\(\int \frac {(e+f x) \text {sech}^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx\) [359]

Optimal result

Integrand size = 32, antiderivative size = 711 \[ \int \frac {(e+f x) \text {sech}^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {(e+f x) \arctan \left (e^{c+d x}\right )}{b d}-\frac {2 a^2 b (e+f x) \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac {a^2 (e+f x) \arctan \left (e^{c+d x}\right )}{b \left (a^2+b^2\right ) d}-\frac {a b^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}-\frac {a b^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a b^2 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 b d^2}+\frac {i a^2 b f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i a^2 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 b \left (a^2+b^2\right ) d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 b d^2}-\frac {i a^2 b f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i a^2 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 b \left (a^2+b^2\right ) d^2}-\frac {a b^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {a b^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {a b^2 f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d^2}+\frac {f \text {sech}(c+d x)}{2 b d^2}-\frac {a^2 f \text {sech}(c+d x)}{2 b \left (a^2+b^2\right ) d^2}-\frac {a (e+f x) \text {sech}^2(c+d x)}{2 \left (a^2+b^2\right ) d}+\frac {a f \tanh (c+d x)}{2 \left (a^2+b^2\right ) d^2}+\frac {(e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b d}-\frac {a^2 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b \left (a^2+b^2\right ) d} \]

[Out]

(f*x+e)*arctan(exp(d*x+c))/b/d-2*a^2*b*(f*x+e)*arctan(exp(d*x+c))/(a^2+b^2)^2/d-a^2*(f*x+e)*arctan(exp(d*x+c))
/b/(a^2+b^2)/d+a*b^2*(f*x+e)*ln(1+exp(2*d*x+2*c))/(a^2+b^2)^2/d-a*b^2*(f*x+e)*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(
1/2)))/(a^2+b^2)^2/d-a*b^2*(f*x+e)*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/(a^2+b^2)^2/d+1/2*I*f*polylog(2,I*ex
p(d*x+c))/b/d^2-I*a^2*b*f*polylog(2,I*exp(d*x+c))/(a^2+b^2)^2/d^2-1/2*I*a^2*f*polylog(2,I*exp(d*x+c))/b/(a^2+b
^2)/d^2+I*a^2*b*f*polylog(2,-I*exp(d*x+c))/(a^2+b^2)^2/d^2-1/2*I*f*polylog(2,-I*exp(d*x+c))/b/d^2+1/2*I*a^2*f*
polylog(2,-I*exp(d*x+c))/b/(a^2+b^2)/d^2+1/2*a*b^2*f*polylog(2,-exp(2*d*x+2*c))/(a^2+b^2)^2/d^2-a*b^2*f*polylo
g(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/(a^2+b^2)^2/d^2-a*b^2*f*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/(a
^2+b^2)^2/d^2+1/2*f*sech(d*x+c)/b/d^2-1/2*a^2*f*sech(d*x+c)/b/(a^2+b^2)/d^2-1/2*a*(f*x+e)*sech(d*x+c)^2/(a^2+b
^2)/d+1/2*a*f*tanh(d*x+c)/(a^2+b^2)/d^2+1/2*(f*x+e)*sech(d*x+c)*tanh(d*x+c)/b/d-1/2*a^2*(f*x+e)*sech(d*x+c)*ta
nh(d*x+c)/b/(a^2+b^2)/d

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 711, normalized size of antiderivative = 1.00, number of steps used = 38, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.406, Rules used = {5702, 4270, 4265, 2317, 2438, 5692, 5680, 2221, 6874, 3799, 5559, 3852, 8} \[ \int \frac {(e+f x) \text {sech}^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a^2 (e+f x) \arctan \left (e^{c+d x}\right )}{b d \left (a^2+b^2\right )}-\frac {2 a^2 b (e+f x) \arctan \left (e^{c+d x}\right )}{d \left (a^2+b^2\right )^2}+\frac {i a^2 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 b d^2 \left (a^2+b^2\right )}+\frac {i a^2 b f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac {i a^2 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 b d^2 \left (a^2+b^2\right )}-\frac {i a^2 b f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac {a b^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac {a b^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2 \left (a^2+b^2\right )^2}+\frac {a b^2 f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d^2 \left (a^2+b^2\right )^2}+\frac {a f \tanh (c+d x)}{2 d^2 \left (a^2+b^2\right )}-\frac {a^2 f \text {sech}(c+d x)}{2 b d^2 \left (a^2+b^2\right )}-\frac {a b^2 (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{d \left (a^2+b^2\right )^2}-\frac {a b^2 (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{d \left (a^2+b^2\right )^2}+\frac {a b^2 (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{d \left (a^2+b^2\right )^2}-\frac {a (e+f x) \text {sech}^2(c+d x)}{2 d \left (a^2+b^2\right )}-\frac {a^2 (e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 b d \left (a^2+b^2\right )}+\frac {(e+f x) \arctan \left (e^{c+d x}\right )}{b d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 b d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 b d^2}+\frac {f \text {sech}(c+d x)}{2 b d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 b d} \]

[In]

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

[Out]

((e + f*x)*ArcTan[E^(c + d*x)])/(b*d) - (2*a^2*b*(e + f*x)*ArcTan[E^(c + d*x)])/((a^2 + b^2)^2*d) - (a^2*(e +
f*x)*ArcTan[E^(c + d*x)])/(b*(a^2 + b^2)*d) - (a*b^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])
/((a^2 + b^2)^2*d) - (a*b^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/((a^2 + b^2)^2*d) + (a*b
^2*(e + f*x)*Log[1 + E^(2*(c + d*x))])/((a^2 + b^2)^2*d) - ((I/2)*f*PolyLog[2, (-I)*E^(c + d*x)])/(b*d^2) + (I
*a^2*b*f*PolyLog[2, (-I)*E^(c + d*x)])/((a^2 + b^2)^2*d^2) + ((I/2)*a^2*f*PolyLog[2, (-I)*E^(c + d*x)])/(b*(a^
2 + b^2)*d^2) + ((I/2)*f*PolyLog[2, I*E^(c + d*x)])/(b*d^2) - (I*a^2*b*f*PolyLog[2, I*E^(c + d*x)])/((a^2 + b^
2)^2*d^2) - ((I/2)*a^2*f*PolyLog[2, I*E^(c + d*x)])/(b*(a^2 + b^2)*d^2) - (a*b^2*f*PolyLog[2, -((b*E^(c + d*x)
)/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^2) - (a*b^2*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]
)/((a^2 + b^2)^2*d^2) + (a*b^2*f*PolyLog[2, -E^(2*(c + d*x))])/(2*(a^2 + b^2)^2*d^2) + (f*Sech[c + d*x])/(2*b*
d^2) - (a^2*f*Sech[c + d*x])/(2*b*(a^2 + b^2)*d^2) - (a*(e + f*x)*Sech[c + d*x]^2)/(2*(a^2 + b^2)*d) + (a*f*Ta
nh[c + d*x])/(2*(a^2 + b^2)*d^2) + ((e + f*x)*Sech[c + d*x]*Tanh[c + d*x])/(2*b*d) - (a^2*(e + f*x)*Sech[c + d
*x]*Tanh[c + d*x])/(2*b*(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 5702

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

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

Mathematica [A] (verified)

Time = 9.12 (sec) , antiderivative size = 821, normalized size of antiderivative = 1.15 \[ \int \frac {(e+f x) \text {sech}^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a b^2 \left (-2 d e (c+d x)+2 c f (c+d x)-f (c+d x)^2+\frac {4 a \sqrt {a^2+b^2} d e \arctan \left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-\left (a^2+b^2\right )^2}}-\frac {4 a \sqrt {-\left (a^2+b^2\right )^2} d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 c f \log \left (b-2 a e^{c+d x}-b e^{2 (c+d x)}\right )+2 d e \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )+2 f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{2 \left (a^2+b^2\right )^2 d^2}+\frac {b \left (-2 a b d e (c+d x)+2 a b c f (c+d x)-a b f (c+d x)^2-2 a^2 d e \arctan \left (e^{c+d x}\right )+2 b^2 d e \arctan \left (e^{c+d x}\right )+2 a^2 c f \arctan \left (e^{c+d x}\right )-2 b^2 c f \arctan \left (e^{c+d x}\right )-i a^2 f (c+d x) \log \left (1-i e^{c+d x}\right )+i b^2 f (c+d x) \log \left (1-i e^{c+d x}\right )+i a^2 f (c+d x) \log \left (1+i e^{c+d x}\right )-i b^2 f (c+d x) \log \left (1+i e^{c+d x}\right )+2 a b d e \log \left (1+e^{2 (c+d x)}\right )-2 a b c f \log \left (1+e^{2 (c+d x)}\right )+2 a b f (c+d x) \log \left (1+e^{2 (c+d x)}\right )+i \left (a^2-b^2\right ) f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )-i \left (a^2-b^2\right ) f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )+a b f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )\right )}{2 \left (a^2+b^2\right )^2 d^2}+\frac {\text {sech}(c+d x) (b f+a f \sinh (c+d x))}{2 \left (a^2+b^2\right ) d^2}+\frac {\text {sech}^2(c+d x) (-a d e+a c f-a f (c+d x)+b d e \sinh (c+d x)-b c f \sinh (c+d x)+b f (c+d x) \sinh (c+d x))}{2 \left (a^2+b^2\right ) d^2} \]

[In]

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

[Out]

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2073 vs. \(2 (656 ) = 1312\).

Time = 13.61 (sec) , antiderivative size = 2074, normalized size of antiderivative = 2.92

method	result	size
risch	\(\text {Expression too large to display}\)	\(2074\)

[In]

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

[Out]

1/(a^2+b^2)^(3/2)/d^2*b^2*c*f/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))*a^2+2*b^2/d^2/(a
^2+b^2)*f/(2*a^2+2*b^2)*dilog(1+I*exp(d*x+c))*a+2*b^2/d^2/(a^2+b^2)*f/(2*a^2+2*b^2)*dilog(1-I*exp(d*x+c))*a+I*
b^3/d^2/(a^2+b^2)*f/(2*a^2+2*b^2)*dilog(1-I*exp(d*x+c))-2*b/d/(a^2+b^2)*a^2*e/(2*a^2+2*b^2)*arctan(exp(d*x+c))
-2*b^2/d/(a^2+b^2)*e/(2*a^2+2*b^2)*a*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)+2*b^2/d/(a^2+b^2)*e/(2*a^2+2*b^2)*a
*ln(1+exp(2*d*x+2*c))-1/(a^2+b^2)^(3/2)/d*b^4*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)^(1/2)/d*b^2*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)^(1/2)/
d^2*b^2*c*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/(a^2+b^2)^(3/2)/d^2*b^4*c*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/(a^2+b^2)^(3/2)/d*b^2*e/(2*a^2+2*b^2)*arctanh(1/
2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))*a^2+2*b^3/d/(a^2+b^2)*e/(2*a^2+2*b^2)*arctan(exp(d*x+c))+2*b^2/d^2/(a^
2+b^2)*f/(2*a^2+2*b^2)*ln(1-I*exp(d*x+c))*a*c-2*b^2/d/(a^2+b^2)*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)))*a*x-2*b^2/d/(a^2+b^2)*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)))*a*x+I*b^3/d/(a^2+b^2)*f/(2*a^2+2*b^2)*ln(1-I*exp(d*x+c))*x+2*b^2/d^2/(a^2+b^2)*c*f/(2*a^2+2*b^2)
*a*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)-2*b^2/d^2/(a^2+b^2)*c*f/(2*a^2+2*b^2)*a*ln(1+exp(2*d*x+2*c))+I*b/d^2/
(a^2+b^2)*a^2*f/(2*a^2+2*b^2)*dilog(1+I*exp(d*x+c))-I*b^3/d^2/(a^2+b^2)*f/(2*a^2+2*b^2)*ln(1+I*exp(d*x+c))*c-I
*b^3/d/(a^2+b^2)*f/(2*a^2+2*b^2)*ln(1+I*exp(d*x+c))*x-I*b/d^2/(a^2+b^2)*a^2*f/(2*a^2+2*b^2)*dilog(1-I*exp(d*x+
c))+2*b/d^2/(a^2+b^2)*c*a^2*f/(2*a^2+2*b^2)*arctan(exp(d*x+c))-2*b^2/d^2/(a^2+b^2)*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)))*a*c-2*b^2/d^2/(a^2+b^2)*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)))*a*c+2*b^2/d/(a^2+b^2)*f/(2*a^2+2*b^2)*ln(1+I*exp(d*x+c))*a*x+2*b^2/d/(a^2+b
^2)*f/(2*a^2+2*b^2)*ln(1-I*exp(d*x+c))*a*x+I*b^3/d^2/(a^2+b^2)*f/(2*a^2+2*b^2)*ln(1-I*exp(d*x+c))*c+2*b^2/d^2/
(a^2+b^2)*f/(2*a^2+2*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*exp(d*x+c)-f*b
*exp(d*x+c)+a*f)/d^2/(a^2+b^2)/(1+exp(2*d*x+2*c))^2-2*b^3/d^2/(a^2+b^2)*c*f/(2*a^2+2*b^2)*arctan(exp(d*x+c))-2
*b^2/d^2/(a^2+b^2)*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)))*a-2*b^2/d^2/(
a^2+b^2)*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)))*a-I*b^3/d^2/(a^2+b^2)*f/(
2*a^2+2*b^2)*dilog(1+I*exp(d*x+c))+I*b/d/(a^2+b^2)*a^2*f/(2*a^2+2*b^2)*ln(1+I*exp(d*x+c))*x+I*b/d^2/(a^2+b^2)*
a^2*f/(2*a^2+2*b^2)*ln(1+I*exp(d*x+c))*c-I*b/d^2/(a^2+b^2)*a^2*f/(2*a^2+2*b^2)*ln(1-I*exp(d*x+c))*c-I*b/d/(a^2
+b^2)*a^2*f/(2*a^2+2*b^2)*ln(1-I*exp(d*x+c))*x

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. 4993 vs. \(2 (635) = 1270\).

Time = 0.38 (sec) , antiderivative size = 4993, normalized size of antiderivative = 7.02 \[ \int \frac {(e+f x) \text {sech}^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/2*(2*((a^2*b + b^3)*d*f*x + (a^2*b + b^3)*d*e + (a^2*b + b^3)*f)*cosh(d*x + c)^3 + 2*((a^2*b + b^3)*d*f*x +
(a^2*b + b^3)*d*e + (a^2*b + b^3)*f)*sinh(d*x + c)^3 - 2*(2*(a^3 + a*b^2)*d*f*x + 2*(a^3 + a*b^2)*d*e + (a^3 +
a*b^2)*f)*cosh(d*x + c)^2 - 2*(2*(a^3 + a*b^2)*d*f*x + 2*(a^3 + a*b^2)*d*e + (a^3 + a*b^2)*f - 3*((a^2*b + b^
3)*d*f*x + (a^2*b + b^3)*d*e + (a^2*b + b^3)*f)*cosh(d*x + c))*sinh(d*x + c)^2 - 2*(a^3 + a*b^2)*f - 2*((a^2*b
+ b^3)*d*f*x + (a^2*b + b^3)*d*e - (a^2*b + b^3)*f)*cosh(d*x + c) - 2*(a*b^2*f*cosh(d*x + c)^4 + 4*a*b^2*f*co
sh(d*x + c)*sinh(d*x + c)^3 + a*b^2*f*sinh(d*x + c)^4 + 2*a*b^2*f*cosh(d*x + c)^2 + a*b^2*f + 2*(3*a*b^2*f*cos
h(d*x + c)^2 + a*b^2*f)*sinh(d*x + c)^2 + 4*(a*b^2*f*cosh(d*x + c)^3 + a*b^2*f*cosh(d*x + c))*sinh(d*x + c))*d
ilog((a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1
) - 2*(a*b^2*f*cosh(d*x + c)^4 + 4*a*b^2*f*cosh(d*x + c)*sinh(d*x + c)^3 + a*b^2*f*sinh(d*x + c)^4 + 2*a*b^2*f
*cosh(d*x + c)^2 + a*b^2*f + 2*(3*a*b^2*f*cosh(d*x + c)^2 + a*b^2*f)*sinh(d*x + c)^2 + 4*(a*b^2*f*cosh(d*x + c
)^3 + a*b^2*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*si
nh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + ((2*a*b^2*f - I*(a^2*b - b^3)*f)*cosh(d*x + c)^4 + 4*(2*a*b^2
*f - I*(a^2*b - b^3)*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*a*b^2*f - I*(a^2*b - b^3)*f)*sinh(d*x + c)^4 + 2*a*
b^2*f + 2*(2*a*b^2*f - I*(a^2*b - b^3)*f)*cosh(d*x + c)^2 + 2*(2*a*b^2*f + 3*(2*a*b^2*f - I*(a^2*b - b^3)*f)*c
osh(d*x + c)^2 - I*(a^2*b - b^3)*f)*sinh(d*x + c)^2 - I*(a^2*b - b^3)*f + 4*((2*a*b^2*f - I*(a^2*b - b^3)*f)*c
osh(d*x + c)^3 + (2*a*b^2*f - I*(a^2*b - b^3)*f)*cosh(d*x + c))*sinh(d*x + c))*dilog(I*cosh(d*x + c) + I*sinh(
d*x + c)) + ((2*a*b^2*f + I*(a^2*b - b^3)*f)*cosh(d*x + c)^4 + 4*(2*a*b^2*f + I*(a^2*b - b^3)*f)*cosh(d*x + c)
*sinh(d*x + c)^3 + (2*a*b^2*f + I*(a^2*b - b^3)*f)*sinh(d*x + c)^4 + 2*a*b^2*f + 2*(2*a*b^2*f + I*(a^2*b - b^3
)*f)*cosh(d*x + c)^2 + 2*(2*a*b^2*f + 3*(2*a*b^2*f + I*(a^2*b - b^3)*f)*cosh(d*x + c)^2 + I*(a^2*b - b^3)*f)*s
inh(d*x + c)^2 + I*(a^2*b - b^3)*f + 4*((2*a*b^2*f + I*(a^2*b - b^3)*f)*cosh(d*x + c)^3 + (2*a*b^2*f + I*(a^2*
b - b^3)*f)*cosh(d*x + c))*sinh(d*x + c))*dilog(-I*cosh(d*x + c) - I*sinh(d*x + c)) - 2*(a*b^2*d*e - a*b^2*c*f
+ (a*b^2*d*e - a*b^2*c*f)*cosh(d*x + c)^4 + 4*(a*b^2*d*e - a*b^2*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (a*b^2*
d*e - a*b^2*c*f)*sinh(d*x + c)^4 + 2*(a*b^2*d*e - a*b^2*c*f)*cosh(d*x + c)^2 + 2*(a*b^2*d*e - a*b^2*c*f + 3*(a
*b^2*d*e - a*b^2*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((a*b^2*d*e - a*b^2*c*f)*cosh(d*x + c)^3 + (a*b^2*d
*e - a*b^2*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*(a*b^2*d*e - a*b^2*c*f + (a*b^2*d*e - a*b^2*c*f)*cosh(d*x + c)^4 + 4*(a*b^2*d*e - a*b^2*c*f)*
cosh(d*x + c)*sinh(d*x + c)^3 + (a*b^2*d*e - a*b^2*c*f)*sinh(d*x + c)^4 + 2*(a*b^2*d*e - a*b^2*c*f)*cosh(d*x +
c)^2 + 2*(a*b^2*d*e - a*b^2*c*f + 3*(a*b^2*d*e - a*b^2*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((a*b^2*d*e
- a*b^2*c*f)*cosh(d*x + c)^3 + (a*b^2*d*e - a*b^2*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*(a*b^2*d*f*x + a*b^2*c*f + (a*b^2*d*f*x + a*b^2*c*f)*c
osh(d*x + c)^4 + 4*(a*b^2*d*f*x + a*b^2*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (a*b^2*d*f*x + a*b^2*c*f)*sinh(d*
x + c)^4 + 2*(a*b^2*d*f*x + a*b^2*c*f)*cosh(d*x + c)^2 + 2*(a*b^2*d*f*x + a*b^2*c*f + 3*(a*b^2*d*f*x + a*b^2*c
*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((a*b^2*d*f*x + a*b^2*c*f)*cosh(d*x + c)^3 + (a*b^2*d*f*x + a*b^2*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*b^2*d*f*x + a*b^2*c*f + (a*b^2*d*f*x + a*b^2*c*f)*cosh(d*x + c)^4 + 4*(a*
b^2*d*f*x + a*b^2*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (a*b^2*d*f*x + a*b^2*c*f)*sinh(d*x + c)^4 + 2*(a*b^2*d*
f*x + a*b^2*c*f)*cosh(d*x + c)^2 + 2*(a*b^2*d*f*x + a*b^2*c*f + 3*(a*b^2*d*f*x + a*b^2*c*f)*cosh(d*x + c)^2)*s
inh(d*x + c)^2 + 4*((a*b^2*d*f*x + a*b^2*c*f)*cosh(d*x + c)^3 + (a*b^2*d*f*x + a*b^2*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*b^2*d*e - 2*a*b^2*c*f + (2*a*b^2*d*e - 2*a*b^2*c*f - I*(a^2*b - b^3)*d*e + I*(a^2*b - b^3)*c*f)
*cosh(d*x + c)^4 + 4*(2*a*b^2*d*e - 2*a*b^2*c*f - I*(a^2*b - b^3)*d*e + I*(a^2*b - b^3)*c*f)*cosh(d*x + c)*sin
h(d*x + c)^3 + (2*a*b^2*d*e - 2*a*b^2*c*f - I*(a^2*b - b^3)*d*e + I*(a^2*b - b^3)*c*f)*sinh(d*x + c)^4 - I*(a^
2*b - b^3)*d*e + I*(a^2*b - b^3)*c*f + 2*(2*a*b^2*d*e - 2*a*b^2*c*f - I*(a^2*b - b^3)*d*e + I*(a^2*b - b^3)*c*
f)*cosh(d*x + c)^2 + 2*(2*a*b^2*d*e - 2*a*b^2*c*f - I*(a^2*b - b^3)*d*e + I*(a^2*b - b^3)*c*f + 3*(2*a*b^2*d*e
- 2*a*b^2*c*f - I*(a^2*b - b^3)*d*e + I*(a^2*b - b^3)*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((2*a*b^2*d*e
- 2*a*b^2*c*f - I*(a^2*b - b^3)*d*e + I*(a^2*b - b^3)*c*f)*cosh(d*x + c)^3 + (2*a*b^2*d*e - 2*a*b^2*c*f - I*(
a^2*b - b^3)*d*e + I*(a^2*b - b^3)*c*f)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + I) +
(2*a*b^2*d*e - 2*a*b^2*c*f + (2*a*b^2*d*e - 2*a*b^2*c*f + I*(a^2*b - b^3)*d*e - I*(a^2*b - b^3)*c*f)*cosh(d*x
+ c)^4 + 4*(2*a*b^2*d*e - 2*a*b^2*c*f + I*(a^2*b - b^3)*d*e - I*(a^2*b - b^3)*c*f)*cosh(d*x + c)*sinh(d*x + c
)^3 + (2*a*b^2*d*e - 2*a*b^2*c*f + I*(a^2*b - b^3)*d*e - I*(a^2*b - b^3)*c*f)*sinh(d*x + c)^4 + I*(a^2*b - b^3
)*d*e - I*(a^2*b - b^3)*c*f + 2*(2*a*b^2*d*e - 2*a*b^2*c*f + I*(a^2*b - b^3)*d*e - I*(a^2*b - b^3)*c*f)*cosh(d
*x + c)^2 + 2*(2*a*b^2*d*e - 2*a*b^2*c*f + I*(a^2*b - b^3)*d*e - I*(a^2*b - b^3)*c*f + 3*(2*a*b^2*d*e - 2*a*b^
2*c*f + I*(a^2*b - b^3)*d*e - I*(a^2*b - b^3)*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((2*a*b^2*d*e - 2*a*b^
2*c*f + I*(a^2*b - b^3)*d*e - I*(a^2*b - b^3)*c*f)*cosh(d*x + c)^3 + (2*a*b^2*d*e - 2*a*b^2*c*f + I*(a^2*b - b
^3)*d*e - I*(a^2*b - b^3)*c*f)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - I) + (2*a*b^2
*d*f*x + 2*a*b^2*c*f + (2*a*b^2*d*f*x + 2*a*b^2*c*f + I*(a^2*b - b^3)*d*f*x + I*(a^2*b - b^3)*c*f)*cosh(d*x +
c)^4 + 4*(2*a*b^2*d*f*x + 2*a*b^2*c*f + I*(a^2*b - b^3)*d*f*x + I*(a^2*b - b^3)*c*f)*cosh(d*x + c)*sinh(d*x +
c)^3 + (2*a*b^2*d*f*x + 2*a*b^2*c*f + I*(a^2*b - b^3)*d*f*x + I*(a^2*b - b^3)*c*f)*sinh(d*x + c)^4 + I*(a^2*b
- b^3)*d*f*x + I*(a^2*b - b^3)*c*f + 2*(2*a*b^2*d*f*x + 2*a*b^2*c*f + I*(a^2*b - b^3)*d*f*x + I*(a^2*b - b^3)*
c*f)*cosh(d*x + c)^2 + 2*(2*a*b^2*d*f*x + 2*a*b^2*c*f + I*(a^2*b - b^3)*d*f*x + I*(a^2*b - b^3)*c*f + 3*(2*a*b
^2*d*f*x + 2*a*b^2*c*f + I*(a^2*b - b^3)*d*f*x + I*(a^2*b - b^3)*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((2
*a*b^2*d*f*x + 2*a*b^2*c*f + I*(a^2*b - b^3)*d*f*x + I*(a^2*b - b^3)*c*f)*cosh(d*x + c)^3 + (2*a*b^2*d*f*x + 2
*a*b^2*c*f + I*(a^2*b - b^3)*d*f*x + I*(a^2*b - 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*b^2*d*f*x + 2*a*b^2*c*f + (2*a*b^2*d*f*x + 2*a*b^2*c*f - I*(a^2*b - b^3)*d*f*x - I
*(a^2*b - b^3)*c*f)*cosh(d*x + c)^4 + 4*(2*a*b^2*d*f*x + 2*a*b^2*c*f - I*(a^2*b - b^3)*d*f*x - I*(a^2*b - b^3)
*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*a*b^2*d*f*x + 2*a*b^2*c*f - I*(a^2*b - b^3)*d*f*x - I*(a^2*b - b^3)*c
*f)*sinh(d*x + c)^4 - I*(a^2*b - b^3)*d*f*x - I*(a^2*b - b^3)*c*f + 2*(2*a*b^2*d*f*x + 2*a*b^2*c*f - I*(a^2*b
- b^3)*d*f*x - I*(a^2*b - b^3)*c*f)*cosh(d*x + c)^2 + 2*(2*a*b^2*d*f*x + 2*a*b^2*c*f - I*(a^2*b - b^3)*d*f*x -
I*(a^2*b - b^3)*c*f + 3*(2*a*b^2*d*f*x + 2*a*b^2*c*f - I*(a^2*b - b^3)*d*f*x - I*(a^2*b - b^3)*c*f)*cosh(d*x
+ c)^2)*sinh(d*x + c)^2 + 4*((2*a*b^2*d*f*x + 2*a*b^2*c*f - I*(a^2*b - b^3)*d*f*x - I*(a^2*b - b^3)*c*f)*cosh(
d*x + c)^3 + (2*a*b^2*d*f*x + 2*a*b^2*c*f - I*(a^2*b - b^3)*d*f*x - I*(a^2*b - 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^2*b + b^3)*d*f*x + (a^2*b + b^3)*d*e - 3*((a^2*b
+ b^3)*d*f*x + (a^2*b + b^3)*d*e + (a^2*b + b^3)*f)*cosh(d*x + c)^2 - (a^2*b + b^3)*f + 2*(2*(a^3 + a*b^2)*d*f
*x + 2*(a^3 + a*b^2)*d*e + (a^3 + a*b^2)*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]

\[ \int \frac {(e+f x) \text {sech}^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right ) \tanh {\left (c + d x \right )} \operatorname {sech}^{2}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {(e+f x) \text {sech}^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \operatorname {sech}\left (d x + c\right )^{2} \tanh \left (d x + c\right )}{b \sinh \left (d x + c\right ) + a} \,d x } \]

[In]

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

[Out]

-(a*b^2*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) - a*b^2*log(e^(-2*d*x - 2*
c) + 1)/((a^4 + 2*a^2*b^2 + b^4)*d) - (a^2*b - b^3)*arctan(e^(-d*x - c))/((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))*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)) + 4*integrate(-1/2*(a^2*b^2*x*e^(d*x + c) - a*b^3*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) - 4*integrate(1/4*(2*a*b^2*x + (a^2*b*e^c - b^3*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. \[ \int \frac {(e+f x) \text {sech}^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x) \text {sech}^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\mathrm {tanh}\left (c+d\,x\right )\,\left (e+f\,x\right )}{{\mathrm {cosh}\left (c+d\,x\right )}^2\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]

[In]

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

[Out]

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