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

Optimal. Leaf size=499 \[ \frac{a^2 f \left (a^2+b^2\right ) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^5 d^2}+\frac{a^2 f \left (a^2+b^2\right ) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^5 d^2}+\frac{a^3 f \cosh (c+d x)}{b^4 d^2}-\frac{a^2 f \sinh (c+d x) \cosh (c+d x)}{4 b^3 d^2}+\frac{a^2 \left (a^2+b^2\right ) (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b^5 d}+\frac{a^2 \left (a^2+b^2\right ) (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b^5 d}+\frac{a^2 (e+f x) \sinh ^2(c+d x)}{2 b^3 d}-\frac{a^3 (e+f x) \sinh (c+d x)}{b^4 d}+\frac{a^2 f x}{4 b^3 d}-\frac{a^2 \left (a^2+b^2\right ) (e+f x)^2}{2 b^5 f}+\frac{a f \cosh ^3(c+d x)}{9 b^2 d^2}+\frac{2 a f \cosh (c+d x)}{3 b^2 d^2}-\frac{2 a (e+f x) \sinh (c+d x)}{3 b^2 d}-\frac{a (e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 b^2 d}-\frac{f \sinh (c+d x) \cosh ^3(c+d x)}{16 b d^2}-\frac{3 f \sinh (c+d x) \cosh (c+d x)}{32 b d^2}+\frac{(e+f x) \cosh ^4(c+d x)}{4 b d}-\frac{3 f x}{32 b d} \]

[Out]

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

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Rubi [A]  time = 0.675401, antiderivative size = 499, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 13, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.382, Rules used = {5579, 5447, 2635, 8, 3310, 3296, 2638, 5565, 5446, 5561, 2190, 2279, 2391} \[ \frac{a^2 f \left (a^2+b^2\right ) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^5 d^2}+\frac{a^2 f \left (a^2+b^2\right ) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^5 d^2}+\frac{a^3 f \cosh (c+d x)}{b^4 d^2}-\frac{a^2 f \sinh (c+d x) \cosh (c+d x)}{4 b^3 d^2}+\frac{a^2 \left (a^2+b^2\right ) (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b^5 d}+\frac{a^2 \left (a^2+b^2\right ) (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b^5 d}+\frac{a^2 (e+f x) \sinh ^2(c+d x)}{2 b^3 d}-\frac{a^3 (e+f x) \sinh (c+d x)}{b^4 d}+\frac{a^2 f x}{4 b^3 d}-\frac{a^2 \left (a^2+b^2\right ) (e+f x)^2}{2 b^5 f}+\frac{a f \cosh ^3(c+d x)}{9 b^2 d^2}+\frac{2 a f \cosh (c+d x)}{3 b^2 d^2}-\frac{2 a (e+f x) \sinh (c+d x)}{3 b^2 d}-\frac{a (e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 b^2 d}-\frac{f \sinh (c+d x) \cosh ^3(c+d x)}{16 b d^2}-\frac{3 f \sinh (c+d x) \cosh (c+d x)}{32 b d^2}+\frac{(e+f x) \cosh ^4(c+d x)}{4 b d}-\frac{3 f x}{32 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*f*x)/(4*b^3*d) - (3*f*x)/(32*b*d) - (a^2*(a^2 + b^2)*(e + f*x)^2)/(2*b^5*f) + (a^3*f*Cosh[c + d*x])/(b^4*
d^2) + (2*a*f*Cosh[c + d*x])/(3*b^2*d^2) + (a*f*Cosh[c + d*x]^3)/(9*b^2*d^2) + ((e + f*x)*Cosh[c + d*x]^4)/(4*
b*d) + (a^2*(a^2 + b^2)*(e + f*x)*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)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^5*d) + (a^2*(a^2 + b^2)*f*PolyLog[2, -((b*E^(c + d
*x))/(a - Sqrt[a^2 + b^2]))])/(b^5*d^2) + (a^2*(a^2 + b^2)*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]
))])/(b^5*d^2) - (a^3*(e + f*x)*Sinh[c + d*x])/(b^4*d) - (2*a*(e + f*x)*Sinh[c + d*x])/(3*b^2*d) - (a^2*f*Cosh
[c + d*x]*Sinh[c + d*x])/(4*b^3*d^2) - (3*f*Cosh[c + d*x]*Sinh[c + d*x])/(32*b*d^2) - (a*(e + f*x)*Cosh[c + d*
x]^2*Sinh[c + d*x])/(3*b^2*d) - (f*Cosh[c + d*x]^3*Sinh[c + d*x])/(16*b*d^2) + (a^2*(e + f*x)*Sinh[c + d*x]^2)
/(2*b^3*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 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 2635

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

Rule 8

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

Rule 3310

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

Rule 3296

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

Rule 2638

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

Rule 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 5446

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 5561

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

Mathematica [A]  time = 3.20973, size = 853, normalized size = 1.71 \[ \frac{-144 d e \log (a+b \sinh (c+d x)) b^4+72 f \left (d x \left (d x-2 \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right )-2 \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right )\right )-2 \text{PolyLog}\left (2,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )-2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )\right ) b^4+72 d e \left (2 b^2 \sinh ^2(c+d x)-4 a b \sinh (c+d x)+\left (4 a^2+b^2\right ) \log (a+b \sinh (c+d x))\right ) b^2+36 f \left (2 d x \cosh (2 (c+d x)) b^2-\sinh (2 (c+d x)) b^2+8 a \cosh (c+d x) b-8 a d x \sinh (c+d x) b+2 \left (4 a^2+b^2\right ) \left (-\frac{1}{2} (c+d x)^2+\log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) (c+d x)+\log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) (c+d x)-c \log (a+b \sinh (c+d x))+\text{PolyLog}\left (2,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )+\text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )\right )\right ) b^2+24 d e \left (12 b^4 \sinh ^4(c+d x)-16 a b^3 \sinh ^3(c+d x)+6 b^2 \left (4 a^2+3 b^2\right ) \sinh ^2(c+d x)-12 a b \left (4 a^2+3 b^2\right ) \sinh (c+d x)+3 \left (16 a^4+12 b^2 a^2+b^4\right ) \log (a+b \sinh (c+d x))\right )+f \left (36 d x \cosh (4 (c+d x)) b^4-9 \sinh (4 (c+d x)) b^4+32 a \cosh (3 (c+d x)) b^3-96 a d x \sinh (3 (c+d x)) b^3+72 \left (4 a^2+b^2\right ) d x \cosh (2 (c+d x)) b^2-36 \left (4 a^2+b^2\right ) \sinh (2 (c+d x)) b^2+576 a \left (2 a^2+b^2\right ) \cosh (c+d x) b-576 a \left (2 a^2+b^2\right ) d x \sinh (c+d x) b+72 \left (16 a^4+12 b^2 a^2+b^4\right ) \left (-\frac{1}{2} (c+d x)^2+\log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) (c+d x)+\log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) (c+d x)-c \log (a+b \sinh (c+d x))+\text{PolyLog}\left (2,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )+\text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )\right )\right )}{1152 b^5 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-144*b^4*d*e*Log[a + b*Sinh[c + d*x]] + 72*b^4*f*(d*x*(d*x - 2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])]
- 2*Log[1 + (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])] -
2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]) + 72*b^2*d*e*((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) + 24*d*e*(3*(16*a^4 + 12*a^2*b^2 + b^4)*Log[a + b*Sinh[c + d*x]]
- 12*a*b*(4*a^2 + 3*b^2)*Sinh[c + d*x] + 6*b^2*(4*a^2 + 3*b^2)*Sinh[c + d*x]^2 - 16*a*b^3*Sinh[c + d*x]^3 + 1
2*b^4*Sinh[c + d*x]^4) + 36*b^2*f*(8*a*b*Cosh[c + d*x] + 2*b^2*d*x*Cosh[2*(c + d*x)] + 2*(4*a^2 + b^2)*(-(c +
d*x)^2/2 + (c + d*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + (c + d*x)*Log[1 + (b*E^(c + d*x))/(a + S
qrt[a^2 + b^2])] - c*Log[a + b*Sinh[c + d*x]] + PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^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)]) + f*(576*a*b*(
2*a^2 + b^2)*Cosh[c + d*x] + 72*b^2*(4*a^2 + b^2)*d*x*Cosh[2*(c + d*x)] + 32*a*b^3*Cosh[3*(c + d*x)] + 36*b^4*
d*x*Cosh[4*(c + d*x)] + 72*(16*a^4 + 12*a^2*b^2 + b^4)*(-(c + d*x)^2/2 + (c + d*x)*Log[1 + (b*E^(c + d*x))/(a
- Sqrt[a^2 + b^2])] + (c + d*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - c*Log[a + b*Sinh[c + d*x]] +
PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]) - 5
76*a*b*(2*a^2 + b^2)*d*x*Sinh[c + d*x] - 36*b^2*(4*a^2 + b^2)*Sinh[2*(c + d*x)] - 96*a*b^3*d*x*Sinh[3*(c + d*x
)] - 9*b^4*Sinh[4*(c + d*x)]))/(1152*b^5*d^2)

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Maple [B]  time = 0.118, size = 1217, normalized size = 2.4 \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)^3*sinh(d*x+c)^2/(a+b*sinh(d*x+c)),x)

[Out]

a^4/b^5/d^2*f*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c+a^4/b^5/d*f*ln((b*exp(d*x+c)+(a^2+b
^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x+a^4/b^5/d^2*f*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c+a
^4/b^5/d*f*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x-a^4/b^5/d^2*f*c*ln(b*exp(2*d*x+2*c)+2*
a*exp(d*x+c)-b)+2*a^4/b^5/d^2*f*c*ln(exp(d*x+c))-2*a^4/b^5/d*f*c*x-1/2*a^4*f*x^2/b^5-1/2*a^2*f*x^2/b^3+1/8*a*(
4*a^2+3*b^2)*(d*f*x+d*e+f)/b^4/d^2*exp(-d*x-c)-1/b^3/d^2*a^2*f*c*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)+2/b^3/d
^2*a^2*f*c*ln(exp(d*x+c))+1/b^3/d*a^2*f*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x+1/b^3/d^2*a
^2*f*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c+1/b^3/d*a^2*f*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2
)-a)/(-a+(a^2+b^2)^(1/2)))*x+1/b^3/d^2*a^2*f*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c-2/b^
3/d*a^2*f*c*x+a^4*e*x/b^5+a^2*e*x/b^3+1/256*(4*d*f*x+4*d*e-f)/d^2/b*exp(4*d*x+4*c)+1/32*(4*a^2*d*f*x+2*b^2*d*f
*x+4*a^2*d*e+2*b^2*d*e-2*a^2*f-b^2*f)/b^3/d^2*exp(2*d*x+2*c)-1/72*a*(3*d*f*x+3*d*e-f)/b^2/d^2*exp(3*d*x+3*c)-1
/b^3/d^2*a^2*f*c^2+1/b^3/d^2*a^2*f*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))+1/b^3/d^2*a^2*f
*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))+1/b^3/d*a^2*e*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c
)-b)-2/b^3/d*a^2*e*ln(exp(d*x+c))+1/256*(4*d*f*x+4*d*e+f)/d^2/b*exp(-4*d*x-4*c)+1/32*(2*a^2+b^2)*(2*d*f*x+2*d*
e+f)/b^3/d^2*exp(-2*d*x-2*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+3*b^2*d*f*x+4
*a^2*d*e+3*b^2*d*e-4*a^2*f-3*b^2*f)/b^4/d^2*exp(d*x+c)-a^4/b^5/d^2*f*c^2+a^4/b^5/d*e*ln(b*exp(2*d*x+2*c)+2*a*e
xp(d*x+c)-b)-2*a^4/b^5/d*e*ln(exp(d*x+c))+a^4/b^5/d^2*f*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1
/2)))+a^4/b^5/d^2*f*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \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)^3*sinh(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

-1/192*e*((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)) + 1/2304*f*((1152*(a^4*d^2*e^(4*c) + a^2*b
^2*d^2*e^(4*c))*x^2 + 9*(4*b^4*d*x*e^(8*c) - b^4*e^(8*c))*e^(4*d*x) - 32*(3*a*b^3*d*x*e^(7*c) - a*b^3*e^(7*c))
*e^(3*d*x) - 72*(2*a^2*b^2*e^(6*c) + b^4*e^(6*c) - 2*(2*a^2*b^2*d*e^(6*c) + b^4*d*e^(6*c))*x)*e^(2*d*x) + 288*
(4*a^3*b*e^(5*c) + 3*a*b^3*e^(5*c) - (4*a^3*b*d*e^(5*c) + 3*a*b^3*d*e^(5*c))*x)*e^(d*x) + 288*(4*a^3*b*e^(3*c)
+ 3*a*b^3*e^(3*c) + (4*a^3*b*d*e^(3*c) + 3*a*b^3*d*e^(3*c))*x)*e^(-d*x) + 72*(2*a^2*b^2*e^(2*c) + b^4*e^(2*c)
+ 2*(2*a^2*b^2*d*e^(2*c) + b^4*d*e^(2*c))*x)*e^(-2*d*x) + 32*(3*a*b^3*d*x*e^c + a*b^3*e^c)*e^(-3*d*x) + 9*(4*
b^4*d*x + b^4)*e^(-4*d*x))*e^(-4*c)/(b^5*d^2) - 72*integrate(64*((a^5*e^c + a^3*b^2*e^c)*x*e^(d*x) - (a^4*b +
a^2*b^3)*x)/(b^6*e^(2*d*x + 2*c) + 2*a*b^5*e^(d*x + c) - b^6), x))

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Fricas [B]  time = 2.75812, size = 8818, normalized size = 17.67 \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)^3*sinh(d*x+c)^2/(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 + 72*(2*(2*a^2*b^2 + b^4
)*d*f*x + 2*(2*a^2*b^2 + b^4)*d*e - (2*a^2*b^2 + b^4)*f)*cosh(d*x + c)^6 + 4*(36*(2*a^2*b^2 + b^4)*d*f*x + 36*
(2*a^2*b^2 + b^4)*d*e + 63*(4*b^4*d*f*x + 4*b^4*d*e - b^4*f)*cosh(d*x + c)^2 - 18*(2*a^2*b^2 + b^4)*f - 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 + 3*a*b^3)*d*
f*x + (4*a^3*b + 3*a*b^3)*d*e - (4*a^3*b + 3*a*b^3)*f)*cosh(d*x + c)^5 - 24*(12*(4*a^3*b + 3*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 + 3*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 + 3*a*b^3)*f - 18*(2*(2*a^2*b^2 + b^4)*d*f*x + 2*(2*a^2*b^2 + b^4
)*d*e - (2*a^2*b^2 + b^4)*f)*cosh(d*x + c))*sinh(d*x + c)^5 + 9*b^4*f - 1152*((a^4 + a^2*b^2)*d^2*f*x^2 + 2*(a
^4 + a^2*b^2)*d^2*e*x + 4*(a^4 + a^2*b^2)*c*d*e - 2*(a^4 + a^2*b^2)*c^2*f)*cosh(d*x + c)^4 - 2*(576*(a^4 + a^2
*b^2)*d^2*f*x^2 + 1152*(a^4 + a^2*b^2)*d^2*e*x - 315*(4*b^4*d*f*x + 4*b^4*d*e - b^4*f)*cosh(d*x + c)^4 + 2304*
(a^4 + a^2*b^2)*c*d*e - 1152*(a^4 + a^2*b^2)*c^2*f + 560*(3*a*b^3*d*f*x + 3*a*b^3*d*e - a*b^3*f)*cosh(d*x + c)
^3 - 540*(2*(2*a^2*b^2 + b^4)*d*f*x + 2*(2*a^2*b^2 + b^4)*d*e - (2*a^2*b^2 + b^4)*f)*cosh(d*x + c)^2 + 720*((4
*a^3*b + 3*a*b^3)*d*f*x + (4*a^3*b + 3*a*b^3)*d*e - (4*a^3*b + 3*a*b^3)*f)*cosh(d*x + c))*sinh(d*x + c)^4 + 28
8*((4*a^3*b + 3*a*b^3)*d*f*x + (4*a^3*b + 3*a*b^3)*d*e + (4*a^3*b + 3*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 + 3*a*b^3)*d*f*x + 180*(2*(2*a^2*b^2 + b^4)*d*f*x + 2*(2*a^2*b^2 + b^4)*d*e - (2*a^2*b^2 + b^4)*f)*c
osh(d*x + c)^3 + 36*(4*a^3*b + 3*a*b^3)*d*e - 360*((4*a^3*b + 3*a*b^3)*d*f*x + (4*a^3*b + 3*a*b^3)*d*e - (4*a^
3*b + 3*a*b^3)*f)*cosh(d*x + c)^2 + 36*(4*a^3*b + 3*a*b^3)*f - 576*((a^4 + a^2*b^2)*d^2*f*x^2 + 2*(a^4 + a^2*b
^2)*d^2*e*x + 4*(a^4 + a^2*b^2)*c*d*e - 2*(a^4 + a^2*b^2)*c^2*f)*cosh(d*x + c))*sinh(d*x + c)^3 + 72*(2*(2*a^2
*b^2 + b^4)*d*f*x + 2*(2*a^2*b^2 + b^4)*d*e + (2*a^2*b^2 + b^4)*f)*cosh(d*x + c)^2 + 12*(21*(4*b^4*d*f*x + 4*b
^4*d*e - b^4*f)*cosh(d*x + c)^6 - 56*(3*a*b^3*d*f*x + 3*a*b^3*d*e - a*b^3*f)*cosh(d*x + c)^5 + 90*(2*(2*a^2*b^
2 + b^4)*d*f*x + 2*(2*a^2*b^2 + b^4)*d*e - (2*a^2*b^2 + b^4)*f)*cosh(d*x + c)^4 + 12*(2*a^2*b^2 + b^4)*d*f*x -
240*((4*a^3*b + 3*a*b^3)*d*f*x + (4*a^3*b + 3*a*b^3)*d*e - (4*a^3*b + 3*a*b^3)*f)*cosh(d*x + c)^3 + 12*(2*a^2
*b^2 + b^4)*d*e - 576*((a^4 + a^2*b^2)*d^2*f*x^2 + 2*(a^4 + a^2*b^2)*d^2*e*x + 4*(a^4 + a^2*b^2)*c*d*e - 2*(a^
4 + a^2*b^2)*c^2*f)*cosh(d*x + c)^2 + 6*(2*a^2*b^2 + b^4)*f + 72*((4*a^3*b + 3*a*b^3)*d*f*x + (4*a^3*b + 3*a*b
^3)*d*e + (4*a^3*b + 3*a*b^3)*f)*cosh(d*x + c))*sinh(d*x + c)^2 + 32*(3*a*b^3*d*f*x + 3*a*b^3*d*e + a*b^3*f)*c
osh(d*x + c) + 2304*((a^4 + a^2*b^2)*f*cosh(d*x + c)^4 + 4*(a^4 + a^2*b^2)*f*cosh(d*x + c)^3*sinh(d*x + c) + 6
*(a^4 + a^2*b^2)*f*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*(a^4 + a^2*b^2)*f*cosh(d*x + c)*sinh(d*x + c)^3 + (a^4
+ a^2*b^2)*f*sinh(d*x + c)^4)*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*s
qrt((a^2 + b^2)/b^2) - b)/b + 1) + 2304*((a^4 + a^2*b^2)*f*cosh(d*x + c)^4 + 4*(a^4 + a^2*b^2)*f*cosh(d*x + c)
^3*sinh(d*x + c) + 6*(a^4 + a^2*b^2)*f*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*(a^4 + a^2*b^2)*f*cosh(d*x + c)*sin
h(d*x + c)^3 + (a^4 + a^2*b^2)*f*sinh(d*x + c)^4)*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^4 + a^2*b^2)*d*e - (a^4 + a^2*b^2)*c*f)*cosh(
d*x + c)^4 + 4*((a^4 + a^2*b^2)*d*e - (a^4 + a^2*b^2)*c*f)*cosh(d*x + c)^3*sinh(d*x + c) + 6*((a^4 + a^2*b^2)*
d*e - (a^4 + a^2*b^2)*c*f)*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*((a^4 + a^2*b^2)*d*e - (a^4 + a^2*b^2)*c*f)*cos
h(d*x + c)*sinh(d*x + c)^3 + ((a^4 + a^2*b^2)*d*e - (a^4 + a^2*b^2)*c*f)*sinh(d*x + c)^4)*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^4 + a^2*b^2)*d*e - (a^4 + a^2*b^2)*c*f)*c
osh(d*x + c)^4 + 4*((a^4 + a^2*b^2)*d*e - (a^4 + a^2*b^2)*c*f)*cosh(d*x + c)^3*sinh(d*x + c) + 6*((a^4 + a^2*b
^2)*d*e - (a^4 + a^2*b^2)*c*f)*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*((a^4 + a^2*b^2)*d*e - (a^4 + a^2*b^2)*c*f)
*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^4 + a^2*b^2)*d*e - (a^4 + a^2*b^2)*c*f)*sinh(d*x + c)^4)*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^4 + a^2*b^2)*d*f*x + (a^4 + a^2*b^2)*
c*f)*cosh(d*x + c)^4 + 4*((a^4 + a^2*b^2)*d*f*x + (a^4 + a^2*b^2)*c*f)*cosh(d*x + c)^3*sinh(d*x + c) + 6*((a^4
+ a^2*b^2)*d*f*x + (a^4 + a^2*b^2)*c*f)*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*((a^4 + a^2*b^2)*d*f*x + (a^4 + a
^2*b^2)*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^4 + a^2*b^2)*d*f*x + (a^4 + a^2*b^2)*c*f)*sinh(d*x + c)^4)*lo
g(-(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) + 23
04*(((a^4 + a^2*b^2)*d*f*x + (a^4 + a^2*b^2)*c*f)*cosh(d*x + c)^4 + 4*((a^4 + a^2*b^2)*d*f*x + (a^4 + a^2*b^2)
*c*f)*cosh(d*x + c)^3*sinh(d*x + c) + 6*((a^4 + a^2*b^2)*d*f*x + (a^4 + a^2*b^2)*c*f)*cosh(d*x + c)^2*sinh(d*x
+ c)^2 + 4*((a^4 + a^2*b^2)*d*f*x + (a^4 + a^2*b^2)*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^4 + a^2*b^2)*d*f
*x + (a^4 + a^2*b^2)*c*f)*sinh(d*x + c)^4)*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) + 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 + 54*(2*(2*a^2*b^2 + b^4)*d*f
*x + 2*(2*a^2*b^2 + b^4)*d*e - (2*a^2*b^2 + b^4)*f)*cosh(d*x + c)^5 + 4*a*b^3*f - 180*((4*a^3*b + 3*a*b^3)*d*f
*x + (4*a^3*b + 3*a*b^3)*d*e - (4*a^3*b + 3*a*b^3)*f)*cosh(d*x + c)^4 - 576*((a^4 + a^2*b^2)*d^2*f*x^2 + 2*(a^
4 + a^2*b^2)*d^2*e*x + 4*(a^4 + a^2*b^2)*c*d*e - 2*(a^4 + a^2*b^2)*c^2*f)*cosh(d*x + c)^3 + 108*((4*a^3*b + 3*
a*b^3)*d*f*x + (4*a^3*b + 3*a*b^3)*d*e + (4*a^3*b + 3*a*b^3)*f)*cosh(d*x + c)^2 + 18*(2*(2*a^2*b^2 + b^4)*d*f*
x + 2*(2*a^2*b^2 + b^4)*d*e + (2*a^2*b^2 + b^4)*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)*si
nh(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)**3*sinh(d*x+c)**2/(a+b*sinh(d*x+c)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (f x + e\right )} \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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