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

Optimal. Leaf size=560 \[ \frac{b^3 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{d^2 \left (a^2+b^2\right )^2}+\frac{b^3 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac{b^3 f \text{PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d^2 \left (a^2+b^2\right )^2}-\frac{i a b^2 f \text{PolyLog}\left (2,-i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )^2}+\frac{i a b^2 f \text{PolyLog}\left (2,i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac{i a f \text{PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2 \left (a^2+b^2\right )}+\frac{i a f \text{PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2 \left (a^2+b^2\right )}-\frac{b f \tanh (c+d x)}{2 d^2 \left (a^2+b^2\right )}+\frac{a f \text{sech}(c+d x)}{2 d^2 \left (a^2+b^2\right )}+\frac{b^3 (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{d \left (a^2+b^2\right )^2}+\frac{b^3 (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{d \left (a^2+b^2\right )^2}-\frac{b^3 (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{d \left (a^2+b^2\right )^2}+\frac{2 a b^2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{d \left (a^2+b^2\right )^2}+\frac{a (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{d \left (a^2+b^2\right )}+\frac{b (e+f x) \text{sech}^2(c+d x)}{2 d \left (a^2+b^2\right )}+\frac{a (e+f x) \tanh (c+d x) \text{sech}(c+d x)}{2 d \left (a^2+b^2\right )} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.938346, antiderivative size = 560, normalized size of antiderivative = 1., number of steps used = 31, number of rules used = 12, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {5573, 5561, 2190, 2279, 2391, 6742, 4180, 3718, 4185, 5451, 3767, 8} \[ \frac{b^3 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{d^2 \left (a^2+b^2\right )^2}+\frac{b^3 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac{b^3 f \text{PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d^2 \left (a^2+b^2\right )^2}-\frac{i a b^2 f \text{PolyLog}\left (2,-i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )^2}+\frac{i a b^2 f \text{PolyLog}\left (2,i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac{i a f \text{PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2 \left (a^2+b^2\right )}+\frac{i a f \text{PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2 \left (a^2+b^2\right )}-\frac{b f \tanh (c+d x)}{2 d^2 \left (a^2+b^2\right )}+\frac{a f \text{sech}(c+d x)}{2 d^2 \left (a^2+b^2\right )}+\frac{b^3 (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{d \left (a^2+b^2\right )^2}+\frac{b^3 (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{d \left (a^2+b^2\right )^2}-\frac{b^3 (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{d \left (a^2+b^2\right )^2}+\frac{2 a b^2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{d \left (a^2+b^2\right )^2}+\frac{a (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{d \left (a^2+b^2\right )}+\frac{b (e+f x) \text{sech}^2(c+d x)}{2 d \left (a^2+b^2\right )}+\frac{a (e+f x) \tanh (c+d x) \text{sech}(c+d x)}{2 d \left (a^2+b^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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]

Rule 8

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

Rubi steps

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

Mathematica [A]  time = 7.07565, size = 588, normalized size = 1.05 \[ \frac{-i a f \left (a^2+3 b^2\right ) \text{PolyLog}\left (2,-i e^{c+d x}\right )+i a f \left (a^2+3 b^2\right ) \text{PolyLog}\left (2,i e^{c+d x}\right )+2 b^3 f \text{PolyLog}\left (2,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )+2 b^3 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )-b^3 f \text{PolyLog}\left (2,-e^{2 (c+d x)}\right )+d \left (a^2+b^2\right ) (e+f x) \text{sech}^2(c+d x) (a \sinh (c+d x)+b)+2 b^3 f (c+d x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )+2 b^3 f (c+d x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )+f \left (a^2+b^2\right ) \text{sech}(c+d x) (a-b \sinh (c+d x))+2 a^3 d e \tan ^{-1}\left (e^{c+d x}\right )+i a^3 f (c+d x) \log \left (1-i e^{c+d x}\right )-i a^3 f (c+d x) \log \left (1+i e^{c+d x}\right )-2 a^3 c f \tan ^{-1}\left (e^{c+d x}\right )+6 a b^2 d e \tan ^{-1}\left (e^{c+d x}\right )+2 b^3 d e \log (a+b \sinh (c+d x))+3 i a b^2 f (c+d x) \log \left (1-i e^{c+d x}\right )-3 i a b^2 f (c+d x) \log \left (1+i e^{c+d x}\right )-6 a b^2 c f \tan ^{-1}\left (e^{c+d x}\right )-2 b^3 c f \log (a+b \sinh (c+d x))+2 b^3 d e (c+d x)-2 b^3 d e \log \left (e^{2 (c+d x)}+1\right )-2 b^3 c f (c+d x)+2 b^3 c f \log \left (e^{2 (c+d x)}+1\right )-2 b^3 f (c+d x) \log \left (e^{2 (c+d x)}+1\right )}{2 d^2 \left (a^2+b^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.176, size = 2051, normalized size = 3.7 \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)*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x)

[Out]

2/(a^2+b^2)/d^2*b^3*f/(2*a^2+2*b^2)*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))-2/(a^2+b^2)/d*
b^3*e/(2*a^2+2*b^2)*ln(1+exp(2*d*x+2*c))+2/(a^2+b^2)/d*b^3*e/(2*a^2+2*b^2)*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-
b)+2/(a^2+b^2)/d*a^3*e/(2*a^2+2*b^2)*arctan(exp(d*x+c))-2/(a^2+b^2)/d^2*b^3*f/(2*a^2+2*b^2)*dilog(1+I*exp(d*x+
c))-2/(a^2+b^2)/d^2*b^3*f/(2*a^2+2*b^2)*dilog(1-I*exp(d*x+c))+2/(a^2+b^2)/d^2*b^3*f/(2*a^2+2*b^2)*dilog((-b*ex
p(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))-2/(a^2+b^2)/d*b^3*f/(2*a^2+2*b^2)*ln(1+I*exp(d*x+c))*x-2/(a^
2+b^2)/d^2*b^3*f/(2*a^2+2*b^2)*ln(1+I*exp(d*x+c))*c-2/(a^2+b^2)/d*b^3*f/(2*a^2+2*b^2)*ln(1-I*exp(d*x+c))*x-2/(
a^2+b^2)/d^2*b^3*f/(2*a^2+2*b^2)*ln(1-I*exp(d*x+c))*c+2/(a^2+b^2)/d^2*b^3*f*c/(2*a^2+2*b^2)*ln(1+exp(2*d*x+2*c
))-2/(a^2+b^2)/d^2*b^3*f*c/(2*a^2+2*b^2)*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)+I/(a^2+b^2)/d^2*a^3*f/(2*a^2+2*
b^2)*dilog(1-I*exp(d*x+c))-2/(a^2+b^2)/d^2*a^3*f*c/(2*a^2+2*b^2)*arctan(exp(d*x+c))-I/(a^2+b^2)/d^2*a^3*f/(2*a
^2+2*b^2)*dilog(1+I*exp(d*x+c))+1/(a^2+b^2)^(3/2)/d^2*a*b^3*f*c/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)
/(a^2+b^2)^(1/2))+1/(a^2+b^2)^(3/2)/d^2*a^3*b*f*c/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/
2))-1/(a^2+b^2)^(1/2)/d^2*a*b*f*c/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+3*I/(a^2+b^2
)/d^2*a*b^2*f/(2*a^2+2*b^2)*ln(1-I*exp(d*x+c))*c-3*I/(a^2+b^2)/d*a*b^2*f/(2*a^2+2*b^2)*ln(1+I*exp(d*x+c))*x-3*
I/(a^2+b^2)/d^2*a*b^2*f/(2*a^2+2*b^2)*ln(1+I*exp(d*x+c))*c+3*I/(a^2+b^2)/d*a*b^2*f/(2*a^2+2*b^2)*ln(1-I*exp(d*
x+c))*x-3*I/(a^2+b^2)/d^2*a*b^2*f/(2*a^2+2*b^2)*dilog(1+I*exp(d*x+c))+3*I/(a^2+b^2)/d^2*a*b^2*f/(2*a^2+2*b^2)*
dilog(1-I*exp(d*x+c))-I/(a^2+b^2)/d*a^3*f/(2*a^2+2*b^2)*ln(1+I*exp(d*x+c))*x-I/(a^2+b^2)/d^2*a^3*f/(2*a^2+2*b^
2)*ln(1+I*exp(d*x+c))*c-1/(a^2+b^2)^(3/2)/d*a*b^3*e/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(
1/2))-1/(a^2+b^2)^(3/2)/d*a^3*b*e/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-6/(a^2+b^2)/
d^2*a*b^2*f*c/(2*a^2+2*b^2)*arctan(exp(d*x+c))+I/(a^2+b^2)/d*a^3*f/(2*a^2+2*b^2)*ln(1-I*exp(d*x+c))*x+I/(a^2+b
^2)/d^2*a^3*f/(2*a^2+2*b^2)*ln(1-I*exp(d*x+c))*c+1/(a^2+b^2)^(1/2)/d*a*b*e/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(
d*x+c)+2*a)/(a^2+b^2)^(1/2))+6/(a^2+b^2)/d*a*b^2*e/(2*a^2+2*b^2)*arctan(exp(d*x+c))+2/(a^2+b^2)/d*b^3*f/(2*a^2
+2*b^2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x+2/(a^2+b^2)/d^2*b^3*f/(2*a^2+2*b^2)*ln((-
b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c+2/(a^2+b^2)/d*b^3*f/(2*a^2+2*b^2)*ln((b*exp(d*x+c)+(a^
2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x+2/(a^2+b^2)/d^2*b^3*f/(2*a^2+2*b^2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)
/(a+(a^2+b^2)^(1/2)))*c+(a*d*f*x*exp(3*d*x+3*c)+a*d*e*exp(3*d*x+3*c)+2*b*d*f*x*exp(2*d*x+2*c)-a*d*f*x*exp(d*x+
c)+a*f*exp(3*d*x+3*c)+2*b*d*e*exp(2*d*x+2*c)-a*d*e*exp(d*x+c)+b*f*exp(2*d*x+2*c)+a*f*exp(d*x+c)+b*f)/d^2/(a^2+
b^2)/(1+exp(2*d*x+2*c))^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\left (\frac{b^{3} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} - \frac{b^{3} \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} - \frac{{\left (a^{3} + 3 \, a b^{2}\right )} \arctan \left (e^{\left (-d x - c\right )}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} + \frac{a e^{\left (-d x - c\right )} + 2 \, b e^{\left (-2 \, d x - 2 \, c\right )} - a e^{\left (-3 \, d x - 3 \, c\right )}}{{\left (a^{2} + b^{2} + 2 \,{\left (a^{2} + b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} +{\left (a^{2} + b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d}\right )} e + f{\left (\frac{{\left (a d x e^{\left (3 \, c\right )} + a e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} +{\left (2 \, b d x e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} -{\left (a d x e^{c} - a e^{c}\right )} e^{\left (d x\right )} + b}{a^{2} d^{2} + b^{2} d^{2} +{\left (a^{2} d^{2} e^{\left (4 \, c\right )} + b^{2} d^{2} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \,{\left (a^{2} d^{2} e^{\left (2 \, c\right )} + b^{2} d^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}} - 8 \, \int -\frac{a b^{3} x e^{\left (d x + c\right )} - b^{4} x}{4 \,{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5} -{\left (a^{4} b e^{\left (2 \, c\right )} + 2 \, a^{2} b^{3} e^{\left (2 \, c\right )} + b^{5} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} - 2 \,{\left (a^{5} e^{c} + 2 \, a^{3} b^{2} e^{c} + a b^{4} e^{c}\right )} e^{\left (d x\right )}\right )}}\,{d x} + 8 \, \int \frac{2 \, b^{3} x +{\left (a^{3} e^{c} + 3 \, a b^{2} e^{c}\right )} x e^{\left (d x\right )}}{8 \,{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4} +{\left (a^{4} e^{\left (2 \, c\right )} + 2 \, a^{2} b^{2} e^{\left (2 \, c\right )} + b^{4} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}}\,{d x}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 3.51499, size = 11385, normalized size = 20.33 \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)*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e + f x\right ) \operatorname{sech}^{3}{\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)*sech(d*x+c)**3/(a+b*sinh(d*x+c)),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (f x + e\right )} \operatorname{sech}\left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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