\(\int \frac {(e+f x) \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx\) [379]

Optimal result

Integrand size = 30, antiderivative size = 516 \[ \int \frac {(e+f x) \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {(e+f x)^2}{2 b f}-\frac {2 a (e+f x) \arctan \left (e^{c+d x}\right )}{b^2 d}+\frac {2 a^3 (e+f x) \arctan \left (e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d}+\frac {(e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b d}-\frac {a^2 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b \left (a^2+b^2\right ) d}+\frac {i a f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b^2 d^2}-\frac {i a^3 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {i a f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b^2 d^2}+\frac {i a^3 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^2}+\frac {a^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {a^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 b d^2}-\frac {a^2 f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 b \left (a^2+b^2\right ) d^2} \]

[Out]

-1/2*(f*x+e)^2/b/f-2*a*(f*x+e)*arctan(exp(d*x+c))/b^2/d+2*a^3*(f*x+e)*arctan(exp(d*x+c))/b^2/(a^2+b^2)/d+(f*x+
e)*ln(1+exp(2*d*x+2*c))/b/d-a^2*(f*x+e)*ln(1+exp(2*d*x+2*c))/b/(a^2+b^2)/d+a^2*(f*x+e)*ln(1+b*exp(d*x+c)/(a-(a
^2+b^2)^(1/2)))/b/(a^2+b^2)/d+a^2*(f*x+e)*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b/(a^2+b^2)/d+I*a*f*polylog(2
,-I*exp(d*x+c))/b^2/d^2-I*a^3*f*polylog(2,-I*exp(d*x+c))/b^2/(a^2+b^2)/d^2-I*a*f*polylog(2,I*exp(d*x+c))/b^2/d
^2+I*a^3*f*polylog(2,I*exp(d*x+c))/b^2/(a^2+b^2)/d^2+1/2*f*polylog(2,-exp(2*d*x+2*c))/b/d^2-1/2*a^2*f*polylog(
2,-exp(2*d*x+2*c))/b/(a^2+b^2)/d^2+a^2*f*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b/(a^2+b^2)/d^2+a^2*f*po
lylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b/(a^2+b^2)/d^2

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 516, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5700, 3799, 2221, 2317, 2438, 5686, 4265, 5692, 5680, 6874} \[ \int \frac {(e+f x) \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {a^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2 \left (a^2+b^2\right )}+\frac {a^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2 \left (a^2+b^2\right )}-\frac {a^2 f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 b d^2 \left (a^2+b^2\right )}+\frac {a^2 (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d \left (a^2+b^2\right )}+\frac {a^2 (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d \left (a^2+b^2\right )}-\frac {a^2 (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{b d \left (a^2+b^2\right )}+\frac {2 a^3 (e+f x) \arctan \left (e^{c+d x}\right )}{b^2 d \left (a^2+b^2\right )}-\frac {i a^3 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b^2 d^2 \left (a^2+b^2\right )}+\frac {i a^3 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b^2 d^2 \left (a^2+b^2\right )}-\frac {2 a (e+f x) \arctan \left (e^{c+d x}\right )}{b^2 d}+\frac {i a f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b^2 d^2}-\frac {i a f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b^2 d^2}+\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 b d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f} \]

[In]

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

[Out]

-1/2*(e + f*x)^2/(b*f) - (2*a*(e + f*x)*ArcTan[E^(c + d*x)])/(b^2*d) + (2*a^3*(e + f*x)*ArcTan[E^(c + d*x)])/(
b^2*(a^2 + b^2)*d) + (a^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)*d) + (a^2*(
e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)*d) + ((e + f*x)*Log[1 + E^(2*(c + d*x)
)])/(b*d) - (a^2*(e + f*x)*Log[1 + E^(2*(c + d*x))])/(b*(a^2 + b^2)*d) + (I*a*f*PolyLog[2, (-I)*E^(c + d*x)])/
(b^2*d^2) - (I*a^3*f*PolyLog[2, (-I)*E^(c + d*x)])/(b^2*(a^2 + b^2)*d^2) - (I*a*f*PolyLog[2, I*E^(c + d*x)])/(
b^2*d^2) + (I*a^3*f*PolyLog[2, I*E^(c + d*x)])/(b^2*(a^2 + b^2)*d^2) + (a^2*f*PolyLog[2, -((b*E^(c + d*x))/(a
- Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)*d^2) + (a^2*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*(a
^2 + b^2)*d^2) + (f*PolyLog[2, -E^(2*(c + d*x))])/(2*b*d^2) - (a^2*f*PolyLog[2, -E^(2*(c + d*x))])/(2*b*(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 5686

Int[(((e_.) + (f_.)*(x_))^(m_.)*Tanh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[1/b, Int[(e + f*x)^m*Sech[c + d*x]*Tanh[c + d*x]^(n - 1), x], x] - Dist[a/b, Int[(e + f*x)^m*Sech
[c + d*x]*(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]

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 5700

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(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*Sinh[c + d*x]^(p - 1)*Tanh[c + d*x]^n, x], x]
- Dist[a/b, Int[(e + f*x)^m*Sinh[c + d*x]^(p - 1)*(Tanh[c + d*x]^n/(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) \tanh (c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b} \\ & = -\frac {(e+f x)^2}{2 b f}-\frac {a \int (e+f x) \text {sech}(c+d x) \, dx}{b^2}+\frac {a^2 \int \frac {(e+f x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}+\frac {2 \int \frac {e^{2 (c+d x)} (e+f x)}{1+e^{2 (c+d x)}} \, dx}{b} \\ & = -\frac {(e+f x)^2}{2 b f}-\frac {2 a (e+f x) \arctan \left (e^{c+d x}\right )}{b^2 d}+\frac {(e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b d}+\frac {a^2 \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2+b^2}+\frac {a^2 \int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x)) \, dx}{b^2 \left (a^2+b^2\right )}+\frac {(i a f) \int \log \left (1-i e^{c+d x}\right ) \, dx}{b^2 d}-\frac {(i a f) \int \log \left (1+i e^{c+d x}\right ) \, dx}{b^2 d}-\frac {f \int \log \left (1+e^{2 (c+d x)}\right ) \, dx}{b d} \\ & = -\frac {(e+f x)^2}{2 b f}-\frac {a^2 (e+f x)^2}{2 b \left (a^2+b^2\right ) f}-\frac {2 a (e+f x) \arctan \left (e^{c+d x}\right )}{b^2 d}+\frac {(e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b d}+\frac {a^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 {a^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 {a^2 \int (a (e+f x) \text {sech}(c+d x)-b (e+f x) \tanh (c+d x)) \, dx}{b^2 \left (a^2+b^2\right )}+\frac {(i a f) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{b^2 d^2}-\frac {(i a f) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{b^2 d^2}-\frac {f \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 b d^2} \\ & = -\frac {(e+f x)^2}{2 b f}-\frac {a^2 (e+f x)^2}{2 b \left (a^2+b^2\right ) f}-\frac {2 a (e+f x) \arctan \left (e^{c+d x}\right )}{b^2 d}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d}+\frac {(e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b d}+\frac {i a f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b^2 d^2}-\frac {i a f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b^2 d^2}+\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 b d^2}+\frac {a^3 \int (e+f x) \text {sech}(c+d x) \, dx}{b^2 \left (a^2+b^2\right )}-\frac {a^2 \int (e+f x) \tanh (c+d x) \, dx}{b \left (a^2+b^2\right )}-\frac {\left (a^2 f\right ) \int \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{b \left (a^2+b^2\right ) d}-\frac {\left (a^2 f\right ) \int \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{b \left (a^2+b^2\right ) d} \\ & = -\frac {(e+f x)^2}{2 b f}-\frac {2 a (e+f x) \arctan \left (e^{c+d x}\right )}{b^2 d}+\frac {2 a^3 (e+f x) \arctan \left (e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d}+\frac {(e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b d}+\frac {i a f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b^2 d^2}-\frac {i a f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b^2 d^2}+\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 b d^2}-\frac {\left (2 a^2\right ) \int \frac {e^{2 (c+d x)} (e+f x)}{1+e^{2 (c+d x)}} \, dx}{b \left (a^2+b^2\right )}-\frac {\left (a^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 )}{b \left (a^2+b^2\right ) d^2}-\frac {\left (a^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 )}{b \left (a^2+b^2\right ) d^2}-\frac {\left (i a^3 f\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{b^2 \left (a^2+b^2\right ) d}+\frac {\left (i a^3 f\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{b^2 \left (a^2+b^2\right ) d} \\ & = -\frac {(e+f x)^2}{2 b f}-\frac {2 a (e+f x) \arctan \left (e^{c+d x}\right )}{b^2 d}+\frac {2 a^3 (e+f x) \arctan \left (e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d}+\frac {(e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b d}-\frac {a^2 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b \left (a^2+b^2\right ) d}+\frac {i a f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b^2 d^2}-\frac {i a f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b^2 d^2}+\frac {a^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {a^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 b d^2}-\frac {\left (i a^3 f\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^2}+\frac {\left (i a^3 f\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^2}+\frac {\left (a^2 f\right ) \int \log \left (1+e^{2 (c+d x)}\right ) \, dx}{b \left (a^2+b^2\right ) d} \\ & = -\frac {(e+f x)^2}{2 b f}-\frac {2 a (e+f x) \arctan \left (e^{c+d x}\right )}{b^2 d}+\frac {2 a^3 (e+f x) \arctan \left (e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d}+\frac {(e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b d}-\frac {a^2 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b \left (a^2+b^2\right ) d}+\frac {i a f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b^2 d^2}-\frac {i a^3 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {i a f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b^2 d^2}+\frac {i a^3 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^2}+\frac {a^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {a^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 b d^2}+\frac {\left (a^2 f\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 b \left (a^2+b^2\right ) d^2} \\ & = -\frac {(e+f x)^2}{2 b f}-\frac {2 a (e+f x) \arctan \left (e^{c+d x}\right )}{b^2 d}+\frac {2 a^3 (e+f x) \arctan \left (e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d}+\frac {(e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b d}-\frac {a^2 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b \left (a^2+b^2\right ) d}+\frac {i a f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b^2 d^2}-\frac {i a^3 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {i a f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b^2 d^2}+\frac {i a^3 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^2}+\frac {a^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {a^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 b d^2}-\frac {a^2 f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 b \left (a^2+b^2\right ) d^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.92 (sec) , antiderivative size = 578, normalized size of antiderivative = 1.12 \[ \int \frac {(e+f x) \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {-2 b d e (c+d x)+2 b c f (c+d x)-b f (c+d x)^2-4 a d e \arctan \left (e^{c+d x}\right )+4 a c f \arctan \left (e^{c+d x}\right )-2 i a f (c+d x) \log \left (1-i e^{c+d x}\right )+2 i a f (c+d x) \log \left (1+i e^{c+d x}\right )+2 b d e \log \left (1+e^{2 (c+d x)}\right )-2 b c f \log \left (1+e^{2 (c+d x)}\right )+2 b f (c+d x) \log \left (1+e^{2 (c+d x)}\right )+2 i a f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )-2 i a f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )+\frac {a^2 \left (-2 d e (c+d x)+2 c f (c+d x)-f (c+d x)^2+\frac {4 a \sqrt {a^2+b^2} d e \arctan \left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-\left (a^2+b^2\right )^2}}-\frac {4 a \sqrt {-\left (a^2+b^2\right )^2} d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 c f \log \left (b-2 a e^{c+d x}-b e^{2 (c+d x)}\right )+2 d e \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )+2 f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{b}+b f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right ) d^2} \]

[In]

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

[Out]

(-2*b*d*e*(c + d*x) + 2*b*c*f*(c + d*x) - b*f*(c + d*x)^2 - 4*a*d*e*ArcTan[E^(c + d*x)] + 4*a*c*f*ArcTan[E^(c
+ d*x)] - (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*d*e*Lo
g[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*I)*a
*f*PolyLog[2, (-I)*E^(c + d*x)] - (2*I)*a*f*PolyLog[2, I*E^(c + d*x)] + (a^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))/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*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2
+ b^2]))]))/b + 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. 3881 vs. \(2 (484 ) = 968\).

Time = 2.07 (sec) , antiderivative size = 3882, normalized size of antiderivative = 7.52

method	result	size
risch	\(\text {Expression too large to display}\)	\(3882\)

[In]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 682, normalized size of antiderivative = 1.32 \[ \int \frac {(e+f x) \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {{\left (a^{2} + b^{2}\right )} d^{2} f x^{2} + 2 \, {\left (a^{2} + b^{2}\right )} d^{2} e x - 2 \, a^{2} 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 ) - 2 \, a^{2} 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 ) + 2 \, {\left (i \, a b f - b^{2} f\right )} {\rm Li}_2\left (i \, \cosh \left (d x + c\right ) + i \, \sinh \left (d x + c\right )\right ) + 2 \, {\left (-i \, a b f - b^{2} f\right )} {\rm Li}_2\left (-i \, \cosh \left (d x + c\right ) - i \, \sinh \left (d x + c\right )\right ) - 2 \, {\left (a^{2} d e - a^{2} 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 ) - 2 \, {\left (a^{2} d e - a^{2} 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 ) - 2 \, {\left (a^{2} d f x + a^{2} 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 ) - 2 \, {\left (a^{2} d f x + a^{2} 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 ) + 2 \, {\left (i \, a b d e - b^{2} d e - i \, a b c f + b^{2} c f\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + i\right ) + 2 \, {\left (-i \, a b d e - b^{2} d e + i \, a b c f + b^{2} c f\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - i\right ) + 2 \, {\left (-i \, a b d f x - b^{2} d f x - i \, a b c f - b^{2} c f\right )} \log \left (i \, \cosh \left (d x + c\right ) + i \, \sinh \left (d x + c\right ) + 1\right ) + 2 \, {\left (i \, a b d f x - b^{2} d f x + i \, a b c f - b^{2} c f\right )} \log \left (-i \, \cosh \left (d x + c\right ) - i \, \sinh \left (d x + c\right ) + 1\right )}{2 \, {\left (a^{2} b + b^{3}\right )} d^{2}} \]

[In]

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

[Out]

-1/2*((a^2 + b^2)*d^2*f*x^2 + 2*(a^2 + b^2)*d^2*e*x - 2*a^2*f*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) + (b*co
sh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) - 2*a^2*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) + 2*(I*a*b*f - b^2*f)*dilog(I*co
sh(d*x + c) + I*sinh(d*x + c)) + 2*(-I*a*b*f - b^2*f)*dilog(-I*cosh(d*x + c) - I*sinh(d*x + c)) - 2*(a^2*d*e -
a^2*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) - 2*(a^2*d*e - a^2*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) - 2*(a^2*d*f*x + a^2*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) - 2*(a^2
*d*f*x + a^2*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) + 2*(I*a*b*d*e - b^2*d*e - I*a*b*c*f + b^2*c*f)*log(cosh(d*x + c) + sinh(d*x + c) + I) + 2*(-I
*a*b*d*e - b^2*d*e + I*a*b*c*f + b^2*c*f)*log(cosh(d*x + c) + sinh(d*x + c) - I) + 2*(-I*a*b*d*f*x - b^2*d*f*x
- I*a*b*c*f - b^2*c*f)*log(I*cosh(d*x + c) + I*sinh(d*x + c) + 1) + 2*(I*a*b*d*f*x - b^2*d*f*x + I*a*b*c*f -
b^2*c*f)*log(-I*cosh(d*x + c) - I*sinh(d*x + c) + 1))/((a^2*b + b^3)*d^2)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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