\(\int \frac {(e+f x)^3 \text {sech}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx\) [277]

Optimal result

Integrand size = 31, antiderivative size = 450 \[ \int \frac {(e+f x)^3 \text {sech}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {2 (e+f x)^3}{3 a d}-\frac {i f (e+f x)^2 \arctan \left (e^{c+d x}\right )}{a d^2}+\frac {i f^3 \arctan (\sinh (c+d x))}{a d^4}-\frac {2 f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a d^2}+\frac {f^3 \log (\cosh (c+d x))}{a d^4}-\frac {f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}+\frac {f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a d^3}-\frac {2 f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{a d^3}+\frac {f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}-\frac {f^3 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{a d^4}+\frac {f^3 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{a d^4}-\frac {i f^2 (e+f x) \text {sech}(c+d x)}{a d^3}+\frac {f (e+f x)^2 \text {sech}^2(c+d x)}{2 a d^2}+\frac {i (e+f x)^3 \text {sech}^3(c+d x)}{3 a d}-\frac {f^2 (e+f x) \tanh (c+d x)}{a d^3}+\frac {2 (e+f x)^3 \tanh (c+d x)}{3 a d}-\frac {i f (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 a d^2}+\frac {(e+f x)^3 \text {sech}^2(c+d x) \tanh (c+d x)}{3 a d} \]

[Out]

2/3*(f*x+e)^3/a/d-I*f^2*(f*x+e)*sech(d*x+c)/a/d^3+I*f^3*arctan(sinh(d*x+c))/a/d^4-2*f*(f*x+e)^2*ln(1+exp(2*d*x
+2*c))/a/d^2+f^3*ln(cosh(d*x+c))/a/d^4-f^2*(f*x+e)*polylog(2,-I*exp(d*x+c))/a/d^3+f^2*(f*x+e)*polylog(2,I*exp(
d*x+c))/a/d^3-2*f^2*(f*x+e)*polylog(2,-exp(2*d*x+2*c))/a/d^3+f^3*polylog(3,-I*exp(d*x+c))/a/d^4-f^3*polylog(3,
I*exp(d*x+c))/a/d^4+f^3*polylog(3,-exp(2*d*x+2*c))/a/d^4+1/3*I*(f*x+e)^3*sech(d*x+c)^3/a/d+1/2*f*(f*x+e)^2*sec
h(d*x+c)^2/a/d^2-1/2*I*f*(f*x+e)^2*sech(d*x+c)*tanh(d*x+c)/a/d^2-f^2*(f*x+e)*tanh(d*x+c)/a/d^3+2/3*(f*x+e)^3*t
anh(d*x+c)/a/d-I*f*(f*x+e)^2*arctan(exp(d*x+c))/a/d^2+1/3*(f*x+e)^3*sech(d*x+c)^2*tanh(d*x+c)/a/d

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 450, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.387, Rules used = {5690, 4271, 4269, 3556, 3799, 2221, 2611, 2320, 6724, 5559, 3855, 4265} \[ \int \frac {(e+f x)^3 \text {sech}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {i f^3 \arctan (\sinh (c+d x))}{a d^4}-\frac {i f (e+f x)^2 \arctan \left (e^{c+d x}\right )}{a d^2}+\frac {f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}-\frac {f^3 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{a d^4}+\frac {f^3 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{a d^4}+\frac {f^3 \log (\cosh (c+d x))}{a d^4}-\frac {f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}+\frac {f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a d^3}-\frac {2 f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{a d^3}-\frac {f^2 (e+f x) \tanh (c+d x)}{a d^3}-\frac {i f^2 (e+f x) \text {sech}(c+d x)}{a d^3}-\frac {2 f (e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{a d^2}+\frac {f (e+f x)^2 \text {sech}^2(c+d x)}{2 a d^2}-\frac {i f (e+f x)^2 \tanh (c+d x) \text {sech}(c+d x)}{2 a d^2}+\frac {2 (e+f x)^3 \tanh (c+d x)}{3 a d}+\frac {i (e+f x)^3 \text {sech}^3(c+d x)}{3 a d}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}^2(c+d x)}{3 a d}+\frac {2 (e+f x)^3}{3 a d} \]

[In]

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

[Out]

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

Rule 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 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 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 3556

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

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 3855

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

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]

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

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1078\) vs. \(2(450)=900\).

Time = 8.48 (sec) , antiderivative size = 1078, normalized size of antiderivative = 2.40 \[ \int \frac {(e+f x)^3 \text {sech}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {i f \left (\frac {(e+f x)^3}{f}+\frac {3 \left (1-i e^c\right ) (e+f x)^2 \log \left (1+i e^{-c-d x}\right )}{d}+\frac {6 i \left (i+e^c\right ) f \left (d (e+f x) \operatorname {PolyLog}\left (2,-i e^{-c-d x}\right )+f \operatorname {PolyLog}\left (3,-i e^{-c-d x}\right )\right )}{d^3}\right )}{2 a d \left (i+e^c\right )}+\frac {i f \left (15 d^2 e^2 x-12 f^2 x-3 \left (1+i e^c\right ) \left (5 d^2 e^2-4 f^2\right ) x+15 d^2 e f x^2+5 d^2 f^2 x^3+30 d e \left (1+i e^c\right ) f x \log \left (1-i e^{-c-d x}\right )+15 d \left (1+i e^c\right ) f^2 x^2 \log \left (1-i e^{-c-d x}\right )+\frac {3 \left (1+i e^c\right ) \left (5 d^2 e^2-4 f^2\right ) \log \left (i-e^{c+d x}\right )}{d}-30 e \left (1+i e^c\right ) f \operatorname {PolyLog}\left (2,i e^{-c-d x}\right )-30 \left (1+i e^c\right ) f^2 x \operatorname {PolyLog}\left (2,i e^{-c-d x}\right )-\frac {30 \left (1+i e^c\right ) f^2 \operatorname {PolyLog}\left (3,i e^{-c-d x}\right )}{d}\right )}{6 a d^3 \left (-i+e^c\right )}+\frac {e^3 \sinh \left (\frac {d x}{2}\right )+3 e^2 f x \sinh \left (\frac {d x}{2}\right )+3 e f^2 x^2 \sinh \left (\frac {d x}{2}\right )+f^3 x^3 \sinh \left (\frac {d x}{2}\right )}{2 a d \left (\cosh \left (\frac {c}{2}\right )-i \sinh \left (\frac {c}{2}\right )\right ) \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )-i \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}+\frac {e^3 \sinh \left (\frac {d x}{2}\right )+3 e^2 f x \sinh \left (\frac {d x}{2}\right )+3 e f^2 x^2 \sinh \left (\frac {d x}{2}\right )+f^3 x^3 \sinh \left (\frac {d x}{2}\right )}{3 a d \left (\cosh \left (\frac {c}{2}\right )+i \sinh \left (\frac {c}{2}\right )\right ) \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )+i \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^3}+\frac {i d e^3 \cosh \left (\frac {c}{2}\right )+3 e^2 f \cosh \left (\frac {c}{2}\right )+3 i d e^2 f x \cosh \left (\frac {c}{2}\right )+6 e f^2 x \cosh \left (\frac {c}{2}\right )+3 i d e f^2 x^2 \cosh \left (\frac {c}{2}\right )+3 f^3 x^2 \cosh \left (\frac {c}{2}\right )+i d f^3 x^3 \cosh \left (\frac {c}{2}\right )+d e^3 \sinh \left (\frac {c}{2}\right )+3 i e^2 f \sinh \left (\frac {c}{2}\right )+3 d e^2 f x \sinh \left (\frac {c}{2}\right )+6 i e f^2 x \sinh \left (\frac {c}{2}\right )+3 d e f^2 x^2 \sinh \left (\frac {c}{2}\right )+3 i f^3 x^2 \sinh \left (\frac {c}{2}\right )+d f^3 x^3 \sinh \left (\frac {c}{2}\right )}{6 a d^2 \left (\cosh \left (\frac {c}{2}\right )+i \sinh \left (\frac {c}{2}\right )\right ) \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )+i \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^2}+\frac {5 d^2 e^3 \sinh \left (\frac {d x}{2}\right )-12 e f^2 \sinh \left (\frac {d x}{2}\right )+15 d^2 e^2 f x \sinh \left (\frac {d x}{2}\right )-12 f^3 x \sinh \left (\frac {d x}{2}\right )+15 d^2 e f^2 x^2 \sinh \left (\frac {d x}{2}\right )+5 d^2 f^3 x^3 \sinh \left (\frac {d x}{2}\right )}{6 a d^3 \left (\cosh \left (\frac {c}{2}\right )+i \sinh \left (\frac {c}{2}\right )\right ) \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )+i \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )} \]

[In]

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

[Out]

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1020 vs. \(2 (424 ) = 848\).

Time = 89.80 (sec) , antiderivative size = 1021, normalized size of antiderivative = 2.27

method	result	size
risch	\(-\frac {3 f^{3} \ln \left (1-i {\mathrm e}^{d x +c}\right ) x^{2}}{2 a \,d^{2}}+\frac {4 f^{2} e \,c^{2}}{a \,d^{3}}+\frac {4 f^{2} e \,x^{2}}{a d}+\frac {4 f^{3} x^{3}}{3 a d}+\frac {f^{3} \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{a \,d^{4}}-\frac {2 f^{3} \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{4}}+\frac {i \left (-4 i d^{2} e^{3}+8 d^{2} f^{3} x^{3} {\mathrm e}^{d x +c}-3 d \,f^{3} x^{2} {\mathrm e}^{3 d x +3 c}+6 i e \,f^{2}+24 d^{2} e \,f^{2} x^{2} {\mathrm e}^{d x +c}-6 d e \,f^{2} x \,{\mathrm e}^{3 d x +3 c}-12 i d^{2} e^{2} f x +6 i f^{3} x \,{\mathrm e}^{2 d x +2 c}+24 d^{2} e^{2} f x \,{\mathrm e}^{d x +c}-3 d \,e^{2} f \,{\mathrm e}^{3 d x +3 c}-3 d \,f^{3} x^{2} {\mathrm e}^{d x +c}-6 f^{3} x \,{\mathrm e}^{3 d x +3 c}-4 i d^{2} x^{3} f^{3}+6 i e \,f^{2} {\mathrm e}^{2 d x +2 c}+8 d^{2} e^{3} {\mathrm e}^{d x +c}-6 d e \,f^{2} x \,{\mathrm e}^{d x +c}-6 e \,f^{2} {\mathrm e}^{3 d x +3 c}+6 i f^{3} x -3 d \,e^{2} f \,{\mathrm e}^{d x +c}-6 f^{3} x \,{\mathrm e}^{d x +c}-12 i d^{2} e \,f^{2} x^{2}-6 e \,f^{2} {\mathrm e}^{d x +c}\right )}{3 \left ({\mathrm e}^{d x +c}+i\right ) \left ({\mathrm e}^{d x +c}-i\right )^{3} d^{3} a}-\frac {3 f^{2} e \ln \left (1-i {\mathrm e}^{d x +c}\right ) x}{a \,d^{2}}+\frac {8 f^{2} e c x}{a \,d^{2}}-\frac {3 f^{2} e \ln \left (1-i {\mathrm e}^{d x +c}\right ) c}{a \,d^{3}}-\frac {i f^{3} c^{2} \arctan \left ({\mathrm e}^{d x +c}\right )}{a \,d^{4}}-\frac {8 f^{2} e c \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{3}}-\frac {i f \,e^{2} \arctan \left ({\mathrm e}^{d x +c}\right )}{a \,d^{2}}-\frac {5 f^{2} e \ln \left (1+i {\mathrm e}^{d x +c}\right ) c}{d^{3} a}-\frac {5 f^{2} e \ln \left (1+i {\mathrm e}^{d x +c}\right ) x}{d^{2} a}+\frac {3 f^{3} \operatorname {polylog}\left (3, i {\mathrm e}^{d x +c}\right )}{a \,d^{4}}+\frac {5 f^{3} \operatorname {polylog}\left (3, -i {\mathrm e}^{d x +c}\right )}{a \,d^{4}}-\frac {2 f^{3} c^{2} \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{d^{4} a}+\frac {4 c^{2} f^{3} \ln \left ({\mathrm e}^{d x +c}\right )}{d^{4} a}-\frac {2 e^{2} f \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{d^{2} a}+\frac {4 e^{2} f \ln \left ({\mathrm e}^{d x +c}\right )}{d^{2} a}-\frac {4 f^{3} x \,c^{2}}{d^{3} a}-\frac {8 f^{3} c^{3}}{3 d^{4} a}+\frac {4 e \,f^{2} c \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{d^{3} a}+\frac {2 i f^{2} c e \arctan \left ({\mathrm e}^{d x +c}\right )}{a \,d^{3}}-\frac {3 f^{3} \operatorname {polylog}\left (2, i {\mathrm e}^{d x +c}\right ) x}{a \,d^{3}}+\frac {3 f^{3} \ln \left (1-i {\mathrm e}^{d x +c}\right ) c^{2}}{2 a \,d^{4}}-\frac {3 f^{2} e \operatorname {polylog}\left (2, i {\mathrm e}^{d x +c}\right )}{a \,d^{3}}+\frac {2 i f^{3} \arctan \left ({\mathrm e}^{d x +c}\right )}{a \,d^{4}}-\frac {5 f^{2} e \operatorname {polylog}\left (2, -i {\mathrm e}^{d x +c}\right )}{d^{3} a}+\frac {5 f^{3} \ln \left (1+i {\mathrm e}^{d x +c}\right ) c^{2}}{2 d^{4} a}-\frac {5 f^{3} \ln \left (1+i {\mathrm e}^{d x +c}\right ) x^{2}}{2 d^{2} a}-\frac {5 f^{3} \operatorname {polylog}\left (2, -i {\mathrm e}^{d x +c}\right ) x}{d^{3} a}\)	\(1021\)

[In]

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

[Out]

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

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. 1405 vs. \(2 (412) = 824\).

Time = 0.27 (sec) , antiderivative size = 1405, normalized size of antiderivative = 3.12 \[ \int \frac {(e+f x)^3 \text {sech}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.49 (sec) , antiderivative size = 730, normalized size of antiderivative = 1.62 \[ \int \frac {(e+f x)^3 \text {sech}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {1}{2} \, e^{2} f {\left (\frac {24 \, {\left (4 i \, d x e^{\left (4 \, d x + 4 \, c\right )} + {\left (8 \, d x e^{\left (3 \, c\right )} + e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} + e^{\left (d x + c\right )}\right )}}{12 i \, a d^{2} e^{\left (4 \, d x + 4 \, c\right )} + 24 \, a d^{2} e^{\left (3 \, d x + 3 \, c\right )} + 24 \, a d^{2} e^{\left (d x + c\right )} - 12 i \, a d^{2}} - \frac {3 \, \log \left ({\left (e^{\left (d x + c\right )} + i\right )} e^{\left (-c\right )}\right )}{a d^{2}} - \frac {5 \, \log \left (-i \, {\left (i \, e^{\left (d x + c\right )} + 1\right )} e^{\left (-c\right )}\right )}{a d^{2}}\right )} + \frac {4}{3} \, e^{3} {\left (\frac {2 \, e^{\left (-d x - c\right )}}{{\left (2 \, a e^{\left (-d x - c\right )} + 2 \, a e^{\left (-3 \, d x - 3 \, c\right )} - i \, a e^{\left (-4 \, d x - 4 \, c\right )} + i \, a\right )} d} + \frac {i}{{\left (2 \, a e^{\left (-d x - c\right )} + 2 \, a e^{\left (-3 \, d x - 3 \, c\right )} - i \, a e^{\left (-4 \, d x - 4 \, c\right )} + i \, a\right )} d}\right )} + \frac {4 i \, d^{2} f^{3} x^{3} + 12 i \, d^{2} e f^{2} x^{2} - 6 i \, f^{3} x - 6 i \, e f^{2} + 3 \, {\left (d f^{3} x^{2} e^{\left (3 \, c\right )} + 2 \, e f^{2} e^{\left (3 \, c\right )} + 2 \, {\left (d e f^{2} + f^{3}\right )} x e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} - 6 \, {\left (i \, f^{3} x e^{\left (2 \, c\right )} + i \, e f^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} - {\left (8 \, d^{2} f^{3} x^{3} e^{c} - 6 \, e f^{2} e^{c} + 3 \, {\left (8 \, d^{2} e f^{2} - d f^{3}\right )} x^{2} e^{c} - 6 \, {\left (d e f^{2} + f^{3}\right )} x e^{c}\right )} e^{\left (d x\right )}}{3 i \, a d^{3} e^{\left (4 \, d x + 4 \, c\right )} + 6 \, a d^{3} e^{\left (3 \, d x + 3 \, c\right )} + 6 \, a d^{3} e^{\left (d x + c\right )} - 3 i \, a d^{3}} - \frac {5 \, {\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 )} e f^{2}}{a d^{3}} - \frac {3 \, {\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 )} e f^{2}}{a d^{3}} - \frac {2 \, f^{3} x}{a d^{3}} - \frac {5 \, {\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^{3}}{2 \, a d^{4}} - \frac {3 \, {\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^{3}}{2 \, a d^{4}} + \frac {2 \, f^{3} \log \left (e^{\left (d x + c\right )} - i\right )}{a d^{4}} + \frac {4 \, {\left (d^{3} f^{3} x^{3} + 3 \, d^{3} e f^{2} x^{2}\right )}}{3 \, a d^{4}} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x)^3 \text {sech}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int \frac {{\left (e+f\,x\right )}^3}{{\mathrm {cosh}\left (c+d\,x\right )}^2\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]

[In]

int((e + f*x)^3/(cosh(c + d*x)^2*(a + a*sinh(c + d*x)*1i)),x)

[Out]

int((e + f*x)^3/(cosh(c + d*x)^2*(a + a*sinh(c + d*x)*1i)), x)