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

Optimal. Leaf size=516 \[ -\frac{i a^3 f \text{PolyLog}\left (2,-i e^{c+d x}\right )}{b^2 d^2 \left (a^2+b^2\right )}+\frac{i a^3 f \text{PolyLog}\left (2,i e^{c+d x}\right )}{b^2 d^2 \left (a^2+b^2\right )}+\frac{a^2 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b d^2 \left (a^2+b^2\right )}+\frac{a^2 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b d^2 \left (a^2+b^2\right )}-\frac{a^2 f \text{PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 b d^2 \left (a^2+b^2\right )}+\frac{i a f \text{PolyLog}\left (2,-i e^{c+d x}\right )}{b^2 d^2}-\frac{i a f \text{PolyLog}\left (2,i e^{c+d x}\right )}{b^2 d^2}+\frac{f \text{PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 b d^2}+\frac{a^2 (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b d \left (a^2+b^2\right )}+\frac{a^2 (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b d \left (a^2+b^2\right )}-\frac{a^2 (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{b d \left (a^2+b^2\right )}+\frac{2 a^3 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d \left (a^2+b^2\right )}-\frac{2 a (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d}+\frac{(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{b d}-\frac{(e+f x)^2}{2 b f} \]

[Out]

-(e + f*x)^2/(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)

________________________________________________________________________________________

Rubi [A]  time = 0.775852, antiderivative size = 516, normalized size of antiderivative = 1., number of steps used = 30, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5581, 3718, 2190, 2279, 2391, 5567, 4180, 5573, 5561, 6742} \[ -\frac{i a^3 f \text{PolyLog}\left (2,-i e^{c+d x}\right )}{b^2 d^2 \left (a^2+b^2\right )}+\frac{i a^3 f \text{PolyLog}\left (2,i e^{c+d x}\right )}{b^2 d^2 \left (a^2+b^2\right )}+\frac{a^2 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b d^2 \left (a^2+b^2\right )}+\frac{a^2 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b d^2 \left (a^2+b^2\right )}-\frac{a^2 f \text{PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 b d^2 \left (a^2+b^2\right )}+\frac{i a f \text{PolyLog}\left (2,-i e^{c+d x}\right )}{b^2 d^2}-\frac{i a f \text{PolyLog}\left (2,i e^{c+d x}\right )}{b^2 d^2}+\frac{f \text{PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 b d^2}+\frac{a^2 (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b d \left (a^2+b^2\right )}+\frac{a^2 (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b d \left (a^2+b^2\right )}-\frac{a^2 (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{b d \left (a^2+b^2\right )}+\frac{2 a^3 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d \left (a^2+b^2\right )}-\frac{2 a (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d}+\frac{(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{b d}-\frac{(e+f x)^2}{2 b f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(e + f*x)^2/(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 5581

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 3718

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> -Simp[(I*(c + d*x)^(m +
1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(1 + E^(2*(-(I*e) + f*fz*x))), x],
x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 5567

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*Sec
h[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 4180

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c
+ d*x)^m*ArcTanh[E^(-(I*e) + f*fz*x)/E^(I*k*Pi)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*
Log[1 - E^(-(I*e) + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e)
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 5573

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[b^2/(a^2 + b^2), Int[((e + f*x)^m*Sech[c + d*x]^(n - 2))/(a + b*Sinh[c + d*x]), x], x] + Dist[1/(
a^2 + b^2), Int[(e + f*x)^m*Sech[c + d*x]^n*(a - b*Sinh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && I
GtQ[m, 0] && NeQ[a^2 + b^2, 0] && IGtQ[n, 0]

Rule 5561

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :
> -Simp[(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[((e + f*x)^m*E^(c + d*x))/(a - Rt[a^2 + b^2, 2] + b*E^(c +
d*x)), x] + Int[((e + f*x)^m*E^(c + d*x))/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x)), x]) /; FreeQ[{a, b, c, d, e,
f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]

Rule 6742

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

Rubi steps

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

Mathematica [A]  time = 2.87536, size = 438, normalized size = 0.85 \[ \frac{\frac{a^2 \left (f \text{PolyLog}\left (2,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )+f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )+f (c+d x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )+f (c+d x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )+d e \log (a+b \sinh (c+d x))-c f \log (a+b \sinh (c+d x))-\frac{1}{2} f (c+d x)^2\right )}{b}+i a f \text{PolyLog}(2,-i (\sinh (c+d x)+\cosh (c+d x)))-i a f \text{PolyLog}(2,i (\sinh (c+d x)+\cosh (c+d x)))+\frac{1}{2} b f \text{PolyLog}(2,-\sinh (2 (c+d x))-\cosh (2 (c+d x)))-2 a d e \tan ^{-1}(\sinh (c+d x)+\cosh (c+d x))+2 a c f \tan ^{-1}(\sinh (c+d x)+\cosh (c+d x))-2 a f (c+d x) \tan ^{-1}(\sinh (c+d x)+\cosh (c+d x))-b d e (c+d x)+b d e \log (\sinh (2 (c+d x))+\cosh (2 (c+d x))+1)-\frac{1}{2} b f (c+d x)^2+b c f (c+d x)-b c f \log (\sinh (2 (c+d x))+\cosh (2 (c+d x))+1)+b f (c+d x) \log (\sinh (2 (c+d x))+\cosh (2 (c+d x))+1)}{d^2 \left (a^2+b^2\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.225, size = 3882, normalized size = 7.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Fricas [A]  time = 2.5035, size = 1767, normalized size = 3.42 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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