3.5.82 \(\int \frac {(e+f x)^2 \coth ^2(c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx\) [482]

Optimal. Leaf size=714 \[ \frac {b (e+f x)^2}{a^2 d}-\frac {(e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}-\frac {f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}+\frac {b (e+f x)^2 \coth (c+d x)}{a^2 d}-\frac {f (e+f x) \text {csch}(c+d x)}{a d^2}-\frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {b \sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {b \sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {2 b f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}-\frac {f (e+f x) \text {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}-\frac {2 b^2 f (e+f x) \text {PolyLog}\left (2,-e^{c+d x}\right )}{a^3 d^2}+\frac {f (e+f x) \text {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}+\frac {2 b^2 f (e+f x) \text {PolyLog}\left (2,e^{c+d x}\right )}{a^3 d^2}-\frac {2 b \sqrt {a^2+b^2} f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {2 b \sqrt {a^2+b^2} f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^2}-\frac {b f^2 \text {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a^2 d^3}+\frac {f^2 \text {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {2 b^2 f^2 \text {PolyLog}\left (3,-e^{c+d x}\right )}{a^3 d^3}-\frac {f^2 \text {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac {2 b^2 f^2 \text {PolyLog}\left (3,e^{c+d x}\right )}{a^3 d^3}+\frac {2 b \sqrt {a^2+b^2} f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^3}-\frac {2 b \sqrt {a^2+b^2} f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^3} \]

[Out]

b*(f*x+e)^2/a^2/d-(f*x+e)^2*arctanh(exp(d*x+c))/a/d-2*b^2*(f*x+e)^2*arctanh(exp(d*x+c))/a^3/d-f^2*arctanh(cosh
(d*x+c))/a/d^3+b*(f*x+e)^2*coth(d*x+c)/a^2/d-f*(f*x+e)*csch(d*x+c)/a/d^2-1/2*(f*x+e)^2*coth(d*x+c)*csch(d*x+c)
/a/d-2*b*f*(f*x+e)*ln(1-exp(2*d*x+2*c))/a^2/d^2-f*(f*x+e)*polylog(2,-exp(d*x+c))/a/d^2-2*b^2*f*(f*x+e)*polylog
(2,-exp(d*x+c))/a^3/d^2+f*(f*x+e)*polylog(2,exp(d*x+c))/a/d^2+2*b^2*f*(f*x+e)*polylog(2,exp(d*x+c))/a^3/d^2-b*
f^2*polylog(2,exp(2*d*x+2*c))/a^2/d^3+f^2*polylog(3,-exp(d*x+c))/a/d^3+2*b^2*f^2*polylog(3,-exp(d*x+c))/a^3/d^
3-f^2*polylog(3,exp(d*x+c))/a/d^3-2*b^2*f^2*polylog(3,exp(d*x+c))/a^3/d^3-b*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a-(a^
2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/a^3/d+b*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/a^3/d-
2*b*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/a^3/d^2+2*b*f*(f*x+e)*polylog(2,-b*
exp(d*x+c)/(a+(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/a^3/d^2+2*b*f^2*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))*(
a^2+b^2)^(1/2)/a^3/d^3-2*b*f^2*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/a^3/d^3

________________________________________________________________________________________

Rubi [A]
time = 1.20, antiderivative size = 714, normalized size of antiderivative = 1.00, number of steps used = 52, number of rules used = 22, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.647, Rules used = {5706, 5565, 4267, 2611, 2320, 6724, 4271, 3855, 5688, 3801, 3797, 2221, 2317, 2438, 32, 5704, 5558, 3377, 2718, 5684, 3403, 2296} \begin {gather*} \frac {2 b^2 f^2 \text {Li}_3\left (-e^{c+d x}\right )}{a^3 d^3}-\frac {2 b^2 f^2 \text {Li}_3\left (e^{c+d x}\right )}{a^3 d^3}-\frac {2 b^2 f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a^3 d^2}+\frac {2 b^2 f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a^3 d^2}-\frac {2 b^2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}-\frac {b f^2 \text {Li}_2\left (e^{2 (c+d x)}\right )}{a^2 d^3}-\frac {2 b f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}+\frac {b (e+f x)^2 \coth (c+d x)}{a^2 d}+\frac {b (e+f x)^2}{a^2 d}+\frac {2 b f^2 \sqrt {a^2+b^2} \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^3}-\frac {2 b f^2 \sqrt {a^2+b^2} \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^3}-\frac {2 b f \sqrt {a^2+b^2} (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {2 b f \sqrt {a^2+b^2} (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^2}-\frac {b \sqrt {a^2+b^2} (e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a^3 d}+\frac {b \sqrt {a^2+b^2} (e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a^3 d}+\frac {f^2 \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}-\frac {f^2 \text {Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac {f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}-\frac {f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac {f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac {f (e+f x) \text {csch}(c+d x)}{a d^2}-\frac {(e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(e + f*x)^2)/(a^2*d) - ((e + f*x)^2*ArcTanh[E^(c + d*x)])/(a*d) - (2*b^2*(e + f*x)^2*ArcTanh[E^(c + d*x)])/
(a^3*d) - (f^2*ArcTanh[Cosh[c + d*x]])/(a*d^3) + (b*(e + f*x)^2*Coth[c + d*x])/(a^2*d) - (f*(e + f*x)*Csch[c +
d*x])/(a*d^2) - ((e + f*x)^2*Coth[c + d*x]*Csch[c + d*x])/(2*a*d) - (b*Sqrt[a^2 + b^2]*(e + f*x)^2*Log[1 + (b
*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^3*d) + (b*Sqrt[a^2 + b^2]*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sq
rt[a^2 + b^2])])/(a^3*d) - (2*b*f*(e + f*x)*Log[1 - E^(2*(c + d*x))])/(a^2*d^2) - (f*(e + f*x)*PolyLog[2, -E^(
c + d*x)])/(a*d^2) - (2*b^2*f*(e + f*x)*PolyLog[2, -E^(c + d*x)])/(a^3*d^2) + (f*(e + f*x)*PolyLog[2, E^(c + d
*x)])/(a*d^2) + (2*b^2*f*(e + f*x)*PolyLog[2, E^(c + d*x)])/(a^3*d^2) - (2*b*Sqrt[a^2 + b^2]*f*(e + f*x)*PolyL
og[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*d^2) + (2*b*Sqrt[a^2 + b^2]*f*(e + f*x)*PolyLog[2, -((b*
E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*d^2) - (b*f^2*PolyLog[2, E^(2*(c + d*x))])/(a^2*d^3) + (f^2*PolyLog
[3, -E^(c + d*x)])/(a*d^3) + (2*b^2*f^2*PolyLog[3, -E^(c + d*x)])/(a^3*d^3) - (f^2*PolyLog[3, E^(c + d*x)])/(a
*d^3) - (2*b^2*f^2*PolyLog[3, E^(c + d*x)])/(a^3*d^3) + (2*b*Sqrt[a^2 + b^2]*f^2*PolyLog[3, -((b*E^(c + d*x))/
(a - Sqrt[a^2 + b^2]))])/(a^3*d^3) - (2*b*Sqrt[a^2 + b^2]*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2
]))])/(a^3*d^3)

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 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 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 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

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 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 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 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((
c + d*x)^(m + 1)/(d*(m + 1))), x] + Dist[2*I, Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 3801

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(c + d*x)^m*((b*Tan[e
+ f*x])^(n - 1)/(f*(n - 1))), x] + (-Dist[b*d*(m/(f*(n - 1))), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && GtQ[m, 0]

Rule 3855

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

Rule 4267

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(Ar
cTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*
fz*x)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && IGtQ[m, 0]

Rule 4271

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-b^2)*(c + d*x)^m*Cot[e
+ f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (Dist[b^2*d^2*m*((m - 1)/(f^2*(n - 1)*(n - 2))), Int[(c +
d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Dist[b^2*((n - 2)/(n - 1)), Int[(c + d*x)^m*(b*Csc[e + f*x])^
(n - 2), x], x] - Simp[b^2*d*m*(c + d*x)^(m - 1)*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x]) /; Free
Q[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

Rule 5558

Int[Cosh[(a_.) + (b_.)*(x_)]^(n_.)*Coth[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int
[(c + d*x)^m*Cosh[a + b*x]^n*Coth[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Cosh[a + b*x]^(n - 2)*Coth[a + b*x]^p
, x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]

Rule 5565

Int[Coth[(a_.) + (b_.)*(x_)]^(p_)*Csch[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(c + d
*x)^m*Csch[a + b*x]*Coth[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Csch[a + b*x]^3*Coth[a + b*x]^(p - 2), x] /; F
reeQ[{a, b, c, d, m}, x] && IGtQ[p/2, 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 5688

Int[(Coth[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[1/a, Int[(e + f*x)^m*Coth[c + d*x]^n, x], x] - Dist[b/a, Int[(e + f*x)^m*Cosh[c + d*x]*(Coth[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 5704

Int[(Cosh[(c_.) + (d_.)*(x_)]^(p_.)*Coth[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[(e + f*x)^m*Cosh[c + d*x]^p*Coth[c + d*x]^n, x], x] - Dis
t[b/a, Int[(e + f*x)^m*Cosh[c + d*x]^(p + 1)*(Coth[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 5706

Int[(Coth[(c_.) + (d_.)*(x_)]^(n_.)*Csch[(c_.) + (d_.)*(x_)]^(p_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[(e + f*x)^m*Csch[c + d*x]^p*Coth[c + d*x]^n, x], x] - Dis
t[b/a, Int[(e + f*x)^m*Csch[c + d*x]^(p - 1)*(Coth[c + d*x]^n/(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 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*} \int \frac {(e+f x)^2 \coth ^2(c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^2 \coth ^2(c+d x) \text {csch}(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^2 \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=\frac {\int (e+f x)^2 \text {csch}(c+d x) \, dx}{a}+\frac {\int (e+f x)^2 \text {csch}^3(c+d x) \, dx}{a}-\frac {b \int (e+f x)^2 \coth ^2(c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x)^2 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2}\\ &=-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {b (e+f x)^2 \coth (c+d x)}{a^2 d}-\frac {f (e+f x) \text {csch}(c+d x)}{a d^2}-\frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {\int (e+f x)^2 \text {csch}(c+d x) \, dx}{2 a}-\frac {b \int (e+f x)^2 \, dx}{a^2}+\frac {b^2 \int (e+f x)^2 \cosh (c+d x) \coth (c+d x) \, dx}{a^3}-\frac {b^3 \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^3}-\frac {(2 f) \int (e+f x) \log \left (1-e^{c+d x}\right ) \, dx}{a d}+\frac {(2 f) \int (e+f x) \log \left (1+e^{c+d x}\right ) \, dx}{a d}-\frac {(2 b f) \int (e+f x) \coth (c+d x) \, dx}{a^2 d}+\frac {f^2 \int \text {csch}(c+d x) \, dx}{a d^2}\\ &=\frac {b (e+f x)^2}{a^2 d}-\frac {b (e+f x)^3}{3 a^2 f}-\frac {(e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}+\frac {b (e+f x)^2 \coth (c+d x)}{a^2 d}-\frac {f (e+f x) \text {csch}(c+d x)}{a d^2}-\frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {2 f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac {2 f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac {b \int (e+f x)^2 \, dx}{a^2}+\frac {b^2 \int (e+f x)^2 \text {csch}(c+d x) \, dx}{a^3}-\frac {\left (b \left (a^2+b^2\right )\right ) \int \frac {(e+f x)^2}{a+b \sinh (c+d x)} \, dx}{a^3}+\frac {f \int (e+f x) \log \left (1-e^{c+d x}\right ) \, dx}{a d}-\frac {f \int (e+f x) \log \left (1+e^{c+d x}\right ) \, dx}{a d}+\frac {(4 b f) \int \frac {e^{2 (c+d x)} (e+f x)}{1-e^{2 (c+d x)}} \, dx}{a^2 d}+\frac {\left (2 f^2\right ) \int \text {Li}_2\left (-e^{c+d x}\right ) \, dx}{a d^2}-\frac {\left (2 f^2\right ) \int \text {Li}_2\left (e^{c+d x}\right ) \, dx}{a d^2}\\ &=\frac {b (e+f x)^2}{a^2 d}-\frac {(e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}-\frac {f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}+\frac {b (e+f x)^2 \coth (c+d x)}{a^2 d}-\frac {f (e+f x) \text {csch}(c+d x)}{a d^2}-\frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {2 b f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}-\frac {f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac {f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac {\left (2 b \left (a^2+b^2\right )\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}{a^3}-\frac {\left (2 b^2 f\right ) \int (e+f x) \log \left (1-e^{c+d x}\right ) \, dx}{a^3 d}+\frac {\left (2 b^2 f\right ) \int (e+f x) \log \left (1+e^{c+d x}\right ) \, dx}{a^3 d}+\frac {\left (2 f^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}-\frac {\left (2 f^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}-\frac {f^2 \int \text {Li}_2\left (-e^{c+d x}\right ) \, dx}{a d^2}+\frac {f^2 \int \text {Li}_2\left (e^{c+d x}\right ) \, dx}{a d^2}+\frac {\left (2 b f^2\right ) \int \log \left (1-e^{2 (c+d x)}\right ) \, dx}{a^2 d^2}\\ &=\frac {b (e+f x)^2}{a^2 d}-\frac {(e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}-\frac {f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}+\frac {b (e+f x)^2 \coth (c+d x)}{a^2 d}-\frac {f (e+f x) \text {csch}(c+d x)}{a d^2}-\frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {2 b f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}-\frac {f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac {2 b^2 f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a^3 d^2}+\frac {f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac {2 b^2 f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a^3 d^2}+\frac {2 f^2 \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}-\frac {2 f^2 \text {Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac {\left (2 b^2 \sqrt {a^2+b^2}\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}{a^3}+\frac {\left (2 b^2 \sqrt {a^2+b^2}\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}{a^3}-\frac {f^2 \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}+\frac {f^2 \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}+\frac {\left (b f^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{a^2 d^3}+\frac {\left (2 b^2 f^2\right ) \int \text {Li}_2\left (-e^{c+d x}\right ) \, dx}{a^3 d^2}-\frac {\left (2 b^2 f^2\right ) \int \text {Li}_2\left (e^{c+d x}\right ) \, dx}{a^3 d^2}\\ &=\frac {b (e+f x)^2}{a^2 d}-\frac {(e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}-\frac {f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}+\frac {b (e+f x)^2 \coth (c+d x)}{a^2 d}-\frac {f (e+f x) \text {csch}(c+d x)}{a d^2}-\frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {b \sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {b \sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {2 b f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}-\frac {f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac {2 b^2 f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a^3 d^2}+\frac {f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac {2 b^2 f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a^3 d^2}-\frac {b f^2 \text {Li}_2\left (e^{2 (c+d x)}\right )}{a^2 d^3}+\frac {f^2 \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}-\frac {f^2 \text {Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac {\left (2 b \sqrt {a^2+b^2} 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}{a^3 d}-\frac {\left (2 b \sqrt {a^2+b^2} 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}{a^3 d}+\frac {\left (2 b^2 f^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^3}-\frac {\left (2 b^2 f^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^3}\\ &=\frac {b (e+f x)^2}{a^2 d}-\frac {(e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}-\frac {f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}+\frac {b (e+f x)^2 \coth (c+d x)}{a^2 d}-\frac {f (e+f x) \text {csch}(c+d x)}{a d^2}-\frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {b \sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {b \sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {2 b f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}-\frac {f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac {2 b^2 f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a^3 d^2}+\frac {f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac {2 b^2 f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a^3 d^2}-\frac {2 b \sqrt {a^2+b^2} f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {2 b \sqrt {a^2+b^2} f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^2}-\frac {b f^2 \text {Li}_2\left (e^{2 (c+d x)}\right )}{a^2 d^3}+\frac {f^2 \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac {2 b^2 f^2 \text {Li}_3\left (-e^{c+d x}\right )}{a^3 d^3}-\frac {f^2 \text {Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac {2 b^2 f^2 \text {Li}_3\left (e^{c+d x}\right )}{a^3 d^3}+\frac {\left (2 b \sqrt {a^2+b^2} 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}{a^3 d^2}-\frac {\left (2 b \sqrt {a^2+b^2} 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}{a^3 d^2}\\ &=\frac {b (e+f x)^2}{a^2 d}-\frac {(e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}-\frac {f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}+\frac {b (e+f x)^2 \coth (c+d x)}{a^2 d}-\frac {f (e+f x) \text {csch}(c+d x)}{a d^2}-\frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {b \sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {b \sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {2 b f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}-\frac {f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac {2 b^2 f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a^3 d^2}+\frac {f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac {2 b^2 f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a^3 d^2}-\frac {2 b \sqrt {a^2+b^2} f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {2 b \sqrt {a^2+b^2} f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^2}-\frac {b f^2 \text {Li}_2\left (e^{2 (c+d x)}\right )}{a^2 d^3}+\frac {f^2 \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac {2 b^2 f^2 \text {Li}_3\left (-e^{c+d x}\right )}{a^3 d^3}-\frac {f^2 \text {Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac {2 b^2 f^2 \text {Li}_3\left (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 {\text {Li}_2\left (\frac {b x}{-a+\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 {\text {Li}_2\left (-\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^3}\\ &=\frac {b (e+f x)^2}{a^2 d}-\frac {(e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}-\frac {f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}+\frac {b (e+f x)^2 \coth (c+d x)}{a^2 d}-\frac {f (e+f x) \text {csch}(c+d x)}{a d^2}-\frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {b \sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {b \sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {2 b f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}-\frac {f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac {2 b^2 f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a^3 d^2}+\frac {f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac {2 b^2 f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a^3 d^2}-\frac {2 b \sqrt {a^2+b^2} f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {2 b \sqrt {a^2+b^2} f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^2}-\frac {b f^2 \text {Li}_2\left (e^{2 (c+d x)}\right )}{a^2 d^3}+\frac {f^2 \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac {2 b^2 f^2 \text {Li}_3\left (-e^{c+d x}\right )}{a^3 d^3}-\frac {f^2 \text {Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac {2 b^2 f^2 \text {Li}_3\left (e^{c+d x}\right )}{a^3 d^3}+\frac {2 b \sqrt {a^2+b^2} f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^3}-\frac {2 b \sqrt {a^2+b^2} f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^3}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1803\) vs. \(2(714)=1428\).
time = 21.67, size = 1803, normalized size = 2.53 \begin {gather*} \frac {8 a b d^2 e e^{2 c} f x+4 a b d^2 e^{2 c} f^2 x^2+2 a^2 d^2 e^2 \tanh ^{-1}\left (e^{c+d x}\right )+4 b^2 d^2 e^2 \tanh ^{-1}\left (e^{c+d x}\right )-2 a^2 d^2 e^2 e^{2 c} \tanh ^{-1}\left (e^{c+d x}\right )-4 b^2 d^2 e^2 e^{2 c} \tanh ^{-1}\left (e^{c+d x}\right )+4 a^2 f^2 \tanh ^{-1}\left (e^{c+d x}\right )-4 a^2 e^{2 c} f^2 \tanh ^{-1}\left (e^{c+d x}\right )-2 a^2 d^2 e f x \log \left (1-e^{c+d x}\right )-4 b^2 d^2 e f x \log \left (1-e^{c+d x}\right )+2 a^2 d^2 e e^{2 c} f x \log \left (1-e^{c+d x}\right )+4 b^2 d^2 e e^{2 c} f x \log \left (1-e^{c+d x}\right )-a^2 d^2 f^2 x^2 \log \left (1-e^{c+d x}\right )-2 b^2 d^2 f^2 x^2 \log \left (1-e^{c+d x}\right )+a^2 d^2 e^{2 c} f^2 x^2 \log \left (1-e^{c+d x}\right )+2 b^2 d^2 e^{2 c} f^2 x^2 \log \left (1-e^{c+d x}\right )+2 a^2 d^2 e f x \log \left (1+e^{c+d x}\right )+4 b^2 d^2 e f x \log \left (1+e^{c+d x}\right )-2 a^2 d^2 e e^{2 c} f x \log \left (1+e^{c+d x}\right )-4 b^2 d^2 e e^{2 c} f x \log \left (1+e^{c+d x}\right )+a^2 d^2 f^2 x^2 \log \left (1+e^{c+d x}\right )+2 b^2 d^2 f^2 x^2 \log \left (1+e^{c+d x}\right )-a^2 d^2 e^{2 c} f^2 x^2 \log \left (1+e^{c+d x}\right )-2 b^2 d^2 e^{2 c} f^2 x^2 \log \left (1+e^{c+d x}\right )+4 a b d e f \log \left (1-e^{2 (c+d x)}\right )-4 a b d e e^{2 c} f \log \left (1-e^{2 (c+d x)}\right )+4 a b d f^2 x \log \left (1-e^{2 (c+d x)}\right )-4 a b d e^{2 c} f^2 x \log \left (1-e^{2 (c+d x)}\right )-2 \left (a^2+2 b^2\right ) d \left (-1+e^{2 c}\right ) f (e+f x) \text {PolyLog}\left (2,-e^{c+d x}\right )+2 \left (a^2+2 b^2\right ) d \left (-1+e^{2 c}\right ) f (e+f x) \text {PolyLog}\left (2,e^{c+d x}\right )+2 a b f^2 \text {PolyLog}\left (2,e^{2 (c+d x)}\right )-2 a b e^{2 c} f^2 \text {PolyLog}\left (2,e^{2 (c+d x)}\right )-2 a^2 f^2 \text {PolyLog}\left (3,-e^{c+d x}\right )-4 b^2 f^2 \text {PolyLog}\left (3,-e^{c+d x}\right )+2 a^2 e^{2 c} f^2 \text {PolyLog}\left (3,-e^{c+d x}\right )+4 b^2 e^{2 c} f^2 \text {PolyLog}\left (3,-e^{c+d x}\right )+2 a^2 f^2 \text {PolyLog}\left (3,e^{c+d x}\right )+4 b^2 f^2 \text {PolyLog}\left (3,e^{c+d x}\right )-2 a^2 e^{2 c} f^2 \text {PolyLog}\left (3,e^{c+d x}\right )-4 b^2 e^{2 c} f^2 \text {PolyLog}\left (3,e^{c+d x}\right )}{2 a^3 d^3 \left (-1+e^{2 c}\right )}-\frac {b \left (a^2+b^2\right ) \left (\frac {2 d^2 e^2 \text {ArcTan}\left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}+\frac {2 d^2 e e^c f x \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{\sqrt {\left (a^2+b^2\right ) e^{2 c}}}+\frac {d^2 e^c f^2 x^2 \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{\sqrt {\left (a^2+b^2\right ) e^{2 c}}}-\frac {2 d^2 e e^c f x \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{\sqrt {\left (a^2+b^2\right ) e^{2 c}}}-\frac {d^2 e^c f^2 x^2 \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{\sqrt {\left (a^2+b^2\right ) e^{2 c}}}+\frac {2 d e^c f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{\sqrt {\left (a^2+b^2\right ) e^{2 c}}}-\frac {2 d e^c f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{\sqrt {\left (a^2+b^2\right ) e^{2 c}}}-\frac {2 e^c f^2 \text {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{\sqrt {\left (a^2+b^2\right ) e^{2 c}}}+\frac {2 e^c f^2 \text {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{a^3 d^3}+\frac {\text {csch}(c) \text {csch}^2(c+d x) \left (2 b d e^2 \cosh (c)+4 b d e f x \cosh (c)+2 b d f^2 x^2 \cosh (c)+2 a e f \cosh (d x)+2 a f^2 x \cosh (d x)-2 a e f \cosh (2 c+d x)-2 a f^2 x \cosh (2 c+d x)-2 b d e^2 \cosh (c+2 d x)-4 b d e f x \cosh (c+2 d x)-2 b d f^2 x^2 \cosh (c+2 d x)+a d e^2 \sinh (d x)+2 a d e f x \sinh (d x)+a d f^2 x^2 \sinh (d x)-a d e^2 \sinh (2 c+d x)-2 a d e f x \sinh (2 c+d x)-a d f^2 x^2 \sinh (2 c+d x)\right )}{4 a^2 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(8*a*b*d^2*e*E^(2*c)*f*x + 4*a*b*d^2*E^(2*c)*f^2*x^2 + 2*a^2*d^2*e^2*ArcTanh[E^(c + d*x)] + 4*b^2*d^2*e^2*ArcT
anh[E^(c + d*x)] - 2*a^2*d^2*e^2*E^(2*c)*ArcTanh[E^(c + d*x)] - 4*b^2*d^2*e^2*E^(2*c)*ArcTanh[E^(c + d*x)] + 4
*a^2*f^2*ArcTanh[E^(c + d*x)] - 4*a^2*E^(2*c)*f^2*ArcTanh[E^(c + d*x)] - 2*a^2*d^2*e*f*x*Log[1 - E^(c + d*x)]
- 4*b^2*d^2*e*f*x*Log[1 - E^(c + d*x)] + 2*a^2*d^2*e*E^(2*c)*f*x*Log[1 - E^(c + d*x)] + 4*b^2*d^2*e*E^(2*c)*f*
x*Log[1 - E^(c + d*x)] - a^2*d^2*f^2*x^2*Log[1 - E^(c + d*x)] - 2*b^2*d^2*f^2*x^2*Log[1 - E^(c + d*x)] + a^2*d
^2*E^(2*c)*f^2*x^2*Log[1 - E^(c + d*x)] + 2*b^2*d^2*E^(2*c)*f^2*x^2*Log[1 - E^(c + d*x)] + 2*a^2*d^2*e*f*x*Log
[1 + E^(c + d*x)] + 4*b^2*d^2*e*f*x*Log[1 + E^(c + d*x)] - 2*a^2*d^2*e*E^(2*c)*f*x*Log[1 + E^(c + d*x)] - 4*b^
2*d^2*e*E^(2*c)*f*x*Log[1 + E^(c + d*x)] + a^2*d^2*f^2*x^2*Log[1 + E^(c + d*x)] + 2*b^2*d^2*f^2*x^2*Log[1 + E^
(c + d*x)] - a^2*d^2*E^(2*c)*f^2*x^2*Log[1 + E^(c + d*x)] - 2*b^2*d^2*E^(2*c)*f^2*x^2*Log[1 + E^(c + d*x)] + 4
*a*b*d*e*f*Log[1 - E^(2*(c + d*x))] - 4*a*b*d*e*E^(2*c)*f*Log[1 - E^(2*(c + d*x))] + 4*a*b*d*f^2*x*Log[1 - E^(
2*(c + d*x))] - 4*a*b*d*E^(2*c)*f^2*x*Log[1 - E^(2*(c + d*x))] - 2*(a^2 + 2*b^2)*d*(-1 + E^(2*c))*f*(e + f*x)*
PolyLog[2, -E^(c + d*x)] + 2*(a^2 + 2*b^2)*d*(-1 + E^(2*c))*f*(e + f*x)*PolyLog[2, E^(c + d*x)] + 2*a*b*f^2*Po
lyLog[2, E^(2*(c + d*x))] - 2*a*b*E^(2*c)*f^2*PolyLog[2, E^(2*(c + d*x))] - 2*a^2*f^2*PolyLog[3, -E^(c + d*x)]
- 4*b^2*f^2*PolyLog[3, -E^(c + d*x)] + 2*a^2*E^(2*c)*f^2*PolyLog[3, -E^(c + d*x)] + 4*b^2*E^(2*c)*f^2*PolyLog
[3, -E^(c + d*x)] + 2*a^2*f^2*PolyLog[3, E^(c + d*x)] + 4*b^2*f^2*PolyLog[3, E^(c + d*x)] - 2*a^2*E^(2*c)*f^2*
PolyLog[3, E^(c + d*x)] - 4*b^2*E^(2*c)*f^2*PolyLog[3, E^(c + d*x)])/(2*a^3*d^3*(-1 + E^(2*c))) - (b*(a^2 + b^
2)*((2*d^2*e^2*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]])/Sqrt[-a^2 - b^2] + (2*d^2*e*E^c*f*x*Log[1 + (b*E^
(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/Sqrt[(a^2 + b^2)*E^(2*c)] + (d^2*E^c*f^2*x^2*Log[1 + (b*E^(
2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/Sqrt[(a^2 + b^2)*E^(2*c)] - (2*d^2*e*E^c*f*x*Log[1 + (b*E^(2
*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/Sqrt[(a^2 + b^2)*E^(2*c)] - (d^2*E^c*f^2*x^2*Log[1 + (b*E^(2*
c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/Sqrt[(a^2 + b^2)*E^(2*c)] + (2*d*E^c*f*(e + f*x)*PolyLog[2, -(
(b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/Sqrt[(a^2 + b^2)*E^(2*c)] - (2*d*E^c*f*(e + f*x)*Poly
Log[2, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/Sqrt[(a^2 + b^2)*E^(2*c)] - (2*E^c*f^2*PolyL
og[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/Sqrt[(a^2 + b^2)*E^(2*c)] + (2*E^c*f^2*PolyLo
g[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/Sqrt[(a^2 + b^2)*E^(2*c)]))/(a^3*d^3) + (Csch[
c]*Csch[c + d*x]^2*(2*b*d*e^2*Cosh[c] + 4*b*d*e*f*x*Cosh[c] + 2*b*d*f^2*x^2*Cosh[c] + 2*a*e*f*Cosh[d*x] + 2*a*
f^2*x*Cosh[d*x] - 2*a*e*f*Cosh[2*c + d*x] - 2*a*f^2*x*Cosh[2*c + d*x] - 2*b*d*e^2*Cosh[c + 2*d*x] - 4*b*d*e*f*
x*Cosh[c + 2*d*x] - 2*b*d*f^2*x^2*Cosh[c + 2*d*x] + a*d*e^2*Sinh[d*x] + 2*a*d*e*f*x*Sinh[d*x] + a*d*f^2*x^2*Si
nh[d*x] - a*d*e^2*Sinh[2*c + d*x] - 2*a*d*e*f*x*Sinh[2*c + d*x] - a*d*f^2*x^2*Sinh[2*c + d*x]))/(4*a^2*d^2)

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Maple [F]
time = 3.35, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{2} \left (\coth ^{2}\left (d x +c \right )\right ) \mathrm {csch}\left (d x +c \right )}{a +b \sinh \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*(a*e^(-d*x - c) + 2*b*e^(-2*d*x - 2*c) + a*e^(-3*d*x - 3*c) - 2*b)/((2*a^2*e^(-2*d*x - 2*c) - a^2*e^(-4
*d*x - 4*c) - a^2)*d) - (a^2 + 2*b^2)*log(e^(-d*x - c) + 1)/(a^3*d) + (a^2 + 2*b^2)*log(e^(-d*x - c) - 1)/(a^3
*d) - 2*(a^2*b + b^3)*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/(sqrt
(a^2 + b^2)*a^3*d))*e^2 - (2*b*d*f^2*x^2 + 4*b*d*f*x*e + (a*d*f^2*x^2*e^(3*c) + 2*a*f*e^(3*c + 1) + 2*(a*f^2*e
^(3*c) + a*d*f*e^(3*c + 1))*x)*e^(3*d*x) - 2*(b*d*f^2*x^2*e^(2*c) + 2*b*d*f*x*e^(2*c + 1))*e^(2*d*x) + (a*d*f^
2*x^2*e^c - 2*a*f*e^(c + 1) + 2*(a*d*f*e^(c + 1) - a*f^2*e^c)*x)*e^(d*x))/(a^2*d^2*e^(4*d*x + 4*c) - 2*a^2*d^2
*e^(2*d*x + 2*c) + a^2*d^2) + (2*b*d*f*e + a*f^2)*x/(a^2*d^2) + (2*b*d*f*e - a*f^2)*x/(a^2*d^2) - (2*b*d*f*e +
a*f^2)*log(e^(d*x + c) + 1)/(a^2*d^3) - (2*b*d*f*e - a*f^2)*log(e^(d*x + c) - 1)/(a^2*d^3) - 1/2*(d^2*x^2*log
(e^(d*x + c) + 1) + 2*d*x*dilog(-e^(d*x + c)) - 2*polylog(3, -e^(d*x + c)))*(a^2*f^2 + 2*b^2*f^2)/(a^3*d^3) +
1/2*(d^2*x^2*log(-e^(d*x + c) + 1) + 2*d*x*dilog(e^(d*x + c)) - 2*polylog(3, e^(d*x + c)))*(a^2*f^2 + 2*b^2*f^
2)/(a^3*d^3) - (2*a*b*f^2 + (a^2*d*f + 2*b^2*d*f)*e)*(d*x*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))/(a^3*d^3
) - (2*a*b*f^2 - (a^2*d*f + 2*b^2*d*f)*e)*(d*x*log(-e^(d*x + c) + 1) + dilog(e^(d*x + c)))/(a^3*d^3) + 1/6*((a
^2*f^2 + 2*b^2*f^2)*d^3*x^3 + 3*(2*a*b*f^2 + (a^2*d*f + 2*b^2*d*f)*e)*d^2*x^2)/(a^3*d^3) - 1/6*((a^2*f^2 + 2*b
^2*f^2)*d^3*x^3 - 3*(2*a*b*f^2 - (a^2*d*f + 2*b^2*d*f)*e)*d^2*x^2)/(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*f*e^c + b^3*f*e^c)*x*e)*e^(d*x)/(a^3*b*e^(2*d*x + 2*c) + 2*a^4*e^(d*x + c) - a^3*b)
, x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 11068 vs. \(2 (680) = 1360\).
time = 0.45, size = 11068, normalized size = 15.50 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(4*a*b*c^2*f^2 - 8*a*b*c*d*f*cosh(1) + 4*a*b*d^2*cosh(1)^2 + 4*a*b*d^2*sinh(1)^2 - 4*(a*b*d^2*f^2*x^2 - a
*b*c^2*f^2 + 2*(a*b*d^2*f*x + a*b*c*d*f)*cosh(1) + 2*(a*b*d^2*f*x + a*b*c*d*f)*sinh(1))*cosh(d*x + c)^4 - 4*(a
*b*d^2*f^2*x^2 - a*b*c^2*f^2 + 2*(a*b*d^2*f*x + a*b*c*d*f)*cosh(1) + 2*(a*b*d^2*f*x + a*b*c*d*f)*sinh(1))*sinh
(d*x + c)^4 + 2*(a^2*d^2*f^2*x^2 + 2*a^2*d*f^2*x + a^2*d^2*cosh(1)^2 + a^2*d^2*sinh(1)^2 + 2*(a^2*d^2*f*x + a^
2*d*f)*cosh(1) + 2*(a^2*d^2*f*x + a^2*d^2*cosh(1) + a^2*d*f)*sinh(1))*cosh(d*x + c)^3 + 2*(a^2*d^2*f^2*x^2 + 2
*a^2*d*f^2*x + a^2*d^2*cosh(1)^2 + a^2*d^2*sinh(1)^2 + 2*(a^2*d^2*f*x + a^2*d*f)*cosh(1) - 8*(a*b*d^2*f^2*x^2
- a*b*c^2*f^2 + 2*(a*b*d^2*f*x + a*b*c*d*f)*cosh(1) + 2*(a*b*d^2*f*x + a*b*c*d*f)*sinh(1))*cosh(d*x + c) + 2*(
a^2*d^2*f*x + a^2*d^2*cosh(1) + a^2*d*f)*sinh(1))*sinh(d*x + c)^3 + 4*(a*b*d^2*f^2*x^2 - 2*a*b*c^2*f^2 - a*b*d
^2*cosh(1)^2 - a*b*d^2*sinh(1)^2 + 2*(a*b*d^2*f*x + 2*a*b*c*d*f)*cosh(1) + 2*(a*b*d^2*f*x + 2*a*b*c*d*f - a*b*
d^2*cosh(1))*sinh(1))*cosh(d*x + c)^2 + 2*(2*a*b*d^2*f^2*x^2 - 4*a*b*c^2*f^2 - 2*a*b*d^2*cosh(1)^2 - 2*a*b*d^2
*sinh(1)^2 - 12*(a*b*d^2*f^2*x^2 - a*b*c^2*f^2 + 2*(a*b*d^2*f*x + a*b*c*d*f)*cosh(1) + 2*(a*b*d^2*f*x + a*b*c*
d*f)*sinh(1))*cosh(d*x + c)^2 + 4*(a*b*d^2*f*x + 2*a*b*c*d*f)*cosh(1) + 3*(a^2*d^2*f^2*x^2 + 2*a^2*d*f^2*x + a
^2*d^2*cosh(1)^2 + a^2*d^2*sinh(1)^2 + 2*(a^2*d^2*f*x + a^2*d*f)*cosh(1) + 2*(a^2*d^2*f*x + a^2*d^2*cosh(1) +
a^2*d*f)*sinh(1))*cosh(d*x + c) + 4*(a*b*d^2*f*x + 2*a*b*c*d*f - a*b*d^2*cosh(1))*sinh(1))*sinh(d*x + c)^2 + 4
*(b^2*d*f^2*x + b^2*d*f*cosh(1) + (b^2*d*f^2*x + b^2*d*f*cosh(1) + b^2*d*f*sinh(1))*cosh(d*x + c)^4 + b^2*d*f*
sinh(1) + 4*(b^2*d*f^2*x + b^2*d*f*cosh(1) + b^2*d*f*sinh(1))*cosh(d*x + c)*sinh(d*x + c)^3 + (b^2*d*f^2*x + b
^2*d*f*cosh(1) + b^2*d*f*sinh(1))*sinh(d*x + c)^4 - 2*(b^2*d*f^2*x + b^2*d*f*cosh(1) + b^2*d*f*sinh(1))*cosh(d
*x + c)^2 - 2*(b^2*d*f^2*x + b^2*d*f*cosh(1) + b^2*d*f*sinh(1) - 3*(b^2*d*f^2*x + b^2*d*f*cosh(1) + b^2*d*f*si
nh(1))*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^2*d*f^2*x + b^2*d*f*cosh(1) + b^2*d*f*sinh(1))*cosh(d*x + c)^3
- (b^2*d*f^2*x + b^2*d*f*cosh(1) + b^2*d*f*sinh(1))*cosh(d*x + c))*sinh(d*x + c))*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) -
4*(b^2*d*f^2*x + b^2*d*f*cosh(1) + (b^2*d*f^2*x + b^2*d*f*cosh(1) + b^2*d*f*sinh(1))*cosh(d*x + c)^4 + b^2*d*f
*sinh(1) + 4*(b^2*d*f^2*x + b^2*d*f*cosh(1) + b^2*d*f*sinh(1))*cosh(d*x + c)*sinh(d*x + c)^3 + (b^2*d*f^2*x +
b^2*d*f*cosh(1) + b^2*d*f*sinh(1))*sinh(d*x + c)^4 - 2*(b^2*d*f^2*x + b^2*d*f*cosh(1) + b^2*d*f*sinh(1))*cosh(
d*x + c)^2 - 2*(b^2*d*f^2*x + b^2*d*f*cosh(1) + b^2*d*f*sinh(1) - 3*(b^2*d*f^2*x + b^2*d*f*cosh(1) + b^2*d*f*s
inh(1))*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^2*d*f^2*x + b^2*d*f*cosh(1) + b^2*d*f*sinh(1))*cosh(d*x + c)^
3 - (b^2*d*f^2*x + b^2*d*f*cosh(1) + b^2*d*f*sinh(1))*cosh(d*x + c))*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2)*dilo
g((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) -
2*(b^2*c^2*f^2 - 2*b^2*c*d*f*cosh(1) + b^2*d^2*cosh(1)^2 + b^2*d^2*sinh(1)^2 + (b^2*c^2*f^2 - 2*b^2*c*d*f*cos
h(1) + b^2*d^2*cosh(1)^2 + b^2*d^2*sinh(1)^2 - 2*(b^2*c*d*f - b^2*d^2*cosh(1))*sinh(1))*cosh(d*x + c)^4 + 4*(b
^2*c^2*f^2 - 2*b^2*c*d*f*cosh(1) + b^2*d^2*cosh(1)^2 + b^2*d^2*sinh(1)^2 - 2*(b^2*c*d*f - b^2*d^2*cosh(1))*sin
h(1))*cosh(d*x + c)*sinh(d*x + c)^3 + (b^2*c^2*f^2 - 2*b^2*c*d*f*cosh(1) + b^2*d^2*cosh(1)^2 + b^2*d^2*sinh(1)
^2 - 2*(b^2*c*d*f - b^2*d^2*cosh(1))*sinh(1))*sinh(d*x + c)^4 - 2*(b^2*c^2*f^2 - 2*b^2*c*d*f*cosh(1) + b^2*d^2
*cosh(1)^2 + b^2*d^2*sinh(1)^2 - 2*(b^2*c*d*f - b^2*d^2*cosh(1))*sinh(1))*cosh(d*x + c)^2 - 2*(b^2*c^2*f^2 - 2
*b^2*c*d*f*cosh(1) + b^2*d^2*cosh(1)^2 + b^2*d^2*sinh(1)^2 - 3*(b^2*c^2*f^2 - 2*b^2*c*d*f*cosh(1) + b^2*d^2*co
sh(1)^2 + b^2*d^2*sinh(1)^2 - 2*(b^2*c*d*f - b^2*d^2*cosh(1))*sinh(1))*cosh(d*x + c)^2 - 2*(b^2*c*d*f - b^2*d^
2*cosh(1))*sinh(1))*sinh(d*x + c)^2 - 2*(b^2*c*d*f - b^2*d^2*cosh(1))*sinh(1) + 4*((b^2*c^2*f^2 - 2*b^2*c*d*f*
cosh(1) + b^2*d^2*cosh(1)^2 + b^2*d^2*sinh(1)^2 - 2*(b^2*c*d*f - b^2*d^2*cosh(1))*sinh(1))*cosh(d*x + c)^3 - (
b^2*c^2*f^2 - 2*b^2*c*d*f*cosh(1) + b^2*d^2*cosh(1)^2 + b^2*d^2*sinh(1)^2 - 2*(b^2*c*d*f - b^2*d^2*cosh(1))*si
nh(1))*cosh(d*x + c))*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqr
t((a^2 + b^2)/b^2) + 2*a) + 2*(b^2*c^2*f^2 - 2*b^2*c*d*f*cosh(1) + b^2*d^2*cosh(1)^2 + b^2*d^2*sinh(1)^2 + (b^
2*c^2*f^2 - 2*b^2*c*d*f*cosh(1) + b^2*d^2*cosh(1)^2 + b^2*d^2*sinh(1)^2 - 2*(b^2*c*d*f - b^2*d^2*cosh(1))*sinh
(1))*cosh(d*x + c)^4 + 4*(b^2*c^2*f^2 - 2*b^2*c*d*f*cosh(1) + b^2*d^2*cosh(1)^2 + b^2*d^2*sinh(1)^2 - 2*(b^2*c
*d*f - b^2*d^2*cosh(1))*sinh(1))*cosh(d*x + c)*sinh(d*x + c)^3 + (b^2*c^2*f^2 - 2*b^2*c*d*f*cosh(1) + b^2*d^2*
cosh(1)^2 + b^2*d^2*sinh(1)^2 - 2*(b^2*c*d*f - b^2*d^2*cosh(1))*sinh(1))*sinh(d*x + c)^4 - 2*(b^2*c^2*f^2 - 2*
b^2*c*d*f*cosh(1) + b^2*d^2*cosh(1)^2 + b^2*d^2...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {coth}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^2}{\mathrm {sinh}\left (c+d\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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