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

Optimal. Leaf size=423 \[ -\frac{3 i f (e+f x) \text{PolyLog}\left (2,-i e^{c+d x}\right )}{4 a d^2}+\frac{3 i f (e+f x) \text{PolyLog}\left (2,i e^{c+d x}\right )}{4 a d^2}+\frac{3 i f^2 \text{PolyLog}\left (3,-i e^{c+d x}\right )}{4 a d^3}-\frac{3 i f^2 \text{PolyLog}\left (3,i e^{c+d x}\right )}{4 a d^3}-\frac{i f (e+f x) \tanh (c+d x)}{3 a d^2}+\frac{f (e+f x) \text{sech}^3(c+d x)}{6 a d^2}+\frac{3 f (e+f x) \text{sech}(c+d x)}{4 a d^2}-\frac{i f (e+f x) \tanh (c+d x) \text{sech}^2(c+d x)}{6 a d^2}-\frac{i f^2 \text{sech}^2(c+d x)}{12 a d^3}-\frac{5 f^2 \tan ^{-1}(\sinh (c+d x))}{6 a d^3}+\frac{i f^2 \log (\cosh (c+d x))}{3 a d^3}-\frac{f^2 \tanh (c+d x) \text{sech}(c+d x)}{12 a d^3}+\frac{3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{4 a d}+\frac{i (e+f x)^2 \text{sech}^4(c+d x)}{4 a d}+\frac{(e+f x)^2 \tanh (c+d x) \text{sech}^3(c+d x)}{4 a d}+\frac{3 (e+f x)^2 \tanh (c+d x) \text{sech}(c+d x)}{8 a d} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.39833, antiderivative size = 423, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 12, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.387, Rules used = {5571, 4186, 3768, 3770, 4180, 2531, 2282, 6589, 5451, 4185, 4184, 3475} \[ -\frac{3 i f (e+f x) \text{PolyLog}\left (2,-i e^{c+d x}\right )}{4 a d^2}+\frac{3 i f (e+f x) \text{PolyLog}\left (2,i e^{c+d x}\right )}{4 a d^2}+\frac{3 i f^2 \text{PolyLog}\left (3,-i e^{c+d x}\right )}{4 a d^3}-\frac{3 i f^2 \text{PolyLog}\left (3,i e^{c+d x}\right )}{4 a d^3}-\frac{i f (e+f x) \tanh (c+d x)}{3 a d^2}+\frac{f (e+f x) \text{sech}^3(c+d x)}{6 a d^2}+\frac{3 f (e+f x) \text{sech}(c+d x)}{4 a d^2}-\frac{i f (e+f x) \tanh (c+d x) \text{sech}^2(c+d x)}{6 a d^2}-\frac{i f^2 \text{sech}^2(c+d x)}{12 a d^3}-\frac{5 f^2 \tan ^{-1}(\sinh (c+d x))}{6 a d^3}+\frac{i f^2 \log (\cosh (c+d x))}{3 a d^3}-\frac{f^2 \tanh (c+d x) \text{sech}(c+d x)}{12 a d^3}+\frac{3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{4 a d}+\frac{i (e+f x)^2 \text{sech}^4(c+d x)}{4 a d}+\frac{(e+f x)^2 \tanh (c+d x) \text{sech}^3(c+d x)}{4 a d}+\frac{3 (e+f x)^2 \tanh (c+d x) \text{sech}(c+d x)}{8 a d} \]

Antiderivative was successfully verified.

[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 5571

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 4186

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]) /; FreeQ[
{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

Rule 3768

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 3770

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

Rule 4180

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 2531

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

Rule 2282

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

Rule 6589

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

Rule 5451

Int[((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.)*Tanh[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> -Si
mp[((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 4185

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] && G
tQ[n, 1] && NeQ[n, 2]

Rule 4184

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

Rule 3475

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

Rubi steps

\begin{align*} \int \frac{(e+f x)^2 \text{sech}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx &=-\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 \tan ^{-1}\left (e^{c+d x}\right )}{4 a d}-\frac{5 f^2 \tan ^{-1}(\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 \tan ^{-1}\left (e^{c+d x}\right )}{4 a d}-\frac{5 f^2 \tan ^{-1}(\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) \text{Li}_2\left (-i e^{c+d x}\right )}{4 a d^2}+\frac{3 i f (e+f x) \text{Li}_2\left (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 \text{Li}_2\left (-i e^{c+d x}\right ) \, dx}{4 a d^2}-\frac{\left (3 i f^2\right ) \int \text{Li}_2\left (i e^{c+d x}\right ) \, dx}{4 a d^2}\\ &=\frac{3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{4 a d}-\frac{5 f^2 \tan ^{-1}(\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) \text{Li}_2\left (-i e^{c+d x}\right )}{4 a d^2}+\frac{3 i f (e+f x) \text{Li}_2\left (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 ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{c+d x}\right )}{4 a d^3}-\frac{\left (3 i f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{c+d x}\right )}{4 a d^3}\\ &=\frac{3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{4 a d}-\frac{5 f^2 \tan ^{-1}(\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) \text{Li}_2\left (-i e^{c+d x}\right )}{4 a d^2}+\frac{3 i f (e+f x) \text{Li}_2\left (i e^{c+d x}\right )}{4 a d^2}+\frac{3 i f^2 \text{Li}_3\left (-i e^{c+d x}\right )}{4 a d^3}-\frac{3 i f^2 \text{Li}_3\left (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]  time = 12.9861, size = 1284, normalized size = 3.04 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((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)*(d*x - 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)] + PolyL
og[3, I*E^(-c - d*x)]/d))/(24*a*d^2*(-I + E^c)) - (3*d^2*e^2*x - 4*f^2*x + 3*d^2*e*f*x^2 + d^2*f^2*x^3 - (6*I)
*d*e*f*x*Log[1 + I*Cosh[c + d*x] - I*Sinh[c + d*x]]*(I + Cosh[c] + Sinh[c]) - (3*I)*d*f^2*x^2*Log[1 + I*Cosh[c
+ d*x] - I*Sinh[c + d*x]]*(I + Cosh[c] + Sinh[c]) + (I*(3*d^2*e^2 - 4*f^2)*(d*x - Log[I + Cosh[c + d*x] + Sin
h[c + d*x]])*(I + Cosh[c] + Sinh[c]))/d + (6*I)*e*f*PolyLog[2, (-I)*(Cosh[c + d*x] - Sinh[c + d*x])]*(I + Cosh
[c] + Sinh[c]) + ((6*I)*f^2*(d*x*PolyLog[2, (-I)*(Cosh[c + d*x] - Sinh[c + d*x])] + PolyLog[3, (-I)*(Cosh[c +
d*x] - Sinh[c + d*x])])*(I + Cosh[c] + Sinh[c]))/d)/(8*a*d^2*(I + Cosh[c] + Sinh[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]  time = 0.208, size = 1044, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(-2*f^2*exp(5*d*x+5*c)-4*f^2*exp(3*d*x+3*c)-2*f^2*exp(d*x+c)-44*I*d*e*f*exp(2*d*x+2*c)+12*d^2*e*f*x*exp(3
*d*x+3*c)+18*d^2*e*f*x*exp(5*d*x+5*c)+6*d^2*f^2*x^2*exp(3*d*x+3*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)+9*d^2*x^2*f^2*exp(d*x+c)-2*d*f^2*x*exp(d*x+c)-2*d*e*f*exp(d*x+c)+16*d*f^2*x*exp(3*d*x+3*c)+16*d*e*
f*exp(3*d*x+3*c)-36*I*d*e*f*exp(4*d*x+4*c)-18*I*d^2*f^2*x^2*exp(4*d*x+4*c)+18*I*d^2*f^2*x^2*exp(2*d*x+2*c)-44*
I*d*f^2*x*exp(2*d*x+2*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)-8*I*d*f*e-8*I*d*f^2*x+6*d
^2*e^2*exp(3*d*x+3*c)+9*d^2*e^2*exp(5*d*x+5*c)+18*d^2*e*f*x*exp(d*x+c)-36*I*d*f^2*x*exp(4*d*x+4*c)+9*d^2*e^2*e
xp(d*x+c)+18*d*e*f*exp(5*d*x+5*c)-18*I*d^2*e^2*exp(4*d*x+4*c)+18*I*d^2*e^2*exp(2*d*x+2*c))/(exp(d*x+c)+I)^2/(e
xp(d*x+c)-I)^4/d^3/a-3/4*I/a/d^2*e*f*c*ln(exp(d*x+c)+I)-3/8*I/a/d*ln(1+I*exp(d*x+c))*f^2*x^2-3/4*I/a/d^2*polyl
og(2,-I*exp(d*x+c))*f^2*x+3/4*I/a/d^2*e*f*c*ln(exp(d*x+c)-I)+3/8*I/a/d^3*ln(1+I*exp(d*x+c))*c^2*f^2+3/8*I/a/d*
ln(1-I*exp(d*x+c))*f^2*x^2-3/8*I/a/d*e^2*ln(exp(d*x+c)-I)-3/4*I/a/d*ln(1+I*exp(d*x+c))*e*f*x+3/4*I/a/d^2*polyl
og(2,I*exp(d*x+c))*f^2*x-3/8*I/a/d^3*c^2*f^2*ln(exp(d*x+c)-I)-3/4*I*f^2*polylog(3,I*exp(d*x+c))/a/d^3+3/4*I/a/
d^2*ln(1-I*exp(d*x+c))*c*e*f-3/4*I/a/d^2*ln(1+I*exp(d*x+c))*c*e*f+3/4*I*f^2*polylog(3,-I*exp(d*x+c))/a/d^3+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))-3/4*I/a/d^2*e*f*polylog(2,-I*exp(d*x+c))-3
/8*I/a/d^3*ln(1-I*exp(d*x+c))*c^2*f^2+3/8*I/a/d*e^2*ln(exp(d*x+c)+I)+7/6*I/a/d^3*f^2*ln(exp(d*x+c)-I)+3/8*I/a/
d^3*c^2*f^2*ln(exp(d*x+c)+I)-1/2*I/a/d^3*f^2*ln(exp(d*x+c)+I)+3/4*I/a/d*ln(1-I*exp(d*x+c))*e*f*x

________________________________________________________________________________________

Maxima [B]  time = 3.73354, size = 1081, normalized size = 2.56 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*e^2*(64*(3*e^(-d*x - c) - 6*I*e^(-2*d*x - 2*c) + 2*e^(-3*d*x - 3*c) + 6*I*e^(-4*d*x - 4*c) + 3*e^(-5*d*x
- 5*c))/((64*I*a*e^(-d*x - c) - 32*a*e^(-2*d*x - 2*c) + 128*I*a*e^(-3*d*x - 3*c) + 32*a*e^(-4*d*x - 4*c) + 64*
I*a*e^(-5*d*x - 5*c) + 32*a*e^(-6*d*x - 6*c) - 32*a)*d) + 3*I*log(e^(-d*x - c) + I)/(a*d) - 3*I*log(e^(-d*x -
c) - I)/(a*d)) + (-8*I*d*f^2*x - 8*I*d*e*f + (9*d^2*f^2*x^2*e^(5*c) + 18*(d^2*e*f + d*f^2)*x*e^(5*c) + 2*(9*d*
e*f - f^2)*e^(5*c))*e^(5*d*x) + (-18*I*d^2*f^2*x^2*e^(4*c) - 36*I*d*e*f*e^(4*c) + (-36*I*d^2*e*f - 36*I*d*f^2)
*x*e^(4*c))*e^(4*d*x) + 2*(3*d^2*f^2*x^2*e^(3*c) + 2*(3*d^2*e*f + 4*d*f^2)*x*e^(3*c) + 2*(4*d*e*f - f^2)*e^(3*
c))*e^(3*d*x) + (18*I*d^2*f^2*x^2*e^(2*c) - 44*I*d*e*f*e^(2*c) + (36*I*d^2*e*f - 44*I*d*f^2)*x*e^(2*c))*e^(2*d
*x) + (9*d^2*f^2*x^2*e^c + 2*(9*d^2*e*f - d*f^2)*x*e^c - 2*(d*e*f + f^2)*e^c)*e^(d*x))/(12*a*d^3*e^(6*d*x + 6*
c) - 24*I*a*d^3*e^(5*d*x + 5*c) + 12*a*d^3*e^(4*d*x + 4*c) - 48*I*a*d^3*e^(3*d*x + 3*c) - 12*a*d^3*e^(2*d*x +
2*c) - 24*I*a*d^3*e^(d*x + c) - 12*a*d^3) - 3/4*I*(d*x*log(I*e^(d*x + c) + 1) + dilog(-I*e^(d*x + c)))*e*f/(a*
d^2) + 3/4*I*(d*x*log(-I*e^(d*x + c) + 1) + dilog(I*e^(d*x + c)))*e*f/(a*d^2) - 2/3*I*f^2*x/(a*d^2) - 3/8*I*(d
^2*x^2*log(I*e^(d*x + c) + 1) + 2*d*x*dilog(-I*e^(d*x + c)) - 2*polylog(3, -I*e^(d*x + c)))*f^2/(a*d^3) + 3/8*
I*(d^2*x^2*log(-I*e^(d*x + c) + 1) + 2*d*x*dilog(I*e^(d*x + c)) - 2*polylog(3, I*e^(d*x + c)))*f^2/(a*d^3) + 7
/6*I*f^2*log(I*e^(d*x + c) + 1)/(a*d^3) - 1/2*I*f^2*log(I*e^(d*x + c) - 1)/(a*d^3)

________________________________________________________________________________________

Fricas [C]  time = 2.77009, size = 5145, normalized size = 12.16 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-16*I*d*e*f + 16*I*c*f^2 + (-18*I*d*f^2*x - 18*I*d*e*f + (18*I*d*f^2*x + 18*I*d*e*f)*e^(6*d*x + 6*c) + 36*(d*
f^2*x + d*e*f)*e^(5*d*x + 5*c) + (18*I*d*f^2*x + 18*I*d*e*f)*e^(4*d*x + 4*c) + 72*(d*f^2*x + d*e*f)*e^(3*d*x +
3*c) + (-18*I*d*f^2*x - 18*I*d*e*f)*e^(2*d*x + 2*c) + 36*(d*f^2*x + d*e*f)*e^(d*x + c))*dilog(I*e^(d*x + c))
+ (18*I*d*f^2*x + 18*I*d*e*f + (-18*I*d*f^2*x - 18*I*d*e*f)*e^(6*d*x + 6*c) - 36*(d*f^2*x + d*e*f)*e^(5*d*x +
5*c) + (-18*I*d*f^2*x - 18*I*d*e*f)*e^(4*d*x + 4*c) - 72*(d*f^2*x + d*e*f)*e^(3*d*x + 3*c) + (18*I*d*f^2*x + 1
8*I*d*e*f)*e^(2*d*x + 2*c) - 36*(d*f^2*x + d*e*f)*e^(d*x + c))*dilog(-I*e^(d*x + c)) + (-16*I*d*f^2*x - 16*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) + (-36*I*d^2*f^2*x^2 - 36*I*d^2*e^2 - 72*I*d*e*f - 16*I*c*f^2 + (-72*I*d^2*e*f - 88*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) + (36*I*d^2*f^2*x^2 + 36*I*d^2*e^2 - 88*I*d*e*f + 16*I*c*f^2 + (72*I*d^2*e*f - 72*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) +
(-9*I*d^2*e^2 + 18*I*c*d*e*f + (-9*I*c^2 + 12*I)*f^2 + (9*I*d^2*e^2 - 18*I*c*d*e*f + (9*I*c^2 - 12*I)*f^2)*e^(
6*d*x + 6*c) + 6*(3*d^2*e^2 - 6*c*d*e*f + (3*c^2 - 4)*f^2)*e^(5*d*x + 5*c) + (9*I*d^2*e^2 - 18*I*c*d*e*f + (9*
I*c^2 - 12*I)*f^2)*e^(4*d*x + 4*c) + 12*(3*d^2*e^2 - 6*c*d*e*f + (3*c^2 - 4)*f^2)*e^(3*d*x + 3*c) + (-9*I*d^2*
e^2 + 18*I*c*d*e*f + (-9*I*c^2 + 12*I)*f^2)*e^(2*d*x + 2*c) + 6*(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 + 18*I*d^2*e*f*x + 18*I*
c*d*e*f - 9*I*c^2*f^2 + (-9*I*d^2*f^2*x^2 - 18*I*d^2*e*f*x - 18*I*c*d*e*f + 9*I*c^2*f^2)*e^(6*d*x + 6*c) - 18*
(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) + (-9*I*d^2*f^2*x^2 - 18*I*d^2*e*f*x - 18*I*
c*d*e*f + 9*I*c^2*f^2)*e^(4*d*x + 4*c) - 36*(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)
+ (9*I*d^2*f^2*x^2 + 18*I*d^2*e*f*x + 18*I*c*d*e*f - 9*I*c^2*f^2)*e^(2*d*x + 2*c) - 18*(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 - 18*I*d^2*e*f*x - 18*I*c*d
*e*f + 9*I*c^2*f^2 + (9*I*d^2*f^2*x^2 + 18*I*d^2*e*f*x + 18*I*c*d*e*f - 9*I*c^2*f^2)*e^(6*d*x + 6*c) + 18*(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) + (9*I*d^2*f^2*x^2 + 18*I*d^2*e*f*x + 18*I*c*d*e
*f - 9*I*c^2*f^2)*e^(4*d*x + 4*c) + 36*(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) + (-9
*I*d^2*f^2*x^2 - 18*I*d^2*e*f*x - 18*I*c*d*e*f + 9*I*c^2*f^2)*e^(2*d*x + 2*c) + 18*(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) - 36*f^2*e^(5*d*x + 5
*c) - 18*I*f^2*e^(4*d*x + 4*c) - 72*f^2*e^(3*d*x + 3*c) + 18*I*f^2*e^(2*d*x + 2*c) - 36*f^2*e^(d*x + c) + 18*I
*f^2)*polylog(3, I*e^(d*x + c)) + (18*I*f^2*e^(6*d*x + 6*c) + 36*f^2*e^(5*d*x + 5*c) + 18*I*f^2*e^(4*d*x + 4*c
) + 72*f^2*e^(3*d*x + 3*c) - 18*I*f^2*e^(2*d*x + 2*c) + 36*f^2*e^(d*x + c) - 18*I*f^2)*polylog(3, -I*e^(d*x +
c)))/(24*a*d^3*e^(6*d*x + 6*c) - 48*I*a*d^3*e^(5*d*x + 5*c) + 24*a*d^3*e^(4*d*x + 4*c) - 96*I*a*d^3*e^(3*d*x +
3*c) - 24*a*d^3*e^(2*d*x + 2*c) - 48*I*a*d^3*e^(d*x + c) - 24*a*d^3)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)