\(\int \frac {(e+f x)^2 \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [314]

Optimal result

Integrand size = 28, antiderivative size = 928 \[ \int \frac {(e+f x)^2 \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {2 a b^2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a (e+f x)^2 \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {a f^2 \arctan (\sinh (c+d x))}{\left (a^2+b^2\right ) d^3}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}-\frac {b^3 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}+\frac {b f^2 \log (\cosh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac {2 i a b^2 f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i a f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {2 i a b^2 f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i a f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {2 b^3 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {2 b^3 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {b^3 f (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {2 i a b^2 f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {i a f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}-\frac {2 i a b^2 f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {i a f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}-\frac {2 b^3 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {2 b^3 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {b^3 f^2 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d^3}+\frac {a f (e+f x) \text {sech}(c+d x)}{\left (a^2+b^2\right ) d^2}+\frac {b (e+f x)^2 \text {sech}^2(c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {b f (e+f x) \tanh (c+d x)}{\left (a^2+b^2\right ) d^2}+\frac {a (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d} \]

[Out]

a*(f*x+e)^2*arctan(exp(d*x+c))/(a^2+b^2)/d-2*I*a*b^2*f*(f*x+e)*polylog(2,-I*exp(d*x+c))/(a^2+b^2)^2/d^2+2*a*b^
2*(f*x+e)^2*arctan(exp(d*x+c))/(a^2+b^2)^2/d+1/2*a*(f*x+e)^2*sech(d*x+c)*tanh(d*x+c)/(a^2+b^2)/d+2*b^3*f*(f*x+
e)*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/(a^2+b^2)^2/d^2+2*b^3*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a+(a^
2+b^2)^(1/2)))/(a^2+b^2)^2/d^2-I*a*f^2*polylog(3,I*exp(d*x+c))/(a^2+b^2)/d^3+1/2*b^3*f^2*polylog(3,-exp(2*d*x+
2*c))/(a^2+b^2)^2/d^3+1/2*b*(f*x+e)^2*sech(d*x+c)^2/(a^2+b^2)/d-2*b^3*f^2*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)
^(1/2)))/(a^2+b^2)^2/d^3-2*b^3*f^2*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/(a^2+b^2)^2/d^3+2*I*a*b^2*f^2*
polylog(3,-I*exp(d*x+c))/(a^2+b^2)^2/d^3+I*a*f*(f*x+e)*polylog(2,I*exp(d*x+c))/(a^2+b^2)/d^2+2*I*a*b^2*f*(f*x+
e)*polylog(2,I*exp(d*x+c))/(a^2+b^2)^2/d^2-a*f^2*arctan(sinh(d*x+c))/(a^2+b^2)/d^3-b^3*f*(f*x+e)*polylog(2,-ex
p(2*d*x+2*c))/(a^2+b^2)^2/d^2+I*a*f^2*polylog(3,-I*exp(d*x+c))/(a^2+b^2)/d^3+a*f*(f*x+e)*sech(d*x+c)/(a^2+b^2)
/d^2-b*f*(f*x+e)*tanh(d*x+c)/(a^2+b^2)/d^2-b^3*(f*x+e)^2*ln(1+exp(2*d*x+2*c))/(a^2+b^2)^2/d+b*f^2*ln(cosh(d*x+
c))/(a^2+b^2)/d^3+b^3*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/(a^2+b^2)^2/d+b^3*(f*x+e)^2*ln(1+b*exp(
d*x+c)/(a+(a^2+b^2)^(1/2)))/(a^2+b^2)^2/d-I*a*f*(f*x+e)*polylog(2,-I*exp(d*x+c))/(a^2+b^2)/d^2-2*I*a*b^2*f^2*p
olylog(3,I*exp(d*x+c))/(a^2+b^2)^2/d^3

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 928, normalized size of antiderivative = 1.00, number of steps used = 39, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5692, 5680, 2221, 2611, 2320, 6724, 6874, 4265, 3799, 4271, 3855, 5559, 4269, 3556} \[ \int \frac {(e+f x)^2 \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {(e+f x)^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) b^3}{\left (a^2+b^2\right )^2 d}+\frac {(e+f x)^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) b^3}{\left (a^2+b^2\right )^2 d}-\frac {(e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) b^3}{\left (a^2+b^2\right )^2 d}+\frac {2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^3}{\left (a^2+b^2\right )^2 d^2}+\frac {2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^3}{\left (a^2+b^2\right )^2 d^2}-\frac {f (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right ) b^3}{\left (a^2+b^2\right )^2 d^2}-\frac {2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^3}{\left (a^2+b^2\right )^2 d^3}-\frac {2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^3}{\left (a^2+b^2\right )^2 d^3}+\frac {f^2 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right ) b^3}{2 \left (a^2+b^2\right )^2 d^3}+\frac {2 a (e+f x)^2 \arctan \left (e^{c+d x}\right ) b^2}{\left (a^2+b^2\right )^2 d}-\frac {2 i a f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) b^2}{\left (a^2+b^2\right )^2 d^2}+\frac {2 i a f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) b^2}{\left (a^2+b^2\right )^2 d^2}+\frac {2 i a f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right ) b^2}{\left (a^2+b^2\right )^2 d^3}-\frac {2 i a f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right ) b^2}{\left (a^2+b^2\right )^2 d^3}+\frac {(e+f x)^2 \text {sech}^2(c+d x) b}{2 \left (a^2+b^2\right ) d}+\frac {f^2 \log (\cosh (c+d x)) b}{\left (a^2+b^2\right ) d^3}-\frac {f (e+f x) \tanh (c+d x) b}{\left (a^2+b^2\right ) d^2}+\frac {a (e+f x)^2 \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {a f^2 \arctan (\sinh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac {i a f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {i a f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {i a f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}-\frac {i a f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {a f (e+f x) \text {sech}(c+d x)}{\left (a^2+b^2\right ) d^2}+\frac {a (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d} \]

[In]

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

[Out]

(2*a*b^2*(e + f*x)^2*ArcTan[E^(c + d*x)])/((a^2 + b^2)^2*d) + (a*(e + f*x)^2*ArcTan[E^(c + d*x)])/((a^2 + b^2)
*d) - (a*f^2*ArcTan[Sinh[c + d*x]])/((a^2 + b^2)*d^3) + (b^3*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2
+ b^2])])/((a^2 + b^2)^2*d) + (b^3*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/((a^2 + b^2)^2
*d) - (b^3*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/((a^2 + b^2)^2*d) + (b*f^2*Log[Cosh[c + d*x]])/((a^2 + b^2)*d
^3) - ((2*I)*a*b^2*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/((a^2 + b^2)^2*d^2) - (I*a*f*(e + f*x)*PolyLog[2,
(-I)*E^(c + d*x)])/((a^2 + b^2)*d^2) + ((2*I)*a*b^2*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/((a^2 + b^2)^2*d^2
) + (I*a*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/((a^2 + b^2)*d^2) + (2*b^3*f*(e + f*x)*PolyLog[2, -((b*E^(c +
d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^2) + (2*b^3*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt
[a^2 + b^2]))])/((a^2 + b^2)^2*d^2) - (b^3*f*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/((a^2 + b^2)^2*d^2) + ((2
*I)*a*b^2*f^2*PolyLog[3, (-I)*E^(c + d*x)])/((a^2 + b^2)^2*d^3) + (I*a*f^2*PolyLog[3, (-I)*E^(c + d*x)])/((a^2
+ b^2)*d^3) - ((2*I)*a*b^2*f^2*PolyLog[3, I*E^(c + d*x)])/((a^2 + b^2)^2*d^3) - (I*a*f^2*PolyLog[3, I*E^(c +
d*x)])/((a^2 + b^2)*d^3) - (2*b^3*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^3
) - (2*b^3*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^3) + (b^3*f^2*PolyLog[3,
-E^(2*(c + d*x))])/(2*(a^2 + b^2)^2*d^3) + (a*f*(e + f*x)*Sech[c + d*x])/((a^2 + b^2)*d^2) + (b*(e + f*x)^2*S
ech[c + d*x]^2)/(2*(a^2 + b^2)*d) - (b*f*(e + f*x)*Tanh[c + d*x])/((a^2 + b^2)*d^2) + (a*(e + f*x)^2*Sech[c +
d*x]*Tanh[c + d*x])/(2*(a^2 + b^2)*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 5680

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :
> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 + b^2, 2] + b*E^(c + d
*x))), x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x))), x]) /; FreeQ[{a, b, c, d, e,
f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]

Rule 5692

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[b^2/(a^2 + b^2), Int[(e + f*x)^m*(Sech[c + d*x]^(n - 2)/(a + b*Sinh[c + d*x])), x], x] + Dist[1/(
a^2 + b^2), Int[(e + f*x)^m*Sech[c + d*x]^n*(a - b*Sinh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && I
GtQ[m, 0] && NeQ[a^2 + b^2, 0] && IGtQ[n, 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 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x)^2 \text {sech}^3(c+d x) (a-b \sinh (c+d x)) \, dx}{a^2+b^2}+\frac {b^2 \int \frac {(e+f x)^2 \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2+b^2} \\ & = \frac {b^2 \int (e+f x)^2 \text {sech}(c+d x) (a-b \sinh (c+d x)) \, dx}{\left (a^2+b^2\right )^2}+\frac {b^4 \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{\left (a^2+b^2\right )^2}+\frac {\int \left (a (e+f x)^2 \text {sech}^3(c+d x)-b (e+f x)^2 \text {sech}^2(c+d x) \tanh (c+d x)\right ) \, dx}{a^2+b^2} \\ & = -\frac {b^3 (e+f x)^3}{3 \left (a^2+b^2\right )^2 f}+\frac {b^2 \int \left (a (e+f x)^2 \text {sech}(c+d x)-b (e+f x)^2 \tanh (c+d x)\right ) \, dx}{\left (a^2+b^2\right )^2}+\frac {b^4 \int \frac {e^{c+d x} (e+f x)^2}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{\left (a^2+b^2\right )^2}+\frac {b^4 \int \frac {e^{c+d x} (e+f x)^2}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{\left (a^2+b^2\right )^2}+\frac {a \int (e+f x)^2 \text {sech}^3(c+d x) \, dx}{a^2+b^2}-\frac {b \int (e+f x)^2 \text {sech}^2(c+d x) \tanh (c+d x) \, dx}{a^2+b^2} \\ & = -\frac {b^3 (e+f x)^3}{3 \left (a^2+b^2\right )^2 f}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a f (e+f x) \text {sech}(c+d x)}{\left (a^2+b^2\right ) d^2}+\frac {b (e+f x)^2 \text {sech}^2(c+d x)}{2 \left (a^2+b^2\right ) d}+\frac {a (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}+\frac {\left (a b^2\right ) \int (e+f x)^2 \text {sech}(c+d x) \, dx}{\left (a^2+b^2\right )^2}-\frac {b^3 \int (e+f x)^2 \tanh (c+d x) \, dx}{\left (a^2+b^2\right )^2}+\frac {a \int (e+f x)^2 \text {sech}(c+d x) \, dx}{2 \left (a^2+b^2\right )}-\frac {\left (2 b^3 f\right ) \int (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^2 d}-\frac {\left (2 b^3 f\right ) \int (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^2 d}-\frac {(b f) \int (e+f x) \text {sech}^2(c+d x) \, dx}{\left (a^2+b^2\right ) d}-\frac {\left (a f^2\right ) \int \text {sech}(c+d x) \, dx}{\left (a^2+b^2\right ) d^2} \\ & = \frac {2 a b^2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a (e+f x)^2 \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {a f^2 \arctan (\sinh (c+d x))}{\left (a^2+b^2\right ) d^3}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac {2 b^3 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {2 b^3 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {a f (e+f x) \text {sech}(c+d x)}{\left (a^2+b^2\right ) d^2}+\frac {b (e+f x)^2 \text {sech}^2(c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {b f (e+f x) \tanh (c+d x)}{\left (a^2+b^2\right ) d^2}+\frac {a (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {\left (2 b^3\right ) \int \frac {e^{2 (c+d x)} (e+f x)^2}{1+e^{2 (c+d x)}} \, dx}{\left (a^2+b^2\right )^2}-\frac {\left (2 i a b^2 f\right ) \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right )^2 d}+\frac {\left (2 i a b^2 f\right ) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right )^2 d}-\frac {(i a f) \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d}+\frac {(i a f) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d}-\frac {\left (2 b^3 f^2\right ) \int \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^2 d^2}-\frac {\left (2 b^3 f^2\right ) \int \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^2 d^2}+\frac {\left (b f^2\right ) \int \tanh (c+d x) \, dx}{\left (a^2+b^2\right ) d^2} \\ & = \frac {2 a b^2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a (e+f x)^2 \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {a f^2 \arctan (\sinh (c+d x))}{\left (a^2+b^2\right ) d^3}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}-\frac {b^3 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}+\frac {b f^2 \log (\cosh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac {2 i a b^2 f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i a f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {2 i a b^2 f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i a f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {2 b^3 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {2 b^3 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {a f (e+f x) \text {sech}(c+d x)}{\left (a^2+b^2\right ) d^2}+\frac {b (e+f x)^2 \text {sech}^2(c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {b f (e+f x) \tanh (c+d x)}{\left (a^2+b^2\right ) d^2}+\frac {a (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}+\frac {\left (2 b^3 f\right ) \int (e+f x) \log \left (1+e^{2 (c+d x)}\right ) \, dx}{\left (a^2+b^2\right )^2 d}-\frac {\left (2 b^3 f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {b x}{-a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {\left (2 b^3 f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {\left (2 i a b^2 f^2\right ) \int \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right )^2 d^2}-\frac {\left (2 i a b^2 f^2\right ) \int \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right )^2 d^2}+\frac {\left (i a f^2\right ) \int \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d^2}-\frac {\left (i a f^2\right ) \int \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d^2} \\ & = \frac {2 a b^2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a (e+f x)^2 \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {a f^2 \arctan (\sinh (c+d x))}{\left (a^2+b^2\right ) d^3}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}-\frac {b^3 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}+\frac {b f^2 \log (\cosh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac {2 i a b^2 f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i a f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {2 i a b^2 f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i a f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {2 b^3 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {2 b^3 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {b^3 f (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {2 b^3 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {2 b^3 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {a f (e+f x) \text {sech}(c+d x)}{\left (a^2+b^2\right ) d^2}+\frac {b (e+f x)^2 \text {sech}^2(c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {b f (e+f x) \tanh (c+d x)}{\left (a^2+b^2\right ) d^2}+\frac {a (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}+\frac {\left (2 i a b^2 f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {\left (2 i a b^2 f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {\left (i a f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}-\frac {\left (i a f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {\left (b^3 f^2\right ) \int \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right ) \, dx}{\left (a^2+b^2\right )^2 d^2} \\ & = \frac {2 a b^2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a (e+f x)^2 \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {a f^2 \arctan (\sinh (c+d x))}{\left (a^2+b^2\right ) d^3}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}-\frac {b^3 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}+\frac {b f^2 \log (\cosh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac {2 i a b^2 f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i a f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {2 i a b^2 f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i a f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {2 b^3 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {2 b^3 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {b^3 f (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {2 i a b^2 f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {i a f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}-\frac {2 i a b^2 f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {i a f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}-\frac {2 b^3 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {2 b^3 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {a f (e+f x) \text {sech}(c+d x)}{\left (a^2+b^2\right ) d^2}+\frac {b (e+f x)^2 \text {sech}^2(c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {b f (e+f x) \tanh (c+d x)}{\left (a^2+b^2\right ) d^2}+\frac {a (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}+\frac {\left (b^3 f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d^3} \\ & = \frac {2 a b^2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a (e+f x)^2 \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {a f^2 \arctan (\sinh (c+d x))}{\left (a^2+b^2\right ) d^3}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}-\frac {b^3 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}+\frac {b f^2 \log (\cosh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac {2 i a b^2 f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i a f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {2 i a b^2 f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i a f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {2 b^3 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {2 b^3 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {b^3 f (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {2 i a b^2 f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {i a f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}-\frac {2 i a b^2 f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {i a f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}-\frac {2 b^3 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {2 b^3 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {b^3 f^2 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d^3}+\frac {a f (e+f x) \text {sech}(c+d x)}{\left (a^2+b^2\right ) d^2}+\frac {b (e+f x)^2 \text {sech}^2(c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {b f (e+f x) \tanh (c+d x)}{\left (a^2+b^2\right ) d^2}+\frac {a (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) 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. \(3368\) vs. \(2(928)=1856\).

Time = 11.82 (sec) , antiderivative size = 3368, normalized size of antiderivative = 3.63 \[ \int \frac {(e+f x)^2 \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

(12*b^3*d^3*e^2*E^(2*c)*x - 12*a^2*b*d*E^(2*c)*f^2*x - 12*b^3*d*E^(2*c)*f^2*x + 12*b^3*d^3*e*E^(2*c)*f*x^2 + 4
*b^3*d^3*E^(2*c)*f^2*x^3 + 6*a^3*d^2*e^2*ArcTan[E^(c + d*x)] + 18*a*b^2*d^2*e^2*ArcTan[E^(c + d*x)] + 6*a^3*d^
2*e^2*E^(2*c)*ArcTan[E^(c + d*x)] + 18*a*b^2*d^2*e^2*E^(2*c)*ArcTan[E^(c + d*x)] - 12*a^3*f^2*ArcTan[E^(c + d*
x)] - 12*a*b^2*f^2*ArcTan[E^(c + d*x)] - 12*a^3*E^(2*c)*f^2*ArcTan[E^(c + d*x)] - 12*a*b^2*E^(2*c)*f^2*ArcTan[
E^(c + d*x)] + (6*I)*a^3*d^2*e*f*x*Log[1 - I*E^(c + d*x)] + (18*I)*a*b^2*d^2*e*f*x*Log[1 - I*E^(c + d*x)] + (6
*I)*a^3*d^2*e*E^(2*c)*f*x*Log[1 - I*E^(c + d*x)] + (18*I)*a*b^2*d^2*e*E^(2*c)*f*x*Log[1 - I*E^(c + d*x)] + (3*
I)*a^3*d^2*f^2*x^2*Log[1 - I*E^(c + d*x)] + (9*I)*a*b^2*d^2*f^2*x^2*Log[1 - I*E^(c + d*x)] + (3*I)*a^3*d^2*E^(
2*c)*f^2*x^2*Log[1 - I*E^(c + d*x)] + (9*I)*a*b^2*d^2*E^(2*c)*f^2*x^2*Log[1 - I*E^(c + d*x)] - (6*I)*a^3*d^2*e
*f*x*Log[1 + I*E^(c + d*x)] - (18*I)*a*b^2*d^2*e*f*x*Log[1 + I*E^(c + d*x)] - (6*I)*a^3*d^2*e*E^(2*c)*f*x*Log[
1 + I*E^(c + d*x)] - (18*I)*a*b^2*d^2*e*E^(2*c)*f*x*Log[1 + I*E^(c + d*x)] - (3*I)*a^3*d^2*f^2*x^2*Log[1 + I*E
^(c + d*x)] - (9*I)*a*b^2*d^2*f^2*x^2*Log[1 + I*E^(c + d*x)] - (3*I)*a^3*d^2*E^(2*c)*f^2*x^2*Log[1 + I*E^(c +
d*x)] - (9*I)*a*b^2*d^2*E^(2*c)*f^2*x^2*Log[1 + I*E^(c + d*x)] - 6*b^3*d^2*e^2*Log[1 + E^(2*(c + d*x))] - 6*b^
3*d^2*e^2*E^(2*c)*Log[1 + E^(2*(c + d*x))] + 6*a^2*b*f^2*Log[1 + E^(2*(c + d*x))] + 6*b^3*f^2*Log[1 + E^(2*(c
+ d*x))] + 6*a^2*b*E^(2*c)*f^2*Log[1 + E^(2*(c + d*x))] + 6*b^3*E^(2*c)*f^2*Log[1 + E^(2*(c + d*x))] - 12*b^3*
d^2*e*f*x*Log[1 + E^(2*(c + d*x))] - 12*b^3*d^2*e*E^(2*c)*f*x*Log[1 + E^(2*(c + d*x))] - 6*b^3*d^2*f^2*x^2*Log
[1 + E^(2*(c + d*x))] - 6*b^3*d^2*E^(2*c)*f^2*x^2*Log[1 + E^(2*(c + d*x))] - (6*I)*a*(a^2 + 3*b^2)*d*(1 + E^(2
*c))*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)] + (6*I)*a*(a^2 + 3*b^2)*d*(1 + E^(2*c))*f*(e + f*x)*PolyLog[2, I
*E^(c + d*x)] - 6*b^3*d*e*f*PolyLog[2, -E^(2*(c + d*x))] - 6*b^3*d*e*E^(2*c)*f*PolyLog[2, -E^(2*(c + d*x))] -
6*b^3*d*f^2*x*PolyLog[2, -E^(2*(c + d*x))] - 6*b^3*d*E^(2*c)*f^2*x*PolyLog[2, -E^(2*(c + d*x))] + (6*I)*a^3*f^
2*PolyLog[3, (-I)*E^(c + d*x)] + (18*I)*a*b^2*f^2*PolyLog[3, (-I)*E^(c + d*x)] + (6*I)*a^3*E^(2*c)*f^2*PolyLog
[3, (-I)*E^(c + d*x)] + (18*I)*a*b^2*E^(2*c)*f^2*PolyLog[3, (-I)*E^(c + d*x)] - (6*I)*a^3*f^2*PolyLog[3, I*E^(
c + d*x)] - (18*I)*a*b^2*f^2*PolyLog[3, I*E^(c + d*x)] - (6*I)*a^3*E^(2*c)*f^2*PolyLog[3, I*E^(c + d*x)] - (18
*I)*a*b^2*E^(2*c)*f^2*PolyLog[3, I*E^(c + d*x)] + 3*b^3*f^2*PolyLog[3, -E^(2*(c + d*x))] + 3*b^3*E^(2*c)*f^2*P
olyLog[3, -E^(2*(c + d*x))])/(6*(a^2 + b^2)^2*d^3*(1 + E^(2*c))) - (b^3*(6*e^2*E^(2*c)*x + 6*e*E^(2*c)*f*x^2 +
2*E^(2*c)*f^2*x^3 + (6*a*Sqrt[a^2 + b^2]*e^2*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]])/(Sqrt[-(a^2 + b^2)
^2]*d) + (6*a*Sqrt[-(a^2 + b^2)^2]*e^2*E^(2*c)*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]])/((a^2 + b^2)^(3/2
)*d) - (6*a*Sqrt[-(a^2 + b^2)^2]*e^2*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]])/((-a^2 - b^2)^(3/2)*d) + (6
*a*Sqrt[-(a^2 + b^2)^2]*e^2*E^(2*c)*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]])/((-a^2 - b^2)^(3/2)*d) + (3*
e^2*Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))])/d - (3*e^2*E^(2*c)*Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c
+ d*x)))])/d + (6*e*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*e*E^(2*c)*f*x*L
og[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (3*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c
- Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (3*E^(2*c)*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(
2*c)])])/d + (6*e*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*e*E^(2*c)*f*x*Log
[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (3*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c +
Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (3*E^(2*c)*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*
c)])])/d - (6*(-1 + E^(2*c))*f*(e + f*x)*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])
/d^2 - (6*(-1 + E^(2*c))*f*(e + f*x)*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^2
- (6*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 + (6*E^(2*c)*f^2*PolyLog[3
, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 - (6*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^
c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 + (6*E^(2*c)*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2
)*E^(2*c)]))])/d^3))/(3*(a^2 + b^2)^2*(-1 + E^(2*c))) + (Csch[c]*Sech[c]*Sech[c + d*x]^2*(-6*a^2*b*e*f - 6*b^3
*e*f + 12*b^3*d^2*e^2*x - 6*a^2*b*f^2*x - 6*b^3*f^2*x + 12*b^3*d^2*e*f*x^2 + 4*b^3*d^2*f^2*x^3 + 6*a^2*b*e*f*C
osh[2*c] + 6*b^3*e*f*Cosh[2*c] + 6*a^2*b*f^2*x*Cosh[2*c] + 6*b^3*f^2*x*Cosh[2*c] + 6*a^2*b*e*f*Cosh[2*d*x] + 6
*b^3*e*f*Cosh[2*d*x] + 6*a^2*b*f^2*x*Cosh[2*d*x] + 6*b^3*f^2*x*Cosh[2*d*x] - 3*a^3*d*e^2*Cosh[c - d*x] - 3*a*b
^2*d*e^2*Cosh[c - d*x] - 6*a^3*d*e*f*x*Cosh[c - d*x] - 6*a*b^2*d*e*f*x*Cosh[c - d*x] - 3*a^3*d*f^2*x^2*Cosh[c
- d*x] - 3*a*b^2*d*f^2*x^2*Cosh[c - d*x] + 3*a^3*d*e^2*Cosh[3*c + d*x] + 3*a*b^2*d*e^2*Cosh[3*c + d*x] + 6*a^3
*d*e*f*x*Cosh[3*c + d*x] + 6*a*b^2*d*e*f*x*Cosh[3*c + d*x] + 3*a^3*d*f^2*x^2*Cosh[3*c + d*x] + 3*a*b^2*d*f^2*x
^2*Cosh[3*c + d*x] - 6*a^2*b*e*f*Cosh[2*c + 2*d*x] - 6*b^3*e*f*Cosh[2*c + 2*d*x] + 12*b^3*d^2*e^2*x*Cosh[2*c +
2*d*x] - 6*a^2*b*f^2*x*Cosh[2*c + 2*d*x] - 6*b^3*f^2*x*Cosh[2*c + 2*d*x] + 12*b^3*d^2*e*f*x^2*Cosh[2*c + 2*d*
x] + 4*b^3*d^2*f^2*x^3*Cosh[2*c + 2*d*x] + 6*a^2*b*d*e^2*Sinh[2*c] + 6*b^3*d*e^2*Sinh[2*c] + 12*a^2*b*d*e*f*x*
Sinh[2*c] + 12*b^3*d*e*f*x*Sinh[2*c] + 6*a^2*b*d*f^2*x^2*Sinh[2*c] + 6*b^3*d*f^2*x^2*Sinh[2*c] + 6*a^3*e*f*Sin
h[c - d*x] + 6*a*b^2*e*f*Sinh[c - d*x] + 6*a^3*f^2*x*Sinh[c - d*x] + 6*a*b^2*f^2*x*Sinh[c - d*x] + 6*a^3*e*f*S
inh[3*c + d*x] + 6*a*b^2*e*f*Sinh[3*c + d*x] + 6*a^3*f^2*x*Sinh[3*c + d*x] + 6*a*b^2*f^2*x*Sinh[3*c + d*x]))/(
24*(a^2 + b^2)^2*d^2)

Maple [F]

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

[In]

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

[Out]

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

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. 10642 vs. \(2 (855) = 1710\).

Time = 0.44 (sec) , antiderivative size = 10642, normalized size of antiderivative = 11.47 \[ \int \frac {(e+f x)^2 \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

a^3*d^2*f^2*integrate(x^2*e^(d*x + c)/(a^4*d^2*e^(2*d*x + 2*c) + 2*a^2*b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*
d*x + 2*c) + a^4*d^2 + 2*a^2*b^2*d^2 + b^4*d^2), x) + 3*a*b^2*d^2*f^2*integrate(x^2*e^(d*x + c)/(a^4*d^2*e^(2*
d*x + 2*c) + 2*a^2*b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*d*x + 2*c) + a^4*d^2 + 2*a^2*b^2*d^2 + b^4*d^2), x)
+ 2*b^3*d^2*f^2*integrate(x^2/(a^4*d^2*e^(2*d*x + 2*c) + 2*a^2*b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*d*x + 2*
c) + a^4*d^2 + 2*a^2*b^2*d^2 + b^4*d^2), x) + 2*a^3*d^2*e*f*integrate(x*e^(d*x + c)/(a^4*d^2*e^(2*d*x + 2*c) +
2*a^2*b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*d*x + 2*c) + a^4*d^2 + 2*a^2*b^2*d^2 + b^4*d^2), x) + 6*a*b^2*d^
2*e*f*integrate(x*e^(d*x + c)/(a^4*d^2*e^(2*d*x + 2*c) + 2*a^2*b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*d*x + 2*
c) + a^4*d^2 + 2*a^2*b^2*d^2 + b^4*d^2), x) + 4*b^3*d^2*e*f*integrate(x/(a^4*d^2*e^(2*d*x + 2*c) + 2*a^2*b^2*d
^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*d*x + 2*c) + a^4*d^2 + 2*a^2*b^2*d^2 + b^4*d^2), x) - a^2*b*f^2*(2*(d*x + c)
/((a^4 + 2*a^2*b^2 + b^4)*d^3) - log(e^(2*d*x + 2*c) + 1)/((a^4 + 2*a^2*b^2 + b^4)*d^3)) - b^3*f^2*(2*(d*x + c
)/((a^4 + 2*a^2*b^2 + b^4)*d^3) - log(e^(2*d*x + 2*c) + 1)/((a^4 + 2*a^2*b^2 + b^4)*d^3)) + (b^3*log(-2*a*e^(-
d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^4 + 2*a^2*b^2 + b^4)*d) - b^3*log(e^(-2*d*x - 2*c) + 1)/((a^4 + 2*a^2*b
^2 + b^4)*d) - (a^3 + 3*a*b^2)*arctan(e^(-d*x - c))/((a^4 + 2*a^2*b^2 + b^4)*d) + (a*e^(-d*x - c) + 2*b*e^(-2*
d*x - 2*c) - a*e^(-3*d*x - 3*c))/((a^2 + b^2 + 2*(a^2 + b^2)*e^(-2*d*x - 2*c) + (a^2 + b^2)*e^(-4*d*x - 4*c))*
d))*e^2 - 2*a^3*f^2*arctan(e^(d*x + c))/((a^4 + 2*a^2*b^2 + b^4)*d^3) - 2*a*b^2*f^2*arctan(e^(d*x + c))/((a^4
+ 2*a^2*b^2 + b^4)*d^3) + (2*b*f^2*x + 2*b*e*f + (a*d*f^2*x^2*e^(3*c) + 2*a*e*f*e^(3*c) + 2*(d*e*f + f^2)*a*x*
e^(3*c))*e^(3*d*x) + 2*(b*d*f^2*x^2*e^(2*c) + b*e*f*e^(2*c) + (2*d*e*f + f^2)*b*x*e^(2*c))*e^(2*d*x) - (a*d*f^
2*x^2*e^c - 2*a*e*f*e^c + 2*(d*e*f - f^2)*a*x*e^c)*e^(d*x))/(a^2*d^2 + b^2*d^2 + (a^2*d^2*e^(4*c) + b^2*d^2*e^
(4*c))*e^(4*d*x) + 2*(a^2*d^2*e^(2*c) + b^2*d^2*e^(2*c))*e^(2*d*x)) - integrate(2*(b^4*f^2*x^2 + 2*b^4*e*f*x -
(a*b^3*f^2*x^2*e^c + 2*a*b^3*e*f*x*e^c)*e^(d*x))/(a^4*b + 2*a^2*b^3 + b^5 - (a^4*b*e^(2*c) + 2*a^2*b^3*e^(2*c
) + b^5*e^(2*c))*e^(2*d*x) - 2*(a^5*e^c + 2*a^3*b^2*e^c + a*b^4*e^c)*e^(d*x)), x)

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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