\(\int \frac {(e+f x)^3 \text {sech}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx\) [283]

Optimal result

Integrand size = 31, antiderivative size = 667 \[ \int \frac {(e+f x)^3 \text {sech}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {i f (e+f x)^2}{2 a d^2}-\frac {5 f^2 (e+f x) \arctan \left (e^{c+d x}\right )}{a d^3}+\frac {3 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{4 a d}+\frac {i f^2 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a d^3}+\frac {5 i f^3 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 a d^4}-\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{8 a d^2}-\frac {5 i f^3 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 a d^4}+\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{8 a d^2}+\frac {i f^3 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 a d^4}+\frac {9 i f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{4 a d^3}-\frac {9 i f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{4 a d^3}-\frac {9 i f^3 \operatorname {PolyLog}\left (4,-i e^{c+d x}\right )}{4 a d^4}+\frac {9 i f^3 \operatorname {PolyLog}\left (4,i e^{c+d x}\right )}{4 a d^4}-\frac {f^3 \text {sech}(c+d x)}{4 a d^4}+\frac {9 f (e+f x)^2 \text {sech}(c+d x)}{8 a d^2}-\frac {i f^2 (e+f x) \text {sech}^2(c+d x)}{4 a d^3}+\frac {f (e+f x)^2 \text {sech}^3(c+d x)}{4 a d^2}+\frac {i (e+f x)^3 \text {sech}^4(c+d x)}{4 a d}+\frac {i f^3 \tanh (c+d x)}{4 a d^4}-\frac {i f (e+f x)^2 \tanh (c+d x)}{2 a d^2}-\frac {f^2 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{4 a d^3}+\frac {3 (e+f x)^3 \text {sech}(c+d x) \tanh (c+d x)}{8 a d}-\frac {i f (e+f x)^2 \text {sech}^2(c+d x) \tanh (c+d x)}{4 a d^2}+\frac {(e+f x)^3 \text {sech}^3(c+d x) \tanh (c+d x)}{4 a d} \]

[Out]

I*f^2*(f*x+e)*ln(1+exp(2*d*x+2*c))/a/d^3-5*f^2*(f*x+e)*arctan(exp(d*x+c))/a/d^3+3/4*(f*x+e)^3*arctan(exp(d*x+c
))/a/d-9/4*I*f^3*polylog(4,-I*exp(d*x+c))/a/d^4-1/2*I*f*(f*x+e)^2/a/d^2-9/8*I*f*(f*x+e)^2*polylog(2,-I*exp(d*x
+c))/a/d^2-9/4*I*f^2*(f*x+e)*polylog(3,I*exp(d*x+c))/a/d^3-1/4*I*f*(f*x+e)^2*sech(d*x+c)^2*tanh(d*x+c)/a/d^2+1
/4*I*f^3*tanh(d*x+c)/a/d^4-1/4*I*f^2*(f*x+e)*sech(d*x+c)^2/a/d^3-5/2*I*f^3*polylog(2,I*exp(d*x+c))/a/d^4+9/8*I
*f*(f*x+e)^2*polylog(2,I*exp(d*x+c))/a/d^2+1/4*I*(f*x+e)^3*sech(d*x+c)^4/a/d-1/4*f^3*sech(d*x+c)/a/d^4+9/8*f*(
f*x+e)^2*sech(d*x+c)/a/d^2+9/4*I*f^2*(f*x+e)*polylog(3,-I*exp(d*x+c))/a/d^3+1/4*f*(f*x+e)^2*sech(d*x+c)^3/a/d^
2+9/4*I*f^3*polylog(4,I*exp(d*x+c))/a/d^4+1/2*I*f^3*polylog(2,-exp(2*d*x+2*c))/a/d^4+5/2*I*f^3*polylog(2,-I*ex
p(d*x+c))/a/d^4-1/4*f^2*(f*x+e)*sech(d*x+c)*tanh(d*x+c)/a/d^3+3/8*(f*x+e)^3*sech(d*x+c)*tanh(d*x+c)/a/d-1/2*I*
f*(f*x+e)^2*tanh(d*x+c)/a/d^2+1/4*(f*x+e)^3*sech(d*x+c)^3*tanh(d*x+c)/a/d

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 667, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.516, Rules used = {5690, 4271, 4270, 4265, 2317, 2438, 2611, 6744, 2320, 6724, 5559, 3852, 8, 4269, 3799, 2221} \[ \int \frac {(e+f x)^3 \text {sech}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {5 f^2 (e+f x) \arctan \left (e^{c+d x}\right )}{a d^3}+\frac {3 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{4 a d}+\frac {5 i f^3 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 a d^4}-\frac {5 i f^3 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 a d^4}+\frac {i f^3 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 a d^4}-\frac {9 i f^3 \operatorname {PolyLog}\left (4,-i e^{c+d x}\right )}{4 a d^4}+\frac {9 i f^3 \operatorname {PolyLog}\left (4,i e^{c+d x}\right )}{4 a d^4}+\frac {i f^3 \tanh (c+d x)}{4 a d^4}-\frac {f^3 \text {sech}(c+d x)}{4 a d^4}+\frac {9 i f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{4 a d^3}-\frac {9 i f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{4 a d^3}+\frac {i f^2 (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{a d^3}-\frac {i f^2 (e+f x) \text {sech}^2(c+d x)}{4 a d^3}-\frac {f^2 (e+f x) \tanh (c+d x) \text {sech}(c+d x)}{4 a d^3}-\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{8 a d^2}+\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{8 a d^2}-\frac {i f (e+f x)^2 \tanh (c+d x)}{2 a d^2}+\frac {f (e+f x)^2 \text {sech}^3(c+d x)}{4 a d^2}+\frac {9 f (e+f x)^2 \text {sech}(c+d x)}{8 a d^2}-\frac {i f (e+f x)^2 \tanh (c+d x) \text {sech}^2(c+d x)}{4 a d^2}+\frac {i (e+f x)^3 \text {sech}^4(c+d x)}{4 a d}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}^3(c+d x)}{4 a d}+\frac {3 (e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{8 a d}-\frac {i f (e+f x)^2}{2 a d^2} \]

[In]

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

[Out]

((-1/2*I)*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 8

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

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 3852

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 4265

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c +
d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^(I*k*Pi)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*
Log[1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 4269

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

Rule 4270

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

Rule 4271

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

Rule 5559

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

Rule 5690

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[1/a, Int[(e + f*x)^m*Sech[c + d*x]^(n + 2), x], x] + Dist[1/b, Int[(e + f*x)^m*Sech[c + d*x]^(n +
1)*Tanh[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && EqQ[a^2 + b^2, 0]

Rule 6724

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

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*} \text {integral}& = -\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) \arctan \left (e^{c+d x}\right )}{a d^3}+\frac {3 (e+f x)^3 \arctan \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 \text {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) \arctan \left (e^{c+d x}\right )}{a d^3}+\frac {3 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{4 a d}-\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{8 a d^2}+\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,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) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) \, dx}{4 a d^2}-\frac {\left (9 i f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) \, dx}{4 a d^2}+\frac {\left (i f^3\right ) \text {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 ) \text {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 ) \text {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 ) \text {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) \arctan \left (e^{c+d x}\right )}{a d^3}+\frac {3 (e+f x)^3 \arctan \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 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 a d^4}-\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{8 a d^2}-\frac {5 i f^3 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 a d^4}+\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{8 a d^2}+\frac {9 i f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{4 a d^3}-\frac {9 i f^2 (e+f x) \operatorname {PolyLog}\left (3,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 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right ) \, dx}{4 a d^3}+\frac {\left (9 i f^3\right ) \int \operatorname {PolyLog}\left (3,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) \arctan \left (e^{c+d x}\right )}{a d^3}+\frac {3 (e+f x)^3 \arctan \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 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 a d^4}-\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{8 a d^2}-\frac {5 i f^3 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 a d^4}+\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{8 a d^2}+\frac {9 i f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{4 a d^3}-\frac {9 i f^2 (e+f x) \operatorname {PolyLog}\left (3,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 ) \text {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 ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-i x)}{x} \, dx,x,e^{c+d x}\right )}{4 a d^4}+\frac {\left (9 i f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(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) \arctan \left (e^{c+d x}\right )}{a d^3}+\frac {3 (e+f x)^3 \arctan \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 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 a d^4}-\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{8 a d^2}-\frac {5 i f^3 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 a d^4}+\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{8 a d^2}+\frac {i f^3 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 a d^4}+\frac {9 i f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{4 a d^3}-\frac {9 i f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{4 a d^3}-\frac {9 i f^3 \operatorname {PolyLog}\left (4,-i e^{c+d x}\right )}{4 a d^4}+\frac {9 i f^3 \operatorname {PolyLog}\left (4,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] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2008\) vs. \(2(667)=1334\).

Time = 9.27 (sec) , antiderivative size = 2008, normalized size of antiderivative = 3.01 \[ \int \frac {(e+f x)^3 \text {sech}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

(-3*E^c*((d^2*e^3*x)/E^c - (4*e*f^2*x)/E^c - (e*(1 - I*E^c)*(d^2*e^2 - 4*f^2)*x)/E^c + (3*d^2*e^2*f*x^2)/(2*E^
c) - (2*f^3*x^2)/E^c + (d^2*e*f^2*x^3)/E^c + (d^2*f^3*x^4)/(4*E^c) + ((1 - I*E^c)*f*(3*d^2*e^2 - 4*f^2)*x*Log[
1 + I*E^(-c - d*x)])/(d*E^c) + (3*d*e*(1 - I*E^c)*f^2*x^2*Log[1 + I*E^(-c - d*x)])/E^c + (d*(1 - I*E^c)*f^3*x^
3*Log[1 + I*E^(-c - d*x)])/E^c + (e*(1 - I*E^c)*(d^2*e^2 - 4*f^2)*Log[I + E^(c + d*x)])/(d*E^c) - ((1 - I*E^c)
*f*(3*d^2*e^2 - 4*f^2)*PolyLog[2, (-I)*E^(-c - d*x)])/(d^2*E^c) - (6*e*(1 - I*E^c)*f^2*x*PolyLog[2, (-I)*E^(-c
- d*x)])/E^c - (3*(1 - I*E^c)*f^3*x^2*PolyLog[2, (-I)*E^(-c - d*x)])/E^c - (6*e*(1 - I*E^c)*f^2*PolyLog[3, (-
I)*E^(-c - d*x)])/(d*E^c) - (6*(1 - I*E^c)*f^3*x*PolyLog[3, (-I)*E^(-c - d*x)])/(d*E^c) - (6*(1 - I*E^c)*f^3*P
olyLog[4, (-I)*E^(-c - d*x)])/(d^2*E^c)))/(8*a*d^2*(I + E^c)) - (-12*d^2*e*(1 + I*E^c)*f*(3*d^2*e^2 - 28*f^2)*
x + (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)] + 108
*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)*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^2*e*(1 + I*E^c)*f^3*x*PolyLog[2, I*E^(-c - d*x)] - 108*d^2*(1 + I*E^c)*f^4*x^2*Poly
Log[2, I*E^(-c - d*x)] - 216*d*e*(1 + I*E^c)*f^3*PolyLog[3, I*E^(-c - d*x)] - 216*d*(1 + I*E^c)*f^4*x*PolyLog[
3, I*E^(-c - d*x)] - 216*(1 + I*E^c)*f^4*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*Sinh[(d*x)/2] + f^3*x^2*Sinh[(d*x)/2]))/(a*d^2*(Cosh[c/2] - I*Sin
h[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*Sinh[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*Co
sh[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*(Co
sh[c/2] + I*Sinh[c/2])*(Cosh[c/2 + (d*x)/2] + I*Sinh[c/2 + (d*x)/2]))

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1908 vs. \(2 (589 ) = 1178\).

Time = 188.52 (sec) , antiderivative size = 1909, normalized size of antiderivative = 2.86

method	result	size
risch	\(\text {Expression too large to display}\)	\(1909\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3854 vs. \(2 (560) = 1120\).

Time = 0.29 (sec) , antiderivative size = 3854, normalized size of antiderivative = 5.78 \[ \int \frac {(e+f x)^3 \text {sech}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [F]

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

[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)

Mupad [F(-1)]

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

[In]

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

[Out]

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