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

Optimal. Leaf size=667 \[ \frac{9 i f^2 (e+f x) \text{PolyLog}\left (3,-i e^{c+d x}\right )}{4 a d^3}-\frac{9 i f^2 (e+f x) \text{PolyLog}\left (3,i e^{c+d x}\right )}{4 a d^3}-\frac{9 i f (e+f x)^2 \text{PolyLog}\left (2,-i e^{c+d x}\right )}{8 a d^2}+\frac{9 i f (e+f x)^2 \text{PolyLog}\left (2,i e^{c+d x}\right )}{8 a d^2}+\frac{5 i f^3 \text{PolyLog}\left (2,-i e^{c+d x}\right )}{2 a d^4}-\frac{5 i f^3 \text{PolyLog}\left (2,i e^{c+d x}\right )}{2 a d^4}+\frac{i f^3 \text{PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 a d^4}-\frac{9 i f^3 \text{PolyLog}\left (4,-i e^{c+d x}\right )}{4 a d^4}+\frac{9 i f^3 \text{PolyLog}\left (4,i e^{c+d x}\right )}{4 a d^4}+\frac{i f^2 (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{a d^3}-\frac{5 f^2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a d^3}-\frac{i f^2 (e+f x) \text{sech}^2(c+d x)}{4 a d^3}-\frac{f^2 (e+f x) \tanh (c+d x) \text{sech}(c+d x)}{4 a d^3}-\frac{i f (e+f x)^2 \tanh (c+d x)}{2 a d^2}+\frac{f (e+f x)^2 \text{sech}^3(c+d x)}{4 a d^2}+\frac{9 f (e+f x)^2 \text{sech}(c+d x)}{8 a d^2}-\frac{i f (e+f x)^2 \tanh (c+d x) \text{sech}^2(c+d x)}{4 a d^2}+\frac{i f^3 \tanh (c+d x)}{4 a d^4}-\frac{f^3 \text{sech}(c+d x)}{4 a d^4}+\frac{3 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{4 a d}+\frac{i (e+f x)^3 \text{sech}^4(c+d x)}{4 a d}+\frac{(e+f x)^3 \tanh (c+d x) \text{sech}^3(c+d x)}{4 a d}+\frac{3 (e+f x)^3 \tanh (c+d x) \text{sech}(c+d x)}{8 a d}-\frac{i f (e+f x)^2}{2 a d^2} \]

[Out]

((-I/2)*f*(e + f*x)^2)/(a*d^2) - (5*f^2*(e + f*x)*ArcTan[E^(c + d*x)])/(a*d^3) + (3*(e + f*x)^3*ArcTan[E^(c +
d*x)])/(4*a*d) + (I*f^2*(e + f*x)*Log[1 + E^(2*(c + d*x))])/(a*d^3) + (((5*I)/2)*f^3*PolyLog[2, (-I)*E^(c + d*
x)])/(a*d^4) - (((9*I)/8)*f*(e + f*x)^2*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^2) - (((5*I)/2)*f^3*PolyLog[2, I*E^
(c + d*x)])/(a*d^4) + (((9*I)/8)*f*(e + f*x)^2*PolyLog[2, I*E^(c + d*x)])/(a*d^2) + ((I/2)*f^3*PolyLog[2, -E^(
2*(c + d*x))])/(a*d^4) + (((9*I)/4)*f^2*(e + f*x)*PolyLog[3, (-I)*E^(c + d*x)])/(a*d^3) - (((9*I)/4)*f^2*(e +
f*x)*PolyLog[3, I*E^(c + d*x)])/(a*d^3) - (((9*I)/4)*f^3*PolyLog[4, (-I)*E^(c + d*x)])/(a*d^4) + (((9*I)/4)*f^
3*PolyLog[4, I*E^(c + d*x)])/(a*d^4) - (f^3*Sech[c + d*x])/(4*a*d^4) + (9*f*(e + f*x)^2*Sech[c + d*x])/(8*a*d^
2) - ((I/4)*f^2*(e + f*x)*Sech[c + d*x]^2)/(a*d^3) + (f*(e + f*x)^2*Sech[c + d*x]^3)/(4*a*d^2) + ((I/4)*(e + f
*x)^3*Sech[c + d*x]^4)/(a*d) + ((I/4)*f^3*Tanh[c + d*x])/(a*d^4) - ((I/2)*f*(e + f*x)^2*Tanh[c + d*x])/(a*d^2)
- (f^2*(e + f*x)*Sech[c + d*x]*Tanh[c + d*x])/(4*a*d^3) + (3*(e + f*x)^3*Sech[c + d*x]*Tanh[c + d*x])/(8*a*d)
- ((I/4)*f*(e + f*x)^2*Sech[c + d*x]^2*Tanh[c + d*x])/(a*d^2) + ((e + f*x)^3*Sech[c + d*x]^3*Tanh[c + d*x])/(
4*a*d)

________________________________________________________________________________________

Rubi [A]  time = 0.71112, antiderivative size = 667, normalized size of antiderivative = 1., number of steps used = 32, number of rules used = 16, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.516, Rules used = {5571, 4186, 4185, 4180, 2279, 2391, 2531, 6609, 2282, 6589, 5451, 3767, 8, 4184, 3718, 2190} \[ \frac{9 i f^2 (e+f x) \text{PolyLog}\left (3,-i e^{c+d x}\right )}{4 a d^3}-\frac{9 i f^2 (e+f x) \text{PolyLog}\left (3,i e^{c+d x}\right )}{4 a d^3}-\frac{9 i f (e+f x)^2 \text{PolyLog}\left (2,-i e^{c+d x}\right )}{8 a d^2}+\frac{9 i f (e+f x)^2 \text{PolyLog}\left (2,i e^{c+d x}\right )}{8 a d^2}+\frac{5 i f^3 \text{PolyLog}\left (2,-i e^{c+d x}\right )}{2 a d^4}-\frac{5 i f^3 \text{PolyLog}\left (2,i e^{c+d x}\right )}{2 a d^4}+\frac{i f^3 \text{PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 a d^4}-\frac{9 i f^3 \text{PolyLog}\left (4,-i e^{c+d x}\right )}{4 a d^4}+\frac{9 i f^3 \text{PolyLog}\left (4,i e^{c+d x}\right )}{4 a d^4}+\frac{i f^2 (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{a d^3}-\frac{5 f^2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a d^3}-\frac{i f^2 (e+f x) \text{sech}^2(c+d x)}{4 a d^3}-\frac{f^2 (e+f x) \tanh (c+d x) \text{sech}(c+d x)}{4 a d^3}-\frac{i f (e+f x)^2 \tanh (c+d x)}{2 a d^2}+\frac{f (e+f x)^2 \text{sech}^3(c+d x)}{4 a d^2}+\frac{9 f (e+f x)^2 \text{sech}(c+d x)}{8 a d^2}-\frac{i f (e+f x)^2 \tanh (c+d x) \text{sech}^2(c+d x)}{4 a d^2}+\frac{i f^3 \tanh (c+d x)}{4 a d^4}-\frac{f^3 \text{sech}(c+d x)}{4 a d^4}+\frac{3 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{4 a d}+\frac{i (e+f x)^3 \text{sech}^4(c+d x)}{4 a d}+\frac{(e+f x)^3 \tanh (c+d x) \text{sech}^3(c+d x)}{4 a d}+\frac{3 (e+f x)^3 \tanh (c+d x) \text{sech}(c+d x)}{8 a d}-\frac{i f (e+f x)^2}{2 a d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-I/2)*f*(e + f*x)^2)/(a*d^2) - (5*f^2*(e + f*x)*ArcTan[E^(c + d*x)])/(a*d^3) + (3*(e + f*x)^3*ArcTan[E^(c +
d*x)])/(4*a*d) + (I*f^2*(e + f*x)*Log[1 + E^(2*(c + d*x))])/(a*d^3) + (((5*I)/2)*f^3*PolyLog[2, (-I)*E^(c + d*
x)])/(a*d^4) - (((9*I)/8)*f*(e + f*x)^2*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^2) - (((5*I)/2)*f^3*PolyLog[2, I*E^
(c + d*x)])/(a*d^4) + (((9*I)/8)*f*(e + f*x)^2*PolyLog[2, I*E^(c + d*x)])/(a*d^2) + ((I/2)*f^3*PolyLog[2, -E^(
2*(c + d*x))])/(a*d^4) + (((9*I)/4)*f^2*(e + f*x)*PolyLog[3, (-I)*E^(c + d*x)])/(a*d^3) - (((9*I)/4)*f^2*(e +
f*x)*PolyLog[3, I*E^(c + d*x)])/(a*d^3) - (((9*I)/4)*f^3*PolyLog[4, (-I)*E^(c + d*x)])/(a*d^4) + (((9*I)/4)*f^
3*PolyLog[4, I*E^(c + d*x)])/(a*d^4) - (f^3*Sech[c + d*x])/(4*a*d^4) + (9*f*(e + f*x)^2*Sech[c + d*x])/(8*a*d^
2) - ((I/4)*f^2*(e + f*x)*Sech[c + d*x]^2)/(a*d^3) + (f*(e + f*x)^2*Sech[c + d*x]^3)/(4*a*d^2) + ((I/4)*(e + f
*x)^3*Sech[c + d*x]^4)/(a*d) + ((I/4)*f^3*Tanh[c + d*x])/(a*d^4) - ((I/2)*f*(e + f*x)^2*Tanh[c + d*x])/(a*d^2)
- (f^2*(e + f*x)*Sech[c + d*x]*Tanh[c + d*x])/(4*a*d^3) + (3*(e + f*x)^3*Sech[c + d*x]*Tanh[c + d*x])/(8*a*d)
- ((I/4)*f*(e + f*x)^2*Sech[c + d*x]^2*Tanh[c + d*x])/(a*d^2) + ((e + f*x)^3*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 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 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 2279

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 2391

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 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 6609

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

Rule 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 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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 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]

Rubi steps

\begin{align*} \int \frac{(e+f x)^3 \text{sech}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx &=-\frac{i \int (e+f x)^3 \text{sech}^4(c+d x) \tanh (c+d x) \, dx}{a}+\frac{\int (e+f x)^3 \text{sech}^5(c+d x) \, dx}{a}\\ &=\frac{f (e+f x)^2 \text{sech}^3(c+d x)}{4 a d^2}+\frac{i (e+f x)^3 \text{sech}^4(c+d x)}{4 a d}+\frac{(e+f x)^3 \text{sech}^3(c+d x) \tanh (c+d x)}{4 a d}+\frac{3 \int (e+f x)^3 \text{sech}^3(c+d x) \, dx}{4 a}-\frac{(3 i f) \int (e+f x)^2 \text{sech}^4(c+d x) \, dx}{4 a d}-\frac{f^2 \int (e+f x) \text{sech}^3(c+d x) \, dx}{2 a d^2}\\ &=-\frac{f^3 \text{sech}(c+d x)}{4 a d^4}+\frac{9 f (e+f x)^2 \text{sech}(c+d x)}{8 a d^2}-\frac{i f^2 (e+f x) \text{sech}^2(c+d x)}{4 a d^3}+\frac{f (e+f x)^2 \text{sech}^3(c+d x)}{4 a d^2}+\frac{i (e+f x)^3 \text{sech}^4(c+d x)}{4 a d}-\frac{f^2 (e+f x) \text{sech}(c+d x) \tanh (c+d x)}{4 a d^3}+\frac{3 (e+f x)^3 \text{sech}(c+d x) \tanh (c+d x)}{8 a d}-\frac{i f (e+f x)^2 \text{sech}^2(c+d x) \tanh (c+d x)}{4 a d^2}+\frac{(e+f x)^3 \text{sech}^3(c+d x) \tanh (c+d x)}{4 a d}+\frac{3 \int (e+f x)^3 \text{sech}(c+d x) \, dx}{8 a}-\frac{(i f) \int (e+f x)^2 \text{sech}^2(c+d x) \, dx}{2 a d}-\frac{f^2 \int (e+f x) \text{sech}(c+d x) \, dx}{4 a d^2}-\frac{\left (9 f^2\right ) \int (e+f x) \text{sech}(c+d x) \, dx}{4 a d^2}+\frac{\left (i f^3\right ) \int \text{sech}^2(c+d x) \, dx}{4 a d^3}\\ &=-\frac{5 f^2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a d^3}+\frac{3 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{4 a d}-\frac{f^3 \text{sech}(c+d x)}{4 a d^4}+\frac{9 f (e+f x)^2 \text{sech}(c+d x)}{8 a d^2}-\frac{i f^2 (e+f x) \text{sech}^2(c+d x)}{4 a d^3}+\frac{f (e+f x)^2 \text{sech}^3(c+d x)}{4 a d^2}+\frac{i (e+f x)^3 \text{sech}^4(c+d x)}{4 a d}-\frac{i f (e+f x)^2 \tanh (c+d x)}{2 a d^2}-\frac{f^2 (e+f x) \text{sech}(c+d x) \tanh (c+d x)}{4 a d^3}+\frac{3 (e+f x)^3 \text{sech}(c+d x) \tanh (c+d x)}{8 a d}-\frac{i f (e+f x)^2 \text{sech}^2(c+d x) \tanh (c+d x)}{4 a d^2}+\frac{(e+f x)^3 \text{sech}^3(c+d x) \tanh (c+d x)}{4 a d}-\frac{(9 i f) \int (e+f x)^2 \log \left (1-i e^{c+d x}\right ) \, dx}{8 a d}+\frac{(9 i f) \int (e+f x)^2 \log \left (1+i e^{c+d x}\right ) \, dx}{8 a d}+\frac{\left (i f^2\right ) \int (e+f x) \tanh (c+d x) \, dx}{a d^2}-\frac{f^3 \operatorname{Subst}(\int 1 \, dx,x,-i \tanh (c+d x))}{4 a d^4}+\frac{\left (i f^3\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{4 a d^3}-\frac{\left (i f^3\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{4 a d^3}+\frac{\left (9 i f^3\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{4 a d^3}-\frac{\left (9 i f^3\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{4 a d^3}\\ &=-\frac{i f (e+f x)^2}{2 a d^2}-\frac{5 f^2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a d^3}+\frac{3 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{4 a d}-\frac{9 i f (e+f x)^2 \text{Li}_2\left (-i e^{c+d x}\right )}{8 a d^2}+\frac{9 i f (e+f x)^2 \text{Li}_2\left (i e^{c+d x}\right )}{8 a d^2}-\frac{f^3 \text{sech}(c+d x)}{4 a d^4}+\frac{9 f (e+f x)^2 \text{sech}(c+d x)}{8 a d^2}-\frac{i f^2 (e+f x) \text{sech}^2(c+d x)}{4 a d^3}+\frac{f (e+f x)^2 \text{sech}^3(c+d x)}{4 a d^2}+\frac{i (e+f x)^3 \text{sech}^4(c+d x)}{4 a d}+\frac{i f^3 \tanh (c+d x)}{4 a d^4}-\frac{i f (e+f x)^2 \tanh (c+d x)}{2 a d^2}-\frac{f^2 (e+f x) \text{sech}(c+d x) \tanh (c+d x)}{4 a d^3}+\frac{3 (e+f x)^3 \text{sech}(c+d x) \tanh (c+d x)}{8 a d}-\frac{i f (e+f x)^2 \text{sech}^2(c+d x) \tanh (c+d x)}{4 a d^2}+\frac{(e+f x)^3 \text{sech}^3(c+d x) \tanh (c+d x)}{4 a d}+\frac{\left (2 i f^2\right ) \int \frac{e^{2 (c+d x)} (e+f x)}{1+e^{2 (c+d x)}} \, dx}{a d^2}+\frac{\left (9 i f^2\right ) \int (e+f x) \text{Li}_2\left (-i e^{c+d x}\right ) \, dx}{4 a d^2}-\frac{\left (9 i f^2\right ) \int (e+f x) \text{Li}_2\left (i e^{c+d x}\right ) \, dx}{4 a d^2}+\frac{\left (i f^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{4 a d^4}-\frac{\left (i f^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{4 a d^4}+\frac{\left (9 i f^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{4 a d^4}-\frac{\left (9 i f^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{4 a d^4}\\ &=-\frac{i f (e+f x)^2}{2 a d^2}-\frac{5 f^2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a d^3}+\frac{3 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{4 a d}+\frac{i f^2 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a d^3}+\frac{5 i f^3 \text{Li}_2\left (-i e^{c+d x}\right )}{2 a d^4}-\frac{9 i f (e+f x)^2 \text{Li}_2\left (-i e^{c+d x}\right )}{8 a d^2}-\frac{5 i f^3 \text{Li}_2\left (i e^{c+d x}\right )}{2 a d^4}+\frac{9 i f (e+f x)^2 \text{Li}_2\left (i e^{c+d x}\right )}{8 a d^2}+\frac{9 i f^2 (e+f x) \text{Li}_3\left (-i e^{c+d x}\right )}{4 a d^3}-\frac{9 i f^2 (e+f x) \text{Li}_3\left (i e^{c+d x}\right )}{4 a d^3}-\frac{f^3 \text{sech}(c+d x)}{4 a d^4}+\frac{9 f (e+f x)^2 \text{sech}(c+d x)}{8 a d^2}-\frac{i f^2 (e+f x) \text{sech}^2(c+d x)}{4 a d^3}+\frac{f (e+f x)^2 \text{sech}^3(c+d x)}{4 a d^2}+\frac{i (e+f x)^3 \text{sech}^4(c+d x)}{4 a d}+\frac{i f^3 \tanh (c+d x)}{4 a d^4}-\frac{i f (e+f x)^2 \tanh (c+d x)}{2 a d^2}-\frac{f^2 (e+f x) \text{sech}(c+d x) \tanh (c+d x)}{4 a d^3}+\frac{3 (e+f x)^3 \text{sech}(c+d x) \tanh (c+d x)}{8 a d}-\frac{i f (e+f x)^2 \text{sech}^2(c+d x) \tanh (c+d x)}{4 a d^2}+\frac{(e+f x)^3 \text{sech}^3(c+d x) \tanh (c+d x)}{4 a d}-\frac{\left (i f^3\right ) \int \log \left (1+e^{2 (c+d x)}\right ) \, dx}{a d^3}-\frac{\left (9 i f^3\right ) \int \text{Li}_3\left (-i e^{c+d x}\right ) \, dx}{4 a d^3}+\frac{\left (9 i f^3\right ) \int \text{Li}_3\left (i e^{c+d x}\right ) \, dx}{4 a d^3}\\ &=-\frac{i f (e+f x)^2}{2 a d^2}-\frac{5 f^2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a d^3}+\frac{3 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{4 a d}+\frac{i f^2 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a d^3}+\frac{5 i f^3 \text{Li}_2\left (-i e^{c+d x}\right )}{2 a d^4}-\frac{9 i f (e+f x)^2 \text{Li}_2\left (-i e^{c+d x}\right )}{8 a d^2}-\frac{5 i f^3 \text{Li}_2\left (i e^{c+d x}\right )}{2 a d^4}+\frac{9 i f (e+f x)^2 \text{Li}_2\left (i e^{c+d x}\right )}{8 a d^2}+\frac{9 i f^2 (e+f x) \text{Li}_3\left (-i e^{c+d x}\right )}{4 a d^3}-\frac{9 i f^2 (e+f x) \text{Li}_3\left (i e^{c+d x}\right )}{4 a d^3}-\frac{f^3 \text{sech}(c+d x)}{4 a d^4}+\frac{9 f (e+f x)^2 \text{sech}(c+d x)}{8 a d^2}-\frac{i f^2 (e+f x) \text{sech}^2(c+d x)}{4 a d^3}+\frac{f (e+f x)^2 \text{sech}^3(c+d x)}{4 a d^2}+\frac{i (e+f x)^3 \text{sech}^4(c+d x)}{4 a d}+\frac{i f^3 \tanh (c+d x)}{4 a d^4}-\frac{i f (e+f x)^2 \tanh (c+d x)}{2 a d^2}-\frac{f^2 (e+f x) \text{sech}(c+d x) \tanh (c+d x)}{4 a d^3}+\frac{3 (e+f x)^3 \text{sech}(c+d x) \tanh (c+d x)}{8 a d}-\frac{i f (e+f x)^2 \text{sech}^2(c+d x) \tanh (c+d x)}{4 a d^2}+\frac{(e+f x)^3 \text{sech}^3(c+d x) \tanh (c+d x)}{4 a d}-\frac{\left (i f^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 a d^4}-\frac{\left (9 i f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-i x)}{x} \, dx,x,e^{c+d x}\right )}{4 a d^4}+\frac{\left (9 i f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(i x)}{x} \, dx,x,e^{c+d x}\right )}{4 a d^4}\\ &=-\frac{i f (e+f x)^2}{2 a d^2}-\frac{5 f^2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a d^3}+\frac{3 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{4 a d}+\frac{i f^2 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a d^3}+\frac{5 i f^3 \text{Li}_2\left (-i e^{c+d x}\right )}{2 a d^4}-\frac{9 i f (e+f x)^2 \text{Li}_2\left (-i e^{c+d x}\right )}{8 a d^2}-\frac{5 i f^3 \text{Li}_2\left (i e^{c+d x}\right )}{2 a d^4}+\frac{9 i f (e+f x)^2 \text{Li}_2\left (i e^{c+d x}\right )}{8 a d^2}+\frac{i f^3 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{2 a d^4}+\frac{9 i f^2 (e+f x) \text{Li}_3\left (-i e^{c+d x}\right )}{4 a d^3}-\frac{9 i f^2 (e+f x) \text{Li}_3\left (i e^{c+d x}\right )}{4 a d^3}-\frac{9 i f^3 \text{Li}_4\left (-i e^{c+d x}\right )}{4 a d^4}+\frac{9 i f^3 \text{Li}_4\left (i e^{c+d x}\right )}{4 a d^4}-\frac{f^3 \text{sech}(c+d x)}{4 a d^4}+\frac{9 f (e+f x)^2 \text{sech}(c+d x)}{8 a d^2}-\frac{i f^2 (e+f x) \text{sech}^2(c+d x)}{4 a d^3}+\frac{f (e+f x)^2 \text{sech}^3(c+d x)}{4 a d^2}+\frac{i (e+f x)^3 \text{sech}^4(c+d x)}{4 a d}+\frac{i f^3 \tanh (c+d x)}{4 a d^4}-\frac{i f (e+f x)^2 \tanh (c+d x)}{2 a d^2}-\frac{f^2 (e+f x) \text{sech}(c+d x) \tanh (c+d x)}{4 a d^3}+\frac{3 (e+f x)^3 \text{sech}(c+d x) \tanh (c+d x)}{8 a d}-\frac{i f (e+f x)^2 \text{sech}^2(c+d x) \tanh (c+d x)}{4 a d^2}+\frac{(e+f x)^3 \text{sech}^3(c+d x) \tanh (c+d x)}{4 a d}\\ \end{align*}

Mathematica [B]  time = 13.394, size = 1804, normalized size = 2.7 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-3*(4*d^2*e*(d^2*e^2 - 4*f^2)*x + 2*d^2*f*(3*d^2*e^2 - 4*f^2)*x^2 + 4*d^4*e*f^2*x^3 + d^4*f^3*x^4 + 4*d*(1 -
I*E^c)*f*(3*d^2*e^2 - 4*f^2)*x*Log[1 + I*E^(-c - d*x)] + 12*d^3*e*(1 - I*E^c)*f^2*x^2*Log[1 + I*E^(-c - d*x)]
+ 4*d^3*(1 - I*E^c)*f^3*x^3*Log[1 + I*E^(-c - d*x)] + (4*I)*d*e*(I + E^c)*(d^2*e^2 - 4*f^2)*(d*x - Log[I + E^(
c + d*x)]) + 4*(1 - I*E^c)*f*(-3*d^2*e^2 + 4*f^2)*PolyLog[2, (-I)*E^(-c - d*x)] + (24*I)*d*e*(I + E^c)*f^2*(d*
x*PolyLog[2, (-I)*E^(-c - d*x)] + PolyLog[3, (-I)*E^(-c - d*x)]) + (12*I)*(I + E^c)*f^3*(d^2*x^2*PolyLog[2, (-
I)*E^(-c - d*x)] + 2*(d*x*PolyLog[3, (-I)*E^(-c - d*x)] + PolyLog[4, (-I)*E^(-c - d*x)]))))/(32*a*d^4*(I + E^c
)) - ((28*f^2 - 3*d^2*(e + f*x)^2)^2 + 12*d*(1 + I*E^c)*f^2*(9*d^2*e^2 - 28*f^2)*x*Log[1 - I*E^(-c - d*x)] + 1
08*d^3*e*(1 + I*E^c)*f^3*x^2*Log[1 - I*E^(-c - d*x)] + 36*d^3*(1 + I*E^c)*f^4*x^3*Log[1 - I*E^(-c - d*x)] - 12
*d*e*(1 + I*E^c)*f*(3*d^2*e^2 - 28*f^2)*(d*x - Log[I - E^(c + d*x)]) + 12*(1 + I*E^c)*f^2*(-9*d^2*e^2 + 28*f^2
)*PolyLog[2, I*E^(-c - d*x)] - 216*d*e*(1 + I*E^c)*f^3*(d*x*PolyLog[2, I*E^(-c - d*x)] + PolyLog[3, I*E^(-c -
d*x)]) - 108*(1 + I*E^c)*f^4*(d^2*x^2*PolyLog[2, I*E^(-c - d*x)] + 2*(d*x*PolyLog[3, I*E^(-c - d*x)] + PolyLog
[4, I*E^(-c - d*x)])))/(96*a*d^4*(-I + E^c)*f) + ((3*e^3*x*Cosh[c])/(4*a) + (3*e^3*x*Sinh[c])/(4*a))/(1 + Cosh
[2*c] + Sinh[2*c]) + ((9*e^2*f*x^2*Cosh[c])/(8*a) + (9*e^2*f*x^2*Sinh[c])/(8*a))/(1 + Cosh[2*c] + Sinh[2*c]) +
((3*e*f^2*x^3*Cosh[c])/(4*a) + (3*e*f^2*x^3*Sinh[c])/(4*a))/(1 + Cosh[2*c] + Sinh[2*c]) + ((3*f^3*x^4*Cosh[c]
)/(16*a) + (3*f^3*x^4*Sinh[c])/(16*a))/(1 + Cosh[2*c] + Sinh[2*c]) - ((I/8)*(e^3 + 3*e^2*f*x + 3*e*f^2*x^2 + f
^3*x^3))/(a*d*(Cosh[c/2 + (d*x)/2] - I*Sinh[c/2 + (d*x)/2])^2) + (((3*I)/4)*(e^2*f*Sinh[(d*x)/2] + 2*e*f^2*x*S
inh[(d*x)/2] + f^3*x^2*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^3 + 3*e^2*f*x + 3*e*f^2*x^2 + f^3*x^3))/(a*d*(Cosh[c/2 + (d*x)/2] + I*Sinh[c/2 + (d*x)/2]
)^4) - ((I/4)*(e^2*f*Sinh[(d*x)/2] + 2*e*f^2*x*Sinh[(d*x)/2] + f^3*x^2*Sinh[(d*x)/2]))/(a*d^2*(Cosh[c/2] + I*S
inh[c/2])*(Cosh[c/2 + (d*x)/2] + I*Sinh[c/2 + (d*x)/2])^3) + ((2*I)*d^2*e^3*Cosh[c/2] + d*e^2*f*Cosh[c/2] - (2
*I)*e*f^2*Cosh[c/2] + (6*I)*d^2*e^2*f*x*Cosh[c/2] + 2*d*e*f^2*x*Cosh[c/2] - (2*I)*f^3*x*Cosh[c/2] + (6*I)*d^2*
e*f^2*x^2*Cosh[c/2] + d*f^3*x^2*Cosh[c/2] + (2*I)*d^2*f^3*x^3*Cosh[c/2] - 2*d^2*e^3*Sinh[c/2] - I*d*e^2*f*Sinh
[c/2] + 2*e*f^2*Sinh[c/2] - 6*d^2*e^2*f*x*Sinh[c/2] - (2*I)*d*e*f^2*x*Sinh[c/2] + 2*f^3*x*Sinh[c/2] - 6*d^2*e*
f^2*x^2*Sinh[c/2] - I*d*f^3*x^2*Sinh[c/2] - 2*d^2*f^3*x^3*Sinh[c/2])/(8*a*d^3*(Cosh[c/2] + I*Sinh[c/2])*(Cosh[
c/2 + (d*x)/2] + I*Sinh[c/2 + (d*x)/2])^2) - ((I/4)*(7*d^2*e^2*f*Sinh[(d*x)/2] - 2*f^3*Sinh[(d*x)/2] + 14*d^2*
e*f^2*x*Sinh[(d*x)/2] + 7*d^2*f^3*x^2*Sinh[(d*x)/2]))/(a*d^4*(Cosh[c/2] + I*Sinh[c/2])*(Cosh[c/2 + (d*x)/2] +
I*Sinh[c/2 + (d*x)/2]))

________________________________________________________________________________________

Maple [B]  time = 0.259, size = 2026, normalized size = 3. \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)^3/(a+I*a*sinh(d*x+c)),x)

[Out]

9/8*I/a/d^2*ln(1-I*exp(d*x+c))*c*e^2*f-9/8*I/a/d*ln(1+I*exp(d*x+c))*e^2*f*x-9/8*I/a/d^2*ln(1+I*exp(d*x+c))*c*e
^2*f+9/8*I/a/d^2*e^2*f*c*ln(exp(d*x+c)-I)+9/8*I/a/d^3*ln(1+I*exp(d*x+c))*c^2*e*f^2-9/8*I/a/d^3*e*f^2*c^2*ln(ex
p(d*x+c)-I)+9/8*I/a/d*ln(1-I*exp(d*x+c))*e*f^2*x^2+9/4*I/a/d^2*polylog(2,I*exp(d*x+c))*e*f^2*x+9/8*I/a/d*ln(1-
I*exp(d*x+c))*e^2*f*x-9/4*I*f^3*polylog(4,-I*exp(d*x+c))/a/d^4+9/4*I*f^3*polylog(4,I*exp(d*x+c))/a/d^4-I/a/d^4
*f^3*c^2+3/8*I/a/d*e^3*ln(exp(d*x+c)+I)+7/2*I/a/d^4*f^3*polylog(2,-I*exp(d*x+c))-3/2*I/a/d^4*f^3*polylog(2,I*e
xp(d*x+c))-3/8*I/a/d*e^3*ln(exp(d*x+c)-I)-I/a/d^2*f^3*x^2-2*I/a/d^3*e*f^2*ln(exp(d*x+c))-3/2*I/a/d^3*f^3*ln(1-
I*exp(d*x+c))*x-3/2*I/a/d^4*f^3*ln(1-I*exp(d*x+c))*c-3/8*I/a/d*f^3*ln(1+I*exp(d*x+c))*x^3-3/8*I/a/d^4*f^3*ln(1
+I*exp(d*x+c))*c^3-9/8*I/a/d^2*f^3*polylog(2,-I*exp(d*x+c))*x^2+9/4*I/a/d^3*f^3*polylog(3,-I*exp(d*x+c))*x+3/8
*I/a/d*f^3*ln(1-I*exp(d*x+c))*x^3+3/8*I/a/d^4*f^3*ln(1-I*exp(d*x+c))*c^3+9/8*I/a/d^2*f^3*polylog(2,I*exp(d*x+c
))*x^2-9/4*I/a/d^3*f^3*polylog(3,I*exp(d*x+c))*x+7/2*I/a/d^3*f^3*ln(1+I*exp(d*x+c))*x+7/2*I/a/d^4*f^3*ln(1+I*e
xp(d*x+c))*c+9/4*I/a/d^3*e*f^2*polylog(3,-I*exp(d*x+c))-9/4*I/a/d^3*e*f^2*polylog(3,I*exp(d*x+c))+7/2*I/a/d^3*
e*f^2*ln(exp(d*x+c)-I)-3/2*I/a/d^3*e*f^2*ln(exp(d*x+c)+I)+2*I/a/d^4*f^3*c*ln(exp(d*x+c))-9/8*I/a/d^2*e^2*f*pol
ylog(2,-I*exp(d*x+c))+9/8*I/a/d^2*e^2*f*polylog(2,I*exp(d*x+c))-7/2*I/a/d^4*f^3*c*ln(exp(d*x+c)-I)+3/2*I/a/d^4
*f^3*c*ln(exp(d*x+c)+I)+3/8*I/a/d^4*f^3*c^3*ln(exp(d*x+c)-I)-3/8*I/a/d^4*f^3*c^3*ln(exp(d*x+c)+I)-2*I/a/d^3*f^
3*c*x+9/8*I/a/d^3*e*f^2*c^2*ln(exp(d*x+c)+I)-9/8*I/a/d^3*ln(1-I*exp(d*x+c))*c^2*e*f^2-9/8*I/a/d*ln(1+I*exp(d*x
+c))*e*f^2*x^2-9/4*I/a/d^2*polylog(2,-I*exp(d*x+c))*e*f^2*x-9/8*I/a/d^2*e^2*f*c*ln(exp(d*x+c)+I)+1/4*(2*I*f^3-
22*I*d^2*f^3*x^2*exp(2*d*x+2*c)-22*I*d^2*e^2*f*exp(2*d*x+2*c)+9*d^3*e*f^2*x^2*exp(5*d*x+5*c)+9*d^3*e^2*f*x*exp
(5*d*x+5*c)+18*d^2*e*f^2*x*exp(5*d*x+5*c)+16*d^2*e*f^2*x*exp(3*d*x+3*c)-8*I*d^2*e*f^2*x-44*I*d^2*e*f^2*x*exp(2
*d*x+2*c)-d^2*f^3*x^2*exp(d*x+c)-d^2*e^2*f*exp(d*x+c)+3*d^3*f^3*x^3*exp(d*x+c)-2*d*f^3*x*exp(d*x+c)-2*d*e*f^2*
exp(d*x+c)+6*I*d^3*f^3*x^3*exp(2*d*x+2*c)-6*I*d^3*f^3*x^3*exp(4*d*x+4*c)-18*I*d^2*f^3*x^2*exp(4*d*x+4*c)-18*I*
d^2*e^2*f*exp(4*d*x+4*c)-18*I*d^3*e^2*f*x*exp(4*d*x+4*c)+18*I*d^3*e*f^2*x^2*exp(2*d*x+2*c)-18*I*d^3*e*f^2*x^2*
exp(4*d*x+4*c)+18*I*d^3*e^2*f*x*exp(2*d*x+2*c)-36*I*d^2*e*f^2*x*exp(4*d*x+4*c)+3*d^3*e^3*exp(5*d*x+5*c)+2*I*f^
3*exp(4*d*x+4*c)+4*I*f^3*exp(2*d*x+2*c)+2*d^3*e^3*exp(3*d*x+3*c)+9*d^3*e*f^2*x^2*exp(d*x+c)+9*d^3*e^2*f*x*exp(
d*x+c)-2*d^2*e*f^2*x*exp(d*x+c)-2*f^3*exp(d*x+c)+3*d^3*e^3*exp(d*x+c)-4*I*d^2*e^2*f-4*I*d^2*f^3*x^2-2*f^3*exp(
5*d*x+5*c)-4*f^3*exp(3*d*x+3*c)+6*d^3*e*f^2*x^2*exp(3*d*x+3*c)+6*d^3*e^2*f*x*exp(3*d*x+3*c)+2*d^3*f^3*x^3*exp(
3*d*x+3*c)+6*I*d^3*e^3*exp(2*d*x+2*c)-6*I*d^3*e^3*exp(4*d*x+4*c)+9*d^2*e^2*f*exp(5*d*x+5*c)+8*d^2*e^2*f*exp(3*
d*x+3*c)-2*d*f^3*x*exp(5*d*x+5*c)-4*d*f^3*x*exp(3*d*x+3*c)-2*d*e*f^2*exp(5*d*x+5*c)-4*d*e*f^2*exp(3*d*x+3*c)+3
*d^3*f^3*x^3*exp(5*d*x+5*c)+9*d^2*f^3*x^2*exp(5*d*x+5*c)+8*d^2*f^3*x^2*exp(3*d*x+3*c))/(exp(d*x+c)+I)^2/(exp(d
*x+c)-I)^4/d^4/a

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Maxima [B]  time = 6.25254, size = 1800, normalized size = 2.7 \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)^3/(a+I*a*sinh(d*x+c)),x, algorithm="maxima")

[Out]

-1/8*e^3*(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)) - 2*I*e*f^2*x/(a*d^2) + (-4*I*d^2*f^3*x^2 - 8*I*d^2*e*f^2*x - 4*I*d^2*e^2*f + 2*I*f^3 + (3*d^3*
f^3*x^3*e^(5*c) + 9*(d^3*e*f^2 + d^2*f^3)*x^2*e^(5*c) + (9*d^3*e^2*f + 18*d^2*e*f^2 - 2*d*f^3)*x*e^(5*c) + (9*
d^2*e^2*f - 2*d*e*f^2 - 2*f^3)*e^(5*c))*e^(5*d*x) + (-6*I*d^3*f^3*x^3*e^(4*c) + (-18*I*d^3*e*f^2 - 18*I*d^2*f^
3)*x^2*e^(4*c) + (-18*I*d^3*e^2*f - 36*I*d^2*e*f^2)*x*e^(4*c) + (-18*I*d^2*e^2*f + 2*I*f^3)*e^(4*c))*e^(4*d*x)
+ 2*(d^3*f^3*x^3*e^(3*c) + (3*d^3*e*f^2 + 4*d^2*f^3)*x^2*e^(3*c) + (3*d^3*e^2*f + 8*d^2*e*f^2 - 2*d*f^3)*x*e^
(3*c) + 2*(2*d^2*e^2*f - d*e*f^2 - f^3)*e^(3*c))*e^(3*d*x) + (6*I*d^3*f^3*x^3*e^(2*c) + (18*I*d^3*e*f^2 - 22*I
*d^2*f^3)*x^2*e^(2*c) + (18*I*d^3*e^2*f - 44*I*d^2*e*f^2)*x*e^(2*c) + (-22*I*d^2*e^2*f + 4*I*f^3)*e^(2*c))*e^(
2*d*x) + (3*d^3*f^3*x^3*e^c + (9*d^3*e*f^2 - d^2*f^3)*x^2*e^c + (9*d^3*e^2*f - 2*d^2*e*f^2 - 2*d*f^3)*x*e^c -
(d^2*e^2*f + 2*d*e*f^2 + 2*f^3)*e^c)*e^(d*x))/(4*a*d^4*e^(6*d*x + 6*c) - 8*I*a*d^4*e^(5*d*x + 5*c) + 4*a*d^4*e
^(4*d*x + 4*c) - 16*I*a*d^4*e^(3*d*x + 3*c) - 4*a*d^4*e^(2*d*x + 2*c) - 8*I*a*d^4*e^(d*x + c) - 4*a*d^4) - 9/8
*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)))*e*f^2/(a*d^3)
+ 9/8*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)))*e*f^2/(a
*d^3) + 7/2*I*e*f^2*log(I*e^(d*x + c) + 1)/(a*d^3) - 3/2*I*e*f^2*log(I*e^(d*x + c) - 1)/(a*d^3) - 3/8*I*(d^3*x
^3*log(I*e^(d*x + c) + 1) + 3*d^2*x^2*dilog(-I*e^(d*x + c)) - 6*d*x*polylog(3, -I*e^(d*x + c)) + 6*polylog(4,
-I*e^(d*x + c)))*f^3/(a*d^4) + 3/8*I*(d^3*x^3*log(-I*e^(d*x + c) + 1) + 3*d^2*x^2*dilog(I*e^(d*x + c)) - 6*d*x
*polylog(3, I*e^(d*x + c)) + 6*polylog(4, I*e^(d*x + c)))*f^3/(a*d^4) - 1/8*I*(9*d^2*e^2*f - 28*f^3)*(d*x*log(
I*e^(d*x + c) + 1) + dilog(-I*e^(d*x + c)))/(a*d^4) + 3/8*I*(3*d^2*e^2*f - 4*f^3)*(d*x*log(-I*e^(d*x + c) + 1)
+ dilog(I*e^(d*x + c)))/(a*d^4) - 1/32*(3*I*d^4*f^3*x^4 + 12*I*d^4*e*f^2*x^3 + (18*I*d^2*e^2*f - 24*I*f^3)*d^
2*x^2)/(a*d^4) + 1/32*(3*I*d^4*f^3*x^4 + 12*I*d^4*e*f^2*x^3 + (18*I*d^2*e^2*f - 56*I*f^3)*d^2*x^2)/(a*d^4)

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Fricas [C]  time = 2.89648, size = 9095, normalized size = 13.64 \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)^3/(a+I*a*sinh(d*x+c)),x, algorithm="fricas")

[Out]

(-8*I*d^2*e^2*f + 16*I*c*d*e*f^2 + (-8*I*c^2 + 4*I)*f^3 + (-9*I*d^2*f^3*x^2 - 18*I*d^2*e*f^2*x - 9*I*d^2*e^2*f
+ 12*I*f^3 + (9*I*d^2*f^3*x^2 + 18*I*d^2*e*f^2*x + 9*I*d^2*e^2*f - 12*I*f^3)*e^(6*d*x + 6*c) + 6*(3*d^2*f^3*x
^2 + 6*d^2*e*f^2*x + 3*d^2*e^2*f - 4*f^3)*e^(5*d*x + 5*c) + (9*I*d^2*f^3*x^2 + 18*I*d^2*e*f^2*x + 9*I*d^2*e^2*
f - 12*I*f^3)*e^(4*d*x + 4*c) + 12*(3*d^2*f^3*x^2 + 6*d^2*e*f^2*x + 3*d^2*e^2*f - 4*f^3)*e^(3*d*x + 3*c) + (-9
*I*d^2*f^3*x^2 - 18*I*d^2*e*f^2*x - 9*I*d^2*e^2*f + 12*I*f^3)*e^(2*d*x + 2*c) + 6*(3*d^2*f^3*x^2 + 6*d^2*e*f^2
*x + 3*d^2*e^2*f - 4*f^3)*e^(d*x + c))*dilog(I*e^(d*x + c)) + (9*I*d^2*f^3*x^2 + 18*I*d^2*e*f^2*x + 9*I*d^2*e^
2*f - 28*I*f^3 + (-9*I*d^2*f^3*x^2 - 18*I*d^2*e*f^2*x - 9*I*d^2*e^2*f + 28*I*f^3)*e^(6*d*x + 6*c) - 2*(9*d^2*f
^3*x^2 + 18*d^2*e*f^2*x + 9*d^2*e^2*f - 28*f^3)*e^(5*d*x + 5*c) + (-9*I*d^2*f^3*x^2 - 18*I*d^2*e*f^2*x - 9*I*d
^2*e^2*f + 28*I*f^3)*e^(4*d*x + 4*c) - 4*(9*d^2*f^3*x^2 + 18*d^2*e*f^2*x + 9*d^2*e^2*f - 28*f^3)*e^(3*d*x + 3*
c) + (9*I*d^2*f^3*x^2 + 18*I*d^2*e*f^2*x + 9*I*d^2*e^2*f - 28*I*f^3)*e^(2*d*x + 2*c) - 2*(9*d^2*f^3*x^2 + 18*d
^2*e*f^2*x + 9*d^2*e^2*f - 28*f^3)*e^(d*x + c))*dilog(-I*e^(d*x + c)) + (-8*I*d^2*f^3*x^2 - 16*I*d^2*e*f^2*x -
16*I*c*d*e*f^2 + 8*I*c^2*f^3)*e^(6*d*x + 6*c) + 2*(3*d^3*f^3*x^3 + 3*d^3*e^3 + 9*d^2*e^2*f - 2*(8*c + 1)*d*e*
f^2 + 2*(4*c^2 - 1)*f^3 + (9*d^3*e*f^2 + d^2*f^3)*x^2 + (9*d^3*e^2*f + 2*d^2*e*f^2 - 2*d*f^3)*x)*e^(5*d*x + 5*
c) + (-12*I*d^3*f^3*x^3 - 12*I*d^3*e^3 - 36*I*d^2*e^2*f - 16*I*c*d*e*f^2 + (8*I*c^2 + 4*I)*f^3 + (-36*I*d^3*e*
f^2 - 44*I*d^2*f^3)*x^2 + (-36*I*d^3*e^2*f - 88*I*d^2*e*f^2)*x)*e^(4*d*x + 4*c) + 4*(d^3*f^3*x^3 + d^3*e^3 + 4
*d^2*e^2*f - 2*(8*c + 1)*d*e*f^2 + 2*(4*c^2 - 1)*f^3 + (3*d^3*e*f^2 - 4*d^2*f^3)*x^2 + (3*d^3*e^2*f - 8*d^2*e*
f^2 - 2*d*f^3)*x)*e^(3*d*x + 3*c) + (12*I*d^3*f^3*x^3 + 12*I*d^3*e^3 - 44*I*d^2*e^2*f + 16*I*c*d*e*f^2 + (-8*I
*c^2 + 8*I)*f^3 + (36*I*d^3*e*f^2 - 36*I*d^2*f^3)*x^2 + (36*I*d^3*e^2*f - 72*I*d^2*e*f^2)*x)*e^(2*d*x + 2*c) +
2*(3*d^3*f^3*x^3 + 3*d^3*e^3 - d^2*e^2*f - 2*(8*c + 1)*d*e*f^2 + 2*(4*c^2 - 1)*f^3 + 9*(d^3*e*f^2 - d^2*f^3)*
x^2 + (9*d^3*e^2*f - 18*d^2*e*f^2 - 2*d*f^3)*x)*e^(d*x + c) + (-3*I*d^3*e^3 + 9*I*c*d^2*e^2*f + (-9*I*c^2 + 12
*I)*d*e*f^2 + (3*I*c^3 - 12*I*c)*f^3 + (3*I*d^3*e^3 - 9*I*c*d^2*e^2*f + (9*I*c^2 - 12*I)*d*e*f^2 + (-3*I*c^3 +
12*I*c)*f^3)*e^(6*d*x + 6*c) + 6*(d^3*e^3 - 3*c*d^2*e^2*f + (3*c^2 - 4)*d*e*f^2 - (c^3 - 4*c)*f^3)*e^(5*d*x +
5*c) + (3*I*d^3*e^3 - 9*I*c*d^2*e^2*f + (9*I*c^2 - 12*I)*d*e*f^2 + (-3*I*c^3 + 12*I*c)*f^3)*e^(4*d*x + 4*c) +
12*(d^3*e^3 - 3*c*d^2*e^2*f + (3*c^2 - 4)*d*e*f^2 - (c^3 - 4*c)*f^3)*e^(3*d*x + 3*c) + (-3*I*d^3*e^3 + 9*I*c*
d^2*e^2*f + (-9*I*c^2 + 12*I)*d*e*f^2 + (3*I*c^3 - 12*I*c)*f^3)*e^(2*d*x + 2*c) + 6*(d^3*e^3 - 3*c*d^2*e^2*f +
(3*c^2 - 4)*d*e*f^2 - (c^3 - 4*c)*f^3)*e^(d*x + c))*log(e^(d*x + c) + I) + (3*I*d^3*e^3 - 9*I*c*d^2*e^2*f + (
9*I*c^2 - 28*I)*d*e*f^2 + (-3*I*c^3 + 28*I*c)*f^3 + (-3*I*d^3*e^3 + 9*I*c*d^2*e^2*f + (-9*I*c^2 + 28*I)*d*e*f^
2 + (3*I*c^3 - 28*I*c)*f^3)*e^(6*d*x + 6*c) - 2*(3*d^3*e^3 - 9*c*d^2*e^2*f + (9*c^2 - 28)*d*e*f^2 - (3*c^3 - 2
8*c)*f^3)*e^(5*d*x + 5*c) + (-3*I*d^3*e^3 + 9*I*c*d^2*e^2*f + (-9*I*c^2 + 28*I)*d*e*f^2 + (3*I*c^3 - 28*I*c)*f
^3)*e^(4*d*x + 4*c) - 4*(3*d^3*e^3 - 9*c*d^2*e^2*f + (9*c^2 - 28)*d*e*f^2 - (3*c^3 - 28*c)*f^3)*e^(3*d*x + 3*c
) + (3*I*d^3*e^3 - 9*I*c*d^2*e^2*f + (9*I*c^2 - 28*I)*d*e*f^2 + (-3*I*c^3 + 28*I*c)*f^3)*e^(2*d*x + 2*c) - 2*(
3*d^3*e^3 - 9*c*d^2*e^2*f + (9*c^2 - 28)*d*e*f^2 - (3*c^3 - 28*c)*f^3)*e^(d*x + c))*log(e^(d*x + c) - I) + (3*
I*d^3*f^3*x^3 + 9*I*d^3*e*f^2*x^2 + 9*I*c*d^2*e^2*f - 9*I*c^2*d*e*f^2 + (3*I*c^3 - 28*I*c)*f^3 + (9*I*d^3*e^2*
f - 28*I*d*f^3)*x + (-3*I*d^3*f^3*x^3 - 9*I*d^3*e*f^2*x^2 - 9*I*c*d^2*e^2*f + 9*I*c^2*d*e*f^2 + (-3*I*c^3 + 28
*I*c)*f^3 + (-9*I*d^3*e^2*f + 28*I*d*f^3)*x)*e^(6*d*x + 6*c) - 2*(3*d^3*f^3*x^3 + 9*d^3*e*f^2*x^2 + 9*c*d^2*e^
2*f - 9*c^2*d*e*f^2 + (3*c^3 - 28*c)*f^3 + (9*d^3*e^2*f - 28*d*f^3)*x)*e^(5*d*x + 5*c) + (-3*I*d^3*f^3*x^3 - 9
*I*d^3*e*f^2*x^2 - 9*I*c*d^2*e^2*f + 9*I*c^2*d*e*f^2 + (-3*I*c^3 + 28*I*c)*f^3 + (-9*I*d^3*e^2*f + 28*I*d*f^3)
*x)*e^(4*d*x + 4*c) - 4*(3*d^3*f^3*x^3 + 9*d^3*e*f^2*x^2 + 9*c*d^2*e^2*f - 9*c^2*d*e*f^2 + (3*c^3 - 28*c)*f^3
+ (9*d^3*e^2*f - 28*d*f^3)*x)*e^(3*d*x + 3*c) + (3*I*d^3*f^3*x^3 + 9*I*d^3*e*f^2*x^2 + 9*I*c*d^2*e^2*f - 9*I*c
^2*d*e*f^2 + (3*I*c^3 - 28*I*c)*f^3 + (9*I*d^3*e^2*f - 28*I*d*f^3)*x)*e^(2*d*x + 2*c) - 2*(3*d^3*f^3*x^3 + 9*d
^3*e*f^2*x^2 + 9*c*d^2*e^2*f - 9*c^2*d*e*f^2 + (3*c^3 - 28*c)*f^3 + (9*d^3*e^2*f - 28*d*f^3)*x)*e^(d*x + c))*l
og(I*e^(d*x + c) + 1) + (-3*I*d^3*f^3*x^3 - 9*I*d^3*e*f^2*x^2 - 9*I*c*d^2*e^2*f + 9*I*c^2*d*e*f^2 + (-3*I*c^3
+ 12*I*c)*f^3 + (-9*I*d^3*e^2*f + 12*I*d*f^3)*x + (3*I*d^3*f^3*x^3 + 9*I*d^3*e*f^2*x^2 + 9*I*c*d^2*e^2*f - 9*I
*c^2*d*e*f^2 + (3*I*c^3 - 12*I*c)*f^3 + (9*I*d^3*e^2*f - 12*I*d*f^3)*x)*e^(6*d*x + 6*c) + 6*(d^3*f^3*x^3 + 3*d
^3*e*f^2*x^2 + 3*c*d^2*e^2*f - 3*c^2*d*e*f^2 + (c^3 - 4*c)*f^3 + (3*d^3*e^2*f - 4*d*f^3)*x)*e^(5*d*x + 5*c) +
(3*I*d^3*f^3*x^3 + 9*I*d^3*e*f^2*x^2 + 9*I*c*d^2*e^2*f - 9*I*c^2*d*e*f^2 + (3*I*c^3 - 12*I*c)*f^3 + (9*I*d^3*e
^2*f - 12*I*d*f^3)*x)*e^(4*d*x + 4*c) + 12*(d^3*f^3*x^3 + 3*d^3*e*f^2*x^2 + 3*c*d^2*e^2*f - 3*c^2*d*e*f^2 + (c
^3 - 4*c)*f^3 + (3*d^3*e^2*f - 4*d*f^3)*x)*e^(3*d*x + 3*c) + (-3*I*d^3*f^3*x^3 - 9*I*d^3*e*f^2*x^2 - 9*I*c*d^2
*e^2*f + 9*I*c^2*d*e*f^2 + (-3*I*c^3 + 12*I*c)*f^3 + (-9*I*d^3*e^2*f + 12*I*d*f^3)*x)*e^(2*d*x + 2*c) + 6*(d^3
*f^3*x^3 + 3*d^3*e*f^2*x^2 + 3*c*d^2*e^2*f - 3*c^2*d*e*f^2 + (c^3 - 4*c)*f^3 + (3*d^3*e^2*f - 4*d*f^3)*x)*e^(d
*x + c))*log(-I*e^(d*x + c) + 1) + (18*I*f^3*e^(6*d*x + 6*c) + 36*f^3*e^(5*d*x + 5*c) + 18*I*f^3*e^(4*d*x + 4*
c) + 72*f^3*e^(3*d*x + 3*c) - 18*I*f^3*e^(2*d*x + 2*c) + 36*f^3*e^(d*x + c) - 18*I*f^3)*polylog(4, I*e^(d*x +
c)) + (-18*I*f^3*e^(6*d*x + 6*c) - 36*f^3*e^(5*d*x + 5*c) - 18*I*f^3*e^(4*d*x + 4*c) - 72*f^3*e^(3*d*x + 3*c)
+ 18*I*f^3*e^(2*d*x + 2*c) - 36*f^3*e^(d*x + c) + 18*I*f^3)*polylog(4, -I*e^(d*x + c)) + (18*I*d*f^3*x + 18*I*
d*e*f^2 + (-18*I*d*f^3*x - 18*I*d*e*f^2)*e^(6*d*x + 6*c) - 36*(d*f^3*x + d*e*f^2)*e^(5*d*x + 5*c) + (-18*I*d*f
^3*x - 18*I*d*e*f^2)*e^(4*d*x + 4*c) - 72*(d*f^3*x + d*e*f^2)*e^(3*d*x + 3*c) + (18*I*d*f^3*x + 18*I*d*e*f^2)*
e^(2*d*x + 2*c) - 36*(d*f^3*x + d*e*f^2)*e^(d*x + c))*polylog(3, I*e^(d*x + c)) + (-18*I*d*f^3*x - 18*I*d*e*f^
2 + (18*I*d*f^3*x + 18*I*d*e*f^2)*e^(6*d*x + 6*c) + 36*(d*f^3*x + d*e*f^2)*e^(5*d*x + 5*c) + (18*I*d*f^3*x + 1
8*I*d*e*f^2)*e^(4*d*x + 4*c) + 72*(d*f^3*x + d*e*f^2)*e^(3*d*x + 3*c) + (-18*I*d*f^3*x - 18*I*d*e*f^2)*e^(2*d*
x + 2*c) + 36*(d*f^3*x + d*e*f^2)*e^(d*x + c))*polylog(3, -I*e^(d*x + c)))/(8*a*d^4*e^(6*d*x + 6*c) - 16*I*a*d
^4*e^(5*d*x + 5*c) + 8*a*d^4*e^(4*d*x + 4*c) - 32*I*a*d^4*e^(3*d*x + 3*c) - 8*a*d^4*e^(2*d*x + 2*c) - 16*I*a*d
^4*e^(d*x + c) - 8*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)**3/(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 )^{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)^3*sech(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="giac")

[Out]

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