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

Optimal. Leaf size=752 \[ \frac {3 b^2 f^3 \text {Li}_4\left (e^{2 (c+d x)}\right )}{4 a^3 d^4}-\frac {3 b^2 f^2 (e+f x) \text {Li}_3\left (e^{2 (c+d x)}\right )}{2 a^3 d^3}+\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (e^{2 (c+d x)}\right )}{2 a^3 d^2}+\frac {b^2 (e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a^3 d}-\frac {6 b f^3 \text {Li}_3\left (-e^{c+d x}\right )}{a^2 d^4}+\frac {6 b f^3 \text {Li}_3\left (e^{c+d x}\right )}{a^2 d^4}+\frac {6 b f^2 (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a^2 d^3}-\frac {6 b f^2 (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a^2 d^3}+\frac {6 b f (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d^2}+\frac {b (e+f x)^3 \text {csch}(c+d x)}{a^2 d}-\frac {6 b^2 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^4}-\frac {6 b^2 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^4}+\frac {6 b^2 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^3}+\frac {6 b^2 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^3}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^2}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^2}-\frac {b^2 (e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a^3 d}-\frac {b^2 (e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a^3 d}+\frac {3 f^3 \text {Li}_2\left (e^{2 (c+d x)}\right )}{2 a d^4}+\frac {3 f^2 (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^3}-\frac {3 f (e+f x)^2 \coth (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \text {csch}^2(c+d x)}{2 a d}-\frac {3 f (e+f x)^2}{2 a d^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.35, antiderivative size = 752, normalized size of antiderivative = 1.00, number of steps used = 34, number of rules used = 14, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {5587, 5452, 4184, 3716, 2190, 2279, 2391, 4182, 2531, 2282, 6589, 5569, 6609, 5561} \[ \frac {6 b^2 f^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^3}+\frac {6 b^2 f^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}\right )}{a^3 d^3}-\frac {3 b^2 f^2 (e+f x) \text {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a^3 d^3}-\frac {3 b^2 f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^2}-\frac {3 b^2 f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}\right )}{a^3 d^2}+\frac {3 b^2 f (e+f x)^2 \text {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a^3 d^2}-\frac {6 b^2 f^3 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^4}-\frac {6 b^2 f^3 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}\right )}{a^3 d^4}+\frac {3 b^2 f^3 \text {PolyLog}\left (4,e^{2 (c+d x)}\right )}{4 a^3 d^4}+\frac {6 b f^2 (e+f x) \text {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^3}-\frac {6 b f^2 (e+f x) \text {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^3}-\frac {6 b f^3 \text {PolyLog}\left (3,-e^{c+d x}\right )}{a^2 d^4}+\frac {6 b f^3 \text {PolyLog}\left (3,e^{c+d x}\right )}{a^2 d^4}+\frac {3 f^3 \text {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a d^4}-\frac {b^2 (e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a^3 d}-\frac {b^2 (e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a^3 d}+\frac {b^2 (e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a^3 d}+\frac {6 b f (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d^2}+\frac {b (e+f x)^3 \text {csch}(c+d x)}{a^2 d}+\frac {3 f^2 (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^3}-\frac {3 f (e+f x)^2 \coth (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \text {csch}^2(c+d x)}{2 a d}-\frac {3 f (e+f x)^2}{2 a d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*f*(e + f*x)^2)/(2*a*d^2) + (6*b*f*(e + f*x)^2*ArcTanh[E^(c + d*x)])/(a^2*d^2) - (3*f*(e + f*x)^2*Coth[c +
d*x])/(2*a*d^2) + (b*(e + f*x)^3*Csch[c + d*x])/(a^2*d) - ((e + f*x)^3*Csch[c + d*x]^2)/(2*a*d) - (b^2*(e + f*
x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^3*d) - (b^2*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + S
qrt[a^2 + b^2])])/(a^3*d) + (3*f^2*(e + f*x)*Log[1 - E^(2*(c + d*x))])/(a*d^3) + (b^2*(e + f*x)^3*Log[1 - E^(2
*(c + d*x))])/(a^3*d) + (6*b*f^2*(e + f*x)*PolyLog[2, -E^(c + d*x)])/(a^2*d^3) - (6*b*f^2*(e + f*x)*PolyLog[2,
 E^(c + d*x)])/(a^2*d^3) - (3*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^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*d^2) + (3*f^3*PolyLog[2, E
^(2*(c + d*x))])/(2*a*d^4) + (3*b^2*f*(e + f*x)^2*PolyLog[2, E^(2*(c + d*x))])/(2*a^3*d^2) - (6*b*f^3*PolyLog[
3, -E^(c + d*x)])/(a^2*d^4) + (6*b*f^3*PolyLog[3, E^(c + d*x)])/(a^2*d^4) + (6*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^2*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sq
rt[a^2 + b^2]))])/(a^3*d^3) - (3*b^2*f^2*(e + f*x)*PolyLog[3, E^(2*(c + d*x))])/(2*a^3*d^3) - (6*b^2*f^3*PolyL
og[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*d^4) - (6*b^2*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt
[a^2 + b^2]))])/(a^3*d^4) + (3*b^2*f^3*PolyLog[4, E^(2*(c + d*x))])/(4*a^3*d^4)

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 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 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 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 4184

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

Rule 5452

Int[Coth[(a_.) + (b_.)*(x_)]^(p_.)*Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Si
mp[((c + d*x)^m*Csch[a + b*x]^n)/(b*n), x] + Dist[(d*m)/(b*n), Int[(c + d*x)^(m - 1)*Csch[a + b*x]^n, x], x] /
; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

Rule 5561

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :
> -Simp[(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[((e + f*x)^m*E^(c + d*x))/(a - Rt[a^2 + b^2, 2] + b*E^(c +
d*x)), x] + Int[((e + f*x)^m*E^(c + d*x))/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x)), x]) /; FreeQ[{a, b, c, d, e,
 f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]

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

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Mathematica [B]  time = 69.32, size = 3043, normalized size = 4.05 \[ \text {Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(b*(e + f*x)^3*Csch[c])/(a^2*d) - ((e + f*x)^3*Csch[(c + d*x)/2]^2)/(8*a*d) - (8*b^2*d^4*e^3*E^(2*c)*x + 24*a^
2*d^2*e*E^(2*c)*f^2*x + 12*b^2*d^4*e^2*E^(2*c)*f*x^2 + 12*a^2*d^2*E^(2*c)*f^3*x^2 + 8*b^2*d^4*e*E^(2*c)*f^2*x^
3 + 2*b^2*d^4*E^(2*c)*f^3*x^4 + 24*a*b*d^2*e^2*f*ArcTanh[E^(c + d*x)] - 24*a*b*d^2*e^2*E^(2*c)*f*ArcTanh[E^(c
+ d*x)] - 24*a*b*d^2*e*f^2*x*Log[1 - E^(c + d*x)] + 24*a*b*d^2*e*E^(2*c)*f^2*x*Log[1 - E^(c + d*x)] - 12*a*b*d
^2*f^3*x^2*Log[1 - E^(c + d*x)] + 12*a*b*d^2*E^(2*c)*f^3*x^2*Log[1 - E^(c + d*x)] + 24*a*b*d^2*e*f^2*x*Log[1 +
 E^(c + d*x)] - 24*a*b*d^2*e*E^(2*c)*f^2*x*Log[1 + E^(c + d*x)] + 12*a*b*d^2*f^3*x^2*Log[1 + E^(c + d*x)] - 12
*a*b*d^2*E^(2*c)*f^3*x^2*Log[1 + E^(c + d*x)] + 4*b^2*d^3*e^3*Log[1 - E^(2*(c + d*x))] - 4*b^2*d^3*e^3*E^(2*c)
*Log[1 - E^(2*(c + d*x))] + 12*a^2*d*e*f^2*Log[1 - E^(2*(c + d*x))] - 12*a^2*d*e*E^(2*c)*f^2*Log[1 - E^(2*(c +
 d*x))] + 12*b^2*d^3*e^2*f*x*Log[1 - E^(2*(c + d*x))] - 12*b^2*d^3*e^2*E^(2*c)*f*x*Log[1 - E^(2*(c + d*x))] +
12*a^2*d*f^3*x*Log[1 - E^(2*(c + d*x))] - 12*a^2*d*E^(2*c)*f^3*x*Log[1 - E^(2*(c + d*x))] + 12*b^2*d^3*e*f^2*x
^2*Log[1 - E^(2*(c + d*x))] - 12*b^2*d^3*e*E^(2*c)*f^2*x^2*Log[1 - E^(2*(c + d*x))] + 4*b^2*d^3*f^3*x^3*Log[1
- E^(2*(c + d*x))] - 4*b^2*d^3*E^(2*c)*f^3*x^3*Log[1 - E^(2*(c + d*x))] - 24*a*b*d*(-1 + E^(2*c))*f^2*(e + f*x
)*PolyLog[2, -E^(c + d*x)] + 24*a*b*d*(-1 + E^(2*c))*f^2*(e + f*x)*PolyLog[2, E^(c + d*x)] + 6*b^2*d^2*e^2*f*P
olyLog[2, E^(2*(c + d*x))] - 6*b^2*d^2*e^2*E^(2*c)*f*PolyLog[2, E^(2*(c + d*x))] + 6*a^2*f^3*PolyLog[2, E^(2*(
c + d*x))] - 6*a^2*E^(2*c)*f^3*PolyLog[2, E^(2*(c + d*x))] + 12*b^2*d^2*e*f^2*x*PolyLog[2, E^(2*(c + d*x))] -
12*b^2*d^2*e*E^(2*c)*f^2*x*PolyLog[2, E^(2*(c + d*x))] + 6*b^2*d^2*f^3*x^2*PolyLog[2, E^(2*(c + d*x))] - 6*b^2
*d^2*E^(2*c)*f^3*x^2*PolyLog[2, E^(2*(c + d*x))] - 24*a*b*f^3*PolyLog[3, -E^(c + d*x)] + 24*a*b*E^(2*c)*f^3*Po
lyLog[3, -E^(c + d*x)] + 24*a*b*f^3*PolyLog[3, E^(c + d*x)] - 24*a*b*E^(2*c)*f^3*PolyLog[3, E^(c + d*x)] - 6*b
^2*d*e*f^2*PolyLog[3, E^(2*(c + d*x))] + 6*b^2*d*e*E^(2*c)*f^2*PolyLog[3, E^(2*(c + d*x))] - 6*b^2*d*f^3*x*Pol
yLog[3, E^(2*(c + d*x))] + 6*b^2*d*E^(2*c)*f^3*x*PolyLog[3, E^(2*(c + d*x))] + 3*b^2*f^3*PolyLog[4, E^(2*(c +
d*x))] - 3*b^2*E^(2*c)*f^3*PolyLog[4, E^(2*(c + d*x))])/(4*a^3*d^4*(-1 + E^(2*c))) + (b^2*(4*e^3*E^(2*c)*x + 6
*e^2*E^(2*c)*f*x^2 + 4*e*E^(2*c)*f^2*x^3 + E^(2*c)*f^3*x^4 + (4*a*Sqrt[-(a^2 + b^2)^2]*e^3*E^(2*c)*ArcTan[(a +
 b*E^(c + d*x))/Sqrt[-a^2 - b^2]])/((a^2 + b^2)^(3/2)*d) + (4*a*Sqrt[-(a^2 + b^2)^2]*e^3*E^(2*c)*ArcTanh[(a +
b*E^(c + d*x))/Sqrt[a^2 + b^2]])/((-a^2 - b^2)^(3/2)*d) + (2*e^3*Log[b - 2*a*E^(c + d*x) - b*E^(2*(c + d*x))])
/d - (2*e^3*E^(2*c)*Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))])/d + (6*e^2*f*x*Log[1 + (b*E^(2*c + d*x))/
(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*e^2*E^(2*c)*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^
2)*E^(2*c)])])/d + (6*e*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*e*E^(2*
c)*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (2*f^3*x^3*Log[1 + (b*E^(2*c +
d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (2*E^(2*c)*f^3*x^3*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^
2 + b^2)*E^(2*c)])])/d + (6*e^2*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*e^2
*E^(2*c)*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (6*e*f^2*x^2*Log[1 + (b*E^(2*
c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*e*E^(2*c)*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sq
rt[(a^2 + b^2)*E^(2*c)])])/d + (2*f^3*x^3*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d -
(2*E^(2*c)*f^3*x^3*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*(-1 + E^(2*c))*f*(e
+ f*x)^2*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^2 - (6*(-1 + E^(2*c))*f*(e +
f*x)^2*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^2 - (12*e*f^2*PolyLog[3, -((b*E
^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 + (12*e*E^(2*c)*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a
*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 - (12*f^3*x*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*
E^(2*c)]))])/d^3 + (12*E^(2*c)*f^3*x*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3
 - (12*e*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 + (12*e*E^(2*c)*f^2*Pol
yLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 - (12*f^3*x*PolyLog[3, -((b*E^(2*c + d*
x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 + (12*E^(2*c)*f^3*x*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqr
t[(a^2 + b^2)*E^(2*c)]))])/d^3 + (12*f^3*PolyLog[4, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])
/d^4 - (12*E^(2*c)*f^3*PolyLog[4, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^4 + (12*f^3*Pol
yLog[4, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^4 - (12*E^(2*c)*f^3*PolyLog[4, -((b*E^(2*
c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^4))/(2*a^3*(-1 + E^(2*c))) + ((e + f*x)^3*Sech[(c + d*x)/2]
^2)/(8*a*d) + ((e + f*x)^2*(3*a*f - 2*b*d*(e + f*x))*Csch[c/2]*Csch[(c + d*x)/2]*Sinh[(d*x)/2])/(4*a^2*d^2) -
((e + f*x)^2*(3*a*f + 2*b*d*(e + f*x))*Sech[c/2]*Sech[(c + d*x)/2]*Sinh[(d*x)/2])/(4*a^2*d^2)

________________________________________________________________________________________

fricas [C]  time = 1.19, size = 11595, normalized size = 15.42 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^3*coth(d*x+c)*csch(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

(3*a^2*d^2*e^2*f - 6*a^2*c*d*e*f^2 + 3*a^2*c^2*f^3 - 3*(a^2*d^2*f^3*x^2 + 2*a^2*d^2*e*f^2*x + 2*a^2*c*d*e*f^2
- a^2*c^2*f^3)*cosh(d*x + c)^4 - 3*(a^2*d^2*f^3*x^2 + 2*a^2*d^2*e*f^2*x + 2*a^2*c*d*e*f^2 - a^2*c^2*f^3)*sinh(
d*x + c)^4 + 2*(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)*cosh(d*x + c)^3 + 2*(
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^2*d^2*f^3*x^2 + 2*a^2*d^2*e*f^2
*x + 2*a^2*c*d*e*f^2 - a^2*c^2*f^3)*cosh(d*x + c))*sinh(d*x + c)^3 - (2*a^2*d^3*f^3*x^3 + 2*a^2*d^3*e^3 + 3*a^
2*d^2*e^2*f - 12*a^2*c*d*e*f^2 + 6*a^2*c^2*f^3 + 3*(2*a^2*d^3*e*f^2 - a^2*d^2*f^3)*x^2 + 6*(a^2*d^3*e^2*f - a^
2*d^2*e*f^2)*x)*cosh(d*x + c)^2 - (2*a^2*d^3*f^3*x^3 + 2*a^2*d^3*e^3 + 3*a^2*d^2*e^2*f - 12*a^2*c*d*e*f^2 + 6*
a^2*c^2*f^3 + 3*(2*a^2*d^3*e*f^2 - a^2*d^2*f^3)*x^2 + 18*(a^2*d^2*f^3*x^2 + 2*a^2*d^2*e*f^2*x + 2*a^2*c*d*e*f^
2 - a^2*c^2*f^3)*cosh(d*x + c)^2 + 6*(a^2*d^3*e^2*f - a^2*d^2*e*f^2)*x - 6*(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)*cosh(d*x + c))*sinh(d*x + c)^2 - 2*(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)*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 +
 (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)*sin
h(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))*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) - 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 + (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))*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) + 3*(b^2*d^2*f^3*x^2 + b^2*d
^2*e^2*f - 2*a*b*d*e*f^2 + a^2*f^3 + (b^2*d^2*f^3*x^2 + b^2*d^2*e^2*f - 2*a*b*d*e*f^2 + a^2*f^3 + 2*(b^2*d^2*e
*f^2 - a*b*d*f^3)*x)*cosh(d*x + c)^4 + 4*(b^2*d^2*f^3*x^2 + b^2*d^2*e^2*f - 2*a*b*d*e*f^2 + a^2*f^3 + 2*(b^2*d
^2*e*f^2 - a*b*d*f^3)*x)*cosh(d*x + c)*sinh(d*x + c)^3 + (b^2*d^2*f^3*x^2 + b^2*d^2*e^2*f - 2*a*b*d*e*f^2 + a^
2*f^3 + 2*(b^2*d^2*e*f^2 - a*b*d*f^3)*x)*sinh(d*x + c)^4 - 2*(b^2*d^2*f^3*x^2 + b^2*d^2*e^2*f - 2*a*b*d*e*f^2
+ a^2*f^3 + 2*(b^2*d^2*e*f^2 - a*b*d*f^3)*x)*cosh(d*x + c)^2 - 2*(b^2*d^2*f^3*x^2 + b^2*d^2*e^2*f - 2*a*b*d*e*
f^2 + a^2*f^3 - 3*(b^2*d^2*f^3*x^2 + b^2*d^2*e^2*f - 2*a*b*d*e*f^2 + a^2*f^3 + 2*(b^2*d^2*e*f^2 - a*b*d*f^3)*x
)*cosh(d*x + c)^2 + 2*(b^2*d^2*e*f^2 - a*b*d*f^3)*x)*sinh(d*x + c)^2 + 2*(b^2*d^2*e*f^2 - a*b*d*f^3)*x + 4*((b
^2*d^2*f^3*x^2 + b^2*d^2*e^2*f - 2*a*b*d*e*f^2 + a^2*f^3 + 2*(b^2*d^2*e*f^2 - a*b*d*f^3)*x)*cosh(d*x + c)^3 -
(b^2*d^2*f^3*x^2 + b^2*d^2*e^2*f - 2*a*b*d*e*f^2 + a^2*f^3 + 2*(b^2*d^2*e*f^2 - a*b*d*f^3)*x)*cosh(d*x + c))*s
inh(d*x + c))*dilog(cosh(d*x + c) + sinh(d*x + c)) + 3*(b^2*d^2*f^3*x^2 + b^2*d^2*e^2*f + 2*a*b*d*e*f^2 + a^2*
f^3 + (b^2*d^2*f^3*x^2 + b^2*d^2*e^2*f + 2*a*b*d*e*f^2 + a^2*f^3 + 2*(b^2*d^2*e*f^2 + a*b*d*f^3)*x)*cosh(d*x +
 c)^4 + 4*(b^2*d^2*f^3*x^2 + b^2*d^2*e^2*f + 2*a*b*d*e*f^2 + a^2*f^3 + 2*(b^2*d^2*e*f^2 + a*b*d*f^3)*x)*cosh(d
*x + c)*sinh(d*x + c)^3 + (b^2*d^2*f^3*x^2 + b^2*d^2*e^2*f + 2*a*b*d*e*f^2 + a^2*f^3 + 2*(b^2*d^2*e*f^2 + a*b*
d*f^3)*x)*sinh(d*x + c)^4 - 2*(b^2*d^2*f^3*x^2 + b^2*d^2*e^2*f + 2*a*b*d*e*f^2 + a^2*f^3 + 2*(b^2*d^2*e*f^2 +
a*b*d*f^3)*x)*cosh(d*x + c)^2 - 2*(b^2*d^2*f^3*x^2 + b^2*d^2*e^2*f + 2*a*b*d*e*f^2 + a^2*f^3 - 3*(b^2*d^2*f^3*
x^2 + b^2*d^2*e^2*f + 2*a*b*d*e*f^2 + a^2*f^3 + 2*(b^2*d^2*e*f^2 + a*b*d*f^3)*x)*cosh(d*x + c)^2 + 2*(b^2*d^2*
e*f^2 + a*b*d*f^3)*x)*sinh(d*x + c)^2 + 2*(b^2*d^2*e*f^2 + a*b*d*f^3)*x + 4*((b^2*d^2*f^3*x^2 + b^2*d^2*e^2*f
+ 2*a*b*d*e*f^2 + a^2*f^3 + 2*(b^2*d^2*e*f^2 + a*b*d*f^3)*x)*cosh(d*x + c)^3 - (b^2*d^2*f^3*x^2 + b^2*d^2*e^2*
f + 2*a*b*d*e*f^2 + a^2*f^3 + 2*(b^2*d^2*e*f^2 + a*b*d*f^3)*x)*cosh(d*x + c))*sinh(d*x + c))*dilog(-cosh(d*x +
 c) - sinh(d*x + c)) - (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))*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2
)/b^2) + 2*a) - (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))*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2)
+ 2*a) - (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))*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) - (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))*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) + (b^2*d^3*f^3*x^3 + b^2*d^3*e^3 + 3*a*b*d^2*e^2*f + 3*a^2*d*e*f^2 + (b^2*d^3*f^3*x^3 + b^2*d^3*e^3 + 3*a*b
*d^2*e^2*f + 3*a^2*d*e*f^2 + 3*(b^2*d^3*e*f^2 + a*b*d^2*f^3)*x^2 + 3*(b^2*d^3*e^2*f + 2*a*b*d^2*e*f^2 + a^2*d*
f^3)*x)*cosh(d*x + c)^4 + 4*(b^2*d^3*f^3*x^3 + b^2*d^3*e^3 + 3*a*b*d^2*e^2*f + 3*a^2*d*e*f^2 + 3*(b^2*d^3*e*f^
2 + a*b*d^2*f^3)*x^2 + 3*(b^2*d^3*e^2*f + 2*a*b*d^2*e*f^2 + a^2*d*f^3)*x)*cosh(d*x + c)*sinh(d*x + c)^3 + (b^2
*d^3*f^3*x^3 + b^2*d^3*e^3 + 3*a*b*d^2*e^2*f + 3*a^2*d*e*f^2 + 3*(b^2*d^3*e*f^2 + a*b*d^2*f^3)*x^2 + 3*(b^2*d^
3*e^2*f + 2*a*b*d^2*e*f^2 + a^2*d*f^3)*x)*sinh(d*x + c)^4 + 3*(b^2*d^3*e*f^2 + a*b*d^2*f^3)*x^2 - 2*(b^2*d^3*f
^3*x^3 + b^2*d^3*e^3 + 3*a*b*d^2*e^2*f + 3*a^2*d*e*f^2 + 3*(b^2*d^3*e*f^2 + a*b*d^2*f^3)*x^2 + 3*(b^2*d^3*e^2*
f + 2*a*b*d^2*e*f^2 + a^2*d*f^3)*x)*cosh(d*x + c)^2 - 2*(b^2*d^3*f^3*x^3 + b^2*d^3*e^3 + 3*a*b*d^2*e^2*f + 3*a
^2*d*e*f^2 + 3*(b^2*d^3*e*f^2 + a*b*d^2*f^3)*x^2 - 3*(b^2*d^3*f^3*x^3 + b^2*d^3*e^3 + 3*a*b*d^2*e^2*f + 3*a^2*
d*e*f^2 + 3*(b^2*d^3*e*f^2 + a*b*d^2*f^3)*x^2 + 3*(b^2*d^3*e^2*f + 2*a*b*d^2*e*f^2 + a^2*d*f^3)*x)*cosh(d*x +
c)^2 + 3*(b^2*d^3*e^2*f + 2*a*b*d^2*e*f^2 + a^2*d*f^3)*x)*sinh(d*x + c)^2 + 3*(b^2*d^3*e^2*f + 2*a*b*d^2*e*f^2
 + a^2*d*f^3)*x + 4*((b^2*d^3*f^3*x^3 + b^2*d^3*e^3 + 3*a*b*d^2*e^2*f + 3*a^2*d*e*f^2 + 3*(b^2*d^3*e*f^2 + a*b
*d^2*f^3)*x^2 + 3*(b^2*d^3*e^2*f + 2*a*b*d^2*e*f^2 + a^2*d*f^3)*x)*cosh(d*x + c)^3 - (b^2*d^3*f^3*x^3 + b^2*d^
3*e^3 + 3*a*b*d^2*e^2*f + 3*a^2*d*e*f^2 + 3*(b^2*d^3*e*f^2 + a*b*d^2*f^3)*x^2 + 3*(b^2*d^3*e^2*f + 2*a*b*d^2*e
*f^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) + (b^2*d^3*e^3 - 3*(
b^2*c + a*b)*d^2*e^2*f + 3*(b^2*c^2 + 2*a*b*c + a^2)*d*e*f^2 + (b^2*d^3*e^3 - 3*(b^2*c + a*b)*d^2*e^2*f + 3*(b
^2*c^2 + 2*a*b*c + a^2)*d*e*f^2 - (b^2*c^3 + 3*a*b*c^2 + 3*a^2*c)*f^3)*cosh(d*x + c)^4 + 4*(b^2*d^3*e^3 - 3*(b
^2*c + a*b)*d^2*e^2*f + 3*(b^2*c^2 + 2*a*b*c + a^2)*d*e*f^2 - (b^2*c^3 + 3*a*b*c^2 + 3*a^2*c)*f^3)*cosh(d*x +
c)*sinh(d*x + c)^3 + (b^2*d^3*e^3 - 3*(b^2*c + a*b)*d^2*e^2*f + 3*(b^2*c^2 + 2*a*b*c + a^2)*d*e*f^2 - (b^2*c^3
 + 3*a*b*c^2 + 3*a^2*c)*f^3)*sinh(d*x + c)^4 - (b^2*c^3 + 3*a*b*c^2 + 3*a^2*c)*f^3 - 2*(b^2*d^3*e^3 - 3*(b^2*c
 + a*b)*d^2*e^2*f + 3*(b^2*c^2 + 2*a*b*c + a^2)*d*e*f^2 - (b^2*c^3 + 3*a*b*c^2 + 3*a^2*c)*f^3)*cosh(d*x + c)^2
 - 2*(b^2*d^3*e^3 - 3*(b^2*c + a*b)*d^2*e^2*f + 3*(b^2*c^2 + 2*a*b*c + a^2)*d*e*f^2 - (b^2*c^3 + 3*a*b*c^2 + 3
*a^2*c)*f^3 - 3*(b^2*d^3*e^3 - 3*(b^2*c + a*b)*d^2*e^2*f + 3*(b^2*c^2 + 2*a*b*c + a^2)*d*e*f^2 - (b^2*c^3 + 3*
a*b*c^2 + 3*a^2*c)*f^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^2*d^3*e^3 - 3*(b^2*c + a*b)*d^2*e^2*f + 3*(b^
2*c^2 + 2*a*b*c + a^2)*d*e*f^2 - (b^2*c^3 + 3*a*b*c^2 + 3*a^2*c)*f^3)*cosh(d*x + c)^3 - (b^2*d^3*e^3 - 3*(b^2*
c + a*b)*d^2*e^2*f + 3*(b^2*c^2 + 2*a*b*c + a^2)*d*e*f^2 - (b^2*c^3 + 3*a*b*c^2 + 3*a^2*c)*f^3)*cosh(d*x + c))
*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - 1) + (b^2*d^3*f^3*x^3 + 3*b^2*c*d^2*e^2*f - 3*(b^2*c^2 + 2
*a*b*c)*d*e*f^2 + (b^2*d^3*f^3*x^3 + 3*b^2*c*d^2*e^2*f - 3*(b^2*c^2 + 2*a*b*c)*d*e*f^2 + (b^2*c^3 + 3*a*b*c^2
+ 3*a^2*c)*f^3 + 3*(b^2*d^3*e*f^2 - a*b*d^2*f^3)*x^2 + 3*(b^2*d^3*e^2*f - 2*a*b*d^2*e*f^2 + a^2*d*f^3)*x)*cosh
(d*x + c)^4 + 4*(b^2*d^3*f^3*x^3 + 3*b^2*c*d^2*e^2*f - 3*(b^2*c^2 + 2*a*b*c)*d*e*f^2 + (b^2*c^3 + 3*a*b*c^2 +
3*a^2*c)*f^3 + 3*(b^2*d^3*e*f^2 - a*b*d^2*f^3)*x^2 + 3*(b^2*d^3*e^2*f - 2*a*b*d^2*e*f^2 + a^2*d*f^3)*x)*cosh(d
*x + c)*sinh(d*x + c)^3 + (b^2*d^3*f^3*x^3 + 3*b^2*c*d^2*e^2*f - 3*(b^2*c^2 + 2*a*b*c)*d*e*f^2 + (b^2*c^3 + 3*
a*b*c^2 + 3*a^2*c)*f^3 + 3*(b^2*d^3*e*f^2 - a*b*d^2*f^3)*x^2 + 3*(b^2*d^3*e^2*f - 2*a*b*d^2*e*f^2 + a^2*d*f^3)
*x)*sinh(d*x + c)^4 + (b^2*c^3 + 3*a*b*c^2 + 3*a^2*c)*f^3 + 3*(b^2*d^3*e*f^2 - a*b*d^2*f^3)*x^2 - 2*(b^2*d^3*f
^3*x^3 + 3*b^2*c*d^2*e^2*f - 3*(b^2*c^2 + 2*a*b*c)*d*e*f^2 + (b^2*c^3 + 3*a*b*c^2 + 3*a^2*c)*f^3 + 3*(b^2*d^3*
e*f^2 - a*b*d^2*f^3)*x^2 + 3*(b^2*d^3*e^2*f - 2*a*b*d^2*e*f^2 + a^2*d*f^3)*x)*cosh(d*x + c)^2 - 2*(b^2*d^3*f^3
*x^3 + 3*b^2*c*d^2*e^2*f - 3*(b^2*c^2 + 2*a*b*c)*d*e*f^2 + (b^2*c^3 + 3*a*b*c^2 + 3*a^2*c)*f^3 + 3*(b^2*d^3*e*
f^2 - a*b*d^2*f^3)*x^2 - 3*(b^2*d^3*f^3*x^3 + 3*b^2*c*d^2*e^2*f - 3*(b^2*c^2 + 2*a*b*c)*d*e*f^2 + (b^2*c^3 + 3
*a*b*c^2 + 3*a^2*c)*f^3 + 3*(b^2*d^3*e*f^2 - a*b*d^2*f^3)*x^2 + 3*(b^2*d^3*e^2*f - 2*a*b*d^2*e*f^2 + a^2*d*f^3
)*x)*cosh(d*x + c)^2 + 3*(b^2*d^3*e^2*f - 2*a*b*d^2*e*f^2 + a^2*d*f^3)*x)*sinh(d*x + c)^2 + 3*(b^2*d^3*e^2*f -
 2*a*b*d^2*e*f^2 + a^2*d*f^3)*x + 4*((b^2*d^3*f^3*x^3 + 3*b^2*c*d^2*e^2*f - 3*(b^2*c^2 + 2*a*b*c)*d*e*f^2 + (b
^2*c^3 + 3*a*b*c^2 + 3*a^2*c)*f^3 + 3*(b^2*d^3*e*f^2 - a*b*d^2*f^3)*x^2 + 3*(b^2*d^3*e^2*f - 2*a*b*d^2*e*f^2 +
 a^2*d*f^3)*x)*cosh(d*x + c)^3 - (b^2*d^3*f^3*x^3 + 3*b^2*c*d^2*e^2*f - 3*(b^2*c^2 + 2*a*b*c)*d*e*f^2 + (b^2*c
^3 + 3*a*b*c^2 + 3*a^2*c)*f^3 + 3*(b^2*d^3*e*f^2 - a*b*d^2*f^3)*x^2 + 3*(b^2*d^3*e^2*f - 2*a*b*d^2*e*f^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*(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))*si
nh(d*x + c))*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) - 6*(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*c
osh(d*x + c)^3 - b^2*f^3*cosh(d*x + c))*sinh(d*x + c))*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) + 6*(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))*polylog(4, co
sh(d*x + c) + sinh(d*x + c)) + 6*(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))*polylog(4, -cosh(d*x + c) - sinh(d*x +
c)) + 6*(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)
)*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
) + 6*(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)*c
osh(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))*
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)
- 6*(b^2*d*f^3*x + b^2*d*e*f^2 - a*b*f^3 + (b^2*d*f^3*x + b^2*d*e*f^2 - a*b*f^3)*cosh(d*x + c)^4 + 4*(b^2*d*f^
3*x + b^2*d*e*f^2 - a*b*f^3)*cosh(d*x + c)*sinh(d*x + c)^3 + (b^2*d*f^3*x + b^2*d*e*f^2 - a*b*f^3)*sinh(d*x +
c)^4 - 2*(b^2*d*f^3*x + b^2*d*e*f^2 - a*b*f^3)*cosh(d*x + c)^2 - 2*(b^2*d*f^3*x + b^2*d*e*f^2 - a*b*f^3 - 3*(b
^2*d*f^3*x + b^2*d*e*f^2 - a*b*f^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^2*d*f^3*x + b^2*d*e*f^2 - a*b*f^3
)*cosh(d*x + c)^3 - (b^2*d*f^3*x + b^2*d*e*f^2 - a*b*f^3)*cosh(d*x + c))*sinh(d*x + c))*polylog(3, cosh(d*x +
c) + sinh(d*x + c)) - 6*(b^2*d*f^3*x + b^2*d*e*f^2 + a*b*f^3 + (b^2*d*f^3*x + b^2*d*e*f^2 + a*b*f^3)*cosh(d*x
+ c)^4 + 4*(b^2*d*f^3*x + b^2*d*e*f^2 + a*b*f^3)*cosh(d*x + c)*sinh(d*x + c)^3 + (b^2*d*f^3*x + b^2*d*e*f^2 +
a*b*f^3)*sinh(d*x + c)^4 - 2*(b^2*d*f^3*x + b^2*d*e*f^2 + a*b*f^3)*cosh(d*x + c)^2 - 2*(b^2*d*f^3*x + b^2*d*e*
f^2 + a*b*f^3 - 3*(b^2*d*f^3*x + b^2*d*e*f^2 + a*b*f^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^2*d*f^3*x + b
^2*d*e*f^2 + a*b*f^3)*cosh(d*x + c)^3 - (b^2*d*f^3*x + b^2*d*e*f^2 + a*b*f^3)*cosh(d*x + c))*sinh(d*x + c))*po
lylog(3, -cosh(d*x + c) - sinh(d*x + c)) - 2*(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^2*d^2*f^3*x^2 + 2*a^2*d^2*e*f^2*x + 2*a^2*c*d*e*f^2 - a^2*c^2*f^3)*cosh(d*x + c)^3 - 3*(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)*cosh(d*x + c)^2 + (2*a^2*d^3*f^3*x^3 + 2*a^2
*d^3*e^3 + 3*a^2*d^2*e^2*f - 12*a^2*c*d*e*f^2 + 6*a^2*c^2*f^3 + 3*(2*a^2*d^3*e*f^2 - a^2*d^2*f^3)*x^2 + 6*(a^2
*d^3*e^2*f - a^2*d^2*e*f^2)*x)*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)*csch(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 2.17, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x +e \right )^{3} \coth \left (d x +c \right ) \mathrm {csch}\left (d x +c \right )^{2}}{a +b \sinh \left (d x +c \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

-e^3*(2*(b*e^(-d*x - c) - a*e^(-2*d*x - 2*c) - b*e^(-3*d*x - 3*c))/((2*a^2*e^(-2*d*x - 2*c) - a^2*e^(-4*d*x -
4*c) - a^2)*d) + b^2*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(a^3*d) - b^2*log(e^(-d*x - c) + 1)/(a^3*
d) - b^2*log(e^(-d*x - c) - 1)/(a^3*d)) + (3*a*f^3*x^2 + 6*a*e*f^2*x + 3*a*e^2*f + 2*(b*d*f^3*x^3*e^(3*c) + 3*
b*d*e*f^2*x^2*e^(3*c) + 3*b*d*e^2*f*x*e^(3*c))*e^(3*d*x) - (2*a*d*f^3*x^3*e^(2*c) + 3*a*e^2*f*e^(2*c) + 3*(2*d
*e*f^2 + f^3)*a*x^2*e^(2*c) + 6*(d*e^2*f + e*f^2)*a*x*e^(2*c))*e^(2*d*x) - 2*(b*d*f^3*x^3*e^c + 3*b*d*e*f^2*x^
2*e^c + 3*b*d*e^2*f*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) + (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)))*b^2*f^3/(a^3*d^4) + (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)))*b^2*f^3/(a^3*d^4) - 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) + 3*(b^2*d*e*f^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*(b^2*d*e*f^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*(b^2*d^2*e^2*f + 2*a*b*d*e*f^2 + a^2*f^3)*(d*x*
log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))/(a^3*d^4) + 3*(b^2*d^2*e^2*f - 2*a*b*d*e*f^2 + a^2*f^3)*(d*x*log(-
e^(d*x + c) + 1) + dilog(e^(d*x + c)))/(a^3*d^4) - 1/4*(b^2*d^4*f^3*x^4 + 4*(b^2*d*e*f^2 + a*b*f^3)*d^3*x^3 +
6*(b^2*d^2*e^2*f + 2*a*b*d*e*f^2 + a^2*f^3)*d^2*x^2)/(a^3*d^4) - 1/4*(b^2*d^4*f^3*x^4 + 4*(b^2*d*e*f^2 - a*b*f
^3)*d^3*x^3 + 6*(b^2*d^2*e^2*f - 2*a*b*d*e*f^2 + a^2*f^3)*d^2*x^2)/(a^3*d^4) + integrate(-2*(b^3*f^3*x^3 + 3*b
^3*e*f^2*x^2 + 3*b^3*e^2*f*x - (a*b^2*f^3*x^3*e^c + 3*a*b^2*e*f^2*x^2*e^c + 3*a*b^2*e^2*f*x*e^c)*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 \[ \int \frac {\mathrm {coth}\left (c+d\,x\right )\,{\left (e+f\,x\right )}^3}{{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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