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

Optimal. Leaf size=1256 \[ \frac {(e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right ) a^3}{b^2 \left (a^2+b^2\right ) d}+\frac {2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right ) a^3}{\left (a^2+b^2\right )^2 d}-\frac {f^2 \tan ^{-1}(\sinh (c+d x)) a^3}{b^2 \left (a^2+b^2\right ) d^3}-\frac {i f (e+f x) \text {Li}_2\left (-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) \text {Li}_2\left (-i e^{c+d x}\right ) a^3}{\left (a^2+b^2\right )^2 d^2}+\frac {i f (e+f x) \text {Li}_2\left (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) \text {Li}_2\left (i e^{c+d x}\right ) a^3}{\left (a^2+b^2\right )^2 d^2}+\frac {i f^2 \text {Li}_3\left (-i e^{c+d x}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^3}+\frac {2 i f^2 \text {Li}_3\left (-i e^{c+d x}\right ) a^3}{\left (a^2+b^2\right )^2 d^3}-\frac {i f^2 \text {Li}_3\left (i e^{c+d x}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^3}-\frac {2 i f^2 \text {Li}_3\left (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) \text {Li}_2\left (-\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) \text {Li}_2\left (-\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) \text {Li}_2\left (-e^{2 (c+d x)}\right ) a^2}{\left (a^2+b^2\right )^2 d^2}-\frac {2 b f^2 \text {Li}_3\left (-\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 \text {Li}_3\left (-\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 \text {Li}_3\left (-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 \tan ^{-1}\left (e^{c+d x}\right ) a}{b^2 d}+\frac {f^2 \tan ^{-1}(\sinh (c+d x)) a}{b^2 d^3}+\frac {i f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right ) a}{b^2 d^2}-\frac {i f (e+f x) \text {Li}_2\left (i e^{c+d x}\right ) a}{b^2 d^2}-\frac {i f^2 \text {Li}_3\left (-i e^{c+d x}\right ) a}{b^2 d^3}+\frac {i f^2 \text {Li}_3\left (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} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.97, antiderivative size = 1256, normalized size of antiderivative = 1.00, number of steps used = 53, number of rules used = 15, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.441, Rules used = {5583, 5451, 4184, 3475, 4186, 3770, 4180, 2531, 2282, 6589, 5573, 5561, 2190, 6742, 3718} \[ \frac {(e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right ) a^3}{b^2 \left (a^2+b^2\right ) d}+\frac {2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right ) a^3}{\left (a^2+b^2\right )^2 d}-\frac {f^2 \tan ^{-1}(\sinh (c+d x)) a^3}{b^2 \left (a^2+b^2\right ) d^3}-\frac {i f (e+f x) \text {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) \text {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) \text {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) \text {PolyLog}\left (2,i e^{c+d x}\right ) a^3}{\left (a^2+b^2\right )^2 d^2}+\frac {i f^2 \text {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 \text {PolyLog}\left (3,-i e^{c+d x}\right ) a^3}{\left (a^2+b^2\right )^2 d^3}-\frac {i f^2 \text {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 \text {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) \text {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) \text {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) \text {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 \text {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 \text {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 \text {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 \tan ^{-1}\left (e^{c+d x}\right ) a}{b^2 d}+\frac {f^2 \tan ^{-1}(\sinh (c+d x)) a}{b^2 d^3}+\frac {i f (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right ) a}{b^2 d^2}-\frac {i f (e+f x) \text {PolyLog}\left (2,i e^{c+d x}\right ) a}{b^2 d^2}-\frac {i f^2 \text {PolyLog}\left (3,-i e^{c+d x}\right ) a}{b^2 d^3}+\frac {i f^2 \text {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} \]

Antiderivative was successfully verified.

[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 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2282

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 2531

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 3475

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

Rule 3718

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 3770

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

Rule 4180

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 4184

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 4186

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]) /; FreeQ[
{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

Rule 5451

Int[((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.)*Tanh[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> -Si
mp[((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 5561

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 5573

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 5583

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 6589

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 6742

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

Rubi steps

\begin {align*} \int \frac {(e+f x)^2 \text {sech}(c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=\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 \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d}+\frac {a f^2 \tan ^{-1}(\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 \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d}+\frac {a f^2 \tan ^{-1}(\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) \text {Li}_2\left (-i e^{c+d x}\right )}{b^2 d^2}-\frac {i a f (e+f x) \text {Li}_2\left (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 \text {Li}_2\left (-i e^{c+d x}\right ) \, dx}{b^2 d^2}+\frac {\left (i a f^2\right ) \int \text {Li}_2\left (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 \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d}+\frac {a f^2 \tan ^{-1}(\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) \text {Li}_2\left (-i e^{c+d x}\right )}{b^2 d^2}-\frac {i a f (e+f x) \text {Li}_2\left (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 ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{c+d x}\right )}{b^2 d^3}+\frac {\left (i a f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_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 \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d}+\frac {2 a^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d}+\frac {a f^2 \tan ^{-1}(\sinh (c+d x))}{b^2 d^3}-\frac {a^3 f^2 \tan ^{-1}(\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) \text {Li}_2\left (-i e^{c+d x}\right )}{b^2 d^2}-\frac {i a f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b^2 d^2}+\frac {2 a^2 b f (e+f x) \text {Li}_2\left (-\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) \text {Li}_2\left (-\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 \text {Li}_3\left (-i e^{c+d x}\right )}{b^2 d^3}+\frac {i a f^2 \text {Li}_3\left (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 \text {Li}_2\left (-\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 \text {Li}_2\left (-\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}\\ &=-\frac {a (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d}+\frac {2 a^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d}+\frac {a f^2 \tan ^{-1}(\sinh (c+d x))}{b^2 d^3}-\frac {a^3 f^2 \tan ^{-1}(\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) \text {Li}_2\left (-i e^{c+d x}\right )}{b^2 d^2}-\frac {2 i a^3 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i a^3 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {i a f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b^2 d^2}+\frac {2 i a^3 f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i a^3 f (e+f x) \text {Li}_2\left (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) \text {Li}_2\left (-\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) \text {Li}_2\left (-\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 \text {Li}_3\left (-i e^{c+d x}\right )}{b^2 d^3}+\frac {i a f^2 \text {Li}_3\left (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 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 a^2 b f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\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 a^2 b f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\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^3 f^2\right ) \int \text {Li}_2\left (-i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right )^2 d^2}-\frac {\left (2 i a^3 f^2\right ) \int \text {Li}_2\left (i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right )^2 d^2}+\frac {\left (i a^3 f^2\right ) \int \text {Li}_2\left (-i e^{c+d x}\right ) \, dx}{b^2 \left (a^2+b^2\right ) d^2}-\frac {\left (i a^3 f^2\right ) \int \text {Li}_2\left (i e^{c+d x}\right ) \, dx}{b^2 \left (a^2+b^2\right ) d^2}\\ &=-\frac {a (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d}+\frac {2 a^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d}+\frac {a f^2 \tan ^{-1}(\sinh (c+d x))}{b^2 d^3}-\frac {a^3 f^2 \tan ^{-1}(\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) \text {Li}_2\left (-i e^{c+d x}\right )}{b^2 d^2}-\frac {2 i a^3 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i a^3 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {i a f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b^2 d^2}+\frac {2 i a^3 f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i a^3 f (e+f x) \text {Li}_2\left (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) \text {Li}_2\left (-\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) \text {Li}_2\left (-\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) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i a f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{b^2 d^3}+\frac {i a f^2 \text {Li}_3\left (i e^{c+d x}\right )}{b^2 d^3}-\frac {2 a^2 b f^2 \text {Li}_3\left (-\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 \text {Li}_3\left (-\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)}{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 i a^3 f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_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^3 f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {\left (i a^3 f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^3}-\frac {\left (i a^3 f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac {\left (a^2 b f^2\right ) \int \text {Li}_2\left (-e^{2 (c+d x)}\right ) \, dx}{\left (a^2+b^2\right )^2 d^2}\\ &=-\frac {a (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d}+\frac {2 a^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d}+\frac {a f^2 \tan ^{-1}(\sinh (c+d x))}{b^2 d^3}-\frac {a^3 f^2 \tan ^{-1}(\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) \text {Li}_2\left (-i e^{c+d x}\right )}{b^2 d^2}-\frac {2 i a^3 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i a^3 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {i a f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b^2 d^2}+\frac {2 i a^3 f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i a^3 f (e+f x) \text {Li}_2\left (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) \text {Li}_2\left (-\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) \text {Li}_2\left (-\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) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i a f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{b^2 d^3}+\frac {2 i a^3 f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {i a^3 f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac {i a f^2 \text {Li}_3\left (i e^{c+d x}\right )}{b^2 d^3}-\frac {2 i a^3 f^2 \text {Li}_3\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {i a^3 f^2 \text {Li}_3\left (i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^3}-\frac {2 a^2 b f^2 \text {Li}_3\left (-\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 \text {Li}_3\left (-\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)}{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 (a^2 b f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d^3}\\ &=-\frac {a (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d}+\frac {2 a^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d}+\frac {a f^2 \tan ^{-1}(\sinh (c+d x))}{b^2 d^3}-\frac {a^3 f^2 \tan ^{-1}(\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) \text {Li}_2\left (-i e^{c+d x}\right )}{b^2 d^2}-\frac {2 i a^3 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i a^3 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {i a f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b^2 d^2}+\frac {2 i a^3 f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i a^3 f (e+f x) \text {Li}_2\left (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) \text {Li}_2\left (-\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) \text {Li}_2\left (-\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) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i a f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{b^2 d^3}+\frac {2 i a^3 f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {i a^3 f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac {i a f^2 \text {Li}_3\left (i e^{c+d x}\right )}{b^2 d^3}-\frac {2 i a^3 f^2 \text {Li}_3\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {i a^3 f^2 \text {Li}_3\left (i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^3}-\frac {2 a^2 b f^2 \text {Li}_3\left (-\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 \text {Li}_3\left (-\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 \text {Li}_3\left (-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}\\ \end {align*}

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Mathematica [B]  time = 27.87, size = 3124, normalized size = 2.49 \[ \text {Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/6*(-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*ArcTan[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
)*PolyLog[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*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)] - 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))])/((a^2 + b^2)^2*d^3*(1 + E^(2*c))) - (a^2*b*(6*d^3*e^2*E^(2*
c)*x + 6*d^3*e*E^(2*c)*f*x^2 + 2*d^3*E^(2*c)*f^2*x^3 + 3*d^2*e^2*Log[b - 2*a*E^(c + d*x) - b*E^(2*(c + d*x))]
- 3*d^2*e^2*E^(2*c)*Log[b - 2*a*E^(c + d*x) - b*E^(2*(c + d*x))] + 6*d^2*e*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^
c - Sqrt[(a^2 + b^2)*E^(2*c)])] - 6*d^2*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)])] + 3*d^2*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] - 3*d^2*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)])] + 6*d^2*e*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c
+ Sqrt[(a^2 + b^2)*E^(2*c)])] - 6*d^2*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
)])] + 3*d^2*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] - 3*d^2*E^(2*c)*f^2*x^2*Lo
g[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] - 6*d*(-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)]))] - 6*d*(-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)]))] - 6*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*
E^(2*c)]))] + 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)]))] - 6*f^2*PolyL
og[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))] + 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)]))]))/(3*(a^2 + b^2)^2*d^3*(-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)

________________________________________________________________________________________

fricas [C]  time = 1.03, size = 10892, normalized size = 8.67 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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]

1/2*(4*((a^2*b + b^3)*d*f^2*x + (a^2*b + b^3)*c*f^2)*cosh(d*x + c)^4 + 4*((a^2*b + b^3)*d*f^2*x + (a^2*b + b^3
)*c*f^2)*sinh(d*x + c)^4 - 4*(a^2*b + b^3)*d*e*f + 4*(a^2*b + b^3)*c*f^2 - 2*((a^3 + a*b^2)*d^2*f^2*x^2 + (a^3
 + a*b^2)*d^2*e^2 + 2*(a^3 + a*b^2)*d*e*f + 2*((a^3 + a*b^2)*d^2*e*f + (a^3 + a*b^2)*d*f^2)*x)*cosh(d*x + c)^3
 - 2*((a^3 + a*b^2)*d^2*f^2*x^2 + (a^3 + a*b^2)*d^2*e^2 + 2*(a^3 + a*b^2)*d*e*f + 2*((a^3 + a*b^2)*d^2*e*f + (
a^3 + a*b^2)*d*f^2)*x - 8*((a^2*b + b^3)*d*f^2*x + (a^2*b + b^3)*c*f^2)*cosh(d*x + c))*sinh(d*x + c)^3 - 4*((a
^2*b + b^3)*d^2*f^2*x^2 + (a^2*b + b^3)*d^2*e^2 + (a^2*b + b^3)*d*e*f - 2*(a^2*b + b^3)*c*f^2 + (2*(a^2*b + b^
3)*d^2*e*f - (a^2*b + b^3)*d*f^2)*x)*cosh(d*x + c)^2 - 2*(2*(a^2*b + b^3)*d^2*f^2*x^2 + 2*(a^2*b + b^3)*d^2*e^
2 + 2*(a^2*b + b^3)*d*e*f - 4*(a^2*b + b^3)*c*f^2 - 12*((a^2*b + b^3)*d*f^2*x + (a^2*b + b^3)*c*f^2)*cosh(d*x
+ c)^2 + 2*(2*(a^2*b + b^3)*d^2*e*f - (a^2*b + b^3)*d*f^2)*x + 3*((a^3 + a*b^2)*d^2*f^2*x^2 + (a^3 + a*b^2)*d^
2*e^2 + 2*(a^3 + a*b^2)*d*e*f + 2*((a^3 + a*b^2)*d^2*e*f + (a^3 + a*b^2)*d*f^2)*x)*cosh(d*x + c))*sinh(d*x + c
)^2 + 2*((a^3 + a*b^2)*d^2*f^2*x^2 + (a^3 + a*b^2)*d^2*e^2 - 2*(a^3 + a*b^2)*d*e*f + 2*((a^3 + a*b^2)*d^2*e*f
- (a^3 + a*b^2)*d*f^2)*x)*cosh(d*x + c) + 4*(a^2*b*d*f^2*x + a^2*b*d*e*f + (a^2*b*d*f^2*x + a^2*b*d*e*f)*cosh(
d*x + c)^4 + 4*(a^2*b*d*f^2*x + a^2*b*d*e*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2*b*d*f^2*x + a^2*b*d*e*f)*sin
h(d*x + c)^4 + 2*(a^2*b*d*f^2*x + a^2*b*d*e*f)*cosh(d*x + c)^2 + 2*(a^2*b*d*f^2*x + a^2*b*d*e*f + 3*(a^2*b*d*f
^2*x + a^2*b*d*e*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((a^2*b*d*f^2*x + a^2*b*d*e*f)*cosh(d*x + c)^3 + (a^2
*b*d*f^2*x + a^2*b*d*e*f)*cosh(d*x + c))*sinh(d*x + c))*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x
 + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + 4*(a^2*b*d*f^2*x + a^2*b*d*e*f + (a^2*b*d*f^2*x +
 a^2*b*d*e*f)*cosh(d*x + c)^4 + 4*(a^2*b*d*f^2*x + a^2*b*d*e*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2*b*d*f^2*x
 + a^2*b*d*e*f)*sinh(d*x + c)^4 + 2*(a^2*b*d*f^2*x + a^2*b*d*e*f)*cosh(d*x + c)^2 + 2*(a^2*b*d*f^2*x + a^2*b*d
*e*f + 3*(a^2*b*d*f^2*x + a^2*b*d*e*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((a^2*b*d*f^2*x + a^2*b*d*e*f)*cos
h(d*x + c)^3 + (a^2*b*d*f^2*x + a^2*b*d*e*f)*cosh(d*x + c))*sinh(d*x + c))*dilog((a*cosh(d*x + c) + a*sinh(d*x
 + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) - (4*a^2*b*d*f^2*x + 4*a^2*b*d*e
*f - 2*I*(a^3 - a*b^2)*d*f^2*x + (4*a^2*b*d*f^2*x + 4*a^2*b*d*e*f - 2*I*(a^3 - a*b^2)*d*f^2*x - 2*I*(a^3 - a*b
^2)*d*e*f)*cosh(d*x + c)^4 + (16*a^2*b*d*f^2*x + 16*a^2*b*d*e*f - 8*I*(a^3 - a*b^2)*d*f^2*x - 8*I*(a^3 - a*b^2
)*d*e*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (4*a^2*b*d*f^2*x + 4*a^2*b*d*e*f - 2*I*(a^3 - a*b^2)*d*f^2*x - 2*I*(a
^3 - a*b^2)*d*e*f)*sinh(d*x + c)^4 - 2*I*(a^3 - a*b^2)*d*e*f + (8*a^2*b*d*f^2*x + 8*a^2*b*d*e*f - 4*I*(a^3 - a
*b^2)*d*f^2*x - 4*I*(a^3 - a*b^2)*d*e*f)*cosh(d*x + c)^2 + (8*a^2*b*d*f^2*x + 8*a^2*b*d*e*f - 4*I*(a^3 - a*b^2
)*d*f^2*x - 4*I*(a^3 - a*b^2)*d*e*f + (24*a^2*b*d*f^2*x + 24*a^2*b*d*e*f - 12*I*(a^3 - a*b^2)*d*f^2*x - 12*I*(
a^3 - a*b^2)*d*e*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((16*a^2*b*d*f^2*x + 16*a^2*b*d*e*f - 8*I*(a^3 - a*b^2)
*d*f^2*x - 8*I*(a^3 - a*b^2)*d*e*f)*cosh(d*x + c)^3 + (16*a^2*b*d*f^2*x + 16*a^2*b*d*e*f - 8*I*(a^3 - a*b^2)*d
*f^2*x - 8*I*(a^3 - a*b^2)*d*e*f)*cosh(d*x + c))*sinh(d*x + c))*dilog(I*cosh(d*x + c) + I*sinh(d*x + c)) - (4*
a^2*b*d*f^2*x + 4*a^2*b*d*e*f + 2*I*(a^3 - a*b^2)*d*f^2*x + (4*a^2*b*d*f^2*x + 4*a^2*b*d*e*f + 2*I*(a^3 - a*b^
2)*d*f^2*x + 2*I*(a^3 - a*b^2)*d*e*f)*cosh(d*x + c)^4 + (16*a^2*b*d*f^2*x + 16*a^2*b*d*e*f + 8*I*(a^3 - a*b^2)
*d*f^2*x + 8*I*(a^3 - a*b^2)*d*e*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (4*a^2*b*d*f^2*x + 4*a^2*b*d*e*f + 2*I*(a^
3 - a*b^2)*d*f^2*x + 2*I*(a^3 - a*b^2)*d*e*f)*sinh(d*x + c)^4 + 2*I*(a^3 - a*b^2)*d*e*f + (8*a^2*b*d*f^2*x + 8
*a^2*b*d*e*f + 4*I*(a^3 - a*b^2)*d*f^2*x + 4*I*(a^3 - a*b^2)*d*e*f)*cosh(d*x + c)^2 + (8*a^2*b*d*f^2*x + 8*a^2
*b*d*e*f + 4*I*(a^3 - a*b^2)*d*f^2*x + 4*I*(a^3 - a*b^2)*d*e*f + (24*a^2*b*d*f^2*x + 24*a^2*b*d*e*f + 12*I*(a^
3 - a*b^2)*d*f^2*x + 12*I*(a^3 - a*b^2)*d*e*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((16*a^2*b*d*f^2*x + 16*a^2*
b*d*e*f + 8*I*(a^3 - a*b^2)*d*f^2*x + 8*I*(a^3 - a*b^2)*d*e*f)*cosh(d*x + c)^3 + (16*a^2*b*d*f^2*x + 16*a^2*b*
d*e*f + 8*I*(a^3 - a*b^2)*d*f^2*x + 8*I*(a^3 - a*b^2)*d*e*f)*cosh(d*x + c))*sinh(d*x + c))*dilog(-I*cosh(d*x +
 c) - I*sinh(d*x + c)) + 2*(a^2*b*d^2*e^2 - 2*a^2*b*c*d*e*f + a^2*b*c^2*f^2 + (a^2*b*d^2*e^2 - 2*a^2*b*c*d*e*f
 + a^2*b*c^2*f^2)*cosh(d*x + c)^4 + 4*(a^2*b*d^2*e^2 - 2*a^2*b*c*d*e*f + a^2*b*c^2*f^2)*cosh(d*x + c)*sinh(d*x
 + c)^3 + (a^2*b*d^2*e^2 - 2*a^2*b*c*d*e*f + a^2*b*c^2*f^2)*sinh(d*x + c)^4 + 2*(a^2*b*d^2*e^2 - 2*a^2*b*c*d*e
*f + a^2*b*c^2*f^2)*cosh(d*x + c)^2 + 2*(a^2*b*d^2*e^2 - 2*a^2*b*c*d*e*f + a^2*b*c^2*f^2 + 3*(a^2*b*d^2*e^2 -
2*a^2*b*c*d*e*f + a^2*b*c^2*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((a^2*b*d^2*e^2 - 2*a^2*b*c*d*e*f + a^2*
b*c^2*f^2)*cosh(d*x + c)^3 + (a^2*b*d^2*e^2 - 2*a^2*b*c*d*e*f + a^2*b*c^2*f^2)*cosh(d*x + c))*sinh(d*x + c))*l
og(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + 2*(a^2*b*d^2*e^2 - 2*a^2*b*c*d*e
*f + a^2*b*c^2*f^2 + (a^2*b*d^2*e^2 - 2*a^2*b*c*d*e*f + a^2*b*c^2*f^2)*cosh(d*x + c)^4 + 4*(a^2*b*d^2*e^2 - 2*
a^2*b*c*d*e*f + a^2*b*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2*b*d^2*e^2 - 2*a^2*b*c*d*e*f + a^2*b*c^2*f^
2)*sinh(d*x + c)^4 + 2*(a^2*b*d^2*e^2 - 2*a^2*b*c*d*e*f + a^2*b*c^2*f^2)*cosh(d*x + c)^2 + 2*(a^2*b*d^2*e^2 -
2*a^2*b*c*d*e*f + a^2*b*c^2*f^2 + 3*(a^2*b*d^2*e^2 - 2*a^2*b*c*d*e*f + a^2*b*c^2*f^2)*cosh(d*x + c)^2)*sinh(d*
x + c)^2 + 4*((a^2*b*d^2*e^2 - 2*a^2*b*c*d*e*f + a^2*b*c^2*f^2)*cosh(d*x + c)^3 + (a^2*b*d^2*e^2 - 2*a^2*b*c*d
*e*f + a^2*b*c^2*f^2)*cosh(d*x + c))*sinh(d*x + c))*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2
+ b^2)/b^2) + 2*a) + 2*(a^2*b*d^2*f^2*x^2 + 2*a^2*b*d^2*e*f*x + 2*a^2*b*c*d*e*f - a^2*b*c^2*f^2 + (a^2*b*d^2*f
^2*x^2 + 2*a^2*b*d^2*e*f*x + 2*a^2*b*c*d*e*f - a^2*b*c^2*f^2)*cosh(d*x + c)^4 + 4*(a^2*b*d^2*f^2*x^2 + 2*a^2*b
*d^2*e*f*x + 2*a^2*b*c*d*e*f - a^2*b*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2*b*d^2*f^2*x^2 + 2*a^2*b*d^2
*e*f*x + 2*a^2*b*c*d*e*f - a^2*b*c^2*f^2)*sinh(d*x + c)^4 + 2*(a^2*b*d^2*f^2*x^2 + 2*a^2*b*d^2*e*f*x + 2*a^2*b
*c*d*e*f - a^2*b*c^2*f^2)*cosh(d*x + c)^2 + 2*(a^2*b*d^2*f^2*x^2 + 2*a^2*b*d^2*e*f*x + 2*a^2*b*c*d*e*f - a^2*b
*c^2*f^2 + 3*(a^2*b*d^2*f^2*x^2 + 2*a^2*b*d^2*e*f*x + 2*a^2*b*c*d*e*f - a^2*b*c^2*f^2)*cosh(d*x + c)^2)*sinh(d
*x + c)^2 + 4*((a^2*b*d^2*f^2*x^2 + 2*a^2*b*d^2*e*f*x + 2*a^2*b*c*d*e*f - a^2*b*c^2*f^2)*cosh(d*x + c)^3 + (a^
2*b*d^2*f^2*x^2 + 2*a^2*b*d^2*e*f*x + 2*a^2*b*c*d*e*f - a^2*b*c^2*f^2)*cosh(d*x + c))*sinh(d*x + c))*log(-(a*c
osh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) + 2*(a^2*b*
d^2*f^2*x^2 + 2*a^2*b*d^2*e*f*x + 2*a^2*b*c*d*e*f - a^2*b*c^2*f^2 + (a^2*b*d^2*f^2*x^2 + 2*a^2*b*d^2*e*f*x + 2
*a^2*b*c*d*e*f - a^2*b*c^2*f^2)*cosh(d*x + c)^4 + 4*(a^2*b*d^2*f^2*x^2 + 2*a^2*b*d^2*e*f*x + 2*a^2*b*c*d*e*f -
 a^2*b*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2*b*d^2*f^2*x^2 + 2*a^2*b*d^2*e*f*x + 2*a^2*b*c*d*e*f - a^2
*b*c^2*f^2)*sinh(d*x + c)^4 + 2*(a^2*b*d^2*f^2*x^2 + 2*a^2*b*d^2*e*f*x + 2*a^2*b*c*d*e*f - a^2*b*c^2*f^2)*cosh
(d*x + c)^2 + 2*(a^2*b*d^2*f^2*x^2 + 2*a^2*b*d^2*e*f*x + 2*a^2*b*c*d*e*f - a^2*b*c^2*f^2 + 3*(a^2*b*d^2*f^2*x^
2 + 2*a^2*b*d^2*e*f*x + 2*a^2*b*c*d*e*f - a^2*b*c^2*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((a^2*b*d^2*f^2*
x^2 + 2*a^2*b*d^2*e*f*x + 2*a^2*b*c*d*e*f - a^2*b*c^2*f^2)*cosh(d*x + c)^3 + (a^2*b*d^2*f^2*x^2 + 2*a^2*b*d^2*
e*f*x + 2*a^2*b*c*d*e*f - a^2*b*c^2*f^2)*cosh(d*x + c))*sinh(d*x + c))*log(-(a*cosh(d*x + c) + a*sinh(d*x + c)
 - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) - (2*a^2*b*d^2*e^2 - 4*a^2*b*c*d*e*f - I*
(a^3 - a*b^2)*d^2*e^2 + 2*I*(a^3 - a*b^2)*c*d*e*f + (2*a^2*b*d^2*e^2 - 4*a^2*b*c*d*e*f - I*(a^3 - a*b^2)*d^2*e
^2 + 2*I*(a^3 - a*b^2)*c*d*e*f + 2*(a^2*b*c^2 + a^2*b + b^3)*f^2 - I*(2*a^3 + 2*a*b^2 + (a^3 - a*b^2)*c^2)*f^2
)*cosh(d*x + c)^4 + (8*a^2*b*d^2*e^2 - 16*a^2*b*c*d*e*f - 4*I*(a^3 - a*b^2)*d^2*e^2 + 8*I*(a^3 - a*b^2)*c*d*e*
f + 8*(a^2*b*c^2 + a^2*b + b^3)*f^2 - 4*I*(2*a^3 + 2*a*b^2 + (a^3 - a*b^2)*c^2)*f^2)*cosh(d*x + c)*sinh(d*x +
c)^3 + (2*a^2*b*d^2*e^2 - 4*a^2*b*c*d*e*f - I*(a^3 - a*b^2)*d^2*e^2 + 2*I*(a^3 - a*b^2)*c*d*e*f + 2*(a^2*b*c^2
 + a^2*b + b^3)*f^2 - I*(2*a^3 + 2*a*b^2 + (a^3 - a*b^2)*c^2)*f^2)*sinh(d*x + c)^4 + 2*(a^2*b*c^2 + a^2*b + b^
3)*f^2 - I*(2*a^3 + 2*a*b^2 + (a^3 - a*b^2)*c^2)*f^2 + (4*a^2*b*d^2*e^2 - 8*a^2*b*c*d*e*f - 2*I*(a^3 - a*b^2)*
d^2*e^2 + 4*I*(a^3 - a*b^2)*c*d*e*f + 4*(a^2*b*c^2 + a^2*b + b^3)*f^2 - 2*I*(2*a^3 + 2*a*b^2 + (a^3 - a*b^2)*c
^2)*f^2)*cosh(d*x + c)^2 + (4*a^2*b*d^2*e^2 - 8*a^2*b*c*d*e*f - 2*I*(a^3 - a*b^2)*d^2*e^2 + 4*I*(a^3 - a*b^2)*
c*d*e*f + 4*(a^2*b*c^2 + a^2*b + b^3)*f^2 - 2*I*(2*a^3 + 2*a*b^2 + (a^3 - a*b^2)*c^2)*f^2 + (12*a^2*b*d^2*e^2
- 24*a^2*b*c*d*e*f - 6*I*(a^3 - a*b^2)*d^2*e^2 + 12*I*(a^3 - a*b^2)*c*d*e*f + 12*(a^2*b*c^2 + a^2*b + b^3)*f^2
 - 6*I*(2*a^3 + 2*a*b^2 + (a^3 - a*b^2)*c^2)*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((8*a^2*b*d^2*e^2 - 16*a^
2*b*c*d*e*f - 4*I*(a^3 - a*b^2)*d^2*e^2 + 8*I*(a^3 - a*b^2)*c*d*e*f + 8*(a^2*b*c^2 + a^2*b + b^3)*f^2 - 4*I*(2
*a^3 + 2*a*b^2 + (a^3 - a*b^2)*c^2)*f^2)*cosh(d*x + c)^3 + (8*a^2*b*d^2*e^2 - 16*a^2*b*c*d*e*f - 4*I*(a^3 - a*
b^2)*d^2*e^2 + 8*I*(a^3 - a*b^2)*c*d*e*f + 8*(a^2*b*c^2 + a^2*b + b^3)*f^2 - 4*I*(2*a^3 + 2*a*b^2 + (a^3 - a*b
^2)*c^2)*f^2)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + I) - (2*a^2*b*d^2*e^2 - 4*a^2*
b*c*d*e*f + I*(a^3 - a*b^2)*d^2*e^2 - 2*I*(a^3 - a*b^2)*c*d*e*f + (2*a^2*b*d^2*e^2 - 4*a^2*b*c*d*e*f + I*(a^3
- a*b^2)*d^2*e^2 - 2*I*(a^3 - a*b^2)*c*d*e*f + 2*(a^2*b*c^2 + a^2*b + b^3)*f^2 + I*(2*a^3 + 2*a*b^2 + (a^3 - a
*b^2)*c^2)*f^2)*cosh(d*x + c)^4 + (8*a^2*b*d^2*e^2 - 16*a^2*b*c*d*e*f + 4*I*(a^3 - a*b^2)*d^2*e^2 - 8*I*(a^3 -
 a*b^2)*c*d*e*f + 8*(a^2*b*c^2 + a^2*b + b^3)*f^2 + 4*I*(2*a^3 + 2*a*b^2 + (a^3 - a*b^2)*c^2)*f^2)*cosh(d*x +
c)*sinh(d*x + c)^3 + (2*a^2*b*d^2*e^2 - 4*a^2*b*c*d*e*f + I*(a^3 - a*b^2)*d^2*e^2 - 2*I*(a^3 - a*b^2)*c*d*e*f
+ 2*(a^2*b*c^2 + a^2*b + b^3)*f^2 + I*(2*a^3 + 2*a*b^2 + (a^3 - a*b^2)*c^2)*f^2)*sinh(d*x + c)^4 + 2*(a^2*b*c^
2 + a^2*b + b^3)*f^2 + I*(2*a^3 + 2*a*b^2 + (a^3 - a*b^2)*c^2)*f^2 + (4*a^2*b*d^2*e^2 - 8*a^2*b*c*d*e*f + 2*I*
(a^3 - a*b^2)*d^2*e^2 - 4*I*(a^3 - a*b^2)*c*d*e*f + 4*(a^2*b*c^2 + a^2*b + b^3)*f^2 + 2*I*(2*a^3 + 2*a*b^2 + (
a^3 - a*b^2)*c^2)*f^2)*cosh(d*x + c)^2 + (4*a^2*b*d^2*e^2 - 8*a^2*b*c*d*e*f + 2*I*(a^3 - a*b^2)*d^2*e^2 - 4*I*
(a^3 - a*b^2)*c*d*e*f + 4*(a^2*b*c^2 + a^2*b + b^3)*f^2 + 2*I*(2*a^3 + 2*a*b^2 + (a^3 - a*b^2)*c^2)*f^2 + (12*
a^2*b*d^2*e^2 - 24*a^2*b*c*d*e*f + 6*I*(a^3 - a*b^2)*d^2*e^2 - 12*I*(a^3 - a*b^2)*c*d*e*f + 12*(a^2*b*c^2 + a^
2*b + b^3)*f^2 + 6*I*(2*a^3 + 2*a*b^2 + (a^3 - a*b^2)*c^2)*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((8*a^2*b*d
^2*e^2 - 16*a^2*b*c*d*e*f + 4*I*(a^3 - a*b^2)*d^2*e^2 - 8*I*(a^3 - a*b^2)*c*d*e*f + 8*(a^2*b*c^2 + a^2*b + b^3
)*f^2 + 4*I*(2*a^3 + 2*a*b^2 + (a^3 - a*b^2)*c^2)*f^2)*cosh(d*x + c)^3 + (8*a^2*b*d^2*e^2 - 16*a^2*b*c*d*e*f +
 4*I*(a^3 - a*b^2)*d^2*e^2 - 8*I*(a^3 - a*b^2)*c*d*e*f + 8*(a^2*b*c^2 + a^2*b + b^3)*f^2 + 4*I*(2*a^3 + 2*a*b^
2 + (a^3 - a*b^2)*c^2)*f^2)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - I) - (2*a^2*b*d^
2*f^2*x^2 + 4*a^2*b*d^2*e*f*x + 4*a^2*b*c*d*e*f - 2*a^2*b*c^2*f^2 + I*(a^3 - a*b^2)*d^2*f^2*x^2 + 2*I*(a^3 - a
*b^2)*d^2*e*f*x + 2*I*(a^3 - a*b^2)*c*d*e*f - I*(a^3 - a*b^2)*c^2*f^2 + (2*a^2*b*d^2*f^2*x^2 + 4*a^2*b*d^2*e*f
*x + 4*a^2*b*c*d*e*f - 2*a^2*b*c^2*f^2 + I*(a^3 - a*b^2)*d^2*f^2*x^2 + 2*I*(a^3 - a*b^2)*d^2*e*f*x + 2*I*(a^3
- a*b^2)*c*d*e*f - I*(a^3 - a*b^2)*c^2*f^2)*cosh(d*x + c)^4 + (8*a^2*b*d^2*f^2*x^2 + 16*a^2*b*d^2*e*f*x + 16*a
^2*b*c*d*e*f - 8*a^2*b*c^2*f^2 + 4*I*(a^3 - a*b^2)*d^2*f^2*x^2 + 8*I*(a^3 - a*b^2)*d^2*e*f*x + 8*I*(a^3 - a*b^
2)*c*d*e*f - 4*I*(a^3 - a*b^2)*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*a^2*b*d^2*f^2*x^2 + 4*a^2*b*d^2*e*f
*x + 4*a^2*b*c*d*e*f - 2*a^2*b*c^2*f^2 + I*(a^3 - a*b^2)*d^2*f^2*x^2 + 2*I*(a^3 - a*b^2)*d^2*e*f*x + 2*I*(a^3
- a*b^2)*c*d*e*f - I*(a^3 - a*b^2)*c^2*f^2)*sinh(d*x + c)^4 + (4*a^2*b*d^2*f^2*x^2 + 8*a^2*b*d^2*e*f*x + 8*a^2
*b*c*d*e*f - 4*a^2*b*c^2*f^2 + 2*I*(a^3 - a*b^2)*d^2*f^2*x^2 + 4*I*(a^3 - a*b^2)*d^2*e*f*x + 4*I*(a^3 - a*b^2)
*c*d*e*f - 2*I*(a^3 - a*b^2)*c^2*f^2)*cosh(d*x + c)^2 + (4*a^2*b*d^2*f^2*x^2 + 8*a^2*b*d^2*e*f*x + 8*a^2*b*c*d
*e*f - 4*a^2*b*c^2*f^2 + 2*I*(a^3 - a*b^2)*d^2*f^2*x^2 + 4*I*(a^3 - a*b^2)*d^2*e*f*x + 4*I*(a^3 - a*b^2)*c*d*e
*f - 2*I*(a^3 - a*b^2)*c^2*f^2 + (12*a^2*b*d^2*f^2*x^2 + 24*a^2*b*d^2*e*f*x + 24*a^2*b*c*d*e*f - 12*a^2*b*c^2*
f^2 + 6*I*(a^3 - a*b^2)*d^2*f^2*x^2 + 12*I*(a^3 - a*b^2)*d^2*e*f*x + 12*I*(a^3 - a*b^2)*c*d*e*f - 6*I*(a^3 - a
*b^2)*c^2*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((8*a^2*b*d^2*f^2*x^2 + 16*a^2*b*d^2*e*f*x + 16*a^2*b*c*d*e*
f - 8*a^2*b*c^2*f^2 + 4*I*(a^3 - a*b^2)*d^2*f^2*x^2 + 8*I*(a^3 - a*b^2)*d^2*e*f*x + 8*I*(a^3 - a*b^2)*c*d*e*f
- 4*I*(a^3 - a*b^2)*c^2*f^2)*cosh(d*x + c)^3 + (8*a^2*b*d^2*f^2*x^2 + 16*a^2*b*d^2*e*f*x + 16*a^2*b*c*d*e*f -
8*a^2*b*c^2*f^2 + 4*I*(a^3 - a*b^2)*d^2*f^2*x^2 + 8*I*(a^3 - a*b^2)*d^2*e*f*x + 8*I*(a^3 - a*b^2)*c*d*e*f - 4*
I*(a^3 - a*b^2)*c^2*f^2)*cosh(d*x + c))*sinh(d*x + c))*log(I*cosh(d*x + c) + I*sinh(d*x + c) + 1) - (2*a^2*b*d
^2*f^2*x^2 + 4*a^2*b*d^2*e*f*x + 4*a^2*b*c*d*e*f - 2*a^2*b*c^2*f^2 - I*(a^3 - a*b^2)*d^2*f^2*x^2 - 2*I*(a^3 -
a*b^2)*d^2*e*f*x - 2*I*(a^3 - a*b^2)*c*d*e*f + I*(a^3 - a*b^2)*c^2*f^2 + (2*a^2*b*d^2*f^2*x^2 + 4*a^2*b*d^2*e*
f*x + 4*a^2*b*c*d*e*f - 2*a^2*b*c^2*f^2 - I*(a^3 - a*b^2)*d^2*f^2*x^2 - 2*I*(a^3 - a*b^2)*d^2*e*f*x - 2*I*(a^3
 - a*b^2)*c*d*e*f + I*(a^3 - a*b^2)*c^2*f^2)*cosh(d*x + c)^4 + (8*a^2*b*d^2*f^2*x^2 + 16*a^2*b*d^2*e*f*x + 16*
a^2*b*c*d*e*f - 8*a^2*b*c^2*f^2 - 4*I*(a^3 - a*b^2)*d^2*f^2*x^2 - 8*I*(a^3 - a*b^2)*d^2*e*f*x - 8*I*(a^3 - a*b
^2)*c*d*e*f + 4*I*(a^3 - a*b^2)*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*a^2*b*d^2*f^2*x^2 + 4*a^2*b*d^2*e*
f*x + 4*a^2*b*c*d*e*f - 2*a^2*b*c^2*f^2 - I*(a^3 - a*b^2)*d^2*f^2*x^2 - 2*I*(a^3 - a*b^2)*d^2*e*f*x - 2*I*(a^3
 - a*b^2)*c*d*e*f + I*(a^3 - a*b^2)*c^2*f^2)*sinh(d*x + c)^4 + (4*a^2*b*d^2*f^2*x^2 + 8*a^2*b*d^2*e*f*x + 8*a^
2*b*c*d*e*f - 4*a^2*b*c^2*f^2 - 2*I*(a^3 - a*b^2)*d^2*f^2*x^2 - 4*I*(a^3 - a*b^2)*d^2*e*f*x - 4*I*(a^3 - a*b^2
)*c*d*e*f + 2*I*(a^3 - a*b^2)*c^2*f^2)*cosh(d*x + c)^2 + (4*a^2*b*d^2*f^2*x^2 + 8*a^2*b*d^2*e*f*x + 8*a^2*b*c*
d*e*f - 4*a^2*b*c^2*f^2 - 2*I*(a^3 - a*b^2)*d^2*f^2*x^2 - 4*I*(a^3 - a*b^2)*d^2*e*f*x - 4*I*(a^3 - a*b^2)*c*d*
e*f + 2*I*(a^3 - a*b^2)*c^2*f^2 + (12*a^2*b*d^2*f^2*x^2 + 24*a^2*b*d^2*e*f*x + 24*a^2*b*c*d*e*f - 12*a^2*b*c^2
*f^2 - 6*I*(a^3 - a*b^2)*d^2*f^2*x^2 - 12*I*(a^3 - a*b^2)*d^2*e*f*x - 12*I*(a^3 - a*b^2)*c*d*e*f + 6*I*(a^3 -
a*b^2)*c^2*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((8*a^2*b*d^2*f^2*x^2 + 16*a^2*b*d^2*e*f*x + 16*a^2*b*c*d*e
*f - 8*a^2*b*c^2*f^2 - 4*I*(a^3 - a*b^2)*d^2*f^2*x^2 - 8*I*(a^3 - a*b^2)*d^2*e*f*x - 8*I*(a^3 - a*b^2)*c*d*e*f
 + 4*I*(a^3 - a*b^2)*c^2*f^2)*cosh(d*x + c)^3 + (8*a^2*b*d^2*f^2*x^2 + 16*a^2*b*d^2*e*f*x + 16*a^2*b*c*d*e*f -
 8*a^2*b*c^2*f^2 - 4*I*(a^3 - a*b^2)*d^2*f^2*x^2 - 8*I*(a^3 - a*b^2)*d^2*e*f*x - 8*I*(a^3 - a*b^2)*c*d*e*f + 4
*I*(a^3 - a*b^2)*c^2*f^2)*cosh(d*x + c))*sinh(d*x + c))*log(-I*cosh(d*x + c) - I*sinh(d*x + c) + 1) - 4*(a^2*b
*f^2*cosh(d*x + c)^4 + 4*a^2*b*f^2*cosh(d*x + c)*sinh(d*x + c)^3 + a^2*b*f^2*sinh(d*x + c)^4 + 2*a^2*b*f^2*cos
h(d*x + c)^2 + a^2*b*f^2 + 2*(3*a^2*b*f^2*cosh(d*x + c)^2 + a^2*b*f^2)*sinh(d*x + c)^2 + 4*(a^2*b*f^2*cosh(d*x
 + c)^3 + a^2*b*f^2*cosh(d*x + c))*sinh(d*x + c))*polylog(3, (a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x
+ c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) - 4*(a^2*b*f^2*cosh(d*x + c)^4 + 4*a^2*b*f^2*cosh(d*x + c)*s
inh(d*x + c)^3 + a^2*b*f^2*sinh(d*x + c)^4 + 2*a^2*b*f^2*cosh(d*x + c)^2 + a^2*b*f^2 + 2*(3*a^2*b*f^2*cosh(d*x
 + c)^2 + a^2*b*f^2)*sinh(d*x + c)^2 + 4*(a^2*b*f^2*cosh(d*x + c)^3 + a^2*b*f^2*cosh(d*x + c))*sinh(d*x + c))*
polylog(3, (a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b)
+ (4*a^2*b*f^2 + (4*a^2*b*f^2 - 2*I*(a^3 - a*b^2)*f^2)*cosh(d*x + c)^4 + (16*a^2*b*f^2 - 8*I*(a^3 - a*b^2)*f^2
)*cosh(d*x + c)*sinh(d*x + c)^3 + (4*a^2*b*f^2 - 2*I*(a^3 - a*b^2)*f^2)*sinh(d*x + c)^4 - 2*I*(a^3 - a*b^2)*f^
2 + (8*a^2*b*f^2 - 4*I*(a^3 - a*b^2)*f^2)*cosh(d*x + c)^2 + (8*a^2*b*f^2 - 4*I*(a^3 - a*b^2)*f^2 + (24*a^2*b*f
^2 - 12*I*(a^3 - a*b^2)*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((16*a^2*b*f^2 - 8*I*(a^3 - a*b^2)*f^2)*cosh(d
*x + c)^3 + (16*a^2*b*f^2 - 8*I*(a^3 - a*b^2)*f^2)*cosh(d*x + c))*sinh(d*x + c))*polylog(3, I*cosh(d*x + c) +
I*sinh(d*x + c)) + (4*a^2*b*f^2 + (4*a^2*b*f^2 + 2*I*(a^3 - a*b^2)*f^2)*cosh(d*x + c)^4 + (16*a^2*b*f^2 + 8*I*
(a^3 - a*b^2)*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (4*a^2*b*f^2 + 2*I*(a^3 - a*b^2)*f^2)*sinh(d*x + c)^4 + 2*I
*(a^3 - a*b^2)*f^2 + (8*a^2*b*f^2 + 4*I*(a^3 - a*b^2)*f^2)*cosh(d*x + c)^2 + (8*a^2*b*f^2 + 4*I*(a^3 - a*b^2)*
f^2 + (24*a^2*b*f^2 + 12*I*(a^3 - a*b^2)*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((16*a^2*b*f^2 + 8*I*(a^3 - a
*b^2)*f^2)*cosh(d*x + c)^3 + (16*a^2*b*f^2 + 8*I*(a^3 - a*b^2)*f^2)*cosh(d*x + c))*sinh(d*x + c))*polylog(3, -
I*cosh(d*x + c) - I*sinh(d*x + c)) + 2*((a^3 + a*b^2)*d^2*f^2*x^2 + (a^3 + a*b^2)*d^2*e^2 - 2*(a^3 + a*b^2)*d*
e*f + 8*((a^2*b + b^3)*d*f^2*x + (a^2*b + b^3)*c*f^2)*cosh(d*x + c)^3 - 3*((a^3 + a*b^2)*d^2*f^2*x^2 + (a^3 +
a*b^2)*d^2*e^2 + 2*(a^3 + a*b^2)*d*e*f + 2*((a^3 + a*b^2)*d^2*e*f + (a^3 + a*b^2)*d*f^2)*x)*cosh(d*x + c)^2 +
2*((a^3 + a*b^2)*d^2*e*f - (a^3 + a*b^2)*d*f^2)*x - 4*((a^2*b + b^3)*d^2*f^2*x^2 + (a^2*b + b^3)*d^2*e^2 + (a^
2*b + b^3)*d*e*f - 2*(a^2*b + b^3)*c*f^2 + (2*(a^2*b + b^3)*d^2*e*f - (a^2*b + b^3)*d*f^2)*x)*cosh(d*x + c))*s
inh(d*x + c))/((a^4 + 2*a^2*b^2 + b^4)*d^3*cosh(d*x + c)^4 + 4*(a^4 + 2*a^2*b^2 + b^4)*d^3*cosh(d*x + c)*sinh(
d*x + c)^3 + (a^4 + 2*a^2*b^2 + b^4)*d^3*sinh(d*x + c)^4 + 2*(a^4 + 2*a^2*b^2 + b^4)*d^3*cosh(d*x + c)^2 + (a^
4 + 2*a^2*b^2 + b^4)*d^3 + 2*(3*(a^4 + 2*a^2*b^2 + b^4)*d^3*cosh(d*x + c)^2 + (a^4 + 2*a^2*b^2 + b^4)*d^3)*sin
h(d*x + c)^2 + 4*((a^4 + 2*a^2*b^2 + b^4)*d^3*cosh(d*x + c)^3 + (a^4 + 2*a^2*b^2 + b^4)*d^3*cosh(d*x + c))*sin
h(d*x + c))

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [F]  time = 1.07, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x +e \right )^{2} \mathrm {sech}\left (d x +c \right ) \left (\tanh ^{2}\left (d x +c \right )\right )}{a +b \sinh \left (d x +c \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {tanh}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^2}{\mathrm {cosh}\left (c+d\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e + f x\right )^{2} \tanh ^{2}{\left (c + d x \right )} \operatorname {sech}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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