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

Optimal. Leaf size=648 \[ -\frac{2 a b f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{d^2 \left (a^2+b^2\right )^{3/2}}+\frac{2 a b f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{d^2 \left (a^2+b^2\right )^{3/2}}+\frac{a^2 f^2 \text{PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b d^3 \left (a^2+b^2\right )}-\frac{2 i a f^2 \text{PolyLog}\left (2,-i e^{c+d x}\right )}{d^3 \left (a^2+b^2\right )}+\frac{2 i a f^2 \text{PolyLog}\left (2,i e^{c+d x}\right )}{d^3 \left (a^2+b^2\right )}+\frac{2 a b f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{d^3 \left (a^2+b^2\right )^{3/2}}-\frac{2 a b f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{d^3 \left (a^2+b^2\right )^{3/2}}-\frac{f^2 \text{PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b d^3}+\frac{2 a^2 f (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{b d^2 \left (a^2+b^2\right )}+\frac{4 a f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )}-\frac{a b (e+f x)^2 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{d \left (a^2+b^2\right )^{3/2}}+\frac{a b (e+f x)^2 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{d \left (a^2+b^2\right )^{3/2}}-\frac{a^2 (e+f x)^2 \tanh (c+d x)}{b d \left (a^2+b^2\right )}-\frac{a (e+f x)^2 \text{sech}(c+d x)}{d \left (a^2+b^2\right )}-\frac{a^2 (e+f x)^2}{b d \left (a^2+b^2\right )}-\frac{2 f (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{b d^2}+\frac{(e+f x)^2 \tanh (c+d x)}{b d}+\frac{(e+f x)^2}{b d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.32148, antiderivative size = 648, normalized size of antiderivative = 1., number of steps used = 30, number of rules used = 15, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.469, Rules used = {5583, 4184, 3718, 2190, 2279, 2391, 5573, 3322, 2264, 2531, 2282, 6589, 6742, 5451, 4180} \[ -\frac{2 a b f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{d^2 \left (a^2+b^2\right )^{3/2}}+\frac{2 a b f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{d^2 \left (a^2+b^2\right )^{3/2}}+\frac{a^2 f^2 \text{PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b d^3 \left (a^2+b^2\right )}-\frac{2 i a f^2 \text{PolyLog}\left (2,-i e^{c+d x}\right )}{d^3 \left (a^2+b^2\right )}+\frac{2 i a f^2 \text{PolyLog}\left (2,i e^{c+d x}\right )}{d^3 \left (a^2+b^2\right )}+\frac{2 a b f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{d^3 \left (a^2+b^2\right )^{3/2}}-\frac{2 a b f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{d^3 \left (a^2+b^2\right )^{3/2}}-\frac{f^2 \text{PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b d^3}+\frac{2 a^2 f (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{b d^2 \left (a^2+b^2\right )}+\frac{4 a f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )}-\frac{a b (e+f x)^2 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{d \left (a^2+b^2\right )^{3/2}}+\frac{a b (e+f x)^2 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{d \left (a^2+b^2\right )^{3/2}}-\frac{a^2 (e+f x)^2 \tanh (c+d x)}{b d \left (a^2+b^2\right )}-\frac{a (e+f x)^2 \text{sech}(c+d x)}{d \left (a^2+b^2\right )}-\frac{a^2 (e+f x)^2}{b d \left (a^2+b^2\right )}-\frac{2 f (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{b d^2}+\frac{(e+f x)^2 \tanh (c+d x)}{b d}+\frac{(e+f x)^2}{b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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 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 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 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 3322

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

Rule 2264

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

Rule 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 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 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]]

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 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]

Rubi steps

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

Mathematica [A]  time = 8.18767, size = 906, normalized size = 1.4 \[ \frac{2 a \left (-\frac{2 \tan ^{-1}\left (\frac{\sinh (c)+\cosh (c) \tanh \left (\frac{d x}{2}\right )}{\sqrt{\cosh ^2(c)-\sinh ^2(c)}}\right ) \tanh ^{-1}(\coth (c))}{\sqrt{\cosh ^2(c)-\sinh ^2(c)}}-\frac{i \text{csch}(c) \left (i \left (d x+\tanh ^{-1}(\coth (c))\right ) \left (\log \left (1-e^{-d x-\tanh ^{-1}(\coth (c))}\right )-\log \left (1+e^{-d x-\tanh ^{-1}(\coth (c))}\right )\right )+i \left (\text{PolyLog}\left (2,-e^{-d x-\tanh ^{-1}(\coth (c))}\right )-\text{PolyLog}\left (2,e^{-d x-\tanh ^{-1}(\coth (c))}\right )\right )\right )}{\sqrt{1-\coth ^2(c)}}\right ) f^2}{\left (a^2+b^2\right ) d^3}+\frac{b \text{csch}(c) \left (\frac{i \coth (c) \left (-d x \left (2 i \tanh ^{-1}(\coth (c))-\pi \right )-\pi \log \left (1+e^{2 d x}\right )-2 \left (i d x+i \tanh ^{-1}(\coth (c))\right ) \log \left (1-e^{2 i \left (i d x+i \tanh ^{-1}(\coth (c))\right )}\right )+\pi \log (\cosh (d x))+2 i \tanh ^{-1}(\coth (c)) \log \left (i \sinh \left (d x+\tanh ^{-1}(\coth (c))\right )\right )+i \text{PolyLog}\left (2,e^{2 i \left (i d x+i \tanh ^{-1}(\coth (c))\right )}\right )\right )}{\sqrt{1-\coth ^2(c)}}-d^2 e^{-\tanh ^{-1}(\coth (c))} x^2\right ) \text{sech}(c) f^2}{\left (a^2+b^2\right ) d^3 \sqrt{\text{csch}^2(c) \left (\sinh ^2(c)-\cosh ^2(c)\right )}}+\frac{4 a e \tan ^{-1}\left (\frac{\sinh (c)+\cosh (c) \tanh \left (\frac{d x}{2}\right )}{\sqrt{\cosh ^2(c)-\sinh ^2(c)}}\right ) f}{\left (a^2+b^2\right ) d^2 \sqrt{\cosh ^2(c)-\sinh ^2(c)}}-\frac{2 b e \text{sech}(c) (\cosh (c) \log (\cosh (c) \cosh (d x)+\sinh (c) \sinh (d x))-d x \sinh (c)) f}{\left (a^2+b^2\right ) d^2 \left (\cosh ^2(c)-\sinh ^2(c)\right )}+\frac{a b \left (2 e^2 \tanh ^{-1}\left (\frac{a+b e^{c+d x}}{\sqrt{a^2+b^2}}\right ) d^2-f^2 x^2 \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) d^2-2 e f x \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) d^2+f^2 x^2 \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) d^2+2 e f x \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) d^2-2 f (e+f x) \text{PolyLog}\left (2,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right ) d+2 f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) d+2 f^2 \text{PolyLog}\left (3,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )-2 f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )\right )}{\left (a^2+b^2\right )^{3/2} d^3}+\frac{\text{sech}(c) \text{sech}(c+d x) \left (-a \cosh (c) e^2+b \sinh (d x) e^2-2 a f x \cosh (c) e+2 b f x \sinh (d x) e-a f^2 x^2 \cosh (c)+b f^2 x^2 \sinh (d x)\right )}{\left (a^2+b^2\right ) d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a*b*(2*d^2*e^2*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] - 2*d^2*e*f*x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a
^2 + b^2])] - d^2*f^2*x^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 2*d^2*e*f*x*Log[1 + (b*E^(c + d*x))
/(a + Sqrt[a^2 + b^2])] + d^2*f^2*x^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - 2*d*f*(e + f*x)*PolyLog
[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 2*d*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2
]))] + 2*f^2*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt
[a^2 + b^2]))]))/((a^2 + b^2)^(3/2)*d^3) - (2*b*e*f*Sech[c]*(Cosh[c]*Log[Cosh[c]*Cosh[d*x] + Sinh[c]*Sinh[d*x]
] - d*x*Sinh[c]))/((a^2 + b^2)*d^2*(Cosh[c]^2 - Sinh[c]^2)) + (4*a*e*f*ArcTan[(Sinh[c] + Cosh[c]*Tanh[(d*x)/2]
)/Sqrt[Cosh[c]^2 - Sinh[c]^2]])/((a^2 + b^2)*d^2*Sqrt[Cosh[c]^2 - Sinh[c]^2]) + (b*f^2*Csch[c]*(-((d^2*x^2)/E^
ArcTanh[Coth[c]]) + (I*Coth[c]*(-(d*x*(-Pi + (2*I)*ArcTanh[Coth[c]])) - Pi*Log[1 + E^(2*d*x)] - 2*(I*d*x + I*A
rcTanh[Coth[c]])*Log[1 - E^((2*I)*(I*d*x + I*ArcTanh[Coth[c]]))] + Pi*Log[Cosh[d*x]] + (2*I)*ArcTanh[Coth[c]]*
Log[I*Sinh[d*x + ArcTanh[Coth[c]]]] + I*PolyLog[2, E^((2*I)*(I*d*x + I*ArcTanh[Coth[c]]))]))/Sqrt[1 - Coth[c]^
2])*Sech[c])/((a^2 + b^2)*d^3*Sqrt[Csch[c]^2*(-Cosh[c]^2 + Sinh[c]^2)]) + (2*a*f^2*(((-I)*Csch[c]*(I*(d*x + Ar
cTanh[Coth[c]])*(Log[1 - E^(-(d*x) - ArcTanh[Coth[c]])] - Log[1 + E^(-(d*x) - ArcTanh[Coth[c]])]) + I*(PolyLog
[2, -E^(-(d*x) - ArcTanh[Coth[c]])] - PolyLog[2, E^(-(d*x) - ArcTanh[Coth[c]])])))/Sqrt[1 - Coth[c]^2] - (2*Ar
cTan[(Sinh[c] + Cosh[c]*Tanh[(d*x)/2])/Sqrt[Cosh[c]^2 - Sinh[c]^2]]*ArcTanh[Coth[c]])/Sqrt[Cosh[c]^2 - Sinh[c]
^2]))/((a^2 + b^2)*d^3) + (Sech[c]*Sech[c + d*x]*(-(a*e^2*Cosh[c]) - 2*a*e*f*x*Cosh[c] - a*f^2*x^2*Cosh[c] + b
*e^2*Sinh[d*x] + 2*b*e*f*x*Sinh[d*x] + b*f^2*x^2*Sinh[d*x]))/((a^2 + b^2)*d)

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Maple [F]  time = 0.931, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{2}{\rm sech} \left (dx+c\right )\tanh \left ( dx+c \right ) }{a+b\sinh \left ( dx+c \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [C]  time = 3.44481, size = 8627, normalized size = 13.31 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(4*(a^2*b + b^3)*d^2*e^2 - 8*(a^2*b + b^3)*c*d*e*f + 4*(a^2*b + b^3)*c^2*f^2 - 4*((a^2*b + b^3)*d^2*f^2*x
^2 + 2*(a^2*b + b^3)*d^2*e*f*x + 2*(a^2*b + b^3)*c*d*e*f - (a^2*b + b^3)*c^2*f^2)*cosh(d*x + c)^2 - 4*((a^2*b
+ b^3)*d^2*f^2*x^2 + 2*(a^2*b + b^3)*d^2*e*f*x + 2*(a^2*b + b^3)*c*d*e*f - (a^2*b + b^3)*c^2*f^2)*sinh(d*x + c
)^2 + 4*(a*b^2*d*f^2*x + a*b^2*d*e*f + (a*b^2*d*f^2*x + a*b^2*d*e*f)*cosh(d*x + c)^2 + 2*(a*b^2*d*f^2*x + a*b^
2*d*e*f)*cosh(d*x + c)*sinh(d*x + c) + (a*b^2*d*f^2*x + a*b^2*d*e*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*di
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 + 1)
- 4*(a*b^2*d*f^2*x + a*b^2*d*e*f + (a*b^2*d*f^2*x + a*b^2*d*e*f)*cosh(d*x + c)^2 + 2*(a*b^2*d*f^2*x + a*b^2*d
*e*f)*cosh(d*x + c)*sinh(d*x + c) + (a*b^2*d*f^2*x + a*b^2*d*e*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*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) -
2*(a*b^2*d^2*e^2 - 2*a*b^2*c*d*e*f + a*b^2*c^2*f^2 + (a*b^2*d^2*e^2 - 2*a*b^2*c*d*e*f + a*b^2*c^2*f^2)*cosh(d*
x + c)^2 + 2*(a*b^2*d^2*e^2 - 2*a*b^2*c*d*e*f + a*b^2*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c) + (a*b^2*d^2*e^2 -
2*a*b^2*c*d*e*f + a*b^2*c^2*f^2)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*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*b^2*d^2*e^2 - 2*a*b^2*c*d*e*f + a*b^2*c^2*f^2 + (a*b^2*d^2*e^2 -
2*a*b^2*c*d*e*f + a*b^2*c^2*f^2)*cosh(d*x + c)^2 + 2*(a*b^2*d^2*e^2 - 2*a*b^2*c*d*e*f + a*b^2*c^2*f^2)*cosh(d
*x + c)*sinh(d*x + c) + (a*b^2*d^2*e^2 - 2*a*b^2*c*d*e*f + a*b^2*c^2*f^2)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^
2)*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*b^2*d^2*f^2*x^2 + 2*a*b
^2*d^2*e*f*x + 2*a*b^2*c*d*e*f - a*b^2*c^2*f^2 + (a*b^2*d^2*f^2*x^2 + 2*a*b^2*d^2*e*f*x + 2*a*b^2*c*d*e*f - a*
b^2*c^2*f^2)*cosh(d*x + c)^2 + 2*(a*b^2*d^2*f^2*x^2 + 2*a*b^2*d^2*e*f*x + 2*a*b^2*c*d*e*f - a*b^2*c^2*f^2)*cos
h(d*x + c)*sinh(d*x + c) + (a*b^2*d^2*f^2*x^2 + 2*a*b^2*d^2*e*f*x + 2*a*b^2*c*d*e*f - a*b^2*c^2*f^2)*sinh(d*x
+ c)^2)*sqrt((a^2 + b^2)/b^2)*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sq
rt((a^2 + b^2)/b^2) - b)/b) - 2*(a*b^2*d^2*f^2*x^2 + 2*a*b^2*d^2*e*f*x + 2*a*b^2*c*d*e*f - a*b^2*c^2*f^2 + (a*
b^2*d^2*f^2*x^2 + 2*a*b^2*d^2*e*f*x + 2*a*b^2*c*d*e*f - a*b^2*c^2*f^2)*cosh(d*x + c)^2 + 2*(a*b^2*d^2*f^2*x^2
+ 2*a*b^2*d^2*e*f*x + 2*a*b^2*c*d*e*f - a*b^2*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c) + (a*b^2*d^2*f^2*x^2 + 2*a*
b^2*d^2*e*f*x + 2*a*b^2*c*d*e*f - a*b^2*c^2*f^2)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*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) - 4*(a*b^2*f^2*cosh(d*x
+ c)^2 + 2*a*b^2*f^2*cosh(d*x + c)*sinh(d*x + c) + a*b^2*f^2*sinh(d*x + c)^2 + a*b^2*f^2)*sqrt((a^2 + b^2)/b^2
)*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*b^2*f^2*cosh(d*x + c)^2 + 2*a*b^2*f^2*cosh(d*x + c)*sinh(d*x + c) + a*b^2*f^2*sinh(d*x + c)^2 + a*b^2
*f^2)*sqrt((a^2 + b^2)/b^2)*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^3 + a*b^2)*d^2*f^2*x^2 + 2*(a^3 + a*b^2)*d^2*e*f*x + (a^3 + a*b^2)*d^2*e^2
)*cosh(d*x + c) - (4*I*(a^3 + a*b^2)*f^2 - 4*(a^2*b + b^3)*f^2 + (4*I*(a^3 + a*b^2)*f^2 - 4*(a^2*b + b^3)*f^2)
*cosh(d*x + c)^2 + (8*I*(a^3 + a*b^2)*f^2 - 8*(a^2*b + b^3)*f^2)*cosh(d*x + c)*sinh(d*x + c) + (4*I*(a^3 + a*b
^2)*f^2 - 4*(a^2*b + b^3)*f^2)*sinh(d*x + c)^2)*dilog(I*cosh(d*x + c) + I*sinh(d*x + c)) - (-4*I*(a^3 + a*b^2)
*f^2 - 4*(a^2*b + b^3)*f^2 + (-4*I*(a^3 + a*b^2)*f^2 - 4*(a^2*b + b^3)*f^2)*cosh(d*x + c)^2 + (-8*I*(a^3 + a*b
^2)*f^2 - 8*(a^2*b + b^3)*f^2)*cosh(d*x + c)*sinh(d*x + c) + (-4*I*(a^3 + a*b^2)*f^2 - 4*(a^2*b + b^3)*f^2)*si
nh(d*x + c)^2)*dilog(-I*cosh(d*x + c) - I*sinh(d*x + c)) - (4*I*(a^3 + a*b^2)*d*e*f - 4*(a^2*b + b^3)*d*e*f -
4*I*(a^3 + a*b^2)*c*f^2 + 4*(a^2*b + b^3)*c*f^2 + (4*I*(a^3 + a*b^2)*d*e*f - 4*(a^2*b + b^3)*d*e*f - 4*I*(a^3
+ a*b^2)*c*f^2 + 4*(a^2*b + b^3)*c*f^2)*cosh(d*x + c)^2 + (8*I*(a^3 + a*b^2)*d*e*f - 8*(a^2*b + b^3)*d*e*f - 8
*I*(a^3 + a*b^2)*c*f^2 + 8*(a^2*b + b^3)*c*f^2)*cosh(d*x + c)*sinh(d*x + c) + (4*I*(a^3 + a*b^2)*d*e*f - 4*(a^
2*b + b^3)*d*e*f - 4*I*(a^3 + a*b^2)*c*f^2 + 4*(a^2*b + b^3)*c*f^2)*sinh(d*x + c)^2)*log(cosh(d*x + c) + sinh(
d*x + c) + I) - (-4*I*(a^3 + a*b^2)*d*e*f - 4*(a^2*b + b^3)*d*e*f + 4*I*(a^3 + a*b^2)*c*f^2 + 4*(a^2*b + b^3)*
c*f^2 + (-4*I*(a^3 + a*b^2)*d*e*f - 4*(a^2*b + b^3)*d*e*f + 4*I*(a^3 + a*b^2)*c*f^2 + 4*(a^2*b + b^3)*c*f^2)*c
osh(d*x + c)^2 + (-8*I*(a^3 + a*b^2)*d*e*f - 8*(a^2*b + b^3)*d*e*f + 8*I*(a^3 + a*b^2)*c*f^2 + 8*(a^2*b + b^3)
*c*f^2)*cosh(d*x + c)*sinh(d*x + c) + (-4*I*(a^3 + a*b^2)*d*e*f - 4*(a^2*b + b^3)*d*e*f + 4*I*(a^3 + a*b^2)*c*
f^2 + 4*(a^2*b + b^3)*c*f^2)*sinh(d*x + c)^2)*log(cosh(d*x + c) + sinh(d*x + c) - I) - (-4*I*(a^3 + a*b^2)*d*f
^2*x - 4*(a^2*b + b^3)*d*f^2*x - 4*I*(a^3 + a*b^2)*c*f^2 - 4*(a^2*b + b^3)*c*f^2 + (-4*I*(a^3 + a*b^2)*d*f^2*x
- 4*(a^2*b + b^3)*d*f^2*x - 4*I*(a^3 + a*b^2)*c*f^2 - 4*(a^2*b + b^3)*c*f^2)*cosh(d*x + c)^2 + (-8*I*(a^3 + a
*b^2)*d*f^2*x - 8*(a^2*b + b^3)*d*f^2*x - 8*I*(a^3 + a*b^2)*c*f^2 - 8*(a^2*b + b^3)*c*f^2)*cosh(d*x + c)*sinh(
d*x + c) + (-4*I*(a^3 + a*b^2)*d*f^2*x - 4*(a^2*b + b^3)*d*f^2*x - 4*I*(a^3 + a*b^2)*c*f^2 - 4*(a^2*b + b^3)*c
*f^2)*sinh(d*x + c)^2)*log(I*cosh(d*x + c) + I*sinh(d*x + c) + 1) - (4*I*(a^3 + a*b^2)*d*f^2*x - 4*(a^2*b + b^
3)*d*f^2*x + 4*I*(a^3 + a*b^2)*c*f^2 - 4*(a^2*b + b^3)*c*f^2 + (4*I*(a^3 + a*b^2)*d*f^2*x - 4*(a^2*b + b^3)*d*
f^2*x + 4*I*(a^3 + a*b^2)*c*f^2 - 4*(a^2*b + b^3)*c*f^2)*cosh(d*x + c)^2 + (8*I*(a^3 + a*b^2)*d*f^2*x - 8*(a^2
*b + b^3)*d*f^2*x + 8*I*(a^3 + a*b^2)*c*f^2 - 8*(a^2*b + b^3)*c*f^2)*cosh(d*x + c)*sinh(d*x + c) + (4*I*(a^3 +
a*b^2)*d*f^2*x - 4*(a^2*b + b^3)*d*f^2*x + 4*I*(a^3 + a*b^2)*c*f^2 - 4*(a^2*b + b^3)*c*f^2)*sinh(d*x + c)^2)*
log(-I*cosh(d*x + c) - I*sinh(d*x + c) + 1) + 4*((a^3 + a*b^2)*d^2*f^2*x^2 + 2*(a^3 + a*b^2)*d^2*e*f*x + (a^3
+ a*b^2)*d^2*e^2 - 2*((a^2*b + b^3)*d^2*f^2*x^2 + 2*(a^2*b + b^3)*d^2*e*f*x + 2*(a^2*b + b^3)*c*d*e*f - (a^2*b
+ b^3)*c^2*f^2)*cosh(d*x + c))*sinh(d*x + c))/((a^4 + 2*a^2*b^2 + b^4)*d^3*cosh(d*x + c)^2 + 2*(a^4 + 2*a^2*b
^2 + b^4)*d^3*cosh(d*x + c)*sinh(d*x + c) + (a^4 + 2*a^2*b^2 + b^4)*d^3*sinh(d*x + c)^2 + (a^4 + 2*a^2*b^2 + b
^4)*d^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out