\(\int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx\) [218]

Optimal result

Integrand size = 31, antiderivative size = 368 \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {2 i (e+f x)^2}{a d}+\frac {3 (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {f^2 \text {arctanh}(\cosh (c+d x))}{a d^3}+\frac {i (e+f x)^2 \coth (c+d x)}{a d}-\frac {f (e+f x) \text {csch}(c+d x)}{a d^2}-\frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {4 i f (e+f x) \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac {2 i f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}-\frac {4 i f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {3 f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac {i f^2 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {3 f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {3 f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}+\frac {i (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d} \]

[Out]

2*I*(f*x+e)^2/a/d+3*(f*x+e)^2*arctanh(exp(d*x+c))/a/d-f^2*arctanh(cosh(d*x+c))/a/d^3+I*(f*x+e)^2*coth(d*x+c)/a
/d-f*(f*x+e)*csch(d*x+c)/a/d^2-1/2*(f*x+e)^2*coth(d*x+c)*csch(d*x+c)/a/d-4*I*f*(f*x+e)*ln(1+I*exp(d*x+c))/a/d^
2-2*I*f*(f*x+e)*ln(1-exp(2*d*x+2*c))/a/d^2+3*f*(f*x+e)*polylog(2,-exp(d*x+c))/a/d^2-4*I*f^2*polylog(2,-I*exp(d
*x+c))/a/d^3-3*f*(f*x+e)*polylog(2,exp(d*x+c))/a/d^2-I*f^2*polylog(2,exp(2*d*x+2*c))/a/d^3-3*f^2*polylog(3,-ex
p(d*x+c))/a/d^3+3*f^2*polylog(3,exp(d*x+c))/a/d^3+I*(f*x+e)^2*tanh(1/2*c+1/4*I*Pi+1/2*d*x)/a/d

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {5694, 4271, 3855, 4267, 2611, 2320, 6724, 4269, 3797, 2221, 2317, 2438, 3399} \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {f^2 \text {arctanh}(\cosh (c+d x))}{a d^3}+\frac {3 (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {4 i f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {i f^2 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {3 f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {3 f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}+\frac {3 f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}-\frac {3 f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac {4 i f (e+f x) \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac {2 i f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^2}-\frac {f (e+f x) \text {csch}(c+d x)}{a d^2}+\frac {i (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{a d}+\frac {i (e+f x)^2 \coth (c+d x)}{a d}-\frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 a d}+\frac {2 i (e+f x)^2}{a d} \]

[In]

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

[Out]

((2*I)*(e + f*x)^2)/(a*d) + (3*(e + f*x)^2*ArcTanh[E^(c + d*x)])/(a*d) - (f^2*ArcTanh[Cosh[c + d*x]])/(a*d^3)
+ (I*(e + f*x)^2*Coth[c + d*x])/(a*d) - (f*(e + f*x)*Csch[c + d*x])/(a*d^2) - ((e + f*x)^2*Coth[c + d*x]*Csch[
c + d*x])/(2*a*d) - ((4*I)*f*(e + f*x)*Log[1 + I*E^(c + d*x)])/(a*d^2) - ((2*I)*f*(e + f*x)*Log[1 - E^(2*(c +
d*x))])/(a*d^2) + (3*f*(e + f*x)*PolyLog[2, -E^(c + d*x)])/(a*d^2) - ((4*I)*f^2*PolyLog[2, (-I)*E^(c + d*x)])/
(a*d^3) - (3*f*(e + f*x)*PolyLog[2, E^(c + d*x)])/(a*d^2) - (I*f^2*PolyLog[2, E^(2*(c + d*x))])/(a*d^3) - (3*f
^2*PolyLog[3, -E^(c + d*x)])/(a*d^3) + (3*f^2*PolyLog[3, E^(c + d*x)])/(a*d^3) + (I*(e + f*x)^2*Tanh[c/2 + (I/
4)*Pi + (d*x)/2])/(a*d)

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 2317

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

Rule 2320

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

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2611

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

Rule 3399

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[(2*a)^n, Int[(c
+ d*x)^m*Sin[(1/2)*(e + Pi*(a/(2*b))) + f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2
- b^2, 0] && IntegerQ[n] && (GtQ[n, 0] || IGtQ[m, 0])

Rule 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((
c + d*x)^(m + 1)/(d*(m + 1))), x] + Dist[2*I, Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

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 4269

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 4271

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-b^2)*(c + d*x)^m*Cot[e
+ f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (Dist[b^2*d^2*m*((m - 1)/(f^2*(n - 1)*(n - 2))), Int[(c +
d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Dist[b^2*((n - 2)/(n - 1)), Int[(c + d*x)^m*(b*Csc[e + f*x])^
(n - 2), x], x] - Simp[b^2*d*m*(c + d*x)^(m - 1)*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x]) /; Free
Q[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

Rule 5694

Int[(Csch[(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*Csch[c + d*x]^n, x], x] - Dist[b/a, Int[(e + f*x)^m*(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]

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]

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

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1496\) vs. \(2(368)=736\).

Time = 8.80 (sec) , antiderivative size = 1496, normalized size of antiderivative = 4.07 \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {4 e^c f \left (\frac {e^{-c} (e+f x)^2}{2 f}+\frac {\left (i+e^{-c}\right ) (e+f x) \log \left (1-i e^{-c-d x}\right )}{d}-\frac {e^{-c} \left (1+i e^c\right ) f \operatorname {PolyLog}\left (2,i e^{-c-d x}\right )}{d^2}\right )}{a d \left (-i+e^c\right )}+\frac {-d \left (-1+e^{2 c}\right ) \left (3 d^2 e^2-4 i d e f-2 f^2\right ) x+d \left (-1+e^{2 c}\right ) \left (3 d^2 e^2+4 i d e f-2 f^2\right ) x+4 i d^2 (e+f x)^2-2 d \left (-1+e^{2 c}\right ) (3 d e+2 i f) f x \log \left (1-e^{-c-d x}\right )-3 d^2 \left (-1+e^{2 c}\right ) f^2 x^2 \log \left (1-e^{-c-d x}\right )+2 d \left (-1+e^{2 c}\right ) (3 d e-2 i f) f x \log \left (1+e^{-c-d x}\right )+3 d^2 \left (-1+e^{2 c}\right ) f^2 x^2 \log \left (1+e^{-c-d x}\right )-\left (-1+e^{2 c}\right ) \left (3 d^2 e^2+4 i d e f-2 f^2\right ) \log \left (1-e^{c+d x}\right )+\left (-1+e^{2 c}\right ) \left (3 d^2 e^2-4 i d e f-2 f^2\right ) \log \left (1+e^{c+d x}\right )-2 \left (-1+e^{2 c}\right ) (3 d e-2 i f) f \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )-6 d \left (-1+e^{2 c}\right ) f^2 x \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )+2 \left (-1+e^{2 c}\right ) (3 d e+2 i f) f \operatorname {PolyLog}\left (2,e^{-c-d x}\right )+6 d \left (-1+e^{2 c}\right ) f^2 x \operatorname {PolyLog}\left (2,e^{-c-d x}\right )-6 \left (-1+e^{2 c}\right ) f^2 \operatorname {PolyLog}\left (3,-e^{-c-d x}\right )+6 \left (-1+e^{2 c}\right ) f^2 \operatorname {PolyLog}\left (3,e^{-c-d x}\right )}{2 a d^3 \left (-1+e^{2 c}\right )}+\frac {\text {csch}(c) \text {csch}^2(c+d x) \left (2 e f \cosh \left (\frac {d x}{2}\right )+2 f^2 x \cosh \left (\frac {d x}{2}\right )+2 e f \cosh \left (\frac {3 d x}{2}\right )+2 f^2 x \cosh \left (\frac {3 d x}{2}\right )+5 i d e^2 \cosh \left (c-\frac {d x}{2}\right )+10 i d e f x \cosh \left (c-\frac {d x}{2}\right )+5 i d f^2 x^2 \cosh \left (c-\frac {d x}{2}\right )-i d e^2 \cosh \left (c+\frac {d x}{2}\right )-2 i d e f x \cosh \left (c+\frac {d x}{2}\right )-i d f^2 x^2 \cosh \left (c+\frac {d x}{2}\right )-2 e f \cosh \left (2 c+\frac {d x}{2}\right )-2 f^2 x \cosh \left (2 c+\frac {d x}{2}\right )+i d e^2 \cosh \left (c+\frac {3 d x}{2}\right )+2 i d e f x \cosh \left (c+\frac {3 d x}{2}\right )+i d f^2 x^2 \cosh \left (c+\frac {3 d x}{2}\right )-2 e f \cosh \left (2 c+\frac {3 d x}{2}\right )-2 f^2 x \cosh \left (2 c+\frac {3 d x}{2}\right )-3 i d e^2 \cosh \left (3 c+\frac {3 d x}{2}\right )-6 i d e f x \cosh \left (3 c+\frac {3 d x}{2}\right )-3 i d f^2 x^2 \cosh \left (3 c+\frac {3 d x}{2}\right )-4 i d e^2 \cosh \left (c+\frac {5 d x}{2}\right )-8 i d e f x \cosh \left (c+\frac {5 d x}{2}\right )-4 i d f^2 x^2 \cosh \left (c+\frac {5 d x}{2}\right )+2 i d e^2 \cosh \left (3 c+\frac {5 d x}{2}\right )+4 i d e f x \cosh \left (3 c+\frac {5 d x}{2}\right )+2 i d f^2 x^2 \cosh \left (3 c+\frac {5 d x}{2}\right )-d e^2 \sinh \left (\frac {d x}{2}\right )-2 d e f x \sinh \left (\frac {d x}{2}\right )-d f^2 x^2 \sinh \left (\frac {d x}{2}\right )-d e^2 \sinh \left (\frac {3 d x}{2}\right )-2 d e f x \sinh \left (\frac {3 d x}{2}\right )-d f^2 x^2 \sinh \left (\frac {3 d x}{2}\right )+2 i e f \sinh \left (c-\frac {d x}{2}\right )+2 i f^2 x \sinh \left (c-\frac {d x}{2}\right )+2 i e f \sinh \left (c+\frac {d x}{2}\right )+2 i f^2 x \sinh \left (c+\frac {d x}{2}\right )-3 d e^2 \sinh \left (2 c+\frac {d x}{2}\right )-6 d e f x \sinh \left (2 c+\frac {d x}{2}\right )-3 d f^2 x^2 \sinh \left (2 c+\frac {d x}{2}\right )+2 i e f \sinh \left (c+\frac {3 d x}{2}\right )+2 i f^2 x \sinh \left (c+\frac {3 d x}{2}\right )-d e^2 \sinh \left (2 c+\frac {3 d x}{2}\right )-2 d e f x \sinh \left (2 c+\frac {3 d x}{2}\right )-d f^2 x^2 \sinh \left (2 c+\frac {3 d x}{2}\right )-2 i e f \sinh \left (3 c+\frac {3 d x}{2}\right )-2 i f^2 x \sinh \left (3 c+\frac {3 d x}{2}\right )+2 d e^2 \sinh \left (2 c+\frac {5 d x}{2}\right )+4 d e f x \sinh \left (2 c+\frac {5 d x}{2}\right )+2 d f^2 x^2 \sinh \left (2 c+\frac {5 d x}{2}\right )\right )}{8 a d^2 \left (\cosh \left (\frac {c}{2}\right )+i \sinh \left (\frac {c}{2}\right )\right ) \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )+i \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )} \]

[In]

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

[Out]

(-4*E^c*f*((e + f*x)^2/(2*E^c*f) + ((I + E^(-c))*(e + f*x)*Log[1 - I*E^(-c - d*x)])/d - ((1 + I*E^c)*f*PolyLog
[2, I*E^(-c - d*x)])/(d^2*E^c)))/(a*d*(-I + E^c)) + (-(d*(-1 + E^(2*c))*(3*d^2*e^2 - (4*I)*d*e*f - 2*f^2)*x) +
d*(-1 + E^(2*c))*(3*d^2*e^2 + (4*I)*d*e*f - 2*f^2)*x + (4*I)*d^2*(e + f*x)^2 - 2*d*(-1 + E^(2*c))*(3*d*e + (2
*I)*f)*f*x*Log[1 - E^(-c - d*x)] - 3*d^2*(-1 + E^(2*c))*f^2*x^2*Log[1 - E^(-c - d*x)] + 2*d*(-1 + E^(2*c))*(3*
d*e - (2*I)*f)*f*x*Log[1 + E^(-c - d*x)] + 3*d^2*(-1 + E^(2*c))*f^2*x^2*Log[1 + E^(-c - d*x)] - (-1 + E^(2*c))
*(3*d^2*e^2 + (4*I)*d*e*f - 2*f^2)*Log[1 - E^(c + d*x)] + (-1 + E^(2*c))*(3*d^2*e^2 - (4*I)*d*e*f - 2*f^2)*Log
[1 + E^(c + d*x)] - 2*(-1 + E^(2*c))*(3*d*e - (2*I)*f)*f*PolyLog[2, -E^(-c - d*x)] - 6*d*(-1 + E^(2*c))*f^2*x*
PolyLog[2, -E^(-c - d*x)] + 2*(-1 + E^(2*c))*(3*d*e + (2*I)*f)*f*PolyLog[2, E^(-c - d*x)] + 6*d*(-1 + E^(2*c))
*f^2*x*PolyLog[2, E^(-c - d*x)] - 6*(-1 + E^(2*c))*f^2*PolyLog[3, -E^(-c - d*x)] + 6*(-1 + E^(2*c))*f^2*PolyLo
g[3, E^(-c - d*x)])/(2*a*d^3*(-1 + E^(2*c))) + (Csch[c]*Csch[c + d*x]^2*(2*e*f*Cosh[(d*x)/2] + 2*f^2*x*Cosh[(d
*x)/2] + 2*e*f*Cosh[(3*d*x)/2] + 2*f^2*x*Cosh[(3*d*x)/2] + (5*I)*d*e^2*Cosh[c - (d*x)/2] + (10*I)*d*e*f*x*Cosh
[c - (d*x)/2] + (5*I)*d*f^2*x^2*Cosh[c - (d*x)/2] - I*d*e^2*Cosh[c + (d*x)/2] - (2*I)*d*e*f*x*Cosh[c + (d*x)/2
] - I*d*f^2*x^2*Cosh[c + (d*x)/2] - 2*e*f*Cosh[2*c + (d*x)/2] - 2*f^2*x*Cosh[2*c + (d*x)/2] + I*d*e^2*Cosh[c +
(3*d*x)/2] + (2*I)*d*e*f*x*Cosh[c + (3*d*x)/2] + I*d*f^2*x^2*Cosh[c + (3*d*x)/2] - 2*e*f*Cosh[2*c + (3*d*x)/2
] - 2*f^2*x*Cosh[2*c + (3*d*x)/2] - (3*I)*d*e^2*Cosh[3*c + (3*d*x)/2] - (6*I)*d*e*f*x*Cosh[3*c + (3*d*x)/2] -
(3*I)*d*f^2*x^2*Cosh[3*c + (3*d*x)/2] - (4*I)*d*e^2*Cosh[c + (5*d*x)/2] - (8*I)*d*e*f*x*Cosh[c + (5*d*x)/2] -
(4*I)*d*f^2*x^2*Cosh[c + (5*d*x)/2] + (2*I)*d*e^2*Cosh[3*c + (5*d*x)/2] + (4*I)*d*e*f*x*Cosh[3*c + (5*d*x)/2]
+ (2*I)*d*f^2*x^2*Cosh[3*c + (5*d*x)/2] - d*e^2*Sinh[(d*x)/2] - 2*d*e*f*x*Sinh[(d*x)/2] - d*f^2*x^2*Sinh[(d*x)
/2] - d*e^2*Sinh[(3*d*x)/2] - 2*d*e*f*x*Sinh[(3*d*x)/2] - d*f^2*x^2*Sinh[(3*d*x)/2] + (2*I)*e*f*Sinh[c - (d*x)
/2] + (2*I)*f^2*x*Sinh[c - (d*x)/2] + (2*I)*e*f*Sinh[c + (d*x)/2] + (2*I)*f^2*x*Sinh[c + (d*x)/2] - 3*d*e^2*Si
nh[2*c + (d*x)/2] - 6*d*e*f*x*Sinh[2*c + (d*x)/2] - 3*d*f^2*x^2*Sinh[2*c + (d*x)/2] + (2*I)*e*f*Sinh[c + (3*d*
x)/2] + (2*I)*f^2*x*Sinh[c + (3*d*x)/2] - d*e^2*Sinh[2*c + (3*d*x)/2] - 2*d*e*f*x*Sinh[2*c + (3*d*x)/2] - d*f^
2*x^2*Sinh[2*c + (3*d*x)/2] - (2*I)*e*f*Sinh[3*c + (3*d*x)/2] - (2*I)*f^2*x*Sinh[3*c + (3*d*x)/2] + 2*d*e^2*Si
nh[2*c + (5*d*x)/2] + 4*d*e*f*x*Sinh[2*c + (5*d*x)/2] + 2*d*f^2*x^2*Sinh[2*c + (5*d*x)/2]))/(8*a*d^2*(Cosh[c/2
] + I*Sinh[c/2])*(Cosh[c/2 + (d*x)/2] + I*Sinh[c/2 + (d*x)/2]))

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1146 vs. \(2 (343 ) = 686\).

Time = 2.78 (sec) , antiderivative size = 1147, normalized size of antiderivative = 3.12

method	result	size
risch	\(\text {Expression too large to display}\)	\(1147\)

[In]

int((f*x+e)^2*csch(d*x+c)^3/(a+I*a*sinh(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-4*I*f^2*polylog(2,-I*exp(d*x+c))/a/d^3-3*f^2*polylog(3,-exp(d*x+c))/a/d^3+3*f^2*polylog(3,exp(d*x+c))/a/d^3-3
/2/a/d*e^2*ln(exp(d*x+c)-1)+3/2/a/d*e^2*ln(exp(d*x+c)+1)-3/a/d^2*e*f*ln(1-exp(d*x+c))*c-3/a/d*e*f*ln(1-exp(d*x
+c))*x+3/a/d*e*f*ln(exp(d*x+c)+1)*x+3/a/d^2*e*c*f*ln(exp(d*x+c)-1)+2*I/a/d^3*f^2*c*ln(1+exp(2*d*x+2*c))+8*I/a/
d^2*e*f*ln(exp(d*x+c))-2*I/a/d^2*e*f*ln(exp(d*x+c)-1)-2*I/a/d^2*e*f*ln(exp(d*x+c)+1)-2*I/a/d^2*e*f*ln(1+exp(2*
d*x+2*c))-2*I/a/d^2*f^2*ln(1-exp(d*x+c))*x-2*I/a/d^2*f^2*ln(exp(d*x+c)+1)*x-4*I/a/d^2*f^2*ln(1+I*exp(d*x+c))*x
+8*I/a/d^2*f^2*c*x-8*I/a/d^3*c*f^2*ln(exp(d*x+c))-4*I/a/d^3*f^2*ln(1+I*exp(d*x+c))*c-2*I/a/d^3*f^2*ln(1-exp(d*
x+c))*c+2*I/a/d^3*f^2*c*ln(exp(d*x+c)-1)-3/a/d^2*e*f*polylog(2,exp(d*x+c))+3/a/d^2*e*f*polylog(2,-exp(d*x+c))+
3/2/a/d^3*f^2*ln(1-exp(d*x+c))*c^2-3/2/a/d*f^2*ln(1-exp(d*x+c))*x^2-3/a/d^2*f^2*polylog(2,exp(d*x+c))*x+3/2/a/
d*f^2*ln(exp(d*x+c)+1)*x^2+3/a/d^2*f^2*polylog(2,-exp(d*x+c))*x-4/a/d^3*c*f^2*arctan(exp(d*x+c))+4/a/d^2*e*f*a
rctan(exp(d*x+c))-2*I/a/d^3*f^2*polylog(2,-exp(d*x+c))+4*I/a/d^3*c^2*f^2+4*I/a/d*f^2*x^2-2*I/a/d^3*f^2*polylog
(2,exp(d*x+c))-3/2/a/d^3*c^2*f^2*ln(exp(d*x+c)-1)-(4*d*e^2+3*d*e^2*exp(4*d*x+4*c)+2*f^2*x*exp(4*d*x+4*c)+2*e*f
*exp(4*d*x+4*c)-2*f^2*x*exp(2*d*x+2*c)-2*e*f*exp(2*d*x+2*c)+4*d*f^2*x^2+3*d*f^2*x^2*exp(4*d*x+4*c)+2*I*f^2*x*e
xp(d*x+c)-5*f^2*x^2*d*exp(2*d*x+2*c)+6*d*e*f*x*exp(4*d*x+4*c)-3*I*d*f^2*x^2*exp(3*d*x+3*c)-2*I*e*f*exp(3*d*x+3
*c)+2*I*d*e*f*x*exp(d*x+c)-3*I*d*e^2*exp(3*d*x+3*c)+I*d*e^2*exp(d*x+c)+2*I*exp(d*x+c)*e*f-6*I*d*e*f*x*exp(3*d*
x+3*c)-2*I*f^2*x*exp(3*d*x+3*c)+I*d*f^2*x^2*exp(d*x+c)-10*e*f*x*d*exp(2*d*x+2*c)+8*d*e*f*x-5*e^2*d*exp(2*d*x+2
*c))/(exp(2*d*x+2*c)-1)^2/d^2/(exp(d*x+c)-I)/a+1/a/d^3*f^2*ln(exp(d*x+c)-1)-1/a/d^3*f^2*ln(exp(d*x+c)+1)

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. 2215 vs. \(2 (329) = 658\).

Time = 0.28 (sec) , antiderivative size = 2215, normalized size of antiderivative = 6.02 \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/2*(8*d^2*e^2 - 16*c*d*e*f + 8*c^2*f^2 + 8*(I*f^2*e^(5*d*x + 5*c) + f^2*e^(4*d*x + 4*c) - 2*I*f^2*e^(3*d*x +
3*c) - 2*f^2*e^(2*d*x + 2*c) + I*f^2*e^(d*x + c) + f^2)*dilog(-I*e^(d*x + c)) + 2*(3*I*d*f^2*x + 3*I*d*e*f +
2*f^2 - (3*d*f^2*x + 3*d*e*f - 2*I*f^2)*e^(5*d*x + 5*c) + (3*I*d*f^2*x + 3*I*d*e*f + 2*f^2)*e^(4*d*x + 4*c) +
2*(3*d*f^2*x + 3*d*e*f - 2*I*f^2)*e^(3*d*x + 3*c) + 2*(-3*I*d*f^2*x - 3*I*d*e*f - 2*f^2)*e^(2*d*x + 2*c) - (3*
d*f^2*x + 3*d*e*f - 2*I*f^2)*e^(d*x + c))*dilog(-e^(d*x + c)) + 2*(-3*I*d*f^2*x - 3*I*d*e*f + 2*f^2 + (3*d*f^2
*x + 3*d*e*f + 2*I*f^2)*e^(5*d*x + 5*c) + (-3*I*d*f^2*x - 3*I*d*e*f + 2*f^2)*e^(4*d*x + 4*c) - 2*(3*d*f^2*x +
3*d*e*f + 2*I*f^2)*e^(3*d*x + 3*c) + 2*(3*I*d*f^2*x + 3*I*d*e*f - 2*f^2)*e^(2*d*x + 2*c) + (3*d*f^2*x + 3*d*e*
f + 2*I*f^2)*e^(d*x + c))*dilog(e^(d*x + c)) + 8*(-I*d^2*f^2*x^2 - 2*I*d^2*e*f*x - 2*I*c*d*e*f + I*c^2*f^2)*e^
(5*d*x + 5*c) - 2*(d^2*f^2*x^2 - 3*d^2*e^2 + 2*(4*c - 1)*d*e*f - 4*c^2*f^2 + 2*(d^2*e*f - d*f^2)*x)*e^(4*d*x +
4*c) + 2*(5*I*d^2*f^2*x^2 - 3*I*d^2*e^2 + 2*(8*I*c - I)*d*e*f - 8*I*c^2*f^2 + 2*(5*I*d^2*e*f - I*d*f^2)*x)*e^
(3*d*x + 3*c) + 2*(3*d^2*f^2*x^2 - 5*d^2*e^2 + 2*(8*c - 1)*d*e*f - 8*c^2*f^2 + 2*(3*d^2*e*f - d*f^2)*x)*e^(2*d
*x + 2*c) + 2*(-3*I*d^2*f^2*x^2 + I*d^2*e^2 + 2*(-4*I*c + I)*d*e*f + 4*I*c^2*f^2 + 2*(-3*I*d^2*e*f + I*d*f^2)*
x)*e^(d*x + c) - (-3*I*d^2*f^2*x^2 - 3*I*d^2*e^2 - 4*d*e*f + 2*I*f^2 - 2*(3*I*d^2*e*f + 2*d*f^2)*x + (3*d^2*f^
2*x^2 + 3*d^2*e^2 - 4*I*d*e*f - 2*f^2 + 2*(3*d^2*e*f - 2*I*d*f^2)*x)*e^(5*d*x + 5*c) + (-3*I*d^2*f^2*x^2 - 3*I
*d^2*e^2 - 4*d*e*f + 2*I*f^2 - 2*(3*I*d^2*e*f + 2*d*f^2)*x)*e^(4*d*x + 4*c) - 2*(3*d^2*f^2*x^2 + 3*d^2*e^2 - 4
*I*d*e*f - 2*f^2 + 2*(3*d^2*e*f - 2*I*d*f^2)*x)*e^(3*d*x + 3*c) - 2*(-3*I*d^2*f^2*x^2 - 3*I*d^2*e^2 - 4*d*e*f
+ 2*I*f^2 + 2*(-3*I*d^2*e*f - 2*d*f^2)*x)*e^(2*d*x + 2*c) + (3*d^2*f^2*x^2 + 3*d^2*e^2 - 4*I*d*e*f - 2*f^2 + 2
*(3*d^2*e*f - 2*I*d*f^2)*x)*e^(d*x + c))*log(e^(d*x + c) + 1) + 8*(d*e*f - c*f^2 + (I*d*e*f - I*c*f^2)*e^(5*d*
x + 5*c) + (d*e*f - c*f^2)*e^(4*d*x + 4*c) + 2*(-I*d*e*f + I*c*f^2)*e^(3*d*x + 3*c) - 2*(d*e*f - c*f^2)*e^(2*d
*x + 2*c) + (I*d*e*f - I*c*f^2)*e^(d*x + c))*log(e^(d*x + c) - I) - (3*I*d^2*e^2 - 2*(3*I*c + 2)*d*e*f + (3*I*
c^2 + 4*c - 2*I)*f^2 - (3*d^2*e^2 - 2*(3*c - 2*I)*d*e*f + (3*c^2 - 4*I*c - 2)*f^2)*e^(5*d*x + 5*c) + (3*I*d^2*
e^2 - 2*(3*I*c + 2)*d*e*f + (3*I*c^2 + 4*c - 2*I)*f^2)*e^(4*d*x + 4*c) + 2*(3*d^2*e^2 - 2*(3*c - 2*I)*d*e*f +
(3*c^2 - 4*I*c - 2)*f^2)*e^(3*d*x + 3*c) - 2*(3*I*d^2*e^2 + 2*(-3*I*c - 2)*d*e*f + (3*I*c^2 + 4*c - 2*I)*f^2)*
e^(2*d*x + 2*c) - (3*d^2*e^2 - 2*(3*c - 2*I)*d*e*f + (3*c^2 - 4*I*c - 2)*f^2)*e^(d*x + c))*log(e^(d*x + c) - 1
) + 8*(d*f^2*x + c*f^2 + (I*d*f^2*x + I*c*f^2)*e^(5*d*x + 5*c) + (d*f^2*x + c*f^2)*e^(4*d*x + 4*c) + 2*(-I*d*f
^2*x - I*c*f^2)*e^(3*d*x + 3*c) - 2*(d*f^2*x + c*f^2)*e^(2*d*x + 2*c) + (I*d*f^2*x + I*c*f^2)*e^(d*x + c))*log
(I*e^(d*x + c) + 1) - (3*I*d^2*f^2*x^2 + 6*I*c*d*e*f + (-3*I*c^2 - 4*c)*f^2 - 2*(-3*I*d^2*e*f + 2*d*f^2)*x - (
3*d^2*f^2*x^2 + 6*c*d*e*f - (3*c^2 - 4*I*c)*f^2 + 2*(3*d^2*e*f + 2*I*d*f^2)*x)*e^(5*d*x + 5*c) + (3*I*d^2*f^2*
x^2 + 6*I*c*d*e*f + (-3*I*c^2 - 4*c)*f^2 - 2*(-3*I*d^2*e*f + 2*d*f^2)*x)*e^(4*d*x + 4*c) + 2*(3*d^2*f^2*x^2 +
6*c*d*e*f - (3*c^2 - 4*I*c)*f^2 + 2*(3*d^2*e*f + 2*I*d*f^2)*x)*e^(3*d*x + 3*c) - 2*(3*I*d^2*f^2*x^2 + 6*I*c*d*
e*f + (-3*I*c^2 - 4*c)*f^2 + 2*(3*I*d^2*e*f - 2*d*f^2)*x)*e^(2*d*x + 2*c) - (3*d^2*f^2*x^2 + 6*c*d*e*f - (3*c^
2 - 4*I*c)*f^2 + 2*(3*d^2*e*f + 2*I*d*f^2)*x)*e^(d*x + c))*log(-e^(d*x + c) + 1) + 6*(f^2*e^(5*d*x + 5*c) - I*
f^2*e^(4*d*x + 4*c) - 2*f^2*e^(3*d*x + 3*c) + 2*I*f^2*e^(2*d*x + 2*c) + f^2*e^(d*x + c) - I*f^2)*polylog(3, -e
^(d*x + c)) - 6*(f^2*e^(5*d*x + 5*c) - I*f^2*e^(4*d*x + 4*c) - 2*f^2*e^(3*d*x + 3*c) + 2*I*f^2*e^(2*d*x + 2*c)
+ f^2*e^(d*x + c) - I*f^2)*polylog(3, e^(d*x + c)))/(a*d^3*e^(5*d*x + 5*c) - I*a*d^3*e^(4*d*x + 4*c) - 2*a*d^
3*e^(3*d*x + 3*c) + 2*I*a*d^3*e^(2*d*x + 2*c) + a*d^3*e^(d*x + c) - I*a*d^3)

Sympy [F]

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

[In]

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

[Out]

-I*(Integral(e**2*csch(c + d*x)**3/(sinh(c + d*x) - I), x) + Integral(f**2*x**2*csch(c + d*x)**3/(sinh(c + d*x
) - I), x) + Integral(2*e*f*x*csch(c + d*x)**3/(sinh(c + d*x) - I), x))/a

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 863 vs. \(2 (329) = 658\).

Time = 0.46 (sec) , antiderivative size = 863, normalized size of antiderivative = 2.35 \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/2*e^2*(2*(-I*e^(-d*x - c) - 5*e^(-2*d*x - 2*c) + 3*I*e^(-3*d*x - 3*c) + 3*e^(-4*d*x - 4*c) + 4)/((a*e^(-d*x
- c) - 2*I*a*e^(-2*d*x - 2*c) - 2*a*e^(-3*d*x - 3*c) + I*a*e^(-4*d*x - 4*c) + a*e^(-5*d*x - 5*c) + I*a)*d) -
3*log(e^(-d*x - c) + 1)/(a*d) + 3*log(e^(-d*x - c) - 1)/(a*d)) + 2*I*f^2*x^2/(a*d) + 4*I*e*f*x/(a*d) - (4*d*f^
2*x^2 + 8*d*e*f*x + (3*d*f^2*x^2*e^(4*c) + 2*e*f*e^(4*c) + 2*(3*d*e*f + f^2)*x*e^(4*c))*e^(4*d*x) + (-3*I*d*f^
2*x^2*e^(3*c) - 2*I*e*f*e^(3*c) - 2*(3*I*d*e*f + I*f^2)*x*e^(3*c))*e^(3*d*x) - (5*d*f^2*x^2*e^(2*c) + 2*e*f*e^
(2*c) + 2*(5*d*e*f + f^2)*x*e^(2*c))*e^(2*d*x) + (I*d*f^2*x^2*e^c + 2*I*e*f*e^c - 2*(-I*d*e*f - I*f^2)*x*e^c)*
e^(d*x))/(a*d^2*e^(5*d*x + 5*c) - I*a*d^2*e^(4*d*x + 4*c) - 2*a*d^2*e^(3*d*x + 3*c) + 2*I*a*d^2*e^(2*d*x + 2*c
) + a*d^2*e^(d*x + c) - I*a*d^2) - 4*I*e*f*log(I*e^(d*x + c) + 1)/(a*d^2) + 3/2*(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)))*f^2/(a*d^3) - 3/2*(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)))*f^2/(a*d^3) - 4*I*(d*x*log(I*e^(d*x + c) + 1) + dilog(-I*
e^(d*x + c)))*f^2/(a*d^3) + (2*I*d*e*f + f^2)*x/(a*d^2) + (2*I*d*e*f - f^2)*x/(a*d^2) + (3*d*e*f - 2*I*f^2)*(d
*x*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))/(a*d^3) - (3*d*e*f + 2*I*f^2)*(d*x*log(-e^(d*x + c) + 1) + dilo
g(e^(d*x + c)))/(a*d^3) - (2*I*d*e*f + f^2)*log(e^(d*x + c) + 1)/(a*d^3) - (2*I*d*e*f - f^2)*log(e^(d*x + c) -
1)/(a*d^3) + 1/2*(d^3*f^2*x^3 + (3*d*e*f + 2*I*f^2)*d^2*x^2)/(a*d^3) - 1/2*(d^3*f^2*x^3 + (3*d*e*f - 2*I*f^2)
*d^2*x^2)/(a*d^3)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int \frac {{\left (e+f\,x\right )}^2}{{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]

[In]

int((e + f*x)^2/(sinh(c + d*x)^3*(a + a*sinh(c + d*x)*1i)),x)

[Out]

int((e + f*x)^2/(sinh(c + d*x)^3*(a + a*sinh(c + d*x)*1i)), x)