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

Optimal. Leaf size=641 \[ -\frac{a^3 f \left (a^2+b^2\right ) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^6 d^2}-\frac{a^3 f \left (a^2+b^2\right ) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^6 d^2}-\frac{a^2 f \cosh ^3(c+d x)}{9 b^3 d^2}-\frac{a^4 f \cosh (c+d x)}{b^5 d^2}-\frac{2 a^2 f \cosh (c+d x)}{3 b^3 d^2}+\frac{a^3 f \sinh (c+d x) \cosh (c+d x)}{4 b^4 d^2}-\frac{a^3 \left (a^2+b^2\right ) (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b^6 d}-\frac{a^3 \left (a^2+b^2\right ) (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b^6 d}-\frac{a^3 (e+f x) \sinh ^2(c+d x)}{2 b^4 d}+\frac{a^4 (e+f x) \sinh (c+d x)}{b^5 d}+\frac{2 a^2 (e+f x) \sinh (c+d x)}{3 b^3 d}+\frac{a^2 (e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 b^3 d}-\frac{a^3 f x}{4 b^4 d}+\frac{a^3 \left (a^2+b^2\right ) (e+f x)^2}{2 b^6 f}+\frac{a f \sinh (c+d x) \cosh ^3(c+d x)}{16 b^2 d^2}+\frac{3 a f \sinh (c+d x) \cosh (c+d x)}{32 b^2 d^2}-\frac{a (e+f x) \cosh ^4(c+d x)}{4 b^2 d}+\frac{3 a f x}{32 b^2 d}+\frac{f \cosh (c+d x)}{8 b d^2}-\frac{f \cosh (3 c+3 d x)}{144 b d^2}-\frac{f \cosh (5 c+5 d x)}{400 b d^2}-\frac{(e+f x) \sinh (c+d x)}{8 b d}+\frac{(e+f x) \sinh (3 c+3 d x)}{48 b d}+\frac{(e+f x) \sinh (5 c+5 d x)}{80 b d} \]

[Out]

-(a^3*f*x)/(4*b^4*d) + (3*a*f*x)/(32*b^2*d) + (a^3*(a^2 + b^2)*(e + f*x)^2)/(2*b^6*f) - (a^4*f*Cosh[c + d*x])/
(b^5*d^2) - (2*a^2*f*Cosh[c + d*x])/(3*b^3*d^2) + (f*Cosh[c + d*x])/(8*b*d^2) - (a^2*f*Cosh[c + d*x]^3)/(9*b^3
*d^2) - (a*(e + f*x)*Cosh[c + d*x]^4)/(4*b^2*d) - (f*Cosh[3*c + 3*d*x])/(144*b*d^2) - (f*Cosh[5*c + 5*d*x])/(4
00*b*d^2) - (a^3*(a^2 + b^2)*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^6*d) - (a^3*(a^2 + b
^2)*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^6*d) - (a^3*(a^2 + b^2)*f*PolyLog[2, -((b*E^(
c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^6*d^2) - (a^3*(a^2 + b^2)*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 +
 b^2]))])/(b^6*d^2) + (a^4*(e + f*x)*Sinh[c + d*x])/(b^5*d) + (2*a^2*(e + f*x)*Sinh[c + d*x])/(3*b^3*d) - ((e
+ f*x)*Sinh[c + d*x])/(8*b*d) + (a^3*f*Cosh[c + d*x]*Sinh[c + d*x])/(4*b^4*d^2) + (3*a*f*Cosh[c + d*x]*Sinh[c
+ d*x])/(32*b^2*d^2) + (a^2*(e + f*x)*Cosh[c + d*x]^2*Sinh[c + d*x])/(3*b^3*d) + (a*f*Cosh[c + d*x]^3*Sinh[c +
 d*x])/(16*b^2*d^2) - (a^3*(e + f*x)*Sinh[c + d*x]^2)/(2*b^4*d) + ((e + f*x)*Sinh[3*c + 3*d*x])/(48*b*d) + ((e
 + f*x)*Sinh[5*c + 5*d*x])/(80*b*d)

________________________________________________________________________________________

Rubi [A]  time = 0.944258, antiderivative size = 641, normalized size of antiderivative = 1., number of steps used = 31, 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, 2635, 8, 3310, 5565, 5446, 5561, 2190, 2279, 2391} \[ -\frac{a^3 f \left (a^2+b^2\right ) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^6 d^2}-\frac{a^3 f \left (a^2+b^2\right ) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^6 d^2}-\frac{a^2 f \cosh ^3(c+d x)}{9 b^3 d^2}-\frac{a^4 f \cosh (c+d x)}{b^5 d^2}-\frac{2 a^2 f \cosh (c+d x)}{3 b^3 d^2}+\frac{a^3 f \sinh (c+d x) \cosh (c+d x)}{4 b^4 d^2}-\frac{a^3 \left (a^2+b^2\right ) (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b^6 d}-\frac{a^3 \left (a^2+b^2\right ) (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b^6 d}-\frac{a^3 (e+f x) \sinh ^2(c+d x)}{2 b^4 d}+\frac{a^4 (e+f x) \sinh (c+d x)}{b^5 d}+\frac{2 a^2 (e+f x) \sinh (c+d x)}{3 b^3 d}+\frac{a^2 (e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 b^3 d}-\frac{a^3 f x}{4 b^4 d}+\frac{a^3 \left (a^2+b^2\right ) (e+f x)^2}{2 b^6 f}+\frac{a f \sinh (c+d x) \cosh ^3(c+d x)}{16 b^2 d^2}+\frac{3 a f \sinh (c+d x) \cosh (c+d x)}{32 b^2 d^2}-\frac{a (e+f x) \cosh ^4(c+d x)}{4 b^2 d}+\frac{3 a f x}{32 b^2 d}+\frac{f \cosh (c+d x)}{8 b d^2}-\frac{f \cosh (3 c+3 d x)}{144 b d^2}-\frac{f \cosh (5 c+5 d x)}{400 b d^2}-\frac{(e+f x) \sinh (c+d x)}{8 b d}+\frac{(e+f x) \sinh (3 c+3 d x)}{48 b d}+\frac{(e+f x) \sinh (5 c+5 d x)}{80 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^3*f*x)/(4*b^4*d) + (3*a*f*x)/(32*b^2*d) + (a^3*(a^2 + b^2)*(e + f*x)^2)/(2*b^6*f) - (a^4*f*Cosh[c + d*x])/
(b^5*d^2) - (2*a^2*f*Cosh[c + d*x])/(3*b^3*d^2) + (f*Cosh[c + d*x])/(8*b*d^2) - (a^2*f*Cosh[c + d*x]^3)/(9*b^3
*d^2) - (a*(e + f*x)*Cosh[c + d*x]^4)/(4*b^2*d) - (f*Cosh[3*c + 3*d*x])/(144*b*d^2) - (f*Cosh[5*c + 5*d*x])/(4
00*b*d^2) - (a^3*(a^2 + b^2)*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^6*d) - (a^3*(a^2 + b
^2)*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^6*d) - (a^3*(a^2 + b^2)*f*PolyLog[2, -((b*E^(
c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^6*d^2) - (a^3*(a^2 + b^2)*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 +
 b^2]))])/(b^6*d^2) + (a^4*(e + f*x)*Sinh[c + d*x])/(b^5*d) + (2*a^2*(e + f*x)*Sinh[c + d*x])/(3*b^3*d) - ((e
+ f*x)*Sinh[c + d*x])/(8*b*d) + (a^3*f*Cosh[c + d*x]*Sinh[c + d*x])/(4*b^4*d^2) + (3*a*f*Cosh[c + d*x]*Sinh[c
+ d*x])/(32*b^2*d^2) + (a^2*(e + f*x)*Cosh[c + d*x]^2*Sinh[c + d*x])/(3*b^3*d) + (a*f*Cosh[c + d*x]^3*Sinh[c +
 d*x])/(16*b^2*d^2) - (a^3*(e + f*x)*Sinh[c + d*x]^2)/(2*b^4*d) + ((e + f*x)*Sinh[3*c + 3*d*x])/(48*b*d) + ((e
 + f*x)*Sinh[5*c + 5*d*x])/(80*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 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 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 ^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac{\int (e+f x) \cosh ^3(c+d x) \sinh ^2(c+d x) \, dx}{b}-\frac{a \int \frac{(e+f x) \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=-\frac{a \int (e+f x) \cosh ^3(c+d x) \sinh (c+d x) \, dx}{b^2}+\frac{a^2 \int \frac{(e+f x) \cosh ^3(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) \cosh (c+d x)+\frac{1}{16} (e+f x) \cosh (3 c+3 d x)+\frac{1}{16} (e+f x) \cosh (5 c+5 d x)\right ) \, dx}{b}\\ &=-\frac{a (e+f x) \cosh ^4(c+d x)}{4 b^2 d}+\frac{a^2 \int (e+f x) \cosh ^3(c+d x) \, dx}{b^3}-\frac{a^3 \int \frac{(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx}{b^3}+\frac{\int (e+f x) \cosh (3 c+3 d x) \, dx}{16 b}+\frac{\int (e+f x) \cosh (5 c+5 d x) \, dx}{16 b}-\frac{\int (e+f x) \cosh (c+d x) \, dx}{8 b}+\frac{(a f) \int \cosh ^4(c+d x) \, dx}{4 b^2 d}\\ &=-\frac{a^2 f \cosh ^3(c+d x)}{9 b^3 d^2}-\frac{a (e+f x) \cosh ^4(c+d x)}{4 b^2 d}-\frac{(e+f x) \sinh (c+d x)}{8 b d}+\frac{a^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{3 b^3 d}+\frac{a f \cosh ^3(c+d x) \sinh (c+d x)}{16 b^2 d^2}+\frac{(e+f x) \sinh (3 c+3 d x)}{48 b d}+\frac{(e+f x) \sinh (5 c+5 d x)}{80 b d}+\frac{a^4 \int (e+f x) \cosh (c+d x) \, dx}{b^5}-\frac{a^3 \int (e+f x) \cosh (c+d x) \sinh (c+d x) \, dx}{b^4}+\frac{\left (2 a^2\right ) \int (e+f x) \cosh (c+d x) \, dx}{3 b^3}-\frac{\left (a^3 \left (a^2+b^2\right )\right ) \int \frac{(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^5}+\frac{(3 a f) \int \cosh ^2(c+d x) \, dx}{16 b^2 d}-\frac{f \int \sinh (5 c+5 d x) \, dx}{80 b d}-\frac{f \int \sinh (3 c+3 d x) \, dx}{48 b d}+\frac{f \int \sinh (c+d x) \, dx}{8 b d}\\ &=\frac{a^3 \left (a^2+b^2\right ) (e+f x)^2}{2 b^6 f}+\frac{f \cosh (c+d x)}{8 b d^2}-\frac{a^2 f \cosh ^3(c+d x)}{9 b^3 d^2}-\frac{a (e+f x) \cosh ^4(c+d x)}{4 b^2 d}-\frac{f \cosh (3 c+3 d x)}{144 b d^2}-\frac{f \cosh (5 c+5 d x)}{400 b d^2}+\frac{a^4 (e+f x) \sinh (c+d x)}{b^5 d}+\frac{2 a^2 (e+f x) \sinh (c+d x)}{3 b^3 d}-\frac{(e+f x) \sinh (c+d x)}{8 b d}+\frac{3 a f \cosh (c+d x) \sinh (c+d x)}{32 b^2 d^2}+\frac{a^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{3 b^3 d}+\frac{a f \cosh ^3(c+d x) \sinh (c+d x)}{16 b^2 d^2}-\frac{a^3 (e+f x) \sinh ^2(c+d x)}{2 b^4 d}+\frac{(e+f x) \sinh (3 c+3 d x)}{48 b d}+\frac{(e+f x) \sinh (5 c+5 d x)}{80 b d}-\frac{\left (a^3 \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^5}-\frac{\left (a^3 \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^5}-\frac{\left (a^4 f\right ) \int \sinh (c+d x) \, dx}{b^5 d}+\frac{\left (a^3 f\right ) \int \sinh ^2(c+d x) \, dx}{2 b^4 d}-\frac{\left (2 a^2 f\right ) \int \sinh (c+d x) \, dx}{3 b^3 d}+\frac{(3 a f) \int 1 \, dx}{32 b^2 d}\\ &=\frac{3 a f x}{32 b^2 d}+\frac{a^3 \left (a^2+b^2\right ) (e+f x)^2}{2 b^6 f}-\frac{a^4 f \cosh (c+d x)}{b^5 d^2}-\frac{2 a^2 f \cosh (c+d x)}{3 b^3 d^2}+\frac{f \cosh (c+d x)}{8 b d^2}-\frac{a^2 f \cosh ^3(c+d x)}{9 b^3 d^2}-\frac{a (e+f x) \cosh ^4(c+d x)}{4 b^2 d}-\frac{f \cosh (3 c+3 d x)}{144 b d^2}-\frac{f \cosh (5 c+5 d x)}{400 b d^2}-\frac{a^3 \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^6 d}-\frac{a^3 \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^6 d}+\frac{a^4 (e+f x) \sinh (c+d x)}{b^5 d}+\frac{2 a^2 (e+f x) \sinh (c+d x)}{3 b^3 d}-\frac{(e+f x) \sinh (c+d x)}{8 b d}+\frac{a^3 f \cosh (c+d x) \sinh (c+d x)}{4 b^4 d^2}+\frac{3 a f \cosh (c+d x) \sinh (c+d x)}{32 b^2 d^2}+\frac{a^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{3 b^3 d}+\frac{a f \cosh ^3(c+d x) \sinh (c+d x)}{16 b^2 d^2}-\frac{a^3 (e+f x) \sinh ^2(c+d x)}{2 b^4 d}+\frac{(e+f x) \sinh (3 c+3 d x)}{48 b d}+\frac{(e+f x) \sinh (5 c+5 d x)}{80 b d}-\frac{\left (a^3 f\right ) \int 1 \, dx}{4 b^4 d}+\frac{\left (a^3 \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^6 d}+\frac{\left (a^3 \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^6 d}\\ &=-\frac{a^3 f x}{4 b^4 d}+\frac{3 a f x}{32 b^2 d}+\frac{a^3 \left (a^2+b^2\right ) (e+f x)^2}{2 b^6 f}-\frac{a^4 f \cosh (c+d x)}{b^5 d^2}-\frac{2 a^2 f \cosh (c+d x)}{3 b^3 d^2}+\frac{f \cosh (c+d x)}{8 b d^2}-\frac{a^2 f \cosh ^3(c+d x)}{9 b^3 d^2}-\frac{a (e+f x) \cosh ^4(c+d x)}{4 b^2 d}-\frac{f \cosh (3 c+3 d x)}{144 b d^2}-\frac{f \cosh (5 c+5 d x)}{400 b d^2}-\frac{a^3 \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^6 d}-\frac{a^3 \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^6 d}+\frac{a^4 (e+f x) \sinh (c+d x)}{b^5 d}+\frac{2 a^2 (e+f x) \sinh (c+d x)}{3 b^3 d}-\frac{(e+f x) \sinh (c+d x)}{8 b d}+\frac{a^3 f \cosh (c+d x) \sinh (c+d x)}{4 b^4 d^2}+\frac{3 a f \cosh (c+d x) \sinh (c+d x)}{32 b^2 d^2}+\frac{a^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{3 b^3 d}+\frac{a f \cosh ^3(c+d x) \sinh (c+d x)}{16 b^2 d^2}-\frac{a^3 (e+f x) \sinh ^2(c+d x)}{2 b^4 d}+\frac{(e+f x) \sinh (3 c+3 d x)}{48 b d}+\frac{(e+f x) \sinh (5 c+5 d x)}{80 b d}+\frac{\left (a^3 \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^6 d^2}+\frac{\left (a^3 \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^6 d^2}\\ &=-\frac{a^3 f x}{4 b^4 d}+\frac{3 a f x}{32 b^2 d}+\frac{a^3 \left (a^2+b^2\right ) (e+f x)^2}{2 b^6 f}-\frac{a^4 f \cosh (c+d x)}{b^5 d^2}-\frac{2 a^2 f \cosh (c+d x)}{3 b^3 d^2}+\frac{f \cosh (c+d x)}{8 b d^2}-\frac{a^2 f \cosh ^3(c+d x)}{9 b^3 d^2}-\frac{a (e+f x) \cosh ^4(c+d x)}{4 b^2 d}-\frac{f \cosh (3 c+3 d x)}{144 b d^2}-\frac{f \cosh (5 c+5 d x)}{400 b d^2}-\frac{a^3 \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^6 d}-\frac{a^3 \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^6 d}-\frac{a^3 \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^6 d^2}-\frac{a^3 \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^6 d^2}+\frac{a^4 (e+f x) \sinh (c+d x)}{b^5 d}+\frac{2 a^2 (e+f x) \sinh (c+d x)}{3 b^3 d}-\frac{(e+f x) \sinh (c+d x)}{8 b d}+\frac{a^3 f \cosh (c+d x) \sinh (c+d x)}{4 b^4 d^2}+\frac{3 a f \cosh (c+d x) \sinh (c+d x)}{32 b^2 d^2}+\frac{a^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{3 b^3 d}+\frac{a f \cosh ^3(c+d x) \sinh (c+d x)}{16 b^2 d^2}-\frac{a^3 (e+f x) \sinh ^2(c+d x)}{2 b^4 d}+\frac{(e+f x) \sinh (3 c+3 d x)}{48 b d}+\frac{(e+f x) \sinh (5 c+5 d x)}{80 b d}\\ \end{align*}

Mathematica [A]  time = 4.34045, size = 958, normalized size = 1.49 \[ -\frac{-14400 d^2 f x^2 a^5-14400 c^2 f a^5-28800 c d f x a^5+28800 c f \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) a^5+28800 d f x \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) a^5+28800 c f \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) a^5+28800 d f x \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) a^5+28800 d e \log (a+b \sinh (c+d x)) a^5-28800 c f \log (a+b \sinh (c+d x)) a^5+28800 b f \cosh (c+d x) a^4-28800 b d e \sinh (c+d x) a^4-28800 b d f x \sinh (c+d x) a^4-14400 b^2 d^2 f x^2 a^3-14400 b^2 c^2 f a^3-28800 b^2 c d f x a^3+7200 b^2 d e \cosh (2 (c+d x)) a^3+7200 b^2 d f x \cosh (2 (c+d x)) a^3+28800 b^2 c f \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) a^3+28800 b^2 d f x \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) a^3+28800 b^2 c f \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) a^3+28800 b^2 d f x \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) a^3+28800 b^2 d e \log (a+b \sinh (c+d x)) a^3-28800 b^2 c f \log (a+b \sinh (c+d x)) a^3+28800 \left (a^2+b^2\right ) f \text{PolyLog}\left (2,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right ) a^3+28800 \left (a^2+b^2\right ) f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) a^3-3600 b^2 f \sinh (2 (c+d x)) a^3+21600 b^3 f \cosh (c+d x) a^2+800 b^3 f \cosh (3 (c+d x)) a^2-21600 b^3 d e \sinh (c+d x) a^2-21600 b^3 d f x \sinh (c+d x) a^2-2400 b^3 d e \sinh (3 (c+d x)) a^2-2400 b^3 d f x \sinh (3 (c+d x)) a^2+3600 b^4 d e \cosh (2 (c+d x)) a+3600 b^4 d f x \cosh (2 (c+d x)) a+900 b^4 d e \cosh (4 (c+d x)) a+900 b^4 d f x \cosh (4 (c+d x)) a-1800 b^4 f \sinh (2 (c+d x)) a-225 b^4 f \sinh (4 (c+d x)) a-3600 b^5 f \cosh (c+d x)+200 b^5 f \cosh (3 (c+d x))+72 b^5 f \cosh (5 (c+d x))+3600 b^5 d e \sinh (c+d x)+3600 b^5 d f x \sinh (c+d x)-600 b^5 d e \sinh (3 (c+d x))-600 b^5 d f x \sinh (3 (c+d x))-360 b^5 d e \sinh (5 (c+d x))-360 b^5 d f x \sinh (5 (c+d x))}{28800 b^6 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-14400*a^5*c^2*f - 14400*a^3*b^2*c^2*f - 28800*a^5*c*d*f*x - 28800*a^3*b^2*c*d*f*x - 14400*a^5*d^2*f*x^2 - 1
4400*a^3*b^2*d^2*f*x^2 + 28800*a^4*b*f*Cosh[c + d*x] + 21600*a^2*b^3*f*Cosh[c + d*x] - 3600*b^5*f*Cosh[c + d*x
] + 7200*a^3*b^2*d*e*Cosh[2*(c + d*x)] + 3600*a*b^4*d*e*Cosh[2*(c + d*x)] + 7200*a^3*b^2*d*f*x*Cosh[2*(c + d*x
)] + 3600*a*b^4*d*f*x*Cosh[2*(c + d*x)] + 800*a^2*b^3*f*Cosh[3*(c + d*x)] + 200*b^5*f*Cosh[3*(c + d*x)] + 900*
a*b^4*d*e*Cosh[4*(c + d*x)] + 900*a*b^4*d*f*x*Cosh[4*(c + d*x)] + 72*b^5*f*Cosh[5*(c + d*x)] + 28800*a^5*c*f*L
og[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 28800*a^3*b^2*c*f*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]
)] + 28800*a^5*d*f*x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 28800*a^3*b^2*d*f*x*Log[1 + (b*E^(c + d*
x))/(a - Sqrt[a^2 + b^2])] + 28800*a^5*c*f*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 28800*a^3*b^2*c*f*
Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 28800*a^5*d*f*x*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])
] + 28800*a^3*b^2*d*f*x*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 28800*a^5*d*e*Log[a + b*Sinh[c + d*x]
] + 28800*a^3*b^2*d*e*Log[a + b*Sinh[c + d*x]] - 28800*a^5*c*f*Log[a + b*Sinh[c + d*x]] - 28800*a^3*b^2*c*f*Lo
g[a + b*Sinh[c + d*x]] + 28800*a^3*(a^2 + b^2)*f*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 28800*a^
3*(a^2 + b^2)*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] - 28800*a^4*b*d*e*Sinh[c + d*x] - 21600*a
^2*b^3*d*e*Sinh[c + d*x] + 3600*b^5*d*e*Sinh[c + d*x] - 28800*a^4*b*d*f*x*Sinh[c + d*x] - 21600*a^2*b^3*d*f*x*
Sinh[c + d*x] + 3600*b^5*d*f*x*Sinh[c + d*x] - 3600*a^3*b^2*f*Sinh[2*(c + d*x)] - 1800*a*b^4*f*Sinh[2*(c + d*x
)] - 2400*a^2*b^3*d*e*Sinh[3*(c + d*x)] - 600*b^5*d*e*Sinh[3*(c + d*x)] - 2400*a^2*b^3*d*f*x*Sinh[3*(c + d*x)]
 - 600*b^5*d*f*x*Sinh[3*(c + d*x)] - 225*a*b^4*f*Sinh[4*(c + d*x)] - 360*b^5*d*e*Sinh[5*(c + d*x)] - 360*b^5*d
*f*x*Sinh[5*(c + d*x)])/(28800*b^6*d^2)

________________________________________________________________________________________

Maple [B]  time = 0.115, size = 1363, normalized size = 2.1 \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)^3/(a+b*sinh(d*x+c)),x)

[Out]

a^5/b^6/d^2*f*c*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)-2*a^5/b^6/d^2*f*c*ln(exp(d*x+c))-a^5/b^6/d*f*ln((-b*exp(
d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x-a^5/b^6/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^5/b^6/d*f*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x-a^5/b^6/d^2*f*ln((b*exp(
d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c+2*a^5/b^6/d*f*c*x+1/2*a^3*f*x^2/b^4-1/32*a*(2*a^2+b^2)*(2*d*f
*x+2*d*e+f)/b^4/d^2*exp(-2*d*x-2*c)-a^3/b^4/d*f*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x-a
^3/b^4/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^3/b^4/d*f*ln((b*exp(d*x+c)+(a^2+b^
2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x-a^3/b^4/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^
3/b^4/d^2*f*c*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)-2*a^3/b^4/d^2*f*c*ln(exp(d*x+c))+2*a^3/b^4/d*f*c*x-a^3*e*x
/b^4+1/800*(5*d*f*x+5*d*e-f)/d^2/b*exp(5*d*x+5*c)+1/288*(12*a^2*d*f*x+3*b^2*d*f*x+12*a^2*d*e+3*b^2*d*e-4*a^2*f
-b^2*f)/b^3/d^2*exp(3*d*x+3*c)+a^5/b^6/d^2*f*c^2+1/16*(8*a^4*d*f*x+6*a^2*b^2*d*f*x-b^4*d*f*x+8*a^4*d*e+6*a^2*b
^2*d*e-b^4*d*e-8*a^4*f-6*a^2*b^2*f+b^4*f)/b^5/d^2*exp(d*x+c)-1/800*(5*d*f*x+5*d*e+f)/d^2/b*exp(-5*d*x-5*c)-1/1
6*(8*a^4+6*a^2*b^2-b^4)*(d*f*x+d*e+f)/b^5/d^2*exp(-d*x-c)-1/288*(4*a^2+b^2)*(3*d*f*x+3*d*e+f)/b^3/d^2*exp(-3*d
*x-3*c)-1/256*a*(4*d*f*x+4*d*e+f)/b^2/d^2*exp(-4*d*x-4*c)-1/256*a*(4*d*f*x+4*d*e-f)/b^2/d^2*exp(4*d*x+4*c)-1/3
2*a*(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^4/d^2*exp(2*d*x+2*c)+a^3/b^4/d^2*f*c^2-a^3/b
^4/d^2*f*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))-a^3/b^4/d^2*f*dilog((-b*exp(d*x+c)+(a^2+b
^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))-a^3/b^4/d*e*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)+2*a^3/b^4/d*e*ln(exp(d*x+
c))+1/2*a^5/b^6*f*x^2-a^5/b^6*e*x-a^5/b^6/d*e*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)+2*a^5/b^6/d*e*ln(exp(d*x+c
))-a^5/b^6/d^2*f*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))-a^5/b^6/d^2*f*dilog((-b*exp(d*x+c
)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))

________________________________________________________________________________________

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

[Out]

-1/960*e*((15*a*b^3*e^(-d*x - c) - 6*b^4 - 10*(4*a^2*b^2 + b^4)*e^(-2*d*x - 2*c) + 60*(2*a^3*b + a*b^3)*e^(-3*
d*x - 3*c) - 60*(8*a^4 + 6*a^2*b^2 - b^4)*e^(-4*d*x - 4*c))*e^(5*d*x + 5*c)/(b^5*d) + 960*(a^5 + a^3*b^2)*(d*x
 + c)/(b^6*d) + (15*a*b^3*e^(-4*d*x - 4*c) + 6*b^4*e^(-5*d*x - 5*c) + 60*(8*a^4 + 6*a^2*b^2 - b^4)*e^(-d*x - c
) + 60*(2*a^3*b + a*b^3)*e^(-2*d*x - 2*c) + 10*(4*a^2*b^2 + b^4)*e^(-3*d*x - 3*c))/(b^5*d) + 960*(a^5 + a^3*b^
2)*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(b^6*d)) - 1/57600*f*((28800*(a^5*d^2*e^(5*c) + a^3*b^2*d^2
*e^(5*c))*x^2 - 72*(5*b^5*d*x*e^(10*c) - b^5*e^(10*c))*e^(5*d*x) + 225*(4*a*b^4*d*x*e^(9*c) - a*b^4*e^(9*c))*e
^(4*d*x) + 200*(4*a^2*b^3*e^(8*c) + b^5*e^(8*c) - 3*(4*a^2*b^3*d*e^(8*c) + b^5*d*e^(8*c))*x)*e^(3*d*x) - 1800*
(2*a^3*b^2*e^(7*c) + a*b^4*e^(7*c) - 2*(2*a^3*b^2*d*e^(7*c) + a*b^4*d*e^(7*c))*x)*e^(2*d*x) + 3600*(8*a^4*b*e^
(6*c) + 6*a^2*b^3*e^(6*c) - b^5*e^(6*c) - (8*a^4*b*d*e^(6*c) + 6*a^2*b^3*d*e^(6*c) - b^5*d*e^(6*c))*x)*e^(d*x)
 + 3600*(8*a^4*b*e^(4*c) + 6*a^2*b^3*e^(4*c) - b^5*e^(4*c) + (8*a^4*b*d*e^(4*c) + 6*a^2*b^3*d*e^(4*c) - b^5*d*
e^(4*c))*x)*e^(-d*x) + 1800*(2*a^3*b^2*e^(3*c) + a*b^4*e^(3*c) + 2*(2*a^3*b^2*d*e^(3*c) + a*b^4*d*e^(3*c))*x)*
e^(-2*d*x) + 200*(4*a^2*b^3*e^(2*c) + b^5*e^(2*c) + 3*(4*a^2*b^3*d*e^(2*c) + b^5*d*e^(2*c))*x)*e^(-3*d*x) + 22
5*(4*a*b^4*d*x*e^c + a*b^4*e^c)*e^(-4*d*x) + 72*(5*b^5*d*x + b^5)*e^(-5*d*x))*e^(-5*c)/(b^6*d^2) - 900*integra
te(128*((a^6*e^c + a^4*b^2*e^c)*x*e^(d*x) - (a^5*b + a^3*b^3)*x)/(b^7*e^(2*d*x + 2*c) + 2*a*b^6*e^(d*x + c) -
b^7), x))

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Fricas [B]  time = 4.19557, size = 12709, normalized size = 19.83 \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)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

1/57600*(72*(5*b^5*d*f*x + 5*b^5*d*e - b^5*f)*cosh(d*x + c)^10 + 72*(5*b^5*d*f*x + 5*b^5*d*e - b^5*f)*sinh(d*x
 + c)^10 - 225*(4*a*b^4*d*f*x + 4*a*b^4*d*e - a*b^4*f)*cosh(d*x + c)^9 - 45*(20*a*b^4*d*f*x + 20*a*b^4*d*e - 5
*a*b^4*f - 16*(5*b^5*d*f*x + 5*b^5*d*e - b^5*f)*cosh(d*x + c))*sinh(d*x + c)^9 + 200*(3*(4*a^2*b^3 + b^5)*d*f*
x + 3*(4*a^2*b^3 + b^5)*d*e - (4*a^2*b^3 + b^5)*f)*cosh(d*x + c)^8 + 5*(120*(4*a^2*b^3 + b^5)*d*f*x + 120*(4*a
^2*b^3 + b^5)*d*e + 648*(5*b^5*d*f*x + 5*b^5*d*e - b^5*f)*cosh(d*x + c)^2 - 40*(4*a^2*b^3 + b^5)*f - 405*(4*a*
b^4*d*f*x + 4*a*b^4*d*e - a*b^4*f)*cosh(d*x + c))*sinh(d*x + c)^8 - 360*b^5*d*f*x - 1800*(2*(2*a^3*b^2 + a*b^4
)*d*f*x + 2*(2*a^3*b^2 + a*b^4)*d*e - (2*a^3*b^2 + a*b^4)*f)*cosh(d*x + c)^7 - 20*(180*(2*a^3*b^2 + a*b^4)*d*f
*x - 432*(5*b^5*d*f*x + 5*b^5*d*e - b^5*f)*cosh(d*x + c)^3 + 180*(2*a^3*b^2 + a*b^4)*d*e + 405*(4*a*b^4*d*f*x
+ 4*a*b^4*d*e - a*b^4*f)*cosh(d*x + c)^2 - 90*(2*a^3*b^2 + a*b^4)*f - 80*(3*(4*a^2*b^3 + b^5)*d*f*x + 3*(4*a^2
*b^3 + b^5)*d*e - (4*a^2*b^3 + b^5)*f)*cosh(d*x + c))*sinh(d*x + c)^7 - 360*b^5*d*e + 3600*((8*a^4*b + 6*a^2*b
^3 - b^5)*d*f*x + (8*a^4*b + 6*a^2*b^3 - b^5)*d*e - (8*a^4*b + 6*a^2*b^3 - b^5)*f)*cosh(d*x + c)^6 + 20*(756*(
5*b^5*d*f*x + 5*b^5*d*e - b^5*f)*cosh(d*x + c)^4 + 180*(8*a^4*b + 6*a^2*b^3 - b^5)*d*f*x - 945*(4*a*b^4*d*f*x
+ 4*a*b^4*d*e - a*b^4*f)*cosh(d*x + c)^3 + 180*(8*a^4*b + 6*a^2*b^3 - b^5)*d*e + 280*(3*(4*a^2*b^3 + b^5)*d*f*
x + 3*(4*a^2*b^3 + b^5)*d*e - (4*a^2*b^3 + b^5)*f)*cosh(d*x + c)^2 - 180*(8*a^4*b + 6*a^2*b^3 - b^5)*f - 630*(
2*(2*a^3*b^2 + a*b^4)*d*f*x + 2*(2*a^3*b^2 + a*b^4)*d*e - (2*a^3*b^2 + a*b^4)*f)*cosh(d*x + c))*sinh(d*x + c)^
6 - 72*b^5*f + 28800*((a^5 + a^3*b^2)*d^2*f*x^2 + 2*(a^5 + a^3*b^2)*d^2*e*x + 4*(a^5 + a^3*b^2)*c*d*e - 2*(a^5
 + a^3*b^2)*c^2*f)*cosh(d*x + c)^5 + 2*(14400*(a^5 + a^3*b^2)*d^2*f*x^2 + 9072*(5*b^5*d*f*x + 5*b^5*d*e - b^5*
f)*cosh(d*x + c)^5 + 28800*(a^5 + a^3*b^2)*d^2*e*x - 14175*(4*a*b^4*d*f*x + 4*a*b^4*d*e - a*b^4*f)*cosh(d*x +
c)^4 + 57600*(a^5 + a^3*b^2)*c*d*e - 28800*(a^5 + a^3*b^2)*c^2*f + 5600*(3*(4*a^2*b^3 + b^5)*d*f*x + 3*(4*a^2*
b^3 + b^5)*d*e - (4*a^2*b^3 + b^5)*f)*cosh(d*x + c)^3 - 18900*(2*(2*a^3*b^2 + a*b^4)*d*f*x + 2*(2*a^3*b^2 + a*
b^4)*d*e - (2*a^3*b^2 + a*b^4)*f)*cosh(d*x + c)^2 + 10800*((8*a^4*b + 6*a^2*b^3 - b^5)*d*f*x + (8*a^4*b + 6*a^
2*b^3 - b^5)*d*e - (8*a^4*b + 6*a^2*b^3 - b^5)*f)*cosh(d*x + c))*sinh(d*x + c)^5 - 3600*((8*a^4*b + 6*a^2*b^3
- b^5)*d*f*x + (8*a^4*b + 6*a^2*b^3 - b^5)*d*e + (8*a^4*b + 6*a^2*b^3 - b^5)*f)*cosh(d*x + c)^4 + 10*(1512*(5*
b^5*d*f*x + 5*b^5*d*e - b^5*f)*cosh(d*x + c)^6 - 2835*(4*a*b^4*d*f*x + 4*a*b^4*d*e - a*b^4*f)*cosh(d*x + c)^5
+ 1400*(3*(4*a^2*b^3 + b^5)*d*f*x + 3*(4*a^2*b^3 + b^5)*d*e - (4*a^2*b^3 + b^5)*f)*cosh(d*x + c)^4 - 360*(8*a^
4*b + 6*a^2*b^3 - b^5)*d*f*x - 6300*(2*(2*a^3*b^2 + a*b^4)*d*f*x + 2*(2*a^3*b^2 + a*b^4)*d*e - (2*a^3*b^2 + a*
b^4)*f)*cosh(d*x + c)^3 - 360*(8*a^4*b + 6*a^2*b^3 - b^5)*d*e + 5400*((8*a^4*b + 6*a^2*b^3 - b^5)*d*f*x + (8*a
^4*b + 6*a^2*b^3 - b^5)*d*e - (8*a^4*b + 6*a^2*b^3 - b^5)*f)*cosh(d*x + c)^2 - 360*(8*a^4*b + 6*a^2*b^3 - b^5)
*f + 14400*((a^5 + a^3*b^2)*d^2*f*x^2 + 2*(a^5 + a^3*b^2)*d^2*e*x + 4*(a^5 + a^3*b^2)*c*d*e - 2*(a^5 + a^3*b^2
)*c^2*f)*cosh(d*x + c))*sinh(d*x + c)^4 - 1800*(2*(2*a^3*b^2 + a*b^4)*d*f*x + 2*(2*a^3*b^2 + a*b^4)*d*e + (2*a
^3*b^2 + a*b^4)*f)*cosh(d*x + c)^3 + 20*(432*(5*b^5*d*f*x + 5*b^5*d*e - b^5*f)*cosh(d*x + c)^7 - 945*(4*a*b^4*
d*f*x + 4*a*b^4*d*e - a*b^4*f)*cosh(d*x + c)^6 + 560*(3*(4*a^2*b^3 + b^5)*d*f*x + 3*(4*a^2*b^3 + b^5)*d*e - (4
*a^2*b^3 + b^5)*f)*cosh(d*x + c)^5 - 3150*(2*(2*a^3*b^2 + a*b^4)*d*f*x + 2*(2*a^3*b^2 + a*b^4)*d*e - (2*a^3*b^
2 + a*b^4)*f)*cosh(d*x + c)^4 - 180*(2*a^3*b^2 + a*b^4)*d*f*x + 3600*((8*a^4*b + 6*a^2*b^3 - b^5)*d*f*x + (8*a
^4*b + 6*a^2*b^3 - b^5)*d*e - (8*a^4*b + 6*a^2*b^3 - b^5)*f)*cosh(d*x + c)^3 - 180*(2*a^3*b^2 + a*b^4)*d*e + 1
4400*((a^5 + a^3*b^2)*d^2*f*x^2 + 2*(a^5 + a^3*b^2)*d^2*e*x + 4*(a^5 + a^3*b^2)*c*d*e - 2*(a^5 + a^3*b^2)*c^2*
f)*cosh(d*x + c)^2 - 90*(2*a^3*b^2 + a*b^4)*f - 720*((8*a^4*b + 6*a^2*b^3 - b^5)*d*f*x + (8*a^4*b + 6*a^2*b^3
- b^5)*d*e + (8*a^4*b + 6*a^2*b^3 - b^5)*f)*cosh(d*x + c))*sinh(d*x + c)^3 - 200*(3*(4*a^2*b^3 + b^5)*d*f*x +
3*(4*a^2*b^3 + b^5)*d*e + (4*a^2*b^3 + b^5)*f)*cosh(d*x + c)^2 + 20*(162*(5*b^5*d*f*x + 5*b^5*d*e - b^5*f)*cos
h(d*x + c)^8 - 405*(4*a*b^4*d*f*x + 4*a*b^4*d*e - a*b^4*f)*cosh(d*x + c)^7 + 280*(3*(4*a^2*b^3 + b^5)*d*f*x +
3*(4*a^2*b^3 + b^5)*d*e - (4*a^2*b^3 + b^5)*f)*cosh(d*x + c)^6 - 1890*(2*(2*a^3*b^2 + a*b^4)*d*f*x + 2*(2*a^3*
b^2 + a*b^4)*d*e - (2*a^3*b^2 + a*b^4)*f)*cosh(d*x + c)^5 + 2700*((8*a^4*b + 6*a^2*b^3 - b^5)*d*f*x + (8*a^4*b
 + 6*a^2*b^3 - b^5)*d*e - (8*a^4*b + 6*a^2*b^3 - b^5)*f)*cosh(d*x + c)^4 - 30*(4*a^2*b^3 + b^5)*d*f*x + 14400*
((a^5 + a^3*b^2)*d^2*f*x^2 + 2*(a^5 + a^3*b^2)*d^2*e*x + 4*(a^5 + a^3*b^2)*c*d*e - 2*(a^5 + a^3*b^2)*c^2*f)*co
sh(d*x + c)^3 - 30*(4*a^2*b^3 + b^5)*d*e - 1080*((8*a^4*b + 6*a^2*b^3 - b^5)*d*f*x + (8*a^4*b + 6*a^2*b^3 - b^
5)*d*e + (8*a^4*b + 6*a^2*b^3 - b^5)*f)*cosh(d*x + c)^2 - 10*(4*a^2*b^3 + b^5)*f - 270*(2*(2*a^3*b^2 + a*b^4)*
d*f*x + 2*(2*a^3*b^2 + a*b^4)*d*e + (2*a^3*b^2 + a*b^4)*f)*cosh(d*x + c))*sinh(d*x + c)^2 - 225*(4*a*b^4*d*f*x
 + 4*a*b^4*d*e + a*b^4*f)*cosh(d*x + c) - 57600*((a^5 + a^3*b^2)*f*cosh(d*x + c)^5 + 5*(a^5 + a^3*b^2)*f*cosh(
d*x + c)^4*sinh(d*x + c) + 10*(a^5 + a^3*b^2)*f*cosh(d*x + c)^3*sinh(d*x + c)^2 + 10*(a^5 + a^3*b^2)*f*cosh(d*
x + c)^2*sinh(d*x + c)^3 + 5*(a^5 + a^3*b^2)*f*cosh(d*x + c)*sinh(d*x + c)^4 + (a^5 + a^3*b^2)*f*sinh(d*x + c)
^5)*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) - 57600*((a^5 + a^3*b^2)*f*cosh(d*x + c)^5 + 5*(a^5 + a^3*b^2)*f*cosh(d*x + c)^4*sinh(d*x + c) + 10*(a^
5 + a^3*b^2)*f*cosh(d*x + c)^3*sinh(d*x + c)^2 + 10*(a^5 + a^3*b^2)*f*cosh(d*x + c)^2*sinh(d*x + c)^3 + 5*(a^5
 + a^3*b^2)*f*cosh(d*x + c)*sinh(d*x + c)^4 + (a^5 + a^3*b^2)*f*sinh(d*x + c)^5)*dilog((a*cosh(d*x + c) + a*si
nh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) - 57600*(((a^5 + a^3*b^2)*
d*e - (a^5 + a^3*b^2)*c*f)*cosh(d*x + c)^5 + 5*((a^5 + a^3*b^2)*d*e - (a^5 + a^3*b^2)*c*f)*cosh(d*x + c)^4*sin
h(d*x + c) + 10*((a^5 + a^3*b^2)*d*e - (a^5 + a^3*b^2)*c*f)*cosh(d*x + c)^3*sinh(d*x + c)^2 + 10*((a^5 + a^3*b
^2)*d*e - (a^5 + a^3*b^2)*c*f)*cosh(d*x + c)^2*sinh(d*x + c)^3 + 5*((a^5 + a^3*b^2)*d*e - (a^5 + a^3*b^2)*c*f)
*cosh(d*x + c)*sinh(d*x + c)^4 + ((a^5 + a^3*b^2)*d*e - (a^5 + a^3*b^2)*c*f)*sinh(d*x + c)^5)*log(2*b*cosh(d*x
 + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) - 57600*(((a^5 + a^3*b^2)*d*e - (a^5 + a^3*b^2)*c
*f)*cosh(d*x + c)^5 + 5*((a^5 + a^3*b^2)*d*e - (a^5 + a^3*b^2)*c*f)*cosh(d*x + c)^4*sinh(d*x + c) + 10*((a^5 +
 a^3*b^2)*d*e - (a^5 + a^3*b^2)*c*f)*cosh(d*x + c)^3*sinh(d*x + c)^2 + 10*((a^5 + a^3*b^2)*d*e - (a^5 + a^3*b^
2)*c*f)*cosh(d*x + c)^2*sinh(d*x + c)^3 + 5*((a^5 + a^3*b^2)*d*e - (a^5 + a^3*b^2)*c*f)*cosh(d*x + c)*sinh(d*x
 + c)^4 + ((a^5 + a^3*b^2)*d*e - (a^5 + a^3*b^2)*c*f)*sinh(d*x + c)^5)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x +
c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) - 57600*(((a^5 + a^3*b^2)*d*f*x + (a^5 + a^3*b^2)*c*f)*cosh(d*x + c)^5 +
 5*((a^5 + a^3*b^2)*d*f*x + (a^5 + a^3*b^2)*c*f)*cosh(d*x + c)^4*sinh(d*x + c) + 10*((a^5 + a^3*b^2)*d*f*x + (
a^5 + a^3*b^2)*c*f)*cosh(d*x + c)^3*sinh(d*x + c)^2 + 10*((a^5 + a^3*b^2)*d*f*x + (a^5 + a^3*b^2)*c*f)*cosh(d*
x + c)^2*sinh(d*x + c)^3 + 5*((a^5 + a^3*b^2)*d*f*x + (a^5 + a^3*b^2)*c*f)*cosh(d*x + c)*sinh(d*x + c)^4 + ((a
^5 + a^3*b^2)*d*f*x + (a^5 + a^3*b^2)*c*f)*sinh(d*x + c)^5)*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) - 57600*(((a^5 + a^3*b^2)*d*f*x + (a^5 + a^3*b^2)*c*
f)*cosh(d*x + c)^5 + 5*((a^5 + a^3*b^2)*d*f*x + (a^5 + a^3*b^2)*c*f)*cosh(d*x + c)^4*sinh(d*x + c) + 10*((a^5
+ a^3*b^2)*d*f*x + (a^5 + a^3*b^2)*c*f)*cosh(d*x + c)^3*sinh(d*x + c)^2 + 10*((a^5 + a^3*b^2)*d*f*x + (a^5 + a
^3*b^2)*c*f)*cosh(d*x + c)^2*sinh(d*x + c)^3 + 5*((a^5 + a^3*b^2)*d*f*x + (a^5 + a^3*b^2)*c*f)*cosh(d*x + c)*s
inh(d*x + c)^4 + ((a^5 + a^3*b^2)*d*f*x + (a^5 + a^3*b^2)*c*f)*sinh(d*x + c)^5)*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) + 5*(144*(5*b^5*d*f*x + 5*b^5*d*
e - b^5*f)*cosh(d*x + c)^9 - 405*(4*a*b^4*d*f*x + 4*a*b^4*d*e - a*b^4*f)*cosh(d*x + c)^8 - 180*a*b^4*d*f*x + 3
20*(3*(4*a^2*b^3 + b^5)*d*f*x + 3*(4*a^2*b^3 + b^5)*d*e - (4*a^2*b^3 + b^5)*f)*cosh(d*x + c)^7 - 180*a*b^4*d*e
 - 2520*(2*(2*a^3*b^2 + a*b^4)*d*f*x + 2*(2*a^3*b^2 + a*b^4)*d*e - (2*a^3*b^2 + a*b^4)*f)*cosh(d*x + c)^6 - 45
*a*b^4*f + 4320*((8*a^4*b + 6*a^2*b^3 - b^5)*d*f*x + (8*a^4*b + 6*a^2*b^3 - b^5)*d*e - (8*a^4*b + 6*a^2*b^3 -
b^5)*f)*cosh(d*x + c)^5 + 28800*((a^5 + a^3*b^2)*d^2*f*x^2 + 2*(a^5 + a^3*b^2)*d^2*e*x + 4*(a^5 + a^3*b^2)*c*d
*e - 2*(a^5 + a^3*b^2)*c^2*f)*cosh(d*x + c)^4 - 2880*((8*a^4*b + 6*a^2*b^3 - b^5)*d*f*x + (8*a^4*b + 6*a^2*b^3
 - b^5)*d*e + (8*a^4*b + 6*a^2*b^3 - b^5)*f)*cosh(d*x + c)^3 - 1080*(2*(2*a^3*b^2 + a*b^4)*d*f*x + 2*(2*a^3*b^
2 + a*b^4)*d*e + (2*a^3*b^2 + a*b^4)*f)*cosh(d*x + c)^2 - 80*(3*(4*a^2*b^3 + b^5)*d*f*x + 3*(4*a^2*b^3 + b^5)*
d*e + (4*a^2*b^3 + b^5)*f)*cosh(d*x + c))*sinh(d*x + c))/(b^6*d^2*cosh(d*x + c)^5 + 5*b^6*d^2*cosh(d*x + c)^4*
sinh(d*x + c) + 10*b^6*d^2*cosh(d*x + c)^3*sinh(d*x + c)^2 + 10*b^6*d^2*cosh(d*x + c)^2*sinh(d*x + c)^3 + 5*b^
6*d^2*cosh(d*x + c)*sinh(d*x + c)^4 + b^6*d^2*sinh(d*x + c)^5)

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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)**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 )^{3} \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)^3*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

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