3.3.71 \(\int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+i a \sinh (c+d x)} \, dx\) [271]

Optimal. Leaf size=463 \[ -\frac {3 i f (e+f x)^2}{2 a d^2}-\frac {6 f^2 (e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{a d^3}+\frac {(e+f x)^3 \text {ArcTan}\left (e^{c+d x}\right )}{a d}+\frac {3 i f^2 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a d^3}+\frac {3 i f^3 \text {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^4}-\frac {3 i f (e+f x)^2 \text {PolyLog}\left (2,-i e^{c+d x}\right )}{2 a d^2}-\frac {3 i f^3 \text {PolyLog}\left (2,i e^{c+d x}\right )}{a d^4}+\frac {3 i f (e+f x)^2 \text {PolyLog}\left (2,i e^{c+d x}\right )}{2 a d^2}+\frac {3 i f^3 \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 a d^4}+\frac {3 i f^2 (e+f x) \text {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^3}-\frac {3 i f^2 (e+f x) \text {PolyLog}\left (3,i e^{c+d x}\right )}{a d^3}-\frac {3 i f^3 \text {PolyLog}\left (4,-i e^{c+d x}\right )}{a d^4}+\frac {3 i f^3 \text {PolyLog}\left (4,i e^{c+d x}\right )}{a d^4}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 a d^2}+\frac {i (e+f x)^3 \text {sech}^2(c+d x)}{2 a d}-\frac {3 i f (e+f x)^2 \tanh (c+d x)}{2 a d^2}+\frac {(e+f x)^3 \text {sech}(c+d x) \tanh (c+d x)}{2 a d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.35, antiderivative size = 463, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 13, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.448, Rules used = {5690, 4271, 4265, 2317, 2438, 2611, 6744, 2320, 6724, 5559, 4269, 3799, 2221} \begin {gather*} -\frac {6 f^2 (e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{a d^3}+\frac {(e+f x)^3 \text {ArcTan}\left (e^{c+d x}\right )}{a d}+\frac {3 i f^3 \text {Li}_2\left (-i e^{c+d x}\right )}{a d^4}-\frac {3 i f^3 \text {Li}_2\left (i e^{c+d x}\right )}{a d^4}+\frac {3 i f^3 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 a d^4}-\frac {3 i f^3 \text {Li}_4\left (-i e^{c+d x}\right )}{a d^4}+\frac {3 i f^3 \text {Li}_4\left (i e^{c+d x}\right )}{a d^4}+\frac {3 i f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{a d^3}-\frac {3 i f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{a d^3}+\frac {3 i f^2 (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{a d^3}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{2 a d^2}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{2 a d^2}-\frac {3 i f (e+f x)^2 \tanh (c+d x)}{2 a d^2}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 a d^2}+\frac {i (e+f x)^3 \text {sech}^2(c+d x)}{2 a d}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 a d}-\frac {3 i f (e+f x)^2}{2 a d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(((-3*I)/2)*f*(e + f*x)^2)/(a*d^2) - (6*f^2*(e + f*x)*ArcTan[E^(c + d*x)])/(a*d^3) + ((e + f*x)^3*ArcTan[E^(c
+ d*x)])/(a*d) + ((3*I)*f^2*(e + f*x)*Log[1 + E^(2*(c + d*x))])/(a*d^3) + ((3*I)*f^3*PolyLog[2, (-I)*E^(c + d*
x)])/(a*d^4) - (((3*I)/2)*f*(e + f*x)^2*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^2) - ((3*I)*f^3*PolyLog[2, I*E^(c +
d*x)])/(a*d^4) + (((3*I)/2)*f*(e + f*x)^2*PolyLog[2, I*E^(c + d*x)])/(a*d^2) + (((3*I)/2)*f^3*PolyLog[2, -E^(
2*(c + d*x))])/(a*d^4) + ((3*I)*f^2*(e + f*x)*PolyLog[3, (-I)*E^(c + d*x)])/(a*d^3) - ((3*I)*f^2*(e + f*x)*Pol
yLog[3, I*E^(c + d*x)])/(a*d^3) - ((3*I)*f^3*PolyLog[4, (-I)*E^(c + d*x)])/(a*d^4) + ((3*I)*f^3*PolyLog[4, I*E
^(c + d*x)])/(a*d^4) + (3*f*(e + f*x)^2*Sech[c + d*x])/(2*a*d^2) + ((I/2)*(e + f*x)^3*Sech[c + d*x]^2)/(a*d) -
(((3*I)/2)*f*(e + f*x)^2*Tanh[c + d*x])/(a*d^2) + ((e + f*x)^3*Sech[c + d*x]*Tanh[c + d*x])/(2*a*d)

Rule 2221

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

Rule 2317

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

Rule 2320

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

Rule 2438

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

Rule 2611

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

Rule 3799

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 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 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]

Rule 6744

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]

Rubi steps

\begin {align*} \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+i a \sinh (c+d x)} \, dx &=-\frac {i \int (e+f x)^3 \text {sech}^2(c+d x) \tanh (c+d x) \, dx}{a}+\frac {\int (e+f x)^3 \text {sech}^3(c+d x) \, dx}{a}\\ &=\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 a d^2}+\frac {i (e+f x)^3 \text {sech}^2(c+d x)}{2 a d}+\frac {(e+f x)^3 \text {sech}(c+d x) \tanh (c+d x)}{2 a d}+\frac {\int (e+f x)^3 \text {sech}(c+d x) \, dx}{2 a}-\frac {(3 i f) \int (e+f x)^2 \text {sech}^2(c+d x) \, dx}{2 a d}-\frac {\left (3 f^2\right ) \int (e+f x) \text {sech}(c+d x) \, dx}{a d^2}\\ &=-\frac {6 f^2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a d^3}+\frac {(e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 a d^2}+\frac {i (e+f x)^3 \text {sech}^2(c+d x)}{2 a d}-\frac {3 i f (e+f x)^2 \tanh (c+d x)}{2 a d^2}+\frac {(e+f x)^3 \text {sech}(c+d x) \tanh (c+d x)}{2 a d}-\frac {(3 i f) \int (e+f x)^2 \log \left (1-i e^{c+d x}\right ) \, dx}{2 a d}+\frac {(3 i f) \int (e+f x)^2 \log \left (1+i e^{c+d x}\right ) \, dx}{2 a d}+\frac {\left (3 i f^2\right ) \int (e+f x) \tanh (c+d x) \, dx}{a d^2}+\frac {\left (3 i f^3\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{a d^3}-\frac {\left (3 i f^3\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{a d^3}\\ &=-\frac {3 i f (e+f x)^2}{2 a d^2}-\frac {6 f^2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a d^3}+\frac {(e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{2 a d^2}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{2 a d^2}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 a d^2}+\frac {i (e+f x)^3 \text {sech}^2(c+d x)}{2 a d}-\frac {3 i f (e+f x)^2 \tanh (c+d x)}{2 a d^2}+\frac {(e+f x)^3 \text {sech}(c+d x) \tanh (c+d x)}{2 a d}+\frac {\left (3 i f^2\right ) \int (e+f x) \text {Li}_2\left (-i e^{c+d x}\right ) \, dx}{a d^2}-\frac {\left (3 i f^2\right ) \int (e+f x) \text {Li}_2\left (i e^{c+d x}\right ) \, dx}{a d^2}+\frac {\left (6 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 (3 i f^3\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}-\frac {\left (3 i f^3\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}\\ &=-\frac {3 i f (e+f x)^2}{2 a d^2}-\frac {6 f^2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a d^3}+\frac {(e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {3 i f^2 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a d^3}+\frac {3 i f^3 \text {Li}_2\left (-i e^{c+d x}\right )}{a d^4}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{2 a d^2}-\frac {3 i f^3 \text {Li}_2\left (i e^{c+d x}\right )}{a d^4}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{2 a d^2}+\frac {3 i f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{a d^3}-\frac {3 i f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{a d^3}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 a d^2}+\frac {i (e+f x)^3 \text {sech}^2(c+d x)}{2 a d}-\frac {3 i f (e+f x)^2 \tanh (c+d x)}{2 a d^2}+\frac {(e+f x)^3 \text {sech}(c+d x) \tanh (c+d x)}{2 a d}-\frac {\left (3 i f^3\right ) \int \log \left (1+e^{2 (c+d x)}\right ) \, dx}{a d^3}-\frac {\left (3 i f^3\right ) \int \text {Li}_3\left (-i e^{c+d x}\right ) \, dx}{a d^3}+\frac {\left (3 i f^3\right ) \int \text {Li}_3\left (i e^{c+d x}\right ) \, dx}{a d^3}\\ &=-\frac {3 i f (e+f x)^2}{2 a d^2}-\frac {6 f^2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a d^3}+\frac {(e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {3 i f^2 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a d^3}+\frac {3 i f^3 \text {Li}_2\left (-i e^{c+d x}\right )}{a d^4}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{2 a d^2}-\frac {3 i f^3 \text {Li}_2\left (i e^{c+d x}\right )}{a d^4}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{2 a d^2}+\frac {3 i f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{a d^3}-\frac {3 i f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{a d^3}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 a d^2}+\frac {i (e+f x)^3 \text {sech}^2(c+d x)}{2 a d}-\frac {3 i f (e+f x)^2 \tanh (c+d x)}{2 a d^2}+\frac {(e+f x)^3 \text {sech}(c+d x) \tanh (c+d x)}{2 a d}-\frac {\left (3 i f^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 a d^4}-\frac {\left (3 i f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}+\frac {\left (3 i f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}\\ &=-\frac {3 i f (e+f x)^2}{2 a d^2}-\frac {6 f^2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a d^3}+\frac {(e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {3 i f^2 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a d^3}+\frac {3 i f^3 \text {Li}_2\left (-i e^{c+d x}\right )}{a d^4}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{2 a d^2}-\frac {3 i f^3 \text {Li}_2\left (i e^{c+d x}\right )}{a d^4}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{2 a d^2}+\frac {3 i f^3 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 a d^4}+\frac {3 i f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{a d^3}-\frac {3 i f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{a d^3}-\frac {3 i f^3 \text {Li}_4\left (-i e^{c+d x}\right )}{a d^4}+\frac {3 i f^3 \text {Li}_4\left (i e^{c+d x}\right )}{a d^4}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 a d^2}+\frac {i (e+f x)^3 \text {sech}^2(c+d x)}{2 a d}-\frac {3 i f (e+f x)^2 \tanh (c+d x)}{2 a d^2}+\frac {(e+f x)^3 \text {sech}(c+d x) \tanh (c+d x)}{2 a d}\\ \end {align*}

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Mathematica [A]
time = 7.97, size = 866, normalized size = 1.87 \begin {gather*} \frac {-\frac {-4 i d^4 e^3 e^c x+48 i d^2 e e^c f^2 x-6 i d^4 e^2 e^c f x^2+24 i d^2 e^c f^3 x^2-4 i d^4 e e^c f^2 x^3-i d^4 e^c f^3 x^4+4 d^3 e^3 \log \left (i-e^{c+d x}\right )+4 i d^3 e^3 e^c \log \left (i-e^{c+d x}\right )-48 d e f^2 \log \left (i-e^{c+d x}\right )-48 i d e e^c f^2 \log \left (i-e^{c+d x}\right )+12 d^3 e^2 f x \log \left (1+i e^{c+d x}\right )+12 i d^3 e^2 e^c f x \log \left (1+i e^{c+d x}\right )-48 d f^3 x \log \left (1+i e^{c+d x}\right )-48 i d e^c f^3 x \log \left (1+i e^{c+d x}\right )+12 d^3 e f^2 x^2 \log \left (1+i e^{c+d x}\right )+12 i d^3 e e^c f^2 x^2 \log \left (1+i e^{c+d x}\right )+4 d^3 f^3 x^3 \log \left (1+i e^{c+d x}\right )+4 i d^3 e^c f^3 x^3 \log \left (1+i e^{c+d x}\right )+12 \left (1+i e^c\right ) f \left (-4 f^2+d^2 (e+f x)^2\right ) \text {PolyLog}\left (2,-i e^{c+d x}\right )-24 i d \left (-i+e^c\right ) f^2 (e+f x) \text {PolyLog}\left (3,-i e^{c+d x}\right )+24 f^3 \text {PolyLog}\left (4,-i e^{c+d x}\right )+24 i e^c f^3 \text {PolyLog}\left (4,-i e^{c+d x}\right )}{d^4 \left (-i+e^c\right )}+\frac {-i d^3 \left (d e^c x \left (4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right )-4 \left (i+e^c\right ) (e+f x)^3 \log \left (1-i e^{c+d x}\right )\right )+12 i d^2 \left (i+e^c\right ) f (e+f x)^2 \text {PolyLog}\left (2,i e^{c+d x}\right )+24 d \left (1-i e^c\right ) f^2 (e+f x) \text {PolyLog}\left (3,i e^{c+d x}\right )+24 i \left (i+e^c\right ) f^3 \text {PolyLog}\left (4,i e^{c+d x}\right )}{d^4 \left (i+e^c\right )}+x \left (4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right ) \text {sech}(c)+\frac {4 i (e+f x)^3}{d \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {24 i f (e+f x)^2 \sinh \left (\frac {d x}{2}\right )}{d^2 \left (\cosh \left (\frac {c}{2}\right )+i \sinh \left (\frac {c}{2}\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )}}{8 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-(((-4*I)*d^4*e^3*E^c*x + (48*I)*d^2*e*E^c*f^2*x - (6*I)*d^4*e^2*E^c*f*x^2 + (24*I)*d^2*E^c*f^3*x^2 - (4*I)*d
^4*e*E^c*f^2*x^3 - I*d^4*E^c*f^3*x^4 + 4*d^3*e^3*Log[I - E^(c + d*x)] + (4*I)*d^3*e^3*E^c*Log[I - E^(c + d*x)]
- 48*d*e*f^2*Log[I - E^(c + d*x)] - (48*I)*d*e*E^c*f^2*Log[I - E^(c + d*x)] + 12*d^3*e^2*f*x*Log[1 + I*E^(c +
d*x)] + (12*I)*d^3*e^2*E^c*f*x*Log[1 + I*E^(c + d*x)] - 48*d*f^3*x*Log[1 + I*E^(c + d*x)] - (48*I)*d*E^c*f^3*
x*Log[1 + I*E^(c + d*x)] + 12*d^3*e*f^2*x^2*Log[1 + I*E^(c + d*x)] + (12*I)*d^3*e*E^c*f^2*x^2*Log[1 + I*E^(c +
d*x)] + 4*d^3*f^3*x^3*Log[1 + I*E^(c + d*x)] + (4*I)*d^3*E^c*f^3*x^3*Log[1 + I*E^(c + d*x)] + 12*(1 + I*E^c)*
f*(-4*f^2 + d^2*(e + f*x)^2)*PolyLog[2, (-I)*E^(c + d*x)] - (24*I)*d*(-I + E^c)*f^2*(e + f*x)*PolyLog[3, (-I)*
E^(c + d*x)] + 24*f^3*PolyLog[4, (-I)*E^(c + d*x)] + (24*I)*E^c*f^3*PolyLog[4, (-I)*E^(c + d*x)])/(d^4*(-I + E
^c))) + ((-I)*d^3*(d*E^c*x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3) - 4*(I + E^c)*(e + f*x)^3*Log[1 - I*E^(
c + d*x)]) + (12*I)*d^2*(I + E^c)*f*(e + f*x)^2*PolyLog[2, I*E^(c + d*x)] + 24*d*(1 - I*E^c)*f^2*(e + f*x)*Pol
yLog[3, I*E^(c + d*x)] + (24*I)*(I + E^c)*f^3*PolyLog[4, I*E^(c + d*x)])/(d^4*(I + E^c)) + x*(4*e^3 + 6*e^2*f*
x + 4*e*f^2*x^2 + f^3*x^3)*Sech[c] + ((4*I)*(e + f*x)^3)/(d*(Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2])^2) - ((2
4*I)*f*(e + f*x)^2*Sinh[(d*x)/2])/(d^2*(Cosh[c/2] + I*Sinh[c/2])*(Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2])))/(
8*a)

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1151 vs. \(2 (416 ) = 832\).
time = 3.25, size = 1152, normalized size = 2.49

method	result	size
risch	\(\text {Expression too large to display}\)	\(1152\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*I/a/d^3*e*f^2*c^2*ln(exp(d*x+c)+I)+3*I*f^3*polylog(4,I*exp(d*x+c))/a/d^4+1/2*I/a/d*f^3*ln(1-I*exp(d*x+c))*
x^3+3/2*I/a/d^2*f^3*polylog(2,I*exp(d*x+c))*x^2-3*I/a/d^3*f^3*polylog(3,I*exp(d*x+c))*x-6*I/a/d^3*f^3*c*x-1/2*
I/a/d^4*f^3*c^3*ln(1+I*exp(d*x+c))+1/2*I/a/d^4*f^3*c^3*ln(1-I*exp(d*x+c))-1/2*I/a/d*f^3*ln(1+I*exp(d*x+c))*x^3
-3/2*I/a/d^2*f^3*polylog(2,-I*exp(d*x+c))*x^2+6*I/a/d^3*e*f^2*ln(exp(d*x+c)-I)-3/2*I/a/d^2*e^2*f*c*ln(exp(d*x+
c)+I)+3/2*I/a/d*ln(1-I*exp(d*x+c))*e^2*f*x+3/2*I/a/d^2*ln(1-I*exp(d*x+c))*c*e^2*f+3/2*I/a/d*ln(1-I*exp(d*x+c))
*e*f^2*x^2-3/2*I/a/d*ln(1+I*exp(d*x+c))*e*f^2*x^2+1/2*I/a/d*e^3*ln(exp(d*x+c)+I)-1/2*I/a/d*e^3*ln(exp(d*x+c)-I
)+6*I/a/d^4*f^3*c*ln(1+I*exp(d*x+c))+6*I/a/d^4*f^3*c*ln(exp(d*x+c))+6*I/a/d^3*f^3*ln(1+I*exp(d*x+c))*x+3*I/a/d
^3*f^3*polylog(3,-I*exp(d*x+c))*x+3/2*I/a/d^2*e^2*f*c*ln(exp(d*x+c)-I)-3*I/a/d^2*polylog(2,-I*exp(d*x+c))*e*f^
2*x+3*I/a/d^2*polylog(2,I*exp(d*x+c))*e*f^2*x-3/2*I/a/d*ln(1+I*exp(d*x+c))*e^2*f*x-3/2*I/a/d^2*ln(1+I*exp(d*x+
c))*c*e^2*f-3/2*I/a/d^3*e*f^2*c^2*ln(exp(d*x+c)-I)-3/2*I/a/d^3*ln(1-I*exp(d*x+c))*c^2*e*f^2+3/2*I/a/d^3*ln(1+I
*exp(d*x+c))*c^2*e*f^2-3*I*f^3*polylog(4,-I*exp(d*x+c))/a/d^4+6*I/a/d^4*f^3*polylog(2,-I*exp(d*x+c))-3*I/a/d^2
*f^3*x^2-3*I/a/d^4*f^3*c^2-6*I/a/d^4*f^3*c*ln(exp(d*x+c)-I)-1/2*I/a/d^4*f^3*c^3*ln(exp(d*x+c)+I)+1/2*I/a/d^4*f
^3*c^3*ln(exp(d*x+c)-I)-3/2*I/a/d^2*e^2*f*polylog(2,-I*exp(d*x+c))+3/2*I/a/d^2*e^2*f*polylog(2,I*exp(d*x+c))-6
*I/a/d^3*e*f^2*ln(exp(d*x+c))+3*I/a/d^3*e*f^2*polylog(3,-I*exp(d*x+c))-3*I/a/d^3*e*f^2*polylog(3,I*exp(d*x+c))
+(d*f^3*x^3*exp(d*x+c)+3*d*e*f^2*x^2*exp(d*x+c)+3*d*e^2*f*x*exp(d*x+c)+d*e^3*exp(d*x+c)+3*f^3*x^2*exp(d*x+c)-3
*I*f^3*x^2+6*e*f^2*x*exp(d*x+c)-6*I*e*f^2*x+3*e^2*f*exp(d*x+c)-3*I*e^2*f)/(exp(d*x+c)-I)^2/d^2/a

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Maxima [A]
time = 0.44, size = 691, normalized size = 1.49 \begin {gather*} -\frac {1}{2} \, {\left (\frac {4 \, e^{\left (-d x - c\right )}}{-2 \, {\left (-2 i \, a e^{\left (-d x - c\right )} - a e^{\left (-2 \, d x - 2 \, c\right )} + a\right )} d} + \frac {i \, \log \left (e^{\left (-d x - c\right )} + i\right )}{a d} - \frac {i \, \log \left (i \, e^{\left (-d x - c\right )} + 1\right )}{a d}\right )} e^{3} - \frac {6 i \, f^{2} x e}{a d^{2}} + \frac {-3 i \, f^{3} x^{2} - 6 i \, f^{2} x e - 3 i \, f e^{2} + {\left (d f^{3} x^{3} e^{c} + 3 \, {\left (d f^{2} e^{\left (c + 1\right )} + f^{3} e^{c}\right )} x^{2} + 3 \, {\left (d f e^{\left (c + 2\right )} + 2 \, f^{2} e^{\left (c + 1\right )}\right )} x + 3 \, f e^{\left (c + 2\right )}\right )} e^{\left (d x\right )}}{a d^{2} e^{\left (2 \, d x + 2 \, c\right )} - 2 i \, a d^{2} e^{\left (d x + c\right )} - a d^{2}} + \frac {3 i \, {\left (d x \log \left (-i \, e^{\left (d x + c\right )} + 1\right ) + {\rm Li}_2\left (i \, e^{\left (d x + c\right )}\right )\right )} f e^{2}}{2 \, a d^{2}} - \frac {3 i \, {\left (d^{2} x^{2} \log \left (i \, e^{\left (d x + c\right )} + 1\right ) + 2 \, d x {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right ) - 2 \, {\rm Li}_{3}(-i \, e^{\left (d x + c\right )})\right )} f^{2} e}{2 \, a d^{3}} + \frac {3 i \, {\left (d^{2} x^{2} \log \left (-i \, e^{\left (d x + c\right )} + 1\right ) + 2 \, d x {\rm Li}_2\left (i \, e^{\left (d x + c\right )}\right ) - 2 \, {\rm Li}_{3}(i \, e^{\left (d x + c\right )})\right )} f^{2} e}{2 \, a d^{3}} + \frac {6 i \, f^{2} e \log \left (i \, e^{\left (d x + c\right )} + 1\right )}{a d^{3}} - \frac {i \, {\left (d^{3} x^{3} \log \left (i \, e^{\left (d x + c\right )} + 1\right ) + 3 \, d^{2} x^{2} {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right ) - 6 \, d x {\rm Li}_{3}(-i \, e^{\left (d x + c\right )}) + 6 \, {\rm Li}_{4}(-i \, e^{\left (d x + c\right )})\right )} f^{3}}{2 \, a d^{4}} + \frac {i \, {\left (d^{3} x^{3} \log \left (-i \, e^{\left (d x + c\right )} + 1\right ) + 3 \, d^{2} x^{2} {\rm Li}_2\left (i \, e^{\left (d x + c\right )}\right ) - 6 \, d x {\rm Li}_{3}(i \, e^{\left (d x + c\right )}) + 6 \, {\rm Li}_{4}(i \, e^{\left (d x + c\right )})\right )} f^{3}}{2 \, a d^{4}} - \frac {3 i \, {\left (d^{2} f e^{2} - 4 \, f^{3}\right )} {\left (d x \log \left (i \, e^{\left (d x + c\right )} + 1\right ) + {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right )\right )}}{2 \, a d^{4}} - \frac {i \, d^{4} f^{3} x^{4} + 4 i \, d^{4} f^{2} x^{3} e + 6 i \, d^{4} f x^{2} e^{2}}{8 \, a d^{4}} + \frac {i \, d^{4} f^{3} x^{4} + 4 i \, d^{4} f^{2} x^{3} e - 6 \, {\left (-i \, d^{2} f e^{2} + 4 i \, f^{3}\right )} d^{2} x^{2}}{8 \, a d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1462 vs. \(2 (402) = 804\).
time = 0.40, size = 1462, normalized size = 3.16 \begin {gather*} \frac {-6 i \, c^{2} f^{3} + 12 i \, c d f^{2} e - 6 i \, d^{2} f e^{2} - 3 \, {\left (i \, d^{2} f^{3} x^{2} + 2 i \, d^{2} f^{2} x e + i \, d^{2} f e^{2} + {\left (-i \, d^{2} f^{3} x^{2} - 2 i \, d^{2} f^{2} x e - i \, d^{2} f e^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )} - 2 \, {\left (d^{2} f^{3} x^{2} + 2 \, d^{2} f^{2} x e + d^{2} f e^{2}\right )} e^{\left (d x + c\right )}\right )} {\rm Li}_2\left (i \, e^{\left (d x + c\right )}\right ) - 3 \, {\left (-i \, d^{2} f^{3} x^{2} - 2 i \, d^{2} f^{2} x e - i \, d^{2} f e^{2} + 4 i \, f^{3} + {\left (i \, d^{2} f^{3} x^{2} + 2 i \, d^{2} f^{2} x e + i \, d^{2} f e^{2} - 4 i \, f^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, {\left (d^{2} f^{3} x^{2} + 2 \, d^{2} f^{2} x e + d^{2} f e^{2} - 4 \, f^{3}\right )} e^{\left (d x + c\right )}\right )} {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right ) - 6 \, {\left (i \, d^{2} f^{3} x^{2} - i \, c^{2} f^{3} + 2 \, {\left (i \, d^{2} f^{2} x + i \, c d f^{2}\right )} e\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, {\left (d^{3} f^{3} x^{3} - 3 \, d^{2} f^{3} x^{2} + 6 \, c^{2} f^{3} + d^{3} e^{3} + 3 \, {\left (d^{3} f x + d^{2} f\right )} e^{2} + 3 \, {\left (d^{3} f^{2} x^{2} - 2 \, d^{2} f^{2} x - 4 \, c d f^{2}\right )} e\right )} e^{\left (d x + c\right )} + {\left (i \, c^{3} f^{3} - 3 i \, c^{2} d f^{2} e + 3 i \, c d^{2} f e^{2} - i \, d^{3} e^{3} + {\left (-i \, c^{3} f^{3} + 3 i \, c^{2} d f^{2} e - 3 i \, c d^{2} f e^{2} + i \, d^{3} e^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )} - 2 \, {\left (c^{3} f^{3} - 3 \, c^{2} d f^{2} e + 3 \, c d^{2} f e^{2} - d^{3} e^{3}\right )} e^{\left (d x + c\right )}\right )} \log \left (e^{\left (d x + c\right )} + i\right ) + {\left (-3 i \, c d^{2} f e^{2} - 3 \, {\left (-i \, c^{2} + 4 i\right )} d f^{2} e + {\left (-i \, c^{3} + 12 i \, c\right )} f^{3} + i \, d^{3} e^{3} + {\left (3 i \, c d^{2} f e^{2} - 3 \, {\left (i \, c^{2} - 4 i\right )} d f^{2} e + {\left (i \, c^{3} - 12 i \, c\right )} f^{3} - i \, d^{3} e^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, {\left (3 \, c d^{2} f e^{2} - 3 \, {\left (c^{2} - 4\right )} d f^{2} e + {\left (c^{3} - 12 \, c\right )} f^{3} - d^{3} e^{3}\right )} e^{\left (d x + c\right )}\right )} \log \left (e^{\left (d x + c\right )} - i\right ) + {\left (i \, d^{3} f^{3} x^{3} - 12 i \, d f^{3} x + {\left (i \, c^{3} - 12 i \, c\right )} f^{3} - 3 \, {\left (-i \, d^{3} f x - i \, c d^{2} f\right )} e^{2} - 3 \, {\left (-i \, d^{3} f^{2} x^{2} + i \, c^{2} d f^{2}\right )} e + {\left (-i \, d^{3} f^{3} x^{3} + 12 i \, d f^{3} x + {\left (-i \, c^{3} + 12 i \, c\right )} f^{3} - 3 \, {\left (i \, d^{3} f x + i \, c d^{2} f\right )} e^{2} - 3 \, {\left (i \, d^{3} f^{2} x^{2} - i \, c^{2} d f^{2}\right )} e\right )} e^{\left (2 \, d x + 2 \, c\right )} - 2 \, {\left (d^{3} f^{3} x^{3} - 12 \, d f^{3} x + {\left (c^{3} - 12 \, c\right )} f^{3} + 3 \, {\left (d^{3} f x + c d^{2} f\right )} e^{2} + 3 \, {\left (d^{3} f^{2} x^{2} - c^{2} d f^{2}\right )} e\right )} e^{\left (d x + c\right )}\right )} \log \left (i \, e^{\left (d x + c\right )} + 1\right ) + {\left (-i \, d^{3} f^{3} x^{3} - i \, c^{3} f^{3} - 3 \, {\left (i \, d^{3} f x + i \, c d^{2} f\right )} e^{2} - 3 \, {\left (i \, d^{3} f^{2} x^{2} - i \, c^{2} d f^{2}\right )} e + {\left (i \, d^{3} f^{3} x^{3} + i \, c^{3} f^{3} - 3 \, {\left (-i \, d^{3} f x - i \, c d^{2} f\right )} e^{2} - 3 \, {\left (-i \, d^{3} f^{2} x^{2} + i \, c^{2} d f^{2}\right )} e\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, {\left (d^{3} f^{3} x^{3} + c^{3} f^{3} + 3 \, {\left (d^{3} f x + c d^{2} f\right )} e^{2} + 3 \, {\left (d^{3} f^{2} x^{2} - c^{2} d f^{2}\right )} e\right )} e^{\left (d x + c\right )}\right )} \log \left (-i \, e^{\left (d x + c\right )} + 1\right ) - 6 \, {\left (-i \, f^{3} e^{\left (2 \, d x + 2 \, c\right )} - 2 \, f^{3} e^{\left (d x + c\right )} + i \, f^{3}\right )} {\rm polylog}\left (4, i \, e^{\left (d x + c\right )}\right ) - 6 \, {\left (i \, f^{3} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, f^{3} e^{\left (d x + c\right )} - i \, f^{3}\right )} {\rm polylog}\left (4, -i \, e^{\left (d x + c\right )}\right ) - 6 \, {\left (-i \, d f^{3} x - i \, d f^{2} e + {\left (i \, d f^{3} x + i \, d f^{2} e\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, {\left (d f^{3} x + d f^{2} e\right )} e^{\left (d x + c\right )}\right )} {\rm polylog}\left (3, i \, e^{\left (d x + c\right )}\right ) - 6 \, {\left (i \, d f^{3} x + i \, d f^{2} e + {\left (-i \, d f^{3} x - i \, d f^{2} e\right )} e^{\left (2 \, d x + 2 \, c\right )} - 2 \, {\left (d f^{3} x + d f^{2} e\right )} e^{\left (d x + c\right )}\right )} {\rm polylog}\left (3, -i \, e^{\left (d x + c\right )}\right )}{2 \, {\left (a d^{4} e^{\left (2 \, d x + 2 \, c\right )} - 2 i \, a d^{4} e^{\left (d x + c\right )} - a d^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {i \left (\int \frac {e^{3} \operatorname {sech}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx + \int \frac {f^{3} x^{3} \operatorname {sech}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx + \int \frac {3 e f^{2} x^{2} \operatorname {sech}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx + \int \frac {3 e^{2} f x \operatorname {sech}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (e+f\,x\right )}^3}{\mathrm {cosh}\left (c+d\,x\right )\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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