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

Optimal. Leaf size=1185 \[ -\frac {(e+f x)^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) b^4}{a \left (a^2+b^2\right )^2 d}-\frac {(e+f x)^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) b^4}{a \left (a^2+b^2\right )^2 d}+\frac {(e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) b^4}{a \left (a^2+b^2\right )^2 d}-\frac {2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^4}{a \left (a^2+b^2\right )^2 d^2}-\frac {2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^4}{a \left (a^2+b^2\right )^2 d^2}+\frac {f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right ) b^4}{a \left (a^2+b^2\right )^2 d^2}+\frac {2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^4}{a \left (a^2+b^2\right )^2 d^3}+\frac {2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^4}{a \left (a^2+b^2\right )^2 d^3}-\frac {f^2 \text {Li}_3\left (-e^{2 (c+d x)}\right ) b^4}{2 a \left (a^2+b^2\right )^2 d^3}-\frac {2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right ) b^3}{\left (a^2+b^2\right )^2 d}+\frac {2 i f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right ) b^3}{\left (a^2+b^2\right )^2 d^2}-\frac {2 i f (e+f x) \text {Li}_2\left (i e^{c+d x}\right ) b^3}{\left (a^2+b^2\right )^2 d^2}-\frac {2 i f^2 \text {Li}_3\left (-i e^{c+d x}\right ) b^3}{\left (a^2+b^2\right )^2 d^3}+\frac {2 i f^2 \text {Li}_3\left (i e^{c+d x}\right ) b^3}{\left (a^2+b^2\right )^2 d^3}-\frac {(e+f x)^2 \text {sech}^2(c+d x) b^2}{2 a \left (a^2+b^2\right ) d}-\frac {f^2 \log (\cosh (c+d x)) b^2}{a \left (a^2+b^2\right ) d^3}+\frac {f (e+f x) \tanh (c+d x) b^2}{a \left (a^2+b^2\right ) d^2}-\frac {(e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right ) b}{\left (a^2+b^2\right ) d}+\frac {f^2 \tan ^{-1}(\sinh (c+d x)) b}{\left (a^2+b^2\right ) d^3}+\frac {i f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right ) b}{\left (a^2+b^2\right ) d^2}-\frac {i f (e+f x) \text {Li}_2\left (i e^{c+d x}\right ) b}{\left (a^2+b^2\right ) d^2}-\frac {i f^2 \text {Li}_3\left (-i e^{c+d x}\right ) b}{\left (a^2+b^2\right ) d^3}+\frac {i f^2 \text {Li}_3\left (i e^{c+d x}\right ) b}{\left (a^2+b^2\right ) d^3}-\frac {f (e+f x) \text {sech}(c+d x) b}{\left (a^2+b^2\right ) d^2}-\frac {(e+f x)^2 \text {sech}(c+d x) \tanh (c+d x) b}{2 \left (a^2+b^2\right ) d}+\frac {f^2 x^2}{2 a d}-\frac {(e+f x)^2 \tanh ^2(c+d x)}{2 a d}+\frac {e f x}{a d}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}+\frac {f^2 \log (\cosh (c+d x))}{a d^3}-\frac {f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a d^2}+\frac {f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right )}{a d^2}+\frac {f^2 \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a d^3}-\frac {f^2 \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a d^3}-\frac {f (e+f x) \tanh (c+d x)}{a d^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 2.22, antiderivative size = 1185, normalized size of antiderivative = 1.00, number of steps used = 57, number of rules used = 23, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.676, Rules used = {5589, 2620, 14, 5462, 6741, 12, 6742, 2551, 4182, 2531, 2282, 6589, 3720, 3475, 5573, 5561, 2190, 4180, 3718, 4186, 3770, 5451, 4184} \[ -\frac {(e+f x)^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) b^4}{a \left (a^2+b^2\right )^2 d}-\frac {(e+f x)^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) b^4}{a \left (a^2+b^2\right )^2 d}+\frac {(e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) b^4}{a \left (a^2+b^2\right )^2 d}-\frac {2 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^4}{a \left (a^2+b^2\right )^2 d^2}-\frac {2 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^4}{a \left (a^2+b^2\right )^2 d^2}+\frac {f (e+f x) \text {PolyLog}\left (2,-e^{2 (c+d x)}\right ) b^4}{a \left (a^2+b^2\right )^2 d^2}+\frac {2 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^4}{a \left (a^2+b^2\right )^2 d^3}+\frac {2 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^4}{a \left (a^2+b^2\right )^2 d^3}-\frac {f^2 \text {PolyLog}\left (3,-e^{2 (c+d x)}\right ) b^4}{2 a \left (a^2+b^2\right )^2 d^3}-\frac {2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right ) b^3}{\left (a^2+b^2\right )^2 d}+\frac {2 i f (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right ) b^3}{\left (a^2+b^2\right )^2 d^2}-\frac {2 i f (e+f x) \text {PolyLog}\left (2,i e^{c+d x}\right ) b^3}{\left (a^2+b^2\right )^2 d^2}-\frac {2 i f^2 \text {PolyLog}\left (3,-i e^{c+d x}\right ) b^3}{\left (a^2+b^2\right )^2 d^3}+\frac {2 i f^2 \text {PolyLog}\left (3,i e^{c+d x}\right ) b^3}{\left (a^2+b^2\right )^2 d^3}-\frac {(e+f x)^2 \text {sech}^2(c+d x) b^2}{2 a \left (a^2+b^2\right ) d}-\frac {f^2 \log (\cosh (c+d x)) b^2}{a \left (a^2+b^2\right ) d^3}+\frac {f (e+f x) \tanh (c+d x) b^2}{a \left (a^2+b^2\right ) d^2}-\frac {(e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right ) b}{\left (a^2+b^2\right ) d}+\frac {f^2 \tan ^{-1}(\sinh (c+d x)) b}{\left (a^2+b^2\right ) d^3}+\frac {i f (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right ) b}{\left (a^2+b^2\right ) d^2}-\frac {i f (e+f x) \text {PolyLog}\left (2,i e^{c+d x}\right ) b}{\left (a^2+b^2\right ) d^2}-\frac {i f^2 \text {PolyLog}\left (3,-i e^{c+d x}\right ) b}{\left (a^2+b^2\right ) d^3}+\frac {i f^2 \text {PolyLog}\left (3,i e^{c+d x}\right ) b}{\left (a^2+b^2\right ) d^3}-\frac {f (e+f x) \text {sech}(c+d x) b}{\left (a^2+b^2\right ) d^2}-\frac {(e+f x)^2 \text {sech}(c+d x) \tanh (c+d x) b}{2 \left (a^2+b^2\right ) d}+\frac {f^2 x^2}{2 a d}-\frac {(e+f x)^2 \tanh ^2(c+d x)}{2 a d}+\frac {e f x}{a d}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}+\frac {f^2 \log (\cosh (c+d x))}{a d^3}-\frac {f (e+f x) \text {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{a d^2}+\frac {f (e+f x) \text {PolyLog}\left (2,e^{2 c+2 d x}\right )}{a d^2}+\frac {f^2 \text {PolyLog}\left (3,-e^{2 c+2 d x}\right )}{2 a d^3}-\frac {f^2 \text {PolyLog}\left (3,e^{2 c+2 d x}\right )}{2 a d^3}-\frac {f (e+f x) \tanh (c+d x)}{a d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 2551

Int[Log[u_]*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Log[u])/(b*(m + 1)), x] - Dist[1/
(b*(m + 1)), Int[SimplifyIntegrand[((a + b*x)^(m + 1)*D[u, x])/u, x], x], x] /; FreeQ[{a, b, m}, x] && Inverse
FunctionFreeQ[u, x] && NeQ[m, -1]

Rule 2620

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

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 3720

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(c + d*x)^m*(b*Tan[e
+ f*x])^(n - 1))/(f*(n - 1)), x] + (-Dist[(b*d*m)/(f*(n - 1)), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && GtQ[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 4182

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

Rule 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 5462

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

Rule 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 5589

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

Rule 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 6741

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

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 {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^2 \text {csch}(c+d x) \text {sech}^3(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^2 \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=\frac {(e+f x)^2 \log (\tanh (c+d x))}{a d}-\frac {(e+f x)^2 \tanh ^2(c+d x)}{2 a d}-\frac {b \int (e+f x)^2 \text {sech}^3(c+d x) (a-b \sinh (c+d x)) \, dx}{a \left (a^2+b^2\right )}-\frac {b^3 \int \frac {(e+f x)^2 \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a \left (a^2+b^2\right )}-\frac {(2 f) \int (e+f x) \left (\frac {\log (\tanh (c+d x))}{d}-\frac {\tanh ^2(c+d x)}{2 d}\right ) \, dx}{a}\\ &=\frac {(e+f x)^2 \log (\tanh (c+d x))}{a d}-\frac {(e+f x)^2 \tanh ^2(c+d x)}{2 a d}-\frac {b^3 \int (e+f x)^2 \text {sech}(c+d x) (a-b \sinh (c+d x)) \, dx}{a \left (a^2+b^2\right )^2}-\frac {b^5 \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{a \left (a^2+b^2\right )^2}-\frac {b \int \left (a (e+f x)^2 \text {sech}^3(c+d x)-b (e+f x)^2 \text {sech}^2(c+d x) \tanh (c+d x)\right ) \, dx}{a \left (a^2+b^2\right )}-\frac {(2 f) \int \frac {(e+f x) \left (2 \log (\tanh (c+d x))-\tanh ^2(c+d x)\right )}{2 d} \, dx}{a}\\ &=\frac {b^4 (e+f x)^3}{3 a \left (a^2+b^2\right )^2 f}+\frac {(e+f x)^2 \log (\tanh (c+d x))}{a d}-\frac {(e+f x)^2 \tanh ^2(c+d x)}{2 a d}-\frac {b^3 \int \left (a (e+f x)^2 \text {sech}(c+d x)-b (e+f x)^2 \tanh (c+d x)\right ) \, dx}{a \left (a^2+b^2\right )^2}-\frac {b^5 \int \frac {e^{c+d x} (e+f x)^2}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a \left (a^2+b^2\right )^2}-\frac {b^5 \int \frac {e^{c+d x} (e+f x)^2}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a \left (a^2+b^2\right )^2}-\frac {b \int (e+f x)^2 \text {sech}^3(c+d x) \, dx}{a^2+b^2}+\frac {b^2 \int (e+f x)^2 \text {sech}^2(c+d x) \tanh (c+d x) \, dx}{a \left (a^2+b^2\right )}-\frac {f \int (e+f x) \left (2 \log (\tanh (c+d x))-\tanh ^2(c+d x)\right ) \, dx}{a d}\\ &=\frac {b^4 (e+f x)^3}{3 a \left (a^2+b^2\right )^2 f}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {(e+f x)^2 \log (\tanh (c+d x))}{a d}-\frac {b f (e+f x) \text {sech}(c+d x)}{\left (a^2+b^2\right ) d^2}-\frac {b^2 (e+f x)^2 \text {sech}^2(c+d x)}{2 a \left (a^2+b^2\right ) d}-\frac {b (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \tanh ^2(c+d x)}{2 a d}-\frac {b^3 \int (e+f x)^2 \text {sech}(c+d x) \, dx}{\left (a^2+b^2\right )^2}+\frac {b^4 \int (e+f x)^2 \tanh (c+d x) \, dx}{a \left (a^2+b^2\right )^2}-\frac {b \int (e+f x)^2 \text {sech}(c+d x) \, dx}{2 \left (a^2+b^2\right )}-\frac {f \int \left (2 (e+f x) \log (\tanh (c+d x))-(e+f x) \tanh ^2(c+d x)\right ) \, dx}{a d}+\frac {\left (2 b^4 f\right ) \int (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right )^2 d}+\frac {\left (2 b^4 f\right ) \int (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right )^2 d}+\frac {\left (b^2 f\right ) \int (e+f x) \text {sech}^2(c+d x) \, dx}{a \left (a^2+b^2\right ) d}+\frac {\left (b f^2\right ) \int \text {sech}(c+d x) \, dx}{\left (a^2+b^2\right ) d^2}\\ &=-\frac {2 b^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac {b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}+\frac {b f^2 \tan ^{-1}(\sinh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {(e+f x)^2 \log (\tanh (c+d x))}{a d}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {b f (e+f x) \text {sech}(c+d x)}{\left (a^2+b^2\right ) d^2}-\frac {b^2 (e+f x)^2 \text {sech}^2(c+d x)}{2 a \left (a^2+b^2\right ) d}+\frac {b^2 f (e+f x) \tanh (c+d x)}{a \left (a^2+b^2\right ) d^2}-\frac {b (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \tanh ^2(c+d x)}{2 a d}+\frac {\left (2 b^4\right ) \int \frac {e^{2 (c+d x)} (e+f x)^2}{1+e^{2 (c+d x)}} \, dx}{a \left (a^2+b^2\right )^2}+\frac {f \int (e+f x) \tanh ^2(c+d x) \, dx}{a d}-\frac {(2 f) \int (e+f x) \log (\tanh (c+d x)) \, dx}{a d}+\frac {\left (2 i b^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 b^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 {(i b f) \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d}-\frac {(i b f) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d}+\frac {\left (2 b^4 f^2\right ) \int \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right )^2 d^2}+\frac {\left (2 b^4 f^2\right ) \int \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right )^2 d^2}-\frac {\left (b^2 f^2\right ) \int \tanh (c+d x) \, dx}{a \left (a^2+b^2\right ) d^2}\\ &=-\frac {2 b^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac {b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}+\frac {b f^2 \tan ^{-1}(\sinh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {b^4 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d}-\frac {b^2 f^2 \log (\cosh (c+d x))}{a \left (a^2+b^2\right ) d^3}+\frac {2 i b^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 b f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 i b^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 b f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {b f (e+f x) \text {sech}(c+d x)}{\left (a^2+b^2\right ) d^2}-\frac {b^2 (e+f x)^2 \text {sech}^2(c+d x)}{2 a \left (a^2+b^2\right ) d}-\frac {f (e+f x) \tanh (c+d x)}{a d^2}+\frac {b^2 f (e+f x) \tanh (c+d x)}{a \left (a^2+b^2\right ) d^2}-\frac {b (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \tanh ^2(c+d x)}{2 a d}+\frac {\int 2 d (e+f x)^2 \text {csch}(2 c+2 d x) \, dx}{a d}+\frac {f \int (e+f x) \, dx}{a d}-\frac {\left (2 b^4 f\right ) \int (e+f x) \log \left (1+e^{2 (c+d x)}\right ) \, dx}{a \left (a^2+b^2\right )^2 d}+\frac {\left (2 b^4 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 )}{a \left (a^2+b^2\right )^2 d^3}+\frac {\left (2 b^4 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 )}{a \left (a^2+b^2\right )^2 d^3}+\frac {f^2 \int \tanh (c+d x) \, dx}{a d^2}-\frac {\left (2 i b^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 b^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 b f^2\right ) \int \text {Li}_2\left (-i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d^2}+\frac {\left (i b f^2\right ) \int \text {Li}_2\left (i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d^2}\\ &=\frac {e f x}{a d}+\frac {f^2 x^2}{2 a d}-\frac {2 b^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac {b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}+\frac {b f^2 \tan ^{-1}(\sinh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {b^4 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {f^2 \log (\cosh (c+d x))}{a d^3}-\frac {b^2 f^2 \log (\cosh (c+d x))}{a \left (a^2+b^2\right ) d^3}+\frac {2 i b^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 b f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 i b^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 b f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}+\frac {b^4 f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d^2}+\frac {2 b^4 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^3}+\frac {2 b^4 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^3}-\frac {b f (e+f x) \text {sech}(c+d x)}{\left (a^2+b^2\right ) d^2}-\frac {b^2 (e+f x)^2 \text {sech}^2(c+d x)}{2 a \left (a^2+b^2\right ) d}-\frac {f (e+f x) \tanh (c+d x)}{a d^2}+\frac {b^2 f (e+f x) \tanh (c+d x)}{a \left (a^2+b^2\right ) d^2}-\frac {b (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \tanh ^2(c+d x)}{2 a d}+\frac {2 \int (e+f x)^2 \text {csch}(2 c+2 d x) \, dx}{a}-\frac {\left (2 i b^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 b^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 b 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 ) d^3}+\frac {\left (i b 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 ) d^3}-\frac {\left (b^4 f^2\right ) \int \text {Li}_2\left (-e^{2 (c+d x)}\right ) \, dx}{a \left (a^2+b^2\right )^2 d^2}\\ &=\frac {e f x}{a d}+\frac {f^2 x^2}{2 a d}-\frac {2 b^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac {b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}+\frac {b f^2 \tan ^{-1}(\sinh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {b^4 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {f^2 \log (\cosh (c+d x))}{a d^3}-\frac {b^2 f^2 \log (\cosh (c+d x))}{a \left (a^2+b^2\right ) d^3}+\frac {2 i b^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 b f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 i b^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 b f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}+\frac {b^4 f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {2 i b^3 f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {i b f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 i b^3 f^2 \text {Li}_3\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {i b f^2 \text {Li}_3\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 b^4 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^3}+\frac {2 b^4 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^3}-\frac {b f (e+f x) \text {sech}(c+d x)}{\left (a^2+b^2\right ) d^2}-\frac {b^2 (e+f x)^2 \text {sech}^2(c+d x)}{2 a \left (a^2+b^2\right ) d}-\frac {f (e+f x) \tanh (c+d x)}{a d^2}+\frac {b^2 f (e+f x) \tanh (c+d x)}{a \left (a^2+b^2\right ) d^2}-\frac {b (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \tanh ^2(c+d x)}{2 a d}-\frac {(2 f) \int (e+f x) \log \left (1-e^{2 c+2 d x}\right ) \, dx}{a d}+\frac {(2 f) \int (e+f x) \log \left (1+e^{2 c+2 d x}\right ) \, dx}{a d}-\frac {\left (b^4 f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 a \left (a^2+b^2\right )^2 d^3}\\ &=\frac {e f x}{a d}+\frac {f^2 x^2}{2 a d}-\frac {2 b^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac {b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}+\frac {b f^2 \tan ^{-1}(\sinh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {b^4 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {f^2 \log (\cosh (c+d x))}{a d^3}-\frac {b^2 f^2 \log (\cosh (c+d x))}{a \left (a^2+b^2\right ) d^3}+\frac {2 i b^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 b f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 i b^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 b f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}+\frac {b^4 f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a d^2}+\frac {f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right )}{a d^2}-\frac {2 i b^3 f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {i b f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 i b^3 f^2 \text {Li}_3\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {i b f^2 \text {Li}_3\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 b^4 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^3}+\frac {2 b^4 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^3}-\frac {b^4 f^2 \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 a \left (a^2+b^2\right )^2 d^3}-\frac {b f (e+f x) \text {sech}(c+d x)}{\left (a^2+b^2\right ) d^2}-\frac {b^2 (e+f x)^2 \text {sech}^2(c+d x)}{2 a \left (a^2+b^2\right ) d}-\frac {f (e+f x) \tanh (c+d x)}{a d^2}+\frac {b^2 f (e+f x) \tanh (c+d x)}{a \left (a^2+b^2\right ) d^2}-\frac {b (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \tanh ^2(c+d x)}{2 a d}+\frac {f^2 \int \text {Li}_2\left (-e^{2 c+2 d x}\right ) \, dx}{a d^2}-\frac {f^2 \int \text {Li}_2\left (e^{2 c+2 d x}\right ) \, dx}{a d^2}\\ &=\frac {e f x}{a d}+\frac {f^2 x^2}{2 a d}-\frac {2 b^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac {b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}+\frac {b f^2 \tan ^{-1}(\sinh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {b^4 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {f^2 \log (\cosh (c+d x))}{a d^3}-\frac {b^2 f^2 \log (\cosh (c+d x))}{a \left (a^2+b^2\right ) d^3}+\frac {2 i b^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 b f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 i b^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 b f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}+\frac {b^4 f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a d^2}+\frac {f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right )}{a d^2}-\frac {2 i b^3 f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {i b f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 i b^3 f^2 \text {Li}_3\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {i b f^2 \text {Li}_3\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 b^4 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^3}+\frac {2 b^4 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^3}-\frac {b^4 f^2 \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 a \left (a^2+b^2\right )^2 d^3}-\frac {b f (e+f x) \text {sech}(c+d x)}{\left (a^2+b^2\right ) d^2}-\frac {b^2 (e+f x)^2 \text {sech}^2(c+d x)}{2 a \left (a^2+b^2\right ) d}-\frac {f (e+f x) \tanh (c+d x)}{a d^2}+\frac {b^2 f (e+f x) \tanh (c+d x)}{a \left (a^2+b^2\right ) d^2}-\frac {b (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \tanh ^2(c+d x)}{2 a d}+\frac {f^2 \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{2 a d^3}-\frac {f^2 \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{2 a d^3}\\ &=\frac {e f x}{a d}+\frac {f^2 x^2}{2 a d}-\frac {2 b^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac {b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}+\frac {b f^2 \tan ^{-1}(\sinh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {b^4 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {f^2 \log (\cosh (c+d x))}{a d^3}-\frac {b^2 f^2 \log (\cosh (c+d x))}{a \left (a^2+b^2\right ) d^3}+\frac {2 i b^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 b f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 i b^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 b f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}+\frac {b^4 f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a d^2}+\frac {f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right )}{a d^2}-\frac {2 i b^3 f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {i b f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 i b^3 f^2 \text {Li}_3\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {i b f^2 \text {Li}_3\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 b^4 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^3}+\frac {2 b^4 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^3}-\frac {b^4 f^2 \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 a \left (a^2+b^2\right )^2 d^3}+\frac {f^2 \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a d^3}-\frac {f^2 \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a d^3}-\frac {b f (e+f x) \text {sech}(c+d x)}{\left (a^2+b^2\right ) d^2}-\frac {b^2 (e+f x)^2 \text {sech}^2(c+d x)}{2 a \left (a^2+b^2\right ) d}-\frac {f (e+f x) \tanh (c+d x)}{a d^2}+\frac {b^2 f (e+f x) \tanh (c+d x)}{a \left (a^2+b^2\right ) d^2}-\frac {b (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \tanh ^2(c+d x)}{2 a d}\\ \end {align*}

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Mathematica [B]  time = 32.58, size = 3806, normalized size = 3.21 \[ \text {Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(E^(2*c)*((2*(e + f*x)^3)/(E^(2*c)*f) - (3*(1 - E^(-2*c))*(e + f*x)^2*Log[1 - E^(-c - d*x)])/d - (3*(1 -
E^(-2*c))*(e + f*x)^2*Log[1 + E^(-c - d*x)])/d + (6*(-1 + E^(2*c))*f*(d*(e + f*x)*PolyLog[2, -E^(-c - d*x)] +
f*PolyLog[3, -E^(-c - d*x)]))/(d^3*E^(2*c)) + (6*(-1 + E^(2*c))*f*(d*(e + f*x)*PolyLog[2, E^(-c - d*x)] + f*Po
lyLog[3, E^(-c - d*x)]))/(d^3*E^(2*c))))/(a*(-1 + E^(2*c))) - (-12*a^3*d^3*e^2*E^(2*c)*x - 24*a*b^2*d^3*e^2*E^
(2*c)*x + 12*a^3*d*E^(2*c)*f^2*x + 12*a*b^2*d*E^(2*c)*f^2*x - 12*a^3*d^3*e*E^(2*c)*f*x^2 - 24*a*b^2*d^3*e*E^(2
*c)*f*x^2 - 4*a^3*d^3*E^(2*c)*f^2*x^3 - 8*a*b^2*d^3*E^(2*c)*f^2*x^3 + 6*a^2*b*d^2*e^2*ArcTan[E^(c + d*x)] + 18
*b^3*d^2*e^2*ArcTan[E^(c + d*x)] + 6*a^2*b*d^2*e^2*E^(2*c)*ArcTan[E^(c + d*x)] + 18*b^3*d^2*e^2*E^(2*c)*ArcTan
[E^(c + d*x)] - 12*a^2*b*f^2*ArcTan[E^(c + d*x)] - 12*b^3*f^2*ArcTan[E^(c + d*x)] - 12*a^2*b*E^(2*c)*f^2*ArcTa
n[E^(c + d*x)] - 12*b^3*E^(2*c)*f^2*ArcTan[E^(c + d*x)] + (6*I)*a^2*b*d^2*e*f*x*Log[1 - I*E^(c + d*x)] + (18*I
)*b^3*d^2*e*f*x*Log[1 - I*E^(c + d*x)] + (6*I)*a^2*b*d^2*e*E^(2*c)*f*x*Log[1 - I*E^(c + d*x)] + (18*I)*b^3*d^2
*e*E^(2*c)*f*x*Log[1 - I*E^(c + d*x)] + (3*I)*a^2*b*d^2*f^2*x^2*Log[1 - I*E^(c + d*x)] + (9*I)*b^3*d^2*f^2*x^2
*Log[1 - I*E^(c + d*x)] + (3*I)*a^2*b*d^2*E^(2*c)*f^2*x^2*Log[1 - I*E^(c + d*x)] + (9*I)*b^3*d^2*E^(2*c)*f^2*x
^2*Log[1 - I*E^(c + d*x)] - (6*I)*a^2*b*d^2*e*f*x*Log[1 + I*E^(c + d*x)] - (18*I)*b^3*d^2*e*f*x*Log[1 + I*E^(c
+ d*x)] - (6*I)*a^2*b*d^2*e*E^(2*c)*f*x*Log[1 + I*E^(c + d*x)] - (18*I)*b^3*d^2*e*E^(2*c)*f*x*Log[1 + I*E^(c
+ d*x)] - (3*I)*a^2*b*d^2*f^2*x^2*Log[1 + I*E^(c + d*x)] - (9*I)*b^3*d^2*f^2*x^2*Log[1 + I*E^(c + d*x)] - (3*I
)*a^2*b*d^2*E^(2*c)*f^2*x^2*Log[1 + I*E^(c + d*x)] - (9*I)*b^3*d^2*E^(2*c)*f^2*x^2*Log[1 + I*E^(c + d*x)] + 6*
a^3*d^2*e^2*Log[1 + E^(2*(c + d*x))] + 12*a*b^2*d^2*e^2*Log[1 + E^(2*(c + d*x))] + 6*a^3*d^2*e^2*E^(2*c)*Log[1
+ E^(2*(c + d*x))] + 12*a*b^2*d^2*e^2*E^(2*c)*Log[1 + E^(2*(c + d*x))] - 6*a^3*f^2*Log[1 + E^(2*(c + d*x))] -
6*a*b^2*f^2*Log[1 + E^(2*(c + d*x))] - 6*a^3*E^(2*c)*f^2*Log[1 + E^(2*(c + d*x))] - 6*a*b^2*E^(2*c)*f^2*Log[1
+ E^(2*(c + d*x))] + 12*a^3*d^2*e*f*x*Log[1 + E^(2*(c + d*x))] + 24*a*b^2*d^2*e*f*x*Log[1 + E^(2*(c + d*x))]
+ 12*a^3*d^2*e*E^(2*c)*f*x*Log[1 + E^(2*(c + d*x))] + 24*a*b^2*d^2*e*E^(2*c)*f*x*Log[1 + E^(2*(c + d*x))] + 6*
a^3*d^2*f^2*x^2*Log[1 + E^(2*(c + d*x))] + 12*a*b^2*d^2*f^2*x^2*Log[1 + E^(2*(c + d*x))] + 6*a^3*d^2*E^(2*c)*f
^2*x^2*Log[1 + E^(2*(c + d*x))] + 12*a*b^2*d^2*E^(2*c)*f^2*x^2*Log[1 + E^(2*(c + d*x))] - (6*I)*b*(a^2 + 3*b^2
)*d*(1 + E^(2*c))*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)] + (6*I)*b*(a^2 + 3*b^2)*d*(1 + E^(2*c))*f*(e + f*x)
*PolyLog[2, I*E^(c + d*x)] + 6*a^3*d*e*f*PolyLog[2, -E^(2*(c + d*x))] + 12*a*b^2*d*e*f*PolyLog[2, -E^(2*(c + d
*x))] + 6*a^3*d*e*E^(2*c)*f*PolyLog[2, -E^(2*(c + d*x))] + 12*a*b^2*d*e*E^(2*c)*f*PolyLog[2, -E^(2*(c + d*x))]
+ 6*a^3*d*f^2*x*PolyLog[2, -E^(2*(c + d*x))] + 12*a*b^2*d*f^2*x*PolyLog[2, -E^(2*(c + d*x))] + 6*a^3*d*E^(2*c
)*f^2*x*PolyLog[2, -E^(2*(c + d*x))] + 12*a*b^2*d*E^(2*c)*f^2*x*PolyLog[2, -E^(2*(c + d*x))] + (6*I)*a^2*b*f^2
*PolyLog[3, (-I)*E^(c + d*x)] + (18*I)*b^3*f^2*PolyLog[3, (-I)*E^(c + d*x)] + (6*I)*a^2*b*E^(2*c)*f^2*PolyLog[
3, (-I)*E^(c + d*x)] + (18*I)*b^3*E^(2*c)*f^2*PolyLog[3, (-I)*E^(c + d*x)] - (6*I)*a^2*b*f^2*PolyLog[3, I*E^(c
+ d*x)] - (18*I)*b^3*f^2*PolyLog[3, I*E^(c + d*x)] - (6*I)*a^2*b*E^(2*c)*f^2*PolyLog[3, I*E^(c + d*x)] - (18*
I)*b^3*E^(2*c)*f^2*PolyLog[3, I*E^(c + d*x)] - 3*a^3*f^2*PolyLog[3, -E^(2*(c + d*x))] - 6*a*b^2*f^2*PolyLog[3,
-E^(2*(c + d*x))] - 3*a^3*E^(2*c)*f^2*PolyLog[3, -E^(2*(c + d*x))] - 6*a*b^2*E^(2*c)*f^2*PolyLog[3, -E^(2*(c
+ d*x))])/(6*(a^2 + b^2)^2*d^3*(1 + E^(2*c))) + (b^4*(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*Log[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*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)]))]))/
(3*a*(a^2 + b^2)^2*d^3*(-1 + E^(2*c))) + (Csch[c]*Sech[c]*Sech[c + d*x]^2*(-6*a^3*e*f - 6*a*b^2*e*f + 12*a^3*d
^2*e^2*x + 24*a*b^2*d^2*e^2*x - 6*a^3*f^2*x - 6*a*b^2*f^2*x + 12*a^3*d^2*e*f*x^2 + 24*a*b^2*d^2*e*f*x^2 + 4*a^
3*d^2*f^2*x^3 + 8*a*b^2*d^2*f^2*x^3 + 6*a^3*e*f*Cosh[2*c] + 6*a*b^2*e*f*Cosh[2*c] + 6*a^3*f^2*x*Cosh[2*c] + 6*
a*b^2*f^2*x*Cosh[2*c] + 6*a^3*e*f*Cosh[2*d*x] + 6*a*b^2*e*f*Cosh[2*d*x] + 6*a^3*f^2*x*Cosh[2*d*x] + 6*a*b^2*f^
2*x*Cosh[2*d*x] + 3*a^2*b*d*e^2*Cosh[c - d*x] + 3*b^3*d*e^2*Cosh[c - d*x] + 6*a^2*b*d*e*f*x*Cosh[c - d*x] + 6*
b^3*d*e*f*x*Cosh[c - d*x] + 3*a^2*b*d*f^2*x^2*Cosh[c - d*x] + 3*b^3*d*f^2*x^2*Cosh[c - d*x] - 3*a^2*b*d*e^2*Co
sh[3*c + d*x] - 3*b^3*d*e^2*Cosh[3*c + d*x] - 6*a^2*b*d*e*f*x*Cosh[3*c + d*x] - 6*b^3*d*e*f*x*Cosh[3*c + d*x]
- 3*a^2*b*d*f^2*x^2*Cosh[3*c + d*x] - 3*b^3*d*f^2*x^2*Cosh[3*c + d*x] - 6*a^3*e*f*Cosh[2*c + 2*d*x] - 6*a*b^2*
e*f*Cosh[2*c + 2*d*x] + 12*a^3*d^2*e^2*x*Cosh[2*c + 2*d*x] + 24*a*b^2*d^2*e^2*x*Cosh[2*c + 2*d*x] - 6*a^3*f^2*
x*Cosh[2*c + 2*d*x] - 6*a*b^2*f^2*x*Cosh[2*c + 2*d*x] + 12*a^3*d^2*e*f*x^2*Cosh[2*c + 2*d*x] + 24*a*b^2*d^2*e*
f*x^2*Cosh[2*c + 2*d*x] + 4*a^3*d^2*f^2*x^3*Cosh[2*c + 2*d*x] + 8*a*b^2*d^2*f^2*x^3*Cosh[2*c + 2*d*x] + 6*a^3*
d*e^2*Sinh[2*c] + 6*a*b^2*d*e^2*Sinh[2*c] + 12*a^3*d*e*f*x*Sinh[2*c] + 12*a*b^2*d*e*f*x*Sinh[2*c] + 6*a^3*d*f^
2*x^2*Sinh[2*c] + 6*a*b^2*d*f^2*x^2*Sinh[2*c] - 6*a^2*b*e*f*Sinh[c - d*x] - 6*b^3*e*f*Sinh[c - d*x] - 6*a^2*b*
f^2*x*Sinh[c - d*x] - 6*b^3*f^2*x*Sinh[c - d*x] - 6*a^2*b*e*f*Sinh[3*c + d*x] - 6*b^3*e*f*Sinh[3*c + d*x] - 6*
a^2*b*f^2*x*Sinh[3*c + d*x] - 6*b^3*f^2*x*Sinh[3*c + d*x]))/(24*(a^2 + b^2)^2*d^2)

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fricas [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*csch(d*x+c)*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

Timed out

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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*csch(d*x+c)*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 2.93, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x +e \right )^{2} \mathrm {csch}\left (d x +c \right ) \mathrm {sech}\left (d x +c \right )^{3}}{a +b \sinh \left (d x +c \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

-a^2*b*d^2*f^2*integrate(x^2*e^(d*x + c)/(a^4*d^2*e^(2*d*x + 2*c) + 2*a^2*b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^
(2*d*x + 2*c) + a^4*d^2 + 2*a^2*b^2*d^2 + b^4*d^2), x) - 3*b^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)
+ 2*a^3*d^2*f^2*integrate(x^2/(a^4*d^2*e^(2*d*x + 2*c) + 2*a^2*b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*d*x + 2
*c) + a^4*d^2 + 2*a^2*b^2*d^2 + b^4*d^2), x) + 4*a*b^2*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^2*b*d^2*e*f*i
ntegrate(x*e^(d*x + c)/(a^4*d^2*e^(2*d*x + 2*c) + 2*a^2*b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*d*x + 2*c) + a^
4*d^2 + 2*a^2*b^2*d^2 + b^4*d^2), x) - 6*b^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) + 4*a^3*d^2*e*f*int
egrate(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) + 8*a*b^2*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^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*b^2*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^4*log(-2*a*e^(-d*x - c) + b*e^(-2*
d*x - 2*c) - b)/((a^5 + 2*a^3*b^2 + a*b^4)*d) - (a^2*b + 3*b^3)*arctan(e^(-d*x - c))/((a^4 + 2*a^2*b^2 + b^4)*
d) + (a^3 + 2*a*b^2)*log(e^(-2*d*x - 2*c) + 1)/((a^4 + 2*a^2*b^2 + b^4)*d) + (b*e^(-d*x - c) - 2*a*e^(-2*d*x -
2*c) - b*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) -
log(e^(-d*x - c) + 1)/(a*d) - log(e^(-d*x - c) - 1)/(a*d))*e^2 + 2*a^2*b*f^2*arctan(e^(d*x + c))/((a^4 + 2*a^2
*b^2 + b^4)*d^3) + 2*b^3*f^2*arctan(e^(d*x + c))/((a^4 + 2*a^2*b^2 + b^4)*d^3) + (2*a*f^2*x + 2*a*e*f - (b*d*f
^2*x^2*e^(3*c) + 2*b*e*f*e^(3*c) + 2*(d*e*f + f^2)*b*x*e^(3*c))*e^(3*d*x) + 2*(a*d*f^2*x^2*e^(2*c) + a*e*f*e^(
2*c) + (2*d*e*f + f^2)*a*x*e^(2*c))*e^(2*d*x) + (b*d*f^2*x^2*e^c - 2*b*e*f*e^c + 2*(d*e*f - f^2)*b*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)) + 2*(d*x*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))*e*f/(a*d^2) + 2*(d*x*log(-e^(d*x + c) + 1) +
dilog(e^(d*x + c)))*e*f/(a*d^2) + (d^2*x^2*log(e^(d*x + c) + 1) + 2*d*x*dilog(-e^(d*x + c)) - 2*polylog(3, -e
^(d*x + c)))*f^2/(a*d^3) + (d^2*x^2*log(-e^(d*x + c) + 1) + 2*d*x*dilog(e^(d*x + c)) - 2*polylog(3, e^(d*x + c
)))*f^2/(a*d^3) - 2/3*(d^3*f^2*x^3 + 3*d^3*e*f*x^2)/(a*d^3) + integrate(2*(b^5*f^2*x^2 + 2*b^5*e*f*x - (a*b^4*
f^2*x^2*e^c + 2*a*b^4*e*f*x*e^c)*e^(d*x))/(a^5*b + 2*a^3*b^3 + a*b^5 - (a^5*b*e^(2*c) + 2*a^3*b^3*e^(2*c) + a*
b^5*e^(2*c))*e^(2*d*x) - 2*(a^6*e^c + 2*a^4*b^2*e^c + a^2*b^4*e^c)*e^(d*x)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (e+f\,x\right )}^2}{{\mathrm {cosh}\left (c+d\,x\right )}^3\,\mathrm {sinh}\left (c+d\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [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*csch(d*x+c)*sech(d*x+c)**3/(a+b*sinh(d*x+c)),x)

[Out]

Timed out

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