^2*e^2*f + 8*a^2*c^3*d*e*f^2 - 2*a^2*c^4*f^3)*cosh(d*x + c)^2 - 2*(4*a^2*d^4*f^3*x^4 + 16*a^2*d^4*e*f^2*x^3 +
24*a^2*d^4*e^2*f*x^2 + 16*a^2*d^4*e^3*x + 32*a^2*c*d^3*e^3 - 48*a^2*c^2*d^2*e^2*f + 32*a^2*c^3*d*e*f^2 - 8*a^
2*c^4*f^3 - 3*(4*b^2*d^3*f^3*x^3 + 4*b^2*d^3*e^3 - 6*b^2*d^2*e^2*f + 6*b^2*d*e*f^2 - 3*b^2*f^3 + 6*(2*b^2*d^3*
e*f^2 - b^2*d^2*f^3)*x^2 + 6*(2*b^2*d^3*e^2*f - 2*b^2*d^2*e*f^2 + b^2*d*f^3)*x)*cosh(d*x + c)^2 + 24*(a*b*d^3*
f^3*x^3 + a*b*d^3*e^3 - 3*a*b*d^2*e^2*f + 6*a*b*d*e*f^2 - 6*a*b*f^3 + 3*(a*b*d^3*e*f^2 - a*b*d^2*f^3)*x^2 + 3*
(a*b*d^3*e^2*f - 2*a*b*d^2*e*f^2 + 2*a*b*d*f^3)*x)*cosh(d*x + c))*sinh(d*x + c)^2 + 6*(2*b^2*d^3*e^2*f + 2*b^2
*d^2*e*f^2 + b^2*d*f^3)*x + 16*(a*b*d^3*f^3*x^3 + a*b*d^3*e^3 + 3*a*b*d^2*e^2*f + 6*a*b*d*e*f^2 + 6*a*b*f^3 +
3*(a*b*d^3*e*f^2 + a*b*d^2*f^3)*x^2 + 3*(a*b*d^3*e^2*f + 2*a*b*d^2*e*f^2 + 2*a*b*d*f^3)*x)*cosh(d*x + c) + 96*
((a^2*d^2*f^3*x^2 + 2*a^2*d^2*e*f^2*x + a^2*d^2*e^2*f)*cosh(d*x + c)^2 + 2*(a^2*d^2*f^3*x^2 + 2*a^2*d^2*e*f^2*
x + a^2*d^2*e^2*f)*cosh(d*x + c)*sinh(d*x + c) + (a^2*d^2*f^3*x^2 + 2*a^2*d^2*e*f^2*x + a^2*d^2*e^2*f)*sinh(d*
x + c)^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) + 96*((a^2*d^2*f^3*x^2 + 2*a^2*d^2*e*f^2*x + a^2*d^2*e^2*f)*cosh(d*x + c)^2 + 2*(a^2*d^2*f^3*x^2
+ 2*a^2*d^2*e*f^2*x + a^2*d^2*e^2*f)*cosh(d*x + c)*sinh(d*x + c) + (a^2*d^2*f^3*x^2 + 2*a^2*d^2*e*f^2*x + a^2*
d^2*e^2*f)*sinh(d*x + c)^2)*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqr
t((a^2 + b^2)/b^2) - b)/b + 1) + 32*((a^2*d^3*e^3 - 3*a^2*c*d^2*e^2*f + 3*a^2*c^2*d*e*f^2 - a^2*c^3*f^3)*cosh(
d*x + c)^2 + 2*(a^2*d^3*e^3 - 3*a^2*c*d^2*e^2*f + 3*a^2*c^2*d*e*f^2 - a^2*c^3*f^3)*cosh(d*x + c)*sinh(d*x + c)
+ (a^2*d^3*e^3 - 3*a^2*c*d^2*e^2*f + 3*a^2*c^2*d*e*f^2 - a^2*c^3*f^3)*sinh(d*x + c)^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) + 32*((a^2*d^3*e^3 - 3*a^2*c*d^2*e^2*f + 3*a^2*c^2*d*e*
f^2 - a^2*c^3*f^3)*cosh(d*x + c)^2 + 2*(a^2*d^3*e^3 - 3*a^2*c*d^2*e^2*f + 3*a^2*c^2*d*e*f^2 - a^2*c^3*f^3)*cos
h(d*x + c)*sinh(d*x + c) + (a^2*d^3*e^3 - 3*a^2*c*d^2*e^2*f + 3*a^2*c^2*d*e*f^2 - a^2*c^3*f^3)*sinh(d*x + c)^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) + 32*((a^2*d^3*f^3*x^3 + 3*a^2*
d^3*e*f^2*x^2 + 3*a^2*d^3*e^2*f*x + 3*a^2*c*d^2*e^2*f - 3*a^2*c^2*d*e*f^2 + a^2*c^3*f^3)*cosh(d*x + c)^2 + 2*(
a^2*d^3*f^3*x^3 + 3*a^2*d^3*e*f^2*x^2 + 3*a^2*d^3*e^2*f*x + 3*a^2*c*d^2*e^2*f - 3*a^2*c^2*d*e*f^2 + a^2*c^3*f^
3)*cosh(d*x + c)*sinh(d*x + c) + (a^2*d^3*f^3*x^3 + 3*a^2*d^3*e*f^2*x^2 + 3*a^2*d^3*e^2*f*x + 3*a^2*c*d^2*e^2*
f - 3*a^2*c^2*d*e*f^2 + a^2*c^3*f^3)*sinh(d*x + c)^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) + 32*((a^2*d^3*f^3*x^3 + 3*a^2*d^3*e*f^2*x^2 + 3*a^2*d^3*e
^2*f*x + 3*a^2*c*d^2*e^2*f - 3*a^2*c^2*d*e*f^2 + a^2*c^3*f^3)*cosh(d*x + c)^2 + 2*(a^2*d^3*f^3*x^3 + 3*a^2*d^3
*e*f^2*x^2 + 3*a^2*d^3*e^2*f*x + 3*a^2*c*d^2*e^2*f - 3*a^2*c^2*d*e*f^2 + a^2*c^3*f^3)*cosh(d*x + c)*sinh(d*x +
c) + (a^2*d^3*f^3*x^3 + 3*a^2*d^3*e*f^2*x^2 + 3*a^2*d^3*e^2*f*x + 3*a^2*c*d^2*e^2*f - 3*a^2*c^2*d*e*f^2 + a^2
*c^3*f^3)*sinh(d*x + c)^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) + 192*(a^2*f^3*cosh(d*x + c)^2 + 2*a^2*f^3*cosh(d*x + c)*sinh(d*x + c) + a^2*f^3*sinh
(d*x + c)^2)*polylog(4, (a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b
^2)/b^2))/b) + 192*(a^2*f^3*cosh(d*x + c)^2 + 2*a^2*f^3*cosh(d*x + c)*sinh(d*x + c) + a^2*f^3*sinh(d*x + c)^2)
*polylog(4, (a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b)
- 192*((a^2*d*f^3*x + a^2*d*e*f^2)*cosh(d*x + c)^2 + 2*(a^2*d*f^3*x + a^2*d*e*f^2)*cosh(d*x + c)*sinh(d*x + c
) + (a^2*d*f^3*x + a^2*d*e*f^2)*sinh(d*x + c)^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) - 192*((a^2*d*f^3*x + a^2*d*e*f^2)*cosh(d*x + c)^2 + 2*(a^2*d
*f^3*x + a^2*d*e*f^2)*cosh(d*x + c)*sinh(d*x + c) + (a^2*d*f^3*x + a^2*d*e*f^2)*sinh(d*x + c)^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) + 4*(4*a*b*d^
3*f^3*x^3 + 4*a*b*d^3*e^3 + 12*a*b*d^2*e^2*f + 24*a*b*d*e*f^2 + 24*a*b*f^3 + (4*b^2*d^3*f^3*x^3 + 4*b^2*d^3*e^
3 - 6*b^2*d^2*e^2*f + 6*b^2*d*e*f^2 - 3*b^2*f^3 + 6*(2*b^2*d^3*e*f^2 - b^2*d^2*f^3)*x^2 + 6*(2*b^2*d^3*e^2*f -
2*b^2*d^2*e*f^2 + b^2*d*f^3)*x)*cosh(d*x + c)^3 + 12*(a*b*d^3*e*f^2 + a*b*d^2*f^3)*x^2 - 12*(a*b*d^3*f^3*x^3
+ a*b*d^3*e^3 - 3*a*b*d^2*e^2*f + 6*a*b*d*e*f^2 - 6*a*b*f^3 + 3*(a*b*d^3*e*f^2 - a*b*d^2*f^3)*x^2 + 3*(a*b*d^3
*e^2*f - 2*a*b*d^2*e*f^2 + 2*a*b*d*f^3)*x)*cosh(d*x + c)^2 + 12*(a*b*d^3*e^2*f + 2*a*b*d^2*e*f^2 + 2*a*b*d*f^3
)*x - 4*(a^2*d^4*f^3*x^4 + 4*a^2*d^4*e*f^2*x^3 + 6*a^2*d^4*e^2*f*x^2 + 4*a^2*d^4*e^3*x + 8*a^2*c*d^3*e^3 - 12*
a^2*c^2*d^2*e^2*f + 8*a^2*c^3*d*e*f^2 - 2*a^2*c^4*f^3)*cosh(d*x + c))*sinh(d*x + c))/(b^3*d^4*cosh(d*x + c)^2
+ 2*b^3*d^4*cosh(d*x + c)*sinh(d*x + c) + b^3*d^4*sinh(d*x + c)^2)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

1/8*e^3*(8*(d*x + c)*a^2/(b^3*d) - (4*a*e^(-d*x - c) - b)*e^(2*d*x + 2*c)/(b^2*d) + 8*a^2*log(-2*a*e^(-d*x - c
) + b*e^(-2*d*x - 2*c) - b)/(b^3*d) + (4*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c))/(b^2*d)) + 1/32*(8*a^2*d^4*f^3*x
^4*e^(2*c) + 32*a^2*d^4*e*f^2*x^3*e^(2*c) + 48*a^2*d^4*e^2*f*x^2*e^(2*c) + (4*b^2*d^3*f^3*x^3*e^(4*c) + 6*(2*d
^3*e*f^2 - d^2*f^3)*b^2*x^2*e^(4*c) + 6*(2*d^3*e^2*f - 2*d^2*e*f^2 + d*f^3)*b^2*x*e^(4*c) - 3*(2*d^2*e^2*f - 2
*d*e*f^2 + f^3)*b^2*e^(4*c))*e^(2*d*x) - 16*(a*b*d^3*f^3*x^3*e^(3*c) + 3*(d^3*e*f^2 - d^2*f^3)*a*b*x^2*e^(3*c)
+ 3*(d^3*e^2*f - 2*d^2*e*f^2 + 2*d*f^3)*a*b*x*e^(3*c) - 3*(d^2*e^2*f - 2*d*e*f^2 + 2*f^3)*a*b*e^(3*c))*e^(d*x
) + 16*(a*b*d^3*f^3*x^3*e^c + 3*(d^3*e*f^2 + d^2*f^3)*a*b*x^2*e^c + 3*(d^3*e^2*f + 2*d^2*e*f^2 + 2*d*f^3)*a*b*
x*e^c + 3*(d^2*e^2*f + 2*d*e*f^2 + 2*f^3)*a*b*e^c)*e^(-d*x) + (4*b^2*d^3*f^3*x^3 + 6*(2*d^3*e*f^2 + d^2*f^3)*b
^2*x^2 + 6*(2*d^3*e^2*f + 2*d^2*e*f^2 + d*f^3)*b^2*x + 3*(2*d^2*e^2*f + 2*d*e*f^2 + f^3)*b^2)*e^(-2*d*x))*e^(-
2*c)/(b^3*d^4) - integrate(-2*(a^2*b*f^3*x^3 + 3*a^2*b*e*f^2*x^2 + 3*a^2*b*e^2*f*x - (a^3*f^3*x^3*e^c + 3*a^3*
e*f^2*x^2*e^c + 3*a^3*e^2*f*x*e^c)*e^(d*x))/(b^4*e^(2*d*x + 2*c) + 2*a*b^3*e^(d*x + c) - b^4), x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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