\(\int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx\) [22]

Optimal result

Integrand size = 28, antiderivative size = 696 \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\frac {3 e f^2 x}{4 a d^2}+\frac {3 f^3 x^2}{8 a d^2}+\frac {(e+f x)^4}{8 a f}+\frac {b^2 (e+f x)^4}{4 a^3 f}-\frac {6 b f^2 (e+f x) \cosh (c+d x)}{a^2 d^3}-\frac {b (e+f x)^3 \cosh (c+d x)}{a^2 d}-\frac {3 f^3 \cosh ^2(c+d x)}{8 a d^4}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 a d^2}-\frac {b \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {b \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {3 b \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {3 b \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {6 b \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d^3}-\frac {6 b \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d^3}-\frac {6 b \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d^4}+\frac {6 b \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d^4}+\frac {6 b f^3 \sinh (c+d x)}{a^2 d^4}+\frac {3 b f (e+f x)^2 \sinh (c+d x)}{a^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 a d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d} \]

[Out]

3/4*e*f^2*x/a/d^2+3/8*f^3*x^2/a/d^2+1/8*(f*x+e)^4/a/f+1/4*b^2*(f*x+e)^4/a^3/f-6*b*f^2*(f*x+e)*cosh(d*x+c)/a^2/
d^3-b*(f*x+e)^3*cosh(d*x+c)/a^2/d-3/8*f^3*cosh(d*x+c)^2/a/d^4-3/4*f*(f*x+e)^2*cosh(d*x+c)^2/a/d^2+6*b*f^3*sinh
(d*x+c)/a^2/d^4+3*b*f*(f*x+e)^2*sinh(d*x+c)/a^2/d^2+3/4*f^2*(f*x+e)*cosh(d*x+c)*sinh(d*x+c)/a/d^3+1/2*(f*x+e)^
3*cosh(d*x+c)*sinh(d*x+c)/a/d-b*(f*x+e)^3*ln(1+a*exp(d*x+c)/(b-(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/a^3/d+b*(f*x+
e)^3*ln(1+a*exp(d*x+c)/(b+(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/a^3/d-3*b*f*(f*x+e)^2*polylog(2,-a*exp(d*x+c)/(b-(
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,-a*exp(d*x+c)/(b+(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,-a*exp(d*x+c)/(b-(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,-a*exp(d*x+c)/(b+(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/a^3/d^3-6*b*f^3*polylog(4,-a*exp(d*x+c)/(b
-(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/a^3/d^4+6*b*f^3*polylog(4,-a*exp(d*x+c)/(b+(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2
)/a^3/d^4

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 696, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.536, Rules used = {5713, 5698, 3392, 32, 3391, 5684, 3377, 2717, 3403, 2296, 2221, 2611, 6744, 2320, 6724} \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\frac {b^2 (e+f x)^4}{4 a^3 f}+\frac {6 b f^3 \sinh (c+d x)}{a^2 d^4}-\frac {6 b f^2 (e+f x) \cosh (c+d x)}{a^2 d^3}+\frac {3 b f (e+f x)^2 \sinh (c+d x)}{a^2 d^2}-\frac {b (e+f x)^3 \cosh (c+d x)}{a^2 d}-\frac {6 b f^3 \sqrt {a^2+b^2} \operatorname {PolyLog}\left (4,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d^4}+\frac {6 b f^3 \sqrt {a^2+b^2} \operatorname {PolyLog}\left (4,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d^4}+\frac {6 b f^2 \sqrt {a^2+b^2} (e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d^3}-\frac {6 b f^2 \sqrt {a^2+b^2} (e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d^3}-\frac {3 b f \sqrt {a^2+b^2} (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {3 b f \sqrt {a^2+b^2} (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d^2}-\frac {b \sqrt {a^2+b^2} (e+f x)^3 \log \left (\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}+1\right )}{a^3 d}+\frac {b \sqrt {a^2+b^2} (e+f x)^3 \log \left (\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}+1\right )}{a^3 d}-\frac {3 f^3 \cosh ^2(c+d x)}{8 a d^4}+\frac {3 f^2 (e+f x) \sinh (c+d x) \cosh (c+d x)}{4 a d^3}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 a d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 a d}+\frac {3 e f^2 x}{4 a d^2}+\frac {3 f^3 x^2}{8 a d^2}+\frac {(e+f x)^4}{8 a f} \]

[In]

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

[Out]

(3*e*f^2*x)/(4*a*d^2) + (3*f^3*x^2)/(8*a*d^2) + (e + f*x)^4/(8*a*f) + (b^2*(e + f*x)^4)/(4*a^3*f) - (6*b*f^2*(
e + f*x)*Cosh[c + d*x])/(a^2*d^3) - (b*(e + f*x)^3*Cosh[c + d*x])/(a^2*d) - (3*f^3*Cosh[c + d*x]^2)/(8*a*d^4)
- (3*f*(e + f*x)^2*Cosh[c + d*x]^2)/(4*a*d^2) - (b*Sqrt[a^2 + b^2]*(e + f*x)^3*Log[1 + (a*E^(c + d*x))/(b - Sq
rt[a^2 + b^2])])/(a^3*d) + (b*Sqrt[a^2 + b^2]*(e + f*x)^3*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])])/(a^3
*d) - (3*b*Sqrt[a^2 + b^2]*f*(e + f*x)^2*PolyLog[2, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a^3*d^2) + (3*
b*Sqrt[a^2 + b^2]*f*(e + f*x)^2*PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))])/(a^3*d^2) + (6*b*Sqrt[a^
2 + b^2]*f^2*(e + f*x)*PolyLog[3, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a^3*d^3) - (6*b*Sqrt[a^2 + b^2]*
f^2*(e + f*x)*PolyLog[3, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))])/(a^3*d^3) - (6*b*Sqrt[a^2 + b^2]*f^3*PolyL
og[4, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a^3*d^4) + (6*b*Sqrt[a^2 + b^2]*f^3*PolyLog[4, -((a*E^(c + d
*x))/(b + Sqrt[a^2 + b^2]))])/(a^3*d^4) + (6*b*f^3*Sinh[c + d*x])/(a^2*d^4) + (3*b*f*(e + f*x)^2*Sinh[c + d*x]
)/(a^2*d^2) + (3*f^2*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(4*a*d^3) + ((e + f*x)^3*Cosh[c + d*x]*Sinh[c + d*
x])/(2*a*d)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2221

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

Rule 2296

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[2*(c/q), Int[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Dist[2*(c/q), Int[(f + g
*x)^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

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

Rule 3391

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

Rule 3392

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

Rule 3403

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Dist[2,
Int[(c + d*x)^m*(E^((-I)*e + f*fz*x)/((-I)*b + 2*a*E^((-I)*e + f*fz*x) + I*b*E^(2*((-I)*e + f*fz*x)))), x], x]
/; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 5684

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

Rule 5698

Int[(Cosh[(c_.) + (d_.)*(x_)]^(p_.)*((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/b, Int[(e + f*x)^m*Cosh[c + d*x]^p*Sinh[c + d*x]^(n - 1), x], x]
- Dist[a/b, Int[(e + f*x)^m*Cosh[c + d*x]^p*(Sinh[c + d*x]^(n - 1)/(a + b*Sinh[c + d*x])), x], x] /; FreeQ[{a
, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]

Rule 5713

Int[(((e_.) + (f_.)*(x_))^(m_.)*(F_)[(c_.) + (d_.)*(x_)]^(n_.))/(Csch[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Sym
bol] :> Int[(e + f*x)^m*Sinh[c + d*x]*(F[c + d*x]^n/(b + a*Sinh[c + d*x])), x] /; FreeQ[{a, b, c, d, e, f}, x]
&& HyperbolicQ[F] && IntegersQ[m, n]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 6744

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{b+a \sinh (c+d x)} \, dx \\ & = \frac {\int (e+f x)^3 \cosh ^2(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{b+a \sinh (c+d x)} \, dx}{a} \\ & = -\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 a d^2}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {\int (e+f x)^3 \, dx}{2 a}-\frac {b \int (e+f x)^3 \sinh (c+d x) \, dx}{a^2}+\frac {b^2 \int (e+f x)^3 \, dx}{a^3}-\frac {\left (b \left (a^2+b^2\right )\right ) \int \frac {(e+f x)^3}{b+a \sinh (c+d x)} \, dx}{a^3}+\frac {\left (3 f^2\right ) \int (e+f x) \cosh ^2(c+d x) \, dx}{2 a d^2} \\ & = \frac {(e+f x)^4}{8 a f}+\frac {b^2 (e+f x)^4}{4 a^3 f}-\frac {b (e+f x)^3 \cosh (c+d x)}{a^2 d}-\frac {3 f^3 \cosh ^2(c+d x)}{8 a d^4}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 a d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 a d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d}-\frac {\left (2 b \left (a^2+b^2\right )\right ) \int \frac {e^{c+d x} (e+f x)^3}{-a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx}{a^3}+\frac {(3 b f) \int (e+f x)^2 \cosh (c+d x) \, dx}{a^2 d}+\frac {\left (3 f^2\right ) \int (e+f x) \, dx}{4 a d^2} \\ & = \frac {3 e f^2 x}{4 a d^2}+\frac {3 f^3 x^2}{8 a d^2}+\frac {(e+f x)^4}{8 a f}+\frac {b^2 (e+f x)^4}{4 a^3 f}-\frac {b (e+f x)^3 \cosh (c+d x)}{a^2 d}-\frac {3 f^3 \cosh ^2(c+d x)}{8 a d^4}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 a d^2}+\frac {3 b f (e+f x)^2 \sinh (c+d x)}{a^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 a d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d}-\frac {\left (2 b \sqrt {a^2+b^2}\right ) \int \frac {e^{c+d x} (e+f x)^3}{2 b-2 \sqrt {a^2+b^2}+2 a e^{c+d x}} \, dx}{a^2}+\frac {\left (2 b \sqrt {a^2+b^2}\right ) \int \frac {e^{c+d x} (e+f x)^3}{2 b+2 \sqrt {a^2+b^2}+2 a e^{c+d x}} \, dx}{a^2}-\frac {\left (6 b f^2\right ) \int (e+f x) \sinh (c+d x) \, dx}{a^2 d^2} \\ & = \frac {3 e f^2 x}{4 a d^2}+\frac {3 f^3 x^2}{8 a d^2}+\frac {(e+f x)^4}{8 a f}+\frac {b^2 (e+f x)^4}{4 a^3 f}-\frac {6 b f^2 (e+f x) \cosh (c+d x)}{a^2 d^3}-\frac {b (e+f x)^3 \cosh (c+d x)}{a^2 d}-\frac {3 f^3 \cosh ^2(c+d x)}{8 a d^4}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 a d^2}-\frac {b \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {b \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {3 b f (e+f x)^2 \sinh (c+d x)}{a^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 a d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {\left (3 b \sqrt {a^2+b^2} f\right ) \int (e+f x)^2 \log \left (1+\frac {2 a e^{c+d x}}{2 b-2 \sqrt {a^2+b^2}}\right ) \, dx}{a^3 d}-\frac {\left (3 b \sqrt {a^2+b^2} f\right ) \int (e+f x)^2 \log \left (1+\frac {2 a e^{c+d x}}{2 b+2 \sqrt {a^2+b^2}}\right ) \, dx}{a^3 d}+\frac {\left (6 b f^3\right ) \int \cosh (c+d x) \, dx}{a^2 d^3} \\ & = \frac {3 e f^2 x}{4 a d^2}+\frac {3 f^3 x^2}{8 a d^2}+\frac {(e+f x)^4}{8 a f}+\frac {b^2 (e+f x)^4}{4 a^3 f}-\frac {6 b f^2 (e+f x) \cosh (c+d x)}{a^2 d^3}-\frac {b (e+f x)^3 \cosh (c+d x)}{a^2 d}-\frac {3 f^3 \cosh ^2(c+d x)}{8 a d^4}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 a d^2}-\frac {b \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {b \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {3 b \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {3 b \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {6 b f^3 \sinh (c+d x)}{a^2 d^4}+\frac {3 b f (e+f x)^2 \sinh (c+d x)}{a^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 a d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {\left (6 b \sqrt {a^2+b^2} f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {2 a e^{c+d x}}{2 b-2 \sqrt {a^2+b^2}}\right ) \, dx}{a^3 d^2}-\frac {\left (6 b \sqrt {a^2+b^2} f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {2 a e^{c+d x}}{2 b+2 \sqrt {a^2+b^2}}\right ) \, dx}{a^3 d^2} \\ & = \frac {3 e f^2 x}{4 a d^2}+\frac {3 f^3 x^2}{8 a d^2}+\frac {(e+f x)^4}{8 a f}+\frac {b^2 (e+f x)^4}{4 a^3 f}-\frac {6 b f^2 (e+f x) \cosh (c+d x)}{a^2 d^3}-\frac {b (e+f x)^3 \cosh (c+d x)}{a^2 d}-\frac {3 f^3 \cosh ^2(c+d x)}{8 a d^4}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 a d^2}-\frac {b \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {b \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {3 b \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {3 b \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {6 b \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d^3}-\frac {6 b \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d^3}+\frac {6 b f^3 \sinh (c+d x)}{a^2 d^4}+\frac {3 b f (e+f x)^2 \sinh (c+d x)}{a^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 a d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d}-\frac {\left (6 b \sqrt {a^2+b^2} f^3\right ) \int \operatorname {PolyLog}\left (3,-\frac {2 a e^{c+d x}}{2 b-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 \operatorname {PolyLog}\left (3,-\frac {2 a e^{c+d x}}{2 b+2 \sqrt {a^2+b^2}}\right ) \, dx}{a^3 d^3} \\ & = \frac {3 e f^2 x}{4 a d^2}+\frac {3 f^3 x^2}{8 a d^2}+\frac {(e+f x)^4}{8 a f}+\frac {b^2 (e+f x)^4}{4 a^3 f}-\frac {6 b f^2 (e+f x) \cosh (c+d x)}{a^2 d^3}-\frac {b (e+f x)^3 \cosh (c+d x)}{a^2 d}-\frac {3 f^3 \cosh ^2(c+d x)}{8 a d^4}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 a d^2}-\frac {b \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {b \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {3 b \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {3 b \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {6 b \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d^3}-\frac {6 b \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d^3}+\frac {6 b f^3 \sinh (c+d x)}{a^2 d^4}+\frac {3 b f (e+f x)^2 \sinh (c+d x)}{a^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 a d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d}-\frac {\left (6 b \sqrt {a^2+b^2} f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,\frac {a x}{-b+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^4}+\frac {\left (6 b \sqrt {a^2+b^2} f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,-\frac {a x}{b+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^4} \\ & = \frac {3 e f^2 x}{4 a d^2}+\frac {3 f^3 x^2}{8 a d^2}+\frac {(e+f x)^4}{8 a f}+\frac {b^2 (e+f x)^4}{4 a^3 f}-\frac {6 b f^2 (e+f x) \cosh (c+d x)}{a^2 d^3}-\frac {b (e+f x)^3 \cosh (c+d x)}{a^2 d}-\frac {3 f^3 \cosh ^2(c+d x)}{8 a d^4}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 a d^2}-\frac {b \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {b \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {3 b \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {3 b \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {6 b \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d^3}-\frac {6 b \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d^3}-\frac {6 b \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d^4}+\frac {6 b \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d^4}+\frac {6 b f^3 \sinh (c+d x)}{a^2 d^4}+\frac {3 b f (e+f x)^2 \sinh (c+d x)}{a^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 a d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(2218\) vs. \(2(696)=1392\).

Time = 9.94 (sec) , antiderivative size = 2218, normalized size of antiderivative = 3.19 \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

(e^3*(c/d + x - (2*b*ArcTan[(a - b*Tanh[(c + d*x)/2])/Sqrt[-a^2 - b^2]])/(Sqrt[-a^2 - b^2]*d))*Csch[c + d*x]*(
b + a*Sinh[c + d*x]))/(4*a*(a + b*Csch[c + d*x])) + (3*e^2*f*Csch[c + d*x]*(x^2 - (2*b*(d*x*(Log[1 + (a*E^(c +
d*x))/(b - Sqrt[a^2 + b^2])] - Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])]) + PolyLog[2, (a*E^(c + d*x))/(
-b + Sqrt[a^2 + b^2])] - PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))]))/(Sqrt[a^2 + b^2]*d^2))*(b + a*
Sinh[c + d*x]))/(8*a*(a + b*Csch[c + d*x])) + (e*f^2*Csch[c + d*x]*(x^3 - (3*b*(d^2*x^2*Log[1 + (a*E^(c + d*x)
)/(b - Sqrt[a^2 + b^2])] - d^2*x^2*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])] + 2*d*x*PolyLog[2, (a*E^(c +
d*x))/(-b + Sqrt[a^2 + b^2])] - 2*d*x*PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))] - 2*PolyLog[3, (a*
E^(c + d*x))/(-b + Sqrt[a^2 + b^2])] + 2*PolyLog[3, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))]))/(Sqrt[a^2 + b^
2]*d^3))*(b + a*Sinh[c + d*x]))/(4*a*(a + b*Csch[c + d*x])) + (f^3*Csch[c + d*x]*(x^4 - (4*b*(d^3*x^3*Log[1 +
(a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])] - d^3*x^3*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])] + 3*d^2*x^2*Po
lyLog[2, (a*E^(c + d*x))/(-b + Sqrt[a^2 + b^2])] - 3*d^2*x^2*PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]
))] - 6*d*x*PolyLog[3, (a*E^(c + d*x))/(-b + Sqrt[a^2 + b^2])] + 6*d*x*PolyLog[3, -((a*E^(c + d*x))/(b + Sqrt[
a^2 + b^2]))] + 6*PolyLog[4, (a*E^(c + d*x))/(-b + Sqrt[a^2 + b^2])] - 6*PolyLog[4, -((a*E^(c + d*x))/(b + Sqr
t[a^2 + b^2]))]))/(Sqrt[a^2 + b^2]*d^4))*(b + a*Sinh[c + d*x]))/(16*a*(a + b*Csch[c + d*x])) + (e*f^2*Csch[c +
d*x]*(2*(a^2 + 4*b^2)*x^3 - (6*b*(3*a^2 + 4*b^2)*(d^2*x^2*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])] - d^
2*x^2*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])] + 2*d*x*PolyLog[2, (a*E^(c + d*x))/(-b + Sqrt[a^2 + b^2])
] - 2*d*x*PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))] - 2*PolyLog[3, (a*E^(c + d*x))/(-b + Sqrt[a^2 +
b^2])] + 2*PolyLog[3, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))]))/(Sqrt[a^2 + b^2]*d^3) - (24*a*b*Cosh[d*x]*(
(2 + d^2*x^2)*Cosh[c] - 2*d*x*Sinh[c]))/d^3 + (3*a^2*Cosh[2*d*x]*(-2*d*x*Cosh[2*c] + (1 + 2*d^2*x^2)*Sinh[2*c]
))/d^3 - (24*a*b*(-2*d*x*Cosh[c] + (2 + d^2*x^2)*Sinh[c])*Sinh[d*x])/d^3 + (3*a^2*((1 + 2*d^2*x^2)*Cosh[2*c] -
2*d*x*Sinh[2*c])*Sinh[2*d*x])/d^3)*(b + a*Sinh[c + d*x]))/(8*a^3*(a + b*Csch[c + d*x])) + (f^3*Csch[c + d*x]*
((a^2 + 4*b^2)*x^4 - (4*b*(3*a^2 + 4*b^2)*(d^3*x^3*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])] - d^3*x^3*Lo
g[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])] + 3*d^2*x^2*PolyLog[2, (a*E^(c + d*x))/(-b + Sqrt[a^2 + b^2])] -
3*d^2*x^2*PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))] - 6*d*x*PolyLog[3, (a*E^(c + d*x))/(-b + Sqrt[a
^2 + b^2])] + 6*d*x*PolyLog[3, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))] + 6*PolyLog[4, (a*E^(c + d*x))/(-b +
Sqrt[a^2 + b^2])] - 6*PolyLog[4, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))]))/(Sqrt[a^2 + b^2]*d^4) - (16*a*b*C
osh[d*x]*(d*x*(6 + d^2*x^2)*Cosh[c] - 3*(2 + d^2*x^2)*Sinh[c]))/d^4 + (a^2*Cosh[2*d*x]*(-3*(1 + 2*d^2*x^2)*Cos
h[2*c] + 2*d*x*(3 + 2*d^2*x^2)*Sinh[2*c]))/d^4 - (16*a*b*(-3*(2 + d^2*x^2)*Cosh[c] + d*x*(6 + d^2*x^2)*Sinh[c]
)*Sinh[d*x])/d^4 + (a^2*(2*d*x*(3 + 2*d^2*x^2)*Cosh[2*c] - 3*(1 + 2*d^2*x^2)*Sinh[2*c])*Sinh[2*d*x])/d^4)*(b +
a*Sinh[c + d*x]))/(16*a^3*(a + b*Csch[c + d*x])) + (e^3*Csch[c + d*x]*(b + a*Sinh[c + d*x])*((a^2 + 4*b^2)*(c
+ d*x) - (2*b*(3*a^2 + 4*b^2)*ArcTan[(a - b*Tanh[(c + d*x)/2])/Sqrt[-a^2 - b^2]])/Sqrt[-a^2 - b^2] - 4*a*b*Co
sh[c + d*x] + a^2*Sinh[2*(c + d*x)]))/(4*a^3*d*(a + b*Csch[c + d*x])) + (3*e^2*f*Csch[c + d*x]*(b + a*Sinh[c +
d*x])*((a^2 + 4*b^2)*(-c + d*x)*(c + d*x) - 8*a*b*d*x*Cosh[c + d*x] - a^2*Cosh[2*(c + d*x)] - (2*b*(3*a^2 + 4
*b^2)*(2*c*ArcTanh[(b + a*E^(c + d*x))/Sqrt[a^2 + b^2]] + (c + d*x)*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^
2])] - (c + d*x)*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])] + PolyLog[2, (a*E^(c + d*x))/(-b + Sqrt[a^2 +
b^2])] - PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))]))/Sqrt[a^2 + b^2] + 8*a*b*Sinh[c + d*x] + 2*a^2*
d*x*Sinh[2*(c + d*x)]))/(8*a^3*d^2*(a + b*Csch[c + d*x]))

Maple [F]

\[\int \frac {\left (f x +e \right )^{3} \cosh \left (d x +c \right )^{2}}{a +b \,\operatorname {csch}\left (d x +c \right )}d x\]

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3847 vs. \(2 (638) = 1276\).

Time = 0.35 (sec) , antiderivative size = 3847, normalized size of antiderivative = 5.53 \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/32*(4*a^2*d^3*f^3*x^3 + 4*a^2*d^3*e^3 + 6*a^2*d^2*e^2*f + 6*a^2*d*e*f^2 + 3*a^2*f^3 - (4*a^2*d^3*f^3*x^3 +
4*a^2*d^3*e^3 - 6*a^2*d^2*e^2*f + 6*a^2*d*e*f^2 - 3*a^2*f^3 + 6*(2*a^2*d^3*e*f^2 - a^2*d^2*f^3)*x^2 + 6*(2*a^2
*d^3*e^2*f - 2*a^2*d^2*e*f^2 + a^2*d*f^3)*x)*cosh(d*x + c)^4 - (4*a^2*d^3*f^3*x^3 + 4*a^2*d^3*e^3 - 6*a^2*d^2*
e^2*f + 6*a^2*d*e*f^2 - 3*a^2*f^3 + 6*(2*a^2*d^3*e*f^2 - a^2*d^2*f^3)*x^2 + 6*(2*a^2*d^3*e^2*f - 2*a^2*d^2*e*f
^2 + a^2*d*f^3)*x)*sinh(d*x + c)^4 + 16*(a*b*d^3*f^3*x^3 + a*b*d^3*e^3 - 3*a*b*d^2*e^2*f + 6*a*b*d*e*f^2 - 6*a
*b*f^3 + 3*(a*b*d^3*e*f^2 - a*b*d^2*f^3)*x^2 + 3*(a*b*d^3*e^2*f - 2*a*b*d^2*e*f^2 + 2*a*b*d*f^3)*x)*cosh(d*x +
c)^3 + 4*(4*a*b*d^3*f^3*x^3 + 4*a*b*d^3*e^3 - 12*a*b*d^2*e^2*f + 24*a*b*d*e*f^2 - 24*a*b*f^3 + 12*(a*b*d^3*e*
f^2 - a*b*d^2*f^3)*x^2 + 12*(a*b*d^3*e^2*f - 2*a*b*d^2*e*f^2 + 2*a*b*d*f^3)*x - (4*a^2*d^3*f^3*x^3 + 4*a^2*d^3
*e^3 - 6*a^2*d^2*e^2*f + 6*a^2*d*e*f^2 - 3*a^2*f^3 + 6*(2*a^2*d^3*e*f^2 - a^2*d^2*f^3)*x^2 + 6*(2*a^2*d^3*e^2*
f - 2*a^2*d^2*e*f^2 + a^2*d*f^3)*x)*cosh(d*x + c))*sinh(d*x + c)^3 + 6*(2*a^2*d^3*e*f^2 + a^2*d^2*f^3)*x^2 - 4
*((a^2 + 2*b^2)*d^4*f^3*x^4 + 4*(a^2 + 2*b^2)*d^4*e*f^2*x^3 + 6*(a^2 + 2*b^2)*d^4*e^2*f*x^2 + 4*(a^2 + 2*b^2)*
d^4*e^3*x)*cosh(d*x + c)^2 - 2*(2*(a^2 + 2*b^2)*d^4*f^3*x^4 + 8*(a^2 + 2*b^2)*d^4*e*f^2*x^3 + 12*(a^2 + 2*b^2)
*d^4*e^2*f*x^2 + 8*(a^2 + 2*b^2)*d^4*e^3*x + 3*(4*a^2*d^3*f^3*x^3 + 4*a^2*d^3*e^3 - 6*a^2*d^2*e^2*f + 6*a^2*d*
e*f^2 - 3*a^2*f^3 + 6*(2*a^2*d^3*e*f^2 - a^2*d^2*f^3)*x^2 + 6*(2*a^2*d^3*e^2*f - 2*a^2*d^2*e*f^2 + a^2*d*f^3)*
x)*cosh(d*x + c)^2 - 24*(a*b*d^3*f^3*x^3 + a*b*d^3*e^3 - 3*a*b*d^2*e^2*f + 6*a*b*d*e*f^2 - 6*a*b*f^3 + 3*(a*b*
d^3*e*f^2 - a*b*d^2*f^3)*x^2 + 3*(a*b*d^3*e^2*f - 2*a*b*d^2*e*f^2 + 2*a*b*d*f^3)*x)*cosh(d*x + c))*sinh(d*x +
c)^2 + 96*((a*b*d^2*f^3*x^2 + 2*a*b*d^2*e*f^2*x + a*b*d^2*e^2*f)*cosh(d*x + c)^2 + 2*(a*b*d^2*f^3*x^2 + 2*a*b*
d^2*e*f^2*x + a*b*d^2*e^2*f)*cosh(d*x + c)*sinh(d*x + c) + (a*b*d^2*f^3*x^2 + 2*a*b*d^2*e*f^2*x + a*b*d^2*e^2*
f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/a^2)*dilog((b*cosh(d*x + c) + b*sinh(d*x + c) + (a*cosh(d*x + c) + a*sinh
(d*x + c))*sqrt((a^2 + b^2)/a^2) - a)/a + 1) - 96*((a*b*d^2*f^3*x^2 + 2*a*b*d^2*e*f^2*x + a*b*d^2*e^2*f)*cosh(
d*x + c)^2 + 2*(a*b*d^2*f^3*x^2 + 2*a*b*d^2*e*f^2*x + a*b*d^2*e^2*f)*cosh(d*x + c)*sinh(d*x + c) + (a*b*d^2*f^
3*x^2 + 2*a*b*d^2*e*f^2*x + a*b*d^2*e^2*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/a^2)*dilog((b*cosh(d*x + c) + b*s
inh(d*x + c) - (a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2) - a)/a + 1) - 32*((a*b*d^3*e^3 - 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 + 2*(a*b*d^3*e^3 - 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) + (a*b*d^3*e^3 - 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)^2)*sqrt((a^2 + b^2)/a^2)*log(2*a*cosh(d*x + c) + 2*a*sinh(d*x + c) + 2*a*sqrt((a^
2 + b^2)/a^2) + 2*b) + 32*((a*b*d^3*e^3 - 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
+ 2*(a*b*d^3*e^3 - 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) + (a*b*d^
3*e^3 - 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)^2)*sqrt((a^2 + b^2)/a^2)*log(2*a*co
sh(d*x + c) + 2*a*sinh(d*x + c) - 2*a*sqrt((a^2 + b^2)/a^2) + 2*b) + 32*((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 + 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 + 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) + (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)^2)*sqrt((a^2 + b^2)/a^2)*log(-(b*cosh(d*x + c) + b*sinh(d*x + c) + (a*c
osh(d*x + c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2) - a)/a) - 32*((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 + 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 + 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) + (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)^2)*sqrt((a^2 + b^2)/a^2)*log(-(b*cosh(d*x + c) + b*sinh(d*x + c) - (a*cosh(d
*x + c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2) - a)/a) + 192*(a*b*f^3*cosh(d*x + c)^2 + 2*a*b*f^3*cosh(d*x +
c)*sinh(d*x + c) + a*b*f^3*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/a^2)*polylog(4, (b*cosh(d*x + c) + b*sinh(d*x +
c) + (a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2))/a) - 192*(a*b*f^3*cosh(d*x + c)^2 + 2*a*b*f^3*
cosh(d*x + c)*sinh(d*x + c) + a*b*f^3*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/a^2)*polylog(4, (b*cosh(d*x + c) + b*s
inh(d*x + c) - (a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2))/a) - 192*((a*b*d*f^3*x + a*b*d*e*f^2
)*cosh(d*x + c)^2 + 2*(a*b*d*f^3*x + a*b*d*e*f^2)*cosh(d*x + c)*sinh(d*x + c) + (a*b*d*f^3*x + a*b*d*e*f^2)*si
nh(d*x + c)^2)*sqrt((a^2 + b^2)/a^2)*polylog(3, (b*cosh(d*x + c) + b*sinh(d*x + c) + (a*cosh(d*x + c) + a*sinh
(d*x + c))*sqrt((a^2 + b^2)/a^2))/a) + 192*((a*b*d*f^3*x + a*b*d*e*f^2)*cosh(d*x + c)^2 + 2*(a*b*d*f^3*x + a*b
*d*e*f^2)*cosh(d*x + c)*sinh(d*x + c) + (a*b*d*f^3*x + a*b*d*e*f^2)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/a^2)*pol
ylog(3, (b*cosh(d*x + c) + b*sinh(d*x + c) - (a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2))/a) + 6
*(2*a^2*d^3*e^2*f + 2*a^2*d^2*e*f^2 + a^2*d*f^3)*x + 16*(a*b*d^3*f^3*x^3 + a*b*d^3*e^3 + 3*a*b*d^2*e^2*f + 6*a
*b*d*e*f^2 + 6*a*b*f^3 + 3*(a*b*d^3*e*f^2 + a*b*d^2*f^3)*x^2 + 3*(a*b*d^3*e^2*f + 2*a*b*d^2*e*f^2 + 2*a*b*d*f^
3)*x)*cosh(d*x + c) + 4*(4*a*b*d^3*f^3*x^3 + 4*a*b*d^3*e^3 + 12*a*b*d^2*e^2*f + 24*a*b*d*e*f^2 + 24*a*b*f^3 -
(4*a^2*d^3*f^3*x^3 + 4*a^2*d^3*e^3 - 6*a^2*d^2*e^2*f + 6*a^2*d*e*f^2 - 3*a^2*f^3 + 6*(2*a^2*d^3*e*f^2 - a^2*d^
2*f^3)*x^2 + 6*(2*a^2*d^3*e^2*f - 2*a^2*d^2*e*f^2 + a^2*d*f^3)*x)*cosh(d*x + c)^3 + 12*(a*b*d^3*e*f^2 + a*b*d^
2*f^3)*x^2 + 12*(a*b*d^3*f^3*x^3 + a*b*d^3*e^3 - 3*a*b*d^2*e^2*f + 6*a*b*d*e*f^2 - 6*a*b*f^3 + 3*(a*b*d^3*e*f^
2 - a*b*d^2*f^3)*x^2 + 3*(a*b*d^3*e^2*f - 2*a*b*d^2*e*f^2 + 2*a*b*d*f^3)*x)*cosh(d*x + c)^2 + 12*(a*b*d^3*e^2*
f + 2*a*b*d^2*e*f^2 + 2*a*b*d*f^3)*x - 2*((a^2 + 2*b^2)*d^4*f^3*x^4 + 4*(a^2 + 2*b^2)*d^4*e*f^2*x^3 + 6*(a^2 +
2*b^2)*d^4*e^2*f*x^2 + 4*(a^2 + 2*b^2)*d^4*e^3*x)*cosh(d*x + c))*sinh(d*x + c))/(a^3*d^4*cosh(d*x + c)^2 + 2*
a^3*d^4*cosh(d*x + c)*sinh(d*x + c) + a^3*d^4*sinh(d*x + c)^2)

Sympy [F]

\[ \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int \frac {\left (e + f x\right )^{3} \cosh ^{2}{\left (c + d x \right )}}{a + b \operatorname {csch}{\left (c + d x \right )}}\, dx \]

[In]

integrate((f*x+e)**3*cosh(d*x+c)**2/(a+b*csch(d*x+c)),x)

[Out]

Integral((e + f*x)**3*cosh(c + d*x)**2/(a + b*csch(c + d*x)), x)

Maxima [F]

\[ \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \cosh \left (d x + c\right )^{2}}{b \operatorname {csch}\left (d x + c\right ) + a} \,d x } \]

[In]

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

[Out]

-1/8*e^3*((4*b*e^(-d*x - c) - a)*e^(2*d*x + 2*c)/(a^2*d) - 4*(a^2 + 2*b^2)*(d*x + c)/(a^3*d) + (4*b*e^(-d*x -
c) + a*e^(-2*d*x - 2*c))/(a^2*d) + 8*(a^2*b + b^3)*log((a*e^(-d*x - c) - b - sqrt(a^2 + b^2))/(a*e^(-d*x - c)
- b + sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*a^3*d)) + 1/32*(4*(a^2*d^4*f^3*e^(2*c) + 2*b^2*d^4*f^3*e^(2*c))*x^4 +
16*(a^2*d^4*e*f^2*e^(2*c) + 2*b^2*d^4*e*f^2*e^(2*c))*x^3 + 24*(a^2*d^4*e^2*f*e^(2*c) + 2*b^2*d^4*e^2*f*e^(2*c
))*x^2 + (4*a^2*d^3*f^3*x^3*e^(4*c) + 6*(2*d^3*e*f^2 - d^2*f^3)*a^2*x^2*e^(4*c) + 6*(2*d^3*e^2*f - 2*d^2*e*f^2
+ d*f^3)*a^2*x*e^(4*c) - 3*(2*d^2*e^2*f - 2*d*e*f^2 + f^3)*a^2*e^(4*c))*e^(2*d*x) - 16*(a*b*d^3*f^3*x^3*e^(3*
c) + 3*(d^3*e*f^2 - d^2*f^3)*a*b*x^2*e^(3*c) + 3*(d^3*e^2*f - 2*d^2*e*f^2 + 2*d*f^3)*a*b*x*e^(3*c) - 3*(d^2*e^
2*f - 2*d*e*f^2 + 2*f^3)*a*b*e^(3*c))*e^(d*x) - 16*(a*b*d^3*f^3*x^3*e^c + 3*(d^3*e*f^2 + d^2*f^3)*a*b*x^2*e^c
+ 3*(d^3*e^2*f + 2*d^2*e*f^2 + 2*d*f^3)*a*b*x*e^c + 3*(d^2*e^2*f + 2*d*e*f^2 + 2*f^3)*a*b*e^c)*e^(-d*x) - (4*a
^2*d^3*f^3*x^3 + 6*(2*d^3*e*f^2 + d^2*f^3)*a^2*x^2 + 6*(2*d^3*e^2*f + 2*d^2*e*f^2 + d*f^3)*a^2*x + 3*(2*d^2*e^
2*f + 2*d*e*f^2 + f^3)*a^2)*e^(-2*d*x))*e^(-2*c)/(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^4*e^(2*d*x + 2*c)
+ 2*a^3*b*e^(d*x + c) - a^4), x)

Giac [F]

\[ \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \cosh \left (d x + c\right )^{2}}{b \operatorname {csch}\left (d x + c\right ) + a} \,d x } \]

[In]

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

[Out]

integrate((f*x + e)^3*cosh(d*x + c)^2/(b*csch(d*x + c) + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^3}{a+\frac {b}{\mathrm {sinh}\left (c+d\,x\right )}} \,d x \]

[In]

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

[Out]

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