\(\int \frac {(e+f x)^3 \text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx\) [435]

Optimal result

Integrand size = 32, antiderivative size = 1049 \[ \int \frac {(e+f x)^3 \text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {2 b (e+f x)^3 \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{2 c+2 d x}\right )}{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 \left (a^2+b^2\right ) 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 \left (a^2+b^2\right ) d}+\frac {b^2 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}+\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 a \left (a^2+b^2\right ) d^2}-\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 a d^2}+\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 a d^2}-\frac {6 i b f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {6 i b f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {6 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {6 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {3 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 a \left (a^2+b^2\right ) d^3}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{2 c+2 d x}\right )}{2 a d^3}-\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{2 c+2 d x}\right )}{2 a d^3}+\frac {6 i b f^3 \operatorname {PolyLog}\left (4,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^4}-\frac {6 i b f^3 \operatorname {PolyLog}\left (4,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^4}-\frac {6 b^2 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^4}-\frac {6 b^2 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^4}+\frac {3 b^2 f^3 \operatorname {PolyLog}\left (4,-e^{2 (c+d x)}\right )}{4 a \left (a^2+b^2\right ) d^4}-\frac {3 f^3 \operatorname {PolyLog}\left (4,-e^{2 c+2 d x}\right )}{4 a d^4}+\frac {3 f^3 \operatorname {PolyLog}\left (4,e^{2 c+2 d x}\right )}{4 a d^4} \]

[Out]

3/2*f*(f*x+e)^2*polylog(2,exp(2*d*x+2*c))/a/d^2-3/2*f^2*(f*x+e)*polylog(3,exp(2*d*x+2*c))/a/d^3-3*I*b*f*(f*x+e
)^2*polylog(2,I*exp(d*x+c))/(a^2+b^2)/d^2-6*I*b*f^2*(f*x+e)*polylog(3,-I*exp(d*x+c))/(a^2+b^2)/d^3+3/2*b^2*f*(
f*x+e)^2*polylog(2,-exp(2*d*x+2*c))/a/(a^2+b^2)/d^2-3/2*b^2*f^2*(f*x+e)*polylog(3,-exp(2*d*x+2*c))/a/(a^2+b^2)
/d^3-3*b^2*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a/(a^2+b^2)/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/(a^2+b^2)/d^2+6*b^2*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1
/2)))/a/(a^2+b^2)/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/(a^2+b^2)/d^3+3*I*b*f*(
f*x+e)^2*polylog(2,-I*exp(d*x+c))/(a^2+b^2)/d^2+6*I*b*f^2*(f*x+e)*polylog(3,I*exp(d*x+c))/(a^2+b^2)/d^3+3/4*b^
2*f^3*polylog(4,-exp(2*d*x+2*c))/a/(a^2+b^2)/d^4-6*b^2*f^3*polylog(4,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a/(a^2
+b^2)/d^4-6*b^2*f^3*polylog(4,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a/(a^2+b^2)/d^4-6*I*b*f^3*polylog(4,I*exp(d*x
+c))/(a^2+b^2)/d^4+b^2*(f*x+e)^3*ln(1+exp(2*d*x+2*c))/a/(a^2+b^2)/d-b^2*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a-(a^2+b^
2)^(1/2)))/a/(a^2+b^2)/d-b^2*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a/(a^2+b^2)/d-2*(f*x+e)^3*arctan
h(exp(2*d*x+2*c))/a/d-3/4*f^3*polylog(4,-exp(2*d*x+2*c))/a/d^4+3/4*f^3*polylog(4,exp(2*d*x+2*c))/a/d^4-2*b*(f*
x+e)^3*arctan(exp(d*x+c))/(a^2+b^2)/d-3/2*f*(f*x+e)^2*polylog(2,-exp(2*d*x+2*c))/a/d^2+3/2*f^2*(f*x+e)*polylog
(3,-exp(2*d*x+2*c))/a/d^3+6*I*b*f^3*polylog(4,-I*exp(d*x+c))/(a^2+b^2)/d^4

Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 1049, normalized size of antiderivative = 1.00, number of steps used = 40, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.406, Rules used = {5708, 5569, 4267, 2611, 6744, 2320, 6724, 5692, 5680, 2221, 6874, 4265, 3799} \[ \int \frac {(e+f x)^3 \text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {6 i b \operatorname {PolyLog}\left (4,-i e^{c+d x}\right ) f^3}{\left (a^2+b^2\right ) d^4}-\frac {6 i b \operatorname {PolyLog}\left (4,i e^{c+d x}\right ) f^3}{\left (a^2+b^2\right ) d^4}-\frac {6 b^2 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) f^3}{a \left (a^2+b^2\right ) d^4}-\frac {6 b^2 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) f^3}{a \left (a^2+b^2\right ) d^4}+\frac {3 b^2 \operatorname {PolyLog}\left (4,-e^{2 (c+d x)}\right ) f^3}{4 a \left (a^2+b^2\right ) d^4}-\frac {3 \operatorname {PolyLog}\left (4,-e^{2 c+2 d x}\right ) f^3}{4 a d^4}+\frac {3 \operatorname {PolyLog}\left (4,e^{2 c+2 d x}\right ) f^3}{4 a d^4}-\frac {6 i b (e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right ) f^2}{\left (a^2+b^2\right ) d^3}+\frac {6 i b (e+f x) \operatorname {PolyLog}\left (3,i e^{c+d x}\right ) f^2}{\left (a^2+b^2\right ) d^3}+\frac {6 b^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) f^2}{a \left (a^2+b^2\right ) d^3}+\frac {6 b^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) f^2}{a \left (a^2+b^2\right ) d^3}-\frac {3 b^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right ) f^2}{2 a \left (a^2+b^2\right ) d^3}+\frac {3 (e+f x) \operatorname {PolyLog}\left (3,-e^{2 c+2 d x}\right ) f^2}{2 a d^3}-\frac {3 (e+f x) \operatorname {PolyLog}\left (3,e^{2 c+2 d x}\right ) f^2}{2 a d^3}+\frac {3 i b (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) f}{\left (a^2+b^2\right ) d^2}-\frac {3 i b (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) f}{\left (a^2+b^2\right ) d^2}-\frac {3 b^2 (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) f}{a \left (a^2+b^2\right ) d^2}-\frac {3 b^2 (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) f}{a \left (a^2+b^2\right ) d^2}+\frac {3 b^2 (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right ) f}{2 a \left (a^2+b^2\right ) d^2}-\frac {3 (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right ) f}{2 a d^2}+\frac {3 (e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right ) f}{2 a d^2}-\frac {2 b (e+f x)^3 \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^2 (e+f x)^3 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )}{a \left (a^2+b^2\right ) d}-\frac {b^2 (e+f x)^3 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )}{a \left (a^2+b^2\right ) d}+\frac {b^2 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d} \]

[In]

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

[Out]

(-2*b*(e + f*x)^3*ArcTan[E^(c + d*x)])/((a^2 + b^2)*d) - (2*(e + f*x)^3*ArcTanh[E^(2*c + 2*d*x)])/(a*d) - (b^2
*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a*(a^2 + b^2)*d) - (b^2*(e + f*x)^3*Log[1 + (b*E
^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a*(a^2 + b^2)*d) + (b^2*(e + f*x)^3*Log[1 + E^(2*(c + d*x))])/(a*(a^2 + b
^2)*d) + ((3*I)*b*f*(e + f*x)^2*PolyLog[2, (-I)*E^(c + d*x)])/((a^2 + b^2)*d^2) - ((3*I)*b*f*(e + f*x)^2*PolyL
og[2, I*E^(c + d*x)])/((a^2 + b^2)*d^2) - (3*b^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^
2]))])/(a*(a^2 + b^2)*d^2) - (3*b^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*(a^
2 + b^2)*d^2) + (3*b^2*f*(e + f*x)^2*PolyLog[2, -E^(2*(c + d*x))])/(2*a*(a^2 + b^2)*d^2) - (3*f*(e + f*x)^2*Po
lyLog[2, -E^(2*c + 2*d*x)])/(2*a*d^2) + (3*f*(e + f*x)^2*PolyLog[2, E^(2*c + 2*d*x)])/(2*a*d^2) - ((6*I)*b*f^2
*(e + f*x)*PolyLog[3, (-I)*E^(c + d*x)])/((a^2 + b^2)*d^3) + ((6*I)*b*f^2*(e + f*x)*PolyLog[3, I*E^(c + d*x)])
/((a^2 + b^2)*d^3) + (6*b^2*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)
*d^3) + (6*b^2*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)*d^3) - (3*b^
2*f^2*(e + f*x)*PolyLog[3, -E^(2*(c + d*x))])/(2*a*(a^2 + b^2)*d^3) + (3*f^2*(e + f*x)*PolyLog[3, -E^(2*c + 2*
d*x)])/(2*a*d^3) - (3*f^2*(e + f*x)*PolyLog[3, E^(2*c + 2*d*x)])/(2*a*d^3) + ((6*I)*b*f^3*PolyLog[4, (-I)*E^(c
+ d*x)])/((a^2 + b^2)*d^4) - ((6*I)*b*f^3*PolyLog[4, I*E^(c + d*x)])/((a^2 + b^2)*d^4) - (6*b^2*f^3*PolyLog[4
, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)*d^4) - (6*b^2*f^3*PolyLog[4, -((b*E^(c + d*x))/(a
+ Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)*d^4) + (3*b^2*f^3*PolyLog[4, -E^(2*(c + d*x))])/(4*a*(a^2 + b^2)*d^4) - (
3*f^3*PolyLog[4, -E^(2*c + 2*d*x)])/(4*a*d^4) + (3*f^3*PolyLog[4, E^(2*c + 2*d*x)])/(4*a*d^4)

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 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 3799

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

Rule 4265

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

Rule 4267

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

Rule 5569

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dis
t[2^n, Int[(c + d*x)^m*Csch[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]

Rule 5680

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 5692

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

Rule 5708

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

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

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

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(3862\) vs. \(2(1049)=2098\).

Time = 12.01 (sec) , antiderivative size = 3862, normalized size of antiderivative = 3.68 \[ \int \frac {(e+f x)^3 \text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

2*((a*E^c*((e + f*x)^4/(4*E^c*f) + ((1 + E^(-c))*(e + f*x)^3*Log[1 + E^(-c - d*x)])/d - (3*(1 + E^c)*f*(d^2*(e
+ f*x)^2*PolyLog[2, -E^(-c - d*x)] + 2*f*(d*(e + f*x)*PolyLog[3, -E^(-c - d*x)] + f*PolyLog[4, -E^(-c - d*x)]
)))/(d^4*E^c)))/(2*(a^2 + b^2)*(1 + E^c)) + ((I/2)*a*E^c*((e + f*x)^4/(4*E^c*f) + ((I + E^(-c))*(e + f*x)^3*Lo
g[1 - I*E^(-c - d*x)])/d - (3*(1 + I*E^c)*f*(d^2*(e + f*x)^2*PolyLog[2, I*E^(-c - d*x)] + 2*f*(d*(e + f*x)*Pol
yLog[3, I*E^(-c - d*x)] + f*PolyLog[4, I*E^(-c - d*x)])))/(d^4*E^c)))/((a^2 + b^2)*(-I + E^c)) - (b^2*E^(2*c)*
((e + f*x)^4/(E^(2*c)*f) - (2*(1 - E^(-2*c))*(e + f*x)^3*Log[1 - E^(-c - d*x)])/d - (2*(1 - E^(-2*c))*(e + f*x
)^3*Log[1 + E^(-c - d*x)])/d + (6*(-1 + E^(2*c))*f*(d^2*(e + f*x)^2*PolyLog[2, -E^(-c - d*x)] + 2*f*(d*(e + f*
x)*PolyLog[3, -E^(-c - d*x)] + f*PolyLog[4, -E^(-c - d*x)])))/(d^4*E^(2*c)) + (6*(-1 + E^(2*c))*f*(d^2*(e + f*
x)^2*PolyLog[2, E^(-c - d*x)] + 2*f*(d*(e + f*x)*PolyLog[3, E^(-c - d*x)] + f*PolyLog[4, E^(-c - d*x)])))/(d^4
*E^(2*c))))/(4*a*(a^2 + b^2)*(-1 + E^(2*c))) - ((I/2)*b*((-2*I)*d^3*e^3*ArcTan[E^(c + d*x)] + 3*d^3*e^2*f*x*Lo
g[1 - I*E^(c + d*x)] + 3*d^3*e*f^2*x^2*Log[1 - I*E^(c + d*x)] + d^3*f^3*x^3*Log[1 - I*E^(c + d*x)] - 3*d^3*e^2
*f*x*Log[1 + I*E^(c + d*x)] - 3*d^3*e*f^2*x^2*Log[1 + I*E^(c + d*x)] - d^3*f^3*x^3*Log[1 + I*E^(c + d*x)] - 3*
d^2*f*(e + f*x)^2*PolyLog[2, (-I)*E^(c + d*x)] + 3*d^2*f*(e + f*x)^2*PolyLog[2, I*E^(c + d*x)] + 6*d*e*f^2*Pol
yLog[3, (-I)*E^(c + d*x)] + 6*d*f^3*x*PolyLog[3, (-I)*E^(c + d*x)] - 6*d*e*f^2*PolyLog[3, I*E^(c + d*x)] - 6*d
*f^3*x*PolyLog[3, I*E^(c + d*x)] - 6*f^3*PolyLog[4, (-I)*E^(c + d*x)] + 6*f^3*PolyLog[4, I*E^(c + d*x)]))/((a^
2 + b^2)*d^4) - (a*(-((e + f*x)^3*Log[1 - E^(c + d*x)]) + (e + f*x)^3*Log[1 - I*E^(c + d*x)] + (3*f*(d^2*(e +
f*x)^2*PolyLog[2, I*E^(c + d*x)] - 2*d*f*(e + f*x)*PolyLog[3, I*E^(c + d*x)] + 2*f^2*PolyLog[4, I*E^(c + d*x)]
))/d^3 - (3*f*(d^2*(e + f*x)^2*PolyLog[2, E^(c + d*x)] - 2*d*f*(e + f*x)*PolyLog[3, E^(c + d*x)] + 2*f^2*PolyL
og[4, E^(c + d*x)]))/d^3))/(2*(a^2 + b^2)*d) + (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]*e^3*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]])/(Sqrt[-(a^2 + b^
2)^2]*d) + (4*a*Sqrt[-a^2 - b^2]*e^3*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]])/(Sqrt[-(a^2 + b^2)^2]*d) -
(2*e^3*E^(2*c)*Log[b - 2*a*E^(c + d*x) - b*E^(2*(c + d*x))])/d + (2*e^3*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 + 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*(-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*PolyL
og[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*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*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*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))/(4*
a*(a^2 + b^2)*(-1 + E^(2*c))) - (b^2*x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3)*Csch[c/2]*Sech[c/2]*Sech[c]
)/(32*a*(a^2 + b^2)) + (3*a*e^2*f*x^2*Csch[c/2]*Sech[c/2])/(16*(a^2 + b^2)*(Cosh[c/2] - I*Sinh[c/2])*(Cosh[c/2
] + I*Sinh[c/2])) + (3*b^2*e^2*f*x^2*Csch[c/2]*Sech[c/2])/(16*a*(a^2 + b^2)*(Cosh[c/2] - I*Sinh[c/2])*(Cosh[c/
2] + I*Sinh[c/2])) + (a*e*f^2*x^3*Csch[c/2]*Sech[c/2])/(8*(a^2 + b^2)*(Cosh[c/2] - I*Sinh[c/2])*(Cosh[c/2] + I
*Sinh[c/2])) + (b^2*e*f^2*x^3*Csch[c/2]*Sech[c/2])/(8*a*(a^2 + b^2)*(Cosh[c/2] - I*Sinh[c/2])*(Cosh[c/2] + I*S
inh[c/2])) + (a*f^3*x^4*Csch[c/2]*Sech[c/2])/(32*(a^2 + b^2)*(Cosh[c/2] - I*Sinh[c/2])*(Cosh[c/2] + I*Sinh[c/2
])) + (b^2*f^3*x^4*Csch[c/2]*Sech[c/2])/(32*a*(a^2 + b^2)*(Cosh[c/2] - I*Sinh[c/2])*(Cosh[c/2] + I*Sinh[c/2]))
- (3*a*e^2*f*x^2*Cosh[c]*Csch[c/2]*Sech[c/2])/(16*(a^2 + b^2)*(Cosh[c/2] - I*Sinh[c/2])*(Cosh[c/2] + I*Sinh[c
/2])) - (a*e*f^2*x^3*Cosh[c]*Csch[c/2]*Sech[c/2])/(8*(a^2 + b^2)*(Cosh[c/2] - I*Sinh[c/2])*(Cosh[c/2] + I*Sinh
[c/2])) - (a*f^3*x^4*Cosh[c]*Csch[c/2]*Sech[c/2])/(32*(a^2 + b^2)*(Cosh[c/2] - I*Sinh[c/2])*(Cosh[c/2] + I*Sin
h[c/2])) - (((3*I)/16)*a*e^2*f*x^2*Csch[c/2]*Sech[c/2]*Sinh[c])/((a^2 + b^2)*(Cosh[c/2] - I*Sinh[c/2])*(Cosh[c
/2] + I*Sinh[c/2])) - ((I/8)*a*e*f^2*x^3*Csch[c/2]*Sech[c/2]*Sinh[c])/((a^2 + b^2)*(Cosh[c/2] - I*Sinh[c/2])*(
Cosh[c/2] + I*Sinh[c/2])) - ((I/32)*a*f^3*x^4*Csch[c/2]*Sech[c/2]*Sinh[c])/((a^2 + b^2)*(Cosh[c/2] - I*Sinh[c/
2])*(Cosh[c/2] + I*Sinh[c/2])) - (e^3*x*Csch[c/2]*Sech[c/2]*(-a^2 - b^2 + a^2*Cosh[c] + I*a^2*Sinh[c]))/(8*a*(
a^2 + b^2)*(Cosh[c/2] - I*Sinh[c/2])*(Cosh[c/2] + I*Sinh[c/2])))

Maple [F]

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2448 vs. \(2 (962) = 1924\).

Time = 0.36 (sec) , antiderivative size = 2448, normalized size of antiderivative = 2.33 \[ \int \frac {(e+f x)^3 \text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-(6*b^2*f^3*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*polylog(4, (a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sq
rt((a^2 + b^2)/b^2))/b) - 6*(a^2 + b^2)*f^3*polylog(4, cosh(d*x + c) + sinh(d*x + c)) - 6*(a^2 + b^2)*f^3*poly
log(4, -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)*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)*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*((a^2 + b^2)*d^2*f^3*x^2 + 2*(a^2 + b^2)*d^2*e*f^2*x +
(a^2 + b^2)*d^2*e^2*f)*dilog(cosh(d*x + c) + sinh(d*x + c)) + 3*(a^2*d^2*f^3*x^2 + I*a*b*d^2*f^3*x^2 + 2*a^2*d
^2*e*f^2*x + 2*I*a*b*d^2*e*f^2*x + a^2*d^2*e^2*f + I*a*b*d^2*e^2*f)*dilog(I*cosh(d*x + c) + I*sinh(d*x + c)) +
3*(a^2*d^2*f^3*x^2 - I*a*b*d^2*f^3*x^2 + 2*a^2*d^2*e*f^2*x - 2*I*a*b*d^2*e*f^2*x + a^2*d^2*e^2*f - I*a*b*d^2*
e^2*f)*dilog(-I*cosh(d*x + c) - I*sinh(d*x + c)) - 3*((a^2 + b^2)*d^2*f^3*x^2 + 2*(a^2 + b^2)*d^2*e*f^2*x + (a
^2 + b^2)*d^2*e^2*f)*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)*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)*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)*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqr
t((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)*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) - ((a^2 + b^2)*d^3*f^3*x^3 + 3*(a^2 + b^2)*d^3*e*f^2*x^2 + 3*(a^2 + b^2)*d^3*e^
2*f*x + (a^2 + b^2)*d^3*e^3)*log(cosh(d*x + c) + sinh(d*x + c) + 1) + (a^2*d^3*e^3 + I*a*b*d^3*e^3 - 3*a^2*c*d
^2*e^2*f - 3*I*a*b*c*d^2*e^2*f + 3*a^2*c^2*d*e*f^2 + 3*I*a*b*c^2*d*e*f^2 - a^2*c^3*f^3 - I*a*b*c^3*f^3)*log(co
sh(d*x + c) + sinh(d*x + c) + I) + (a^2*d^3*e^3 - I*a*b*d^3*e^3 - 3*a^2*c*d^2*e^2*f + 3*I*a*b*c*d^2*e^2*f + 3*
a^2*c^2*d*e*f^2 - 3*I*a*b*c^2*d*e*f^2 - a^2*c^3*f^3 + I*a*b*c^3*f^3)*log(cosh(d*x + c) + sinh(d*x + c) - I) -
((a^2 + b^2)*d^3*e^3 - 3*(a^2 + b^2)*c*d^2*e^2*f + 3*(a^2 + b^2)*c^2*d*e*f^2 - (a^2 + b^2)*c^3*f^3)*log(cosh(d
*x + c) + sinh(d*x + c) - 1) + (a^2*d^3*f^3*x^3 - I*a*b*d^3*f^3*x^3 + 3*a^2*d^3*e*f^2*x^2 - 3*I*a*b*d^3*e*f^2*
x^2 + 3*a^2*d^3*e^2*f*x - 3*I*a*b*d^3*e^2*f*x + 3*a^2*c*d^2*e^2*f - 3*I*a*b*c*d^2*e^2*f - 3*a^2*c^2*d*e*f^2 +
3*I*a*b*c^2*d*e*f^2 + a^2*c^3*f^3 - I*a*b*c^3*f^3)*log(I*cosh(d*x + c) + I*sinh(d*x + c) + 1) + (a^2*d^3*f^3*x
^3 + I*a*b*d^3*f^3*x^3 + 3*a^2*d^3*e*f^2*x^2 + 3*I*a*b*d^3*e*f^2*x^2 + 3*a^2*d^3*e^2*f*x + 3*I*a*b*d^3*e^2*f*x
+ 3*a^2*c*d^2*e^2*f + 3*I*a*b*c*d^2*e^2*f - 3*a^2*c^2*d*e*f^2 - 3*I*a*b*c^2*d*e*f^2 + a^2*c^3*f^3 + I*a*b*c^3
*f^3)*log(-I*cosh(d*x + c) - I*sinh(d*x + c) + 1) - ((a^2 + b^2)*d^3*f^3*x^3 + 3*(a^2 + b^2)*d^3*e*f^2*x^2 + 3
*(a^2 + b^2)*d^3*e^2*f*x + 3*(a^2 + b^2)*c*d^2*e^2*f - 3*(a^2 + b^2)*c^2*d*e*f^2 + (a^2 + b^2)*c^3*f^3)*log(-c
osh(d*x + c) - sinh(d*x + c) + 1) + 6*(a^2*f^3 + I*a*b*f^3)*polylog(4, I*cosh(d*x + c) + I*sinh(d*x + c)) + 6*
(a^2*f^3 - I*a*b*f^3)*polylog(4, -I*cosh(d*x + c) - I*sinh(d*x + c)) - 6*(b^2*d*f^3*x + b^2*d*e*f^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) - 6*(b^2*
d*f^3*x + b^2*d*e*f^2)*polylog(3, (a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqr
t((a^2 + b^2)/b^2))/b) + 6*((a^2 + b^2)*d*f^3*x + (a^2 + b^2)*d*e*f^2)*polylog(3, cosh(d*x + c) + sinh(d*x + c
)) - 6*(a^2*d*f^3*x + I*a*b*d*f^3*x + a^2*d*e*f^2 + I*a*b*d*e*f^2)*polylog(3, I*cosh(d*x + c) + I*sinh(d*x + c
)) - 6*(a^2*d*f^3*x - I*a*b*d*f^3*x + a^2*d*e*f^2 - I*a*b*d*e*f^2)*polylog(3, -I*cosh(d*x + c) - I*sinh(d*x +
c)) + 6*((a^2 + b^2)*d*f^3*x + (a^2 + b^2)*d*e*f^2)*polylog(3, -cosh(d*x + c) - sinh(d*x + c)))/((a^3 + a*b^2)
*d^4)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

-e^3*(b^2*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^3 + a*b^2)*d) - 2*b*arctan(e^(-d*x - c))/((a^2 +
b^2)*d) + a*log(e^(-2*d*x - 2*c) + 1)/((a^2 + b^2)*d) - log(e^(-d*x - c) + 1)/(a*d) - log(e^(-d*x - c) - 1)/(
a*d)) + 3*(d*x*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))*e^2*f/(a*d^2) + 3*(d*x*log(-e^(d*x + c) + 1) + dilo
g(e^(d*x + c)))*e^2*f/(a*d^2) + 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)))*e*f^2/(a*d^3) + 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)))*e*f^2/(a*d^3) + (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)))*f^3/(a*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)))*f^3/(a*d^4) - 1/2*(d^4*f^3*x^4 + 4*d^4*e*f^2*
x^3 + 6*d^4*e^2*f*x^2)/(a*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 + a*b^3 - (a^3*b*e^(2*c) + a*b^3*e^(2*c))*e^
(2*d*x) - 2*(a^4*e^c + a^2*b^2*e^c)*e^(d*x)), x) - integrate(-2*(a*f^3*x^3 + 3*a*e*f^2*x^2 + 3*a*e^2*f*x - (b*
f^3*x^3*e^c + 3*b*e*f^2*x^2*e^c + 3*b*e^2*f*x*e^c)*e^(d*x))/(a^2 + b^2 + (a^2*e^(2*c) + b^2*e^(2*c))*e^(2*d*x)
), x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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