\(\int \frac {(e+f x)^2 \text {sech}(c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [387]

Optimal result

Integrand size = 34, antiderivative size = 1256 \[ \int \frac {(e+f x)^2 \text {sech}(c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a (e+f x)^2 \arctan \left (e^{c+d x}\right )}{b^2 d}+\frac {2 a^3 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a^3 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d}+\frac {a f^2 \arctan (\sinh (c+d x))}{b^2 d^3}-\frac {a^3 f^2 \arctan (\sinh (c+d x))}{b^2 \left (a^2+b^2\right ) d^3}+\frac {a^2 b (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^2 b (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^2 b (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}-\frac {f^2 \log (\cosh (c+d x))}{b d^3}+\frac {a^2 f^2 \log (\cosh (c+d x))}{b \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 )}{b^2 d^2}-\frac {2 i a^3 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^3 f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b^2 \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 )}{b^2 d^2}+\frac {2 i a^3 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^3 f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^2}+\frac {2 a^2 b 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 a^2 b 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^2 b f (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i a f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{b^2 d^3}+\frac {2 i a^3 f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {i a^3 f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac {i a f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{b^2 d^3}-\frac {2 i a^3 f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {i a^3 f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^3}-\frac {2 a^2 b 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 a^2 b 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^2 b 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)}{b^2 d^2}+\frac {a^3 f (e+f x) \text {sech}(c+d x)}{b^2 \left (a^2+b^2\right ) d^2}-\frac {(e+f x)^2 \text {sech}^2(c+d x)}{2 b d}+\frac {a^2 (e+f x)^2 \text {sech}^2(c+d x)}{2 b \left (a^2+b^2\right ) d}+\frac {f (e+f x) \tanh (c+d x)}{b d^2}-\frac {a^2 f (e+f x) \tanh (c+d x)}{b \left (a^2+b^2\right ) d^2}-\frac {a (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b^2 d}+\frac {a^3 (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b^2 \left (a^2+b^2\right ) d} \]

[Out]

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

Rubi [A] (verified)

Time = 1.39 (sec) , antiderivative size = 1256, normalized size of antiderivative = 1.00, number of steps used = 53, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.441, Rules used = {5702, 5559, 4269, 3556, 4271, 3855, 4265, 2611, 2320, 6724, 5692, 5680, 2221, 6874, 3799} \[ \int \frac {(e+f x)^2 \text {sech}(c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {(e+f x)^2 \arctan \left (e^{c+d x}\right ) a^3}{b^2 \left (a^2+b^2\right ) d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right ) a^3}{\left (a^2+b^2\right )^2 d}-\frac {f^2 \arctan (\sinh (c+d x)) a^3}{b^2 \left (a^2+b^2\right ) d^3}-\frac {i f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^2}-\frac {2 i f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) a^3}{\left (a^2+b^2\right )^2 d^2}+\frac {i f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^2}+\frac {2 i f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) a^3}{\left (a^2+b^2\right )^2 d^2}+\frac {i f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^3}+\frac {2 i f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right ) a^3}{\left (a^2+b^2\right )^2 d^3}-\frac {i f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^3}-\frac {2 i f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right ) a^3}{\left (a^2+b^2\right )^2 d^3}+\frac {f (e+f x) \text {sech}(c+d x) a^3}{b^2 \left (a^2+b^2\right ) d^2}+\frac {(e+f x)^2 \text {sech}(c+d x) \tanh (c+d x) a^3}{2 b^2 \left (a^2+b^2\right ) d}+\frac {(e+f x)^2 \text {sech}^2(c+d x) a^2}{2 b \left (a^2+b^2\right ) d}+\frac {b (e+f x)^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) a^2}{\left (a^2+b^2\right )^2 d}+\frac {b (e+f x)^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) a^2}{\left (a^2+b^2\right )^2 d}-\frac {b (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) a^2}{\left (a^2+b^2\right )^2 d}+\frac {f^2 \log (\cosh (c+d x)) a^2}{b \left (a^2+b^2\right ) d^3}+\frac {2 b f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) a^2}{\left (a^2+b^2\right )^2 d^2}+\frac {2 b f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) a^2}{\left (a^2+b^2\right )^2 d^2}-\frac {b f (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right ) a^2}{\left (a^2+b^2\right )^2 d^2}-\frac {2 b f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) a^2}{\left (a^2+b^2\right )^2 d^3}-\frac {2 b f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) a^2}{\left (a^2+b^2\right )^2 d^3}+\frac {b f^2 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right ) a^2}{2 \left (a^2+b^2\right )^2 d^3}-\frac {f (e+f x) \tanh (c+d x) a^2}{b \left (a^2+b^2\right ) d^2}-\frac {(e+f x)^2 \arctan \left (e^{c+d x}\right ) a}{b^2 d}+\frac {f^2 \arctan (\sinh (c+d x)) a}{b^2 d^3}+\frac {i f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) a}{b^2 d^2}-\frac {i f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) a}{b^2 d^2}-\frac {i f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right ) a}{b^2 d^3}+\frac {i f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right ) a}{b^2 d^3}-\frac {f (e+f x) \text {sech}(c+d x) a}{b^2 d^2}-\frac {(e+f x)^2 \text {sech}(c+d x) \tanh (c+d x) a}{2 b^2 d}-\frac {(e+f x)^2 \text {sech}^2(c+d x)}{2 b d}-\frac {f^2 \log (\cosh (c+d x))}{b d^3}+\frac {f (e+f x) \tanh (c+d x)}{b d^2} \]

[In]

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

[Out]

-((a*(e + f*x)^2*ArcTan[E^(c + d*x)])/(b^2*d)) + (2*a^3*(e + f*x)^2*ArcTan[E^(c + d*x)])/((a^2 + b^2)^2*d) + (
a^3*(e + f*x)^2*ArcTan[E^(c + d*x)])/(b^2*(a^2 + b^2)*d) + (a*f^2*ArcTan[Sinh[c + d*x]])/(b^2*d^3) - (a^3*f^2*
ArcTan[Sinh[c + d*x]])/(b^2*(a^2 + b^2)*d^3) + (a^2*b*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]
)])/((a^2 + b^2)^2*d) + (a^2*b*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/((a^2 + b^2)^2*d) -
(a^2*b*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/((a^2 + b^2)^2*d) - (f^2*Log[Cosh[c + d*x]])/(b*d^3) + (a^2*f^2*
Log[Cosh[c + d*x]])/(b*(a^2 + b^2)*d^3) + (I*a*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(b^2*d^2) - ((2*I)*a^
3*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/((a^2 + b^2)^2*d^2) - (I*a^3*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*
x)])/(b^2*(a^2 + b^2)*d^2) - (I*a*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(b^2*d^2) + ((2*I)*a^3*f*(e + f*x)*Po
lyLog[2, I*E^(c + d*x)])/((a^2 + b^2)^2*d^2) + (I*a^3*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(b^2*(a^2 + b^2)*
d^2) + (2*a^2*b*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*a^2
*b*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^2) - (a^2*b*f*(e + f*x)*
PolyLog[2, -E^(2*(c + d*x))])/((a^2 + b^2)^2*d^2) - (I*a*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(b^2*d^3) + ((2*I)*
a^3*f^2*PolyLog[3, (-I)*E^(c + d*x)])/((a^2 + b^2)^2*d^3) + (I*a^3*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(b^2*(a^2
+ b^2)*d^3) + (I*a*f^2*PolyLog[3, I*E^(c + d*x)])/(b^2*d^3) - ((2*I)*a^3*f^2*PolyLog[3, I*E^(c + d*x)])/((a^2
+ b^2)^2*d^3) - (I*a^3*f^2*PolyLog[3, I*E^(c + d*x)])/(b^2*(a^2 + b^2)*d^3) - (2*a^2*b*f^2*PolyLog[3, -((b*E^
(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^3) - (2*a^2*b*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[
a^2 + b^2]))])/((a^2 + b^2)^2*d^3) + (a^2*b*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])/(b^2*d^2) + (a^3*f*(e + f*x)*Sech[c + d*x])/(b^2*(a^2 + b^2)*d^2) - ((e + f*x)^2*Sech[c
+ d*x]^2)/(2*b*d) + (a^2*(e + f*x)^2*Sech[c + d*x]^2)/(2*b*(a^2 + b^2)*d) + (f*(e + f*x)*Tanh[c + d*x])/(b*d^2
) - (a^2*f*(e + f*x)*Tanh[c + d*x])/(b*(a^2 + b^2)*d^2) - (a*(e + f*x)^2*Sech[c + d*x]*Tanh[c + d*x])/(2*b^2*d
) + (a^3*(e + f*x)^2*Sech[c + d*x]*Tanh[c + d*x])/(2*b^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 5702

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(p_.)*Tanh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/b, Int[(e + f*x)^m*Sech[c + d*x]^(p + 1)*Tanh[c + d*x]^(n - 1),
x], x] - Dist[a/b, Int[(e + f*x)^m*Sech[c + d*x]^(p + 1)*(Tanh[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 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}^2(c+d x) \tanh (c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x)^2 \text {sech}^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b} \\ & = -\frac {(e+f x)^2 \text {sech}^2(c+d x)}{2 b d}-\frac {a \int (e+f x)^2 \text {sech}^3(c+d x) \, dx}{b^2}+\frac {a^2 \int \frac {(e+f x)^2 \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}+\frac {f \int (e+f x) \text {sech}^2(c+d x) \, dx}{b d} \\ & = -\frac {a f (e+f x) \text {sech}(c+d x)}{b^2 d^2}-\frac {(e+f x)^2 \text {sech}^2(c+d x)}{2 b d}+\frac {f (e+f x) \tanh (c+d x)}{b d^2}-\frac {a (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b^2 d}-\frac {a \int (e+f x)^2 \text {sech}(c+d x) \, dx}{2 b^2}+\frac {a^2 \int \frac {(e+f x)^2 \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2+b^2}+\frac {a^2 \int (e+f x)^2 \text {sech}^3(c+d x) (a-b \sinh (c+d x)) \, dx}{b^2 \left (a^2+b^2\right )}+\frac {\left (a f^2\right ) \int \text {sech}(c+d x) \, dx}{b^2 d^2}-\frac {f^2 \int \tanh (c+d x) \, dx}{b d^2} \\ & = -\frac {a (e+f x)^2 \arctan \left (e^{c+d x}\right )}{b^2 d}+\frac {a f^2 \arctan (\sinh (c+d x))}{b^2 d^3}-\frac {f^2 \log (\cosh (c+d x))}{b d^3}-\frac {a f (e+f x) \text {sech}(c+d x)}{b^2 d^2}-\frac {(e+f x)^2 \text {sech}^2(c+d x)}{2 b d}+\frac {f (e+f x) \tanh (c+d x)}{b d^2}-\frac {a (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b^2 d}+\frac {a^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 {\left (a^2 b^2\right ) \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 {a^2 \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}{b^2 \left (a^2+b^2\right )}+\frac {(i a f) \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{b^2 d}-\frac {(i a f) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{b^2 d} \\ & = -\frac {a^2 b (e+f x)^3}{3 \left (a^2+b^2\right )^2 f}-\frac {a (e+f x)^2 \arctan \left (e^{c+d x}\right )}{b^2 d}+\frac {a f^2 \arctan (\sinh (c+d x))}{b^2 d^3}-\frac {f^2 \log (\cosh (c+d x))}{b d^3}+\frac {i a f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b^2 d^2}-\frac {i a f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b^2 d^2}-\frac {a f (e+f x) \text {sech}(c+d x)}{b^2 d^2}-\frac {(e+f x)^2 \text {sech}^2(c+d x)}{2 b d}+\frac {f (e+f x) \tanh (c+d x)}{b d^2}-\frac {a (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b^2 d}+\frac {a^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 {\left (a^2 b^2\right ) \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 {\left (a^2 b^2\right ) \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^3 \int (e+f x)^2 \text {sech}^3(c+d x) \, dx}{b^2 \left (a^2+b^2\right )}-\frac {a^2 \int (e+f x)^2 \text {sech}^2(c+d x) \tanh (c+d x) \, dx}{b \left (a^2+b^2\right )}-\frac {\left (i a f^2\right ) \int \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) \, dx}{b^2 d^2}+\frac {\left (i a f^2\right ) \int \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) \, dx}{b^2 d^2} \\ & = -\frac {a^2 b (e+f x)^3}{3 \left (a^2+b^2\right )^2 f}-\frac {a (e+f x)^2 \arctan \left (e^{c+d x}\right )}{b^2 d}+\frac {a f^2 \arctan (\sinh (c+d x))}{b^2 d^3}+\frac {a^2 b (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^2 b (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 {f^2 \log (\cosh (c+d x))}{b d^3}+\frac {i a f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b^2 d^2}-\frac {i a f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b^2 d^2}-\frac {a f (e+f x) \text {sech}(c+d x)}{b^2 d^2}+\frac {a^3 f (e+f x) \text {sech}(c+d x)}{b^2 \left (a^2+b^2\right ) d^2}-\frac {(e+f x)^2 \text {sech}^2(c+d x)}{2 b d}+\frac {a^2 (e+f x)^2 \text {sech}^2(c+d x)}{2 b \left (a^2+b^2\right ) d}+\frac {f (e+f x) \tanh (c+d x)}{b d^2}-\frac {a (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b^2 d}+\frac {a^3 (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b^2 \left (a^2+b^2\right ) d}+\frac {a^3 \int (e+f x)^2 \text {sech}(c+d x) \, dx}{\left (a^2+b^2\right )^2}-\frac {\left (a^2 b\right ) \int (e+f x)^2 \tanh (c+d x) \, dx}{\left (a^2+b^2\right )^2}+\frac {a^3 \int (e+f x)^2 \text {sech}(c+d x) \, dx}{2 b^2 \left (a^2+b^2\right )}-\frac {\left (2 a^2 b 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 a^2 b 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 (a^2 f\right ) \int (e+f x) \text {sech}^2(c+d x) \, dx}{b \left (a^2+b^2\right ) d}-\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 )}{b^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 )}{b^2 d^3}-\frac {\left (a^3 f^2\right ) \int \text {sech}(c+d x) \, dx}{b^2 \left (a^2+b^2\right ) d^2} \\ & = -\frac {a (e+f x)^2 \arctan \left (e^{c+d x}\right )}{b^2 d}+\frac {2 a^3 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a^3 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d}+\frac {a f^2 \arctan (\sinh (c+d x))}{b^2 d^3}-\frac {a^3 f^2 \arctan (\sinh (c+d x))}{b^2 \left (a^2+b^2\right ) d^3}+\frac {a^2 b (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^2 b (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 {f^2 \log (\cosh (c+d x))}{b d^3}+\frac {i a f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b^2 d^2}-\frac {i a f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b^2 d^2}+\frac {2 a^2 b 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 a^2 b 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 {i a f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{b^2 d^3}+\frac {i a f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{b^2 d^3}-\frac {a f (e+f x) \text {sech}(c+d x)}{b^2 d^2}+\frac {a^3 f (e+f x) \text {sech}(c+d x)}{b^2 \left (a^2+b^2\right ) d^2}-\frac {(e+f x)^2 \text {sech}^2(c+d x)}{2 b d}+\frac {a^2 (e+f x)^2 \text {sech}^2(c+d x)}{2 b \left (a^2+b^2\right ) d}+\frac {f (e+f x) \tanh (c+d x)}{b d^2}-\frac {a^2 f (e+f x) \tanh (c+d x)}{b \left (a^2+b^2\right ) d^2}-\frac {a (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b^2 d}+\frac {a^3 (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b^2 \left (a^2+b^2\right ) d}-\frac {\left (2 a^2 b\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^3 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^3 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 (i a^3 f\right ) \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{b^2 \left (a^2+b^2\right ) d}+\frac {\left (i a^3 f\right ) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{b^2 \left (a^2+b^2\right ) d}-\frac {\left (2 a^2 b 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 a^2 b 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 (a^2 f^2\right ) \int \tanh (c+d x) \, dx}{b \left (a^2+b^2\right ) d^2} \\ & = \text {Too large to display} \\ \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. \(3390\) vs. \(2(1256)=2512\).

Time = 12.07 (sec) , antiderivative size = 3390, normalized size of antiderivative = 2.70 \[ \int \frac {(e+f x)^2 \text {sech}(c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

(12*a^2*b*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*a^2*b*d^3*e*E^(2*c)*f*x^2
+ 4*a^2*b*d^3*E^(2*c)*f^2*x^3 + 6*a^3*d^2*e^2*ArcTan[E^(c + d*x)] - 6*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)] - 6*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*Arc
Tan[E^(c + d*x)] + (6*I)*a^3*d^2*e*f*x*Log[1 - I*E^(c + d*x)] - (6*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)] - (6*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)] - (3*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)] - (3*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)] + (6*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)] + (6*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)] + (3*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)] + (3*I)*a*b^2*d^2*E^(2*c)*f^2*x^2*Log[1 + I*E^(c + d*x)] - 6*a^2*b*d^2*e^2*Log[1 + E^(2*(c + d*x))] - 6*
a^2*b*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*
a^2*b*d^2*e*f*x*Log[1 + E^(2*(c + d*x))] - 12*a^2*b*d^2*e*E^(2*c)*f*x*Log[1 + E^(2*(c + d*x))] - 6*a^2*b*d^2*f
^2*x^2*Log[1 + E^(2*(c + d*x))] - 6*a^2*b*d^2*E^(2*c)*f^2*x^2*Log[1 + E^(2*(c + d*x))] - (6*I)*a*(a^2 - b^2)*d
*(1 + E^(2*c))*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)] + (6*I)*a*(a^2 - b^2)*d*(1 + E^(2*c))*f*(e + f*x)*Poly
Log[2, I*E^(c + d*x)] - 6*a^2*b*d*e*f*PolyLog[2, -E^(2*(c + d*x))] - 6*a^2*b*d*e*E^(2*c)*f*PolyLog[2, -E^(2*(c
+ d*x))] - 6*a^2*b*d*f^2*x*PolyLog[2, -E^(2*(c + d*x))] - 6*a^2*b*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)] - (6*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)] - (6*I)*a*b^2*E^(2*c)*f^2*PolyLog[3, (-I)*E^(c + d*x)] - (6*I)*a^3*f^2*Po
lyLog[3, I*E^(c + d*x)] + (6*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)] + (6*I)*a*b^2*E^(2*c)*f^2*PolyLog[3, I*E^(c + d*x)] + 3*a^2*b*f^2*PolyLog[3, -E^(2*(c + d*x))] + 3*a^2
*b*E^(2*c)*f^2*PolyLog[3, -E^(2*(c + d*x))])/(6*(a^2 + b^2)^2*d^3*(1 + E^(2*c))) - (a^2*b*(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*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 - Sq
rt[(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*a^2*b*d^2*e^2*x + 6*a^2*b*f^2*x + 6*b^3*f^2*x + 12*a^2*b*d^2*e*f*x^2 + 4*a^2*b*d^2*
f^2*x^3 - 6*a^2*b*e*f*Cosh[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*a
^2*b*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*a^2*b*
d^2*e*f*x^2*Cosh[2*c + 2*d*x] + 4*a^2*b*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*Sinh[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*Sinh[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]

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

[In]

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

[Out]

int((f*x+e)^2*sech(d*x+c)*tanh(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. 10934 vs. \(2 (1156) = 2312\).

Time = 0.46 (sec) , antiderivative size = 10934, normalized size of antiderivative = 8.71 \[ \int \frac {(e+f x)^2 \text {sech}(c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

integrate((f*x+e)^2*sech(d*x+c)*tanh(d*x+c)^2/(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) - 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*a^2*b*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) - 2*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*a^2*b*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)) + (a^2*b*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) - a^2*b*log(e^(-2*d*x - 2*c) + 1)/((a^4 + 2
*a^2*b^2 + b^4)*d) - (a^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*(a^2*b^2*f^2*x^2 + 2*a^2
*b^2*e*f*x - (a^3*b*f^2*x^2*e^c + 2*a^3*b*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. \[ \int \frac {(e+f x)^2 \text {sech}(c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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