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

Optimal. Leaf size=636 \[ -\frac{2 a f \left (a^2+b^2\right ) (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^4 d^2}-\frac{2 a f \left (a^2+b^2\right ) (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^4 d^2}+\frac{2 a f^2 \left (a^2+b^2\right ) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^4 d^3}+\frac{2 a f^2 \left (a^2+b^2\right ) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^4 d^3}-\frac{2 a^2 f (e+f x) \cosh (c+d x)}{b^3 d^2}+\frac{2 a^2 f^2 \sinh (c+d x)}{b^3 d^3}-\frac{a \left (a^2+b^2\right ) (e+f x)^2 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b^4 d}-\frac{a \left (a^2+b^2\right ) (e+f x)^2 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b^4 d}+\frac{a^2 (e+f x)^2 \sinh (c+d x)}{b^3 d}+\frac{a \left (a^2+b^2\right ) (e+f x)^3}{3 b^4 f}+\frac{a f (e+f x) \sinh (c+d x) \cosh (c+d x)}{2 b^2 d^2}-\frac{a f^2 \sinh ^2(c+d x)}{4 b^2 d^3}-\frac{a (e+f x)^2 \sinh ^2(c+d x)}{2 b^2 d}-\frac{a e f x}{2 b^2 d}-\frac{a f^2 x^2}{4 b^2 d}-\frac{2 f (e+f x) \cosh ^3(c+d x)}{9 b d^2}-\frac{4 f (e+f x) \cosh (c+d x)}{3 b d^2}+\frac{2 f^2 \sinh ^3(c+d x)}{27 b d^3}+\frac{14 f^2 \sinh (c+d x)}{9 b d^3}+\frac{2 (e+f x)^2 \sinh (c+d x)}{3 b d}+\frac{(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 b d} \]

[Out]

-(a*e*f*x)/(2*b^2*d) - (a*f^2*x^2)/(4*b^2*d) + (a*(a^2 + b^2)*(e + f*x)^3)/(3*b^4*f) - (2*a^2*f*(e + f*x)*Cosh
[c + d*x])/(b^3*d^2) - (4*f*(e + f*x)*Cosh[c + d*x])/(3*b*d^2) - (2*f*(e + f*x)*Cosh[c + d*x]^3)/(9*b*d^2) - (
a*(a^2 + b^2)*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^4*d) - (a*(a^2 + b^2)*(e + f*x)^2
*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^4*d) - (2*a*(a^2 + b^2)*f*(e + f*x)*PolyLog[2, -((b*E^(c +
d*x))/(a - Sqrt[a^2 + b^2]))])/(b^4*d^2) - (2*a*(a^2 + b^2)*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqr
t[a^2 + b^2]))])/(b^4*d^2) + (2*a*(a^2 + b^2)*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^4*d
^3) + (2*a*(a^2 + b^2)*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^4*d^3) + (2*a^2*f^2*Sinh[c
+ d*x])/(b^3*d^3) + (14*f^2*Sinh[c + d*x])/(9*b*d^3) + (a^2*(e + f*x)^2*Sinh[c + d*x])/(b^3*d) + (2*(e + f*x)
^2*Sinh[c + d*x])/(3*b*d) + (a*f*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(2*b^2*d^2) + ((e + f*x)^2*Cosh[c + d*
x]^2*Sinh[c + d*x])/(3*b*d) - (a*f^2*Sinh[c + d*x]^2)/(4*b^2*d^3) - (a*(e + f*x)^2*Sinh[c + d*x]^2)/(2*b^2*d)
+ (2*f^2*Sinh[c + d*x]^3)/(27*b*d^3)

________________________________________________________________________________________

Rubi [A]  time = 0.875195, antiderivative size = 636, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 13, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.382, Rules used = {5579, 3311, 3296, 2637, 2633, 5565, 5446, 3310, 5561, 2190, 2531, 2282, 6589} \[ -\frac{2 a f \left (a^2+b^2\right ) (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^4 d^2}-\frac{2 a f \left (a^2+b^2\right ) (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^4 d^2}+\frac{2 a f^2 \left (a^2+b^2\right ) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^4 d^3}+\frac{2 a f^2 \left (a^2+b^2\right ) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^4 d^3}-\frac{2 a^2 f (e+f x) \cosh (c+d x)}{b^3 d^2}+\frac{2 a^2 f^2 \sinh (c+d x)}{b^3 d^3}-\frac{a \left (a^2+b^2\right ) (e+f x)^2 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b^4 d}-\frac{a \left (a^2+b^2\right ) (e+f x)^2 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b^4 d}+\frac{a^2 (e+f x)^2 \sinh (c+d x)}{b^3 d}+\frac{a \left (a^2+b^2\right ) (e+f x)^3}{3 b^4 f}+\frac{a f (e+f x) \sinh (c+d x) \cosh (c+d x)}{2 b^2 d^2}-\frac{a f^2 \sinh ^2(c+d x)}{4 b^2 d^3}-\frac{a (e+f x)^2 \sinh ^2(c+d x)}{2 b^2 d}-\frac{a e f x}{2 b^2 d}-\frac{a f^2 x^2}{4 b^2 d}-\frac{2 f (e+f x) \cosh ^3(c+d x)}{9 b d^2}-\frac{4 f (e+f x) \cosh (c+d x)}{3 b d^2}+\frac{2 f^2 \sinh ^3(c+d x)}{27 b d^3}+\frac{14 f^2 \sinh (c+d x)}{9 b d^3}+\frac{2 (e+f x)^2 \sinh (c+d x)}{3 b d}+\frac{(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*e*f*x)/(2*b^2*d) - (a*f^2*x^2)/(4*b^2*d) + (a*(a^2 + b^2)*(e + f*x)^3)/(3*b^4*f) - (2*a^2*f*(e + f*x)*Cosh
[c + d*x])/(b^3*d^2) - (4*f*(e + f*x)*Cosh[c + d*x])/(3*b*d^2) - (2*f*(e + f*x)*Cosh[c + d*x]^3)/(9*b*d^2) - (
a*(a^2 + b^2)*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^4*d) - (a*(a^2 + b^2)*(e + f*x)^2
*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^4*d) - (2*a*(a^2 + b^2)*f*(e + f*x)*PolyLog[2, -((b*E^(c +
d*x))/(a - Sqrt[a^2 + b^2]))])/(b^4*d^2) - (2*a*(a^2 + b^2)*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqr
t[a^2 + b^2]))])/(b^4*d^2) + (2*a*(a^2 + b^2)*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^4*d
^3) + (2*a*(a^2 + b^2)*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^4*d^3) + (2*a^2*f^2*Sinh[c
+ d*x])/(b^3*d^3) + (14*f^2*Sinh[c + d*x])/(9*b*d^3) + (a^2*(e + f*x)^2*Sinh[c + d*x])/(b^3*d) + (2*(e + f*x)
^2*Sinh[c + d*x])/(3*b*d) + (a*f*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(2*b^2*d^2) + ((e + f*x)^2*Cosh[c + d*
x]^2*Sinh[c + d*x])/(3*b*d) - (a*f^2*Sinh[c + d*x]^2)/(4*b^2*d^3) - (a*(e + f*x)^2*Sinh[c + d*x]^2)/(2*b^2*d)
+ (2*f^2*Sinh[c + d*x]^3)/(27*b*d^3)

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 3311

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

Rule 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 2637

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

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 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 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 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 2531

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

Rule 2282

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

Rule 6589

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

Rubi steps

\begin{align*} \int \frac{(e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac{\int (e+f x)^2 \cosh ^3(c+d x) \, dx}{b}-\frac{a \int \frac{(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=-\frac{2 f (e+f x) \cosh ^3(c+d x)}{9 b d^2}+\frac{(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b d}+\frac{a^2 \int (e+f x)^2 \cosh (c+d x) \, dx}{b^3}-\frac{a \int (e+f x)^2 \cosh (c+d x) \sinh (c+d x) \, dx}{b^2}+\frac{2 \int (e+f x)^2 \cosh (c+d x) \, dx}{3 b}-\frac{\left (a \left (a^2+b^2\right )\right ) \int \frac{(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^3}+\frac{\left (2 f^2\right ) \int \cosh ^3(c+d x) \, dx}{9 b d^2}\\ &=\frac{a \left (a^2+b^2\right ) (e+f x)^3}{3 b^4 f}-\frac{2 f (e+f x) \cosh ^3(c+d x)}{9 b d^2}+\frac{a^2 (e+f x)^2 \sinh (c+d x)}{b^3 d}+\frac{2 (e+f x)^2 \sinh (c+d x)}{3 b d}+\frac{(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b d}-\frac{a (e+f x)^2 \sinh ^2(c+d x)}{2 b^2 d}-\frac{\left (a \left (a^2+b^2\right )\right ) \int \frac{e^{c+d x} (e+f x)^2}{a-\sqrt{a^2+b^2}+b e^{c+d x}} \, dx}{b^3}-\frac{\left (a \left (a^2+b^2\right )\right ) \int \frac{e^{c+d x} (e+f x)^2}{a+\sqrt{a^2+b^2}+b e^{c+d x}} \, dx}{b^3}-\frac{\left (2 a^2 f\right ) \int (e+f x) \sinh (c+d x) \, dx}{b^3 d}+\frac{(a f) \int (e+f x) \sinh ^2(c+d x) \, dx}{b^2 d}-\frac{(4 f) \int (e+f x) \sinh (c+d x) \, dx}{3 b d}+\frac{\left (2 i f^2\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-i \sinh (c+d x)\right )}{9 b d^3}\\ &=\frac{a \left (a^2+b^2\right ) (e+f x)^3}{3 b^4 f}-\frac{2 a^2 f (e+f x) \cosh (c+d x)}{b^3 d^2}-\frac{4 f (e+f x) \cosh (c+d x)}{3 b d^2}-\frac{2 f (e+f x) \cosh ^3(c+d x)}{9 b d^2}-\frac{a \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^4 d}-\frac{a \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^4 d}+\frac{2 f^2 \sinh (c+d x)}{9 b d^3}+\frac{a^2 (e+f x)^2 \sinh (c+d x)}{b^3 d}+\frac{2 (e+f x)^2 \sinh (c+d x)}{3 b d}+\frac{a f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^2 d^2}+\frac{(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b d}-\frac{a f^2 \sinh ^2(c+d x)}{4 b^2 d^3}-\frac{a (e+f x)^2 \sinh ^2(c+d x)}{2 b^2 d}+\frac{2 f^2 \sinh ^3(c+d x)}{27 b d^3}-\frac{(a f) \int (e+f x) \, dx}{2 b^2 d}+\frac{\left (2 a \left (a^2+b^2\right ) f\right ) \int (e+f x) \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) \, dx}{b^4 d}+\frac{\left (2 a \left (a^2+b^2\right ) f\right ) \int (e+f x) \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) \, dx}{b^4 d}+\frac{\left (2 a^2 f^2\right ) \int \cosh (c+d x) \, dx}{b^3 d^2}+\frac{\left (4 f^2\right ) \int \cosh (c+d x) \, dx}{3 b d^2}\\ &=-\frac{a e f x}{2 b^2 d}-\frac{a f^2 x^2}{4 b^2 d}+\frac{a \left (a^2+b^2\right ) (e+f x)^3}{3 b^4 f}-\frac{2 a^2 f (e+f x) \cosh (c+d x)}{b^3 d^2}-\frac{4 f (e+f x) \cosh (c+d x)}{3 b d^2}-\frac{2 f (e+f x) \cosh ^3(c+d x)}{9 b d^2}-\frac{a \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^4 d}-\frac{a \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^4 d}-\frac{2 a \left (a^2+b^2\right ) f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^4 d^2}-\frac{2 a \left (a^2+b^2\right ) f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^4 d^2}+\frac{2 a^2 f^2 \sinh (c+d x)}{b^3 d^3}+\frac{14 f^2 \sinh (c+d x)}{9 b d^3}+\frac{a^2 (e+f x)^2 \sinh (c+d x)}{b^3 d}+\frac{2 (e+f x)^2 \sinh (c+d x)}{3 b d}+\frac{a f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^2 d^2}+\frac{(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b d}-\frac{a f^2 \sinh ^2(c+d x)}{4 b^2 d^3}-\frac{a (e+f x)^2 \sinh ^2(c+d x)}{2 b^2 d}+\frac{2 f^2 \sinh ^3(c+d x)}{27 b d^3}+\frac{\left (2 a \left (a^2+b^2\right ) f^2\right ) \int \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) \, dx}{b^4 d^2}+\frac{\left (2 a \left (a^2+b^2\right ) f^2\right ) \int \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) \, dx}{b^4 d^2}\\ &=-\frac{a e f x}{2 b^2 d}-\frac{a f^2 x^2}{4 b^2 d}+\frac{a \left (a^2+b^2\right ) (e+f x)^3}{3 b^4 f}-\frac{2 a^2 f (e+f x) \cosh (c+d x)}{b^3 d^2}-\frac{4 f (e+f x) \cosh (c+d x)}{3 b d^2}-\frac{2 f (e+f x) \cosh ^3(c+d x)}{9 b d^2}-\frac{a \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^4 d}-\frac{a \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^4 d}-\frac{2 a \left (a^2+b^2\right ) f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^4 d^2}-\frac{2 a \left (a^2+b^2\right ) f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^4 d^2}+\frac{2 a^2 f^2 \sinh (c+d x)}{b^3 d^3}+\frac{14 f^2 \sinh (c+d x)}{9 b d^3}+\frac{a^2 (e+f x)^2 \sinh (c+d x)}{b^3 d}+\frac{2 (e+f x)^2 \sinh (c+d x)}{3 b d}+\frac{a f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^2 d^2}+\frac{(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b d}-\frac{a f^2 \sinh ^2(c+d x)}{4 b^2 d^3}-\frac{a (e+f x)^2 \sinh ^2(c+d x)}{2 b^2 d}+\frac{2 f^2 \sinh ^3(c+d x)}{27 b d^3}+\frac{\left (2 a \left (a^2+b^2\right ) f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{b x}{-a+\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^4 d^3}+\frac{\left (2 a \left (a^2+b^2\right ) f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{b x}{a+\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^4 d^3}\\ &=-\frac{a e f x}{2 b^2 d}-\frac{a f^2 x^2}{4 b^2 d}+\frac{a \left (a^2+b^2\right ) (e+f x)^3}{3 b^4 f}-\frac{2 a^2 f (e+f x) \cosh (c+d x)}{b^3 d^2}-\frac{4 f (e+f x) \cosh (c+d x)}{3 b d^2}-\frac{2 f (e+f x) \cosh ^3(c+d x)}{9 b d^2}-\frac{a \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^4 d}-\frac{a \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^4 d}-\frac{2 a \left (a^2+b^2\right ) f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^4 d^2}-\frac{2 a \left (a^2+b^2\right ) f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^4 d^2}+\frac{2 a \left (a^2+b^2\right ) f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^4 d^3}+\frac{2 a \left (a^2+b^2\right ) f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^4 d^3}+\frac{2 a^2 f^2 \sinh (c+d x)}{b^3 d^3}+\frac{14 f^2 \sinh (c+d x)}{9 b d^3}+\frac{a^2 (e+f x)^2 \sinh (c+d x)}{b^3 d}+\frac{2 (e+f x)^2 \sinh (c+d x)}{3 b d}+\frac{a f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^2 d^2}+\frac{(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b d}-\frac{a f^2 \sinh ^2(c+d x)}{4 b^2 d^3}-\frac{a (e+f x)^2 \sinh ^2(c+d x)}{2 b^2 d}+\frac{2 f^2 \sinh ^3(c+d x)}{27 b d^3}\\ \end{align*}

Mathematica [B]  time = 16.7223, size = 3509, normalized size = 5.52 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(f^2*(2*a*x^3*(-1 + Coth[c]) - 2*a*x^3*Coth[c] - (6*a*b^2*(d^2*x^2*Log[1 + ((a - Sqrt[a^2 + b^2])*(Cosh[c + d*
x] - Sinh[c + d*x]))/b] - 2*d*x*PolyLog[2, ((-a + Sqrt[a^2 + b^2])*(Cosh[c + d*x] - Sinh[c + d*x]))/b] - 2*Pol
yLog[3, ((-a + Sqrt[a^2 + b^2])*(Cosh[c + d*x] - Sinh[c + d*x]))/b]))/(Sqrt[a^2 + b^2]*(-a + Sqrt[a^2 + b^2])*
d^3) - (6*a*b^2*(d^2*x^2*Log[1 + ((a + Sqrt[a^2 + b^2])*(Cosh[c + d*x] - Sinh[c + d*x]))/b] - 2*d*x*PolyLog[2,
((a + Sqrt[a^2 + b^2])*(-Cosh[c + d*x] + Sinh[c + d*x]))/b] - 2*PolyLog[3, ((a + Sqrt[a^2 + b^2])*(-Cosh[c +
d*x] + Sinh[c + d*x]))/b]))/(Sqrt[a^2 + b^2]*(a + Sqrt[a^2 + b^2])*d^3) + (6*a^2*(d^2*x^2*Log[1 + (b*(Cosh[c +
d*x] + Sinh[c + d*x]))/(a - Sqrt[a^2 + b^2])] + 2*d*x*PolyLog[2, (b*(Cosh[c + d*x] + Sinh[c + d*x]))/(-a + Sq
rt[a^2 + b^2])] - 2*PolyLog[3, (b*(Cosh[c + d*x] + Sinh[c + d*x]))/(-a + Sqrt[a^2 + b^2])]))/(Sqrt[a^2 + b^2]*
d^3) - (6*a^2*(d^2*x^2*Log[1 + (b*(Cosh[c + d*x] + Sinh[c + d*x]))/(a + Sqrt[a^2 + b^2])] + 2*d*x*PolyLog[2, -
((b*(Cosh[c + d*x] + Sinh[c + d*x]))/(a + Sqrt[a^2 + b^2]))] - 2*PolyLog[3, -((b*(Cosh[c + d*x] + Sinh[c + d*x
]))/(a + Sqrt[a^2 + b^2]))]))/(Sqrt[a^2 + b^2]*d^3) + (6*b*Cosh[d*x]*(-2*d*x*Cosh[c] + (2 + d^2*x^2)*Sinh[c]))
/d^3 + (6*b*((2 + d^2*x^2)*Cosh[c] - 2*d*x*Sinh[c])*Sinh[d*x])/d^3))/(12*b^2) - (e^2*((a*Log[a + b*Sinh[c + d*
x]])/b^2 - Sinh[c + d*x]/b))/(2*d) + (e*f*(-(b*Cosh[c + d*x]) - a*(-(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]))]) + b*d*x*Sinh[c + d*x]))/(b^2*d^2) + (e^2*(-3*a*(2*a^2 + b^2)*Log[a + b*Sinh[c + d*x]] + 3*b*(2*a^2 + b
^2)*Sinh[c + d*x] - 3*a*b^2*Sinh[c + d*x]^2 + 2*b^3*Sinh[c + d*x]^3))/(6*b^4*d) + (e*f*(-18*b*(4*a^2 + b^2)*Co
sh[c + d*x] - 18*a*b^2*d*x*Cosh[2*(c + d*x)] - 2*b^3*Cosh[3*(c + d*x)] - 36*a*(2*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 + 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]))]) + 18*b*(4*a^2 + b^2)*d*x*Sinh[c + d*x] + 9*a*b^2*Sinh[2*(c + d*x)] + 6*b^3*d
*x*Sinh[3*(c + d*x)]))/(36*b^4*d^2) + (f^2*((2*a*(2*a^2 + b^2)*(-1 + Coth[c])*(2*x^3 + (6*a*(d^2*x^2*Log[1 + (
b*(Cosh[c + d*x] + Sinh[c + d*x]))/(a - Sqrt[a^2 + b^2])] + 2*d*x*PolyLog[2, (b*(Cosh[c + d*x] + Sinh[c + d*x]
))/(-a + Sqrt[a^2 + b^2])] - 2*PolyLog[3, (b*(Cosh[c + d*x] + Sinh[c + d*x]))/(-a + Sqrt[a^2 + b^2])])*Sinh[c]
*(Cosh[c] + Sinh[c]))/(Sqrt[a^2 + b^2]*d^3) - (3*b^2*(d^2*x^2*Log[1 + ((a - Sqrt[a^2 + b^2])*(Cosh[c + d*x] -
Sinh[c + d*x]))/b] - 2*d*x*PolyLog[2, ((-a + Sqrt[a^2 + b^2])*(Cosh[c + d*x] - Sinh[c + d*x]))/b] - 2*PolyLog[
3, ((-a + Sqrt[a^2 + b^2])*(Cosh[c + d*x] - Sinh[c + d*x]))/b])*(-1 + Cosh[2*c] + Sinh[2*c]))/(Sqrt[a^2 + b^2]
*(-a + Sqrt[a^2 + b^2])*d^3) - (3*b^2*(d^2*x^2*Log[1 + ((a + Sqrt[a^2 + b^2])*(Cosh[c + d*x] - Sinh[c + d*x]))
/b] - 2*d*x*PolyLog[2, ((a + Sqrt[a^2 + b^2])*(-Cosh[c + d*x] + Sinh[c + d*x]))/b] - 2*PolyLog[3, ((a + Sqrt[a
^2 + b^2])*(-Cosh[c + d*x] + Sinh[c + d*x]))/b])*(-1 + Cosh[2*c] + Sinh[2*c]))/(Sqrt[a^2 + b^2]*(a + Sqrt[a^2
+ b^2])*d^3) - (3*a*(d^2*x^2*Log[1 + (b*(Cosh[c + d*x] + Sinh[c + d*x]))/(a + Sqrt[a^2 + b^2])] + 2*d*x*PolyLo
g[2, -((b*(Cosh[c + d*x] + Sinh[c + d*x]))/(a + Sqrt[a^2 + b^2]))] - 2*PolyLog[3, -((b*(Cosh[c + d*x] + Sinh[c
+ d*x]))/(a + Sqrt[a^2 + b^2]))])*(-1 + Cosh[2*c] + Sinh[2*c]))/(Sqrt[a^2 + b^2]*d^3)))/(3*b^4) + Csch[c]*(Co
sh[3*c + 3*d*x]/(108*b^4*d^3) - Sinh[3*c + 3*d*x]/(108*b^4*d^3))*(27*a*b^2*Cosh[d*x] + 54*a*b^2*d*x*Cosh[d*x]
+ 54*a*b^2*d^2*x^2*Cosh[d*x] - 27*a*b^2*Cosh[2*c + d*x] - 54*a*b^2*d*x*Cosh[2*c + d*x] - 54*a*b^2*d^2*x^2*Cosh
[2*c + d*x] + 432*a^2*b*Cosh[c + 2*d*x] + 108*b^3*Cosh[c + 2*d*x] + 432*a^2*b*d*x*Cosh[c + 2*d*x] + 108*b^3*d*
x*Cosh[c + 2*d*x] + 216*a^2*b*d^2*x^2*Cosh[c + 2*d*x] + 54*b^3*d^2*x^2*Cosh[c + 2*d*x] - 432*a^2*b*Cosh[3*c +
2*d*x] - 108*b^3*Cosh[3*c + 2*d*x] - 432*a^2*b*d*x*Cosh[3*c + 2*d*x] - 108*b^3*d*x*Cosh[3*c + 2*d*x] - 216*a^2
*b*d^2*x^2*Cosh[3*c + 2*d*x] - 54*b^3*d^2*x^2*Cosh[3*c + 2*d*x] - 144*a^3*d^3*x^3*Cosh[2*c + 3*d*x] - 72*a*b^2
*d^3*x^3*Cosh[2*c + 3*d*x] - 144*a^3*d^3*x^3*Cosh[4*c + 3*d*x] - 72*a*b^2*d^3*x^3*Cosh[4*c + 3*d*x] - 432*a^2*
b*Cosh[3*c + 4*d*x] - 108*b^3*Cosh[3*c + 4*d*x] + 432*a^2*b*d*x*Cosh[3*c + 4*d*x] + 108*b^3*d*x*Cosh[3*c + 4*d
*x] - 216*a^2*b*d^2*x^2*Cosh[3*c + 4*d*x] - 54*b^3*d^2*x^2*Cosh[3*c + 4*d*x] + 432*a^2*b*Cosh[5*c + 4*d*x] + 1
08*b^3*Cosh[5*c + 4*d*x] - 432*a^2*b*d*x*Cosh[5*c + 4*d*x] - 108*b^3*d*x*Cosh[5*c + 4*d*x] + 216*a^2*b*d^2*x^2
*Cosh[5*c + 4*d*x] + 54*b^3*d^2*x^2*Cosh[5*c + 4*d*x] + 27*a*b^2*Cosh[4*c + 5*d*x] - 54*a*b^2*d*x*Cosh[4*c + 5
*d*x] + 54*a*b^2*d^2*x^2*Cosh[4*c + 5*d*x] - 27*a*b^2*Cosh[6*c + 5*d*x] + 54*a*b^2*d*x*Cosh[6*c + 5*d*x] - 54*
a*b^2*d^2*x^2*Cosh[6*c + 5*d*x] - 4*b^3*Cosh[5*c + 6*d*x] + 12*b^3*d*x*Cosh[5*c + 6*d*x] - 18*b^3*d^2*x^2*Cosh
[5*c + 6*d*x] + 4*b^3*Cosh[7*c + 6*d*x] - 12*b^3*d*x*Cosh[7*c + 6*d*x] + 18*b^3*d^2*x^2*Cosh[7*c + 6*d*x] - 8*
b^3*Sinh[c] - 24*b^3*d*x*Sinh[c] - 36*b^3*d^2*x^2*Sinh[c] + 27*a*b^2*Sinh[d*x] + 54*a*b^2*d*x*Sinh[d*x] + 54*a
*b^2*d^2*x^2*Sinh[d*x] - 27*a*b^2*Sinh[2*c + d*x] - 54*a*b^2*d*x*Sinh[2*c + d*x] - 54*a*b^2*d^2*x^2*Sinh[2*c +
d*x] + 432*a^2*b*Sinh[c + 2*d*x] + 108*b^3*Sinh[c + 2*d*x] + 432*a^2*b*d*x*Sinh[c + 2*d*x] + 108*b^3*d*x*Sinh
[c + 2*d*x] + 216*a^2*b*d^2*x^2*Sinh[c + 2*d*x] + 54*b^3*d^2*x^2*Sinh[c + 2*d*x] - 432*a^2*b*Sinh[3*c + 2*d*x]
- 108*b^3*Sinh[3*c + 2*d*x] - 432*a^2*b*d*x*Sinh[3*c + 2*d*x] - 108*b^3*d*x*Sinh[3*c + 2*d*x] - 216*a^2*b*d^2
*x^2*Sinh[3*c + 2*d*x] - 54*b^3*d^2*x^2*Sinh[3*c + 2*d*x] - 144*a^3*d^3*x^3*Sinh[2*c + 3*d*x] - 72*a*b^2*d^3*x
^3*Sinh[2*c + 3*d*x] - 144*a^3*d^3*x^3*Sinh[4*c + 3*d*x] - 72*a*b^2*d^3*x^3*Sinh[4*c + 3*d*x] - 432*a^2*b*Sinh
[3*c + 4*d*x] - 108*b^3*Sinh[3*c + 4*d*x] + 432*a^2*b*d*x*Sinh[3*c + 4*d*x] + 108*b^3*d*x*Sinh[3*c + 4*d*x] -
216*a^2*b*d^2*x^2*Sinh[3*c + 4*d*x] - 54*b^3*d^2*x^2*Sinh[3*c + 4*d*x] + 432*a^2*b*Sinh[5*c + 4*d*x] + 108*b^3
*Sinh[5*c + 4*d*x] - 432*a^2*b*d*x*Sinh[5*c + 4*d*x] - 108*b^3*d*x*Sinh[5*c + 4*d*x] + 216*a^2*b*d^2*x^2*Sinh[
5*c + 4*d*x] + 54*b^3*d^2*x^2*Sinh[5*c + 4*d*x] + 27*a*b^2*Sinh[4*c + 5*d*x] - 54*a*b^2*d*x*Sinh[4*c + 5*d*x]
+ 54*a*b^2*d^2*x^2*Sinh[4*c + 5*d*x] - 27*a*b^2*Sinh[6*c + 5*d*x] + 54*a*b^2*d*x*Sinh[6*c + 5*d*x] - 54*a*b^2*
d^2*x^2*Sinh[6*c + 5*d*x] - 4*b^3*Sinh[5*c + 6*d*x] + 12*b^3*d*x*Sinh[5*c + 6*d*x] - 18*b^3*d^2*x^2*Sinh[5*c +
6*d*x] + 4*b^3*Sinh[7*c + 6*d*x] - 12*b^3*d*x*Sinh[7*c + 6*d*x] + 18*b^3*d^2*x^2*Sinh[7*c + 6*d*x])))/8

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Maple [F]  time = 0.191, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{2} \left ( \cosh \left ( dx+c \right ) \right ) ^{3}\sinh \left ( dx+c \right ) }{a+b\sinh \left ( dx+c \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

-1/24*e^2*((3*a*b*e^(-d*x - c) - b^2 - 3*(4*a^2 + 3*b^2)*e^(-2*d*x - 2*c))*e^(3*d*x + 3*c)/(b^3*d) + 24*(a^3 +
a*b^2)*(d*x + c)/(b^4*d) + (3*a*b*e^(-2*d*x - 2*c) + b^2*e^(-3*d*x - 3*c) + 3*(4*a^2 + 3*b^2)*e^(-d*x - c))/(
b^3*d) + 24*(a^3 + a*b^2)*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(b^4*d)) - 1/432*(144*(a^3*d^3*f^2*e
^(3*c) + a*b^2*d^3*f^2*e^(3*c))*x^3 + 432*(a^3*d^3*e*f*e^(3*c) + a*b^2*d^3*e*f*e^(3*c))*x^2 - 2*(9*b^3*d^2*f^2
*x^2*e^(6*c) + 6*(3*d^2*e*f - d*f^2)*b^3*x*e^(6*c) - 2*(3*d*e*f - f^2)*b^3*e^(6*c))*e^(3*d*x) + 27*(2*a*b^2*d^
2*f^2*x^2*e^(5*c) + 2*(2*d^2*e*f - d*f^2)*a*b^2*x*e^(5*c) - (2*d*e*f - f^2)*a*b^2*e^(5*c))*e^(2*d*x) + 54*(8*(
d*e*f - f^2)*a^2*b*e^(4*c) + 6*(d*e*f - f^2)*b^3*e^(4*c) - (4*a^2*b*d^2*f^2*e^(4*c) + 3*b^3*d^2*f^2*e^(4*c))*x
^2 - 2*(4*(d^2*e*f - d*f^2)*a^2*b*e^(4*c) + 3*(d^2*e*f - d*f^2)*b^3*e^(4*c))*x)*e^(d*x) + 54*(8*(d*e*f + f^2)*
a^2*b*e^(2*c) + 6*(d*e*f + f^2)*b^3*e^(2*c) + (4*a^2*b*d^2*f^2*e^(2*c) + 3*b^3*d^2*f^2*e^(2*c))*x^2 + 2*(4*(d^
2*e*f + d*f^2)*a^2*b*e^(2*c) + 3*(d^2*e*f + d*f^2)*b^3*e^(2*c))*x)*e^(-d*x) + 27*(2*a*b^2*d^2*f^2*x^2*e^c + 2*
(2*d^2*e*f + d*f^2)*a*b^2*x*e^c + (2*d*e*f + f^2)*a*b^2*e^c)*e^(-2*d*x) + 2*(9*b^3*d^2*f^2*x^2 + 6*(3*d^2*e*f
+ d*f^2)*b^3*x + 2*(3*d*e*f + f^2)*b^3)*e^(-3*d*x))*e^(-3*c)/(b^4*d^3) + integrate(-2*((a^3*b*f^2 + a*b^3*f^2)
*x^2 + 2*(a^3*b*e*f + a*b^3*e*f)*x - ((a^4*f^2*e^c + a^2*b^2*f^2*e^c)*x^2 + 2*(a^4*e*f*e^c + a^2*b^2*e*f*e^c)*
x)*e^(d*x))/(b^5*e^(2*d*x + 2*c) + 2*a*b^4*e^(d*x + c) - b^5), x)

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Fricas [C]  time = 3.04855, size = 10886, normalized size = 17.12 \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)^2*cosh(d*x+c)^3*sinh(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-1/432*(18*b^3*d^2*f^2*x^2 + 18*b^3*d^2*e^2 - 2*(9*b^3*d^2*f^2*x^2 + 9*b^3*d^2*e^2 - 6*b^3*d*e*f + 2*b^3*f^2 +
6*(3*b^3*d^2*e*f - b^3*d*f^2)*x)*cosh(d*x + c)^6 - 2*(9*b^3*d^2*f^2*x^2 + 9*b^3*d^2*e^2 - 6*b^3*d*e*f + 2*b^3
*f^2 + 6*(3*b^3*d^2*e*f - b^3*d*f^2)*x)*sinh(d*x + c)^6 + 12*b^3*d*e*f + 27*(2*a*b^2*d^2*f^2*x^2 + 2*a*b^2*d^2
*e^2 - 2*a*b^2*d*e*f + a*b^2*f^2 + 2*(2*a*b^2*d^2*e*f - a*b^2*d*f^2)*x)*cosh(d*x + c)^5 + 3*(18*a*b^2*d^2*f^2*
x^2 + 18*a*b^2*d^2*e^2 - 18*a*b^2*d*e*f + 9*a*b^2*f^2 + 18*(2*a*b^2*d^2*e*f - a*b^2*d*f^2)*x - 4*(9*b^3*d^2*f^
2*x^2 + 9*b^3*d^2*e^2 - 6*b^3*d*e*f + 2*b^3*f^2 + 6*(3*b^3*d^2*e*f - b^3*d*f^2)*x)*cosh(d*x + c))*sinh(d*x + c
)^5 + 4*b^3*f^2 - 54*((4*a^2*b + 3*b^3)*d^2*f^2*x^2 + (4*a^2*b + 3*b^3)*d^2*e^2 - 2*(4*a^2*b + 3*b^3)*d*e*f +
2*(4*a^2*b + 3*b^3)*f^2 + 2*((4*a^2*b + 3*b^3)*d^2*e*f - (4*a^2*b + 3*b^3)*d*f^2)*x)*cosh(d*x + c)^4 - 3*(18*(
4*a^2*b + 3*b^3)*d^2*f^2*x^2 + 18*(4*a^2*b + 3*b^3)*d^2*e^2 - 36*(4*a^2*b + 3*b^3)*d*e*f + 36*(4*a^2*b + 3*b^3
)*f^2 + 10*(9*b^3*d^2*f^2*x^2 + 9*b^3*d^2*e^2 - 6*b^3*d*e*f + 2*b^3*f^2 + 6*(3*b^3*d^2*e*f - b^3*d*f^2)*x)*cos
h(d*x + c)^2 + 36*((4*a^2*b + 3*b^3)*d^2*e*f - (4*a^2*b + 3*b^3)*d*f^2)*x - 45*(2*a*b^2*d^2*f^2*x^2 + 2*a*b^2*
d^2*e^2 - 2*a*b^2*d*e*f + a*b^2*f^2 + 2*(2*a*b^2*d^2*e*f - a*b^2*d*f^2)*x)*cosh(d*x + c))*sinh(d*x + c)^4 - 14
4*((a^3 + a*b^2)*d^3*f^2*x^3 + 3*(a^3 + a*b^2)*d^3*e*f*x^2 + 3*(a^3 + a*b^2)*d^3*e^2*x + 6*(a^3 + a*b^2)*c*d^2
*e^2 - 6*(a^3 + a*b^2)*c^2*d*e*f + 2*(a^3 + a*b^2)*c^3*f^2)*cosh(d*x + c)^3 - 2*(72*(a^3 + a*b^2)*d^3*f^2*x^3
+ 216*(a^3 + a*b^2)*d^3*e*f*x^2 + 216*(a^3 + a*b^2)*d^3*e^2*x + 432*(a^3 + a*b^2)*c*d^2*e^2 - 432*(a^3 + a*b^2
)*c^2*d*e*f + 144*(a^3 + a*b^2)*c^3*f^2 + 20*(9*b^3*d^2*f^2*x^2 + 9*b^3*d^2*e^2 - 6*b^3*d*e*f + 2*b^3*f^2 + 6*
(3*b^3*d^2*e*f - b^3*d*f^2)*x)*cosh(d*x + c)^3 - 135*(2*a*b^2*d^2*f^2*x^2 + 2*a*b^2*d^2*e^2 - 2*a*b^2*d*e*f +
a*b^2*f^2 + 2*(2*a*b^2*d^2*e*f - a*b^2*d*f^2)*x)*cosh(d*x + c)^2 + 108*((4*a^2*b + 3*b^3)*d^2*f^2*x^2 + (4*a^2
*b + 3*b^3)*d^2*e^2 - 2*(4*a^2*b + 3*b^3)*d*e*f + 2*(4*a^2*b + 3*b^3)*f^2 + 2*((4*a^2*b + 3*b^3)*d^2*e*f - (4*
a^2*b + 3*b^3)*d*f^2)*x)*cosh(d*x + c))*sinh(d*x + c)^3 + 54*((4*a^2*b + 3*b^3)*d^2*f^2*x^2 + (4*a^2*b + 3*b^3
)*d^2*e^2 + 2*(4*a^2*b + 3*b^3)*d*e*f + 2*(4*a^2*b + 3*b^3)*f^2 + 2*((4*a^2*b + 3*b^3)*d^2*e*f + (4*a^2*b + 3*
b^3)*d*f^2)*x)*cosh(d*x + c)^2 + 6*(9*(4*a^2*b + 3*b^3)*d^2*f^2*x^2 + 9*(4*a^2*b + 3*b^3)*d^2*e^2 - 5*(9*b^3*d
^2*f^2*x^2 + 9*b^3*d^2*e^2 - 6*b^3*d*e*f + 2*b^3*f^2 + 6*(3*b^3*d^2*e*f - b^3*d*f^2)*x)*cosh(d*x + c)^4 + 18*(
4*a^2*b + 3*b^3)*d*e*f + 45*(2*a*b^2*d^2*f^2*x^2 + 2*a*b^2*d^2*e^2 - 2*a*b^2*d*e*f + a*b^2*f^2 + 2*(2*a*b^2*d^
2*e*f - a*b^2*d*f^2)*x)*cosh(d*x + c)^3 + 18*(4*a^2*b + 3*b^3)*f^2 - 54*((4*a^2*b + 3*b^3)*d^2*f^2*x^2 + (4*a^
2*b + 3*b^3)*d^2*e^2 - 2*(4*a^2*b + 3*b^3)*d*e*f + 2*(4*a^2*b + 3*b^3)*f^2 + 2*((4*a^2*b + 3*b^3)*d^2*e*f - (4
*a^2*b + 3*b^3)*d*f^2)*x)*cosh(d*x + c)^2 + 18*((4*a^2*b + 3*b^3)*d^2*e*f + (4*a^2*b + 3*b^3)*d*f^2)*x - 72*((
a^3 + a*b^2)*d^3*f^2*x^3 + 3*(a^3 + a*b^2)*d^3*e*f*x^2 + 3*(a^3 + a*b^2)*d^3*e^2*x + 6*(a^3 + a*b^2)*c*d^2*e^2
- 6*(a^3 + a*b^2)*c^2*d*e*f + 2*(a^3 + a*b^2)*c^3*f^2)*cosh(d*x + c))*sinh(d*x + c)^2 + 12*(3*b^3*d^2*e*f + b
^3*d*f^2)*x + 27*(2*a*b^2*d^2*f^2*x^2 + 2*a*b^2*d^2*e^2 + 2*a*b^2*d*e*f + a*b^2*f^2 + 2*(2*a*b^2*d^2*e*f + a*b
^2*d*f^2)*x)*cosh(d*x + c) + 864*(((a^3 + a*b^2)*d*f^2*x + (a^3 + a*b^2)*d*e*f)*cosh(d*x + c)^3 + 3*((a^3 + a*
b^2)*d*f^2*x + (a^3 + a*b^2)*d*e*f)*cosh(d*x + c)^2*sinh(d*x + c) + 3*((a^3 + a*b^2)*d*f^2*x + (a^3 + a*b^2)*d
*e*f)*cosh(d*x + c)*sinh(d*x + c)^2 + ((a^3 + a*b^2)*d*f^2*x + (a^3 + a*b^2)*d*e*f)*sinh(d*x + c)^3)*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) + 864*
(((a^3 + a*b^2)*d*f^2*x + (a^3 + a*b^2)*d*e*f)*cosh(d*x + c)^3 + 3*((a^3 + a*b^2)*d*f^2*x + (a^3 + a*b^2)*d*e*
f)*cosh(d*x + c)^2*sinh(d*x + c) + 3*((a^3 + a*b^2)*d*f^2*x + (a^3 + a*b^2)*d*e*f)*cosh(d*x + c)*sinh(d*x + c)
^2 + ((a^3 + a*b^2)*d*f^2*x + (a^3 + a*b^2)*d*e*f)*sinh(d*x + c)^3)*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) + 432*(((a^3 + a*b^2)*d^2*e^2 - 2*(a^3
+ a*b^2)*c*d*e*f + (a^3 + a*b^2)*c^2*f^2)*cosh(d*x + c)^3 + 3*((a^3 + a*b^2)*d^2*e^2 - 2*(a^3 + a*b^2)*c*d*e*f
+ (a^3 + a*b^2)*c^2*f^2)*cosh(d*x + c)^2*sinh(d*x + c) + 3*((a^3 + a*b^2)*d^2*e^2 - 2*(a^3 + a*b^2)*c*d*e*f +
(a^3 + a*b^2)*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^2 + ((a^3 + a*b^2)*d^2*e^2 - 2*(a^3 + a*b^2)*c*d*e*f + (a^
3 + a*b^2)*c^2*f^2)*sinh(d*x + c)^3)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2
*a) + 432*(((a^3 + a*b^2)*d^2*e^2 - 2*(a^3 + a*b^2)*c*d*e*f + (a^3 + a*b^2)*c^2*f^2)*cosh(d*x + c)^3 + 3*((a^3
+ a*b^2)*d^2*e^2 - 2*(a^3 + a*b^2)*c*d*e*f + (a^3 + a*b^2)*c^2*f^2)*cosh(d*x + c)^2*sinh(d*x + c) + 3*((a^3 +
a*b^2)*d^2*e^2 - 2*(a^3 + a*b^2)*c*d*e*f + (a^3 + a*b^2)*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^2 + ((a^3 + a*b
^2)*d^2*e^2 - 2*(a^3 + a*b^2)*c*d*e*f + (a^3 + a*b^2)*c^2*f^2)*sinh(d*x + c)^3)*log(2*b*cosh(d*x + c) + 2*b*si
nh(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + 432*(((a^3 + a*b^2)*d^2*f^2*x^2 + 2*(a^3 + a*b^2)*d^2*e*f*x +
2*(a^3 + a*b^2)*c*d*e*f - (a^3 + a*b^2)*c^2*f^2)*cosh(d*x + c)^3 + 3*((a^3 + a*b^2)*d^2*f^2*x^2 + 2*(a^3 + a*
b^2)*d^2*e*f*x + 2*(a^3 + a*b^2)*c*d*e*f - (a^3 + a*b^2)*c^2*f^2)*cosh(d*x + c)^2*sinh(d*x + c) + 3*((a^3 + a*
b^2)*d^2*f^2*x^2 + 2*(a^3 + a*b^2)*d^2*e*f*x + 2*(a^3 + a*b^2)*c*d*e*f - (a^3 + a*b^2)*c^2*f^2)*cosh(d*x + c)*
sinh(d*x + c)^2 + ((a^3 + a*b^2)*d^2*f^2*x^2 + 2*(a^3 + a*b^2)*d^2*e*f*x + 2*(a^3 + a*b^2)*c*d*e*f - (a^3 + a*
b^2)*c^2*f^2)*sinh(d*x + c)^3)*log(-(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) + 432*(((a^3 + a*b^2)*d^2*f^2*x^2 + 2*(a^3 + a*b^2)*d^2*e*f*x + 2*(a^3 + a*b^2)*c
*d*e*f - (a^3 + a*b^2)*c^2*f^2)*cosh(d*x + c)^3 + 3*((a^3 + a*b^2)*d^2*f^2*x^2 + 2*(a^3 + a*b^2)*d^2*e*f*x + 2
*(a^3 + a*b^2)*c*d*e*f - (a^3 + a*b^2)*c^2*f^2)*cosh(d*x + c)^2*sinh(d*x + c) + 3*((a^3 + a*b^2)*d^2*f^2*x^2 +
2*(a^3 + a*b^2)*d^2*e*f*x + 2*(a^3 + a*b^2)*c*d*e*f - (a^3 + a*b^2)*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^2 +
((a^3 + a*b^2)*d^2*f^2*x^2 + 2*(a^3 + a*b^2)*d^2*e*f*x + 2*(a^3 + a*b^2)*c*d*e*f - (a^3 + a*b^2)*c^2*f^2)*sinh
(d*x + c)^3)*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) - 864*((a^3 + a*b^2)*f^2*cosh(d*x + c)^3 + 3*(a^3 + a*b^2)*f^2*cosh(d*x + c)^2*sinh(d*x + c) + 3*(a
^3 + a*b^2)*f^2*cosh(d*x + c)*sinh(d*x + c)^2 + (a^3 + a*b^2)*f^2*sinh(d*x + c)^3)*polylog(3, (a*cosh(d*x + c)
+ a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) - 864*((a^3 + a*b^2)*f^2*co
sh(d*x + c)^3 + 3*(a^3 + a*b^2)*f^2*cosh(d*x + c)^2*sinh(d*x + c) + 3*(a^3 + a*b^2)*f^2*cosh(d*x + c)*sinh(d*x
+ c)^2 + (a^3 + a*b^2)*f^2*sinh(d*x + c)^3)*polylog(3, (a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c)
+ b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) + 3*(18*a*b^2*d^2*f^2*x^2 + 18*a*b^2*d^2*e^2 + 18*a*b^2*d*e*f - 4
*(9*b^3*d^2*f^2*x^2 + 9*b^3*d^2*e^2 - 6*b^3*d*e*f + 2*b^3*f^2 + 6*(3*b^3*d^2*e*f - b^3*d*f^2)*x)*cosh(d*x + c)
^5 + 9*a*b^2*f^2 + 45*(2*a*b^2*d^2*f^2*x^2 + 2*a*b^2*d^2*e^2 - 2*a*b^2*d*e*f + a*b^2*f^2 + 2*(2*a*b^2*d^2*e*f
- a*b^2*d*f^2)*x)*cosh(d*x + c)^4 - 72*((4*a^2*b + 3*b^3)*d^2*f^2*x^2 + (4*a^2*b + 3*b^3)*d^2*e^2 - 2*(4*a^2*b
+ 3*b^3)*d*e*f + 2*(4*a^2*b + 3*b^3)*f^2 + 2*((4*a^2*b + 3*b^3)*d^2*e*f - (4*a^2*b + 3*b^3)*d*f^2)*x)*cosh(d*
x + c)^3 - 144*((a^3 + a*b^2)*d^3*f^2*x^3 + 3*(a^3 + a*b^2)*d^3*e*f*x^2 + 3*(a^3 + a*b^2)*d^3*e^2*x + 6*(a^3 +
a*b^2)*c*d^2*e^2 - 6*(a^3 + a*b^2)*c^2*d*e*f + 2*(a^3 + a*b^2)*c^3*f^2)*cosh(d*x + c)^2 + 18*(2*a*b^2*d^2*e*f
+ a*b^2*d*f^2)*x + 36*((4*a^2*b + 3*b^3)*d^2*f^2*x^2 + (4*a^2*b + 3*b^3)*d^2*e^2 + 2*(4*a^2*b + 3*b^3)*d*e*f
+ 2*(4*a^2*b + 3*b^3)*f^2 + 2*((4*a^2*b + 3*b^3)*d^2*e*f + (4*a^2*b + 3*b^3)*d*f^2)*x)*cosh(d*x + c))*sinh(d*x
+ c))/(b^4*d^3*cosh(d*x + c)^3 + 3*b^4*d^3*cosh(d*x + c)^2*sinh(d*x + c) + 3*b^4*d^3*cosh(d*x + c)*sinh(d*x +
c)^2 + b^4*d^3*sinh(d*x + c)^3)

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

[Out]

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