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

Optimal. Leaf size=450 \[ -\frac{f^2 (e+f x) \text{PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}+\frac{f^2 (e+f x) \text{PolyLog}\left (2,i e^{c+d x}\right )}{a d^3}-\frac{2 f^2 (e+f x) \text{PolyLog}\left (2,-e^{2 (c+d x)}\right )}{a d^3}+\frac{f^3 \text{PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}-\frac{f^3 \text{PolyLog}\left (3,i e^{c+d x}\right )}{a d^4}+\frac{f^3 \text{PolyLog}\left (3,-e^{2 (c+d x)}\right )}{a d^4}-\frac{f^2 (e+f x) \tanh (c+d x)}{a d^3}-\frac{i f^2 (e+f x) \text{sech}(c+d x)}{a d^3}-\frac{2 f (e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{a d^2}-\frac{i f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}+\frac{f (e+f x)^2 \text{sech}^2(c+d x)}{2 a d^2}-\frac{i f (e+f x)^2 \tanh (c+d x) \text{sech}(c+d x)}{2 a d^2}+\frac{i f^3 \tan ^{-1}(\sinh (c+d x))}{a d^4}+\frac{f^3 \log (\cosh (c+d x))}{a d^4}+\frac{2 (e+f x)^3 \tanh (c+d x)}{3 a d}+\frac{i (e+f x)^3 \text{sech}^3(c+d x)}{3 a d}+\frac{(e+f x)^3 \tanh (c+d x) \text{sech}^2(c+d x)}{3 a d}+\frac{2 (e+f x)^3}{3 a d} \]

[Out]

(2*(e + f*x)^3)/(3*a*d) - (I*f*(e + f*x)^2*ArcTan[E^(c + d*x)])/(a*d^2) + (I*f^3*ArcTan[Sinh[c + d*x]])/(a*d^4
) - (2*f*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/(a*d^2) + (f^3*Log[Cosh[c + d*x]])/(a*d^4) - (f^2*(e + f*x)*Pol
yLog[2, (-I)*E^(c + d*x)])/(a*d^3) + (f^2*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(a*d^3) - (2*f^2*(e + f*x)*Poly
Log[2, -E^(2*(c + d*x))])/(a*d^3) + (f^3*PolyLog[3, (-I)*E^(c + d*x)])/(a*d^4) - (f^3*PolyLog[3, I*E^(c + d*x)
])/(a*d^4) + (f^3*PolyLog[3, -E^(2*(c + d*x))])/(a*d^4) - (I*f^2*(e + f*x)*Sech[c + d*x])/(a*d^3) + (f*(e + f*
x)^2*Sech[c + d*x]^2)/(2*a*d^2) + ((I/3)*(e + f*x)^3*Sech[c + d*x]^3)/(a*d) - (f^2*(e + f*x)*Tanh[c + d*x])/(a
*d^3) + (2*(e + f*x)^3*Tanh[c + d*x])/(3*a*d) - ((I/2)*f*(e + f*x)^2*Sech[c + d*x]*Tanh[c + d*x])/(a*d^2) + ((
e + f*x)^3*Sech[c + d*x]^2*Tanh[c + d*x])/(3*a*d)

________________________________________________________________________________________

Rubi [A]  time = 0.589366, antiderivative size = 450, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 12, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.387, Rules used = {5571, 4186, 4184, 3475, 3718, 2190, 2531, 2282, 6589, 5451, 3770, 4180} \[ -\frac{f^2 (e+f x) \text{PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}+\frac{f^2 (e+f x) \text{PolyLog}\left (2,i e^{c+d x}\right )}{a d^3}-\frac{2 f^2 (e+f x) \text{PolyLog}\left (2,-e^{2 (c+d x)}\right )}{a d^3}+\frac{f^3 \text{PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}-\frac{f^3 \text{PolyLog}\left (3,i e^{c+d x}\right )}{a d^4}+\frac{f^3 \text{PolyLog}\left (3,-e^{2 (c+d x)}\right )}{a d^4}-\frac{f^2 (e+f x) \tanh (c+d x)}{a d^3}-\frac{i f^2 (e+f x) \text{sech}(c+d x)}{a d^3}-\frac{2 f (e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{a d^2}-\frac{i f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}+\frac{f (e+f x)^2 \text{sech}^2(c+d x)}{2 a d^2}-\frac{i f (e+f x)^2 \tanh (c+d x) \text{sech}(c+d x)}{2 a d^2}+\frac{i f^3 \tan ^{-1}(\sinh (c+d x))}{a d^4}+\frac{f^3 \log (\cosh (c+d x))}{a d^4}+\frac{2 (e+f x)^3 \tanh (c+d x)}{3 a d}+\frac{i (e+f x)^3 \text{sech}^3(c+d x)}{3 a d}+\frac{(e+f x)^3 \tanh (c+d x) \text{sech}^2(c+d x)}{3 a d}+\frac{2 (e+f x)^3}{3 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(e + f*x)^3)/(3*a*d) - (I*f*(e + f*x)^2*ArcTan[E^(c + d*x)])/(a*d^2) + (I*f^3*ArcTan[Sinh[c + d*x]])/(a*d^4
) - (2*f*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/(a*d^2) + (f^3*Log[Cosh[c + d*x]])/(a*d^4) - (f^2*(e + f*x)*Pol
yLog[2, (-I)*E^(c + d*x)])/(a*d^3) + (f^2*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(a*d^3) - (2*f^2*(e + f*x)*Poly
Log[2, -E^(2*(c + d*x))])/(a*d^3) + (f^3*PolyLog[3, (-I)*E^(c + d*x)])/(a*d^4) - (f^3*PolyLog[3, I*E^(c + d*x)
])/(a*d^4) + (f^3*PolyLog[3, -E^(2*(c + d*x))])/(a*d^4) - (I*f^2*(e + f*x)*Sech[c + d*x])/(a*d^3) + (f*(e + f*
x)^2*Sech[c + d*x]^2)/(2*a*d^2) + ((I/3)*(e + f*x)^3*Sech[c + d*x]^3)/(a*d) - (f^2*(e + f*x)*Tanh[c + d*x])/(a
*d^3) + (2*(e + f*x)^3*Tanh[c + d*x])/(3*a*d) - ((I/2)*f*(e + f*x)^2*Sech[c + d*x]*Tanh[c + d*x])/(a*d^2) + ((
e + f*x)^3*Sech[c + d*x]^2*Tanh[c + d*x])/(3*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 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]

Rule 3718

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 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 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 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]

Rubi steps

\begin{align*} \int \frac{(e+f x)^3 \text{sech}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx &=-\frac{i \int (e+f x)^3 \text{sech}^3(c+d x) \tanh (c+d x) \, dx}{a}+\frac{\int (e+f x)^3 \text{sech}^4(c+d x) \, dx}{a}\\ &=\frac{f (e+f x)^2 \text{sech}^2(c+d x)}{2 a d^2}+\frac{i (e+f x)^3 \text{sech}^3(c+d x)}{3 a d}+\frac{(e+f x)^3 \text{sech}^2(c+d x) \tanh (c+d x)}{3 a d}+\frac{2 \int (e+f x)^3 \text{sech}^2(c+d x) \, dx}{3 a}-\frac{(i f) \int (e+f x)^2 \text{sech}^3(c+d x) \, dx}{a d}-\frac{f^2 \int (e+f x) \text{sech}^2(c+d x) \, dx}{a d^2}\\ &=-\frac{i f^2 (e+f x) \text{sech}(c+d x)}{a d^3}+\frac{f (e+f x)^2 \text{sech}^2(c+d x)}{2 a d^2}+\frac{i (e+f x)^3 \text{sech}^3(c+d x)}{3 a d}-\frac{f^2 (e+f x) \tanh (c+d x)}{a d^3}+\frac{2 (e+f x)^3 \tanh (c+d x)}{3 a d}-\frac{i f (e+f x)^2 \text{sech}(c+d x) \tanh (c+d x)}{2 a d^2}+\frac{(e+f x)^3 \text{sech}^2(c+d x) \tanh (c+d x)}{3 a d}-\frac{(i f) \int (e+f x)^2 \text{sech}(c+d x) \, dx}{2 a d}-\frac{(2 f) \int (e+f x)^2 \tanh (c+d x) \, dx}{a d}+\frac{\left (i f^3\right ) \int \text{sech}(c+d x) \, dx}{a d^3}+\frac{f^3 \int \tanh (c+d x) \, dx}{a d^3}\\ &=\frac{2 (e+f x)^3}{3 a d}-\frac{i f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}+\frac{i f^3 \tan ^{-1}(\sinh (c+d x))}{a d^4}+\frac{f^3 \log (\cosh (c+d x))}{a d^4}-\frac{i f^2 (e+f x) \text{sech}(c+d x)}{a d^3}+\frac{f (e+f x)^2 \text{sech}^2(c+d x)}{2 a d^2}+\frac{i (e+f x)^3 \text{sech}^3(c+d x)}{3 a d}-\frac{f^2 (e+f x) \tanh (c+d x)}{a d^3}+\frac{2 (e+f x)^3 \tanh (c+d x)}{3 a d}-\frac{i f (e+f x)^2 \text{sech}(c+d x) \tanh (c+d x)}{2 a d^2}+\frac{(e+f x)^3 \text{sech}^2(c+d x) \tanh (c+d x)}{3 a d}-\frac{(4 f) \int \frac{e^{2 (c+d x)} (e+f x)^2}{1+e^{2 (c+d x)}} \, dx}{a d}-\frac{f^2 \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{a d^2}+\frac{f^2 \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{a d^2}\\ &=\frac{2 (e+f x)^3}{3 a d}-\frac{i f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}+\frac{i f^3 \tan ^{-1}(\sinh (c+d x))}{a d^4}-\frac{2 f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a d^2}+\frac{f^3 \log (\cosh (c+d x))}{a d^4}-\frac{f^2 (e+f x) \text{Li}_2\left (-i e^{c+d x}\right )}{a d^3}+\frac{f^2 (e+f x) \text{Li}_2\left (i e^{c+d x}\right )}{a d^3}-\frac{i f^2 (e+f x) \text{sech}(c+d x)}{a d^3}+\frac{f (e+f x)^2 \text{sech}^2(c+d x)}{2 a d^2}+\frac{i (e+f x)^3 \text{sech}^3(c+d x)}{3 a d}-\frac{f^2 (e+f x) \tanh (c+d x)}{a d^3}+\frac{2 (e+f x)^3 \tanh (c+d x)}{3 a d}-\frac{i f (e+f x)^2 \text{sech}(c+d x) \tanh (c+d x)}{2 a d^2}+\frac{(e+f x)^3 \text{sech}^2(c+d x) \tanh (c+d x)}{3 a d}+\frac{\left (4 f^2\right ) \int (e+f x) \log \left (1+e^{2 (c+d x)}\right ) \, dx}{a d^2}+\frac{f^3 \int \text{Li}_2\left (-i e^{c+d x}\right ) \, dx}{a d^3}-\frac{f^3 \int \text{Li}_2\left (i e^{c+d x}\right ) \, dx}{a d^3}\\ &=\frac{2 (e+f x)^3}{3 a d}-\frac{i f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}+\frac{i f^3 \tan ^{-1}(\sinh (c+d x))}{a d^4}-\frac{2 f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a d^2}+\frac{f^3 \log (\cosh (c+d x))}{a d^4}-\frac{f^2 (e+f x) \text{Li}_2\left (-i e^{c+d x}\right )}{a d^3}+\frac{f^2 (e+f x) \text{Li}_2\left (i e^{c+d x}\right )}{a d^3}-\frac{2 f^2 (e+f x) \text{Li}_2\left (-e^{2 (c+d x)}\right )}{a d^3}-\frac{i f^2 (e+f x) \text{sech}(c+d x)}{a d^3}+\frac{f (e+f x)^2 \text{sech}^2(c+d x)}{2 a d^2}+\frac{i (e+f x)^3 \text{sech}^3(c+d x)}{3 a d}-\frac{f^2 (e+f x) \tanh (c+d x)}{a d^3}+\frac{2 (e+f x)^3 \tanh (c+d x)}{3 a d}-\frac{i f (e+f x)^2 \text{sech}(c+d x) \tanh (c+d x)}{2 a d^2}+\frac{(e+f x)^3 \text{sech}^2(c+d x) \tanh (c+d x)}{3 a d}+\frac{f^3 \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}-\frac{f^3 \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}+\frac{\left (2 f^3\right ) \int \text{Li}_2\left (-e^{2 (c+d x)}\right ) \, dx}{a d^3}\\ &=\frac{2 (e+f x)^3}{3 a d}-\frac{i f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}+\frac{i f^3 \tan ^{-1}(\sinh (c+d x))}{a d^4}-\frac{2 f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a d^2}+\frac{f^3 \log (\cosh (c+d x))}{a d^4}-\frac{f^2 (e+f x) \text{Li}_2\left (-i e^{c+d x}\right )}{a d^3}+\frac{f^2 (e+f x) \text{Li}_2\left (i e^{c+d x}\right )}{a d^3}-\frac{2 f^2 (e+f x) \text{Li}_2\left (-e^{2 (c+d x)}\right )}{a d^3}+\frac{f^3 \text{Li}_3\left (-i e^{c+d x}\right )}{a d^4}-\frac{f^3 \text{Li}_3\left (i e^{c+d x}\right )}{a d^4}-\frac{i f^2 (e+f x) \text{sech}(c+d x)}{a d^3}+\frac{f (e+f x)^2 \text{sech}^2(c+d x)}{2 a d^2}+\frac{i (e+f x)^3 \text{sech}^3(c+d x)}{3 a d}-\frac{f^2 (e+f x) \tanh (c+d x)}{a d^3}+\frac{2 (e+f x)^3 \tanh (c+d x)}{3 a d}-\frac{i f (e+f x)^2 \text{sech}(c+d x) \tanh (c+d x)}{2 a d^2}+\frac{(e+f x)^3 \text{sech}^2(c+d x) \tanh (c+d x)}{3 a d}+\frac{f^3 \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{a d^4}\\ &=\frac{2 (e+f x)^3}{3 a d}-\frac{i f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}+\frac{i f^3 \tan ^{-1}(\sinh (c+d x))}{a d^4}-\frac{2 f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a d^2}+\frac{f^3 \log (\cosh (c+d x))}{a d^4}-\frac{f^2 (e+f x) \text{Li}_2\left (-i e^{c+d x}\right )}{a d^3}+\frac{f^2 (e+f x) \text{Li}_2\left (i e^{c+d x}\right )}{a d^3}-\frac{2 f^2 (e+f x) \text{Li}_2\left (-e^{2 (c+d x)}\right )}{a d^3}+\frac{f^3 \text{Li}_3\left (-i e^{c+d x}\right )}{a d^4}-\frac{f^3 \text{Li}_3\left (i e^{c+d x}\right )}{a d^4}+\frac{f^3 \text{Li}_3\left (-e^{2 (c+d x)}\right )}{a d^4}-\frac{i f^2 (e+f x) \text{sech}(c+d x)}{a d^3}+\frac{f (e+f x)^2 \text{sech}^2(c+d x)}{2 a d^2}+\frac{i (e+f x)^3 \text{sech}^3(c+d x)}{3 a d}-\frac{f^2 (e+f x) \tanh (c+d x)}{a d^3}+\frac{2 (e+f x)^3 \tanh (c+d x)}{3 a d}-\frac{i f (e+f x)^2 \text{sech}(c+d x) \tanh (c+d x)}{2 a d^2}+\frac{(e+f x)^3 \text{sech}^2(c+d x) \tanh (c+d x)}{3 a d}\\ \end{align*}

Mathematica [B]  time = 12.2762, size = 1049, normalized size = 2.33 \[ -\frac{i f \left (\frac{(e+f x)^3}{f}+\frac{3 \left (1-i e^c\right ) \log \left (1+i e^{-c-d x}\right ) (e+f x)^2}{d}+\frac{6 i \left (i+e^c\right ) f \left (d (e+f x) \text{PolyLog}\left (2,-i e^{-c-d x}\right )+f \text{PolyLog}\left (3,-i e^{-c-d x}\right )\right )}{d^3}\right )}{2 a d \left (i+e^c\right )}+\frac{i f \left (5 d^2 f^2 x^3+15 d^2 e f x^2+15 d \left (1+i e^c\right ) f^2 \log \left (1-i e^{-c-d x}\right ) x^2+3 \left (5 d^2 e^2-4 f^2\right ) x+30 d e \left (1+i e^c\right ) f \log \left (1-i e^{-c-d x}\right ) x-\frac{3 \left (1+i e^c\right ) \left (5 d^2 e^2-4 f^2\right ) \left (d x-\log \left (i-e^{c+d x}\right )\right )}{d}-30 e \left (1+i e^c\right ) f \text{PolyLog}\left (2,i e^{-c-d x}\right )-30 \left (1+i e^c\right ) f^2 \left (x \text{PolyLog}\left (2,i e^{-c-d x}\right )+\frac{\text{PolyLog}\left (3,i e^{-c-d x}\right )}{d}\right )\right )}{6 a d^3 \left (-i+e^c\right )}+\frac{\sinh \left (\frac{d x}{2}\right ) e^3+3 f x \sinh \left (\frac{d x}{2}\right ) e^2+3 f^2 x^2 \sinh \left (\frac{d x}{2}\right ) e+f^3 x^3 \sinh \left (\frac{d x}{2}\right )}{2 a d \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{5 d^2 \sinh \left (\frac{d x}{2}\right ) e^3+15 d^2 f x \sinh \left (\frac{d x}{2}\right ) e^2-12 f^2 \sinh \left (\frac{d x}{2}\right ) e+15 d^2 f^2 x^2 \sinh \left (\frac{d x}{2}\right ) e+5 d^2 f^3 x^3 \sinh \left (\frac{d x}{2}\right )-12 f^3 x \sinh \left (\frac{d x}{2}\right )}{6 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 )}+\frac{i d \cosh \left (\frac{c}{2}\right ) e^3+d \sinh \left (\frac{c}{2}\right ) e^3+3 f \cosh \left (\frac{c}{2}\right ) e^2+3 i d f x \cosh \left (\frac{c}{2}\right ) e^2+3 i f \sinh \left (\frac{c}{2}\right ) e^2+3 d f x \sinh \left (\frac{c}{2}\right ) e^2+3 i d f^2 x^2 \cosh \left (\frac{c}{2}\right ) e+6 f^2 x \cosh \left (\frac{c}{2}\right ) e+3 d f^2 x^2 \sinh \left (\frac{c}{2}\right ) e+6 i f^2 x \sinh \left (\frac{c}{2}\right ) e+i d f^3 x^3 \cosh \left (\frac{c}{2}\right )+3 f^3 x^2 \cosh \left (\frac{c}{2}\right )+d f^3 x^3 \sinh \left (\frac{c}{2}\right )+3 i f^3 x^2 \sinh \left (\frac{c}{2}\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 )^2}+\frac{\sinh \left (\frac{d x}{2}\right ) e^3+3 f x \sinh \left (\frac{d x}{2}\right ) e^2+3 f^2 x^2 \sinh \left (\frac{d x}{2}\right ) e+f^3 x^3 \sinh \left (\frac{d x}{2}\right )}{3 a d \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} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-I/2)*f*((e + f*x)^3/f + (3*(1 - I*E^c)*(e + f*x)^2*Log[1 + I*E^(-c - d*x)])/d + ((6*I)*(I + E^c)*f*(d*(e +
f*x)*PolyLog[2, (-I)*E^(-c - d*x)] + f*PolyLog[3, (-I)*E^(-c - d*x)]))/d^3))/(a*d*(I + E^c)) + ((I/6)*f*(3*(5*
d^2*e^2 - 4*f^2)*x + 15*d^2*e*f*x^2 + 5*d^2*f^2*x^3 + 30*d*e*(1 + I*E^c)*f*x*Log[1 - I*E^(-c - d*x)] + 15*d*(1
+ I*E^c)*f^2*x^2*Log[1 - I*E^(-c - d*x)] - (3*(1 + I*E^c)*(5*d^2*e^2 - 4*f^2)*(d*x - Log[I - E^(c + d*x)]))/d
- 30*e*(1 + I*E^c)*f*PolyLog[2, I*E^(-c - d*x)] - 30*(1 + I*E^c)*f^2*(x*PolyLog[2, I*E^(-c - d*x)] + PolyLog[
3, I*E^(-c - d*x)]/d)))/(a*d^3*(-I + E^c)) + (e^3*Sinh[(d*x)/2] + 3*e^2*f*x*Sinh[(d*x)/2] + 3*e*f^2*x^2*Sinh[(
d*x)/2] + f^3*x^3*Sinh[(d*x)/2])/(2*a*d*(Cosh[c/2] - I*Sinh[c/2])*(Cosh[c/2 + (d*x)/2] - I*Sinh[c/2 + (d*x)/2]
)) + (e^3*Sinh[(d*x)/2] + 3*e^2*f*x*Sinh[(d*x)/2] + 3*e*f^2*x^2*Sinh[(d*x)/2] + f^3*x^3*Sinh[(d*x)/2])/(3*a*d*
(Cosh[c/2] + I*Sinh[c/2])*(Cosh[c/2 + (d*x)/2] + I*Sinh[c/2 + (d*x)/2])^3) + (I*d*e^3*Cosh[c/2] + 3*e^2*f*Cosh
[c/2] + (3*I)*d*e^2*f*x*Cosh[c/2] + 6*e*f^2*x*Cosh[c/2] + (3*I)*d*e*f^2*x^2*Cosh[c/2] + 3*f^3*x^2*Cosh[c/2] +
I*d*f^3*x^3*Cosh[c/2] + d*e^3*Sinh[c/2] + (3*I)*e^2*f*Sinh[c/2] + 3*d*e^2*f*x*Sinh[c/2] + (6*I)*e*f^2*x*Sinh[c
/2] + 3*d*e*f^2*x^2*Sinh[c/2] + (3*I)*f^3*x^2*Sinh[c/2] + d*f^3*x^3*Sinh[c/2])/(6*a*d^2*(Cosh[c/2] + I*Sinh[c/
2])*(Cosh[c/2 + (d*x)/2] + I*Sinh[c/2 + (d*x)/2])^2) + (5*d^2*e^3*Sinh[(d*x)/2] - 12*e*f^2*Sinh[(d*x)/2] + 15*
d^2*e^2*f*x*Sinh[(d*x)/2] - 12*f^3*x*Sinh[(d*x)/2] + 15*d^2*e*f^2*x^2*Sinh[(d*x)/2] + 5*d^2*f^3*x^3*Sinh[(d*x)
/2])/(6*a*d^3*(Cosh[c/2] + I*Sinh[c/2])*(Cosh[c/2 + (d*x)/2] + I*Sinh[c/2 + (d*x)/2]))

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Maple [B]  time = 0.18, size = 1001, normalized size = 2.2 \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)^3*sech(d*x+c)^2/(a+I*a*sinh(d*x+c)),x)

[Out]

-5*f^2/d^2/a*e*ln(1+I*exp(d*x+c))*x-5*f^2/d^3/a*e*ln(1+I*exp(d*x+c))*c-8*f^2/d^3/a*e*c*ln(exp(d*x+c))+5*f^2/d^
3/a*e*c*ln(exp(d*x+c)-I)+8*f^2/d^2/a*e*c*x-8/3*f^3/d^4/a*c^3+4/3*f^3/d/a*x^3+3*f^3*polylog(3,I*exp(d*x+c))/a/d
^4+5*f^3*polylog(3,-I*exp(d*x+c))/a/d^4+1/3*I*(6*I*f^3*x*exp(2*d*x+2*c)+8*d^2*f^3*x^3*exp(d*x+c)-3*d*f^3*x^2*e
xp(3*d*x+3*c)-4*I*d^2*f^3*x^3+24*d^2*e*f^2*x^2*exp(d*x+c)-6*d*e*f^2*x*exp(3*d*x+3*c)+6*I*e*f^2*exp(2*d*x+2*c)+
6*I*e*f^2+24*d^2*e^2*f*x*exp(d*x+c)-3*d*e^2*f*exp(3*d*x+3*c)-3*d*f^3*x^2*exp(d*x+c)-6*f^3*x*exp(3*d*x+3*c)+6*I
*f^3*x-4*I*d^2*e^3+8*d^2*e^3*exp(d*x+c)-6*d*e*f^2*x*exp(d*x+c)-6*e*f^2*exp(3*d*x+3*c)-12*I*d^2*e^2*f*x-3*d*e^2
*f*exp(d*x+c)-6*f^3*x*exp(d*x+c)-12*I*d^2*e*f^2*x^2-6*e*f^2*exp(d*x+c))/(exp(d*x+c)+I)/(exp(d*x+c)-I)^3/d^3/a-
2*f^3/d^4/a*ln(exp(d*x+c))+2*f^3/d^4/a*ln(exp(d*x+c)-I)-4*f^3/d^3/a*c^2*x+4*f/d^2/a*ln(exp(d*x+c))*e^2-5/2*f^3
/d^2/a*ln(1+I*exp(d*x+c))*x^2+5/2*f^3/d^4/a*ln(1+I*exp(d*x+c))*c^2+4*f^2/d^3/a*e*c^2-5/2*f/d^2/a*ln(exp(d*x+c)
-I)*e^2-5/2*f^3/d^4/a*c^2*ln(exp(d*x+c)-I)+4*f^2/d/a*e*x^2+4*f^3/d^4/a*c^2*ln(exp(d*x+c))-5*f^3/d^3/a*polylog(
2,-I*exp(d*x+c))*x-5*f^2/d^3/a*e*polylog(2,-I*exp(d*x+c))-3/2*f^3/d^2/a*ln(1-I*exp(d*x+c))*x^2-3*f^3/d^3/a*pol
ylog(2,I*exp(d*x+c))*x+3/2*f^3/d^4/a*ln(1-I*exp(d*x+c))*c^2-3/2*f/d^2/a*e^2*ln(exp(d*x+c)+I)-3/2*f^3/d^4/a*c^2
*ln(exp(d*x+c)+I)-3*f^2/d^3/a*e*polylog(2,I*exp(d*x+c))+3*f^2/d^3/a*e*c*ln(exp(d*x+c)+I)-3*f^2/d^2/a*ln(1-I*ex
p(d*x+c))*e*x-3*f^2/d^3/a*ln(1-I*exp(d*x+c))*c*e

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Maxima [A]  time = 2.5021, size = 984, normalized size = 2.19 \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)^3*sech(d*x+c)^2/(a+I*a*sinh(d*x+c)),x, algorithm="maxima")

[Out]

1/2*e^2*f*(24*(4*I*d*x*e^(4*d*x + 4*c) + (8*d*x*e^(3*c) + e^(3*c))*e^(3*d*x) + e^(d*x + c))/(12*I*a*d^2*e^(4*d
*x + 4*c) + 24*a*d^2*e^(3*d*x + 3*c) + 24*a*d^2*e^(d*x + c) - 12*I*a*d^2) - 3*log((e^(d*x + c) + I)*e^(-c))/(a
*d^2) - 5*log(-I*(I*e^(d*x + c) + 1)*e^(-c))/(a*d^2)) + 4*e^3*(2*e^(-d*x - c)/((6*a*e^(-d*x - c) + 6*a*e^(-3*d
*x - 3*c) - 3*I*a*e^(-4*d*x - 4*c) + 3*I*a)*d) + I/((6*a*e^(-d*x - c) + 6*a*e^(-3*d*x - 3*c) - 3*I*a*e^(-4*d*x
- 4*c) + 3*I*a)*d)) + (4*I*d^2*f^3*x^3 + 12*I*d^2*e*f^2*x^2 - 6*I*f^3*x - 6*I*e*f^2 + 3*(d*f^3*x^2*e^(3*c) +
2*e*f^2*e^(3*c) + 2*(d*e*f^2 + f^3)*x*e^(3*c))*e^(3*d*x) + (-6*I*f^3*x*e^(2*c) - 6*I*e*f^2*e^(2*c))*e^(2*d*x)
- (8*d^2*f^3*x^3*e^c - 6*e*f^2*e^c + 3*(8*d^2*e*f^2 - d*f^3)*x^2*e^c - 6*(d*e*f^2 + f^3)*x*e^c)*e^(d*x))/(3*I*
a*d^3*e^(4*d*x + 4*c) + 6*a*d^3*e^(3*d*x + 3*c) + 6*a*d^3*e^(d*x + c) - 3*I*a*d^3) - 5*(d*x*log(I*e^(d*x + c)
+ 1) + dilog(-I*e^(d*x + c)))*e*f^2/(a*d^3) - 3*(d*x*log(-I*e^(d*x + c) + 1) + dilog(I*e^(d*x + c)))*e*f^2/(a*
d^3) - 2*f^3*x/(a*d^3) - 5/2*(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^3/(a*d^4) - 3/2*(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^3/(a*d^4) + 2*f^3*log(e^(d*x + c) - I)/(a*d^4) + 4/3*(d^3*f^3*x^3 + 3*d^3*e*f^2*x^2)/(a*d^4)

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Fricas [C]  time = 2.41309, size = 3393, normalized size = 7.54 \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)^3*sech(d*x+c)^2/(a+I*a*sinh(d*x+c)),x, algorithm="fricas")

[Out]

(8*d^3*e^3 - 24*c*d^2*e^2*f + 12*(2*c^2 - 1)*d*e*f^2 - 4*(2*c^3 - 3*c)*f^3 + (18*d*f^3*x + 18*d*e*f^2 - 18*(d*
f^3*x + d*e*f^2)*e^(4*d*x + 4*c) + (36*I*d*f^3*x + 36*I*d*e*f^2)*e^(3*d*x + 3*c) + (36*I*d*f^3*x + 36*I*d*e*f^
2)*e^(d*x + c))*dilog(I*e^(d*x + c)) + (30*d*f^3*x + 30*d*e*f^2 - 30*(d*f^3*x + d*e*f^2)*e^(4*d*x + 4*c) + (60
*I*d*f^3*x + 60*I*d*e*f^2)*e^(3*d*x + 3*c) + (60*I*d*f^3*x + 60*I*d*e*f^2)*e^(d*x + c))*dilog(-I*e^(d*x + c))
+ 4*(2*d^3*f^3*x^3 + 6*d^3*e*f^2*x^2 + 6*c*d^2*e^2*f - 6*c^2*d*e*f^2 + (2*c^3 - 3*c)*f^3 + 3*(2*d^3*e^2*f - d*
f^3)*x)*e^(4*d*x + 4*c) + (-16*I*d^3*f^3*x^3 + (-48*I*c - 6*I)*d^2*e^2*f + (48*I*c^2 - 12*I)*d*e*f^2 + (-16*I*
c^3 + 24*I*c)*f^3 + (-48*I*d^3*e*f^2 - 6*I*d^2*f^3)*x^2 + (-48*I*d^3*e^2*f - 12*I*d^2*e*f^2 + 12*I*d*f^3)*x)*e
^(3*d*x + 3*c) - 12*(d*f^3*x + d*e*f^2)*e^(2*d*x + 2*c) + (-6*I*d^2*f^3*x^2 + 16*I*d^3*e^3 + (-48*I*c - 6*I)*d
^2*e^2*f + (48*I*c^2 - 12*I)*d*e*f^2 + (-16*I*c^3 + 24*I*c)*f^3 + (-12*I*d^2*e*f^2 + 12*I*d*f^3)*x)*e^(d*x + c
) + (9*d^2*e^2*f - 18*c*d*e*f^2 + 9*c^2*f^3 - 9*(d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*e^(4*d*x + 4*c) + (18*I*d^
2*e^2*f - 36*I*c*d*e*f^2 + 18*I*c^2*f^3)*e^(3*d*x + 3*c) + (18*I*d^2*e^2*f - 36*I*c*d*e*f^2 + 18*I*c^2*f^3)*e^
(d*x + c))*log(e^(d*x + c) + I) + (15*d^2*e^2*f - 30*c*d*e*f^2 + 3*(5*c^2 - 4)*f^3 - 3*(5*d^2*e^2*f - 10*c*d*e
*f^2 + (5*c^2 - 4)*f^3)*e^(4*d*x + 4*c) + (30*I*d^2*e^2*f - 60*I*c*d*e*f^2 + (30*I*c^2 - 24*I)*f^3)*e^(3*d*x +
3*c) + (30*I*d^2*e^2*f - 60*I*c*d*e*f^2 + (30*I*c^2 - 24*I)*f^3)*e^(d*x + c))*log(e^(d*x + c) - I) + (15*d^2*
f^3*x^2 + 30*d^2*e*f^2*x + 30*c*d*e*f^2 - 15*c^2*f^3 - 15*(d^2*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3
)*e^(4*d*x + 4*c) + (30*I*d^2*f^3*x^2 + 60*I*d^2*e*f^2*x + 60*I*c*d*e*f^2 - 30*I*c^2*f^3)*e^(3*d*x + 3*c) + (3
0*I*d^2*f^3*x^2 + 60*I*d^2*e*f^2*x + 60*I*c*d*e*f^2 - 30*I*c^2*f^3)*e^(d*x + c))*log(I*e^(d*x + c) + 1) + (9*d
^2*f^3*x^2 + 18*d^2*e*f^2*x + 18*c*d*e*f^2 - 9*c^2*f^3 - 9*(d^2*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^
3)*e^(4*d*x + 4*c) + (18*I*d^2*f^3*x^2 + 36*I*d^2*e*f^2*x + 36*I*c*d*e*f^2 - 18*I*c^2*f^3)*e^(3*d*x + 3*c) + (
18*I*d^2*f^3*x^2 + 36*I*d^2*e*f^2*x + 36*I*c*d*e*f^2 - 18*I*c^2*f^3)*e^(d*x + c))*log(-I*e^(d*x + c) + 1) + (1
8*f^3*e^(4*d*x + 4*c) - 36*I*f^3*e^(3*d*x + 3*c) - 36*I*f^3*e^(d*x + c) - 18*f^3)*polylog(3, I*e^(d*x + c)) +
(30*f^3*e^(4*d*x + 4*c) - 60*I*f^3*e^(3*d*x + 3*c) - 60*I*f^3*e^(d*x + c) - 30*f^3)*polylog(3, -I*e^(d*x + c))
)/(6*a*d^4*e^(4*d*x + 4*c) - 12*I*a*d^4*e^(3*d*x + 3*c) - 12*I*a*d^4*e^(d*x + c) - 6*a*d^4)

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (f x + e\right )}^{3} \operatorname{sech}\left (d x + c\right )^{2}}{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)^3*sech(d*x+c)^2/(a+I*a*sinh(d*x+c)),x, algorithm="giac")

[Out]

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