\(\int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [441]

Optimal result

Integrand size = 34, antiderivative size = 795 \[ \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {b (e+f x)^2}{\left (a^2+b^2\right ) d}-\frac {4 f (e+f x) \arctan \left (e^{c+d x}\right )}{a d^2}+\frac {4 b^2 f (e+f x) \arctan \left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {2 b f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac {2 i f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {2 i b^2 f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {2 i f^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a d^3}+\frac {2 i b^2 f^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {2 f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a 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 )}{a \left (a^2+b^2\right )^{3/2} 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 )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {b f^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}-\frac {2 f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a 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 )}{a \left (a^2+b^2\right )^{3/2} 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 )}{a \left (a^2+b^2\right )^{3/2} d^3}+\frac {(e+f x)^2 \text {sech}(c+d x)}{a d}-\frac {b^2 (e+f x)^2 \text {sech}(c+d x)}{a \left (a^2+b^2\right ) d}-\frac {b (e+f x)^2 \tanh (c+d x)}{\left (a^2+b^2\right ) d} \]

[Out]

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

Rubi [A] (verified)

Time = 1.18 (sec) , antiderivative size = 795, normalized size of antiderivative = 1.00, number of steps used = 44, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.676, Rules used = {5708, 2702, 327, 213, 5570, 6873, 12, 6874, 6408, 4267, 2611, 2320, 6724, 4265, 2317, 2438, 5692, 3403, 2296, 2221, 4269, 3799, 5559} \[ \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(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}{a \left (a^2+b^2\right )^{3/2} d}+\frac {(e+f x)^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) b^3}{a \left (a^2+b^2\right )^{3/2} 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}{a \left (a^2+b^2\right )^{3/2} 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}{a \left (a^2+b^2\right )^{3/2} 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}{a \left (a^2+b^2\right )^{3/2} 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}{a \left (a^2+b^2\right )^{3/2} d^3}+\frac {4 f (e+f x) \arctan \left (e^{c+d x}\right ) b^2}{a \left (a^2+b^2\right ) d^2}-\frac {2 i f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) b^2}{a \left (a^2+b^2\right ) d^3}+\frac {2 i f^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) b^2}{a \left (a^2+b^2\right ) d^3}-\frac {(e+f x)^2 \text {sech}(c+d x) b^2}{a \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 b}{\left (a^2+b^2\right ) d}+\frac {2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right ) b}{\left (a^2+b^2\right ) d^2}+\frac {f^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right ) b}{\left (a^2+b^2\right ) d^3}-\frac {(e+f x)^2 \tanh (c+d x) b}{\left (a^2+b^2\right ) d}-\frac {4 f (e+f x) \arctan \left (e^{c+d x}\right )}{a d^2}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {2 f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac {2 i f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {2 i f^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a d^3}+\frac {2 f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}+\frac {2 f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}-\frac {2 f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}+\frac {(e+f x)^2 \text {sech}(c+d x)}{a d} \]

[In]

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

[Out]

-((b*(e + f*x)^2)/((a^2 + b^2)*d)) - (4*f*(e + f*x)*ArcTan[E^(c + d*x)])/(a*d^2) + (4*b^2*f*(e + f*x)*ArcTan[E
^(c + d*x)])/(a*(a^2 + b^2)*d^2) - (2*(e + f*x)^2*ArcTanh[E^(c + d*x)])/(a*d) - (b^3*(e + f*x)^2*Log[1 + (b*E^
(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a*(a^2 + b^2)^(3/2)*d) + (b^3*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqr
t[a^2 + b^2])])/(a*(a^2 + b^2)^(3/2)*d) + (2*b*f*(e + f*x)*Log[1 + E^(2*(c + d*x))])/((a^2 + b^2)*d^2) - (2*f*
(e + f*x)*PolyLog[2, -E^(c + d*x)])/(a*d^2) + ((2*I)*f^2*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^3) - ((2*I)*b^2*f^
2*PolyLog[2, (-I)*E^(c + d*x)])/(a*(a^2 + b^2)*d^3) - ((2*I)*f^2*PolyLog[2, I*E^(c + d*x)])/(a*d^3) + ((2*I)*b
^2*f^2*PolyLog[2, I*E^(c + d*x)])/(a*(a^2 + b^2)*d^3) + (2*f*(e + f*x)*PolyLog[2, E^(c + d*x)])/(a*d^2) - (2*b
^3*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^(3/2)*d^2) + (2*b^3*f*(e +
f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^(3/2)*d^2) + (b*f^2*PolyLog[2, -E^(
2*(c + d*x))])/((a^2 + b^2)*d^3) + (2*f^2*PolyLog[3, -E^(c + d*x)])/(a*d^3) - (2*f^2*PolyLog[3, E^(c + d*x)])/
(a*d^3) + (2*b^3*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^(3/2)*d^3) - (2*b^3*
f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^(3/2)*d^3) + ((e + f*x)^2*Sech[c + d*
x])/(a*d) - (b^2*(e + f*x)^2*Sech[c + d*x])/(a*(a^2 + b^2)*d) - (b*(e + f*x)^2*Tanh[c + d*x])/((a^2 + b^2)*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, 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 2296

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[2*(c/q), Int[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Dist[2*(c/q), Int[(f + g
*x)^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && 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 2702

Int[csc[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[1/(f*a^n), Subst[Int
[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n
+ 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rule 3403

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Dist[2,
Int[(c + d*x)^m*(E^((-I)*e + f*fz*x)/((-I)*b + 2*a*E^((-I)*e + f*fz*x) + I*b*E^(2*((-I)*e + f*fz*x)))), x], x]
/; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 3799

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

Rule 4265

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

Rule 4267

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(Ar
cTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*
fz*x)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && 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 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 5570

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Wit
h[{u = IntHide[Csch[a + b*x]^n*Sech[a + b*x]^p, x]}, Dist[(c + d*x)^m, u, x] - Dist[d*m, Int[(c + d*x)^(m - 1)
*u, x], x]] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p] && GtQ[m, 0] && NeQ[n, p]

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 5708

Int[(Csch[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(p_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[(e + f*x)^m*Sech[c + d*x]^p*Csch[c + d*x]^n, x], x] - Dis
t[b/a, Int[(e + f*x)^m*Sech[c + d*x]^p*(Csch[c + d*x]^(n - 1)/(a + b*Sinh[c + d*x])), x], x] /; FreeQ[{a, b, c
, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]

Rule 6408

Int[((a_.) + ArcTanh[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m + 1)*((a + b*ArcTan
h[u])/(d*(m + 1))), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[(c + d*x)^(m + 1)*(D[u, x]/(1 - u^2)), x],
x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] &&  !FunctionOfQ[(c + d*x)^(m
+ 1), u, x] && FalseQ[PowerVariableExpn[u, m + 1, x]]

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 6873

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

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

Mathematica [A] (verified)

Time = 8.30 (sec) , antiderivative size = 928, normalized size of antiderivative = 1.17 \[ \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=4 \left (-\frac {f \text {csch}(c+d x) \left (4 b d^2 e e^{2 c} x-4 b d^2 e \left (1+e^{2 c}\right ) x+2 b d^2 e^{2 c} f x^2-2 b d^2 \left (1+e^{2 c}\right ) f x^2+4 a d e \left (1+e^{2 c}\right ) \arctan \left (e^{c+d x}\right )+2 b d e \left (1+e^{2 c}\right ) \left (2 d x-\log \left (1+e^{2 (c+d x)}\right )\right )+2 i a \left (1+e^{2 c}\right ) f \left (d x \left (\log \left (1-i e^{c+d x}\right )-\log \left (1+i e^{c+d x}\right )\right )-\operatorname {PolyLog}\left (2,-i e^{c+d x}\right )+\operatorname {PolyLog}\left (2,i e^{c+d x}\right )\right )+b \left (1+e^{2 c}\right ) f \left (2 d x \left (d x-\log \left (1+e^{2 (c+d x)}\right )\right )-\operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )\right )\right ) (a+b \sinh (c+d x))}{4 \left (a^2+b^2\right ) d^3 \left (1+e^{2 c}\right ) (b+a \text {csch}(c+d x))}+\frac {\text {csch}(c+d x) \left ((e+f x)^2 \log \left (1-e^{c+d x}\right )-(e+f x)^2 \log \left (1+e^{c+d x}\right )-\frac {2 f \left (d (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )-f \operatorname {PolyLog}\left (3,-e^{c+d x}\right )\right )}{d^2}+\frac {2 f \left (d (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )-f \operatorname {PolyLog}\left (3,e^{c+d x}\right )\right )}{d^2}\right ) (a+b \sinh (c+d x))}{4 a d (b+a \text {csch}(c+d x))}-\frac {b^3 \text {csch}(c+d x) \left (-2 d^2 e^2 \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+2 d^2 e f x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+d^2 f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-2 d^2 e f x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-d^2 f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+2 d f (e+f x) \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-2 d f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 f^2 \operatorname {PolyLog}\left (3,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right ) (a+b \sinh (c+d x))}{4 a \left (a^2+b^2\right )^{3/2} d^3 (b+a \text {csch}(c+d x))}+\frac {\text {csch}(c+d x) \text {sech}(c) \text {sech}(c+d x) \left (a e^2 \cosh (c)+2 a e f x \cosh (c)+a f^2 x^2 \cosh (c)-b e^2 \sinh (d x)-2 b e f x \sinh (d x)-b f^2 x^2 \sinh (d x)\right ) (a+b \sinh (c+d x))}{4 \left (a^2+b^2\right ) d (b+a \text {csch}(c+d x))}\right ) \]

[In]

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

[Out]

4*(-1/4*(f*Csch[c + d*x]*(4*b*d^2*e*E^(2*c)*x - 4*b*d^2*e*(1 + E^(2*c))*x + 2*b*d^2*E^(2*c)*f*x^2 - 2*b*d^2*(1
+ E^(2*c))*f*x^2 + 4*a*d*e*(1 + E^(2*c))*ArcTan[E^(c + d*x)] + 2*b*d*e*(1 + E^(2*c))*(2*d*x - Log[1 + E^(2*(c
+ d*x))]) + (2*I)*a*(1 + E^(2*c))*f*(d*x*(Log[1 - I*E^(c + d*x)] - Log[1 + I*E^(c + d*x)]) - PolyLog[2, (-I)*
E^(c + d*x)] + PolyLog[2, I*E^(c + d*x)]) + b*(1 + E^(2*c))*f*(2*d*x*(d*x - Log[1 + E^(2*(c + d*x))]) - PolyLo
g[2, -E^(2*(c + d*x))]))*(a + b*Sinh[c + d*x]))/((a^2 + b^2)*d^3*(1 + E^(2*c))*(b + a*Csch[c + d*x])) + (Csch[
c + d*x]*((e + f*x)^2*Log[1 - E^(c + d*x)] - (e + f*x)^2*Log[1 + E^(c + d*x)] - (2*f*(d*(e + f*x)*PolyLog[2, -
E^(c + d*x)] - f*PolyLog[3, -E^(c + d*x)]))/d^2 + (2*f*(d*(e + f*x)*PolyLog[2, E^(c + d*x)] - f*PolyLog[3, E^(
c + d*x)]))/d^2)*(a + b*Sinh[c + d*x]))/(4*a*d*(b + a*Csch[c + d*x])) - (b^3*Csch[c + d*x]*(-2*d^2*e^2*ArcTanh
[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] + 2*d^2*e*f*x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + d^2*f^2*x
^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - 2*d^2*e*f*x*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])]
- d^2*f^2*x^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 2*d*f*(e + f*x)*PolyLog[2, (b*E^(c + d*x))/(-a
+ Sqrt[a^2 + b^2])] - 2*d*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] - 2*f^2*PolyLog[3,
(b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])*(a + b*
Sinh[c + d*x]))/(4*a*(a^2 + b^2)^(3/2)*d^3*(b + a*Csch[c + d*x])) + (Csch[c + d*x]*Sech[c]*Sech[c + d*x]*(a*e^
2*Cosh[c] + 2*a*e*f*x*Cosh[c] + a*f^2*x^2*Cosh[c] - b*e^2*Sinh[d*x] - 2*b*e*f*x*Sinh[d*x] - b*f^2*x^2*Sinh[d*x
])*(a + b*Sinh[c + d*x]))/(4*(a^2 + b^2)*d*(b + a*Csch[c + d*x])))

Maple [F]

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

[In]

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

[Out]

int((f*x+e)^2*csch(d*x+c)*sech(d*x+c)^2/(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. 5562 vs. \(2 (729) = 1458\).

Time = 0.40 (sec) , antiderivative size = 5562, normalized size of antiderivative = 7.00 \[ \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

-2*b*e*f*(2*(d*x + c)/((a^2 + b^2)*d^2) - log(e^(2*d*x + 2*c) + 1)/((a^2 + b^2)*d^2)) - 4*a*f^2*integrate(x*e^
(d*x + c)/(a^2*d*e^(2*d*x + 2*c) + b^2*d*e^(2*d*x + 2*c) + a^2*d + b^2*d), x) - 4*b*f^2*integrate(x/(a^2*d*e^(
2*d*x + 2*c) + b^2*d*e^(2*d*x + 2*c) + a^2*d + b^2*d), x) - (b^3*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b
*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/((a^3 + a*b^2)*sqrt(a^2 + b^2)*d) - 2*(a*e^(-d*x - c) - b)/((a^2 + b^2 +
(a^2 + b^2)*e^(-2*d*x - 2*c))*d) + log(e^(-d*x - c) + 1)/(a*d) - log(e^(-d*x - c) - 1)/(a*d))*e^2 - 4*a*e*f*a
rctan(e^(d*x + c))/((a^2 + b^2)*d^2) + 2*(b*f^2*x^2 + 2*b*e*f*x + (a*f^2*x^2*e^c + 2*a*e*f*x*e^c)*e^(d*x))/(a^
2*d + b^2*d + (a^2*d*e^(2*c) + b^2*d*e^(2*c))*e^(2*d*x)) - 2*(d*x*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))*
e*f/(a*d^2) + 2*(d*x*log(-e^(d*x + c) + 1) + dilog(e^(d*x + c)))*e*f/(a*d^2) - (d^2*x^2*log(e^(d*x + c) + 1) +
2*d*x*dilog(-e^(d*x + c)) - 2*polylog(3, -e^(d*x + c)))*f^2/(a*d^3) + (d^2*x^2*log(-e^(d*x + c) + 1) + 2*d*x*
dilog(e^(d*x + c)) - 2*polylog(3, e^(d*x + c)))*f^2/(a*d^3) - integrate(-2*(b^3*f^2*x^2*e^c + 2*b^3*e*f*x*e^c)
*e^(d*x)/(a^3*b + a*b^3 - (a^3*b*e^(2*c) + a*b^3*e^(2*c))*e^(2*d*x) - 2*(a^4*e^c + a^2*b^2*e^c)*e^(d*x)), x)

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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