\(\int \frac {(e+f x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx\) [306]

Optimal result

Integrand size = 24, antiderivative size = 334 \[ \int \frac {(e+f x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {2 a (e+f x) \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}+\frac {b (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d}+\frac {b (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d}-\frac {b (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d}-\frac {i a f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {i a f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {b f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^2}+\frac {b f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^2}-\frac {b f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right ) d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5692, 5680, 2221, 2317, 2438, 6874, 4265, 3799} \[ \int \frac {(e+f x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {2 a (e+f x) \arctan \left (e^{c+d x}\right )}{d \left (a^2+b^2\right )}-\frac {i a f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )}+\frac {i a f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )}+\frac {b f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2 \left (a^2+b^2\right )}+\frac {b f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2 \left (a^2+b^2\right )}-\frac {b f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d^2 \left (a^2+b^2\right )}+\frac {b (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{d \left (a^2+b^2\right )}+\frac {b (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{d \left (a^2+b^2\right )}-\frac {b (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{d \left (a^2+b^2\right )} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 2.79 (sec) , antiderivative size = 520, normalized size of antiderivative = 1.56 \[ \int \frac {(e+f x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {4 a d e \arctan \left (e^{c+d x}\right )-4 a c f \arctan \left (e^{c+d x}\right )+\frac {4 a b \sqrt {a^2+b^2} d e \arctan \left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-\left (a^2+b^2\right )^2}}-\frac {4 a b \sqrt {-\left (a^2+b^2\right )^2} d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}+2 i a f (c+d x) \log \left (1-i e^{c+d x}\right )-2 i a f (c+d x) \log \left (1+i e^{c+d x}\right )+2 b f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+2 b f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 b d e \log \left (1+e^{2 (c+d x)}\right )+2 b c f \log \left (1+e^{2 (c+d x)}\right )-2 b f (c+d x) \log \left (1+e^{2 (c+d x)}\right )-2 b c f \log \left (b-2 a e^{c+d x}-b e^{2 (c+d x)}\right )+2 b d e \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )-2 i a f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )+2 i a f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )+2 b f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 b f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-b f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right ) d^2} \]

[In]

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

[Out]

(4*a*d*e*ArcTan[E^(c + d*x)] - 4*a*c*f*ArcTan[E^(c + d*x)] + (4*a*b*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*b*Sqrt[-(a^2 + b^2)^2]*d*e*ArcTanh[(a + b*E^(c + d*x))/Sqr
t[a^2 + b^2]])/(-a^2 - b^2)^(3/2) + (2*I)*a*f*(c + d*x)*Log[1 - I*E^(c + d*x)] - (2*I)*a*f*(c + d*x)*Log[1 + I
*E^(c + d*x)] + 2*b*f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 2*b*f*(c + d*x)*Log[1 + (b*E^
(c + d*x))/(a + Sqrt[a^2 + b^2])] - 2*b*d*e*Log[1 + E^(2*(c + d*x))] + 2*b*c*f*Log[1 + E^(2*(c + d*x))] - 2*b*
f*(c + d*x)*Log[1 + E^(2*(c + d*x))] - 2*b*c*f*Log[b - 2*a*E^(c + d*x) - b*E^(2*(c + d*x))] + 2*b*d*e*Log[2*a*
E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))] - (2*I)*a*f*PolyLog[2, (-I)*E^(c + d*x)] + (2*I)*a*f*PolyLog[2, I*E^(c
+ d*x)] + 2*b*f*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 2*b*f*PolyLog[2, -((b*E^(c + d*x))/(a +
Sqrt[a^2 + b^2]))] - b*f*PolyLog[2, -E^(2*(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. 953 vs. \(2 (313 ) = 626\).

Time = 3.25 (sec) , antiderivative size = 954, normalized size of antiderivative = 2.86

method	result	size
risch	\(-\frac {2 e b \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{d \left (2 a^{2}+2 b^{2}\right )}+\frac {4 e a \arctan \left ({\mathrm e}^{d x +c}\right )}{d \left (2 a^{2}+2 b^{2}\right )}+\frac {2 e b \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d \left (2 a^{2}+2 b^{2}\right )}+\frac {2 f b \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \left (2 a^{2}+2 b^{2}\right )}+\frac {2 f b \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} \left (2 a^{2}+2 b^{2}\right )}+\frac {2 f b \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \left (2 a^{2}+2 b^{2}\right )}+\frac {2 f b \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} \left (2 a^{2}+2 b^{2}\right )}+\frac {2 f b \operatorname {dilog}\left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} \left (2 a^{2}+2 b^{2}\right )}+\frac {2 f b \operatorname {dilog}\left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} \left (2 a^{2}+2 b^{2}\right )}-\frac {2 f \ln \left (1+i {\mathrm e}^{d x +c}\right ) b x}{d \left (2 a^{2}+2 b^{2}\right )}-\frac {2 f \ln \left (1+i {\mathrm e}^{d x +c}\right ) b c}{d^{2} \left (2 a^{2}+2 b^{2}\right )}-\frac {2 i f \operatorname {dilog}\left (1+i {\mathrm e}^{d x +c}\right ) a}{d^{2} \left (2 a^{2}+2 b^{2}\right )}-\frac {2 i f \ln \left (1+i {\mathrm e}^{d x +c}\right ) a c}{d^{2} \left (2 a^{2}+2 b^{2}\right )}-\frac {2 f \ln \left (1-i {\mathrm e}^{d x +c}\right ) b x}{d \left (2 a^{2}+2 b^{2}\right )}-\frac {2 f \ln \left (1-i {\mathrm e}^{d x +c}\right ) b c}{d^{2} \left (2 a^{2}+2 b^{2}\right )}+\frac {2 i f \ln \left (1-i {\mathrm e}^{d x +c}\right ) a c}{d^{2} \left (2 a^{2}+2 b^{2}\right )}-\frac {2 i f \ln \left (1+i {\mathrm e}^{d x +c}\right ) a x}{d \left (2 a^{2}+2 b^{2}\right )}-\frac {2 f \operatorname {dilog}\left (1+i {\mathrm e}^{d x +c}\right ) b}{d^{2} \left (2 a^{2}+2 b^{2}\right )}+\frac {2 i f \ln \left (1-i {\mathrm e}^{d x +c}\right ) a x}{d \left (2 a^{2}+2 b^{2}\right )}-\frac {2 f \operatorname {dilog}\left (1-i {\mathrm e}^{d x +c}\right ) b}{d^{2} \left (2 a^{2}+2 b^{2}\right )}+\frac {2 i f \operatorname {dilog}\left (1-i {\mathrm e}^{d x +c}\right ) a}{d^{2} \left (2 a^{2}+2 b^{2}\right )}+\frac {2 c f b \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{d^{2} \left (2 a^{2}+2 b^{2}\right )}-\frac {4 c f a \arctan \left ({\mathrm e}^{d x +c}\right )}{d^{2} \left (2 a^{2}+2 b^{2}\right )}-\frac {2 c f b \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d^{2} \left (2 a^{2}+2 b^{2}\right )}\)	\(954\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 588, normalized size of antiderivative = 1.76 \[ \int \frac {(e+f x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b f {\rm Li}_2\left (\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} - b}{b} + 1\right ) + b f {\rm Li}_2\left (\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) - {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} - b}{b} + 1\right ) + {\left (i \, a f - b f\right )} {\rm Li}_2\left (i \, \cosh \left (d x + c\right ) + i \, \sinh \left (d x + c\right )\right ) + {\left (-i \, a f - b f\right )} {\rm Li}_2\left (-i \, \cosh \left (d x + c\right ) - i \, \sinh \left (d x + c\right )\right ) + {\left (b d e - b c f\right )} \log \left (2 \, b \cosh \left (d x + c\right ) + 2 \, b \sinh \left (d x + c\right ) + 2 \, b \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} + 2 \, a\right ) + {\left (b d e - b c f\right )} \log \left (2 \, b \cosh \left (d x + c\right ) + 2 \, b \sinh \left (d x + c\right ) - 2 \, b \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} + 2 \, a\right ) + {\left (b d f x + b c f\right )} \log \left (-\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} - b}{b}\right ) + {\left (b d f x + b c f\right )} \log \left (-\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) - {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} - b}{b}\right ) + {\left (i \, a d e - b d e - i \, a c f + b c f\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + i\right ) + {\left (-i \, a d e - b d e + i \, a c f + b c f\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - i\right ) + {\left (-i \, a d f x - b d f x - i \, a c f - b c f\right )} \log \left (i \, \cosh \left (d x + c\right ) + i \, \sinh \left (d x + c\right ) + 1\right ) + {\left (i \, a d f x - b d f x + i \, a c f - b c f\right )} \log \left (-i \, \cosh \left (d x + c\right ) - i \, \sinh \left (d x + c\right ) + 1\right )}{{\left (a^{2} + b^{2}\right )} d^{2}} \]

[In]

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

[Out]

(b*f*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) + b*f*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) + (I*a*f - b*f)*dilog(I*cosh(d*x + c) + I*sinh(d*x + c)) + (-I*a*f - b*f)*dilog(-I*cosh(d*x +
c) - I*sinh(d*x + c)) + (b*d*e - b*c*f)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2)
+ 2*a) + (b*d*e - b*c*f)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + (b*d*
f*x + b*c*f)*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) + (b*d*f*x + b*c*f)*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*s
qrt((a^2 + b^2)/b^2) - b)/b) + (I*a*d*e - b*d*e - I*a*c*f + b*c*f)*log(cosh(d*x + c) + sinh(d*x + c) + I) + (-
I*a*d*e - b*d*e + I*a*c*f + b*c*f)*log(cosh(d*x + c) + sinh(d*x + c) - I) + (-I*a*d*f*x - b*d*f*x - I*a*c*f -
b*c*f)*log(I*cosh(d*x + c) + I*sinh(d*x + c) + 1) + (I*a*d*f*x - b*d*f*x + I*a*c*f - b*c*f)*log(-I*cosh(d*x +
c) - I*sinh(d*x + c) + 1))/((a^2 + b^2)*d^2)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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