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

Optimal. Leaf size=522 \[ -\frac{2 a^3 f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^3 d^2 \sqrt{a^2+b^2}}+\frac{2 a^3 f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^3 d^2 \sqrt{a^2+b^2}}+\frac{2 a^3 f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^3 d^3 \sqrt{a^2+b^2}}-\frac{2 a^3 f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^3 d^3 \sqrt{a^2+b^2}}-\frac{a^3 (e+f x)^2 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b^3 d \sqrt{a^2+b^2}}+\frac{a^3 (e+f x)^2 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b^3 d \sqrt{a^2+b^2}}+\frac{a^2 (e+f x)^3}{3 b^3 f}+\frac{2 a f (e+f x) \sinh (c+d x)}{b^2 d^2}-\frac{2 a f^2 \cosh (c+d x)}{b^2 d^3}-\frac{a (e+f x)^2 \cosh (c+d x)}{b^2 d}-\frac{f (e+f x) \sinh ^2(c+d x)}{2 b d^2}+\frac{f^2 \sinh (c+d x) \cosh (c+d x)}{4 b d^3}+\frac{(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 b d}-\frac{f^2 x}{4 b d^2}-\frac{(e+f x)^3}{6 b f} \]

[Out]

-(f^2*x)/(4*b*d^2) + (a^2*(e + f*x)^3)/(3*b^3*f) - (e + f*x)^3/(6*b*f) - (2*a*f^2*Cosh[c + d*x])/(b^2*d^3) - (
a*(e + f*x)^2*Cosh[c + d*x])/(b^2*d) - (a^3*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^3*S
qrt[a^2 + b^2]*d) + (a^3*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^3*Sqrt[a^2 + b^2]*d) -
(2*a^3*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*Sqrt[a^2 + b^2]*d^2) + (2*a^3*f
*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*Sqrt[a^2 + b^2]*d^2) + (2*a^3*f^2*PolyLo
g[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*Sqrt[a^2 + b^2]*d^3) - (2*a^3*f^2*PolyLog[3, -((b*E^(c +
d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*Sqrt[a^2 + b^2]*d^3) + (2*a*f*(e + f*x)*Sinh[c + d*x])/(b^2*d^2) + (f^2*Co
sh[c + d*x]*Sinh[c + d*x])/(4*b*d^3) + ((e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x])/(2*b*d) - (f*(e + f*x)*Sinh[c
+ d*x]^2)/(2*b*d^2)

________________________________________________________________________________________

Rubi [A]  time = 1.04363, antiderivative size = 522, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 13, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.464, Rules used = {5557, 3311, 32, 2635, 8, 3296, 2638, 3322, 2264, 2190, 2531, 2282, 6589} \[ -\frac{2 a^3 f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^3 d^2 \sqrt{a^2+b^2}}+\frac{2 a^3 f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^3 d^2 \sqrt{a^2+b^2}}+\frac{2 a^3 f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^3 d^3 \sqrt{a^2+b^2}}-\frac{2 a^3 f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^3 d^3 \sqrt{a^2+b^2}}-\frac{a^3 (e+f x)^2 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b^3 d \sqrt{a^2+b^2}}+\frac{a^3 (e+f x)^2 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b^3 d \sqrt{a^2+b^2}}+\frac{a^2 (e+f x)^3}{3 b^3 f}+\frac{2 a f (e+f x) \sinh (c+d x)}{b^2 d^2}-\frac{2 a f^2 \cosh (c+d x)}{b^2 d^3}-\frac{a (e+f x)^2 \cosh (c+d x)}{b^2 d}-\frac{f (e+f x) \sinh ^2(c+d x)}{2 b d^2}+\frac{f^2 \sinh (c+d x) \cosh (c+d x)}{4 b d^3}+\frac{(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 b d}-\frac{f^2 x}{4 b d^2}-\frac{(e+f x)^3}{6 b f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(f^2*x)/(4*b*d^2) + (a^2*(e + f*x)^3)/(3*b^3*f) - (e + f*x)^3/(6*b*f) - (2*a*f^2*Cosh[c + d*x])/(b^2*d^3) - (
a*(e + f*x)^2*Cosh[c + d*x])/(b^2*d) - (a^3*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^3*S
qrt[a^2 + b^2]*d) + (a^3*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^3*Sqrt[a^2 + b^2]*d) -
(2*a^3*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*Sqrt[a^2 + b^2]*d^2) + (2*a^3*f
*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*Sqrt[a^2 + b^2]*d^2) + (2*a^3*f^2*PolyLo
g[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*Sqrt[a^2 + b^2]*d^3) - (2*a^3*f^2*PolyLog[3, -((b*E^(c +
d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*Sqrt[a^2 + b^2]*d^3) + (2*a*f*(e + f*x)*Sinh[c + d*x])/(b^2*d^2) + (f^2*Co
sh[c + d*x]*Sinh[c + d*x])/(4*b*d^3) + ((e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x])/(2*b*d) - (f*(e + f*x)*Sinh[c
+ d*x]^2)/(2*b*d^2)

Rule 5557

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[1/b, Int[(e + f*x)^m*Sinh[c + d*x]^(n - 1), x], x] - Dist[a/b, Int[((e + f*x)^m*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]

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 32

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

Rule 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 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 3322

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Dist[2,
Int[((c + d*x)^m*E^(-(I*e) + f*fz*x))/(-(I*b) + 2*a*E^(-(I*e) + f*fz*x) + I*b*E^(2*(-(I*e) + f*fz*x))), x], x]
/; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2190

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

Rule 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 \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac{\int (e+f x)^2 \sinh ^2(c+d x) \, dx}{b}-\frac{a \int \frac{(e+f x)^2 \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=\frac{(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac{f (e+f x) \sinh ^2(c+d x)}{2 b d^2}-\frac{a \int (e+f x)^2 \sinh (c+d x) \, dx}{b^2}+\frac{a^2 \int \frac{(e+f x)^2 \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}-\frac{\int (e+f x)^2 \, dx}{2 b}+\frac{f^2 \int \sinh ^2(c+d x) \, dx}{2 b d^2}\\ &=-\frac{(e+f x)^3}{6 b f}-\frac{a (e+f x)^2 \cosh (c+d x)}{b^2 d}+\frac{f^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac{(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac{f (e+f x) \sinh ^2(c+d x)}{2 b d^2}+\frac{a^2 \int (e+f x)^2 \, dx}{b^3}-\frac{a^3 \int \frac{(e+f x)^2}{a+b \sinh (c+d x)} \, dx}{b^3}+\frac{(2 a f) \int (e+f x) \cosh (c+d x) \, dx}{b^2 d}-\frac{f^2 \int 1 \, dx}{4 b d^2}\\ &=-\frac{f^2 x}{4 b d^2}+\frac{a^2 (e+f x)^3}{3 b^3 f}-\frac{(e+f x)^3}{6 b f}-\frac{a (e+f x)^2 \cosh (c+d x)}{b^2 d}+\frac{2 a f (e+f x) \sinh (c+d x)}{b^2 d^2}+\frac{f^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac{(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac{f (e+f x) \sinh ^2(c+d x)}{2 b d^2}-\frac{\left (2 a^3\right ) \int \frac{e^{c+d x} (e+f x)^2}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{b^3}-\frac{\left (2 a f^2\right ) \int \sinh (c+d x) \, dx}{b^2 d^2}\\ &=-\frac{f^2 x}{4 b d^2}+\frac{a^2 (e+f x)^3}{3 b^3 f}-\frac{(e+f x)^3}{6 b f}-\frac{2 a f^2 \cosh (c+d x)}{b^2 d^3}-\frac{a (e+f x)^2 \cosh (c+d x)}{b^2 d}+\frac{2 a f (e+f x) \sinh (c+d x)}{b^2 d^2}+\frac{f^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac{(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac{f (e+f x) \sinh ^2(c+d x)}{2 b d^2}-\frac{\left (2 a^3\right ) \int \frac{e^{c+d x} (e+f x)^2}{2 a-2 \sqrt{a^2+b^2}+2 b e^{c+d x}} \, dx}{b^2 \sqrt{a^2+b^2}}+\frac{\left (2 a^3\right ) \int \frac{e^{c+d x} (e+f x)^2}{2 a+2 \sqrt{a^2+b^2}+2 b e^{c+d x}} \, dx}{b^2 \sqrt{a^2+b^2}}\\ &=-\frac{f^2 x}{4 b d^2}+\frac{a^2 (e+f x)^3}{3 b^3 f}-\frac{(e+f x)^3}{6 b f}-\frac{2 a f^2 \cosh (c+d x)}{b^2 d^3}-\frac{a (e+f x)^2 \cosh (c+d x)}{b^2 d}-\frac{a^3 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^3 \sqrt{a^2+b^2} d}+\frac{a^3 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^3 \sqrt{a^2+b^2} d}+\frac{2 a f (e+f x) \sinh (c+d x)}{b^2 d^2}+\frac{f^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac{(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac{f (e+f x) \sinh ^2(c+d x)}{2 b d^2}+\frac{\left (2 a^3 f\right ) \int (e+f x) \log \left (1+\frac{2 b e^{c+d x}}{2 a-2 \sqrt{a^2+b^2}}\right ) \, dx}{b^3 \sqrt{a^2+b^2} d}-\frac{\left (2 a^3 f\right ) \int (e+f x) \log \left (1+\frac{2 b e^{c+d x}}{2 a+2 \sqrt{a^2+b^2}}\right ) \, dx}{b^3 \sqrt{a^2+b^2} d}\\ &=-\frac{f^2 x}{4 b d^2}+\frac{a^2 (e+f x)^3}{3 b^3 f}-\frac{(e+f x)^3}{6 b f}-\frac{2 a f^2 \cosh (c+d x)}{b^2 d^3}-\frac{a (e+f x)^2 \cosh (c+d x)}{b^2 d}-\frac{a^3 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^3 \sqrt{a^2+b^2} d}+\frac{a^3 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^3 \sqrt{a^2+b^2} d}-\frac{2 a^3 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^3 \sqrt{a^2+b^2} d^2}+\frac{2 a^3 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^3 \sqrt{a^2+b^2} d^2}+\frac{2 a f (e+f x) \sinh (c+d x)}{b^2 d^2}+\frac{f^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac{(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac{f (e+f x) \sinh ^2(c+d x)}{2 b d^2}+\frac{\left (2 a^3 f^2\right ) \int \text{Li}_2\left (-\frac{2 b e^{c+d x}}{2 a-2 \sqrt{a^2+b^2}}\right ) \, dx}{b^3 \sqrt{a^2+b^2} d^2}-\frac{\left (2 a^3 f^2\right ) \int \text{Li}_2\left (-\frac{2 b e^{c+d x}}{2 a+2 \sqrt{a^2+b^2}}\right ) \, dx}{b^3 \sqrt{a^2+b^2} d^2}\\ &=-\frac{f^2 x}{4 b d^2}+\frac{a^2 (e+f x)^3}{3 b^3 f}-\frac{(e+f x)^3}{6 b f}-\frac{2 a f^2 \cosh (c+d x)}{b^2 d^3}-\frac{a (e+f x)^2 \cosh (c+d x)}{b^2 d}-\frac{a^3 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^3 \sqrt{a^2+b^2} d}+\frac{a^3 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^3 \sqrt{a^2+b^2} d}-\frac{2 a^3 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^3 \sqrt{a^2+b^2} d^2}+\frac{2 a^3 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^3 \sqrt{a^2+b^2} d^2}+\frac{2 a f (e+f x) \sinh (c+d x)}{b^2 d^2}+\frac{f^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac{(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac{f (e+f x) \sinh ^2(c+d x)}{2 b d^2}+\frac{\left (2 a^3 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^3 \sqrt{a^2+b^2} d^3}-\frac{\left (2 a^3 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^3 \sqrt{a^2+b^2} d^3}\\ &=-\frac{f^2 x}{4 b d^2}+\frac{a^2 (e+f x)^3}{3 b^3 f}-\frac{(e+f x)^3}{6 b f}-\frac{2 a f^2 \cosh (c+d x)}{b^2 d^3}-\frac{a (e+f x)^2 \cosh (c+d x)}{b^2 d}-\frac{a^3 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^3 \sqrt{a^2+b^2} d}+\frac{a^3 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^3 \sqrt{a^2+b^2} d}-\frac{2 a^3 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^3 \sqrt{a^2+b^2} d^2}+\frac{2 a^3 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^3 \sqrt{a^2+b^2} d^2}+\frac{2 a^3 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^3 \sqrt{a^2+b^2} d^3}-\frac{2 a^3 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^3 \sqrt{a^2+b^2} d^3}+\frac{2 a f (e+f x) \sinh (c+d x)}{b^2 d^2}+\frac{f^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac{(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac{f (e+f x) \sinh ^2(c+d x)}{2 b d^2}\\ \end{align*}

Mathematica [A]  time = 4.35971, size = 740, normalized size = 1.42 \[ \frac{-\frac{48 a^3 f (e+f x) \text{PolyLog}\left (2,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )}{d^2 \sqrt{a^2+b^2}}+\frac{48 a^3 f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{d^2 \sqrt{a^2+b^2}}+\frac{48 a^3 f^2 \text{PolyLog}\left (3,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )}{d^3 \sqrt{a^2+b^2}}-\frac{48 a^3 f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{d^3 \sqrt{a^2+b^2}}+\frac{48 a^3 e^2 \tanh ^{-1}\left (\frac{a+b e^{c+d x}}{\sqrt{a^2+b^2}}\right )}{d \sqrt{a^2+b^2}}-\frac{48 a^3 e f x \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{d \sqrt{a^2+b^2}}+\frac{48 a^3 e f x \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{d \sqrt{a^2+b^2}}-\frac{24 a^3 f^2 x^2 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{d \sqrt{a^2+b^2}}+\frac{24 a^3 f^2 x^2 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{d \sqrt{a^2+b^2}}+24 a^2 e^2 x+24 a^2 e f x^2+8 a^2 f^2 x^3+\frac{48 a b e f \sinh (c+d x)}{d^2}+\frac{48 a b f^2 x \sinh (c+d x)}{d^2}-\frac{48 a b f^2 \cosh (c+d x)}{d^3}-\frac{24 a b e^2 \cosh (c+d x)}{d}-\frac{48 a b e f x \cosh (c+d x)}{d}-\frac{24 a b f^2 x^2 \cosh (c+d x)}{d}-\frac{6 b^2 e f \cosh (2 (c+d x))}{d^2}+\frac{3 b^2 f^2 \sinh (2 (c+d x))}{d^3}-\frac{6 b^2 f^2 x \cosh (2 (c+d x))}{d^2}+\frac{6 b^2 e^2 \sinh (2 (c+d x))}{d}+\frac{12 b^2 e f x \sinh (2 (c+d x))}{d}+\frac{6 b^2 f^2 x^2 \sinh (2 (c+d x))}{d}-12 b^2 e^2 x-12 b^2 e f x^2-4 b^2 f^2 x^3}{24 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(24*a^2*e^2*x - 12*b^2*e^2*x + 24*a^2*e*f*x^2 - 12*b^2*e*f*x^2 + 8*a^2*f^2*x^3 - 4*b^2*f^2*x^3 + (48*a^3*e^2*A
rcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]])/(Sqrt[a^2 + b^2]*d) - (24*a*b*e^2*Cosh[c + d*x])/d - (48*a*b*f^2*
Cosh[c + d*x])/d^3 - (48*a*b*e*f*x*Cosh[c + d*x])/d - (24*a*b*f^2*x^2*Cosh[c + d*x])/d - (6*b^2*e*f*Cosh[2*(c
+ d*x)])/d^2 - (6*b^2*f^2*x*Cosh[2*(c + d*x)])/d^2 - (48*a^3*e*f*x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2
])])/(Sqrt[a^2 + b^2]*d) - (24*a^3*f^2*x^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(Sqrt[a^2 + b^2]*d)
+ (48*a^3*e*f*x*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(Sqrt[a^2 + b^2]*d) + (24*a^3*f^2*x^2*Log[1 +
(b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(Sqrt[a^2 + b^2]*d) - (48*a^3*f*(e + f*x)*PolyLog[2, (b*E^(c + d*x))/
(-a + Sqrt[a^2 + b^2])])/(Sqrt[a^2 + b^2]*d^2) + (48*a^3*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^
2 + b^2]))])/(Sqrt[a^2 + b^2]*d^2) + (48*a^3*f^2*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])])/(Sqrt[a^2
+ b^2]*d^3) - (48*a^3*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(Sqrt[a^2 + b^2]*d^3) + (48*a
*b*e*f*Sinh[c + d*x])/d^2 + (48*a*b*f^2*x*Sinh[c + d*x])/d^2 + (6*b^2*e^2*Sinh[2*(c + d*x)])/d + (3*b^2*f^2*Si
nh[2*(c + d*x)])/d^3 + (12*b^2*e*f*x*Sinh[2*(c + d*x)])/d + (6*b^2*f^2*x^2*Sinh[2*(c + d*x)])/d)/(24*b^3)

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Maple [F]  time = 0.086, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{2} \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{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*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x)

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [C]  time = 3.25167, size = 7156, normalized size = 13.71 \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*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-1/48*(6*(a^2*b^2 + b^4)*d^2*f^2*x^2 + 6*(a^2*b^2 + b^4)*d^2*e^2 - 3*(2*(a^2*b^2 + b^4)*d^2*f^2*x^2 + 2*(a^2*b
^2 + b^4)*d^2*e^2 - 2*(a^2*b^2 + b^4)*d*e*f + (a^2*b^2 + b^4)*f^2 + 2*(2*(a^2*b^2 + b^4)*d^2*e*f - (a^2*b^2 +
b^4)*d*f^2)*x)*cosh(d*x + c)^4 - 3*(2*(a^2*b^2 + b^4)*d^2*f^2*x^2 + 2*(a^2*b^2 + b^4)*d^2*e^2 - 2*(a^2*b^2 + b
^4)*d*e*f + (a^2*b^2 + b^4)*f^2 + 2*(2*(a^2*b^2 + b^4)*d^2*e*f - (a^2*b^2 + b^4)*d*f^2)*x)*sinh(d*x + c)^4 + 6
*(a^2*b^2 + b^4)*d*e*f + 24*((a^3*b + a*b^3)*d^2*f^2*x^2 + (a^3*b + a*b^3)*d^2*e^2 - 2*(a^3*b + a*b^3)*d*e*f +
2*(a^3*b + a*b^3)*f^2 + 2*((a^3*b + a*b^3)*d^2*e*f - (a^3*b + a*b^3)*d*f^2)*x)*cosh(d*x + c)^3 + 12*(2*(a^3*b
+ a*b^3)*d^2*f^2*x^2 + 2*(a^3*b + a*b^3)*d^2*e^2 - 4*(a^3*b + a*b^3)*d*e*f + 4*(a^3*b + a*b^3)*f^2 + 4*((a^3*
b + a*b^3)*d^2*e*f - (a^3*b + a*b^3)*d*f^2)*x - (2*(a^2*b^2 + b^4)*d^2*f^2*x^2 + 2*(a^2*b^2 + b^4)*d^2*e^2 - 2
*(a^2*b^2 + b^4)*d*e*f + (a^2*b^2 + b^4)*f^2 + 2*(2*(a^2*b^2 + b^4)*d^2*e*f - (a^2*b^2 + b^4)*d*f^2)*x)*cosh(d
*x + c))*sinh(d*x + c)^3 + 3*(a^2*b^2 + b^4)*f^2 - 8*((2*a^4 + a^2*b^2 - b^4)*d^3*f^2*x^3 + 3*(2*a^4 + a^2*b^2
- b^4)*d^3*e*f*x^2 + 3*(2*a^4 + a^2*b^2 - b^4)*d^3*e^2*x)*cosh(d*x + c)^2 - 2*(4*(2*a^4 + a^2*b^2 - b^4)*d^3*
f^2*x^3 + 12*(2*a^4 + a^2*b^2 - b^4)*d^3*e*f*x^2 + 12*(2*a^4 + a^2*b^2 - b^4)*d^3*e^2*x + 9*(2*(a^2*b^2 + b^4)
*d^2*f^2*x^2 + 2*(a^2*b^2 + b^4)*d^2*e^2 - 2*(a^2*b^2 + b^4)*d*e*f + (a^2*b^2 + b^4)*f^2 + 2*(2*(a^2*b^2 + b^4
)*d^2*e*f - (a^2*b^2 + b^4)*d*f^2)*x)*cosh(d*x + c)^2 - 36*((a^3*b + a*b^3)*d^2*f^2*x^2 + (a^3*b + a*b^3)*d^2*
e^2 - 2*(a^3*b + a*b^3)*d*e*f + 2*(a^3*b + a*b^3)*f^2 + 2*((a^3*b + a*b^3)*d^2*e*f - (a^3*b + a*b^3)*d*f^2)*x)
*cosh(d*x + c))*sinh(d*x + c)^2 + 96*((a^3*b*d*f^2*x + a^3*b*d*e*f)*cosh(d*x + c)^2 + 2*(a^3*b*d*f^2*x + a^3*b
*d*e*f)*cosh(d*x + c)*sinh(d*x + c) + (a^3*b*d*f^2*x + a^3*b*d*e*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*dil
og((a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1)
- 96*((a^3*b*d*f^2*x + a^3*b*d*e*f)*cosh(d*x + c)^2 + 2*(a^3*b*d*f^2*x + a^3*b*d*e*f)*cosh(d*x + c)*sinh(d*x +
c) + (a^3*b*d*f^2*x + a^3*b*d*e*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*dilog((a*cosh(d*x + c) + a*sinh(d*x
+ c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) - 48*((a^3*b*d^2*e^2 - 2*a^3*b*c
*d*e*f + a^3*b*c^2*f^2)*cosh(d*x + c)^2 + 2*(a^3*b*d^2*e^2 - 2*a^3*b*c*d*e*f + a^3*b*c^2*f^2)*cosh(d*x + c)*si
nh(d*x + c) + (a^3*b*d^2*e^2 - 2*a^3*b*c*d*e*f + a^3*b*c^2*f^2)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*log(2*b
*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + 48*((a^3*b*d^2*e^2 - 2*a^3*b*c*d*e*f +
a^3*b*c^2*f^2)*cosh(d*x + c)^2 + 2*(a^3*b*d^2*e^2 - 2*a^3*b*c*d*e*f + a^3*b*c^2*f^2)*cosh(d*x + c)*sinh(d*x +
c) + (a^3*b*d^2*e^2 - 2*a^3*b*c*d*e*f + a^3*b*c^2*f^2)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*log(2*b*cosh(d*
x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + 48*((a^3*b*d^2*f^2*x^2 + 2*a^3*b*d^2*e*f*x + 2
*a^3*b*c*d*e*f - a^3*b*c^2*f^2)*cosh(d*x + c)^2 + 2*(a^3*b*d^2*f^2*x^2 + 2*a^3*b*d^2*e*f*x + 2*a^3*b*c*d*e*f -
a^3*b*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c) + (a^3*b*d^2*f^2*x^2 + 2*a^3*b*d^2*e*f*x + 2*a^3*b*c*d*e*f - a^3*b
*c^2*f^2)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) +
b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) - 48*((a^3*b*d^2*f^2*x^2 + 2*a^3*b*d^2*e*f*x + 2*a^3*b*c*d*e*f
- a^3*b*c^2*f^2)*cosh(d*x + c)^2 + 2*(a^3*b*d^2*f^2*x^2 + 2*a^3*b*d^2*e*f*x + 2*a^3*b*c*d*e*f - a^3*b*c^2*f^2)
*cosh(d*x + c)*sinh(d*x + c) + (a^3*b*d^2*f^2*x^2 + 2*a^3*b*d^2*e*f*x + 2*a^3*b*c*d*e*f - a^3*b*c^2*f^2)*sinh(
d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c)
)*sqrt((a^2 + b^2)/b^2) - b)/b) - 96*(a^3*b*f^2*cosh(d*x + c)^2 + 2*a^3*b*f^2*cosh(d*x + c)*sinh(d*x + c) + a^
3*b*f^2*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*polylog(3, (a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c
) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) + 96*(a^3*b*f^2*cosh(d*x + c)^2 + 2*a^3*b*f^2*cosh(d*x + c)*sin
h(d*x + c) + a^3*b*f^2*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*polylog(3, (a*cosh(d*x + c) + a*sinh(d*x + c) -
(b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) + 6*(2*(a^2*b^2 + b^4)*d^2*e*f + (a^2*b^2 + b^4)
*d*f^2)*x + 24*((a^3*b + a*b^3)*d^2*f^2*x^2 + (a^3*b + a*b^3)*d^2*e^2 + 2*(a^3*b + a*b^3)*d*e*f + 2*(a^3*b + a
*b^3)*f^2 + 2*((a^3*b + a*b^3)*d^2*e*f + (a^3*b + a*b^3)*d*f^2)*x)*cosh(d*x + c) + 4*(6*(a^3*b + a*b^3)*d^2*f^
2*x^2 + 6*(a^3*b + a*b^3)*d^2*e^2 + 12*(a^3*b + a*b^3)*d*e*f - 3*(2*(a^2*b^2 + b^4)*d^2*f^2*x^2 + 2*(a^2*b^2 +
b^4)*d^2*e^2 - 2*(a^2*b^2 + b^4)*d*e*f + (a^2*b^2 + b^4)*f^2 + 2*(2*(a^2*b^2 + b^4)*d^2*e*f - (a^2*b^2 + b^4)
*d*f^2)*x)*cosh(d*x + c)^3 + 12*(a^3*b + a*b^3)*f^2 + 18*((a^3*b + a*b^3)*d^2*f^2*x^2 + (a^3*b + a*b^3)*d^2*e^
2 - 2*(a^3*b + a*b^3)*d*e*f + 2*(a^3*b + a*b^3)*f^2 + 2*((a^3*b + a*b^3)*d^2*e*f - (a^3*b + a*b^3)*d*f^2)*x)*c
osh(d*x + c)^2 + 12*((a^3*b + a*b^3)*d^2*e*f + (a^3*b + a*b^3)*d*f^2)*x - 4*((2*a^4 + a^2*b^2 - b^4)*d^3*f^2*x
^3 + 3*(2*a^4 + a^2*b^2 - b^4)*d^3*e*f*x^2 + 3*(2*a^4 + a^2*b^2 - b^4)*d^3*e^2*x)*cosh(d*x + c))*sinh(d*x + c)
)/((a^2*b^3 + b^5)*d^3*cosh(d*x + c)^2 + 2*(a^2*b^3 + b^5)*d^3*cosh(d*x + c)*sinh(d*x + c) + (a^2*b^3 + b^5)*d
^3*sinh(d*x + c)^2)

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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*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 )}^{2} \sinh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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