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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.97253, antiderivative size = 636, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 14, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5594, 5579, 3311, 3296, 2637, 2633, 5565, 5446, 3310, 5561, 2190, 2531, 2282, 6589} \[ -\frac{2 b f \left (a^2+b^2\right ) (e+f x) \text{PolyLog}\left (2,-\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^4 d^2}-\frac{2 b f \left (a^2+b^2\right ) (e+f x) \text{PolyLog}\left (2,-\frac{a e^{c+d x}}{\sqrt{a^2+b^2}+b}\right )}{a^4 d^2}+\frac{2 b f^2 \left (a^2+b^2\right ) \text{PolyLog}\left (3,-\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^4 d^3}+\frac{2 b f^2 \left (a^2+b^2\right ) \text{PolyLog}\left (3,-\frac{a e^{c+d x}}{\sqrt{a^2+b^2}+b}\right )}{a^4 d^3}-\frac{2 b^2 f (e+f x) \cosh (c+d x)}{a^3 d^2}+\frac{2 b^2 f^2 \sinh (c+d x)}{a^3 d^3}-\frac{b \left (a^2+b^2\right ) (e+f x)^2 \log \left (\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}+1\right )}{a^4 d}-\frac{b \left (a^2+b^2\right ) (e+f x)^2 \log \left (\frac{a e^{c+d x}}{\sqrt{a^2+b^2}+b}+1\right )}{a^4 d}+\frac{b^2 (e+f x)^2 \sinh (c+d x)}{a^3 d}+\frac{b \left (a^2+b^2\right ) (e+f x)^3}{3 a^4 f}+\frac{b f (e+f x) \sinh (c+d x) \cosh (c+d x)}{2 a^2 d^2}-\frac{b f^2 \sinh ^2(c+d x)}{4 a^2 d^3}-\frac{b (e+f x)^2 \sinh ^2(c+d x)}{2 a^2 d}-\frac{b e f x}{2 a^2 d}-\frac{b f^2 x^2}{4 a^2 d}-\frac{2 f (e+f x) \cosh ^3(c+d x)}{9 a d^2}-\frac{4 f (e+f x) \cosh (c+d x)}{3 a d^2}+\frac{2 f^2 \sinh ^3(c+d x)}{27 a d^3}+\frac{14 f^2 \sinh (c+d x)}{9 a d^3}+\frac{2 (e+f x)^2 \sinh (c+d x)}{3 a d}+\frac{(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5594

Int[(((e_.) + (f_.)*(x_))^(m_.)*(F_)[(c_.) + (d_.)*(x_)]^(n_.))/(Csch[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Sym
bol] :> Int[((e + f*x)^m*Sinh[c + d*x]*F[c + d*x]^n)/(b + a*Sinh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x]
&& HyperbolicQ[F] && IntegersQ[m, n]

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

Mathematica [C]  time = 17.0746, size = 3874, normalized size = 6.09 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(f^2*Csch[c + d*x]*(2*b*x^3*(-1 + Coth[c]) - 2*b*x^3*Coth[c] - (6*a^2*b*(d^2*x^2*Log[1 + ((b - Sqrt[a^2 + b^2]
)*(Cosh[c + d*x] - Sinh[c + d*x]))/a] - 2*d*x*PolyLog[2, ((-b + Sqrt[a^2 + b^2])*(Cosh[c + d*x] - Sinh[c + d*x
]))/a] - 2*PolyLog[3, ((-b + Sqrt[a^2 + b^2])*(Cosh[c + d*x] - Sinh[c + d*x]))/a]))/(Sqrt[a^2 + b^2]*(-b + Sqr
t[a^2 + b^2])*d^3) - (6*a^2*b*(d^2*x^2*Log[1 + ((b + Sqrt[a^2 + b^2])*(Cosh[c + d*x] - Sinh[c + d*x]))/a] - 2*
d*x*PolyLog[2, ((b + Sqrt[a^2 + b^2])*(-Cosh[c + d*x] + Sinh[c + d*x]))/a] - 2*PolyLog[3, ((b + Sqrt[a^2 + b^2
])*(-Cosh[c + d*x] + Sinh[c + d*x]))/a]))/(Sqrt[a^2 + b^2]*(b + Sqrt[a^2 + b^2])*d^3) + (6*b^2*(d^2*x^2*Log[1
+ (a*(Cosh[c + d*x] + Sinh[c + d*x]))/(b - Sqrt[a^2 + b^2])] + 2*d*x*PolyLog[2, (a*(Cosh[c + d*x] + Sinh[c + d
*x]))/(-b + Sqrt[a^2 + b^2])] - 2*PolyLog[3, (a*(Cosh[c + d*x] + Sinh[c + d*x]))/(-b + Sqrt[a^2 + b^2])]))/(Sq
rt[a^2 + b^2]*d^3) - (6*b^2*(d^2*x^2*Log[1 + (a*(Cosh[c + d*x] + Sinh[c + d*x]))/(b + Sqrt[a^2 + b^2])] + 2*d*
x*PolyLog[2, -((a*(Cosh[c + d*x] + Sinh[c + d*x]))/(b + Sqrt[a^2 + b^2]))] - 2*PolyLog[3, -((a*(Cosh[c + d*x]
+ Sinh[c + d*x]))/(b + Sqrt[a^2 + b^2]))]))/(Sqrt[a^2 + b^2]*d^3) + (6*a*Cosh[d*x]*(-2*d*x*Cosh[c] + (2 + d^2*
x^2)*Sinh[c]))/d^3 + (6*a*((2 + d^2*x^2)*Cosh[c] - 2*d*x*Sinh[c])*Sinh[d*x])/d^3)*(b + a*Sinh[c + d*x]))/(12*a
^2*(a + b*Csch[c + d*x])) - (e^2*Csch[c + d*x]*((b*Log[b + a*Sinh[c + d*x]])/a^2 - Sinh[c + d*x]/a)*(b + a*Sin
h[c + d*x]))/(2*d*(a + b*Csch[c + d*x])) + (e*f*Csch[c + d*x]*(b + a*Sinh[c + d*x])*(-(a*Cosh[c + d*x]) - b*(c
+ d*x)*Log[b + a*Sinh[c + d*x]] + b*c*Log[1 + (a*Sinh[c + d*x])/b] + I*b*((-I/8)*(2*c + I*Pi + 2*d*x)^2 - (4*
I)*ArcSin[Sqrt[1 + (I*b)/a]/Sqrt[2]]*ArcTan[((I*a + b)*Cot[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[a^2 + b^2]] - (
((-2*I)*c + Pi - (2*I)*d*x + 4*ArcSin[Sqrt[1 + (I*b)/a]/Sqrt[2]])*Log[1 + ((-b + Sqrt[a^2 + b^2])*E^(c + d*x))
/a])/2 - (((-2*I)*c + Pi - (2*I)*d*x - 4*ArcSin[Sqrt[1 + (I*b)/a]/Sqrt[2]])*Log[1 - ((b + Sqrt[a^2 + b^2])*E^(
c + d*x))/a])/2 + (Pi/2 - I*(c + d*x))*Log[b + a*Sinh[c + d*x]] + I*(PolyLog[2, ((b - Sqrt[a^2 + b^2])*E^(c +
d*x))/a] + PolyLog[2, ((b + Sqrt[a^2 + b^2])*E^(c + d*x))/a])) + a*d*x*Sinh[c + d*x]))/(a^2*d^2*(a + b*Csch[c
+ d*x])) + (e^2*Csch[c + d*x]*(b + a*Sinh[c + d*x])*(-3*b*(a^2 + 2*b^2)*Log[b + a*Sinh[c + d*x]] + 3*a*(a^2 +
2*b^2)*Sinh[c + d*x] - 3*a^2*b*Sinh[c + d*x]^2 + 2*a^3*Sinh[c + d*x]^3))/(6*a^4*d*(a + b*Csch[c + d*x])) + (e*
f*Csch[c + d*x]*(b + a*Sinh[c + d*x])*(-18*a*(a^2 + 4*b^2)*Cosh[c + d*x] - 18*a^2*b*d*x*Cosh[2*(c + d*x)] - 2*
a^3*Cosh[3*(c + d*x)] - 36*b*(a^2 + 2*b^2)*(-(c + d*x)^2/2 + (c + d*x)*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 +
b^2])] + (c + d*x)*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])] - c*Log[b + a*Sinh[c + d*x]] + PolyLog[2, (
a*E^(c + d*x))/(-b + Sqrt[a^2 + b^2])] + PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))]) + 18*a*(a^2 + 4
*b^2)*d*x*Sinh[c + d*x] + 9*a^2*b*Sinh[2*(c + d*x)] + 6*a^3*d*x*Sinh[3*(c + d*x)]))/(36*a^4*d^2*(a + b*Csch[c
+ d*x])) + (f^2*Csch[c + d*x]*(b + a*Sinh[c + d*x])*((2*b*(a^2 + 2*b^2)*(-1 + Coth[c])*(2*x^3 + (6*b*(d^2*x^2*
Log[1 + (a*(Cosh[c + d*x] + Sinh[c + d*x]))/(b - Sqrt[a^2 + b^2])] + 2*d*x*PolyLog[2, (a*(Cosh[c + d*x] + Sinh
[c + d*x]))/(-b + Sqrt[a^2 + b^2])] - 2*PolyLog[3, (a*(Cosh[c + d*x] + Sinh[c + d*x]))/(-b + Sqrt[a^2 + b^2])]
)*Sinh[c]*(Cosh[c] + Sinh[c]))/(Sqrt[a^2 + b^2]*d^3) - (3*a^2*(d^2*x^2*Log[1 + ((b - Sqrt[a^2 + b^2])*(Cosh[c
+ d*x] - Sinh[c + d*x]))/a] - 2*d*x*PolyLog[2, ((-b + Sqrt[a^2 + b^2])*(Cosh[c + d*x] - Sinh[c + d*x]))/a] - 2
*PolyLog[3, ((-b + Sqrt[a^2 + b^2])*(Cosh[c + d*x] - Sinh[c + d*x]))/a])*(-1 + Cosh[2*c] + Sinh[2*c]))/(Sqrt[a
^2 + b^2]*(-b + Sqrt[a^2 + b^2])*d^3) - (3*a^2*(d^2*x^2*Log[1 + ((b + Sqrt[a^2 + b^2])*(Cosh[c + d*x] - Sinh[c
+ d*x]))/a] - 2*d*x*PolyLog[2, ((b + Sqrt[a^2 + b^2])*(-Cosh[c + d*x] + Sinh[c + d*x]))/a] - 2*PolyLog[3, ((b
+ Sqrt[a^2 + b^2])*(-Cosh[c + d*x] + Sinh[c + d*x]))/a])*(-1 + Cosh[2*c] + Sinh[2*c]))/(Sqrt[a^2 + b^2]*(b +
Sqrt[a^2 + b^2])*d^3) - (3*b*(d^2*x^2*Log[1 + (a*(Cosh[c + d*x] + Sinh[c + d*x]))/(b + Sqrt[a^2 + b^2])] + 2*d
*x*PolyLog[2, -((a*(Cosh[c + d*x] + Sinh[c + d*x]))/(b + Sqrt[a^2 + b^2]))] - 2*PolyLog[3, -((a*(Cosh[c + d*x]
+ Sinh[c + d*x]))/(b + Sqrt[a^2 + b^2]))])*(-1 + Cosh[2*c] + Sinh[2*c]))/(Sqrt[a^2 + b^2]*d^3)))/(3*a^4) + Cs
ch[c]*(Cosh[3*c + 3*d*x]/(108*a^4*d^3) - Sinh[3*c + 3*d*x]/(108*a^4*d^3))*(27*a^2*b*Cosh[d*x] + 54*a^2*b*d*x*C
osh[d*x] + 54*a^2*b*d^2*x^2*Cosh[d*x] - 27*a^2*b*Cosh[2*c + d*x] - 54*a^2*b*d*x*Cosh[2*c + d*x] - 54*a^2*b*d^2
*x^2*Cosh[2*c + d*x] + 108*a^3*Cosh[c + 2*d*x] + 432*a*b^2*Cosh[c + 2*d*x] + 108*a^3*d*x*Cosh[c + 2*d*x] + 432
*a*b^2*d*x*Cosh[c + 2*d*x] + 54*a^3*d^2*x^2*Cosh[c + 2*d*x] + 216*a*b^2*d^2*x^2*Cosh[c + 2*d*x] - 108*a^3*Cosh
[3*c + 2*d*x] - 432*a*b^2*Cosh[3*c + 2*d*x] - 108*a^3*d*x*Cosh[3*c + 2*d*x] - 432*a*b^2*d*x*Cosh[3*c + 2*d*x]
- 54*a^3*d^2*x^2*Cosh[3*c + 2*d*x] - 216*a*b^2*d^2*x^2*Cosh[3*c + 2*d*x] - 72*a^2*b*d^3*x^3*Cosh[2*c + 3*d*x]
- 144*b^3*d^3*x^3*Cosh[2*c + 3*d*x] - 72*a^2*b*d^3*x^3*Cosh[4*c + 3*d*x] - 144*b^3*d^3*x^3*Cosh[4*c + 3*d*x] -
108*a^3*Cosh[3*c + 4*d*x] - 432*a*b^2*Cosh[3*c + 4*d*x] + 108*a^3*d*x*Cosh[3*c + 4*d*x] + 432*a*b^2*d*x*Cosh[
3*c + 4*d*x] - 54*a^3*d^2*x^2*Cosh[3*c + 4*d*x] - 216*a*b^2*d^2*x^2*Cosh[3*c + 4*d*x] + 108*a^3*Cosh[5*c + 4*d
*x] + 432*a*b^2*Cosh[5*c + 4*d*x] - 108*a^3*d*x*Cosh[5*c + 4*d*x] - 432*a*b^2*d*x*Cosh[5*c + 4*d*x] + 54*a^3*d
^2*x^2*Cosh[5*c + 4*d*x] + 216*a*b^2*d^2*x^2*Cosh[5*c + 4*d*x] + 27*a^2*b*Cosh[4*c + 5*d*x] - 54*a^2*b*d*x*Cos
h[4*c + 5*d*x] + 54*a^2*b*d^2*x^2*Cosh[4*c + 5*d*x] - 27*a^2*b*Cosh[6*c + 5*d*x] + 54*a^2*b*d*x*Cosh[6*c + 5*d
*x] - 54*a^2*b*d^2*x^2*Cosh[6*c + 5*d*x] - 4*a^3*Cosh[5*c + 6*d*x] + 12*a^3*d*x*Cosh[5*c + 6*d*x] - 18*a^3*d^2
*x^2*Cosh[5*c + 6*d*x] + 4*a^3*Cosh[7*c + 6*d*x] - 12*a^3*d*x*Cosh[7*c + 6*d*x] + 18*a^3*d^2*x^2*Cosh[7*c + 6*
d*x] - 8*a^3*Sinh[c] - 24*a^3*d*x*Sinh[c] - 36*a^3*d^2*x^2*Sinh[c] + 27*a^2*b*Sinh[d*x] + 54*a^2*b*d*x*Sinh[d*
x] + 54*a^2*b*d^2*x^2*Sinh[d*x] - 27*a^2*b*Sinh[2*c + d*x] - 54*a^2*b*d*x*Sinh[2*c + d*x] - 54*a^2*b*d^2*x^2*S
inh[2*c + d*x] + 108*a^3*Sinh[c + 2*d*x] + 432*a*b^2*Sinh[c + 2*d*x] + 108*a^3*d*x*Sinh[c + 2*d*x] + 432*a*b^2
*d*x*Sinh[c + 2*d*x] + 54*a^3*d^2*x^2*Sinh[c + 2*d*x] + 216*a*b^2*d^2*x^2*Sinh[c + 2*d*x] - 108*a^3*Sinh[3*c +
2*d*x] - 432*a*b^2*Sinh[3*c + 2*d*x] - 108*a^3*d*x*Sinh[3*c + 2*d*x] - 432*a*b^2*d*x*Sinh[3*c + 2*d*x] - 54*a
^3*d^2*x^2*Sinh[3*c + 2*d*x] - 216*a*b^2*d^2*x^2*Sinh[3*c + 2*d*x] - 72*a^2*b*d^3*x^3*Sinh[2*c + 3*d*x] - 144*
b^3*d^3*x^3*Sinh[2*c + 3*d*x] - 72*a^2*b*d^3*x^3*Sinh[4*c + 3*d*x] - 144*b^3*d^3*x^3*Sinh[4*c + 3*d*x] - 108*a
^3*Sinh[3*c + 4*d*x] - 432*a*b^2*Sinh[3*c + 4*d*x] + 108*a^3*d*x*Sinh[3*c + 4*d*x] + 432*a*b^2*d*x*Sinh[3*c +
4*d*x] - 54*a^3*d^2*x^2*Sinh[3*c + 4*d*x] - 216*a*b^2*d^2*x^2*Sinh[3*c + 4*d*x] + 108*a^3*Sinh[5*c + 4*d*x] +
432*a*b^2*Sinh[5*c + 4*d*x] - 108*a^3*d*x*Sinh[5*c + 4*d*x] - 432*a*b^2*d*x*Sinh[5*c + 4*d*x] + 54*a^3*d^2*x^2
*Sinh[5*c + 4*d*x] + 216*a*b^2*d^2*x^2*Sinh[5*c + 4*d*x] + 27*a^2*b*Sinh[4*c + 5*d*x] - 54*a^2*b*d*x*Sinh[4*c
+ 5*d*x] + 54*a^2*b*d^2*x^2*Sinh[4*c + 5*d*x] - 27*a^2*b*Sinh[6*c + 5*d*x] + 54*a^2*b*d*x*Sinh[6*c + 5*d*x] -
54*a^2*b*d^2*x^2*Sinh[6*c + 5*d*x] - 4*a^3*Sinh[5*c + 6*d*x] + 12*a^3*d*x*Sinh[5*c + 6*d*x] - 18*a^3*d^2*x^2*S
inh[5*c + 6*d*x] + 4*a^3*Sinh[7*c + 6*d*x] - 12*a^3*d*x*Sinh[7*c + 6*d*x] + 18*a^3*d^2*x^2*Sinh[7*c + 6*d*x]))
)/(8*(a + b*Csch[c + d*x]))

________________________________________________________________________________________

Maple [F]  time = 0.355, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{2} \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}{a+b{\rm csch} \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/(a+b*csch(d*x+c)),x)

[Out]

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

________________________________________________________________________________________

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/(a+b*csch(d*x+c)),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [C]  time = 2.42445, 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/(a+b*csch(d*x+c)),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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/(a+b*csch(d*x+c)),x)

[Out]

Timed out

________________________________________________________________________________________

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}}{b \operatorname{csch}\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/(a+b*csch(d*x+c)),x, algorithm="giac")

[Out]

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