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

Optimal. Leaf size=746 \[ -\frac{b^4 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a d^2 \left (a^2+b^2\right )^2}-\frac{b^4 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a d^2 \left (a^2+b^2\right )^2}+\frac{b^4 f \text{PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 a d^2 \left (a^2+b^2\right )^2}+\frac{i b^3 f \text{PolyLog}\left (2,-i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac{i b^3 f \text{PolyLog}\left (2,i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )^2}+\frac{i b f \text{PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2 \left (a^2+b^2\right )}-\frac{i b f \text{PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2 \left (a^2+b^2\right )}-\frac{f \text{PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 a d^2}+\frac{f \text{PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 a d^2}+\frac{b^2 f \tanh (c+d x)}{2 a d^2 \left (a^2+b^2\right )}-\frac{b f \text{sech}(c+d x)}{2 d^2 \left (a^2+b^2\right )}-\frac{b^4 (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a d \left (a^2+b^2\right )^2}-\frac{b^4 (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a d \left (a^2+b^2\right )^2}+\frac{b^4 (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{a d \left (a^2+b^2\right )^2}-\frac{2 b^3 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{d \left (a^2+b^2\right )^2}-\frac{b (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{d \left (a^2+b^2\right )}-\frac{b^2 (e+f x) \text{sech}^2(c+d x)}{2 a d \left (a^2+b^2\right )}-\frac{b (e+f x) \tanh (c+d x) \text{sech}(c+d x)}{2 d \left (a^2+b^2\right )}-\frac{f \tanh (c+d x)}{2 a d^2}-\frac{(e+f x) \tanh ^2(c+d x)}{2 a d}+\frac{(e+f x) \log (\tanh (c+d x))}{a d}-\frac{2 f x \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac{f x \log (\tanh (c+d x))}{a d}+\frac{f x}{2 a d} \]

[Out]

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

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Rubi [A]  time = 1.05909, antiderivative size = 746, normalized size of antiderivative = 1., number of steps used = 43, number of rules used = 20, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {5589, 2620, 14, 5462, 2548, 12, 4182, 2279, 2391, 3473, 8, 5573, 5561, 2190, 6742, 4180, 3718, 4185, 5451, 3767} \[ -\frac{b^4 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a d^2 \left (a^2+b^2\right )^2}-\frac{b^4 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a d^2 \left (a^2+b^2\right )^2}+\frac{b^4 f \text{PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 a d^2 \left (a^2+b^2\right )^2}+\frac{i b^3 f \text{PolyLog}\left (2,-i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac{i b^3 f \text{PolyLog}\left (2,i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )^2}+\frac{i b f \text{PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2 \left (a^2+b^2\right )}-\frac{i b f \text{PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2 \left (a^2+b^2\right )}-\frac{f \text{PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 a d^2}+\frac{f \text{PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 a d^2}+\frac{b^2 f \tanh (c+d x)}{2 a d^2 \left (a^2+b^2\right )}-\frac{b f \text{sech}(c+d x)}{2 d^2 \left (a^2+b^2\right )}-\frac{b^4 (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a d \left (a^2+b^2\right )^2}-\frac{b^4 (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a d \left (a^2+b^2\right )^2}+\frac{b^4 (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{a d \left (a^2+b^2\right )^2}-\frac{2 b^3 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{d \left (a^2+b^2\right )^2}-\frac{b (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{d \left (a^2+b^2\right )}-\frac{b^2 (e+f x) \text{sech}^2(c+d x)}{2 a d \left (a^2+b^2\right )}-\frac{b (e+f x) \tanh (c+d x) \text{sech}(c+d x)}{2 d \left (a^2+b^2\right )}-\frac{f \tanh (c+d x)}{2 a d^2}-\frac{(e+f x) \tanh ^2(c+d x)}{2 a d}+\frac{(e+f x) \log (\tanh (c+d x))}{a d}-\frac{2 f x \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac{f x \log (\tanh (c+d x))}{a d}+\frac{f x}{2 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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 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 2548

Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/u, x], x] /; InverseFunctionFr
eeQ[u, x]

Rule 12

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

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 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 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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 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 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 6742

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

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 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 4185

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> -Simp[(b^2*(c + d*x)*Cot[e + f*x]*
(b*Csc[e + f*x])^(n - 2))/(f*(n - 1)), x] + (Dist[(b^2*(n - 2))/(n - 1), Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2
), x], x] - Simp[(b^2*d*(b*Csc[e + f*x])^(n - 2))/(f^2*(n - 1)*(n - 2)), x]) /; FreeQ[{b, c, d, e, f}, x] && G
tQ[n, 1] && NeQ[n, 2]

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 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps

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

Mathematica [A]  time = 10.8415, size = 886, normalized size = 1.19 \[ -\frac{\left (-\frac{1}{2} f (c+d x)^2+f \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) (c+d x)+f \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) (c+d x)+d e \log (a+b \sinh (c+d x))-c f \log (a+b \sinh (c+d x))+f \text{PolyLog}\left (2,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )+f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )\right ) b^4}{a \left (a^2+b^2\right )^2 d^2}+\frac{e \log (\sinh (c+d x))}{a d}-\frac{c f \log (\sinh (c+d x))}{a d^2}-\frac{i f \left (i (c+d x) \log \left (1-e^{-2 (c+d x)}\right )-\frac{1}{2} i \left (\text{PolyLog}\left (2,e^{-2 (c+d x)}\right )-(c+d x)^2\right )\right )}{a d^2}-\frac{-f (c+d x)^2 a^3-2 d e (c+d x) a^3+2 c f (c+d x) a^3+2 d e \log \left (1+e^{2 (c+d x)}\right ) a^3-2 c f \log \left (1+e^{2 (c+d x)}\right ) a^3+2 f (c+d x) \log \left (1+e^{2 (c+d x)}\right ) a^3+f \text{PolyLog}\left (2,-e^{2 (c+d x)}\right ) a^3+2 b d e \tan ^{-1}\left (e^{c+d x}\right ) a^2-2 b c f \tan ^{-1}\left (e^{c+d x}\right ) a^2+i b f (c+d x) \log \left (1-i e^{c+d x}\right ) a^2-i b f (c+d x) \log \left (1+i e^{c+d x}\right ) a^2-2 b^2 f (c+d x)^2 a-4 b^2 d e (c+d x) a+4 b^2 c f (c+d x) a+4 b^2 d e \log \left (1+e^{2 (c+d x)}\right ) a-4 b^2 c f \log \left (1+e^{2 (c+d x)}\right ) a+4 b^2 f (c+d x) \log \left (1+e^{2 (c+d x)}\right ) a+2 b^2 f \text{PolyLog}\left (2,-e^{2 (c+d x)}\right ) a+6 b^3 d e \tan ^{-1}\left (e^{c+d x}\right )-6 b^3 c f \tan ^{-1}\left (e^{c+d x}\right )+3 i b^3 f (c+d x) \log \left (1-i e^{c+d x}\right )-3 i b^3 f (c+d x) \log \left (1+i e^{c+d x}\right )-i b \left (a^2+3 b^2\right ) f \text{PolyLog}\left (2,-i e^{c+d x}\right )+i b \left (a^2+3 b^2\right ) f \text{PolyLog}\left (2,i e^{c+d x}\right )}{2 \left (a^2+b^2\right )^2 d^2}+\frac{\text{sech}(c+d x) (-b f-a \sinh (c+d x) f)}{2 \left (a^2+b^2\right ) d^2}+\frac{\text{sech}^2(c+d x) (a d e-b d \sinh (c+d x) e-a c f+a f (c+d x)+b c f \sinh (c+d x)-b f (c+d x) \sinh (c+d x))}{2 \left (a^2+b^2\right ) d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e*Log[Sinh[c + d*x]])/(a*d) - (c*f*Log[Sinh[c + d*x]])/(a*d^2) - (I*f*(I*(c + d*x)*Log[1 - E^(-2*(c + d*x))]
- (I/2)*(-(c + d*x)^2 + PolyLog[2, E^(-2*(c + d*x))])))/(a*d^2) - (b^4*(-(f*(c + d*x)^2)/2 + f*(c + d*x)*Log[1
+ (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + d*e*L
og[a + b*Sinh[c + d*x]] - c*f*Log[a + b*Sinh[c + d*x]] + f*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])]
+ f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(a*(a^2 + b^2)^2*d^2) - (-2*a^3*d*e*(c + d*x) - 4*a
*b^2*d*e*(c + d*x) + 2*a^3*c*f*(c + d*x) + 4*a*b^2*c*f*(c + d*x) - a^3*f*(c + d*x)^2 - 2*a*b^2*f*(c + d*x)^2 +
2*a^2*b*d*e*ArcTan[E^(c + d*x)] + 6*b^3*d*e*ArcTan[E^(c + d*x)] - 2*a^2*b*c*f*ArcTan[E^(c + d*x)] - 6*b^3*c*f
*ArcTan[E^(c + d*x)] + I*a^2*b*f*(c + d*x)*Log[1 - I*E^(c + d*x)] + (3*I)*b^3*f*(c + d*x)*Log[1 - I*E^(c + d*x
)] - I*a^2*b*f*(c + d*x)*Log[1 + I*E^(c + d*x)] - (3*I)*b^3*f*(c + d*x)*Log[1 + I*E^(c + d*x)] + 2*a^3*d*e*Log
[1 + E^(2*(c + d*x))] + 4*a*b^2*d*e*Log[1 + E^(2*(c + d*x))] - 2*a^3*c*f*Log[1 + E^(2*(c + d*x))] - 4*a*b^2*c*
f*Log[1 + E^(2*(c + d*x))] + 2*a^3*f*(c + d*x)*Log[1 + E^(2*(c + d*x))] + 4*a*b^2*f*(c + d*x)*Log[1 + E^(2*(c
+ d*x))] - I*b*(a^2 + 3*b^2)*f*PolyLog[2, (-I)*E^(c + d*x)] + I*b*(a^2 + 3*b^2)*f*PolyLog[2, I*E^(c + d*x)] +
a^3*f*PolyLog[2, -E^(2*(c + d*x))] + 2*a*b^2*f*PolyLog[2, -E^(2*(c + d*x))])/(2*(a^2 + b^2)^2*d^2) + (Sech[c +
d*x]*(-(b*f) - a*f*Sinh[c + d*x]))/(2*(a^2 + b^2)*d^2) + (Sech[c + d*x]^2*(a*d*e - a*c*f + a*f*(c + d*x) - b*
d*e*Sinh[c + d*x] + b*c*f*Sinh[c + d*x] - b*f*(c + d*x)*Sinh[c + d*x]))/(2*(a^2 + b^2)*d^2)

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Maple [B]  time = 0.244, size = 2580, normalized size = 3.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(a^2+b^2)/d*a*e*ln(exp(d*x+c)-1)+1/(a^2+b^2)/d*a*e*ln(exp(d*x+c)+1)-1/(a^2+b^2)/d^2*a*f*dilog(exp(d*x+c))+1/
(a^2+b^2)/d^2*a*f*dilog(exp(d*x+c)+1)-8/(a^2+b^2)/d*b^2*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*a*x-8/(a^2+b^2)/d^2
*b^2*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*a*c+8/(a^2+b^2)/d^2*b^2*f*c/(4*a^2+4*b^2)*a*ln(1+exp(2*d*x+2*c))-6*I/(
a^2+b^2)/d*b^3*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*x-6*I/(a^2+b^2)/d^2*b^3*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*c
+6*I/(a^2+b^2)/d*b^3*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*x+6*I/(a^2+b^2)/d^2*b^3*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x
+c))*c+2*I/(a^2+b^2)/d^2*a^2*f/(4*a^2+4*b^2)*dilog(1+I*exp(d*x+c))*b-2*I/(a^2+b^2)/d^2*a^2*f/(4*a^2+4*b^2)*dil
og(1-I*exp(d*x+c))*b-1/2/(a^2+b^2)^(5/2)/d^2*a^2*b^2*f*c*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+4/(
a^2+b^2)/d^2*a^2*f*c/(4*a^2+4*b^2)*b*arctan(exp(d*x+c))-8/(a^2+b^2)/d*b^2*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*a
*x-8/(a^2+b^2)/d^2*b^2*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*a*c+(-b*d*f*x*exp(3*d*x+3*c)+2*a*d*f*x*exp(2*d*x+2*c
)-b*d*e*exp(3*d*x+3*c)+2*a*d*e*exp(2*d*x+2*c)+b*d*f*x*exp(d*x+c)-b*f*exp(3*d*x+3*c)+a*f*exp(2*d*x+2*c)+b*d*e*e
xp(d*x+c)-f*b*exp(d*x+c)+a*f)/d^2/(a^2+b^2)/(1+exp(2*d*x+2*c))^2+1/(a^2+b^2)/d*ln(exp(d*x+c)+1)*a*f*x-1/(a^2+b
^2)/d^2*a*f*c*ln(exp(d*x+c)-1)-1/(a^2+b^2)/d^2*b^2*f/a*dilog(exp(d*x+c))+1/(a^2+b^2)/d^2*b^2*f/a*dilog(exp(d*x
+c)+1)+1/(a^2+b^2)/d*b^2*e/a*ln(exp(d*x+c)-1)+1/(a^2+b^2)/d*b^2*e/a*ln(exp(d*x+c)+1)-1/(a^2+b^2)/d^2*b^2*f*c/a
*ln(exp(d*x+c)-1)+1/(a^2+b^2)/d*b^2*f/a*ln(exp(d*x+c)+1)*x+2*I/(a^2+b^2)/d*a^2*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+
c))*b*x+2*I/(a^2+b^2)/d^2*a^2*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*b*c-2*I/(a^2+b^2)/d*a^2*f/(4*a^2+4*b^2)*ln(1-
I*exp(d*x+c))*b*x-2*I/(a^2+b^2)/d^2*a^2*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*b*c-1/(a^2+b^2)^2/d^2*b^4*f/a*dilog
((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))-1/(a^2+b^2)^2/d^2*b^4*f/a*dilog((b*exp(d*x+c)+(a^2+b^
2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))-1/2/(a^2+b^2)^(3/2)/d*b^2*e*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))
-12/(a^2+b^2)/d*b^3*e/(4*a^2+4*b^2)*arctan(exp(d*x+c))+1/2/(a^2+b^2)^(5/2)/d*b^4*e*arctanh(1/2*(2*b*exp(d*x+c)
+2*a)/(a^2+b^2)^(1/2))-4/(a^2+b^2)/d^2*a^3*f/(4*a^2+4*b^2)*dilog(1+I*exp(d*x+c))-4/(a^2+b^2)/d^2*a^3*f/(4*a^2+
4*b^2)*dilog(1-I*exp(d*x+c))-1/(a^2+b^2)^2/d*b^4*e/a*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)-4/(a^2+b^2)/d*a^3*e
/(4*a^2+4*b^2)*ln(1+exp(2*d*x+2*c))-1/(a^2+b^2)^2/d^2*b^4*f/a*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)
^(1/2)))*c+6*I/(a^2+b^2)/d^2*b^3*f/(4*a^2+4*b^2)*dilog(1+I*exp(d*x+c))-6*I/(a^2+b^2)/d^2*b^3*f/(4*a^2+4*b^2)*d
ilog(1-I*exp(d*x+c))+1/2/(a^2+b^2)^(3/2)/d^2*b^2*f*c*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+12/(a^2
+b^2)/d^2*b^3*f*c/(4*a^2+4*b^2)*arctan(exp(d*x+c))-1/2/(a^2+b^2)^(5/2)/d^2*b^4*f*c*arctanh(1/2*(2*b*exp(d*x+c)
+2*a)/(a^2+b^2)^(1/2))-8/(a^2+b^2)/d^2*b^2*f/(4*a^2+4*b^2)*dilog(1+I*exp(d*x+c))*a-8/(a^2+b^2)/d^2*b^2*f/(4*a^
2+4*b^2)*dilog(1-I*exp(d*x+c))*a+4/(a^2+b^2)/d^2*a^3*f*c/(4*a^2+4*b^2)*ln(1+exp(2*d*x+2*c))+1/(a^2+b^2)^2/d^2*
b^4*f*c/a*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)-4/(a^2+b^2)/d*a^2*e/(4*a^2+4*b^2)*b*arctan(exp(d*x+c))-8/(a^2+
b^2)/d*b^2*e/(4*a^2+4*b^2)*a*ln(1+exp(2*d*x+2*c))+1/2/(a^2+b^2)^(5/2)/d*a^2*b^2*e*arctanh(1/2*(2*b*exp(d*x+c)+
2*a)/(a^2+b^2)^(1/2))-4/(a^2+b^2)/d*a^3*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*x-4/(a^2+b^2)/d^2*a^3*f/(4*a^2+4*b^
2)*ln(1+I*exp(d*x+c))*c-4/(a^2+b^2)/d*a^3*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*x-4/(a^2+b^2)/d^2*a^3*f/(4*a^2+4*
b^2)*ln(1-I*exp(d*x+c))*c-1/(a^2+b^2)^2/d*b^4*f/a*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x
-1/(a^2+b^2)^2/d^2*b^4*f/a*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c-1/(a^2+b^2)^2/d*b^4*f/
a*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x

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Maxima [F]  time = 0., size = 0, normalized size = 0. \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)*csch(d*x+c)*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

-(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 - f*(((b*d*
x*e^(3*c) + b*e^(3*c))*e^(3*d*x) - (2*a*d*x*e^(2*c) + a*e^(2*c))*e^(2*d*x) - (b*d*x*e^c - b*e^c)*e^(d*x) - a)/
(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)) - 16*integrate(-1/8*(a*b^4*x*e^(d*x + c) - b^5*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) + 16*integrate
(1/16*((a^2*b*e^c + 3*b^3*e^c)*x*e^(d*x) - 2*(a^3 + 2*a*b^2)*x)/(a^4 + 2*a^2*b^2 + b^4 + (a^4*e^(2*c) + 2*a^2*
b^2*e^(2*c) + b^4*e^(2*c))*e^(2*d*x)), x) + 16*integrate(1/16*x/(a*e^(d*x + c) + a), x) - 16*integrate(1/16*x/
(a*e^(d*x + c) - a), x))

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Fricas [B]  time = 4.2291, size = 17816, normalized size = 23.88 \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)*csch(d*x+c)*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-1/2*(2*((a^3*b + a*b^3)*d*f*x + (a^3*b + a*b^3)*d*e + (a^3*b + a*b^3)*f)*cosh(d*x + c)^3 + 2*((a^3*b + a*b^3)
*d*f*x + (a^3*b + a*b^3)*d*e + (a^3*b + a*b^3)*f)*sinh(d*x + c)^3 - 2*(2*(a^4 + a^2*b^2)*d*f*x + 2*(a^4 + a^2*
b^2)*d*e + (a^4 + a^2*b^2)*f)*cosh(d*x + c)^2 - 2*(2*(a^4 + a^2*b^2)*d*f*x + 2*(a^4 + a^2*b^2)*d*e + (a^4 + a^
2*b^2)*f - 3*((a^3*b + a*b^3)*d*f*x + (a^3*b + a*b^3)*d*e + (a^3*b + a*b^3)*f)*cosh(d*x + c))*sinh(d*x + c)^2
- 2*(a^4 + a^2*b^2)*f - 2*((a^3*b + a*b^3)*d*f*x + (a^3*b + a*b^3)*d*e - (a^3*b + a*b^3)*f)*cosh(d*x + c) + 2*
(b^4*f*cosh(d*x + c)^4 + 4*b^4*f*cosh(d*x + c)*sinh(d*x + c)^3 + b^4*f*sinh(d*x + c)^4 + 2*b^4*f*cosh(d*x + c)
^2 + b^4*f + 2*(3*b^4*f*cosh(d*x + c)^2 + b^4*f)*sinh(d*x + c)^2 + 4*(b^4*f*cosh(d*x + c)^3 + b^4*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) + 2*(b^4*f*cosh(d*x + c)^4 + 4*b^4*f*cosh(d*x + c)*sinh(d*x + c)^3 + b^4*f*sinh(d*x + c
)^4 + 2*b^4*f*cosh(d*x + c)^2 + b^4*f + 2*(3*b^4*f*cosh(d*x + c)^2 + b^4*f)*sinh(d*x + c)^2 + 4*(b^4*f*cosh(d*
x + c)^3 + b^4*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) - 2*((a^4 + 2*a^2*b^2 + b^4)*f*cosh(d*x + c)^4 + 4*(a^4 + 2*
a^2*b^2 + b^4)*f*cosh(d*x + c)*sinh(d*x + c)^3 + (a^4 + 2*a^2*b^2 + b^4)*f*sinh(d*x + c)^4 + 2*(a^4 + 2*a^2*b^
2 + b^4)*f*cosh(d*x + c)^2 + 2*(3*(a^4 + 2*a^2*b^2 + b^4)*f*cosh(d*x + c)^2 + (a^4 + 2*a^2*b^2 + b^4)*f)*sinh(
d*x + c)^2 + (a^4 + 2*a^2*b^2 + b^4)*f + 4*((a^4 + 2*a^2*b^2 + b^4)*f*cosh(d*x + c)^3 + (a^4 + 2*a^2*b^2 + b^4
)*f*cosh(d*x + c))*sinh(d*x + c))*dilog(cosh(d*x + c) + sinh(d*x + c)) + ((2*(a^4 + 2*a^2*b^2)*f + I*(a^3*b +
3*a*b^3)*f)*cosh(d*x + c)^4 + 4*(2*(a^4 + 2*a^2*b^2)*f + I*(a^3*b + 3*a*b^3)*f)*cosh(d*x + c)*sinh(d*x + c)^3
+ (2*(a^4 + 2*a^2*b^2)*f + I*(a^3*b + 3*a*b^3)*f)*sinh(d*x + c)^4 + 2*(2*(a^4 + 2*a^2*b^2)*f + I*(a^3*b + 3*a*
b^3)*f)*cosh(d*x + c)^2 + 2*(3*(2*(a^4 + 2*a^2*b^2)*f + I*(a^3*b + 3*a*b^3)*f)*cosh(d*x + c)^2 + 2*(a^4 + 2*a^
2*b^2)*f + I*(a^3*b + 3*a*b^3)*f)*sinh(d*x + c)^2 + 2*(a^4 + 2*a^2*b^2)*f + I*(a^3*b + 3*a*b^3)*f + 4*((2*(a^4
+ 2*a^2*b^2)*f + I*(a^3*b + 3*a*b^3)*f)*cosh(d*x + c)^3 + (2*(a^4 + 2*a^2*b^2)*f + I*(a^3*b + 3*a*b^3)*f)*cos
h(d*x + c))*sinh(d*x + c))*dilog(I*cosh(d*x + c) + I*sinh(d*x + c)) + ((2*(a^4 + 2*a^2*b^2)*f - I*(a^3*b + 3*a
*b^3)*f)*cosh(d*x + c)^4 + 4*(2*(a^4 + 2*a^2*b^2)*f - I*(a^3*b + 3*a*b^3)*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (
2*(a^4 + 2*a^2*b^2)*f - I*(a^3*b + 3*a*b^3)*f)*sinh(d*x + c)^4 + 2*(2*(a^4 + 2*a^2*b^2)*f - I*(a^3*b + 3*a*b^3
)*f)*cosh(d*x + c)^2 + (6*(2*(a^4 + 2*a^2*b^2)*f - I*(a^3*b + 3*a*b^3)*f)*cosh(d*x + c)^2 + 4*(a^4 + 2*a^2*b^2
)*f - 2*I*(a^3*b + 3*a*b^3)*f)*sinh(d*x + c)^2 + 2*(a^4 + 2*a^2*b^2)*f - I*(a^3*b + 3*a*b^3)*f + 4*((2*(a^4 +
2*a^2*b^2)*f - I*(a^3*b + 3*a*b^3)*f)*cosh(d*x + c)^3 + (2*(a^4 + 2*a^2*b^2)*f - I*(a^3*b + 3*a*b^3)*f)*cosh(d
*x + c))*sinh(d*x + c))*dilog(-I*cosh(d*x + c) - I*sinh(d*x + c)) - 2*((a^4 + 2*a^2*b^2 + b^4)*f*cosh(d*x + c)
^4 + 4*(a^4 + 2*a^2*b^2 + b^4)*f*cosh(d*x + c)*sinh(d*x + c)^3 + (a^4 + 2*a^2*b^2 + b^4)*f*sinh(d*x + c)^4 + 2
*(a^4 + 2*a^2*b^2 + b^4)*f*cosh(d*x + c)^2 + 2*(3*(a^4 + 2*a^2*b^2 + b^4)*f*cosh(d*x + c)^2 + (a^4 + 2*a^2*b^2
+ b^4)*f)*sinh(d*x + c)^2 + (a^4 + 2*a^2*b^2 + b^4)*f + 4*((a^4 + 2*a^2*b^2 + b^4)*f*cosh(d*x + c)^3 + (a^4 +
2*a^2*b^2 + b^4)*f*cosh(d*x + c))*sinh(d*x + c))*dilog(-cosh(d*x + c) - sinh(d*x + c)) + 2*(b^4*d*e - b^4*c*f
+ (b^4*d*e - b^4*c*f)*cosh(d*x + c)^4 + 4*(b^4*d*e - b^4*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (b^4*d*e - b^4*
c*f)*sinh(d*x + c)^4 + 2*(b^4*d*e - b^4*c*f)*cosh(d*x + c)^2 + 2*(b^4*d*e - b^4*c*f + 3*(b^4*d*e - b^4*c*f)*co
sh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^4*d*e - b^4*c*f)*cosh(d*x + c)^3 + (b^4*d*e - b^4*c*f)*cosh(d*x + c))*s
inh(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*(b^4*d*e - b^4*
c*f + (b^4*d*e - b^4*c*f)*cosh(d*x + c)^4 + 4*(b^4*d*e - b^4*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (b^4*d*e - b
^4*c*f)*sinh(d*x + c)^4 + 2*(b^4*d*e - b^4*c*f)*cosh(d*x + c)^2 + 2*(b^4*d*e - b^4*c*f + 3*(b^4*d*e - b^4*c*f)
*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^4*d*e - b^4*c*f)*cosh(d*x + c)^3 + (b^4*d*e - b^4*c*f)*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*(b^4*d*f*x +
b^4*c*f + (b^4*d*f*x + b^4*c*f)*cosh(d*x + c)^4 + 4*(b^4*d*f*x + b^4*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (b^
4*d*f*x + b^4*c*f)*sinh(d*x + c)^4 + 2*(b^4*d*f*x + b^4*c*f)*cosh(d*x + c)^2 + 2*(b^4*d*f*x + b^4*c*f + 3*(b^4
*d*f*x + b^4*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^4*d*f*x + b^4*c*f)*cosh(d*x + c)^3 + (b^4*d*f*x + b
^4*c*f)*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*(b^4*d*f*x + b^4*c*f + (b^4*d*f*x + b^4*c*f)*cosh(d*x + c)^4 + 4*(b^4*
d*f*x + b^4*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (b^4*d*f*x + b^4*c*f)*sinh(d*x + c)^4 + 2*(b^4*d*f*x + b^4*c*
f)*cosh(d*x + c)^2 + 2*(b^4*d*f*x + b^4*c*f + 3*(b^4*d*f*x + b^4*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b
^4*d*f*x + b^4*c*f)*cosh(d*x + c)^3 + (b^4*d*f*x + b^4*c*f)*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^4 + 2*a^2*b^2
+ b^4)*d*f*x + (a^4 + 2*a^2*b^2 + b^4)*d*e)*cosh(d*x + c)^4 + 4*((a^4 + 2*a^2*b^2 + b^4)*d*f*x + (a^4 + 2*a^2
*b^2 + b^4)*d*e)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^4 + 2*a^2*b^2 + b^4)*d*f*x + (a^4 + 2*a^2*b^2 + b^4)*d*e)
*sinh(d*x + c)^4 + (a^4 + 2*a^2*b^2 + b^4)*d*f*x + (a^4 + 2*a^2*b^2 + b^4)*d*e + 2*((a^4 + 2*a^2*b^2 + b^4)*d*
f*x + (a^4 + 2*a^2*b^2 + b^4)*d*e)*cosh(d*x + c)^2 + 2*((a^4 + 2*a^2*b^2 + b^4)*d*f*x + (a^4 + 2*a^2*b^2 + b^4
)*d*e + 3*((a^4 + 2*a^2*b^2 + b^4)*d*f*x + (a^4 + 2*a^2*b^2 + b^4)*d*e)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*(
((a^4 + 2*a^2*b^2 + b^4)*d*f*x + (a^4 + 2*a^2*b^2 + b^4)*d*e)*cosh(d*x + c)^3 + ((a^4 + 2*a^2*b^2 + b^4)*d*f*x
+ (a^4 + 2*a^2*b^2 + b^4)*d*e)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) + ((2*(a^
4 + 2*a^2*b^2)*d*e + I*(a^3*b + 3*a*b^3)*d*e - 2*(a^4 + 2*a^2*b^2)*c*f - I*(a^3*b + 3*a*b^3)*c*f)*cosh(d*x + c
)^4 + (8*(a^4 + 2*a^2*b^2)*d*e + 4*I*(a^3*b + 3*a*b^3)*d*e - 8*(a^4 + 2*a^2*b^2)*c*f - 4*I*(a^3*b + 3*a*b^3)*c
*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*(a^4 + 2*a^2*b^2)*d*e + I*(a^3*b + 3*a*b^3)*d*e - 2*(a^4 + 2*a^2*b^2)*c
*f - I*(a^3*b + 3*a*b^3)*c*f)*sinh(d*x + c)^4 + 2*(a^4 + 2*a^2*b^2)*d*e + I*(a^3*b + 3*a*b^3)*d*e - 2*(a^4 + 2
*a^2*b^2)*c*f - I*(a^3*b + 3*a*b^3)*c*f + (4*(a^4 + 2*a^2*b^2)*d*e + 2*I*(a^3*b + 3*a*b^3)*d*e - 4*(a^4 + 2*a^
2*b^2)*c*f - 2*I*(a^3*b + 3*a*b^3)*c*f)*cosh(d*x + c)^2 + (4*(a^4 + 2*a^2*b^2)*d*e + 2*I*(a^3*b + 3*a*b^3)*d*e
- 4*(a^4 + 2*a^2*b^2)*c*f - 2*I*(a^3*b + 3*a*b^3)*c*f + (12*(a^4 + 2*a^2*b^2)*d*e + 6*I*(a^3*b + 3*a*b^3)*d*e
- 12*(a^4 + 2*a^2*b^2)*c*f - 6*I*(a^3*b + 3*a*b^3)*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((8*(a^4 + 2*a^2*b
^2)*d*e + 4*I*(a^3*b + 3*a*b^3)*d*e - 8*(a^4 + 2*a^2*b^2)*c*f - 4*I*(a^3*b + 3*a*b^3)*c*f)*cosh(d*x + c)^3 + (
8*(a^4 + 2*a^2*b^2)*d*e + 4*I*(a^3*b + 3*a*b^3)*d*e - 8*(a^4 + 2*a^2*b^2)*c*f - 4*I*(a^3*b + 3*a*b^3)*c*f)*cos
h(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + I) + ((2*(a^4 + 2*a^2*b^2)*d*e - I*(a^3*b + 3*a
*b^3)*d*e - 2*(a^4 + 2*a^2*b^2)*c*f + I*(a^3*b + 3*a*b^3)*c*f)*cosh(d*x + c)^4 + (8*(a^4 + 2*a^2*b^2)*d*e - 4*
I*(a^3*b + 3*a*b^3)*d*e - 8*(a^4 + 2*a^2*b^2)*c*f + 4*I*(a^3*b + 3*a*b^3)*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 +
(2*(a^4 + 2*a^2*b^2)*d*e - I*(a^3*b + 3*a*b^3)*d*e - 2*(a^4 + 2*a^2*b^2)*c*f + I*(a^3*b + 3*a*b^3)*c*f)*sinh(
d*x + c)^4 + 2*(a^4 + 2*a^2*b^2)*d*e - I*(a^3*b + 3*a*b^3)*d*e - 2*(a^4 + 2*a^2*b^2)*c*f + I*(a^3*b + 3*a*b^3)
*c*f + (4*(a^4 + 2*a^2*b^2)*d*e - 2*I*(a^3*b + 3*a*b^3)*d*e - 4*(a^4 + 2*a^2*b^2)*c*f + 2*I*(a^3*b + 3*a*b^3)*
c*f)*cosh(d*x + c)^2 + (4*(a^4 + 2*a^2*b^2)*d*e - 2*I*(a^3*b + 3*a*b^3)*d*e - 4*(a^4 + 2*a^2*b^2)*c*f + 2*I*(a
^3*b + 3*a*b^3)*c*f + (12*(a^4 + 2*a^2*b^2)*d*e - 6*I*(a^3*b + 3*a*b^3)*d*e - 12*(a^4 + 2*a^2*b^2)*c*f + 6*I*(
a^3*b + 3*a*b^3)*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((8*(a^4 + 2*a^2*b^2)*d*e - 4*I*(a^3*b + 3*a*b^3)*d*e
- 8*(a^4 + 2*a^2*b^2)*c*f + 4*I*(a^3*b + 3*a*b^3)*c*f)*cosh(d*x + c)^3 + (8*(a^4 + 2*a^2*b^2)*d*e - 4*I*(a^3*
b + 3*a*b^3)*d*e - 8*(a^4 + 2*a^2*b^2)*c*f + 4*I*(a^3*b + 3*a*b^3)*c*f)*cosh(d*x + c))*sinh(d*x + c))*log(cosh
(d*x + c) + sinh(d*x + c) - I) - 2*(((a^4 + 2*a^2*b^2 + b^4)*d*e - (a^4 + 2*a^2*b^2 + b^4)*c*f)*cosh(d*x + c)^
4 + 4*((a^4 + 2*a^2*b^2 + b^4)*d*e - (a^4 + 2*a^2*b^2 + b^4)*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^4 + 2*a^
2*b^2 + b^4)*d*e - (a^4 + 2*a^2*b^2 + b^4)*c*f)*sinh(d*x + c)^4 + (a^4 + 2*a^2*b^2 + b^4)*d*e - (a^4 + 2*a^2*b
^2 + b^4)*c*f + 2*((a^4 + 2*a^2*b^2 + b^4)*d*e - (a^4 + 2*a^2*b^2 + b^4)*c*f)*cosh(d*x + c)^2 + 2*((a^4 + 2*a^
2*b^2 + b^4)*d*e - (a^4 + 2*a^2*b^2 + b^4)*c*f + 3*((a^4 + 2*a^2*b^2 + b^4)*d*e - (a^4 + 2*a^2*b^2 + b^4)*c*f)
*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*(((a^4 + 2*a^2*b^2 + b^4)*d*e - (a^4 + 2*a^2*b^2 + b^4)*c*f)*cosh(d*x +
c)^3 + ((a^4 + 2*a^2*b^2 + b^4)*d*e - (a^4 + 2*a^2*b^2 + b^4)*c*f)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x
+ c) + sinh(d*x + c) - 1) + ((2*(a^4 + 2*a^2*b^2)*d*f*x - I*(a^3*b + 3*a*b^3)*d*f*x + 2*(a^4 + 2*a^2*b^2)*c*f
- I*(a^3*b + 3*a*b^3)*c*f)*cosh(d*x + c)^4 + (8*(a^4 + 2*a^2*b^2)*d*f*x - 4*I*(a^3*b + 3*a*b^3)*d*f*x + 8*(a^4
+ 2*a^2*b^2)*c*f - 4*I*(a^3*b + 3*a*b^3)*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*(a^4 + 2*a^2*b^2)*d*f*x - I*
(a^3*b + 3*a*b^3)*d*f*x + 2*(a^4 + 2*a^2*b^2)*c*f - I*(a^3*b + 3*a*b^3)*c*f)*sinh(d*x + c)^4 + 2*(a^4 + 2*a^2*
b^2)*d*f*x - I*(a^3*b + 3*a*b^3)*d*f*x + 2*(a^4 + 2*a^2*b^2)*c*f - I*(a^3*b + 3*a*b^3)*c*f + (4*(a^4 + 2*a^2*b
^2)*d*f*x - 2*I*(a^3*b + 3*a*b^3)*d*f*x + 4*(a^4 + 2*a^2*b^2)*c*f - 2*I*(a^3*b + 3*a*b^3)*c*f)*cosh(d*x + c)^2
+ (4*(a^4 + 2*a^2*b^2)*d*f*x - 2*I*(a^3*b + 3*a*b^3)*d*f*x + 4*(a^4 + 2*a^2*b^2)*c*f - 2*I*(a^3*b + 3*a*b^3)*
c*f + (12*(a^4 + 2*a^2*b^2)*d*f*x - 6*I*(a^3*b + 3*a*b^3)*d*f*x + 12*(a^4 + 2*a^2*b^2)*c*f - 6*I*(a^3*b + 3*a*
b^3)*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((8*(a^4 + 2*a^2*b^2)*d*f*x - 4*I*(a^3*b + 3*a*b^3)*d*f*x + 8*(a^
4 + 2*a^2*b^2)*c*f - 4*I*(a^3*b + 3*a*b^3)*c*f)*cosh(d*x + c)^3 + (8*(a^4 + 2*a^2*b^2)*d*f*x - 4*I*(a^3*b + 3*
a*b^3)*d*f*x + 8*(a^4 + 2*a^2*b^2)*c*f - 4*I*(a^3*b + 3*a*b^3)*c*f)*cosh(d*x + c))*sinh(d*x + c))*log(I*cosh(d
*x + c) + I*sinh(d*x + c) + 1) + ((2*(a^4 + 2*a^2*b^2)*d*f*x + I*(a^3*b + 3*a*b^3)*d*f*x + 2*(a^4 + 2*a^2*b^2)
*c*f + I*(a^3*b + 3*a*b^3)*c*f)*cosh(d*x + c)^4 + (8*(a^4 + 2*a^2*b^2)*d*f*x + 4*I*(a^3*b + 3*a*b^3)*d*f*x + 8
*(a^4 + 2*a^2*b^2)*c*f + 4*I*(a^3*b + 3*a*b^3)*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*(a^4 + 2*a^2*b^2)*d*f*x
+ I*(a^3*b + 3*a*b^3)*d*f*x + 2*(a^4 + 2*a^2*b^2)*c*f + I*(a^3*b + 3*a*b^3)*c*f)*sinh(d*x + c)^4 + 2*(a^4 + 2
*a^2*b^2)*d*f*x + I*(a^3*b + 3*a*b^3)*d*f*x + 2*(a^4 + 2*a^2*b^2)*c*f + I*(a^3*b + 3*a*b^3)*c*f + (4*(a^4 + 2*
a^2*b^2)*d*f*x + 2*I*(a^3*b + 3*a*b^3)*d*f*x + 4*(a^4 + 2*a^2*b^2)*c*f + 2*I*(a^3*b + 3*a*b^3)*c*f)*cosh(d*x +
c)^2 + (4*(a^4 + 2*a^2*b^2)*d*f*x + 2*I*(a^3*b + 3*a*b^3)*d*f*x + 4*(a^4 + 2*a^2*b^2)*c*f + 2*I*(a^3*b + 3*a*
b^3)*c*f + (12*(a^4 + 2*a^2*b^2)*d*f*x + 6*I*(a^3*b + 3*a*b^3)*d*f*x + 12*(a^4 + 2*a^2*b^2)*c*f + 6*I*(a^3*b +
3*a*b^3)*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((8*(a^4 + 2*a^2*b^2)*d*f*x + 4*I*(a^3*b + 3*a*b^3)*d*f*x +
8*(a^4 + 2*a^2*b^2)*c*f + 4*I*(a^3*b + 3*a*b^3)*c*f)*cosh(d*x + c)^3 + (8*(a^4 + 2*a^2*b^2)*d*f*x + 4*I*(a^3*b
+ 3*a*b^3)*d*f*x + 8*(a^4 + 2*a^2*b^2)*c*f + 4*I*(a^3*b + 3*a*b^3)*c*f)*cosh(d*x + c))*sinh(d*x + c))*log(-I*
cosh(d*x + c) - I*sinh(d*x + c) + 1) - 2*(((a^4 + 2*a^2*b^2 + b^4)*d*f*x + (a^4 + 2*a^2*b^2 + b^4)*c*f)*cosh(d
*x + c)^4 + 4*((a^4 + 2*a^2*b^2 + b^4)*d*f*x + (a^4 + 2*a^2*b^2 + b^4)*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + ((
a^4 + 2*a^2*b^2 + b^4)*d*f*x + (a^4 + 2*a^2*b^2 + b^4)*c*f)*sinh(d*x + c)^4 + (a^4 + 2*a^2*b^2 + b^4)*d*f*x +
(a^4 + 2*a^2*b^2 + b^4)*c*f + 2*((a^4 + 2*a^2*b^2 + b^4)*d*f*x + (a^4 + 2*a^2*b^2 + b^4)*c*f)*cosh(d*x + c)^2
+ 2*((a^4 + 2*a^2*b^2 + b^4)*d*f*x + (a^4 + 2*a^2*b^2 + b^4)*c*f + 3*((a^4 + 2*a^2*b^2 + b^4)*d*f*x + (a^4 + 2
*a^2*b^2 + b^4)*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*(((a^4 + 2*a^2*b^2 + b^4)*d*f*x + (a^4 + 2*a^2*b^2 +
b^4)*c*f)*cosh(d*x + c)^3 + ((a^4 + 2*a^2*b^2 + b^4)*d*f*x + (a^4 + 2*a^2*b^2 + b^4)*c*f)*cosh(d*x + c))*sinh
(d*x + c))*log(-cosh(d*x + c) - sinh(d*x + c) + 1) - 2*((a^3*b + a*b^3)*d*f*x + (a^3*b + a*b^3)*d*e - 3*((a^3*
b + a*b^3)*d*f*x + (a^3*b + a*b^3)*d*e + (a^3*b + a*b^3)*f)*cosh(d*x + c)^2 - (a^3*b + a*b^3)*f + 2*(2*(a^4 +
a^2*b^2)*d*f*x + 2*(a^4 + a^2*b^2)*d*e + (a^4 + a^2*b^2)*f)*cosh(d*x + c))*sinh(d*x + c))/((a^5 + 2*a^3*b^2 +
a*b^4)*d^2*cosh(d*x + c)^4 + 4*(a^5 + 2*a^3*b^2 + a*b^4)*d^2*cosh(d*x + c)*sinh(d*x + c)^3 + (a^5 + 2*a^3*b^2
+ a*b^4)*d^2*sinh(d*x + c)^4 + 2*(a^5 + 2*a^3*b^2 + a*b^4)*d^2*cosh(d*x + c)^2 + (a^5 + 2*a^3*b^2 + a*b^4)*d^2
+ 2*(3*(a^5 + 2*a^3*b^2 + a*b^4)*d^2*cosh(d*x + c)^2 + (a^5 + 2*a^3*b^2 + a*b^4)*d^2)*sinh(d*x + c)^2 + 4*((a
^5 + 2*a^3*b^2 + a*b^4)*d^2*cosh(d*x + c)^3 + (a^5 + 2*a^3*b^2 + a*b^4)*d^2*cosh(d*x + c))*sinh(d*x + c))

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

[Out]

Timed out

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

[Out]

Timed out