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

Optimal. Leaf size=474 \[ -\frac{a^3 f \sqrt{a^2+b^2} \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^5 d^2}+\frac{a^3 f \sqrt{a^2+b^2} \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^5 d^2}+\frac{a^3 f \sinh (c+d x)}{b^4 d^2}-\frac{a^2 f \cosh ^2(c+d x)}{4 b^3 d^2}-\frac{a^3 \sqrt{a^2+b^2} (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b^5 d}+\frac{a^3 \sqrt{a^2+b^2} (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b^5 d}-\frac{a^3 (e+f x) \cosh (c+d x)}{b^4 d}+\frac{a^2 (e+f x) \sinh (c+d x) \cosh (c+d x)}{2 b^3 d}+\frac{a^4 e x}{b^5}+\frac{a^2 e x}{2 b^3}+\frac{a^4 f x^2}{2 b^5}+\frac{a^2 f x^2}{4 b^3}+\frac{a f \sinh ^3(c+d x)}{9 b^2 d^2}+\frac{a f \sinh (c+d x)}{3 b^2 d^2}-\frac{a (e+f x) \cosh ^3(c+d x)}{3 b^2 d}-\frac{f \cosh (4 c+4 d x)}{128 b d^2}+\frac{(e+f x) \sinh (4 c+4 d x)}{32 b d}-\frac{(e+f x)^2}{16 b f} \]

[Out]

(a^4*e*x)/b^5 + (a^2*e*x)/(2*b^3) + (a^4*f*x^2)/(2*b^5) + (a^2*f*x^2)/(4*b^3) - (e + f*x)^2/(16*b*f) - (a^3*(e
+ f*x)*Cosh[c + d*x])/(b^4*d) - (a^2*f*Cosh[c + d*x]^2)/(4*b^3*d^2) - (a*(e + f*x)*Cosh[c + d*x]^3)/(3*b^2*d)
- (f*Cosh[4*c + 4*d*x])/(128*b*d^2) - (a^3*Sqrt[a^2 + b^2]*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 +
b^2])])/(b^5*d) + (a^3*Sqrt[a^2 + b^2]*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^5*d) - (a^
3*Sqrt[a^2 + b^2]*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^5*d^2) + (a^3*Sqrt[a^2 + b^2]*f*P
olyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^5*d^2) + (a^3*f*Sinh[c + d*x])/(b^4*d^2) + (a*f*Sinh[c
+ d*x])/(3*b^2*d^2) + (a^2*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(2*b^3*d) + (a*f*Sinh[c + d*x]^3)/(9*b^2*d^
2) + ((e + f*x)*Sinh[4*c + 4*d*x])/(32*b*d)

________________________________________________________________________________________

Rubi [A]  time = 0.86452, antiderivative size = 474, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 14, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412, Rules used = {5579, 5448, 3296, 2638, 5447, 2633, 3310, 5565, 2637, 3322, 2264, 2190, 2279, 2391} \[ -\frac{a^3 f \sqrt{a^2+b^2} \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^5 d^2}+\frac{a^3 f \sqrt{a^2+b^2} \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^5 d^2}+\frac{a^3 f \sinh (c+d x)}{b^4 d^2}-\frac{a^2 f \cosh ^2(c+d x)}{4 b^3 d^2}-\frac{a^3 \sqrt{a^2+b^2} (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b^5 d}+\frac{a^3 \sqrt{a^2+b^2} (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b^5 d}-\frac{a^3 (e+f x) \cosh (c+d x)}{b^4 d}+\frac{a^2 (e+f x) \sinh (c+d x) \cosh (c+d x)}{2 b^3 d}+\frac{a^4 e x}{b^5}+\frac{a^2 e x}{2 b^3}+\frac{a^4 f x^2}{2 b^5}+\frac{a^2 f x^2}{4 b^3}+\frac{a f \sinh ^3(c+d x)}{9 b^2 d^2}+\frac{a f \sinh (c+d x)}{3 b^2 d^2}-\frac{a (e+f x) \cosh ^3(c+d x)}{3 b^2 d}-\frac{f \cosh (4 c+4 d x)}{128 b d^2}+\frac{(e+f x) \sinh (4 c+4 d x)}{32 b d}-\frac{(e+f x)^2}{16 b f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*e*x)/b^5 + (a^2*e*x)/(2*b^3) + (a^4*f*x^2)/(2*b^5) + (a^2*f*x^2)/(4*b^3) - (e + f*x)^2/(16*b*f) - (a^3*(e
+ f*x)*Cosh[c + d*x])/(b^4*d) - (a^2*f*Cosh[c + d*x]^2)/(4*b^3*d^2) - (a*(e + f*x)*Cosh[c + d*x]^3)/(3*b^2*d)
- (f*Cosh[4*c + 4*d*x])/(128*b*d^2) - (a^3*Sqrt[a^2 + b^2]*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 +
b^2])])/(b^5*d) + (a^3*Sqrt[a^2 + b^2]*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^5*d) - (a^
3*Sqrt[a^2 + b^2]*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^5*d^2) + (a^3*Sqrt[a^2 + b^2]*f*P
olyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^5*d^2) + (a^3*f*Sinh[c + d*x])/(b^4*d^2) + (a*f*Sinh[c
+ d*x])/(3*b^2*d^2) + (a^2*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(2*b^3*d) + (a*f*Sinh[c + d*x]^3)/(9*b^2*d^
2) + ((e + f*x)*Sinh[4*c + 4*d*x])/(32*b*d)

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 5448

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

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 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 2633

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

Rule 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 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 2637

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

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 2279

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

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{(e+f x) \cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac{\int (e+f x) \cosh ^2(c+d x) \sinh ^2(c+d x) \, dx}{b}-\frac{a \int \frac{(e+f x) \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=-\frac{a \int (e+f x) \cosh ^2(c+d x) \sinh (c+d x) \, dx}{b^2}+\frac{a^2 \int \frac{(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}+\frac{\int \left (\frac{1}{8} (-e-f x)+\frac{1}{8} (e+f x) \cosh (4 c+4 d x)\right ) \, dx}{b}\\ &=-\frac{(e+f x)^2}{16 b f}-\frac{a (e+f x) \cosh ^3(c+d x)}{3 b^2 d}+\frac{a^2 \int (e+f x) \cosh ^2(c+d x) \, dx}{b^3}-\frac{a^3 \int \frac{(e+f x) \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b^3}+\frac{\int (e+f x) \cosh (4 c+4 d x) \, dx}{8 b}+\frac{(a f) \int \cosh ^3(c+d x) \, dx}{3 b^2 d}\\ &=-\frac{(e+f x)^2}{16 b f}-\frac{a^2 f \cosh ^2(c+d x)}{4 b^3 d^2}-\frac{a (e+f x) \cosh ^3(c+d x)}{3 b^2 d}+\frac{a^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^3 d}+\frac{(e+f x) \sinh (4 c+4 d x)}{32 b d}+\frac{a^4 \int (e+f x) \, dx}{b^5}-\frac{a^3 \int (e+f x) \sinh (c+d x) \, dx}{b^4}+\frac{a^2 \int (e+f x) \, dx}{2 b^3}-\frac{\left (a^3 \left (a^2+b^2\right )\right ) \int \frac{e+f x}{a+b \sinh (c+d x)} \, dx}{b^5}+\frac{(i a f) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-i \sinh (c+d x)\right )}{3 b^2 d^2}-\frac{f \int \sinh (4 c+4 d x) \, dx}{32 b d}\\ &=\frac{a^4 e x}{b^5}+\frac{a^2 e x}{2 b^3}+\frac{a^4 f x^2}{2 b^5}+\frac{a^2 f x^2}{4 b^3}-\frac{(e+f x)^2}{16 b f}-\frac{a^3 (e+f x) \cosh (c+d x)}{b^4 d}-\frac{a^2 f \cosh ^2(c+d x)}{4 b^3 d^2}-\frac{a (e+f x) \cosh ^3(c+d x)}{3 b^2 d}-\frac{f \cosh (4 c+4 d x)}{128 b d^2}+\frac{a f \sinh (c+d x)}{3 b^2 d^2}+\frac{a^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^3 d}+\frac{a f \sinh ^3(c+d x)}{9 b^2 d^2}+\frac{(e+f x) \sinh (4 c+4 d x)}{32 b d}-\frac{\left (2 a^3 \left (a^2+b^2\right )\right ) \int \frac{e^{c+d x} (e+f x)}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{b^5}+\frac{\left (a^3 f\right ) \int \cosh (c+d x) \, dx}{b^4 d}\\ &=\frac{a^4 e x}{b^5}+\frac{a^2 e x}{2 b^3}+\frac{a^4 f x^2}{2 b^5}+\frac{a^2 f x^2}{4 b^3}-\frac{(e+f x)^2}{16 b f}-\frac{a^3 (e+f x) \cosh (c+d x)}{b^4 d}-\frac{a^2 f \cosh ^2(c+d x)}{4 b^3 d^2}-\frac{a (e+f x) \cosh ^3(c+d x)}{3 b^2 d}-\frac{f \cosh (4 c+4 d x)}{128 b d^2}+\frac{a^3 f \sinh (c+d x)}{b^4 d^2}+\frac{a f \sinh (c+d x)}{3 b^2 d^2}+\frac{a^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^3 d}+\frac{a f \sinh ^3(c+d x)}{9 b^2 d^2}+\frac{(e+f x) \sinh (4 c+4 d x)}{32 b d}-\frac{\left (2 a^3 \sqrt{a^2+b^2}\right ) \int \frac{e^{c+d x} (e+f x)}{2 a-2 \sqrt{a^2+b^2}+2 b e^{c+d x}} \, dx}{b^4}+\frac{\left (2 a^3 \sqrt{a^2+b^2}\right ) \int \frac{e^{c+d x} (e+f x)}{2 a+2 \sqrt{a^2+b^2}+2 b e^{c+d x}} \, dx}{b^4}\\ &=\frac{a^4 e x}{b^5}+\frac{a^2 e x}{2 b^3}+\frac{a^4 f x^2}{2 b^5}+\frac{a^2 f x^2}{4 b^3}-\frac{(e+f x)^2}{16 b f}-\frac{a^3 (e+f x) \cosh (c+d x)}{b^4 d}-\frac{a^2 f \cosh ^2(c+d x)}{4 b^3 d^2}-\frac{a (e+f x) \cosh ^3(c+d x)}{3 b^2 d}-\frac{f \cosh (4 c+4 d x)}{128 b d^2}-\frac{a^3 \sqrt{a^2+b^2} (e+f x) \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^5 d}+\frac{a^3 \sqrt{a^2+b^2} (e+f x) \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^5 d}+\frac{a^3 f \sinh (c+d x)}{b^4 d^2}+\frac{a f \sinh (c+d x)}{3 b^2 d^2}+\frac{a^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^3 d}+\frac{a f \sinh ^3(c+d x)}{9 b^2 d^2}+\frac{(e+f x) \sinh (4 c+4 d x)}{32 b d}+\frac{\left (a^3 \sqrt{a^2+b^2} f\right ) \int \log \left (1+\frac{2 b e^{c+d x}}{2 a-2 \sqrt{a^2+b^2}}\right ) \, dx}{b^5 d}-\frac{\left (a^3 \sqrt{a^2+b^2} f\right ) \int \log \left (1+\frac{2 b e^{c+d x}}{2 a+2 \sqrt{a^2+b^2}}\right ) \, dx}{b^5 d}\\ &=\frac{a^4 e x}{b^5}+\frac{a^2 e x}{2 b^3}+\frac{a^4 f x^2}{2 b^5}+\frac{a^2 f x^2}{4 b^3}-\frac{(e+f x)^2}{16 b f}-\frac{a^3 (e+f x) \cosh (c+d x)}{b^4 d}-\frac{a^2 f \cosh ^2(c+d x)}{4 b^3 d^2}-\frac{a (e+f x) \cosh ^3(c+d x)}{3 b^2 d}-\frac{f \cosh (4 c+4 d x)}{128 b d^2}-\frac{a^3 \sqrt{a^2+b^2} (e+f x) \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^5 d}+\frac{a^3 \sqrt{a^2+b^2} (e+f x) \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^5 d}+\frac{a^3 f \sinh (c+d x)}{b^4 d^2}+\frac{a f \sinh (c+d x)}{3 b^2 d^2}+\frac{a^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^3 d}+\frac{a f \sinh ^3(c+d x)}{9 b^2 d^2}+\frac{(e+f x) \sinh (4 c+4 d x)}{32 b d}+\frac{\left (a^3 \sqrt{a^2+b^2} f\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 b x}{2 a-2 \sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^5 d^2}-\frac{\left (a^3 \sqrt{a^2+b^2} f\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 b x}{2 a+2 \sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^5 d^2}\\ &=\frac{a^4 e x}{b^5}+\frac{a^2 e x}{2 b^3}+\frac{a^4 f x^2}{2 b^5}+\frac{a^2 f x^2}{4 b^3}-\frac{(e+f x)^2}{16 b f}-\frac{a^3 (e+f x) \cosh (c+d x)}{b^4 d}-\frac{a^2 f \cosh ^2(c+d x)}{4 b^3 d^2}-\frac{a (e+f x) \cosh ^3(c+d x)}{3 b^2 d}-\frac{f \cosh (4 c+4 d x)}{128 b d^2}-\frac{a^3 \sqrt{a^2+b^2} (e+f x) \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^5 d}+\frac{a^3 \sqrt{a^2+b^2} (e+f x) \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^5 d}-\frac{a^3 \sqrt{a^2+b^2} f \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^5 d^2}+\frac{a^3 \sqrt{a^2+b^2} f \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^5 d^2}+\frac{a^3 f \sinh (c+d x)}{b^4 d^2}+\frac{a f \sinh (c+d x)}{3 b^2 d^2}+\frac{a^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^3 d}+\frac{a f \sinh ^3(c+d x)}{9 b^2 d^2}+\frac{(e+f x) \sinh (4 c+4 d x)}{32 b d}\\ \end{align*}

Mathematica [C]  time = 11.8889, size = 2915, normalized size = 6.15 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(e*(c/d + x - (2*a*ArcTan[(b - a*Tanh[(c + d*x)/2])/Sqrt[-a^2 - b^2]])/(Sqrt[-a^2 - b^2]*d)))/(8*b) - (f*(x^2
+ (2*a*((I*Pi*ArcTanh[(-b + a*Tanh[(c + d*x)/2])/Sqrt[a^2 + b^2]])/Sqrt[a^2 + b^2] + (2*((-I)*c + ArcCos[((-I
)*a)/b])*ArcTanh[((a + I*b)*Cot[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[-a^2 - b^2]] + ((-2*I)*c + Pi - (2*I)*d*x)
*ArcTanh[((a - I*b)*Tan[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[-a^2 - b^2]] - (ArcCos[((-I)*a)/b] + (2*I)*ArcTanh
[((a + I*b)*Cot[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[-a^2 - b^2]])*Log[((I*a + b)*(a + I*(b + Sqrt[-a^2 - b^2])
)*(-I + Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))/(b*(I*a + b + I*Sqrt[-a^2 - b^2]*Cot[((2*I)*c + Pi + (2*I)*d*x)/4]
))] - (ArcCos[((-I)*a)/b] - (2*I)*ArcTanh[((a + I*b)*Cot[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[-a^2 - b^2]])*Log
[((I*a + b)*(I*a - b + Sqrt[-a^2 - b^2])*(I + Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))/(b*(a - I*b + Sqrt[-a^2 - b^
2]*Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))] + (ArcCos[((-I)*a)/b] - (2*I)*ArcTanh[((a + I*b)*Cot[((2*I)*c + Pi + (
2*I)*d*x)/4])/Sqrt[-a^2 - b^2]] - (2*I)*ArcTanh[((a - I*b)*Tan[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[-a^2 - b^2]
])*Log[-(((-1)^(3/4)*Sqrt[-a^2 - b^2]*E^(-c/2 - (d*x)/2))/(Sqrt[2]*Sqrt[(-I)*b]*Sqrt[a + b*Sinh[c + d*x]]))] +
(ArcCos[((-I)*a)/b] + (2*I)*(ArcTanh[((a + I*b)*Cot[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[-a^2 - b^2]] + ArcTan
h[((a - I*b)*Tan[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[-a^2 - b^2]]))*Log[((-1)^(1/4)*Sqrt[-a^2 - b^2]*E^((c + d
*x)/2))/(Sqrt[2]*Sqrt[(-I)*b]*Sqrt[a + b*Sinh[c + d*x]])] + I*(PolyLog[2, ((I*a + Sqrt[-a^2 - b^2])*(I*a + b -
I*Sqrt[-a^2 - b^2]*Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))/(b*(I*a + b + I*Sqrt[-a^2 - b^2]*Cot[((2*I)*c + Pi + (
2*I)*d*x)/4]))] - PolyLog[2, ((a + I*Sqrt[-a^2 - b^2])*(-a + I*b + Sqrt[-a^2 - b^2]*Cot[((2*I)*c + Pi + (2*I)*
d*x)/4]))/(b*(I*a + b + I*Sqrt[-a^2 - b^2]*Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))]))/Sqrt[-a^2 - b^2]))/d^2))/(16
*b) - (e*((4*a^2 + b^2)*(c + d*x) - (2*a*(4*a^2 + 3*b^2)*ArcTan[(b - a*Tanh[(c + d*x)/2])/Sqrt[-a^2 - b^2]])/S
qrt[-a^2 - b^2] - 4*a*b*Cosh[c + d*x] + b^2*Sinh[2*(c + d*x)]))/(16*b^3*d) - (f*((4*a^2 + b^2)*(-c + d*x)*(c +
d*x) - 8*a*b*d*x*Cosh[c + d*x] - b^2*Cosh[2*(c + d*x)] - (2*a*(4*a^2 + 3*b^2)*(2*c*ArcTanh[(a + b*Cosh[c + d*
x] + b*Sinh[c + d*x])/Sqrt[a^2 + b^2]] + (c + d*x)*Log[1 + (b*(Cosh[c + d*x] + Sinh[c + d*x]))/(a - Sqrt[a^2 +
b^2])] - (c + d*x)*Log[1 + (b*(Cosh[c + d*x] + Sinh[c + d*x]))/(a + Sqrt[a^2 + b^2])] + PolyLog[2, (b*(Cosh[c
+ d*x] + Sinh[c + d*x]))/(-a + Sqrt[a^2 + b^2])] - PolyLog[2, -((b*(Cosh[c + d*x] + Sinh[c + d*x]))/(a + Sqrt
[a^2 + b^2]))]))/Sqrt[a^2 + b^2] + 8*a*b*Sinh[c + d*x] + 2*b^2*d*x*Sinh[2*(c + d*x)]))/(32*b^3*d^2) + (e*(6*(1
6*a^4 + 12*a^2*b^2 + b^4)*(c + d*x) - (12*a*(16*a^4 + 20*a^2*b^2 + 5*b^4)*ArcTan[(b - a*Tanh[(c + d*x)/2])/Sqr
t[-a^2 - b^2]])/Sqrt[-a^2 - b^2] - 48*a*b*(2*a^2 + b^2)*Cosh[c + d*x] - 8*a*b^3*Cosh[3*(c + d*x)] + 6*b^2*(4*a
^2 + b^2)*Sinh[2*(c + d*x)] + 3*b^4*Sinh[4*(c + d*x)]))/(96*b^5*d) + (f*(-576*a^4*Sqrt[a^2 + b^2]*c^2 - 432*a^
2*b^2*Sqrt[a^2 + b^2]*c^2 - 36*b^4*Sqrt[a^2 + b^2]*c^2 + 576*a^4*Sqrt[a^2 + b^2]*d^2*x^2 + 432*a^2*b^2*Sqrt[a^
2 + b^2]*d^2*x^2 + 36*b^4*Sqrt[a^2 + b^2]*d^2*x^2 - 2304*a^5*c*ArcTanh[(a + b*Cosh[c + d*x] + b*Sinh[c + d*x])
/Sqrt[a^2 + b^2]] - 2880*a^3*b^2*c*ArcTanh[(a + b*Cosh[c + d*x] + b*Sinh[c + d*x])/Sqrt[a^2 + b^2]] - 720*a*b^
4*c*ArcTanh[(a + b*Cosh[c + d*x] + b*Sinh[c + d*x])/Sqrt[a^2 + b^2]] - 1152*a^3*b*Sqrt[a^2 + b^2]*d*x*Cosh[c +
d*x] - 576*a*b^3*Sqrt[a^2 + b^2]*d*x*Cosh[c + d*x] - 144*a^2*b^2*Sqrt[a^2 + b^2]*Cosh[2*(c + d*x)] - 36*b^4*S
qrt[a^2 + b^2]*Cosh[2*(c + d*x)] - 96*a*b^3*Sqrt[a^2 + b^2]*d*x*Cosh[3*(c + d*x)] - 9*b^4*Sqrt[a^2 + b^2]*Cosh
[4*(c + d*x)] - 1152*a^5*c*Log[1 + (b*(Cosh[c + d*x] + Sinh[c + d*x]))/(a - Sqrt[a^2 + b^2])] - 1440*a^3*b^2*c
*Log[1 + (b*(Cosh[c + d*x] + Sinh[c + d*x]))/(a - Sqrt[a^2 + b^2])] - 360*a*b^4*c*Log[1 + (b*(Cosh[c + d*x] +
Sinh[c + d*x]))/(a - Sqrt[a^2 + b^2])] - 1152*a^5*d*x*Log[1 + (b*(Cosh[c + d*x] + Sinh[c + d*x]))/(a - Sqrt[a^
2 + b^2])] - 1440*a^3*b^2*d*x*Log[1 + (b*(Cosh[c + d*x] + Sinh[c + d*x]))/(a - Sqrt[a^2 + b^2])] - 360*a*b^4*d
*x*Log[1 + (b*(Cosh[c + d*x] + Sinh[c + d*x]))/(a - Sqrt[a^2 + b^2])] + 1152*a^5*c*Log[1 + (b*(Cosh[c + d*x] +
Sinh[c + d*x]))/(a + Sqrt[a^2 + b^2])] + 1440*a^3*b^2*c*Log[1 + (b*(Cosh[c + d*x] + Sinh[c + d*x]))/(a + Sqrt
[a^2 + b^2])] + 360*a*b^4*c*Log[1 + (b*(Cosh[c + d*x] + Sinh[c + d*x]))/(a + Sqrt[a^2 + b^2])] + 1152*a^5*d*x*
Log[1 + (b*(Cosh[c + d*x] + Sinh[c + d*x]))/(a + Sqrt[a^2 + b^2])] + 1440*a^3*b^2*d*x*Log[1 + (b*(Cosh[c + d*x
] + Sinh[c + d*x]))/(a + Sqrt[a^2 + b^2])] + 360*a*b^4*d*x*Log[1 + (b*(Cosh[c + d*x] + Sinh[c + d*x]))/(a + Sq
rt[a^2 + b^2])] - 72*a*(16*a^4 + 20*a^2*b^2 + 5*b^4)*PolyLog[2, (b*(Cosh[c + d*x] + Sinh[c + d*x]))/(-a + Sqrt
[a^2 + b^2])] + 72*a*(16*a^4 + 20*a^2*b^2 + 5*b^4)*PolyLog[2, -((b*(Cosh[c + d*x] + Sinh[c + d*x]))/(a + Sqrt[
a^2 + b^2]))] + 1152*a^3*b*Sqrt[a^2 + b^2]*Sinh[c + d*x] + 576*a*b^3*Sqrt[a^2 + b^2]*Sinh[c + d*x] + 288*a^2*b
^2*Sqrt[a^2 + b^2]*d*x*Sinh[2*(c + d*x)] + 72*b^4*Sqrt[a^2 + b^2]*d*x*Sinh[2*(c + d*x)] + 32*a*b^3*Sqrt[a^2 +
b^2]*Sinh[3*(c + d*x)] + 36*b^4*Sqrt[a^2 + b^2]*d*x*Sinh[4*(c + d*x)]))/(1152*b^5*Sqrt[a^2 + b^2]*d^2)

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Maple [B]  time = 0.106, size = 1213, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*f*x^2/b-a^5/b^5/d^2*f/(a^2+b^2)^(1/2)*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))+a^5/
b^5/d^2*f/(a^2+b^2)^(1/2)*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))+2*a^5/b^5/d*e/(a^2+b^2)^
(1/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+1/2*a^4*f*x^2/b^5+1/4*a^2*f*x^2/b^3-1/8*a*(4*a^2+b^2)*
(d*f*x+d*e+f)/b^4/d^2*exp(-d*x-c)+a^4*e*x/b^5+1/2*a^2*e*x/b^3-1/8*e*x/b+2*a^3/b^3/d*e/(a^2+b^2)^(1/2)*arctanh(
1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-a^3/b^3/d^2*f/(a^2+b^2)^(1/2)*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a
)/(-a+(a^2+b^2)^(1/2)))+a^3/b^3/d^2*f/(a^2+b^2)^(1/2)*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2
)))-2*a^5/b^5/d^2*f*c/(a^2+b^2)^(1/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-a^5/b^5/d*f/(a^2+b^2)^
(1/2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x-a^5/b^5/d^2*f/(a^2+b^2)^(1/2)*ln((-b*exp(d*
x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c+a^5/b^5/d*f/(a^2+b^2)^(1/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a
)/(a+(a^2+b^2)^(1/2)))*x+a^5/b^5/d^2*f/(a^2+b^2)^(1/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2))
)*c-a^3/b^3/d*f/(a^2+b^2)^(1/2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x-a^3/b^3/d^2*f/(a^
2+b^2)^(1/2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c+a^3/b^3/d*f/(a^2+b^2)^(1/2)*ln((b*ex
p(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x+a^3/b^3/d^2*f/(a^2+b^2)^(1/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1
/2)+a)/(a+(a^2+b^2)^(1/2)))*c-2*a^3/b^3/d^2*f*c/(a^2+b^2)^(1/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/
2))+1/256*(4*d*f*x+4*d*e-f)/d^2/b*exp(4*d*x+4*c)-1/72*a*(3*d*f*x+3*d*e-f)/b^2/d^2*exp(3*d*x+3*c)-1/8*a*(4*a^2*
d*f*x+b^2*d*f*x+4*a^2*d*e+b^2*d*e-4*a^2*f-b^2*f)/b^4/d^2*exp(d*x+c)-1/256*(4*d*f*x+4*d*e+f)/d^2/b*exp(-4*d*x-4
*c)-1/72*a*(3*d*f*x+3*d*e+f)/b^2/d^2*exp(-3*d*x-3*c)+1/16*a^2*(2*d*f*x+2*d*e-f)/b^3/d^2*exp(2*d*x+2*c)-1/16*a^
2*(2*d*f*x+2*d*e+f)/b^3/d^2*exp(-2*d*x-2*c)

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.79816, size = 7611, normalized size = 16.06 \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)*cosh(d*x+c)^2*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

1/2304*(9*(4*b^4*d*f*x + 4*b^4*d*e - b^4*f)*cosh(d*x + c)^8 + 9*(4*b^4*d*f*x + 4*b^4*d*e - b^4*f)*sinh(d*x + c
)^8 - 32*(3*a*b^3*d*f*x + 3*a*b^3*d*e - a*b^3*f)*cosh(d*x + c)^7 - 8*(12*a*b^3*d*f*x + 12*a*b^3*d*e - 4*a*b^3*
f - 9*(4*b^4*d*f*x + 4*b^4*d*e - b^4*f)*cosh(d*x + c))*sinh(d*x + c)^7 - 36*b^4*d*f*x + 144*(2*a^2*b^2*d*f*x +
2*a^2*b^2*d*e - a^2*b^2*f)*cosh(d*x + c)^6 + 4*(72*a^2*b^2*d*f*x + 72*a^2*b^2*d*e - 36*a^2*b^2*f + 63*(4*b^4*
d*f*x + 4*b^4*d*e - b^4*f)*cosh(d*x + c)^2 - 56*(3*a*b^3*d*f*x + 3*a*b^3*d*e - a*b^3*f)*cosh(d*x + c))*sinh(d*
x + c)^6 - 36*b^4*d*e - 288*((4*a^3*b + a*b^3)*d*f*x + (4*a^3*b + a*b^3)*d*e - (4*a^3*b + a*b^3)*f)*cosh(d*x +
c)^5 - 24*(12*(4*a^3*b + a*b^3)*d*f*x - 21*(4*b^4*d*f*x + 4*b^4*d*e - b^4*f)*cosh(d*x + c)^3 + 12*(4*a^3*b +
a*b^3)*d*e + 28*(3*a*b^3*d*f*x + 3*a*b^3*d*e - a*b^3*f)*cosh(d*x + c)^2 - 12*(4*a^3*b + a*b^3)*f - 36*(2*a^2*b
^2*d*f*x + 2*a^2*b^2*d*e - a^2*b^2*f)*cosh(d*x + c))*sinh(d*x + c)^5 - 9*b^4*f + 144*((8*a^4 + 4*a^2*b^2 - b^4
)*d^2*f*x^2 + 2*(8*a^4 + 4*a^2*b^2 - b^4)*d^2*e*x)*cosh(d*x + c)^4 + 2*(72*(8*a^4 + 4*a^2*b^2 - b^4)*d^2*f*x^2
+ 144*(8*a^4 + 4*a^2*b^2 - b^4)*d^2*e*x + 315*(4*b^4*d*f*x + 4*b^4*d*e - b^4*f)*cosh(d*x + c)^4 - 560*(3*a*b^
3*d*f*x + 3*a*b^3*d*e - a*b^3*f)*cosh(d*x + c)^3 + 1080*(2*a^2*b^2*d*f*x + 2*a^2*b^2*d*e - a^2*b^2*f)*cosh(d*x
+ c)^2 - 720*((4*a^3*b + a*b^3)*d*f*x + (4*a^3*b + a*b^3)*d*e - (4*a^3*b + a*b^3)*f)*cosh(d*x + c))*sinh(d*x
+ c)^4 - 288*((4*a^3*b + a*b^3)*d*f*x + (4*a^3*b + a*b^3)*d*e + (4*a^3*b + a*b^3)*f)*cosh(d*x + c)^3 + 8*(63*(
4*b^4*d*f*x + 4*b^4*d*e - b^4*f)*cosh(d*x + c)^5 - 140*(3*a*b^3*d*f*x + 3*a*b^3*d*e - a*b^3*f)*cosh(d*x + c)^4
- 36*(4*a^3*b + a*b^3)*d*f*x + 360*(2*a^2*b^2*d*f*x + 2*a^2*b^2*d*e - a^2*b^2*f)*cosh(d*x + c)^3 - 36*(4*a^3*
b + a*b^3)*d*e - 360*((4*a^3*b + a*b^3)*d*f*x + (4*a^3*b + a*b^3)*d*e - (4*a^3*b + a*b^3)*f)*cosh(d*x + c)^2 -
36*(4*a^3*b + a*b^3)*f + 72*((8*a^4 + 4*a^2*b^2 - b^4)*d^2*f*x^2 + 2*(8*a^4 + 4*a^2*b^2 - b^4)*d^2*e*x)*cosh(
d*x + c))*sinh(d*x + c)^3 - 144*(2*a^2*b^2*d*f*x + 2*a^2*b^2*d*e + a^2*b^2*f)*cosh(d*x + c)^2 - 12*(24*a^2*b^2
*d*f*x - 21*(4*b^4*d*f*x + 4*b^4*d*e - b^4*f)*cosh(d*x + c)^6 + 24*a^2*b^2*d*e + 56*(3*a*b^3*d*f*x + 3*a*b^3*d
*e - a*b^3*f)*cosh(d*x + c)^5 + 12*a^2*b^2*f - 180*(2*a^2*b^2*d*f*x + 2*a^2*b^2*d*e - a^2*b^2*f)*cosh(d*x + c)
^4 + 240*((4*a^3*b + a*b^3)*d*f*x + (4*a^3*b + a*b^3)*d*e - (4*a^3*b + a*b^3)*f)*cosh(d*x + c)^3 - 72*((8*a^4
+ 4*a^2*b^2 - b^4)*d^2*f*x^2 + 2*(8*a^4 + 4*a^2*b^2 - b^4)*d^2*e*x)*cosh(d*x + c)^2 + 72*((4*a^3*b + a*b^3)*d*
f*x + (4*a^3*b + a*b^3)*d*e + (4*a^3*b + a*b^3)*f)*cosh(d*x + c))*sinh(d*x + c)^2 - 2304*(a^3*b*f*cosh(d*x + c
)^4 + 4*a^3*b*f*cosh(d*x + c)^3*sinh(d*x + c) + 6*a^3*b*f*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*a^3*b*f*cosh(d*x
+ c)*sinh(d*x + c)^3 + a^3*b*f*sinh(d*x + c)^4)*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) + 2304*(a^3*b*f*cosh(d*x + c)^4 + 4*
a^3*b*f*cosh(d*x + c)^3*sinh(d*x + c) + 6*a^3*b*f*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*a^3*b*f*cosh(d*x + c)*si
nh(d*x + c)^3 + a^3*b*f*sinh(d*x + c)^4)*sqrt((a^2 + b^2)/b^2)*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) - (b*c
osh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + 2304*((a^3*b*d*e - a^3*b*c*f)*cosh(d*x + c
)^4 + 4*(a^3*b*d*e - a^3*b*c*f)*cosh(d*x + c)^3*sinh(d*x + c) + 6*(a^3*b*d*e - a^3*b*c*f)*cosh(d*x + c)^2*sinh
(d*x + c)^2 + 4*(a^3*b*d*e - a^3*b*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^3*b*d*e - a^3*b*c*f)*sinh(d*x + c)^
4)*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) - 2304*(
(a^3*b*d*e - a^3*b*c*f)*cosh(d*x + c)^4 + 4*(a^3*b*d*e - a^3*b*c*f)*cosh(d*x + c)^3*sinh(d*x + c) + 6*(a^3*b*d
*e - a^3*b*c*f)*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*(a^3*b*d*e - a^3*b*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (a
^3*b*d*e - a^3*b*c*f)*sinh(d*x + c)^4)*sqrt((a^2 + b^2)/b^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*s
qrt((a^2 + b^2)/b^2) + 2*a) - 2304*((a^3*b*d*f*x + a^3*b*c*f)*cosh(d*x + c)^4 + 4*(a^3*b*d*f*x + a^3*b*c*f)*co
sh(d*x + c)^3*sinh(d*x + c) + 6*(a^3*b*d*f*x + a^3*b*c*f)*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*(a^3*b*d*f*x + a
^3*b*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^3*b*d*f*x + a^3*b*c*f)*sinh(d*x + c)^4)*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) + 230
4*((a^3*b*d*f*x + a^3*b*c*f)*cosh(d*x + c)^4 + 4*(a^3*b*d*f*x + a^3*b*c*f)*cosh(d*x + c)^3*sinh(d*x + c) + 6*(
a^3*b*d*f*x + a^3*b*c*f)*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*(a^3*b*d*f*x + a^3*b*c*f)*cosh(d*x + c)*sinh(d*x
+ c)^3 + (a^3*b*d*f*x + a^3*b*c*f)*sinh(d*x + c)^4)*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) - 32*(3*a*b^3*d*f*x + 3*a*b^3*d*e + a*
b^3*f)*cosh(d*x + c) + 8*(9*(4*b^4*d*f*x + 4*b^4*d*e - b^4*f)*cosh(d*x + c)^7 - 12*a*b^3*d*f*x - 28*(3*a*b^3*d
*f*x + 3*a*b^3*d*e - a*b^3*f)*cosh(d*x + c)^6 - 12*a*b^3*d*e + 108*(2*a^2*b^2*d*f*x + 2*a^2*b^2*d*e - a^2*b^2*
f)*cosh(d*x + c)^5 - 4*a*b^3*f - 180*((4*a^3*b + a*b^3)*d*f*x + (4*a^3*b + a*b^3)*d*e - (4*a^3*b + a*b^3)*f)*c
osh(d*x + c)^4 + 72*((8*a^4 + 4*a^2*b^2 - b^4)*d^2*f*x^2 + 2*(8*a^4 + 4*a^2*b^2 - b^4)*d^2*e*x)*cosh(d*x + c)^
3 - 108*((4*a^3*b + a*b^3)*d*f*x + (4*a^3*b + a*b^3)*d*e + (4*a^3*b + a*b^3)*f)*cosh(d*x + c)^2 - 36*(2*a^2*b^
2*d*f*x + 2*a^2*b^2*d*e + a^2*b^2*f)*cosh(d*x + c))*sinh(d*x + c))/(b^5*d^2*cosh(d*x + c)^4 + 4*b^5*d^2*cosh(d
*x + c)^3*sinh(d*x + c) + 6*b^5*d^2*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*b^5*d^2*cosh(d*x + c)*sinh(d*x + c)^3
+ b^5*d^2*sinh(d*x + c)^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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