3.5.81 \(\int \frac {(e+f x)^3 \coth ^2(c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx\) [481]

Optimal. Leaf size=1038 \[ \frac {b (e+f x)^3}{a^2 d}-\frac {6 f^2 (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^3}-\frac {(e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}+\frac {b (e+f x)^3 \coth (c+d x)}{a^2 d}-\frac {3 f (e+f x)^2 \text {csch}(c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {b \sqrt {a^2+b^2} (e+f x)^3 \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)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {3 b f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}-\frac {3 f^3 \text {PolyLog}\left (2,-e^{c+d x}\right )}{a d^4}-\frac {3 f (e+f x)^2 \text {PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}-\frac {3 b^2 f (e+f x)^2 \text {PolyLog}\left (2,-e^{c+d x}\right )}{a^3 d^2}+\frac {3 f^3 \text {PolyLog}\left (2,e^{c+d x}\right )}{a d^4}+\frac {3 f (e+f x)^2 \text {PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}+\frac {3 b^2 f (e+f x)^2 \text {PolyLog}\left (2,e^{c+d x}\right )}{a^3 d^2}-\frac {3 b \sqrt {a^2+b^2} f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {3 b \sqrt {a^2+b^2} f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^2}-\frac {3 b f^2 (e+f x) \text {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a^2 d^3}+\frac {3 f^2 (e+f x) \text {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {6 b^2 f^2 (e+f x) \text {PolyLog}\left (3,-e^{c+d x}\right )}{a^3 d^3}-\frac {3 f^2 (e+f x) \text {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac {6 b^2 f^2 (e+f x) \text {PolyLog}\left (3,e^{c+d x}\right )}{a^3 d^3}+\frac {6 b \sqrt {a^2+b^2} f^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^3}-\frac {6 b \sqrt {a^2+b^2} f^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^3}+\frac {3 b f^3 \text {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a^2 d^4}-\frac {3 f^3 \text {PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}-\frac {6 b^2 f^3 \text {PolyLog}\left (4,-e^{c+d x}\right )}{a^3 d^4}+\frac {3 f^3 \text {PolyLog}\left (4,e^{c+d x}\right )}{a d^4}+\frac {6 b^2 f^3 \text {PolyLog}\left (4,e^{c+d x}\right )}{a^3 d^4}-\frac {6 b \sqrt {a^2+b^2} f^3 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^4}+\frac {6 b \sqrt {a^2+b^2} f^3 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^4} \]

[Out]

-b*(f*x+e)^3*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)^3*ln(1+b*exp(d*x+c)/(a+(a^
2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/a^3/d-6*b*f^3*polylog(4,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/a^3/
d^4+6*b*f^3*polylog(4,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/a^3/d^4-3*f^3*polylog(2,-exp(d*x+c))/
a/d^4+3*f^3*polylog(2,exp(d*x+c))/a/d^4-3/2*f*(f*x+e)^2*polylog(2,-exp(d*x+c))/a/d^2+3/2*f*(f*x+e)^2*polylog(2
,exp(d*x+c))/a/d^2+3*f^2*(f*x+e)*polylog(3,-exp(d*x+c))/a/d^3-2*b^2*(f*x+e)^3*arctanh(exp(d*x+c))/a^3/d+3/2*b*
f^3*polylog(3,exp(2*d*x+2*c))/a^2/d^4-6*b^2*f^3*polylog(4,-exp(d*x+c))/a^3/d^4+6*b^2*f^3*polylog(4,exp(d*x+c))
/a^3/d^4-6*f^2*(f*x+e)*arctanh(exp(d*x+c))/a/d^3-3/2*f*(f*x+e)^2*csch(d*x+c)/a/d^2-1/2*(f*x+e)^3*coth(d*x+c)*c
sch(d*x+c)/a/d-3*f^2*(f*x+e)*polylog(3,exp(d*x+c))/a/d^3+b*(f*x+e)^3/a^2/d+b*(f*x+e)^3*coth(d*x+c)/a^2/d-3*b^2
*f*(f*x+e)^2*polylog(2,-exp(d*x+c))/a^3/d^2+3*b^2*f*(f*x+e)^2*polylog(2,exp(d*x+c))/a^3/d^2-3*b*f^2*(f*x+e)*po
lylog(2,exp(2*d*x+2*c))/a^2/d^3+6*b^2*f^2*(f*x+e)*polylog(3,-exp(d*x+c))/a^3/d^3-6*b^2*f^2*(f*x+e)*polylog(3,e
xp(d*x+c))/a^3/d^3-(f*x+e)^3*arctanh(exp(d*x+c))/a/d-3*f^3*polylog(4,-exp(d*x+c))/a/d^4+3*f^3*polylog(4,exp(d*
x+c))/a/d^4-3*b*f*(f*x+e)^2*ln(1-exp(2*d*x+2*c))/a^2/d^2-3*b*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^
(1/2)))*(a^2+b^2)^(1/2)/a^3/d^2+3*b*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/a
^3/d^2+6*b*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/a^3/d^3-6*b*f^2*(f*x+e)*po
lylog(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 = 1.52, antiderivative size = 1038, normalized size of antiderivative = 1.00, number of steps used = 67, number of rules used = 22, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.647, Rules used = {5706, 5565, 4267, 2611, 6744, 2320, 6724, 4271, 2317, 2438, 5688, 3801, 3797, 2221, 32, 5704, 5558, 3377, 2717, 5684, 3403, 2296} \begin {gather*} -\frac {3 \text {Li}_2\left (-e^{c+d x}\right ) f^3}{a d^4}+\frac {3 \text {Li}_2\left (e^{c+d x}\right ) f^3}{a d^4}+\frac {3 b \text {Li}_3\left (e^{2 (c+d x)}\right ) f^3}{2 a^2 d^4}-\frac {6 b^2 \text {Li}_4\left (-e^{c+d x}\right ) f^3}{a^3 d^4}-\frac {3 \text {Li}_4\left (-e^{c+d x}\right ) f^3}{a d^4}+\frac {6 b^2 \text {Li}_4\left (e^{c+d x}\right ) f^3}{a^3 d^4}+\frac {3 \text {Li}_4\left (e^{c+d x}\right ) f^3}{a d^4}-\frac {6 b \sqrt {a^2+b^2} \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) f^3}{a^3 d^4}+\frac {6 b \sqrt {a^2+b^2} \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) f^3}{a^3 d^4}-\frac {6 (e+f x) \tanh ^{-1}\left (e^{c+d x}\right ) f^2}{a d^3}-\frac {3 b (e+f x) \text {Li}_2\left (e^{2 (c+d x)}\right ) f^2}{a^2 d^3}+\frac {6 b^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right ) f^2}{a^3 d^3}+\frac {3 (e+f x) \text {Li}_3\left (-e^{c+d x}\right ) f^2}{a d^3}-\frac {6 b^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right ) f^2}{a^3 d^3}-\frac {3 (e+f x) \text {Li}_3\left (e^{c+d x}\right ) f^2}{a d^3}+\frac {6 b \sqrt {a^2+b^2} (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) f^2}{a^3 d^3}-\frac {6 b \sqrt {a^2+b^2} (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) f^2}{a^3 d^3}-\frac {3 (e+f x)^2 \text {csch}(c+d x) f}{2 a d^2}-\frac {3 b (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right ) f}{a^2 d^2}-\frac {3 b^2 (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right ) f}{a^3 d^2}-\frac {3 (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right ) f}{2 a d^2}+\frac {3 b^2 (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right ) f}{a^3 d^2}+\frac {3 (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right ) f}{2 a d^2}-\frac {3 b \sqrt {a^2+b^2} (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) f}{a^3 d^2}+\frac {3 b \sqrt {a^2+b^2} (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) f}{a^3 d^2}+\frac {b (e+f x)^3}{a^2 d}-\frac {2 b^2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}-\frac {(e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {b (e+f x)^3 \coth (c+d x)}{a^2 d}-\frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {b \sqrt {a^2+b^2} (e+f x)^3 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )}{a^3 d}+\frac {b \sqrt {a^2+b^2} (e+f x)^3 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )}{a^3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(e + f*x)^3)/(a^2*d) - (6*f^2*(e + f*x)*ArcTanh[E^(c + d*x)])/(a*d^3) - ((e + f*x)^3*ArcTanh[E^(c + d*x)])/
(a*d) - (2*b^2*(e + f*x)^3*ArcTanh[E^(c + d*x)])/(a^3*d) + (b*(e + f*x)^3*Coth[c + d*x])/(a^2*d) - (3*f*(e + f
*x)^2*Csch[c + d*x])/(2*a*d^2) - ((e + f*x)^3*Coth[c + d*x]*Csch[c + d*x])/(2*a*d) - (b*Sqrt[a^2 + b^2]*(e + f
*x)^3*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)^3*Log[1 + (b*E^(c
+ d*x))/(a + Sqrt[a^2 + b^2])])/(a^3*d) - (3*b*f*(e + f*x)^2*Log[1 - E^(2*(c + d*x))])/(a^2*d^2) - (3*f^3*Pol
yLog[2, -E^(c + d*x)])/(a*d^4) - (3*f*(e + f*x)^2*PolyLog[2, -E^(c + d*x)])/(2*a*d^2) - (3*b^2*f*(e + f*x)^2*P
olyLog[2, -E^(c + d*x)])/(a^3*d^2) + (3*f^3*PolyLog[2, E^(c + d*x)])/(a*d^4) + (3*f*(e + f*x)^2*PolyLog[2, E^(
c + d*x)])/(2*a*d^2) + (3*b^2*f*(e + f*x)^2*PolyLog[2, E^(c + d*x)])/(a^3*d^2) - (3*b*Sqrt[a^2 + b^2]*f*(e + f
*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*d^2) + (3*b*Sqrt[a^2 + b^2]*f*(e + f*x)^2*Pol
yLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*d^2) - (3*b*f^2*(e + f*x)*PolyLog[2, E^(2*(c + d*x))])
/(a^2*d^3) + (3*f^2*(e + f*x)*PolyLog[3, -E^(c + d*x)])/(a*d^3) + (6*b^2*f^2*(e + f*x)*PolyLog[3, -E^(c + d*x)
])/(a^3*d^3) - (3*f^2*(e + f*x)*PolyLog[3, E^(c + d*x)])/(a*d^3) - (6*b^2*f^2*(e + f*x)*PolyLog[3, E^(c + d*x)
])/(a^3*d^3) + (6*b*Sqrt[a^2 + b^2]*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*d
^3) - (6*b*Sqrt[a^2 + b^2]*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*d^3) + (3*
b*f^3*PolyLog[3, E^(2*(c + d*x))])/(2*a^2*d^4) - (3*f^3*PolyLog[4, -E^(c + d*x)])/(a*d^4) - (6*b^2*f^3*PolyLog
[4, -E^(c + d*x)])/(a^3*d^4) + (3*f^3*PolyLog[4, E^(c + d*x)])/(a*d^4) + (6*b^2*f^3*PolyLog[4, E^(c + d*x)])/(
a^3*d^4) - (6*b*Sqrt[a^2 + b^2]*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*d^4) + (6*b*Sqr
t[a^2 + b^2]*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*d^4)

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 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[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 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]

Rule 6744

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

Rubi steps

\begin {align*} \int \frac {(e+f x)^3 \coth ^2(c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^3 \coth ^2(c+d x) \text {csch}(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^3 \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=\frac {\int (e+f x)^3 \text {csch}(c+d x) \, dx}{a}+\frac {\int (e+f x)^3 \text {csch}^3(c+d x) \, dx}{a}-\frac {b \int (e+f x)^3 \coth ^2(c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x)^3 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2}\\ &=-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {b (e+f x)^3 \coth (c+d x)}{a^2 d}-\frac {3 f (e+f x)^2 \text {csch}(c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {\int (e+f x)^3 \text {csch}(c+d x) \, dx}{2 a}-\frac {b \int (e+f x)^3 \, dx}{a^2}+\frac {b^2 \int (e+f x)^3 \cosh (c+d x) \coth (c+d x) \, dx}{a^3}-\frac {b^3 \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^3}-\frac {(3 f) \int (e+f x)^2 \log \left (1-e^{c+d x}\right ) \, dx}{a d}+\frac {(3 f) \int (e+f x)^2 \log \left (1+e^{c+d x}\right ) \, dx}{a d}-\frac {(3 b f) \int (e+f x)^2 \coth (c+d x) \, dx}{a^2 d}+\frac {\left (3 f^2\right ) \int (e+f x) \text {csch}(c+d x) \, dx}{a d^2}\\ &=\frac {b (e+f x)^3}{a^2 d}-\frac {b (e+f x)^4}{4 a^2 f}-\frac {6 f^2 (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^3}-\frac {(e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {b (e+f x)^3 \coth (c+d x)}{a^2 d}-\frac {3 f (e+f x)^2 \text {csch}(c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac {b \int (e+f x)^3 \, dx}{a^2}+\frac {b^2 \int (e+f x)^3 \text {csch}(c+d x) \, dx}{a^3}-\frac {\left (b \left (a^2+b^2\right )\right ) \int \frac {(e+f x)^3}{a+b \sinh (c+d x)} \, dx}{a^3}+\frac {(3 f) \int (e+f x)^2 \log \left (1-e^{c+d x}\right ) \, dx}{2 a d}-\frac {(3 f) \int (e+f x)^2 \log \left (1+e^{c+d x}\right ) \, dx}{2 a d}+\frac {(6 b f) \int \frac {e^{2 (c+d x)} (e+f x)^2}{1-e^{2 (c+d x)}} \, dx}{a^2 d}+\frac {\left (6 f^2\right ) \int (e+f x) \text {Li}_2\left (-e^{c+d x}\right ) \, dx}{a d^2}-\frac {\left (6 f^2\right ) \int (e+f x) \text {Li}_2\left (e^{c+d x}\right ) \, dx}{a d^2}-\frac {\left (3 f^3\right ) \int \log \left (1-e^{c+d x}\right ) \, dx}{a d^3}+\frac {\left (3 f^3\right ) \int \log \left (1+e^{c+d x}\right ) \, dx}{a d^3}\\ &=\frac {b (e+f x)^3}{a^2 d}-\frac {6 f^2 (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^3}-\frac {(e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}+\frac {b (e+f x)^3 \coth (c+d x)}{a^2 d}-\frac {3 f (e+f x)^2 \text {csch}(c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {3 b f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{2 a d^2}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{2 a d^2}+\frac {6 f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}-\frac {6 f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac {\left (2 b \left (a^2+b^2\right )\right ) \int \frac {e^{c+d x} (e+f x)^3}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{a^3}-\frac {\left (3 b^2 f\right ) \int (e+f x)^2 \log \left (1-e^{c+d x}\right ) \, dx}{a^3 d}+\frac {\left (3 b^2 f\right ) \int (e+f x)^2 \log \left (1+e^{c+d x}\right ) \, dx}{a^3 d}-\frac {\left (3 f^2\right ) \int (e+f x) \text {Li}_2\left (-e^{c+d x}\right ) \, dx}{a d^2}+\frac {\left (3 f^2\right ) \int (e+f x) \text {Li}_2\left (e^{c+d x}\right ) \, dx}{a d^2}+\frac {\left (6 b f^2\right ) \int (e+f x) \log \left (1-e^{2 (c+d x)}\right ) \, dx}{a^2 d^2}-\frac {\left (3 f^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}+\frac {\left (3 f^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}-\frac {\left (6 f^3\right ) \int \text {Li}_3\left (-e^{c+d x}\right ) \, dx}{a d^3}+\frac {\left (6 f^3\right ) \int \text {Li}_3\left (e^{c+d x}\right ) \, dx}{a d^3}\\ &=\frac {b (e+f x)^3}{a^2 d}-\frac {6 f^2 (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^3}-\frac {(e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}+\frac {b (e+f x)^3 \coth (c+d x)}{a^2 d}-\frac {3 f (e+f x)^2 \text {csch}(c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {3 b f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}-\frac {3 f^3 \text {Li}_2\left (-e^{c+d x}\right )}{a d^4}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{2 a d^2}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{a^3 d^2}+\frac {3 f^3 \text {Li}_2\left (e^{c+d x}\right )}{a d^4}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{2 a d^2}+\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a^3 d^2}-\frac {3 b f^2 (e+f x) \text {Li}_2\left (e^{2 (c+d x)}\right )}{a^2 d^3}+\frac {3 f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}-\frac {3 f^2 (e+f x) \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)^3}{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)^3}{2 a+2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{a^3}+\frac {\left (6 b^2 f^2\right ) \int (e+f x) \text {Li}_2\left (-e^{c+d x}\right ) \, dx}{a^3 d^2}-\frac {\left (6 b^2 f^2\right ) \int (e+f x) \text {Li}_2\left (e^{c+d x}\right ) \, dx}{a^3 d^2}-\frac {\left (6 f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}+\frac {\left (6 f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}+\frac {\left (3 f^3\right ) \int \text {Li}_3\left (-e^{c+d x}\right ) \, dx}{a d^3}-\frac {\left (3 f^3\right ) \int \text {Li}_3\left (e^{c+d x}\right ) \, dx}{a d^3}+\frac {\left (3 b f^3\right ) \int \text {Li}_2\left (e^{2 (c+d x)}\right ) \, dx}{a^2 d^3}\\ &=\frac {b (e+f x)^3}{a^2 d}-\frac {6 f^2 (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^3}-\frac {(e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}+\frac {b (e+f x)^3 \coth (c+d x)}{a^2 d}-\frac {3 f (e+f x)^2 \text {csch}(c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {b \sqrt {a^2+b^2} (e+f x)^3 \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)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {3 b f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}-\frac {3 f^3 \text {Li}_2\left (-e^{c+d x}\right )}{a d^4}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{2 a d^2}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{a^3 d^2}+\frac {3 f^3 \text {Li}_2\left (e^{c+d x}\right )}{a d^4}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{2 a d^2}+\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a^3 d^2}-\frac {3 b f^2 (e+f x) \text {Li}_2\left (e^{2 (c+d x)}\right )}{a^2 d^3}+\frac {3 f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac {6 b^2 f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a^3 d^3}-\frac {3 f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac {6 b^2 f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a^3 d^3}-\frac {6 f^3 \text {Li}_4\left (-e^{c+d x}\right )}{a d^4}+\frac {6 f^3 \text {Li}_4\left (e^{c+d x}\right )}{a d^4}+\frac {\left (3 b \sqrt {a^2+b^2} f\right ) \int (e+f x)^2 \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 (3 b \sqrt {a^2+b^2} f\right ) \int (e+f x)^2 \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 (3 f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}-\frac {\left (3 f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}+\frac {\left (3 b f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 a^2 d^4}-\frac {\left (6 b^2 f^3\right ) \int \text {Li}_3\left (-e^{c+d x}\right ) \, dx}{a^3 d^3}+\frac {\left (6 b^2 f^3\right ) \int \text {Li}_3\left (e^{c+d x}\right ) \, dx}{a^3 d^3}\\ &=\frac {b (e+f x)^3}{a^2 d}-\frac {6 f^2 (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^3}-\frac {(e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}+\frac {b (e+f x)^3 \coth (c+d x)}{a^2 d}-\frac {3 f (e+f x)^2 \text {csch}(c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {b \sqrt {a^2+b^2} (e+f x)^3 \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)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {3 b f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}-\frac {3 f^3 \text {Li}_2\left (-e^{c+d x}\right )}{a d^4}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{2 a d^2}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{a^3 d^2}+\frac {3 f^3 \text {Li}_2\left (e^{c+d x}\right )}{a d^4}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{2 a d^2}+\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a^3 d^2}-\frac {3 b \sqrt {a^2+b^2} f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {3 b \sqrt {a^2+b^2} f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^2}-\frac {3 b f^2 (e+f x) \text {Li}_2\left (e^{2 (c+d x)}\right )}{a^2 d^3}+\frac {3 f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac {6 b^2 f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a^3 d^3}-\frac {3 f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac {6 b^2 f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a^3 d^3}+\frac {3 b f^3 \text {Li}_3\left (e^{2 (c+d x)}\right )}{2 a^2 d^4}-\frac {3 f^3 \text {Li}_4\left (-e^{c+d x}\right )}{a d^4}+\frac {3 f^3 \text {Li}_4\left (e^{c+d x}\right )}{a d^4}+\frac {\left (6 b \sqrt {a^2+b^2} f^2\right ) \int (e+f x) \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 (6 b \sqrt {a^2+b^2} f^2\right ) \int (e+f x) \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 (6 b^2 f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^4}+\frac {\left (6 b^2 f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^4}\\ &=\frac {b (e+f x)^3}{a^2 d}-\frac {6 f^2 (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^3}-\frac {(e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}+\frac {b (e+f x)^3 \coth (c+d x)}{a^2 d}-\frac {3 f (e+f x)^2 \text {csch}(c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {b \sqrt {a^2+b^2} (e+f x)^3 \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)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {3 b f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}-\frac {3 f^3 \text {Li}_2\left (-e^{c+d x}\right )}{a d^4}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{2 a d^2}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{a^3 d^2}+\frac {3 f^3 \text {Li}_2\left (e^{c+d x}\right )}{a d^4}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{2 a d^2}+\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a^3 d^2}-\frac {3 b \sqrt {a^2+b^2} f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {3 b \sqrt {a^2+b^2} f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^2}-\frac {3 b f^2 (e+f x) \text {Li}_2\left (e^{2 (c+d x)}\right )}{a^2 d^3}+\frac {3 f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac {6 b^2 f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a^3 d^3}-\frac {3 f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac {6 b^2 f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a^3 d^3}+\frac {6 b \sqrt {a^2+b^2} f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^3}-\frac {6 b \sqrt {a^2+b^2} f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^3}+\frac {3 b f^3 \text {Li}_3\left (e^{2 (c+d x)}\right )}{2 a^2 d^4}-\frac {3 f^3 \text {Li}_4\left (-e^{c+d x}\right )}{a d^4}-\frac {6 b^2 f^3 \text {Li}_4\left (-e^{c+d x}\right )}{a^3 d^4}+\frac {3 f^3 \text {Li}_4\left (e^{c+d x}\right )}{a d^4}+\frac {6 b^2 f^3 \text {Li}_4\left (e^{c+d x}\right )}{a^3 d^4}-\frac {\left (6 b \sqrt {a^2+b^2} f^3\right ) \int \text {Li}_3\left (-\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{a^3 d^3}+\frac {\left (6 b \sqrt {a^2+b^2} f^3\right ) \int \text {Li}_3\left (-\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{a^3 d^3}\\ &=\frac {b (e+f x)^3}{a^2 d}-\frac {6 f^2 (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^3}-\frac {(e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}+\frac {b (e+f x)^3 \coth (c+d x)}{a^2 d}-\frac {3 f (e+f x)^2 \text {csch}(c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {b \sqrt {a^2+b^2} (e+f x)^3 \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)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {3 b f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}-\frac {3 f^3 \text {Li}_2\left (-e^{c+d x}\right )}{a d^4}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{2 a d^2}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{a^3 d^2}+\frac {3 f^3 \text {Li}_2\left (e^{c+d x}\right )}{a d^4}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{2 a d^2}+\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a^3 d^2}-\frac {3 b \sqrt {a^2+b^2} f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {3 b \sqrt {a^2+b^2} f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^2}-\frac {3 b f^2 (e+f x) \text {Li}_2\left (e^{2 (c+d x)}\right )}{a^2 d^3}+\frac {3 f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac {6 b^2 f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a^3 d^3}-\frac {3 f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac {6 b^2 f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a^3 d^3}+\frac {6 b \sqrt {a^2+b^2} f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^3}-\frac {6 b \sqrt {a^2+b^2} f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^3}+\frac {3 b f^3 \text {Li}_3\left (e^{2 (c+d x)}\right )}{2 a^2 d^4}-\frac {3 f^3 \text {Li}_4\left (-e^{c+d x}\right )}{a d^4}-\frac {6 b^2 f^3 \text {Li}_4\left (-e^{c+d x}\right )}{a^3 d^4}+\frac {3 f^3 \text {Li}_4\left (e^{c+d x}\right )}{a d^4}+\frac {6 b^2 f^3 \text {Li}_4\left (e^{c+d x}\right )}{a^3 d^4}-\frac {\left (6 b \sqrt {a^2+b^2} f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (\frac {b x}{-a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^4}+\frac {\left (6 b \sqrt {a^2+b^2} f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^4}\\ &=\frac {b (e+f x)^3}{a^2 d}-\frac {6 f^2 (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^3}-\frac {(e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}+\frac {b (e+f x)^3 \coth (c+d x)}{a^2 d}-\frac {3 f (e+f x)^2 \text {csch}(c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {b \sqrt {a^2+b^2} (e+f x)^3 \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)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {3 b f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}-\frac {3 f^3 \text {Li}_2\left (-e^{c+d x}\right )}{a d^4}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{2 a d^2}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{a^3 d^2}+\frac {3 f^3 \text {Li}_2\left (e^{c+d x}\right )}{a d^4}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{2 a d^2}+\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a^3 d^2}-\frac {3 b \sqrt {a^2+b^2} f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {3 b \sqrt {a^2+b^2} f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^2}-\frac {3 b f^2 (e+f x) \text {Li}_2\left (e^{2 (c+d x)}\right )}{a^2 d^3}+\frac {3 f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac {6 b^2 f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a^3 d^3}-\frac {3 f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac {6 b^2 f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a^3 d^3}+\frac {6 b \sqrt {a^2+b^2} f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^3}-\frac {6 b \sqrt {a^2+b^2} f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^3}+\frac {3 b f^3 \text {Li}_3\left (e^{2 (c+d x)}\right )}{2 a^2 d^4}-\frac {3 f^3 \text {Li}_4\left (-e^{c+d x}\right )}{a d^4}-\frac {6 b^2 f^3 \text {Li}_4\left (-e^{c+d x}\right )}{a^3 d^4}+\frac {3 f^3 \text {Li}_4\left (e^{c+d x}\right )}{a d^4}+\frac {6 b^2 f^3 \text {Li}_4\left (e^{c+d x}\right )}{a^3 d^4}-\frac {6 b \sqrt {a^2+b^2} f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^4}+\frac {6 b \sqrt {a^2+b^2} f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^4}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 39.99, size = 5829, normalized size = 5.62 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [F]
time = 3.34, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{3} \left (\coth ^{2}\left (d x +c \right )\right ) \mathrm {csch}\left (d x +c \right )}{a +b \sinh \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((f*x+e)^3*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)^3*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^3 - (2*b*d*f^3*x^3 + 6*b*d*f^2*x^2*e + 6*b*d*f*x*e^2 + (a*d*f^3*x^3*e^(3*c) + 3*(a*f^3*e
^(3*c) + a*d*f^2*e^(3*c + 1))*x^2 + 3*a*f*e^(3*c + 2) + 3*(a*d*f*e^(3*c + 2) + 2*a*f^2*e^(3*c + 1))*x)*e^(3*d*
x) - 2*(b*d*f^3*x^3*e^(2*c) + 3*b*d*f^2*x^2*e^(2*c + 1) + 3*b*d*f*x*e^(2*c + 2))*e^(2*d*x) + (a*d*f^3*x^3*e^c
+ 3*(a*d*f^2*e^(c + 1) - a*f^3*e^c)*x^2 - 3*a*f*e^(c + 2) + 3*(a*d*f*e^(c + 2) - 2*a*f^2*e^(c + 1))*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) + 3*(b*d*f*e^2 + a*f^2*e)*x/(a^2*d^2) + 3*(b
*d*f*e^2 - a*f^2*e)*x/(a^2*d^2) - 3*(b*d*f*e^2 + a*f^2*e)*log(e^(d*x + c) + 1)/(a^2*d^3) - 3*(b*d*f*e^2 - a*f^
2*e)*log(e^(d*x + c) - 1)/(a^2*d^3) - 1/2*(d^3*x^3*log(e^(d*x + c) + 1) + 3*d^2*x^2*dilog(-e^(d*x + c)) - 6*d*
x*polylog(3, -e^(d*x + c)) + 6*polylog(4, -e^(d*x + c)))*(a^2*f^3 + 2*b^2*f^3)/(a^3*d^4) + 1/2*(d^3*x^3*log(-e
^(d*x + c) + 1) + 3*d^2*x^2*dilog(e^(d*x + c)) - 6*d*x*polylog(3, e^(d*x + c)) + 6*polylog(4, e^(d*x + c)))*(a
^2*f^3 + 2*b^2*f^3)/(a^3*d^4) - 3/2*(2*a*b*f^3 + (a^2*d*f^2 + 2*b^2*d*f^2)*e)*(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^3*d^4) - 3/2*(2*a*b*f^3 - (a^2*d*f^2 + 2*b^2*d*f^2)
*e)*(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^3*d^4) - 3/2*(4*
a*b*d*f^2*e + 2*a^2*f^3 + (a^2*d^2*f + 2*b^2*d^2*f)*e^2)*(d*x*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))/(a^3
*d^4) - 3/2*(4*a*b*d*f^2*e - 2*a^2*f^3 - (a^2*d^2*f + 2*b^2*d^2*f)*e^2)*(d*x*log(-e^(d*x + c) + 1) + dilog(e^(
d*x + c)))/(a^3*d^4) + 1/8*((a^2*f^3 + 2*b^2*f^3)*d^4*x^4 + 4*(2*a*b*f^3 + (a^2*d*f^2 + 2*b^2*d*f^2)*e)*d^3*x^
3 + 6*(4*a*b*d*f^2*e + 2*a^2*f^3 + (a^2*d^2*f + 2*b^2*d^2*f)*e^2)*d^2*x^2)/(a^3*d^4) - 1/8*((a^2*f^3 + 2*b^2*f
^3)*d^4*x^4 - 4*(2*a*b*f^3 - (a^2*d*f^2 + 2*b^2*d*f^2)*e)*d^3*x^3 - 6*(4*a*b*d*f^2*e - 2*a^2*f^3 - (a^2*d^2*f
+ 2*b^2*d^2*f)*e^2)*d^2*x^2)/(a^3*d^4) - integrate(2*((a^2*b*f^3*e^c + b^3*f^3*e^c)*x^3 + 3*(a^2*b*f^2*e^c + b
^3*f^2*e^c)*x^2*e + 3*(a^2*b*f*e^c + b^3*f*e^c)*x*e^2)*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. 24038 vs. \(2 (985) = 1970\).
time = 0.62, size = 24038, normalized size = 23.16 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^3*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^3*f^3 - 12*a*b*c^2*d*f^2*cosh(1) + 12*a*b*c*d^2*f*cosh(1)^2 - 4*a*b*d^3*cosh(1)^3 - 4*a*b*d^3*sin
h(1)^3 + 4*(a*b*d^3*f^3*x^3 + a*b*c^3*f^3 + 3*(a*b*d^3*f*x + a*b*c*d^2*f)*cosh(1)^2 + 3*(a*b*d^3*f*x + a*b*c*d
^2*f)*sinh(1)^2 + 3*(a*b*d^3*f^2*x^2 - a*b*c^2*d*f^2)*cosh(1) + 3*(a*b*d^3*f^2*x^2 - a*b*c^2*d*f^2 + 2*(a*b*d^
3*f*x + a*b*c*d^2*f)*cosh(1))*sinh(1))*cosh(d*x + c)^4 + 4*(a*b*d^3*f^3*x^3 + a*b*c^3*f^3 + 3*(a*b*d^3*f*x + a
*b*c*d^2*f)*cosh(1)^2 + 3*(a*b*d^3*f*x + a*b*c*d^2*f)*sinh(1)^2 + 3*(a*b*d^3*f^2*x^2 - a*b*c^2*d*f^2)*cosh(1)
+ 3*(a*b*d^3*f^2*x^2 - a*b*c^2*d*f^2 + 2*(a*b*d^3*f*x + a*b*c*d^2*f)*cosh(1))*sinh(1))*sinh(d*x + c)^4 - 2*(a^
2*d^3*f^3*x^3 + 3*a^2*d^2*f^3*x^2 + a^2*d^3*cosh(1)^3 + a^2*d^3*sinh(1)^3 + 3*(a^2*d^3*f*x + a^2*d^2*f)*cosh(1
)^2 + 3*(a^2*d^3*f*x + a^2*d^3*cosh(1) + a^2*d^2*f)*sinh(1)^2 + 3*(a^2*d^3*f^2*x^2 + 2*a^2*d^2*f^2*x)*cosh(1)
+ 3*(a^2*d^3*f^2*x^2 + 2*a^2*d^2*f^2*x + a^2*d^3*cosh(1)^2 + 2*(a^2*d^3*f*x + a^2*d^2*f)*cosh(1))*sinh(1))*cos
h(d*x + c)^3 - 2*(a^2*d^3*f^3*x^3 + 3*a^2*d^2*f^3*x^2 + a^2*d^3*cosh(1)^3 + a^2*d^3*sinh(1)^3 + 3*(a^2*d^3*f*x
+ a^2*d^2*f)*cosh(1)^2 + 3*(a^2*d^3*f*x + a^2*d^3*cosh(1) + a^2*d^2*f)*sinh(1)^2 + 3*(a^2*d^3*f^2*x^2 + 2*a^2
*d^2*f^2*x)*cosh(1) - 8*(a*b*d^3*f^3*x^3 + a*b*c^3*f^3 + 3*(a*b*d^3*f*x + a*b*c*d^2*f)*cosh(1)^2 + 3*(a*b*d^3*
f*x + a*b*c*d^2*f)*sinh(1)^2 + 3*(a*b*d^3*f^2*x^2 - a*b*c^2*d*f^2)*cosh(1) + 3*(a*b*d^3*f^2*x^2 - a*b*c^2*d*f^
2 + 2*(a*b*d^3*f*x + a*b*c*d^2*f)*cosh(1))*sinh(1))*cosh(d*x + c) + 3*(a^2*d^3*f^2*x^2 + 2*a^2*d^2*f^2*x + a^2
*d^3*cosh(1)^2 + 2*(a^2*d^3*f*x + a^2*d^2*f)*cosh(1))*sinh(1))*sinh(d*x + c)^3 - 4*(a*b*d^3*f^3*x^3 + 2*a*b*c^
3*f^3 - a*b*d^3*cosh(1)^3 - a*b*d^3*sinh(1)^3 + 3*(a*b*d^3*f*x + 2*a*b*c*d^2*f)*cosh(1)^2 + 3*(a*b*d^3*f*x + 2
*a*b*c*d^2*f - a*b*d^3*cosh(1))*sinh(1)^2 + 3*(a*b*d^3*f^2*x^2 - 2*a*b*c^2*d*f^2)*cosh(1) + 3*(a*b*d^3*f^2*x^2
- 2*a*b*c^2*d*f^2 - a*b*d^3*cosh(1)^2 + 2*(a*b*d^3*f*x + 2*a*b*c*d^2*f)*cosh(1))*sinh(1))*cosh(d*x + c)^2 + 1
2*(a*b*c*d^2*f - a*b*d^3*cosh(1))*sinh(1)^2 - 2*(2*a*b*d^3*f^3*x^3 + 4*a*b*c^3*f^3 - 2*a*b*d^3*cosh(1)^3 - 2*a
*b*d^3*sinh(1)^3 + 6*(a*b*d^3*f*x + 2*a*b*c*d^2*f)*cosh(1)^2 - 12*(a*b*d^3*f^3*x^3 + a*b*c^3*f^3 + 3*(a*b*d^3*
f*x + a*b*c*d^2*f)*cosh(1)^2 + 3*(a*b*d^3*f*x + a*b*c*d^2*f)*sinh(1)^2 + 3*(a*b*d^3*f^2*x^2 - a*b*c^2*d*f^2)*c
osh(1) + 3*(a*b*d^3*f^2*x^2 - a*b*c^2*d*f^2 + 2*(a*b*d^3*f*x + a*b*c*d^2*f)*cosh(1))*sinh(1))*cosh(d*x + c)^2
+ 6*(a*b*d^3*f*x + 2*a*b*c*d^2*f - a*b*d^3*cosh(1))*sinh(1)^2 + 6*(a*b*d^3*f^2*x^2 - 2*a*b*c^2*d*f^2)*cosh(1)
+ 3*(a^2*d^3*f^3*x^3 + 3*a^2*d^2*f^3*x^2 + a^2*d^3*cosh(1)^3 + a^2*d^3*sinh(1)^3 + 3*(a^2*d^3*f*x + a^2*d^2*f)
*cosh(1)^2 + 3*(a^2*d^3*f*x + a^2*d^3*cosh(1) + a^2*d^2*f)*sinh(1)^2 + 3*(a^2*d^3*f^2*x^2 + 2*a^2*d^2*f^2*x)*c
osh(1) + 3*(a^2*d^3*f^2*x^2 + 2*a^2*d^2*f^2*x + a^2*d^3*cosh(1)^2 + 2*(a^2*d^3*f*x + a^2*d^2*f)*cosh(1))*sinh(
1))*cosh(d*x + c) + 6*(a*b*d^3*f^2*x^2 - 2*a*b*c^2*d*f^2 - a*b*d^3*cosh(1)^2 + 2*(a*b*d^3*f*x + 2*a*b*c*d^2*f)
*cosh(1))*sinh(1))*sinh(d*x + c)^2 - 6*(b^2*d^2*f^3*x^2 + 2*b^2*d^2*f^2*x*cosh(1) + b^2*d^2*f*cosh(1)^2 + b^2*
d^2*f*sinh(1)^2 + (b^2*d^2*f^3*x^2 + 2*b^2*d^2*f^2*x*cosh(1) + b^2*d^2*f*cosh(1)^2 + b^2*d^2*f*sinh(1)^2 + 2*(
b^2*d^2*f^2*x + b^2*d^2*f*cosh(1))*sinh(1))*cosh(d*x + c)^4 + 4*(b^2*d^2*f^3*x^2 + 2*b^2*d^2*f^2*x*cosh(1) + b
^2*d^2*f*cosh(1)^2 + b^2*d^2*f*sinh(1)^2 + 2*(b^2*d^2*f^2*x + b^2*d^2*f*cosh(1))*sinh(1))*cosh(d*x + c)*sinh(d
*x + c)^3 + (b^2*d^2*f^3*x^2 + 2*b^2*d^2*f^2*x*cosh(1) + b^2*d^2*f*cosh(1)^2 + b^2*d^2*f*sinh(1)^2 + 2*(b^2*d^
2*f^2*x + b^2*d^2*f*cosh(1))*sinh(1))*sinh(d*x + c)^4 - 2*(b^2*d^2*f^3*x^2 + 2*b^2*d^2*f^2*x*cosh(1) + b^2*d^2
*f*cosh(1)^2 + b^2*d^2*f*sinh(1)^2 + 2*(b^2*d^2*f^2*x + b^2*d^2*f*cosh(1))*sinh(1))*cosh(d*x + c)^2 - 2*(b^2*d
^2*f^3*x^2 + 2*b^2*d^2*f^2*x*cosh(1) + b^2*d^2*f*cosh(1)^2 + b^2*d^2*f*sinh(1)^2 - 3*(b^2*d^2*f^3*x^2 + 2*b^2*
d^2*f^2*x*cosh(1) + b^2*d^2*f*cosh(1)^2 + b^2*d^2*f*sinh(1)^2 + 2*(b^2*d^2*f^2*x + b^2*d^2*f*cosh(1))*sinh(1))
*cosh(d*x + c)^2 + 2*(b^2*d^2*f^2*x + b^2*d^2*f*cosh(1))*sinh(1))*sinh(d*x + c)^2 + 2*(b^2*d^2*f^2*x + b^2*d^2
*f*cosh(1))*sinh(1) + 4*((b^2*d^2*f^3*x^2 + 2*b^2*d^2*f^2*x*cosh(1) + b^2*d^2*f*cosh(1)^2 + b^2*d^2*f*sinh(1)^
2 + 2*(b^2*d^2*f^2*x + b^2*d^2*f*cosh(1))*sinh(1))*cosh(d*x + c)^3 - (b^2*d^2*f^3*x^2 + 2*b^2*d^2*f^2*x*cosh(1
) + b^2*d^2*f*cosh(1)^2 + b^2*d^2*f*sinh(1)^2 + 2*(b^2*d^2*f^2*x + b^2*d^2*f*cosh(1))*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) + 6*(b^2*d^2*f^3*x^2 + 2*b^2*d^2*f^2*x*cosh(1) + b^2*d^2*f*cosh(1)^2 +
b^2*d^2*f*sinh(1)^2 + (b^2*d^2*f^3*x^2 + 2*b^2*d^2*f^2*x*cosh(1) + b^2*d^2*f*cosh(1)^2 + b^2*d^2*f*sinh(1)^2
+ 2*(b^2*d^2*f^2*x + b^2*d^2*f*cosh(1))*sinh(1))*cosh(d*x + c)^4 + 4*(b^2*d^2*f^3*x^2 + 2*b^2*d^2*f^2*x*cosh(1
) + b^2*d^2*f*cosh(1)^2 + b^2*d^2*f*sinh(1)^2 + 2*(b^2*d^2*f^2*x + b^2*d^2*f*cosh(1))*sinh(1))*cosh(d*x + c)*s
inh(d*x + c)^3 + (b^2*d^2*f^3*x^2 + 2*b^2*d^2*f...

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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)**3*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(-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)^3*coth(d*x+c)^2*csch(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

Timed out

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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 )}^3}{\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)^3)/(sinh(c + d*x)*(a + b*sinh(c + d*x))),x)

[Out]

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

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