\(\int \frac {(e+f x)^2 \text {sech}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx\) [284]

Optimal result

Integrand size = 31, antiderivative size = 423 \[ \int \frac {(e+f x)^2 \text {sech}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {3 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{4 a d}-\frac {5 f^2 \arctan (\sinh (c+d x))}{6 a d^3}+\frac {i f^2 \log (\cosh (c+d x))}{3 a d^3}-\frac {3 i f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{4 a d^2}+\frac {3 i f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{4 a d^2}+\frac {3 i f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{4 a d^3}-\frac {3 i f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{4 a d^3}+\frac {3 f (e+f x) \text {sech}(c+d x)}{4 a d^2}-\frac {i f^2 \text {sech}^2(c+d x)}{12 a d^3}+\frac {f (e+f x) \text {sech}^3(c+d x)}{6 a d^2}+\frac {i (e+f x)^2 \text {sech}^4(c+d x)}{4 a d}-\frac {i f (e+f x) \tanh (c+d x)}{3 a d^2}-\frac {f^2 \text {sech}(c+d x) \tanh (c+d x)}{12 a d^3}+\frac {3 (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{8 a d}-\frac {i f (e+f x) \text {sech}^2(c+d x) \tanh (c+d x)}{6 a d^2}+\frac {(e+f x)^2 \text {sech}^3(c+d x) \tanh (c+d x)}{4 a d} \]

[Out]

3/4*(f*x+e)^2*arctan(exp(d*x+c))/a/d-5/6*f^2*arctan(sinh(d*x+c))/a/d^3-1/6*I*f*(f*x+e)*sech(d*x+c)^2*tanh(d*x+
c)/a/d^2-3/4*I*f*(f*x+e)*polylog(2,-I*exp(d*x+c))/a/d^2+3/4*I*f*(f*x+e)*polylog(2,I*exp(d*x+c))/a/d^2-1/3*I*f*
(f*x+e)*tanh(d*x+c)/a/d^2+1/3*I*f^2*ln(cosh(d*x+c))/a/d^3+3/4*f*(f*x+e)*sech(d*x+c)/a/d^2+1/4*I*(f*x+e)^2*sech
(d*x+c)^4/a/d+1/6*f*(f*x+e)*sech(d*x+c)^3/a/d^2-1/12*I*f^2*sech(d*x+c)^2/a/d^3-3/4*I*f^2*polylog(3,I*exp(d*x+c
))/a/d^3-1/12*f^2*sech(d*x+c)*tanh(d*x+c)/a/d^3+3/8*(f*x+e)^2*sech(d*x+c)*tanh(d*x+c)/a/d+3/4*I*f^2*polylog(3,
-I*exp(d*x+c))/a/d^3+1/4*(f*x+e)^2*sech(d*x+c)^3*tanh(d*x+c)/a/d

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.387, Rules used = {5690, 4271, 3853, 3855, 4265, 2611, 2320, 6724, 5559, 4270, 4269, 3556} \[ \int \frac {(e+f x)^2 \text {sech}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {5 f^2 \arctan (\sinh (c+d x))}{6 a d^3}+\frac {3 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{4 a d}+\frac {3 i f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{4 a d^3}-\frac {3 i f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{4 a d^3}-\frac {i f^2 \text {sech}^2(c+d x)}{12 a d^3}+\frac {i f^2 \log (\cosh (c+d x))}{3 a d^3}-\frac {f^2 \tanh (c+d x) \text {sech}(c+d x)}{12 a d^3}-\frac {3 i f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{4 a d^2}+\frac {3 i f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{4 a d^2}-\frac {i f (e+f x) \tanh (c+d x)}{3 a d^2}+\frac {f (e+f x) \text {sech}^3(c+d x)}{6 a d^2}+\frac {3 f (e+f x) \text {sech}(c+d x)}{4 a d^2}-\frac {i f (e+f x) \tanh (c+d x) \text {sech}^2(c+d x)}{6 a d^2}+\frac {i (e+f x)^2 \text {sech}^4(c+d x)}{4 a d}+\frac {(e+f x)^2 \tanh (c+d x) \text {sech}^3(c+d x)}{4 a d}+\frac {3 (e+f x)^2 \tanh (c+d x) \text {sech}(c+d x)}{8 a d} \]

[In]

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

[Out]

(3*(e + f*x)^2*ArcTan[E^(c + d*x)])/(4*a*d) - (5*f^2*ArcTan[Sinh[c + d*x]])/(6*a*d^3) + ((I/3)*f^2*Log[Cosh[c
+ d*x]])/(a*d^3) - (((3*I)/4)*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^2) + (((3*I)/4)*f*(e + f*x)*PolyL
og[2, I*E^(c + d*x)])/(a*d^2) + (((3*I)/4)*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(a*d^3) - (((3*I)/4)*f^2*PolyLog[
3, I*E^(c + d*x)])/(a*d^3) + (3*f*(e + f*x)*Sech[c + d*x])/(4*a*d^2) - ((I/12)*f^2*Sech[c + d*x]^2)/(a*d^3) +
(f*(e + f*x)*Sech[c + d*x]^3)/(6*a*d^2) + ((I/4)*(e + f*x)^2*Sech[c + d*x]^4)/(a*d) - ((I/3)*f*(e + f*x)*Tanh[
c + d*x])/(a*d^2) - (f^2*Sech[c + d*x]*Tanh[c + d*x])/(12*a*d^3) + (3*(e + f*x)^2*Sech[c + d*x]*Tanh[c + d*x])
/(8*a*d) - ((I/6)*f*(e + f*x)*Sech[c + d*x]^2*Tanh[c + d*x])/(a*d^2) + ((e + f*x)^2*Sech[c + d*x]^3*Tanh[c + d
*x])/(4*a*d)

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 3556

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

Rule 3853

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

Rule 3855

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

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 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 4270

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

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 5559

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

Rule 5690

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[1/a, Int[(e + f*x)^m*Sech[c + d*x]^(n + 2), x], x] + Dist[1/b, Int[(e + f*x)^m*Sech[c + d*x]^(n +
1)*Tanh[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && EqQ[a^2 + b^2, 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}& = -\frac {i \int (e+f x)^2 \text {sech}^4(c+d x) \tanh (c+d x) \, dx}{a}+\frac {\int (e+f x)^2 \text {sech}^5(c+d x) \, dx}{a} \\ & = \frac {f (e+f x) \text {sech}^3(c+d x)}{6 a d^2}+\frac {i (e+f x)^2 \text {sech}^4(c+d x)}{4 a d}+\frac {(e+f x)^2 \text {sech}^3(c+d x) \tanh (c+d x)}{4 a d}+\frac {3 \int (e+f x)^2 \text {sech}^3(c+d x) \, dx}{4 a}-\frac {(i f) \int (e+f x) \text {sech}^4(c+d x) \, dx}{2 a d}-\frac {f^2 \int \text {sech}^3(c+d x) \, dx}{6 a d^2} \\ & = \frac {3 f (e+f x) \text {sech}(c+d x)}{4 a d^2}-\frac {i f^2 \text {sech}^2(c+d x)}{12 a d^3}+\frac {f (e+f x) \text {sech}^3(c+d x)}{6 a d^2}+\frac {i (e+f x)^2 \text {sech}^4(c+d x)}{4 a d}-\frac {f^2 \text {sech}(c+d x) \tanh (c+d x)}{12 a d^3}+\frac {3 (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{8 a d}-\frac {i f (e+f x) \text {sech}^2(c+d x) \tanh (c+d x)}{6 a d^2}+\frac {(e+f x)^2 \text {sech}^3(c+d x) \tanh (c+d x)}{4 a d}+\frac {3 \int (e+f x)^2 \text {sech}(c+d x) \, dx}{8 a}-\frac {(i f) \int (e+f x) \text {sech}^2(c+d x) \, dx}{3 a d}-\frac {f^2 \int \text {sech}(c+d x) \, dx}{12 a d^2}-\frac {\left (3 f^2\right ) \int \text {sech}(c+d x) \, dx}{4 a d^2} \\ & = \frac {3 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{4 a d}-\frac {5 f^2 \arctan (\sinh (c+d x))}{6 a d^3}+\frac {3 f (e+f x) \text {sech}(c+d x)}{4 a d^2}-\frac {i f^2 \text {sech}^2(c+d x)}{12 a d^3}+\frac {f (e+f x) \text {sech}^3(c+d x)}{6 a d^2}+\frac {i (e+f x)^2 \text {sech}^4(c+d x)}{4 a d}-\frac {i f (e+f x) \tanh (c+d x)}{3 a d^2}-\frac {f^2 \text {sech}(c+d x) \tanh (c+d x)}{12 a d^3}+\frac {3 (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{8 a d}-\frac {i f (e+f x) \text {sech}^2(c+d x) \tanh (c+d x)}{6 a d^2}+\frac {(e+f x)^2 \text {sech}^3(c+d x) \tanh (c+d x)}{4 a d}-\frac {(3 i f) \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{4 a d}+\frac {(3 i f) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{4 a d}+\frac {\left (i f^2\right ) \int \tanh (c+d x) \, dx}{3 a d^2} \\ & = \frac {3 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{4 a d}-\frac {5 f^2 \arctan (\sinh (c+d x))}{6 a d^3}+\frac {i f^2 \log (\cosh (c+d x))}{3 a d^3}-\frac {3 i f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{4 a d^2}+\frac {3 i f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{4 a d^2}+\frac {3 f (e+f x) \text {sech}(c+d x)}{4 a d^2}-\frac {i f^2 \text {sech}^2(c+d x)}{12 a d^3}+\frac {f (e+f x) \text {sech}^3(c+d x)}{6 a d^2}+\frac {i (e+f x)^2 \text {sech}^4(c+d x)}{4 a d}-\frac {i f (e+f x) \tanh (c+d x)}{3 a d^2}-\frac {f^2 \text {sech}(c+d x) \tanh (c+d x)}{12 a d^3}+\frac {3 (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{8 a d}-\frac {i f (e+f x) \text {sech}^2(c+d x) \tanh (c+d x)}{6 a d^2}+\frac {(e+f x)^2 \text {sech}^3(c+d x) \tanh (c+d x)}{4 a d}+\frac {\left (3 i f^2\right ) \int \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) \, dx}{4 a d^2}-\frac {\left (3 i f^2\right ) \int \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) \, dx}{4 a d^2} \\ & = \frac {3 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{4 a d}-\frac {5 f^2 \arctan (\sinh (c+d x))}{6 a d^3}+\frac {i f^2 \log (\cosh (c+d x))}{3 a d^3}-\frac {3 i f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{4 a d^2}+\frac {3 i f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{4 a d^2}+\frac {3 f (e+f x) \text {sech}(c+d x)}{4 a d^2}-\frac {i f^2 \text {sech}^2(c+d x)}{12 a d^3}+\frac {f (e+f x) \text {sech}^3(c+d x)}{6 a d^2}+\frac {i (e+f x)^2 \text {sech}^4(c+d x)}{4 a d}-\frac {i f (e+f x) \tanh (c+d x)}{3 a d^2}-\frac {f^2 \text {sech}(c+d x) \tanh (c+d x)}{12 a d^3}+\frac {3 (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{8 a d}-\frac {i f (e+f x) \text {sech}^2(c+d x) \tanh (c+d x)}{6 a d^2}+\frac {(e+f x)^2 \text {sech}^3(c+d x) \tanh (c+d x)}{4 a d}+\frac {\left (3 i f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{c+d x}\right )}{4 a d^3}-\frac {\left (3 i f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{c+d x}\right )}{4 a d^3} \\ & = \frac {3 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{4 a d}-\frac {5 f^2 \arctan (\sinh (c+d x))}{6 a d^3}+\frac {i f^2 \log (\cosh (c+d x))}{3 a d^3}-\frac {3 i f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{4 a d^2}+\frac {3 i f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{4 a d^2}+\frac {3 i f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{4 a d^3}-\frac {3 i f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{4 a d^3}+\frac {3 f (e+f x) \text {sech}(c+d x)}{4 a d^2}-\frac {i f^2 \text {sech}^2(c+d x)}{12 a d^3}+\frac {f (e+f x) \text {sech}^3(c+d x)}{6 a d^2}+\frac {i (e+f x)^2 \text {sech}^4(c+d x)}{4 a d}-\frac {i f (e+f x) \tanh (c+d x)}{3 a d^2}-\frac {f^2 \text {sech}(c+d x) \tanh (c+d x)}{12 a d^3}+\frac {3 (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{8 a d}-\frac {i f (e+f x) \text {sech}^2(c+d x) \tanh (c+d x)}{6 a d^2}+\frac {(e+f x)^2 \text {sech}^3(c+d x) \tanh (c+d x)}{4 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. \(1363\) vs. \(2(423)=846\).

Time = 8.18 (sec) , antiderivative size = 1363, normalized size of antiderivative = 3.22 \[ \int \frac {(e+f x)^2 \text {sech}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {e^c \left (3 d^2 e^2 e^{-c} x-4 e^{-c} f^2 x-e^{-c} \left (1-i e^c\right ) \left (3 d^2 e^2-4 f^2\right ) x+3 d^2 e e^{-c} f x^2+d^2 e^{-c} f^2 x^3+6 d e e^{-c} \left (1-i e^c\right ) f x \log \left (1+i e^{-c-d x}\right )+3 d e^{-c} \left (1-i e^c\right ) f^2 x^2 \log \left (1+i e^{-c-d x}\right )+\frac {e^{-c} \left (1-i e^c\right ) \left (3 d^2 e^2-4 f^2\right ) \log \left (i+e^{c+d x}\right )}{d}-6 e e^{-c} \left (1-i e^c\right ) f \operatorname {PolyLog}\left (2,-i e^{-c-d x}\right )-6 e^{-c} \left (1-i e^c\right ) f^2 x \operatorname {PolyLog}\left (2,-i e^{-c-d x}\right )-\frac {6 e^{-c} \left (1-i e^c\right ) f^2 \operatorname {PolyLog}\left (3,-i e^{-c-d x}\right )}{d}\right )}{8 a d^2 \left (i+e^c\right )}-\frac {9 d^2 e^2 x-28 f^2 x-\left (1+i e^c\right ) \left (9 d^2 e^2-28 f^2\right ) x+9 d^2 e f x^2+3 d^2 f^2 x^3+18 d e \left (1+i e^c\right ) f x \log \left (1-i e^{-c-d x}\right )+9 d \left (1+i e^c\right ) f^2 x^2 \log \left (1-i e^{-c-d x}\right )+\frac {\left (1+i e^c\right ) \left (9 d^2 e^2-28 f^2\right ) \log \left (i-e^{c+d x}\right )}{d}-18 e \left (1+i e^c\right ) f \operatorname {PolyLog}\left (2,i e^{-c-d x}\right )-18 \left (1+i e^c\right ) f^2 x \operatorname {PolyLog}\left (2,i e^{-c-d x}\right )-\frac {18 \left (1+i e^c\right ) f^2 \operatorname {PolyLog}\left (3,i e^{-c-d x}\right )}{d}}{24 a d^2 \left (-i+e^c\right )}+\frac {\frac {3 e^2 x \cosh (c)}{4 a}+\frac {3 e^2 x \sinh (c)}{4 a}}{1+\cosh (2 c)+\sinh (2 c)}+\frac {\frac {3 e f x^2 \cosh (c)}{4 a}+\frac {3 e f x^2 \sinh (c)}{4 a}}{1+\cosh (2 c)+\sinh (2 c)}+\frac {\frac {f^2 x^3 \cosh (c)}{4 a}+\frac {f^2 x^3 \sinh (c)}{4 a}}{1+\cosh (2 c)+\sinh (2 c)}-\frac {i \left (e^2+2 e f x+f^2 x^2\right )}{8 a d \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )-i \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^2}+\frac {i \left (e f \sinh \left (\frac {d x}{2}\right )+f^2 x \sinh \left (\frac {d x}{2}\right )\right )}{2 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 )}+\frac {i \left (e^2+2 e f x+f^2 x^2\right )}{8 a d \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )+i \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^4}-\frac {i \left (e f \sinh \left (\frac {d x}{2}\right )+f^2 x \sinh \left (\frac {d x}{2}\right )\right )}{6 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 )^3}+\frac {3 i d^2 e^2 \cosh \left (\frac {c}{2}\right )+d e f \cosh \left (\frac {c}{2}\right )-i f^2 \cosh \left (\frac {c}{2}\right )+6 i d^2 e f x \cosh \left (\frac {c}{2}\right )+d f^2 x \cosh \left (\frac {c}{2}\right )+3 i d^2 f^2 x^2 \cosh \left (\frac {c}{2}\right )-3 d^2 e^2 \sinh \left (\frac {c}{2}\right )-i d e f \sinh \left (\frac {c}{2}\right )+f^2 \sinh \left (\frac {c}{2}\right )-6 d^2 e f x \sinh \left (\frac {c}{2}\right )-i d f^2 x \sinh \left (\frac {c}{2}\right )-3 d^2 f^2 x^2 \sinh \left (\frac {c}{2}\right )}{12 a d^3 \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 )^2}-\frac {7 i \left (e f \sinh \left (\frac {d x}{2}\right )+f^2 x \sinh \left (\frac {d x}{2}\right )\right )}{6 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*Sech[c + d*x]^3)/(a + I*a*Sinh[c + d*x]),x]

[Out]

-1/8*(E^c*((3*d^2*e^2*x)/E^c - (4*f^2*x)/E^c - ((1 - I*E^c)*(3*d^2*e^2 - 4*f^2)*x)/E^c + (3*d^2*e*f*x^2)/E^c +
(d^2*f^2*x^3)/E^c + (6*d*e*(1 - I*E^c)*f*x*Log[1 + I*E^(-c - d*x)])/E^c + (3*d*(1 - I*E^c)*f^2*x^2*Log[1 + I*
E^(-c - d*x)])/E^c + ((1 - I*E^c)*(3*d^2*e^2 - 4*f^2)*Log[I + E^(c + d*x)])/(d*E^c) - (6*e*(1 - I*E^c)*f*PolyL
og[2, (-I)*E^(-c - d*x)])/E^c - (6*(1 - I*E^c)*f^2*x*PolyLog[2, (-I)*E^(-c - d*x)])/E^c - (6*(1 - I*E^c)*f^2*P
olyLog[3, (-I)*E^(-c - d*x)])/(d*E^c)))/(a*d^2*(I + E^c)) - (9*d^2*e^2*x - 28*f^2*x - (1 + I*E^c)*(9*d^2*e^2 -
28*f^2)*x + 9*d^2*e*f*x^2 + 3*d^2*f^2*x^3 + 18*d*e*(1 + I*E^c)*f*x*Log[1 - I*E^(-c - d*x)] + 9*d*(1 + I*E^c)*
f^2*x^2*Log[1 - I*E^(-c - d*x)] + ((1 + I*E^c)*(9*d^2*e^2 - 28*f^2)*Log[I - E^(c + d*x)])/d - 18*e*(1 + I*E^c)
*f*PolyLog[2, I*E^(-c - d*x)] - 18*(1 + I*E^c)*f^2*x*PolyLog[2, I*E^(-c - d*x)] - (18*(1 + I*E^c)*f^2*PolyLog[
3, I*E^(-c - d*x)])/d)/(24*a*d^2*(-I + E^c)) + ((3*e^2*x*Cosh[c])/(4*a) + (3*e^2*x*Sinh[c])/(4*a))/(1 + Cosh[2
*c] + Sinh[2*c]) + ((3*e*f*x^2*Cosh[c])/(4*a) + (3*e*f*x^2*Sinh[c])/(4*a))/(1 + Cosh[2*c] + Sinh[2*c]) + ((f^2
*x^3*Cosh[c])/(4*a) + (f^2*x^3*Sinh[c])/(4*a))/(1 + Cosh[2*c] + Sinh[2*c]) - ((I/8)*(e^2 + 2*e*f*x + f^2*x^2))
/(a*d*(Cosh[c/2 + (d*x)/2] - I*Sinh[c/2 + (d*x)/2])^2) + ((I/2)*(e*f*Sinh[(d*x)/2] + f^2*x*Sinh[(d*x)/2]))/(a*
d^2*(Cosh[c/2] - I*Sinh[c/2])*(Cosh[c/2 + (d*x)/2] - I*Sinh[c/2 + (d*x)/2])) + ((I/8)*(e^2 + 2*e*f*x + f^2*x^2
))/(a*d*(Cosh[c/2 + (d*x)/2] + I*Sinh[c/2 + (d*x)/2])^4) - ((I/6)*(e*f*Sinh[(d*x)/2] + f^2*x*Sinh[(d*x)/2]))/(
a*d^2*(Cosh[c/2] + I*Sinh[c/2])*(Cosh[c/2 + (d*x)/2] + I*Sinh[c/2 + (d*x)/2])^3) + ((3*I)*d^2*e^2*Cosh[c/2] +
d*e*f*Cosh[c/2] - I*f^2*Cosh[c/2] + (6*I)*d^2*e*f*x*Cosh[c/2] + d*f^2*x*Cosh[c/2] + (3*I)*d^2*f^2*x^2*Cosh[c/2
] - 3*d^2*e^2*Sinh[c/2] - I*d*e*f*Sinh[c/2] + f^2*Sinh[c/2] - 6*d^2*e*f*x*Sinh[c/2] - I*d*f^2*x*Sinh[c/2] - 3*
d^2*f^2*x^2*Sinh[c/2])/(12*a*d^3*(Cosh[c/2] + I*Sinh[c/2])*(Cosh[c/2 + (d*x)/2] + I*Sinh[c/2 + (d*x)/2])^2) -
(((7*I)/6)*(e*f*Sinh[(d*x)/2] + f^2*x*Sinh[(d*x)/2]))/(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. 960 vs. \(2 (373 ) = 746\).

Time = 87.45 (sec) , antiderivative size = 961, normalized size of antiderivative = 2.27

method	result	size
risch	\(\frac {-4 f^{2} {\mathrm e}^{3 d x +3 c}+9 d^{2} e^{2} {\mathrm e}^{5 d x +5 c}+18 d^{2} e f x \,{\mathrm e}^{d x +c}-2 f^{2} {\mathrm e}^{d x +c}+9 d^{2} x^{2} f^{2} {\mathrm e}^{d x +c}+9 d^{2} e^{2} {\mathrm e}^{d x +c}+16 d \,f^{2} x \,{\mathrm e}^{3 d x +3 c}+18 d^{2} e f x \,{\mathrm e}^{5 d x +5 c}+6 d^{2} e^{2} {\mathrm e}^{3 d x +3 c}-2 f^{2} {\mathrm e}^{5 d x +5 c}-2 \,{\mathrm e}^{d x +c} d \,f^{2} x -2 \,{\mathrm e}^{d x +c} d e f +18 i d^{2} f^{2} x^{2} {\mathrm e}^{2 d x +2 c}-36 i d \,f^{2} x \,{\mathrm e}^{4 d x +4 c}-36 i d e f \,{\mathrm e}^{4 d x +4 c}-44 i d e f \,{\mathrm e}^{2 d x +2 c}-18 i d^{2} f^{2} x^{2} {\mathrm e}^{4 d x +4 c}-44 i d \,f^{2} x \,{\mathrm e}^{2 d x +2 c}+6 d^{2} f^{2} x^{2} {\mathrm e}^{3 d x +3 c}+9 d^{2} f^{2} x^{2} {\mathrm e}^{5 d x +5 c}+18 d \,f^{2} x \,{\mathrm e}^{5 d x +5 c}+18 d e f \,{\mathrm e}^{5 d x +5 c}+12 d^{2} e f x \,{\mathrm e}^{3 d x +3 c}+16 d e f \,{\mathrm e}^{3 d x +3 c}+36 i d^{2} e f x \,{\mathrm e}^{2 d x +2 c}-36 i d^{2} e f x \,{\mathrm e}^{4 d x +4 c}+18 i d^{2} e^{2} {\mathrm e}^{2 d x +2 c}-18 i d^{2} e^{2} {\mathrm e}^{4 d x +4 c}-8 i d e f -8 i d \,f^{2} x}{12 \left ({\mathrm e}^{d x +c}+i\right )^{2} \left ({\mathrm e}^{d x +c}-i\right )^{4} d^{3} a}+\frac {3 i \operatorname {polylog}\left (2, i {\mathrm e}^{d x +c}\right ) f^{2} x}{4 a \,d^{2}}+\frac {3 i \ln \left (1-i {\mathrm e}^{d x +c}\right ) e f x}{4 a d}+\frac {3 e^{2} \arctan \left ({\mathrm e}^{d x +c}\right )}{4 a d}+\frac {3 c^{2} f^{2} \arctan \left ({\mathrm e}^{d x +c}\right )}{4 a \,d^{3}}-\frac {3 i \ln \left (1+i {\mathrm e}^{d x +c}\right ) e f x}{4 a d}+\frac {3 i \ln \left (1-i {\mathrm e}^{d x +c}\right ) f^{2} x^{2}}{8 a d}-\frac {3 i \ln \left (1+i {\mathrm e}^{d x +c}\right ) f^{2} x^{2}}{8 a d}+\frac {3 i f^{2} \ln \left (1+i {\mathrm e}^{d x +c}\right ) c^{2}}{8 d^{3} a}-\frac {3 c e f \arctan \left ({\mathrm e}^{d x +c}\right )}{2 a \,d^{2}}-\frac {3 i e f \ln \left (1+i {\mathrm e}^{d x +c}\right ) c}{4 d^{2} a}-\frac {3 i \operatorname {polylog}\left (2, -i {\mathrm e}^{d x +c}\right ) f^{2} x}{4 a \,d^{2}}+\frac {3 i e f \operatorname {polylog}\left (2, i {\mathrm e}^{d x +c}\right )}{4 a \,d^{2}}-\frac {2 i f^{2} \ln \left ({\mathrm e}^{d x +c}\right )}{3 a \,d^{3}}+\frac {i f^{2} \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{3 a \,d^{3}}-\frac {3 i f^{2} \operatorname {polylog}\left (3, i {\mathrm e}^{d x +c}\right )}{4 a \,d^{3}}-\frac {3 i e f \operatorname {polylog}\left (2, -i {\mathrm e}^{d x +c}\right )}{4 a \,d^{2}}+\frac {3 i \ln \left (1-i {\mathrm e}^{d x +c}\right ) c e f}{4 a \,d^{2}}-\frac {3 i \ln \left (1-i {\mathrm e}^{d x +c}\right ) c^{2} f^{2}}{8 a \,d^{3}}+\frac {3 i f^{2} \operatorname {polylog}\left (3, -i {\mathrm e}^{d x +c}\right )}{4 a \,d^{3}}-\frac {5 f^{2} \arctan \left ({\mathrm e}^{d x +c}\right )}{3 a \,d^{3}}\)	\(961\)

[In]

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

[Out]

1/12*(-4*f^2*exp(3*d*x+3*c)+36*I*d^2*e*f*x*exp(2*d*x+2*c)-36*I*d^2*e*f*x*exp(4*d*x+4*c)+9*d^2*e^2*exp(5*d*x+5*
c)+18*d^2*e*f*x*exp(d*x+c)-2*f^2*exp(d*x+c)+9*d^2*x^2*f^2*exp(d*x+c)+9*d^2*e^2*exp(d*x+c)+18*I*d^2*f^2*x^2*exp
(2*d*x+2*c)-36*I*d*f^2*x*exp(4*d*x+4*c)+16*d*f^2*x*exp(3*d*x+3*c)+18*d^2*e*f*x*exp(5*d*x+5*c)+6*d^2*e^2*exp(3*
d*x+3*c)-2*f^2*exp(5*d*x+5*c)-2*exp(d*x+c)*d*f^2*x-2*exp(d*x+c)*d*e*f-36*I*d*e*f*exp(4*d*x+4*c)+18*I*d^2*e^2*e
xp(2*d*x+2*c)+6*d^2*f^2*x^2*exp(3*d*x+3*c)-44*I*d*e*f*exp(2*d*x+2*c)-18*I*d^2*f^2*x^2*exp(4*d*x+4*c)+9*d^2*f^2
*x^2*exp(5*d*x+5*c)+18*d*f^2*x*exp(5*d*x+5*c)+18*d*e*f*exp(5*d*x+5*c)-18*I*d^2*e^2*exp(4*d*x+4*c)-8*I*d*e*f-8*
I*d*f^2*x+12*d^2*e*f*x*exp(3*d*x+3*c)-44*I*d*f^2*x*exp(2*d*x+2*c)+16*d*e*f*exp(3*d*x+3*c))/(exp(d*x+c)+I)^2/(e
xp(d*x+c)-I)^4/d^3/a+3/4*I/a/d^2*polylog(2,I*exp(d*x+c))*f^2*x+3/4*I/a/d*ln(1-I*exp(d*x+c))*e*f*x+3/4/a/d*e^2*
arctan(exp(d*x+c))+3/4/a/d^3*c^2*f^2*arctan(exp(d*x+c))-3/4*I/a/d*ln(1+I*exp(d*x+c))*e*f*x+3/8*I/a/d*ln(1-I*ex
p(d*x+c))*f^2*x^2-3/8*I/a/d*ln(1+I*exp(d*x+c))*f^2*x^2+3/8*I/a/d^3*ln(1+I*exp(d*x+c))*c^2*f^2-3/2/a/d^2*c*e*f*
arctan(exp(d*x+c))-3/4*I/a/d^2*ln(1+I*exp(d*x+c))*c*e*f-3/4*I/a/d^2*polylog(2,-I*exp(d*x+c))*f^2*x+3/4*I/a/d^2
*e*f*polylog(2,I*exp(d*x+c))-2/3*I/a/d^3*f^2*ln(exp(d*x+c))+1/3*I/a/d^3*f^2*ln(1+exp(2*d*x+2*c))-3/4*I*f^2*pol
ylog(3,I*exp(d*x+c))/a/d^3-3/4*I/a/d^2*e*f*polylog(2,-I*exp(d*x+c))+3/4*I/a/d^2*ln(1-I*exp(d*x+c))*c*e*f-3/8*I
/a/d^3*ln(1-I*exp(d*x+c))*c^2*f^2+3/4*I*f^2*polylog(3,-I*exp(d*x+c))/a/d^3-5/3/a/d^3*f^2*arctan(exp(d*x+c))

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. 2066 vs. \(2 (358) = 716\).

Time = 0.29 (sec) , antiderivative size = 2066, normalized size of antiderivative = 4.88 \[ \int \frac {(e+f x)^2 \text {sech}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/24*(-16*I*d*e*f + 16*I*c*f^2 - 18*(I*d*f^2*x + I*d*e*f + (-I*d*f^2*x - I*d*e*f)*e^(6*d*x + 6*c) - 2*(d*f^2*x
+ d*e*f)*e^(5*d*x + 5*c) + (-I*d*f^2*x - I*d*e*f)*e^(4*d*x + 4*c) - 4*(d*f^2*x + d*e*f)*e^(3*d*x + 3*c) + (I*
d*f^2*x + I*d*e*f)*e^(2*d*x + 2*c) - 2*(d*f^2*x + d*e*f)*e^(d*x + c))*dilog(I*e^(d*x + c)) - 18*(-I*d*f^2*x -
I*d*e*f + (I*d*f^2*x + I*d*e*f)*e^(6*d*x + 6*c) + 2*(d*f^2*x + d*e*f)*e^(5*d*x + 5*c) + (I*d*f^2*x + I*d*e*f)*
e^(4*d*x + 4*c) + 4*(d*f^2*x + d*e*f)*e^(3*d*x + 3*c) + (-I*d*f^2*x - I*d*e*f)*e^(2*d*x + 2*c) + 2*(d*f^2*x +
d*e*f)*e^(d*x + c))*dilog(-I*e^(d*x + c)) - 16*(I*d*f^2*x + I*c*f^2)*e^(6*d*x + 6*c) + 2*(9*d^2*f^2*x^2 + 9*d^
2*e^2 + 18*d*e*f - 2*(8*c + 1)*f^2 + 2*(9*d^2*e*f + d*f^2)*x)*e^(5*d*x + 5*c) - 4*(9*I*d^2*f^2*x^2 + 9*I*d^2*e
^2 + 18*I*d*e*f + 4*I*c*f^2 + 2*(9*I*d^2*e*f + 11*I*d*f^2)*x)*e^(4*d*x + 4*c) + 4*(3*d^2*f^2*x^2 + 3*d^2*e^2 +
8*d*e*f - 2*(8*c + 1)*f^2 + 2*(3*d^2*e*f - 4*d*f^2)*x)*e^(3*d*x + 3*c) - 4*(-9*I*d^2*f^2*x^2 - 9*I*d^2*e^2 +
22*I*d*e*f - 4*I*c*f^2 + 18*(-I*d^2*e*f + I*d*f^2)*x)*e^(2*d*x + 2*c) + 2*(9*d^2*f^2*x^2 + 9*d^2*e^2 - 2*d*e*f
- 2*(8*c + 1)*f^2 + 18*(d^2*e*f - d*f^2)*x)*e^(d*x + c) - 3*(3*I*d^2*e^2 - 6*I*c*d*e*f + (3*I*c^2 - 4*I)*f^2
+ (-3*I*d^2*e^2 + 6*I*c*d*e*f + (-3*I*c^2 + 4*I)*f^2)*e^(6*d*x + 6*c) - 2*(3*d^2*e^2 - 6*c*d*e*f + (3*c^2 - 4)
*f^2)*e^(5*d*x + 5*c) + (-3*I*d^2*e^2 + 6*I*c*d*e*f + (-3*I*c^2 + 4*I)*f^2)*e^(4*d*x + 4*c) - 4*(3*d^2*e^2 - 6
*c*d*e*f + (3*c^2 - 4)*f^2)*e^(3*d*x + 3*c) + (3*I*d^2*e^2 - 6*I*c*d*e*f + (3*I*c^2 - 4*I)*f^2)*e^(2*d*x + 2*c
) - 2*(3*d^2*e^2 - 6*c*d*e*f + (3*c^2 - 4)*f^2)*e^(d*x + c))*log(e^(d*x + c) + I) + (9*I*d^2*e^2 - 18*I*c*d*e*
f + (9*I*c^2 - 28*I)*f^2 + (-9*I*d^2*e^2 + 18*I*c*d*e*f + (-9*I*c^2 + 28*I)*f^2)*e^(6*d*x + 6*c) - 2*(9*d^2*e^
2 - 18*c*d*e*f + (9*c^2 - 28)*f^2)*e^(5*d*x + 5*c) + (-9*I*d^2*e^2 + 18*I*c*d*e*f + (-9*I*c^2 + 28*I)*f^2)*e^(
4*d*x + 4*c) - 4*(9*d^2*e^2 - 18*c*d*e*f + (9*c^2 - 28)*f^2)*e^(3*d*x + 3*c) + (9*I*d^2*e^2 - 18*I*c*d*e*f + (
9*I*c^2 - 28*I)*f^2)*e^(2*d*x + 2*c) - 2*(9*d^2*e^2 - 18*c*d*e*f + (9*c^2 - 28)*f^2)*e^(d*x + c))*log(e^(d*x +
c) - I) - 9*(-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 + (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^(6*d*x + 6*c) + 2*(d^2*f^2*x^2 + 2*d^2*e*f*x + 2*c*d*e*f - c^2*f^2)*e^(5*d*x + 5*c) + (
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^(4*d*x + 4*c) + 4*(d^2*f^2*x^2 + 2*d^2*e*f*x + 2*c*
d*e*f - c^2*f^2)*e^(3*d*x + 3*c) + (-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^(2*d*x + 2*c)
+ 2*(d^2*f^2*x^2 + 2*d^2*e*f*x + 2*c*d*e*f - c^2*f^2)*e^(d*x + c))*log(I*e^(d*x + c) + 1) - 9*(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 + (-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^(6*d*x
+ 6*c) - 2*(d^2*f^2*x^2 + 2*d^2*e*f*x + 2*c*d*e*f - c^2*f^2)*e^(5*d*x + 5*c) + (-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^(4*d*x + 4*c) - 4*(d^2*f^2*x^2 + 2*d^2*e*f*x + 2*c*d*e*f - c^2*f^2)*e^(3*d*x +
3*c) + (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^(2*d*x + 2*c) - 2*(d^2*f^2*x^2 + 2*d^2*e*f*
x + 2*c*d*e*f - c^2*f^2)*e^(d*x + c))*log(-I*e^(d*x + c) + 1) - 18*(I*f^2*e^(6*d*x + 6*c) + 2*f^2*e^(5*d*x + 5
*c) + I*f^2*e^(4*d*x + 4*c) + 4*f^2*e^(3*d*x + 3*c) - I*f^2*e^(2*d*x + 2*c) + 2*f^2*e^(d*x + c) - I*f^2)*polyl
og(3, I*e^(d*x + c)) - 18*(-I*f^2*e^(6*d*x + 6*c) - 2*f^2*e^(5*d*x + 5*c) - I*f^2*e^(4*d*x + 4*c) - 4*f^2*e^(3
*d*x + 3*c) + I*f^2*e^(2*d*x + 2*c) - 2*f^2*e^(d*x + c) + I*f^2)*polylog(3, -I*e^(d*x + c)))/(a*d^3*e^(6*d*x +
6*c) - 2*I*a*d^3*e^(5*d*x + 5*c) + a*d^3*e^(4*d*x + 4*c) - 4*I*a*d^3*e^(3*d*x + 3*c) - a*d^3*e^(2*d*x + 2*c)
- 2*I*a*d^3*e^(d*x + c) - a*d^3)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {(e+f x)^2 \text {sech}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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