3.481 \(\int \frac {(e+f x)^3 \coth ^2(c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx\)
Optimal. Leaf size=1038 \[ -\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} \]
[Out]
-3*b*f*(f*x+e)^2*ln(1-exp(2*d*x+2*c))/a^2/d^2-2*b^2*(f*x+e)^3*arctanh(exp(d*x+c))/a^3/d+3/2*b*f^3*polylog(3,ex
p(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^4-3*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)*poly
log(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,exp
(d*x+c))/a^3/d^3-6*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)*csch(d*x+c)/a/d-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^3-3*f^2*(f*x+e)*polylog(3,exp(d*x+c))/a/d^3-3*f^3*polylog(2,-exp(d*
x+c))/a/d^4+3*f^3*polylog(2,exp(d*x+c))/a/d^4-(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-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*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)*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))*(a^2+
b^2)^(1/2)/a^3/d^3+b*(f*x+e)^3/a^2/d+b*(f*x+e)^3*coth(d*x+c)/a^2/d
________________________________________________________________________________________
Rubi [A] time = 2.24, 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
= {5587, 5457, 4182, 2531, 6609, 2282, 6589, 4186, 2279, 2391, 5569, 3720, 3716, 2190, 32, 5585,
5450, 3296, 2637, 5565, 3322, 2264} \[ -\frac {3 \text {PolyLog}\left (2,-e^{c+d x}\right ) f^3}{a d^4}+\frac {3 \text {PolyLog}\left (2,e^{c+d x}\right ) f^3}{a d^4}+\frac {3 b \text {PolyLog}\left (3,e^{2 (c+d x)}\right ) f^3}{2 a^2 d^4}-\frac {6 b^2 \text {PolyLog}\left (4,-e^{c+d x}\right ) f^3}{a^3 d^4}-\frac {3 \text {PolyLog}\left (4,-e^{c+d x}\right ) f^3}{a d^4}+\frac {6 b^2 \text {PolyLog}\left (4,e^{c+d x}\right ) f^3}{a^3 d^4}+\frac {3 \text {PolyLog}\left (4,e^{c+d x}\right ) f^3}{a d^4}-\frac {6 b \sqrt {a^2+b^2} \text {PolyLog}\left (4,-\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 {PolyLog}\left (4,-\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 {PolyLog}\left (2,e^{2 (c+d x)}\right ) f^2}{a^2 d^3}+\frac {6 b^2 (e+f x) \text {PolyLog}\left (3,-e^{c+d x}\right ) f^2}{a^3 d^3}+\frac {3 (e+f x) \text {PolyLog}\left (3,-e^{c+d x}\right ) f^2}{a d^3}-\frac {6 b^2 (e+f x) \text {PolyLog}\left (3,e^{c+d x}\right ) f^2}{a^3 d^3}-\frac {3 (e+f x) \text {PolyLog}\left (3,e^{c+d x}\right ) f^2}{a d^3}+\frac {6 b \sqrt {a^2+b^2} (e+f x) \text {PolyLog}\left (3,-\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 {PolyLog}\left (3,-\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 {PolyLog}\left (2,-e^{c+d x}\right ) f}{a^3 d^2}-\frac {3 (e+f x)^2 \text {PolyLog}\left (2,-e^{c+d x}\right ) f}{2 a d^2}+\frac {3 b^2 (e+f x)^2 \text {PolyLog}\left (2,e^{c+d x}\right ) f}{a^3 d^2}+\frac {3 (e+f x)^2 \text {PolyLog}\left (2,e^{c+d x}\right ) f}{2 a d^2}-\frac {3 b \sqrt {a^2+b^2} (e+f x)^2 \text {PolyLog}\left (2,-\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 {PolyLog}\left (2,-\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} \]
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 2190
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Rule 2264
Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]
Rule 2279
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 2282
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]
Rule 2391
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 2531
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]
Rule 2637
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Rule 3296
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Rule 3322
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Dist[2,
Int[((c + d*x)^m*E^(-(I*e) + f*fz*x))/(-(I*b) + 2*a*E^(-(I*e) + f*fz*x) + I*b*E^(2*(-(I*e) + f*fz*x))), x], x]
/; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
Rule 3716
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)))/(E^(2*I*k*Pi)*(1 + E^(2*
(-(I*e) + f*fz*x))/E^(2*I*k*Pi))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]
Rule 3720
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 4182
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 4186
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]) /; FreeQ[
{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]
Rule 5450
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 5457
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 5565
Int[(Cosh[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symb
ol] :> -Dist[a/b^2, Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2), x], x] + (Dist[1/b, Int[(e + f*x)^m*Cosh[c + d*x]^(
n - 2)*Sinh[c + d*x], x], x] + Dist[(a^2 + b^2)/b^2, Int[((e + f*x)^m*Cosh[c + d*x]^(n - 2))/(a + b*Sinh[c + d
*x]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0]
Rule 5569
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 5585
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 5587
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 6589
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]
Rule 6609
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 ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}+\frac {\left (3 f^3\right ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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] time = 42.75, size = 2383, normalized size = 2.30 \[ \text {Result too large to show} \]
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]
(e^3*Log[Tanh[(c + d*x)/2]])/(2*a*d) + (b^2*e^3*Log[Tanh[(c + d*x)/2]])/(a^3*d) + (3*e*f^2*Log[Tanh[(c + d*x)/
2]])/(a*d^3) + (3*e^2*f*(-(c*Log[Tanh[(c + d*x)/2]]) - I*((I*c + I*d*x)*(Log[1 - E^(I*(I*c + I*d*x))] - Log[1
+ E^(I*(I*c + I*d*x))]) + I*(PolyLog[2, -E^(I*(I*c + I*d*x))] - PolyLog[2, E^(I*(I*c + I*d*x))]))))/(2*a*d^2)
+ (3*b^2*e^2*f*(-(c*Log[Tanh[(c + d*x)/2]]) - I*((I*c + I*d*x)*(Log[1 - E^(I*(I*c + I*d*x))] - Log[1 + E^(I*(I
*c + I*d*x))]) + I*(PolyLog[2, -E^(I*(I*c + I*d*x))] - PolyLog[2, E^(I*(I*c + I*d*x))]))))/(a^3*d^2) + (3*f^3*
(-(c*Log[Tanh[(c + d*x)/2]]) - I*((I*c + I*d*x)*(Log[1 - E^(I*(I*c + I*d*x))] - Log[1 + E^(I*(I*c + I*d*x))])
+ I*(PolyLog[2, -E^(I*(I*c + I*d*x))] - PolyLog[2, E^(I*(I*c + I*d*x))]))))/(a*d^4) + (b*E^c*f^3*Csch[c]*((2*d
^3*x^3)/E^(2*c) - 3*d^2*(1 - E^(-2*c))*x^2*Log[1 - E^(-c - d*x)] - 3*d^2*(1 - E^(-2*c))*x^2*Log[1 + E^(-c - d*
x)] + 6*(1 - E^(-2*c))*(d*x*PolyLog[2, -E^(-c - d*x)] + PolyLog[3, -E^(-c - d*x)]) + 6*(1 - E^(-2*c))*(d*x*Pol
yLog[2, E^(-c - d*x)] + PolyLog[3, E^(-c - d*x)])))/(2*a^2*d^4) - (3*e*f^2*(d^2*x^2*ArcTanh[Cosh[c + d*x] + Si
nh[c + d*x]] + d*x*PolyLog[2, -Cosh[c + d*x] - Sinh[c + d*x]] - d*x*PolyLog[2, Cosh[c + d*x] + Sinh[c + d*x]]
- PolyLog[3, -Cosh[c + d*x] - Sinh[c + d*x]] + PolyLog[3, Cosh[c + d*x] + Sinh[c + d*x]]))/(a*d^3) - (6*b^2*e*
f^2*(d^2*x^2*ArcTanh[Cosh[c + d*x] + Sinh[c + d*x]] + d*x*PolyLog[2, -Cosh[c + d*x] - Sinh[c + d*x]] - d*x*Pol
yLog[2, Cosh[c + d*x] + Sinh[c + d*x]] - PolyLog[3, -Cosh[c + d*x] - Sinh[c + d*x]] + PolyLog[3, Cosh[c + d*x]
+ Sinh[c + d*x]]))/(a^3*d^3) + (b*Sqrt[a^2 + b^2]*(2*d^3*e^3*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] - 3
*d^3*e^2*f*x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - 3*d^3*e*f^2*x^2*Log[1 + (b*E^(c + d*x))/(a - Sqr
t[a^2 + b^2])] - d^3*f^3*x^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 3*d^3*e^2*f*x*Log[1 + (b*E^(c +
d*x))/(a + Sqrt[a^2 + b^2])] + 3*d^3*e*f^2*x^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + d^3*f^3*x^3*Lo
g[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - 3*d^2*f*(e + f*x)^2*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 +
b^2])] + 3*d^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] + 6*d*e*f^2*PolyLog[3, (b*E
^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 6*d*f^3*x*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 6*d*e*f^2
*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] - 6*d*f^3*x*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 +
b^2]))] - 6*f^3*PolyLog[4, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 6*f^3*PolyLog[4, -((b*E^(c + d*x))/(a +
Sqrt[a^2 + b^2]))]))/(a^3*d^4) + (f^3*(-2*d^3*x^3*ArcTanh[Cosh[c + d*x] + Sinh[c + d*x]] - 3*d^2*x^2*PolyLog[2
, -Cosh[c + d*x] - Sinh[c + d*x]] + 3*d^2*x^2*PolyLog[2, Cosh[c + d*x] + Sinh[c + d*x]] + 6*d*x*PolyLog[3, -Co
sh[c + d*x] - Sinh[c + d*x]] - 6*d*x*PolyLog[3, Cosh[c + d*x] + Sinh[c + d*x]] - 6*PolyLog[4, -Cosh[c + d*x] -
Sinh[c + d*x]] + 6*PolyLog[4, Cosh[c + d*x] + Sinh[c + d*x]]))/(2*a*d^4) + (b^2*f^3*(-2*d^3*x^3*ArcTanh[Cosh[
c + d*x] + Sinh[c + d*x]] - 3*d^2*x^2*PolyLog[2, -Cosh[c + d*x] - Sinh[c + d*x]] + 3*d^2*x^2*PolyLog[2, Cosh[c
+ d*x] + Sinh[c + d*x]] + 6*d*x*PolyLog[3, -Cosh[c + d*x] - Sinh[c + d*x]] - 6*d*x*PolyLog[3, Cosh[c + d*x] +
Sinh[c + d*x]] - 6*PolyLog[4, -Cosh[c + d*x] - Sinh[c + d*x]] + 6*PolyLog[4, Cosh[c + d*x] + Sinh[c + d*x]]))
/(a^3*d^4) + (3*b*e^2*f*Csch[c]*(-(d*x*Cosh[c]) + Log[Cosh[d*x]*Sinh[c] + Cosh[c]*Sinh[d*x]]*Sinh[c]))/(a^2*d^
2*(-Cosh[c]^2 + Sinh[c]^2)) + (Csch[c]*Csch[c + d*x]^2*(2*b*d*e^3*Cosh[c] + 6*b*d*e^2*f*x*Cosh[c] + 6*b*d*e*f^
2*x^2*Cosh[c] + 2*b*d*f^3*x^3*Cosh[c] + 3*a*e^2*f*Cosh[d*x] + 6*a*e*f^2*x*Cosh[d*x] + 3*a*f^3*x^2*Cosh[d*x] -
3*a*e^2*f*Cosh[2*c + d*x] - 6*a*e*f^2*x*Cosh[2*c + d*x] - 3*a*f^3*x^2*Cosh[2*c + d*x] - 2*b*d*e^3*Cosh[c + 2*d
*x] - 6*b*d*e^2*f*x*Cosh[c + 2*d*x] - 6*b*d*e*f^2*x^2*Cosh[c + 2*d*x] - 2*b*d*f^3*x^3*Cosh[c + 2*d*x] + a*d*e^
3*Sinh[d*x] + 3*a*d*e^2*f*x*Sinh[d*x] + 3*a*d*e*f^2*x^2*Sinh[d*x] + a*d*f^3*x^3*Sinh[d*x] - a*d*e^3*Sinh[2*c +
d*x] - 3*a*d*e^2*f*x*Sinh[2*c + d*x] - 3*a*d*e*f^2*x^2*Sinh[2*c + d*x] - a*d*f^3*x^3*Sinh[2*c + d*x]))/(4*a^2
*d^2) + (3*b*e*f^2*Csch[c]*Sech[c]*((d^2*x^2)/E^ArcTanh[Tanh[c]] - (I*(-(d*x*(-Pi + (2*I)*ArcTanh[Tanh[c]])) -
Pi*Log[1 + E^(2*d*x)] - 2*(I*d*x + I*ArcTanh[Tanh[c]])*Log[1 - E^((2*I)*(I*d*x + I*ArcTanh[Tanh[c]]))] + Pi*L
og[Cosh[d*x]] + (2*I)*ArcTanh[Tanh[c]]*Log[I*Sinh[d*x + ArcTanh[Tanh[c]]]] + I*PolyLog[2, E^((2*I)*(I*d*x + I*
ArcTanh[Tanh[c]]))])*Tanh[c])/Sqrt[1 - Tanh[c]^2]))/(a^2*d^3*Sqrt[Sech[c]^2*(Cosh[c]^2 - Sinh[c]^2)])
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fricas [C] time = 1.50, size = 13504, normalized size = 13.01 \[ \text {result too large to display} \]
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*d^3*e^3 - 12*a*b*c*d^2*e^2*f + 12*a*b*c^2*d*e*f^2 - 4*a*b*c^3*f^3 - 4*(a*b*d^3*f^3*x^3 + 3*a*b*d^3
*e*f^2*x^2 + 3*a*b*d^3*e^2*f*x + 3*a*b*c*d^2*e^2*f - 3*a*b*c^2*d*e*f^2 + a*b*c^3*f^3)*cosh(d*x + c)^4 - 4*(a*b
*d^3*f^3*x^3 + 3*a*b*d^3*e*f^2*x^2 + 3*a*b*d^3*e^2*f*x + 3*a*b*c*d^2*e^2*f - 3*a*b*c^2*d*e*f^2 + a*b*c^3*f^3)*
sinh(d*x + c)^4 + 2*(a^2*d^3*f^3*x^3 + a^2*d^3*e^3 + 3*a^2*d^2*e^2*f + 3*(a^2*d^3*e*f^2 + a^2*d^2*f^3)*x^2 + 3
*(a^2*d^3*e^2*f + 2*a^2*d^2*e*f^2)*x)*cosh(d*x + c)^3 + 2*(a^2*d^3*f^3*x^3 + a^2*d^3*e^3 + 3*a^2*d^2*e^2*f + 3
*(a^2*d^3*e*f^2 + a^2*d^2*f^3)*x^2 + 3*(a^2*d^3*e^2*f + 2*a^2*d^2*e*f^2)*x - 8*(a*b*d^3*f^3*x^3 + 3*a*b*d^3*e*
f^2*x^2 + 3*a*b*d^3*e^2*f*x + 3*a*b*c*d^2*e^2*f - 3*a*b*c^2*d*e*f^2 + a*b*c^3*f^3)*cosh(d*x + c))*sinh(d*x + c
)^3 + 4*(a*b*d^3*f^3*x^3 + 3*a*b*d^3*e*f^2*x^2 + 3*a*b*d^3*e^2*f*x - a*b*d^3*e^3 + 6*a*b*c*d^2*e^2*f - 6*a*b*c
^2*d*e*f^2 + 2*a*b*c^3*f^3)*cosh(d*x + c)^2 + 2*(2*a*b*d^3*f^3*x^3 + 6*a*b*d^3*e*f^2*x^2 + 6*a*b*d^3*e^2*f*x -
2*a*b*d^3*e^3 + 12*a*b*c*d^2*e^2*f - 12*a*b*c^2*d*e*f^2 + 4*a*b*c^3*f^3 - 12*(a*b*d^3*f^3*x^3 + 3*a*b*d^3*e*f
^2*x^2 + 3*a*b*d^3*e^2*f*x + 3*a*b*c*d^2*e^2*f - 3*a*b*c^2*d*e*f^2 + a*b*c^3*f^3)*cosh(d*x + c)^2 + 3*(a^2*d^3
*f^3*x^3 + a^2*d^3*e^3 + 3*a^2*d^2*e^2*f + 3*(a^2*d^3*e*f^2 + a^2*d^2*f^3)*x^2 + 3*(a^2*d^3*e^2*f + 2*a^2*d^2*
e*f^2)*x)*cosh(d*x + c))*sinh(d*x + c)^2 + 6*(b^2*d^2*f^3*x^2 + 2*b^2*d^2*e*f^2*x + b^2*d^2*e^2*f + (b^2*d^2*f
^3*x^2 + 2*b^2*d^2*e*f^2*x + b^2*d^2*e^2*f)*cosh(d*x + c)^4 + 4*(b^2*d^2*f^3*x^2 + 2*b^2*d^2*e*f^2*x + b^2*d^2
*e^2*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (b^2*d^2*f^3*x^2 + 2*b^2*d^2*e*f^2*x + b^2*d^2*e^2*f)*sinh(d*x + c)^4
- 2*(b^2*d^2*f^3*x^2 + 2*b^2*d^2*e*f^2*x + b^2*d^2*e^2*f)*cosh(d*x + c)^2 - 2*(b^2*d^2*f^3*x^2 + 2*b^2*d^2*e*f
^2*x + b^2*d^2*e^2*f - 3*(b^2*d^2*f^3*x^2 + 2*b^2*d^2*e*f^2*x + b^2*d^2*e^2*f)*cosh(d*x + c)^2)*sinh(d*x + c)^
2 + 4*((b^2*d^2*f^3*x^2 + 2*b^2*d^2*e*f^2*x + b^2*d^2*e^2*f)*cosh(d*x + c)^3 - (b^2*d^2*f^3*x^2 + 2*b^2*d^2*e*
f^2*x + b^2*d^2*e^2*f)*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
*e*f^2*x + b^2*d^2*e^2*f + (b^2*d^2*f^3*x^2 + 2*b^2*d^2*e*f^2*x + b^2*d^2*e^2*f)*cosh(d*x + c)^4 + 4*(b^2*d^2*
f^3*x^2 + 2*b^2*d^2*e*f^2*x + b^2*d^2*e^2*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (b^2*d^2*f^3*x^2 + 2*b^2*d^2*e*f^
2*x + b^2*d^2*e^2*f)*sinh(d*x + c)^4 - 2*(b^2*d^2*f^3*x^2 + 2*b^2*d^2*e*f^2*x + b^2*d^2*e^2*f)*cosh(d*x + c)^2
- 2*(b^2*d^2*f^3*x^2 + 2*b^2*d^2*e*f^2*x + b^2*d^2*e^2*f - 3*(b^2*d^2*f^3*x^2 + 2*b^2*d^2*e*f^2*x + b^2*d^2*e
^2*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^2*d^2*f^3*x^2 + 2*b^2*d^2*e*f^2*x + b^2*d^2*e^2*f)*cosh(d*x + c
)^3 - (b^2*d^2*f^3*x^2 + 2*b^2*d^2*e*f^2*x + b^2*d^2*e^2*f)*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) - 2*(b^2*d^3*e^3 - 3*b^2*c*d^2*e^2*f + 3*b^2*c^2*d*e*f^2 - b^2*c^3*f^3 + (b^2*d^3*e^3 - 3*b^2*c*d^2*e^2*f
+ 3*b^2*c^2*d*e*f^2 - b^2*c^3*f^3)*cosh(d*x + c)^4 + 4*(b^2*d^3*e^3 - 3*b^2*c*d^2*e^2*f + 3*b^2*c^2*d*e*f^2 -
b^2*c^3*f^3)*cosh(d*x + c)*sinh(d*x + c)^3 + (b^2*d^3*e^3 - 3*b^2*c*d^2*e^2*f + 3*b^2*c^2*d*e*f^2 - b^2*c^3*f
^3)*sinh(d*x + c)^4 - 2*(b^2*d^3*e^3 - 3*b^2*c*d^2*e^2*f + 3*b^2*c^2*d*e*f^2 - b^2*c^3*f^3)*cosh(d*x + c)^2 -
2*(b^2*d^3*e^3 - 3*b^2*c*d^2*e^2*f + 3*b^2*c^2*d*e*f^2 - b^2*c^3*f^3 - 3*(b^2*d^3*e^3 - 3*b^2*c*d^2*e^2*f + 3*
b^2*c^2*d*e*f^2 - b^2*c^3*f^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^2*d^3*e^3 - 3*b^2*c*d^2*e^2*f + 3*b^2*
c^2*d*e*f^2 - b^2*c^3*f^3)*cosh(d*x + c)^3 - (b^2*d^3*e^3 - 3*b^2*c*d^2*e^2*f + 3*b^2*c^2*d*e*f^2 - b^2*c^3*f^
3)*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*sqrt((a
^2 + b^2)/b^2) + 2*a) + 2*(b^2*d^3*e^3 - 3*b^2*c*d^2*e^2*f + 3*b^2*c^2*d*e*f^2 - b^2*c^3*f^3 + (b^2*d^3*e^3 -
3*b^2*c*d^2*e^2*f + 3*b^2*c^2*d*e*f^2 - b^2*c^3*f^3)*cosh(d*x + c)^4 + 4*(b^2*d^3*e^3 - 3*b^2*c*d^2*e^2*f + 3*
b^2*c^2*d*e*f^2 - b^2*c^3*f^3)*cosh(d*x + c)*sinh(d*x + c)^3 + (b^2*d^3*e^3 - 3*b^2*c*d^2*e^2*f + 3*b^2*c^2*d*
e*f^2 - b^2*c^3*f^3)*sinh(d*x + c)^4 - 2*(b^2*d^3*e^3 - 3*b^2*c*d^2*e^2*f + 3*b^2*c^2*d*e*f^2 - b^2*c^3*f^3)*c
osh(d*x + c)^2 - 2*(b^2*d^3*e^3 - 3*b^2*c*d^2*e^2*f + 3*b^2*c^2*d*e*f^2 - b^2*c^3*f^3 - 3*(b^2*d^3*e^3 - 3*b^2
*c*d^2*e^2*f + 3*b^2*c^2*d*e*f^2 - b^2*c^3*f^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^2*d^3*e^3 - 3*b^2*c*d
^2*e^2*f + 3*b^2*c^2*d*e*f^2 - b^2*c^3*f^3)*cosh(d*x + c)^3 - (b^2*d^3*e^3 - 3*b^2*c*d^2*e^2*f + 3*b^2*c^2*d*e
*f^2 - b^2*c^3*f^3)*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*sqrt((a^2 + b^2)/b^2) + 2*a) + 2*(b^2*d^3*f^3*x^3 + 3*b^2*d^3*e*f^2*x^2 + 3*b^2*d^3*e^2*f*x + 3*b^2*
c*d^2*e^2*f - 3*b^2*c^2*d*e*f^2 + b^2*c^3*f^3 + (b^2*d^3*f^3*x^3 + 3*b^2*d^3*e*f^2*x^2 + 3*b^2*d^3*e^2*f*x + 3
*b^2*c*d^2*e^2*f - 3*b^2*c^2*d*e*f^2 + b^2*c^3*f^3)*cosh(d*x + c)^4 + 4*(b^2*d^3*f^3*x^3 + 3*b^2*d^3*e*f^2*x^2
+ 3*b^2*d^3*e^2*f*x + 3*b^2*c*d^2*e^2*f - 3*b^2*c^2*d*e*f^2 + b^2*c^3*f^3)*cosh(d*x + c)*sinh(d*x + c)^3 + (b
^2*d^3*f^3*x^3 + 3*b^2*d^3*e*f^2*x^2 + 3*b^2*d^3*e^2*f*x + 3*b^2*c*d^2*e^2*f - 3*b^2*c^2*d*e*f^2 + b^2*c^3*f^3
)*sinh(d*x + c)^4 - 2*(b^2*d^3*f^3*x^3 + 3*b^2*d^3*e*f^2*x^2 + 3*b^2*d^3*e^2*f*x + 3*b^2*c*d^2*e^2*f - 3*b^2*c
^2*d*e*f^2 + b^2*c^3*f^3)*cosh(d*x + c)^2 - 2*(b^2*d^3*f^3*x^3 + 3*b^2*d^3*e*f^2*x^2 + 3*b^2*d^3*e^2*f*x + 3*b
^2*c*d^2*e^2*f - 3*b^2*c^2*d*e*f^2 + b^2*c^3*f^3 - 3*(b^2*d^3*f^3*x^3 + 3*b^2*d^3*e*f^2*x^2 + 3*b^2*d^3*e^2*f*
x + 3*b^2*c*d^2*e^2*f - 3*b^2*c^2*d*e*f^2 + b^2*c^3*f^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^2*d^3*f^3*x^
3 + 3*b^2*d^3*e*f^2*x^2 + 3*b^2*d^3*e^2*f*x + 3*b^2*c*d^2*e^2*f - 3*b^2*c^2*d*e*f^2 + b^2*c^3*f^3)*cosh(d*x +
c)^3 - (b^2*d^3*f^3*x^3 + 3*b^2*d^3*e*f^2*x^2 + 3*b^2*d^3*e^2*f*x + 3*b^2*c*d^2*e^2*f - 3*b^2*c^2*d*e*f^2 + b^
2*c^3*f^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2)*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) + (b*co
sh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) - 2*(b^2*d^3*f^3*x^3 + 3*b^2*d^3*e*f^2*x^2 + 3*b^
2*d^3*e^2*f*x + 3*b^2*c*d^2*e^2*f - 3*b^2*c^2*d*e*f^2 + b^2*c^3*f^3 + (b^2*d^3*f^3*x^3 + 3*b^2*d^3*e*f^2*x^2 +
3*b^2*d^3*e^2*f*x + 3*b^2*c*d^2*e^2*f - 3*b^2*c^2*d*e*f^2 + b^2*c^3*f^3)*cosh(d*x + c)^4 + 4*(b^2*d^3*f^3*x^3
+ 3*b^2*d^3*e*f^2*x^2 + 3*b^2*d^3*e^2*f*x + 3*b^2*c*d^2*e^2*f - 3*b^2*c^2*d*e*f^2 + b^2*c^3*f^3)*cosh(d*x + c
)*sinh(d*x + c)^3 + (b^2*d^3*f^3*x^3 + 3*b^2*d^3*e*f^2*x^2 + 3*b^2*d^3*e^2*f*x + 3*b^2*c*d^2*e^2*f - 3*b^2*c^2
*d*e*f^2 + b^2*c^3*f^3)*sinh(d*x + c)^4 - 2*(b^2*d^3*f^3*x^3 + 3*b^2*d^3*e*f^2*x^2 + 3*b^2*d^3*e^2*f*x + 3*b^2
*c*d^2*e^2*f - 3*b^2*c^2*d*e*f^2 + b^2*c^3*f^3)*cosh(d*x + c)^2 - 2*(b^2*d^3*f^3*x^3 + 3*b^2*d^3*e*f^2*x^2 + 3
*b^2*d^3*e^2*f*x + 3*b^2*c*d^2*e^2*f - 3*b^2*c^2*d*e*f^2 + b^2*c^3*f^3 - 3*(b^2*d^3*f^3*x^3 + 3*b^2*d^3*e*f^2*
x^2 + 3*b^2*d^3*e^2*f*x + 3*b^2*c*d^2*e^2*f - 3*b^2*c^2*d*e*f^2 + b^2*c^3*f^3)*cosh(d*x + c)^2)*sinh(d*x + c)^
2 + 4*((b^2*d^3*f^3*x^3 + 3*b^2*d^3*e*f^2*x^2 + 3*b^2*d^3*e^2*f*x + 3*b^2*c*d^2*e^2*f - 3*b^2*c^2*d*e*f^2 + b^
2*c^3*f^3)*cosh(d*x + c)^3 - (b^2*d^3*f^3*x^3 + 3*b^2*d^3*e*f^2*x^2 + 3*b^2*d^3*e^2*f*x + 3*b^2*c*d^2*e^2*f -
3*b^2*c^2*d*e*f^2 + b^2*c^3*f^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2)*log(-(a*cosh(d*x + c) + a
*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) + 12*(b^2*f^3*cosh(d*x + c)
^4 + 4*b^2*f^3*cosh(d*x + c)*sinh(d*x + c)^3 + b^2*f^3*sinh(d*x + c)^4 - 2*b^2*f^3*cosh(d*x + c)^2 + b^2*f^3 +
2*(3*b^2*f^3*cosh(d*x + c)^2 - b^2*f^3)*sinh(d*x + c)^2 + 4*(b^2*f^3*cosh(d*x + c)^3 - b^2*f^3*cosh(d*x + c))
*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2)*polylog(4, (a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sin
h(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) - 12*(b^2*f^3*cosh(d*x + c)^4 + 4*b^2*f^3*cosh(d*x + c)*sinh(d*x + c)^3
+ b^2*f^3*sinh(d*x + c)^4 - 2*b^2*f^3*cosh(d*x + c)^2 + b^2*f^3 + 2*(3*b^2*f^3*cosh(d*x + c)^2 - b^2*f^3)*sinh
(d*x + c)^2 + 4*(b^2*f^3*cosh(d*x + c)^3 - b^2*f^3*cosh(d*x + c))*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2)*polylog
(4, (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) - 12*(b
^2*d*f^3*x + b^2*d*e*f^2 + (b^2*d*f^3*x + b^2*d*e*f^2)*cosh(d*x + c)^4 + 4*(b^2*d*f^3*x + b^2*d*e*f^2)*cosh(d*
x + c)*sinh(d*x + c)^3 + (b^2*d*f^3*x + b^2*d*e*f^2)*sinh(d*x + c)^4 - 2*(b^2*d*f^3*x + b^2*d*e*f^2)*cosh(d*x
+ c)^2 - 2*(b^2*d*f^3*x + b^2*d*e*f^2 - 3*(b^2*d*f^3*x + b^2*d*e*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b
^2*d*f^3*x + b^2*d*e*f^2)*cosh(d*x + c)^3 - (b^2*d*f^3*x + b^2*d*e*f^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt((a^
2 + b^2)/b^2)*polylog(3, (a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 +
b^2)/b^2))/b) + 12*(b^2*d*f^3*x + b^2*d*e*f^2 + (b^2*d*f^3*x + b^2*d*e*f^2)*cosh(d*x + c)^4 + 4*(b^2*d*f^3*x +
b^2*d*e*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (b^2*d*f^3*x + b^2*d*e*f^2)*sinh(d*x + c)^4 - 2*(b^2*d*f^3*x + b
^2*d*e*f^2)*cosh(d*x + c)^2 - 2*(b^2*d*f^3*x + b^2*d*e*f^2 - 3*(b^2*d*f^3*x + b^2*d*e*f^2)*cosh(d*x + c)^2)*si
nh(d*x + c)^2 + 4*((b^2*d*f^3*x + b^2*d*e*f^2)*cosh(d*x + c)^3 - (b^2*d*f^3*x + b^2*d*e*f^2)*cosh(d*x + c))*si
nh(d*x + c))*sqrt((a^2 + b^2)/b^2)*polylog(3, (a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d
*x + c))*sqrt((a^2 + b^2)/b^2))/b) + 2*(a^2*d^3*f^3*x^3 + a^2*d^3*e^3 - 3*a^2*d^2*e^2*f + 3*(a^2*d^3*e*f^2 - a
^2*d^2*f^3)*x^2 + 3*(a^2*d^3*e^2*f - 2*a^2*d^2*e*f^2)*x)*cosh(d*x + c) - 3*((a^2 + 2*b^2)*d^2*f^3*x^2 + (a^2 +
2*b^2)*d^2*e^2*f - 4*a*b*d*e*f^2 + 2*a^2*f^3 + ((a^2 + 2*b^2)*d^2*f^3*x^2 + (a^2 + 2*b^2)*d^2*e^2*f - 4*a*b*d
*e*f^2 + 2*a^2*f^3 + 2*((a^2 + 2*b^2)*d^2*e*f^2 - 2*a*b*d*f^3)*x)*cosh(d*x + c)^4 + 4*((a^2 + 2*b^2)*d^2*f^3*x
^2 + (a^2 + 2*b^2)*d^2*e^2*f - 4*a*b*d*e*f^2 + 2*a^2*f^3 + 2*((a^2 + 2*b^2)*d^2*e*f^2 - 2*a*b*d*f^3)*x)*cosh(d
*x + c)*sinh(d*x + c)^3 + ((a^2 + 2*b^2)*d^2*f^3*x^2 + (a^2 + 2*b^2)*d^2*e^2*f - 4*a*b*d*e*f^2 + 2*a^2*f^3 + 2
*((a^2 + 2*b^2)*d^2*e*f^2 - 2*a*b*d*f^3)*x)*sinh(d*x + c)^4 - 2*((a^2 + 2*b^2)*d^2*f^3*x^2 + (a^2 + 2*b^2)*d^2
*e^2*f - 4*a*b*d*e*f^2 + 2*a^2*f^3 + 2*((a^2 + 2*b^2)*d^2*e*f^2 - 2*a*b*d*f^3)*x)*cosh(d*x + c)^2 - 2*((a^2 +
2*b^2)*d^2*f^3*x^2 + (a^2 + 2*b^2)*d^2*e^2*f - 4*a*b*d*e*f^2 + 2*a^2*f^3 - 3*((a^2 + 2*b^2)*d^2*f^3*x^2 + (a^2
+ 2*b^2)*d^2*e^2*f - 4*a*b*d*e*f^2 + 2*a^2*f^3 + 2*((a^2 + 2*b^2)*d^2*e*f^2 - 2*a*b*d*f^3)*x)*cosh(d*x + c)^2
+ 2*((a^2 + 2*b^2)*d^2*e*f^2 - 2*a*b*d*f^3)*x)*sinh(d*x + c)^2 + 2*((a^2 + 2*b^2)*d^2*e*f^2 - 2*a*b*d*f^3)*x
+ 4*(((a^2 + 2*b^2)*d^2*f^3*x^2 + (a^2 + 2*b^2)*d^2*e^2*f - 4*a*b*d*e*f^2 + 2*a^2*f^3 + 2*((a^2 + 2*b^2)*d^2*e
*f^2 - 2*a*b*d*f^3)*x)*cosh(d*x + c)^3 - ((a^2 + 2*b^2)*d^2*f^3*x^2 + (a^2 + 2*b^2)*d^2*e^2*f - 4*a*b*d*e*f^2
+ 2*a^2*f^3 + 2*((a^2 + 2*b^2)*d^2*e*f^2 - 2*a*b*d*f^3)*x)*cosh(d*x + c))*sinh(d*x + c))*dilog(cosh(d*x + c) +
sinh(d*x + c)) + 3*((a^2 + 2*b^2)*d^2*f^3*x^2 + (a^2 + 2*b^2)*d^2*e^2*f + 4*a*b*d*e*f^2 + 2*a^2*f^3 + ((a^2 +
2*b^2)*d^2*f^3*x^2 + (a^2 + 2*b^2)*d^2*e^2*f + 4*a*b*d*e*f^2 + 2*a^2*f^3 + 2*((a^2 + 2*b^2)*d^2*e*f^2 + 2*a*b
*d*f^3)*x)*cosh(d*x + c)^4 + 4*((a^2 + 2*b^2)*d^2*f^3*x^2 + (a^2 + 2*b^2)*d^2*e^2*f + 4*a*b*d*e*f^2 + 2*a^2*f^
3 + 2*((a^2 + 2*b^2)*d^2*e*f^2 + 2*a*b*d*f^3)*x)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^2 + 2*b^2)*d^2*f^3*x^2 +
(a^2 + 2*b^2)*d^2*e^2*f + 4*a*b*d*e*f^2 + 2*a^2*f^3 + 2*((a^2 + 2*b^2)*d^2*e*f^2 + 2*a*b*d*f^3)*x)*sinh(d*x +
c)^4 - 2*((a^2 + 2*b^2)*d^2*f^3*x^2 + (a^2 + 2*b^2)*d^2*e^2*f + 4*a*b*d*e*f^2 + 2*a^2*f^3 + 2*((a^2 + 2*b^2)*d
^2*e*f^2 + 2*a*b*d*f^3)*x)*cosh(d*x + c)^2 - 2*((a^2 + 2*b^2)*d^2*f^3*x^2 + (a^2 + 2*b^2)*d^2*e^2*f + 4*a*b*d*
e*f^2 + 2*a^2*f^3 - 3*((a^2 + 2*b^2)*d^2*f^3*x^2 + (a^2 + 2*b^2)*d^2*e^2*f + 4*a*b*d*e*f^2 + 2*a^2*f^3 + 2*((a
^2 + 2*b^2)*d^2*e*f^2 + 2*a*b*d*f^3)*x)*cosh(d*x + c)^2 + 2*((a^2 + 2*b^2)*d^2*e*f^2 + 2*a*b*d*f^3)*x)*sinh(d*
x + c)^2 + 2*((a^2 + 2*b^2)*d^2*e*f^2 + 2*a*b*d*f^3)*x + 4*(((a^2 + 2*b^2)*d^2*f^3*x^2 + (a^2 + 2*b^2)*d^2*e^2
*f + 4*a*b*d*e*f^2 + 2*a^2*f^3 + 2*((a^2 + 2*b^2)*d^2*e*f^2 + 2*a*b*d*f^3)*x)*cosh(d*x + c)^3 - ((a^2 + 2*b^2)
*d^2*f^3*x^2 + (a^2 + 2*b^2)*d^2*e^2*f + 4*a*b*d*e*f^2 + 2*a^2*f^3 + 2*((a^2 + 2*b^2)*d^2*e*f^2 + 2*a*b*d*f^3)
*x)*cosh(d*x + c))*sinh(d*x + c))*dilog(-cosh(d*x + c) - sinh(d*x + c)) + ((a^2 + 2*b^2)*d^3*f^3*x^3 + (a^2 +
2*b^2)*d^3*e^3 + 6*a*b*d^2*e^2*f + 6*a^2*d*e*f^2 + ((a^2 + 2*b^2)*d^3*f^3*x^3 + (a^2 + 2*b^2)*d^3*e^3 + 6*a*b*
d^2*e^2*f + 6*a^2*d*e*f^2 + 3*((a^2 + 2*b^2)*d^3*e*f^2 + 2*a*b*d^2*f^3)*x^2 + 3*((a^2 + 2*b^2)*d^3*e^2*f + 4*a
*b*d^2*e*f^2 + 2*a^2*d*f^3)*x)*cosh(d*x + c)^4 + 4*((a^2 + 2*b^2)*d^3*f^3*x^3 + (a^2 + 2*b^2)*d^3*e^3 + 6*a*b*
d^2*e^2*f + 6*a^2*d*e*f^2 + 3*((a^2 + 2*b^2)*d^3*e*f^2 + 2*a*b*d^2*f^3)*x^2 + 3*((a^2 + 2*b^2)*d^3*e^2*f + 4*a
*b*d^2*e*f^2 + 2*a^2*d*f^3)*x)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^2 + 2*b^2)*d^3*f^3*x^3 + (a^2 + 2*b^2)*d^3*
e^3 + 6*a*b*d^2*e^2*f + 6*a^2*d*e*f^2 + 3*((a^2 + 2*b^2)*d^3*e*f^2 + 2*a*b*d^2*f^3)*x^2 + 3*((a^2 + 2*b^2)*d^3
*e^2*f + 4*a*b*d^2*e*f^2 + 2*a^2*d*f^3)*x)*sinh(d*x + c)^4 + 3*((a^2 + 2*b^2)*d^3*e*f^2 + 2*a*b*d^2*f^3)*x^2 -
2*((a^2 + 2*b^2)*d^3*f^3*x^3 + (a^2 + 2*b^2)*d^3*e^3 + 6*a*b*d^2*e^2*f + 6*a^2*d*e*f^2 + 3*((a^2 + 2*b^2)*d^3
*e*f^2 + 2*a*b*d^2*f^3)*x^2 + 3*((a^2 + 2*b^2)*d^3*e^2*f + 4*a*b*d^2*e*f^2 + 2*a^2*d*f^3)*x)*cosh(d*x + c)^2 -
2*((a^2 + 2*b^2)*d^3*f^3*x^3 + (a^2 + 2*b^2)*d^3*e^3 + 6*a*b*d^2*e^2*f + 6*a^2*d*e*f^2 + 3*((a^2 + 2*b^2)*d^3
*e*f^2 + 2*a*b*d^2*f^3)*x^2 - 3*((a^2 + 2*b^2)*d^3*f^3*x^3 + (a^2 + 2*b^2)*d^3*e^3 + 6*a*b*d^2*e^2*f + 6*a^2*d
*e*f^2 + 3*((a^2 + 2*b^2)*d^3*e*f^2 + 2*a*b*d^2*f^3)*x^2 + 3*((a^2 + 2*b^2)*d^3*e^2*f + 4*a*b*d^2*e*f^2 + 2*a^
2*d*f^3)*x)*cosh(d*x + c)^2 + 3*((a^2 + 2*b^2)*d^3*e^2*f + 4*a*b*d^2*e*f^2 + 2*a^2*d*f^3)*x)*sinh(d*x + c)^2 +
3*((a^2 + 2*b^2)*d^3*e^2*f + 4*a*b*d^2*e*f^2 + 2*a^2*d*f^3)*x + 4*(((a^2 + 2*b^2)*d^3*f^3*x^3 + (a^2 + 2*b^2)
*d^3*e^3 + 6*a*b*d^2*e^2*f + 6*a^2*d*e*f^2 + 3*((a^2 + 2*b^2)*d^3*e*f^2 + 2*a*b*d^2*f^3)*x^2 + 3*((a^2 + 2*b^2
)*d^3*e^2*f + 4*a*b*d^2*e*f^2 + 2*a^2*d*f^3)*x)*cosh(d*x + c)^3 - ((a^2 + 2*b^2)*d^3*f^3*x^3 + (a^2 + 2*b^2)*d
^3*e^3 + 6*a*b*d^2*e^2*f + 6*a^2*d*e*f^2 + 3*((a^2 + 2*b^2)*d^3*e*f^2 + 2*a*b*d^2*f^3)*x^2 + 3*((a^2 + 2*b^2)*
d^3*e^2*f + 4*a*b*d^2*e*f^2 + 2*a^2*d*f^3)*x)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c)
+ 1) - ((a^2 + 2*b^2)*d^3*e^3 - 3*(2*a*b + (a^2 + 2*b^2)*c)*d^2*e^2*f + 3*(4*a*b*c + (a^2 + 2*b^2)*c^2 + 2*a^2
)*d*e*f^2 + ((a^2 + 2*b^2)*d^3*e^3 - 3*(2*a*b + (a^2 + 2*b^2)*c)*d^2*e^2*f + 3*(4*a*b*c + (a^2 + 2*b^2)*c^2 +
2*a^2)*d*e*f^2 - (6*a*b*c^2 + (a^2 + 2*b^2)*c^3 + 6*a^2*c)*f^3)*cosh(d*x + c)^4 + 4*((a^2 + 2*b^2)*d^3*e^3 - 3
*(2*a*b + (a^2 + 2*b^2)*c)*d^2*e^2*f + 3*(4*a*b*c + (a^2 + 2*b^2)*c^2 + 2*a^2)*d*e*f^2 - (6*a*b*c^2 + (a^2 + 2
*b^2)*c^3 + 6*a^2*c)*f^3)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^2 + 2*b^2)*d^3*e^3 - 3*(2*a*b + (a^2 + 2*b^2)*c)
*d^2*e^2*f + 3*(4*a*b*c + (a^2 + 2*b^2)*c^2 + 2*a^2)*d*e*f^2 - (6*a*b*c^2 + (a^2 + 2*b^2)*c^3 + 6*a^2*c)*f^3)*
sinh(d*x + c)^4 - (6*a*b*c^2 + (a^2 + 2*b^2)*c^3 + 6*a^2*c)*f^3 - 2*((a^2 + 2*b^2)*d^3*e^3 - 3*(2*a*b + (a^2 +
2*b^2)*c)*d^2*e^2*f + 3*(4*a*b*c + (a^2 + 2*b^2)*c^2 + 2*a^2)*d*e*f^2 - (6*a*b*c^2 + (a^2 + 2*b^2)*c^3 + 6*a^
2*c)*f^3)*cosh(d*x + c)^2 - 2*((a^2 + 2*b^2)*d^3*e^3 - 3*(2*a*b + (a^2 + 2*b^2)*c)*d^2*e^2*f + 3*(4*a*b*c + (a
^2 + 2*b^2)*c^2 + 2*a^2)*d*e*f^2 - (6*a*b*c^2 + (a^2 + 2*b^2)*c^3 + 6*a^2*c)*f^3 - 3*((a^2 + 2*b^2)*d^3*e^3 -
3*(2*a*b + (a^2 + 2*b^2)*c)*d^2*e^2*f + 3*(4*a*b*c + (a^2 + 2*b^2)*c^2 + 2*a^2)*d*e*f^2 - (6*a*b*c^2 + (a^2 +
2*b^2)*c^3 + 6*a^2*c)*f^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*(((a^2 + 2*b^2)*d^3*e^3 - 3*(2*a*b + (a^2 + 2*
b^2)*c)*d^2*e^2*f + 3*(4*a*b*c + (a^2 + 2*b^2)*c^2 + 2*a^2)*d*e*f^2 - (6*a*b*c^2 + (a^2 + 2*b^2)*c^3 + 6*a^2*c
)*f^3)*cosh(d*x + c)^3 - ((a^2 + 2*b^2)*d^3*e^3 - 3*(2*a*b + (a^2 + 2*b^2)*c)*d^2*e^2*f + 3*(4*a*b*c + (a^2 +
2*b^2)*c^2 + 2*a^2)*d*e*f^2 - (6*a*b*c^2 + (a^2 + 2*b^2)*c^3 + 6*a^2*c)*f^3)*cosh(d*x + c))*sinh(d*x + c))*log
(cosh(d*x + c) + sinh(d*x + c) - 1) - ((a^2 + 2*b^2)*d^3*f^3*x^3 + 3*(a^2 + 2*b^2)*c*d^2*e^2*f - 3*(4*a*b*c +
(a^2 + 2*b^2)*c^2)*d*e*f^2 + ((a^2 + 2*b^2)*d^3*f^3*x^3 + 3*(a^2 + 2*b^2)*c*d^2*e^2*f - 3*(4*a*b*c + (a^2 + 2*
b^2)*c^2)*d*e*f^2 + (6*a*b*c^2 + (a^2 + 2*b^2)*c^3 + 6*a^2*c)*f^3 + 3*((a^2 + 2*b^2)*d^3*e*f^2 - 2*a*b*d^2*f^3
)*x^2 + 3*((a^2 + 2*b^2)*d^3*e^2*f - 4*a*b*d^2*e*f^2 + 2*a^2*d*f^3)*x)*cosh(d*x + c)^4 + 4*((a^2 + 2*b^2)*d^3*
f^3*x^3 + 3*(a^2 + 2*b^2)*c*d^2*e^2*f - 3*(4*a*b*c + (a^2 + 2*b^2)*c^2)*d*e*f^2 + (6*a*b*c^2 + (a^2 + 2*b^2)*c
^3 + 6*a^2*c)*f^3 + 3*((a^2 + 2*b^2)*d^3*e*f^2 - 2*a*b*d^2*f^3)*x^2 + 3*((a^2 + 2*b^2)*d^3*e^2*f - 4*a*b*d^2*e
*f^2 + 2*a^2*d*f^3)*x)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^2 + 2*b^2)*d^3*f^3*x^3 + 3*(a^2 + 2*b^2)*c*d^2*e^2*
f - 3*(4*a*b*c + (a^2 + 2*b^2)*c^2)*d*e*f^2 + (6*a*b*c^2 + (a^2 + 2*b^2)*c^3 + 6*a^2*c)*f^3 + 3*((a^2 + 2*b^2)
*d^3*e*f^2 - 2*a*b*d^2*f^3)*x^2 + 3*((a^2 + 2*b^2)*d^3*e^2*f - 4*a*b*d^2*e*f^2 + 2*a^2*d*f^3)*x)*sinh(d*x + c)
^4 + (6*a*b*c^2 + (a^2 + 2*b^2)*c^3 + 6*a^2*c)*f^3 + 3*((a^2 + 2*b^2)*d^3*e*f^2 - 2*a*b*d^2*f^3)*x^2 - 2*((a^2
+ 2*b^2)*d^3*f^3*x^3 + 3*(a^2 + 2*b^2)*c*d^2*e^2*f - 3*(4*a*b*c + (a^2 + 2*b^2)*c^2)*d*e*f^2 + (6*a*b*c^2 + (
a^2 + 2*b^2)*c^3 + 6*a^2*c)*f^3 + 3*((a^2 + 2*b^2)*d^3*e*f^2 - 2*a*b*d^2*f^3)*x^2 + 3*((a^2 + 2*b^2)*d^3*e^2*f
- 4*a*b*d^2*e*f^2 + 2*a^2*d*f^3)*x)*cosh(d*x + c)^2 - 2*((a^2 + 2*b^2)*d^3*f^3*x^3 + 3*(a^2 + 2*b^2)*c*d^2*e^
2*f - 3*(4*a*b*c + (a^2 + 2*b^2)*c^2)*d*e*f^2 + (6*a*b*c^2 + (a^2 + 2*b^2)*c^3 + 6*a^2*c)*f^3 + 3*((a^2 + 2*b^
2)*d^3*e*f^2 - 2*a*b*d^2*f^3)*x^2 - 3*((a^2 + 2*b^2)*d^3*f^3*x^3 + 3*(a^2 + 2*b^2)*c*d^2*e^2*f - 3*(4*a*b*c +
(a^2 + 2*b^2)*c^2)*d*e*f^2 + (6*a*b*c^2 + (a^2 + 2*b^2)*c^3 + 6*a^2*c)*f^3 + 3*((a^2 + 2*b^2)*d^3*e*f^2 - 2*a*
b*d^2*f^3)*x^2 + 3*((a^2 + 2*b^2)*d^3*e^2*f - 4*a*b*d^2*e*f^2 + 2*a^2*d*f^3)*x)*cosh(d*x + c)^2 + 3*((a^2 + 2*
b^2)*d^3*e^2*f - 4*a*b*d^2*e*f^2 + 2*a^2*d*f^3)*x)*sinh(d*x + c)^2 + 3*((a^2 + 2*b^2)*d^3*e^2*f - 4*a*b*d^2*e*
f^2 + 2*a^2*d*f^3)*x + 4*(((a^2 + 2*b^2)*d^3*f^3*x^3 + 3*(a^2 + 2*b^2)*c*d^2*e^2*f - 3*(4*a*b*c + (a^2 + 2*b^2
)*c^2)*d*e*f^2 + (6*a*b*c^2 + (a^2 + 2*b^2)*c^3 + 6*a^2*c)*f^3 + 3*((a^2 + 2*b^2)*d^3*e*f^2 - 2*a*b*d^2*f^3)*x
^2 + 3*((a^2 + 2*b^2)*d^3*e^2*f - 4*a*b*d^2*e*f^2 + 2*a^2*d*f^3)*x)*cosh(d*x + c)^3 - ((a^2 + 2*b^2)*d^3*f^3*x
^3 + 3*(a^2 + 2*b^2)*c*d^2*e^2*f - 3*(4*a*b*c + (a^2 + 2*b^2)*c^2)*d*e*f^2 + (6*a*b*c^2 + (a^2 + 2*b^2)*c^3 +
6*a^2*c)*f^3 + 3*((a^2 + 2*b^2)*d^3*e*f^2 - 2*a*b*d^2*f^3)*x^2 + 3*((a^2 + 2*b^2)*d^3*e^2*f - 4*a*b*d^2*e*f^2
+ 2*a^2*d*f^3)*x)*cosh(d*x + c))*sinh(d*x + c))*log(-cosh(d*x + c) - sinh(d*x + c) + 1) - 6*((a^2 + 2*b^2)*f^3
*cosh(d*x + c)^4 + 4*(a^2 + 2*b^2)*f^3*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 + 2*b^2)*f^3*sinh(d*x + c)^4 - 2*(
a^2 + 2*b^2)*f^3*cosh(d*x + c)^2 + (a^2 + 2*b^2)*f^3 + 2*(3*(a^2 + 2*b^2)*f^3*cosh(d*x + c)^2 - (a^2 + 2*b^2)*
f^3)*sinh(d*x + c)^2 + 4*((a^2 + 2*b^2)*f^3*cosh(d*x + c)^3 - (a^2 + 2*b^2)*f^3*cosh(d*x + c))*sinh(d*x + c))*
polylog(4, cosh(d*x + c) + sinh(d*x + c)) + 6*((a^2 + 2*b^2)*f^3*cosh(d*x + c)^4 + 4*(a^2 + 2*b^2)*f^3*cosh(d*
x + c)*sinh(d*x + c)^3 + (a^2 + 2*b^2)*f^3*sinh(d*x + c)^4 - 2*(a^2 + 2*b^2)*f^3*cosh(d*x + c)^2 + (a^2 + 2*b^
2)*f^3 + 2*(3*(a^2 + 2*b^2)*f^3*cosh(d*x + c)^2 - (a^2 + 2*b^2)*f^3)*sinh(d*x + c)^2 + 4*((a^2 + 2*b^2)*f^3*co
sh(d*x + c)^3 - (a^2 + 2*b^2)*f^3*cosh(d*x + c))*sinh(d*x + c))*polylog(4, -cosh(d*x + c) - sinh(d*x + c)) + 6
*((a^2 + 2*b^2)*d*f^3*x + (a^2 + 2*b^2)*d*e*f^2 - 2*a*b*f^3 + ((a^2 + 2*b^2)*d*f^3*x + (a^2 + 2*b^2)*d*e*f^2 -
2*a*b*f^3)*cosh(d*x + c)^4 + 4*((a^2 + 2*b^2)*d*f^3*x + (a^2 + 2*b^2)*d*e*f^2 - 2*a*b*f^3)*cosh(d*x + c)*sinh
(d*x + c)^3 + ((a^2 + 2*b^2)*d*f^3*x + (a^2 + 2*b^2)*d*e*f^2 - 2*a*b*f^3)*sinh(d*x + c)^4 - 2*((a^2 + 2*b^2)*d
*f^3*x + (a^2 + 2*b^2)*d*e*f^2 - 2*a*b*f^3)*cosh(d*x + c)^2 - 2*((a^2 + 2*b^2)*d*f^3*x + (a^2 + 2*b^2)*d*e*f^2
- 2*a*b*f^3 - 3*((a^2 + 2*b^2)*d*f^3*x + (a^2 + 2*b^2)*d*e*f^2 - 2*a*b*f^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2
+ 4*(((a^2 + 2*b^2)*d*f^3*x + (a^2 + 2*b^2)*d*e*f^2 - 2*a*b*f^3)*cosh(d*x + c)^3 - ((a^2 + 2*b^2)*d*f^3*x + (a
^2 + 2*b^2)*d*e*f^2 - 2*a*b*f^3)*cosh(d*x + c))*sinh(d*x + c))*polylog(3, cosh(d*x + c) + sinh(d*x + c)) - 6*(
(a^2 + 2*b^2)*d*f^3*x + (a^2 + 2*b^2)*d*e*f^2 + 2*a*b*f^3 + ((a^2 + 2*b^2)*d*f^3*x + (a^2 + 2*b^2)*d*e*f^2 + 2
*a*b*f^3)*cosh(d*x + c)^4 + 4*((a^2 + 2*b^2)*d*f^3*x + (a^2 + 2*b^2)*d*e*f^2 + 2*a*b*f^3)*cosh(d*x + c)*sinh(d
*x + c)^3 + ((a^2 + 2*b^2)*d*f^3*x + (a^2 + 2*b^2)*d*e*f^2 + 2*a*b*f^3)*sinh(d*x + c)^4 - 2*((a^2 + 2*b^2)*d*f
^3*x + (a^2 + 2*b^2)*d*e*f^2 + 2*a*b*f^3)*cosh(d*x + c)^2 - 2*((a^2 + 2*b^2)*d*f^3*x + (a^2 + 2*b^2)*d*e*f^2 +
2*a*b*f^3 - 3*((a^2 + 2*b^2)*d*f^3*x + (a^2 + 2*b^2)*d*e*f^2 + 2*a*b*f^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 +
4*(((a^2 + 2*b^2)*d*f^3*x + (a^2 + 2*b^2)*d*e*f^2 + 2*a*b*f^3)*cosh(d*x + c)^3 - ((a^2 + 2*b^2)*d*f^3*x + (a^2
+ 2*b^2)*d*e*f^2 + 2*a*b*f^3)*cosh(d*x + c))*sinh(d*x + c))*polylog(3, -cosh(d*x + c) - sinh(d*x + c)) + 2*(a
^2*d^3*f^3*x^3 + a^2*d^3*e^3 - 3*a^2*d^2*e^2*f - 8*(a*b*d^3*f^3*x^3 + 3*a*b*d^3*e*f^2*x^2 + 3*a*b*d^3*e^2*f*x
+ 3*a*b*c*d^2*e^2*f - 3*a*b*c^2*d*e*f^2 + a*b*c^3*f^3)*cosh(d*x + c)^3 + 3*(a^2*d^3*e*f^2 - a^2*d^2*f^3)*x^2 +
3*(a^2*d^3*f^3*x^3 + a^2*d^3*e^3 + 3*a^2*d^2*e^2*f + 3*(a^2*d^3*e*f^2 + a^2*d^2*f^3)*x^2 + 3*(a^2*d^3*e^2*f +
2*a^2*d^2*e*f^2)*x)*cosh(d*x + c)^2 + 3*(a^2*d^3*e^2*f - 2*a^2*d^2*e*f^2)*x + 4*(a*b*d^3*f^3*x^3 + 3*a*b*d^3*
e*f^2*x^2 + 3*a*b*d^3*e^2*f*x - a*b*d^3*e^3 + 6*a*b*c*d^2*e^2*f - 6*a*b*c^2*d*e*f^2 + 2*a*b*c^3*f^3)*cosh(d*x
+ c))*sinh(d*x + c))/(a^3*d^4*cosh(d*x + c)^4 + 4*a^3*d^4*cosh(d*x + c)*sinh(d*x + c)^3 + a^3*d^4*sinh(d*x + c
)^4 - 2*a^3*d^4*cosh(d*x + c)^2 + a^3*d^4 + 2*(3*a^3*d^4*cosh(d*x + c)^2 - a^3*d^4)*sinh(d*x + c)^2 + 4*(a^3*d
^4*cosh(d*x + c)^3 - a^3*d^4*cosh(d*x + c))*sinh(d*x + c))
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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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maple [F] time = 2.50, 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 \[ \text {result too large to display} \]
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*e^3*(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)) - (2*b*d*f^3*x^3 + 6*b*d*e*f^2*x^2 + 6*b*d*e^2*f*x + (a*d*f^3*x^3*e^(3*c) + 3*a*e^2*f*
e^(3*c) + 3*(d*e*f^2 + f^3)*a*x^2*e^(3*c) + 3*(d*e^2*f + 2*e*f^2)*a*x*e^(3*c))*e^(3*d*x) - 2*(b*d*f^3*x^3*e^(2
*c) + 3*b*d*e*f^2*x^2*e^(2*c) + 3*b*d*e^2*f*x*e^(2*c))*e^(2*d*x) + (a*d*f^3*x^3*e^c - 3*a*e^2*f*e^c + 3*(d*e*f
^2 - f^3)*a*x^2*e^c + 3*(d*e^2*f - 2*e*f^2)*a*x*e^c)*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*e^2*f + a*e*f^2)*x/(a^2*d^2) + 3*(b*d*e^2*f - a*e*f^2)*x/(a^2*d^2) - 3*(b*d*e^2*f + a
*e*f^2)*log(e^(d*x + c) + 1)/(a^2*d^3) - 3*(b*d*e^2*f - a*e*f^2)*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*(a^2*d*e*f^2 +
2*b^2*d*e*f^2 + 2*a*b*f^3)*(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*(a^2*d*e*f^2 + 2*b^2*d*e*f^2 - 2*a*b*f^3)*(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*b^2*d^2*e^2*f + 4*a*b*d*e*f^2 + (d^2*e^2*f + 2*f^3
)*a^2)*(d*x*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))/(a^3*d^4) + 3/2*(2*b^2*d^2*e^2*f - 4*a*b*d*e*f^2 + (d^
2*e^2*f + 2*f^3)*a^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*(a^2*d*e*f^2 + 2*b^2*d*e*f^2 + 2*a*b*f^3)*d^3*x^3 + 6*(2*b^2*d^2*e^2*f + 4*a*b*d*e*f^2 + (d^2*e^2*
f + 2*f^3)*a^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*(a^2*d*e*f^2 + 2*b^2*d*e*f^2 - 2*a
*b*f^3)*d^3*x^3 + 6*(2*b^2*d^2*e^2*f - 4*a*b*d*e*f^2 + (d^2*e^2*f + 2*f^3)*a^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*e*f^2*e^c + b^3*e*f^2*e^c)*x^2 + 3*(a^2*b*e^2*f*e^c + b^3*e^2
*f*e^c)*x)*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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {coth}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^3}{\mathrm {sinh}\left (c+d\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]
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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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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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