\(\int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx\) [23]

Optimal result

Integrand size = 28, antiderivative size = 510 \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\frac {f^2 x}{4 a d^2}+\frac {(e+f x)^3}{6 a f}+\frac {b^2 (e+f x)^3}{3 a^3 f}-\frac {2 b f^2 \cosh (c+d x)}{a^2 d^3}-\frac {b (e+f x)^2 \cosh (c+d x)}{a^2 d}-\frac {f (e+f x) \cosh ^2(c+d x)}{2 a d^2}-\frac {b \sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {b \sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {2 b \sqrt {a^2+b^2} f (e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {2 b \sqrt {a^2+b^2} f (e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {2 b \sqrt {a^2+b^2} f^2 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d^3}-\frac {2 b \sqrt {a^2+b^2} f^2 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d^3}+\frac {2 b f (e+f x) \sinh (c+d x)}{a^2 d^2}+\frac {f^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^3}+\frac {(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 a d} \]

[Out]

1/4*f^2*x/a/d^2+1/6*(f*x+e)^3/a/f+1/3*b^2*(f*x+e)^3/a^3/f-2*b*f^2*cosh(d*x+c)/a^2/d^3-b*(f*x+e)^2*cosh(d*x+c)/
a^2/d-1/2*f*(f*x+e)*cosh(d*x+c)^2/a/d^2+2*b*f*(f*x+e)*sinh(d*x+c)/a^2/d^2+1/4*f^2*cosh(d*x+c)*sinh(d*x+c)/a/d^
3+1/2*(f*x+e)^2*cosh(d*x+c)*sinh(d*x+c)/a/d-b*(f*x+e)^2*ln(1+a*exp(d*x+c)/(b-(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)
/a^3/d+b*(f*x+e)^2*ln(1+a*exp(d*x+c)/(b+(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/a^3/d-2*b*f*(f*x+e)*polylog(2,-a*exp
(d*x+c)/(b-(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/a^3/d^2+2*b*f*(f*x+e)*polylog(2,-a*exp(d*x+c)/(b+(a^2+b^2)^(1/2))
)*(a^2+b^2)^(1/2)/a^3/d^2+2*b*f^2*polylog(3,-a*exp(d*x+c)/(b-(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/a^3/d^3-2*b*f^2
*polylog(3,-a*exp(d*x+c)/(b+(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/a^3/d^3

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 510, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.536, Rules used = {5713, 5698, 3392, 32, 2715, 8, 5684, 3377, 2718, 3403, 2296, 2221, 2611, 2320, 6724} \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\frac {b^2 (e+f x)^3}{3 a^3 f}-\frac {2 b f^2 \cosh (c+d x)}{a^2 d^3}+\frac {2 b f (e+f x) \sinh (c+d x)}{a^2 d^2}-\frac {b (e+f x)^2 \cosh (c+d x)}{a^2 d}+\frac {2 b f^2 \sqrt {a^2+b^2} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d^3}-\frac {2 b f^2 \sqrt {a^2+b^2} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d^3}-\frac {2 b f \sqrt {a^2+b^2} (e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {2 b f \sqrt {a^2+b^2} (e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d^2}-\frac {b \sqrt {a^2+b^2} (e+f x)^2 \log \left (\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}+1\right )}{a^3 d}+\frac {b \sqrt {a^2+b^2} (e+f x)^2 \log \left (\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}+1\right )}{a^3 d}+\frac {f^2 \sinh (c+d x) \cosh (c+d x)}{4 a d^3}-\frac {f (e+f x) \cosh ^2(c+d x)}{2 a d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 a d}+\frac {f^2 x}{4 a d^2}+\frac {(e+f x)^3}{6 a f} \]

[In]

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

[Out]

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

Rule 8

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

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 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2296

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 2320

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 2611

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 2715

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] && Integ
erQ[2*n]

Rule 2718

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

Rule 3377

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 3392

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 3403

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 5684

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 5698

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 5713

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 6724

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1216\) vs. \(2(510)=1020\).

Time = 4.93 (sec) , antiderivative size = 1216, normalized size of antiderivative = 2.38 \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\frac {\text {csch}(c+d x) (b+a \sinh (c+d x)) \left (6 a^2 e^2 \left (\frac {c}{d}+x-\frac {2 b \arctan \left (\frac {a-b \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2} d}\right )+6 a^2 e f \left (x^2-\frac {2 b \left (d x \left (\log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )-\log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )\right )+\operatorname {PolyLog}\left (2,\frac {a e^{c+d x}}{-b+\sqrt {a^2+b^2}}\right )-\operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )\right )}{\sqrt {a^2+b^2} d^2}\right )+2 a^2 f^2 \left (x^3-\frac {3 b \left (d^2 x^2 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )-d^2 x^2 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )+2 d x \operatorname {PolyLog}\left (2,\frac {a e^{c+d x}}{-b+\sqrt {a^2+b^2}}\right )-2 d x \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )-2 \operatorname {PolyLog}\left (3,\frac {a e^{c+d x}}{-b+\sqrt {a^2+b^2}}\right )+2 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )\right )}{\sqrt {a^2+b^2} d^3}\right )+f^2 \left (2 \left (a^2+4 b^2\right ) x^3-\frac {6 b \left (3 a^2+4 b^2\right ) \left (d^2 x^2 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )-d^2 x^2 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )+2 d x \operatorname {PolyLog}\left (2,\frac {a e^{c+d x}}{-b+\sqrt {a^2+b^2}}\right )-2 d x \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )-2 \operatorname {PolyLog}\left (3,\frac {a e^{c+d x}}{-b+\sqrt {a^2+b^2}}\right )+2 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )\right )}{\sqrt {a^2+b^2} d^3}-\frac {24 a b \cosh (d x) \left (\left (2+d^2 x^2\right ) \cosh (c)-2 d x \sinh (c)\right )}{d^3}+\frac {3 a^2 \cosh (2 d x) \left (-2 d x \cosh (2 c)+\left (1+2 d^2 x^2\right ) \sinh (2 c)\right )}{d^3}-\frac {24 a b \left (-2 d x \cosh (c)+\left (2+d^2 x^2\right ) \sinh (c)\right ) \sinh (d x)}{d^3}+\frac {3 a^2 \left (\left (1+2 d^2 x^2\right ) \cosh (2 c)-2 d x \sinh (2 c)\right ) \sinh (2 d x)}{d^3}\right )+\frac {6 e^2 \left (\left (a^2+4 b^2\right ) (c+d x)-\frac {2 b \left (3 a^2+4 b^2\right ) \arctan \left (\frac {a-b \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}-4 a b \cosh (c+d x)+a^2 \sinh (2 (c+d x))\right )}{d}+\frac {6 e f \left (\left (a^2+4 b^2\right ) (-c+d x) (c+d x)-8 a b d x \cosh (c+d x)-a^2 \cosh (2 (c+d x))-\frac {2 b \left (3 a^2+4 b^2\right ) \left (2 c \text {arctanh}\left (\frac {b+a e^{c+d x}}{\sqrt {a^2+b^2}}\right )+(c+d x) \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )-(c+d x) \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )+\operatorname {PolyLog}\left (2,\frac {a e^{c+d x}}{-b+\sqrt {a^2+b^2}}\right )-\operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )\right )}{\sqrt {a^2+b^2}}+8 a b \sinh (c+d x)+2 a^2 d x \sinh (2 (c+d x))\right )}{d^2}\right )}{24 a^3 (a+b \text {csch}(c+d x))} \]

[In]

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

[Out]

(Csch[c + d*x]*(b + a*Sinh[c + d*x])*(6*a^2*e^2*(c/d + x - (2*b*ArcTan[(a - b*Tanh[(c + d*x)/2])/Sqrt[-a^2 - b
^2]])/(Sqrt[-a^2 - b^2]*d)) + 6*a^2*e*f*(x^2 - (2*b*(d*x*(Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])] - Log
[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])]) + 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]))]))/(Sqrt[a^2 + b^2]*d^2)) + 2*a^2*f^2*(x^3 - (3*b*(d^2*x^2*Log[1 + (
a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])] - d^2*x^2*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])] + 2*d*x*PolyLog
[2, (a*E^(c + d*x))/(-b + Sqrt[a^2 + b^2])] - 2*d*x*PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))] - 2*P
olyLog[3, (a*E^(c + d*x))/(-b + Sqrt[a^2 + b^2])] + 2*PolyLog[3, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))]))/(
Sqrt[a^2 + b^2]*d^3)) + f^2*(2*(a^2 + 4*b^2)*x^3 - (6*b*(3*a^2 + 4*b^2)*(d^2*x^2*Log[1 + (a*E^(c + d*x))/(b -
Sqrt[a^2 + b^2])] - d^2*x^2*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])] + 2*d*x*PolyLog[2, (a*E^(c + d*x))/
(-b + Sqrt[a^2 + b^2])] - 2*d*x*PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))] - 2*PolyLog[3, (a*E^(c +
d*x))/(-b + Sqrt[a^2 + b^2])] + 2*PolyLog[3, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))]))/(Sqrt[a^2 + b^2]*d^3)
- (24*a*b*Cosh[d*x]*((2 + d^2*x^2)*Cosh[c] - 2*d*x*Sinh[c]))/d^3 + (3*a^2*Cosh[2*d*x]*(-2*d*x*Cosh[2*c] + (1
+ 2*d^2*x^2)*Sinh[2*c]))/d^3 - (24*a*b*(-2*d*x*Cosh[c] + (2 + d^2*x^2)*Sinh[c])*Sinh[d*x])/d^3 + (3*a^2*((1 +
2*d^2*x^2)*Cosh[2*c] - 2*d*x*Sinh[2*c])*Sinh[2*d*x])/d^3) + (6*e^2*((a^2 + 4*b^2)*(c + d*x) - (2*b*(3*a^2 + 4*
b^2)*ArcTan[(a - b*Tanh[(c + d*x)/2])/Sqrt[-a^2 - b^2]])/Sqrt[-a^2 - b^2] - 4*a*b*Cosh[c + d*x] + a^2*Sinh[2*(
c + d*x)]))/d + (6*e*f*((a^2 + 4*b^2)*(-c + d*x)*(c + d*x) - 8*a*b*d*x*Cosh[c + d*x] - a^2*Cosh[2*(c + d*x)] -
(2*b*(3*a^2 + 4*b^2)*(2*c*ArcTanh[(b + a*E^(c + d*x))/Sqrt[a^2 + b^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])] + 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]))]))/Sqrt[a^2 + b^2] + 8*a*b*Sinh[c
+ d*x] + 2*a^2*d*x*Sinh[2*(c + d*x)]))/d^2))/(24*a^3*(a + b*Csch[c + d*x]))

Maple [F]

\[\int \frac {\left (f x +e \right )^{2} \cosh \left (d x +c \right )^{2}}{a +b \,\operatorname {csch}\left (d x +c \right )}d x\]

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2410 vs. \(2 (466) = 932\).

Time = 0.31 (sec) , antiderivative size = 2410, normalized size of antiderivative = 4.73 \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/48*(6*a^2*d^2*f^2*x^2 + 6*a^2*d^2*e^2 + 6*a^2*d*e*f - 3*(2*a^2*d^2*f^2*x^2 + 2*a^2*d^2*e^2 - 2*a^2*d*e*f +
a^2*f^2 + 2*(2*a^2*d^2*e*f - a^2*d*f^2)*x)*cosh(d*x + c)^4 - 3*(2*a^2*d^2*f^2*x^2 + 2*a^2*d^2*e^2 - 2*a^2*d*e*
f + a^2*f^2 + 2*(2*a^2*d^2*e*f - a^2*d*f^2)*x)*sinh(d*x + c)^4 + 3*a^2*f^2 + 24*(a*b*d^2*f^2*x^2 + a*b*d^2*e^2
- 2*a*b*d*e*f + 2*a*b*f^2 + 2*(a*b*d^2*e*f - a*b*d*f^2)*x)*cosh(d*x + c)^3 + 12*(2*a*b*d^2*f^2*x^2 + 2*a*b*d^
2*e^2 - 4*a*b*d*e*f + 4*a*b*f^2 + 4*(a*b*d^2*e*f - a*b*d*f^2)*x - (2*a^2*d^2*f^2*x^2 + 2*a^2*d^2*e^2 - 2*a^2*d
*e*f + a^2*f^2 + 2*(2*a^2*d^2*e*f - a^2*d*f^2)*x)*cosh(d*x + c))*sinh(d*x + c)^3 - 8*((a^2 + 2*b^2)*d^3*f^2*x^
3 + 3*(a^2 + 2*b^2)*d^3*e*f*x^2 + 3*(a^2 + 2*b^2)*d^3*e^2*x)*cosh(d*x + c)^2 - 2*(4*(a^2 + 2*b^2)*d^3*f^2*x^3
+ 12*(a^2 + 2*b^2)*d^3*e*f*x^2 + 12*(a^2 + 2*b^2)*d^3*e^2*x + 9*(2*a^2*d^2*f^2*x^2 + 2*a^2*d^2*e^2 - 2*a^2*d*e
*f + a^2*f^2 + 2*(2*a^2*d^2*e*f - a^2*d*f^2)*x)*cosh(d*x + c)^2 - 36*(a*b*d^2*f^2*x^2 + a*b*d^2*e^2 - 2*a*b*d*
e*f + 2*a*b*f^2 + 2*(a*b*d^2*e*f - a*b*d*f^2)*x)*cosh(d*x + c))*sinh(d*x + c)^2 + 96*((a*b*d*f^2*x + a*b*d*e*f
)*cosh(d*x + c)^2 + 2*(a*b*d*f^2*x + a*b*d*e*f)*cosh(d*x + c)*sinh(d*x + c) + (a*b*d*f^2*x + a*b*d*e*f)*sinh(d
*x + c)^2)*sqrt((a^2 + b^2)/a^2)*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) - 96*((a*b*d*f^2*x + a*b*d*e*f)*cosh(d*x + c)^2 + 2*(a*b*d*f^2*x + a*b*d*e
*f)*cosh(d*x + c)*sinh(d*x + c) + (a*b*d*f^2*x + a*b*d*e*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/a^2)*dilog((b*co
sh(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) - 48*((a
*b*d^2*e^2 - 2*a*b*c*d*e*f + a*b*c^2*f^2)*cosh(d*x + c)^2 + 2*(a*b*d^2*e^2 - 2*a*b*c*d*e*f + a*b*c^2*f^2)*cosh
(d*x + c)*sinh(d*x + c) + (a*b*d^2*e^2 - 2*a*b*c*d*e*f + a*b*c^2*f^2)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/a^2)*l
og(2*a*cosh(d*x + c) + 2*a*sinh(d*x + c) + 2*a*sqrt((a^2 + b^2)/a^2) + 2*b) + 48*((a*b*d^2*e^2 - 2*a*b*c*d*e*f
+ a*b*c^2*f^2)*cosh(d*x + c)^2 + 2*(a*b*d^2*e^2 - 2*a*b*c*d*e*f + a*b*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c) +
(a*b*d^2*e^2 - 2*a*b*c*d*e*f + a*b*c^2*f^2)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/a^2)*log(2*a*cosh(d*x + c) + 2*a
*sinh(d*x + c) - 2*a*sqrt((a^2 + b^2)/a^2) + 2*b) + 48*((a*b*d^2*f^2*x^2 + 2*a*b*d^2*e*f*x + 2*a*b*c*d*e*f - a
*b*c^2*f^2)*cosh(d*x + c)^2 + 2*(a*b*d^2*f^2*x^2 + 2*a*b*d^2*e*f*x + 2*a*b*c*d*e*f - a*b*c^2*f^2)*cosh(d*x + c
)*sinh(d*x + c) + (a*b*d^2*f^2*x^2 + 2*a*b*d^2*e*f*x + 2*a*b*c*d*e*f - a*b*c^2*f^2)*sinh(d*x + c)^2)*sqrt((a^2
+ b^2)/a^2)*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) - 48*((a*b*d^2*f^2*x^2 + 2*a*b*d^2*e*f*x + 2*a*b*c*d*e*f - a*b*c^2*f^2)*cosh(d*x + c)^2 + 2*(a*b*d^
2*f^2*x^2 + 2*a*b*d^2*e*f*x + 2*a*b*c*d*e*f - a*b*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c) + (a*b*d^2*f^2*x^2 + 2*
a*b*d^2*e*f*x + 2*a*b*c*d*e*f - a*b*c^2*f^2)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/a^2)*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) - 96*(a*b*f^2*cosh(d*x + c)^
2 + 2*a*b*f^2*cosh(d*x + c)*sinh(d*x + c) + a*b*f^2*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/a^2)*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) + 96*(a*b*f^2*cosh(
d*x + c)^2 + 2*a*b*f^2*cosh(d*x + c)*sinh(d*x + c) + a*b*f^2*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/a^2)*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) + 6*(2*a^2
*d^2*e*f + a^2*d*f^2)*x + 24*(a*b*d^2*f^2*x^2 + a*b*d^2*e^2 + 2*a*b*d*e*f + 2*a*b*f^2 + 2*(a*b*d^2*e*f + a*b*d
*f^2)*x)*cosh(d*x + c) + 4*(6*a*b*d^2*f^2*x^2 + 6*a*b*d^2*e^2 + 12*a*b*d*e*f + 12*a*b*f^2 - 3*(2*a^2*d^2*f^2*x
^2 + 2*a^2*d^2*e^2 - 2*a^2*d*e*f + a^2*f^2 + 2*(2*a^2*d^2*e*f - a^2*d*f^2)*x)*cosh(d*x + c)^3 + 18*(a*b*d^2*f^
2*x^2 + a*b*d^2*e^2 - 2*a*b*d*e*f + 2*a*b*f^2 + 2*(a*b*d^2*e*f - a*b*d*f^2)*x)*cosh(d*x + c)^2 + 12*(a*b*d^2*e
*f + a*b*d*f^2)*x - 4*((a^2 + 2*b^2)*d^3*f^2*x^3 + 3*(a^2 + 2*b^2)*d^3*e*f*x^2 + 3*(a^2 + 2*b^2)*d^3*e^2*x)*co
sh(d*x + c))*sinh(d*x + c))/(a^3*d^3*cosh(d*x + c)^2 + 2*a^3*d^3*cosh(d*x + c)*sinh(d*x + c) + a^3*d^3*sinh(d*
x + c)^2)

Sympy [F]

\[ \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int \frac {\left (e + f x\right )^{2} \cosh ^{2}{\left (c + d x \right )}}{a + b \operatorname {csch}{\left (c + d x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \cosh \left (d x + c\right )^{2}}{b \operatorname {csch}\left (d x + c\right ) + a} \,d x } \]

[In]

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

[Out]

-1/8*e^2*((4*b*e^(-d*x - c) - a)*e^(2*d*x + 2*c)/(a^2*d) - 4*(a^2 + 2*b^2)*(d*x + c)/(a^3*d) + (4*b*e^(-d*x -
c) + a*e^(-2*d*x - 2*c))/(a^2*d) + 8*(a^2*b + b^3)*log((a*e^(-d*x - c) - b - sqrt(a^2 + b^2))/(a*e^(-d*x - c)
- b + sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*a^3*d)) + 1/48*(8*(a^2*d^3*f^2*e^(2*c) + 2*b^2*d^3*f^2*e^(2*c))*x^3 +
24*(a^2*d^3*e*f*e^(2*c) + 2*b^2*d^3*e*f*e^(2*c))*x^2 + 3*(2*a^2*d^2*f^2*x^2*e^(4*c) + 2*(2*d^2*e*f - d*f^2)*a
^2*x*e^(4*c) - (2*d*e*f - f^2)*a^2*e^(4*c))*e^(2*d*x) - 24*(a*b*d^2*f^2*x^2*e^(3*c) + 2*(d^2*e*f - d*f^2)*a*b*
x*e^(3*c) - 2*(d*e*f - f^2)*a*b*e^(3*c))*e^(d*x) - 24*(a*b*d^2*f^2*x^2*e^c + 2*(d^2*e*f + d*f^2)*a*b*x*e^c + 2
*(d*e*f + f^2)*a*b*e^c)*e^(-d*x) - 3*(2*a^2*d^2*f^2*x^2 + 2*(2*d^2*e*f + d*f^2)*a^2*x + (2*d*e*f + f^2)*a^2)*e
^(-2*d*x))*e^(-2*c)/(a^3*d^3) - integrate(2*((a^2*b*f^2*e^c + b^3*f^2*e^c)*x^2 + 2*(a^2*b*e*f*e^c + b^3*e*f*e^
c)*x)*e^(d*x)/(a^4*e^(2*d*x + 2*c) + 2*a^3*b*e^(d*x + c) - a^4), x)

Giac [F]

\[ \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \cosh \left (d x + c\right )^{2}}{b \operatorname {csch}\left (d x + c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^2}{a+\frac {b}{\mathrm {sinh}\left (c+d\,x\right )}} \,d x \]

[In]

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

[Out]

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