\(\int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [315]

Optimal result

Integrand size = 26, antiderivative size = 560 \[ \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {2 a b^2 (e+f x) \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a (e+f x) \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}+\frac {b^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac {b^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}-\frac {b^3 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}-\frac {i a b^2 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i a f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 \left (a^2+b^2\right ) d^2}+\frac {i a b^2 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i a f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 \left (a^2+b^2\right ) d^2}+\frac {b^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {b^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {b^3 f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d^2}+\frac {a f \text {sech}(c+d x)}{2 \left (a^2+b^2\right ) d^2}+\frac {b (e+f x) \text {sech}^2(c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {b f \tanh (c+d x)}{2 \left (a^2+b^2\right ) d^2}+\frac {a (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d} \]

[Out]

2*a*b^2*(f*x+e)*arctan(exp(d*x+c))/(a^2+b^2)^2/d+a*(f*x+e)*arctan(exp(d*x+c))/(a^2+b^2)/d-b^3*(f*x+e)*ln(1+exp
(2*d*x+2*c))/(a^2+b^2)^2/d+b^3*(f*x+e)*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/(a^2+b^2)^2/d+b^3*(f*x+e)*ln(1+b
*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/(a^2+b^2)^2/d+I*a*b^2*f*polylog(2,I*exp(d*x+c))/(a^2+b^2)^2/d^2-1/2*I*a*f*pol
ylog(2,-I*exp(d*x+c))/(a^2+b^2)/d^2+1/2*I*a*f*polylog(2,I*exp(d*x+c))/(a^2+b^2)/d^2-I*a*b^2*f*polylog(2,-I*exp
(d*x+c))/(a^2+b^2)^2/d^2-1/2*b^3*f*polylog(2,-exp(2*d*x+2*c))/(a^2+b^2)^2/d^2+b^3*f*polylog(2,-b*exp(d*x+c)/(a
-(a^2+b^2)^(1/2)))/(a^2+b^2)^2/d^2+b^3*f*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/(a^2+b^2)^2/d^2+1/2*a*f*
sech(d*x+c)/(a^2+b^2)/d^2+1/2*b*(f*x+e)*sech(d*x+c)^2/(a^2+b^2)/d-1/2*b*f*tanh(d*x+c)/(a^2+b^2)/d^2+1/2*a*(f*x
+e)*sech(d*x+c)*tanh(d*x+c)/(a^2+b^2)/d

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 560, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {5692, 5680, 2221, 2317, 2438, 6874, 4265, 3799, 4270, 5559, 3852, 8} \[ \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {2 a b^2 (e+f x) \arctan \left (e^{c+d x}\right )}{d \left (a^2+b^2\right )^2}+\frac {a (e+f x) \arctan \left (e^{c+d x}\right )}{d \left (a^2+b^2\right )}-\frac {i a b^2 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )^2}+\frac {i a b^2 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac {i a f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2 \left (a^2+b^2\right )}+\frac {i a f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2 \left (a^2+b^2\right )}-\frac {b f \tanh (c+d x)}{2 d^2 \left (a^2+b^2\right )}+\frac {a f \text {sech}(c+d x)}{2 d^2 \left (a^2+b^2\right )}+\frac {b (e+f x) \text {sech}^2(c+d x)}{2 d \left (a^2+b^2\right )}+\frac {a (e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d \left (a^2+b^2\right )}+\frac {b^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2 \left (a^2+b^2\right )^2}+\frac {b^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac {b^3 f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d^2 \left (a^2+b^2\right )^2}+\frac {b^3 (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{d \left (a^2+b^2\right )^2}+\frac {b^3 (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{d \left (a^2+b^2\right )^2}-\frac {b^3 (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{d \left (a^2+b^2\right )^2} \]

[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 8

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

Rule 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2317

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 2438

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 3799

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 3852

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 4265

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 4270

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] &&
GtQ[n, 1] && NeQ[n, 2]

Rule 5559

Int[((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.)*Tanh[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Sim
p[(-(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 5680

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 5692

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 6874

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

Rubi steps \begin{align*} \text {integral}& = \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) \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a (e+f x) \arctan \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 ) \text {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 ) \text {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) \text {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) \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a (e+f x) \arctan \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 \operatorname {PolyLog}\left (2,-\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 \operatorname {PolyLog}\left (2,-\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 ) \text {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 ) \text {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) \text {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) \text {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) \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a (e+f x) \arctan \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 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i a f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 \left (a^2+b^2\right ) d^2}+\frac {i a b^2 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i a f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 \left (a^2+b^2\right ) d^2}+\frac {b^3 f \operatorname {PolyLog}\left (2,-\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 \operatorname {PolyLog}\left (2,-\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 ) \text {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) \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a (e+f x) \arctan \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 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i a f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 \left (a^2+b^2\right ) d^2}+\frac {i a b^2 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i a f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 \left (a^2+b^2\right ) d^2}+\frac {b^3 f \operatorname {PolyLog}\left (2,-\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 \operatorname {PolyLog}\left (2,-\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 \operatorname {PolyLog}\left (2,-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] (verified)

Time = 8.53 (sec) , antiderivative size = 832, normalized size of antiderivative = 1.49 \[ \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b^3 \left (-2 d e (c+d x)+2 c f (c+d x)-f (c+d x)^2+\frac {4 a \sqrt {a^2+b^2} d e \arctan \left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-\left (a^2+b^2\right )^2}}-\frac {4 a \sqrt {-\left (a^2+b^2\right )^2} d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 c f \log \left (b-2 a e^{c+d x}-b e^{2 (c+d x)}\right )+2 d e \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )+2 f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{2 \left (a^2+b^2\right )^2 d^2}-\frac {-2 b^3 d e (c+d x)+2 b^3 c f (c+d x)-b^3 f (c+d x)^2-2 a^3 d e \arctan \left (e^{c+d x}\right )-6 a b^2 d e \arctan \left (e^{c+d x}\right )+2 a^3 c f \arctan \left (e^{c+d x}\right )+6 a b^2 c f \arctan \left (e^{c+d x}\right )-i a^3 f (c+d x) \log \left (1-i e^{c+d x}\right )-3 i a b^2 f (c+d x) \log \left (1-i e^{c+d x}\right )+i a^3 f (c+d x) \log \left (1+i e^{c+d x}\right )+3 i a b^2 f (c+d x) \log \left (1+i e^{c+d x}\right )+2 b^3 d e \log \left (1+e^{2 (c+d x)}\right )-2 b^3 c f \log \left (1+e^{2 (c+d x)}\right )+2 b^3 f (c+d x) \log \left (1+e^{2 (c+d x)}\right )+i a \left (a^2+3 b^2\right ) f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )-i a \left (a^2+3 b^2\right ) f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )+b^3 f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d^2}+\frac {\text {sech}(c+d x) (a f-b f \sinh (c+d x))}{2 \left (a^2+b^2\right ) d^2}+\frac {\text {sech}^2(c+d x) (b d e-b c f+b f (c+d x)+a d e \sinh (c+d x)-a c f \sinh (c+d x)+a f (c+d x) \sinh (c+d x))}{2 \left (a^2+b^2\right ) d^2} \]

[In]

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

[Out]

(b^3*(-2*d*e*(c + d*x) + 2*c*f*(c + d*x) - f*(c + d*x)^2 + (4*a*Sqrt[a^2 + b^2]*d*e*ArcTan[(a + b*E^(c + d*x))
/Sqrt[-a^2 - b^2]])/Sqrt[-(a^2 + b^2)^2] - (4*a*Sqrt[-(a^2 + b^2)^2]*d*e*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2
+ b^2]])/(-a^2 - b^2)^(3/2) + 2*f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 2*f*(c + d*x)*Log
[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - 2*c*f*Log[b - 2*a*E^(c + d*x) - b*E^(2*(c + d*x))] + 2*d*e*Log[2
*a*E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))] + 2*f*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 2*f*Poly
Log[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(2*(a^2 + b^2)^2*d^2) - (-2*b^3*d*e*(c + d*x) + 2*b^3*c*f*(
c + d*x) - b^3*f*(c + d*x)^2 - 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*Arc
Tan[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*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)*L
og[1 + E^(2*(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)] + b^3*f*PolyLog[2, -E^(2*(c + d*x))])/(2*(a^2 + b^2)^2*d^2) + (Sech[c + d*x]*(a*f - b*f*Sinh[c +
d*x]))/(2*(a^2 + b^2)*d^2) + (Sech[c + d*x]^2*(b*d*e - b*c*f + b*f*(c + d*x) + a*d*e*Sinh[c + d*x] - a*c*f*Sin
h[c + d*x] + a*f*(c + d*x)*Sinh[c + d*x]))/(2*(a^2 + b^2)*d^2)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2050 vs. \(2 (520 ) = 1040\).

Time = 20.31 (sec) , antiderivative size = 2051, normalized size of antiderivative = 3.66

method	result	size
risch	\(\text {Expression too large to display}\)	\(2051\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 4729 vs. \(2 (505) = 1010\).

Time = 0.34 (sec) , antiderivative size = 4729, normalized size of antiderivative = 8.44 \[ \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

[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 + 4*(2*b^3*f - 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 + 2*(2*b^3*f - I*(a^3 + 3*a*b^2)
*f)*cosh(d*x + c)^2 + 2*(2*b^3*f + 3*(2*b^3*f - I*(a^3 + 3*a*b^2)*f)*cosh(d*x + c)^2 - I*(a^3 + 3*a*b^2)*f)*si
nh(d*x + c)^2 - I*(a^3 + 3*a*b^2)*f + 4*((2*b^3*f - I*(a^3 + 3*a*b^2)*f)*cosh(d*x + c)^3 + (2*b^3*f - 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 + 4*(2*b^3*f + 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 + 2*(2*b^3*f + I*(a^3 + 3*a*b^2)*f)*cosh(d*x + c)^2 + 2*(2*b^3*f + 3*
(2*b^3*f + I*(a^3 + 3*a*b^2)*f)*cosh(d*x + c)^2 + I*(a^3 + 3*a*b^2)*f)*sinh(d*x + c)^2 + I*(a^3 + 3*a*b^2)*f +
4*((2*b^3*f + I*(a^3 + 3*a*b^2)*f)*cosh(d*x + c)^3 + (2*b^3*f + 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*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)*cosh(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))*lo
g(-(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)*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*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 + 4*(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)*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 + 2*(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)^2 + 2*(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 + 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)*cosh(d*x +
c)^2)*sinh(d*x + c)^2 + 4*((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)^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)*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 + 4*(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)*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 + 2*(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)^2 + 2*(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 + 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)*cosh(d*x + c)^2)*si
nh(d*x + c)^2 + 4*((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)^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)*cosh(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 + 4*(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)*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 + 2*(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)^2 + 2*(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 + 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)*
cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((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)^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)*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 + 4*(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)*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 + 2*
(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)^2 + 2*(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 + 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)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((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)^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)*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)
)

Sympy [F]

\[ \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right ) \operatorname {sech}^{3}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \operatorname {sech}\left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \]

[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))

Giac [F(-1)]

Timed out. \[ \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {e+f\,x}{{\mathrm {cosh}\left (c+d\,x\right )}^3\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]

[In]

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

[Out]

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