3.413 \(\int \frac{(e+f x) \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\)

Optimal. Leaf size=454 \[ -\frac{a^3 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b d^2 \left (a^2+b^2\right )^{3/2}}+\frac{a^3 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b d^2 \left (a^2+b^2\right )^{3/2}}+\frac{a^3 f \tan ^{-1}(\sinh (c+d x))}{b^2 d^2 \left (a^2+b^2\right )}+\frac{a^4 f \log (\cosh (c+d x))}{b^3 d^2 \left (a^2+b^2\right )}-\frac{a^2 f \log (\cosh (c+d x))}{b^3 d^2}-\frac{a^3 (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b d \left (a^2+b^2\right )^{3/2}}+\frac{a^3 (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b d \left (a^2+b^2\right )^{3/2}}-\frac{a^4 (e+f x) \tanh (c+d x)}{b^3 d \left (a^2+b^2\right )}+\frac{a^2 (e+f x) \tanh (c+d x)}{b^3 d}-\frac{a^3 (e+f x) \text{sech}(c+d x)}{b^2 d \left (a^2+b^2\right )}-\frac{a f \tan ^{-1}(\sinh (c+d x))}{b^2 d^2}+\frac{a (e+f x) \text{sech}(c+d x)}{b^2 d}+\frac{f \log (\cosh (c+d x))}{b d^2}-\frac{(e+f x) \tanh (c+d x)}{b d}+\frac{e x}{b}+\frac{f x^2}{2 b} \]

[Out]

(e*x)/b + (f*x^2)/(2*b) - (a*f*ArcTan[Sinh[c + d*x]])/(b^2*d^2) + (a^3*f*ArcTan[Sinh[c + d*x]])/(b^2*(a^2 + b^
2)*d^2) - (a^3*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)^(3/2)*d) + (a^3*(e + f
*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)^(3/2)*d) - (a^2*f*Log[Cosh[c + d*x]])/(b^3*
d^2) + (f*Log[Cosh[c + d*x]])/(b*d^2) + (a^4*f*Log[Cosh[c + d*x]])/(b^3*(a^2 + b^2)*d^2) - (a^3*f*PolyLog[2, -
((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^2) + (a^3*f*PolyLog[2, -((b*E^(c + d*x))/(a +
Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^2) + (a*(e + f*x)*Sech[c + d*x])/(b^2*d) - (a^3*(e + f*x)*Sech[c +
d*x])/(b^2*(a^2 + b^2)*d) + (a^2*(e + f*x)*Tanh[c + d*x])/(b^3*d) - ((e + f*x)*Tanh[c + d*x])/(b*d) - (a^4*(e
+ f*x)*Tanh[c + d*x])/(b^3*(a^2 + b^2)*d)

________________________________________________________________________________________

Rubi [A]  time = 0.891763, antiderivative size = 454, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 15, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.469, Rules used = {5581, 3720, 3475, 5567, 5451, 3770, 5583, 4184, 5573, 3322, 2264, 2190, 2279, 2391, 6742} \[ -\frac{a^3 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b d^2 \left (a^2+b^2\right )^{3/2}}+\frac{a^3 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b d^2 \left (a^2+b^2\right )^{3/2}}+\frac{a^3 f \tan ^{-1}(\sinh (c+d x))}{b^2 d^2 \left (a^2+b^2\right )}+\frac{a^4 f \log (\cosh (c+d x))}{b^3 d^2 \left (a^2+b^2\right )}-\frac{a^2 f \log (\cosh (c+d x))}{b^3 d^2}-\frac{a^3 (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b d \left (a^2+b^2\right )^{3/2}}+\frac{a^3 (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b d \left (a^2+b^2\right )^{3/2}}-\frac{a^4 (e+f x) \tanh (c+d x)}{b^3 d \left (a^2+b^2\right )}+\frac{a^2 (e+f x) \tanh (c+d x)}{b^3 d}-\frac{a^3 (e+f x) \text{sech}(c+d x)}{b^2 d \left (a^2+b^2\right )}-\frac{a f \tan ^{-1}(\sinh (c+d x))}{b^2 d^2}+\frac{a (e+f x) \text{sech}(c+d x)}{b^2 d}+\frac{f \log (\cosh (c+d x))}{b d^2}-\frac{(e+f x) \tanh (c+d x)}{b d}+\frac{e x}{b}+\frac{f x^2}{2 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e*x)/b + (f*x^2)/(2*b) - (a*f*ArcTan[Sinh[c + d*x]])/(b^2*d^2) + (a^3*f*ArcTan[Sinh[c + d*x]])/(b^2*(a^2 + b^
2)*d^2) - (a^3*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)^(3/2)*d) + (a^3*(e + f
*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)^(3/2)*d) - (a^2*f*Log[Cosh[c + d*x]])/(b^3*
d^2) + (f*Log[Cosh[c + d*x]])/(b*d^2) + (a^4*f*Log[Cosh[c + d*x]])/(b^3*(a^2 + b^2)*d^2) - (a^3*f*PolyLog[2, -
((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^2) + (a^3*f*PolyLog[2, -((b*E^(c + d*x))/(a +
Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^2) + (a*(e + f*x)*Sech[c + d*x])/(b^2*d) - (a^3*(e + f*x)*Sech[c +
d*x])/(b^2*(a^2 + b^2)*d) + (a^2*(e + f*x)*Tanh[c + d*x])/(b^3*d) - ((e + f*x)*Tanh[c + d*x])/(b*d) - (a^4*(e
+ f*x)*Tanh[c + d*x])/(b^3*(a^2 + b^2)*d)

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 3720

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(c + d*x)^m*(b*Tan[e
+ f*x])^(n - 1))/(f*(n - 1)), x] + (-Dist[(b*d*m)/(f*(n - 1)), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && GtQ[m, 0]

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

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 5451

Int[((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.)*Tanh[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> -Si
mp[((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 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 5583

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 4184

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp[((c + d*x)^m*Cot[e + f*x])/f, x]
+ Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[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 3322

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

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && 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 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 ^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac{\int (e+f x) \tanh ^2(c+d x) \, dx}{b}-\frac{a \int \frac{(e+f x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=-\frac{(e+f x) \tanh (c+d x)}{b d}-\frac{a \int (e+f x) \text{sech}(c+d x) \tanh (c+d x) \, dx}{b^2}+\frac{a^2 \int \frac{(e+f x) \text{sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}+\frac{\int (e+f x) \, dx}{b}+\frac{f \int \tanh (c+d x) \, dx}{b d}\\ &=\frac{e x}{b}+\frac{f x^2}{2 b}+\frac{f \log (\cosh (c+d x))}{b d^2}+\frac{a (e+f x) \text{sech}(c+d x)}{b^2 d}-\frac{(e+f x) \tanh (c+d x)}{b d}+\frac{a^2 \int (e+f x) \text{sech}^2(c+d x) \, dx}{b^3}-\frac{a^3 \int \frac{(e+f x) \text{sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b^3}-\frac{(a f) \int \text{sech}(c+d x) \, dx}{b^2 d}\\ &=\frac{e x}{b}+\frac{f x^2}{2 b}-\frac{a f \tan ^{-1}(\sinh (c+d x))}{b^2 d^2}+\frac{f \log (\cosh (c+d x))}{b d^2}+\frac{a (e+f x) \text{sech}(c+d x)}{b^2 d}+\frac{a^2 (e+f x) \tanh (c+d x)}{b^3 d}-\frac{(e+f x) \tanh (c+d x)}{b d}-\frac{a^3 \int (e+f x) \text{sech}^2(c+d x) (a-b \sinh (c+d x)) \, dx}{b^3 \left (a^2+b^2\right )}-\frac{a^3 \int \frac{e+f x}{a+b \sinh (c+d x)} \, dx}{b \left (a^2+b^2\right )}-\frac{\left (a^2 f\right ) \int \tanh (c+d x) \, dx}{b^3 d}\\ &=\frac{e x}{b}+\frac{f x^2}{2 b}-\frac{a f \tan ^{-1}(\sinh (c+d x))}{b^2 d^2}-\frac{a^2 f \log (\cosh (c+d x))}{b^3 d^2}+\frac{f \log (\cosh (c+d x))}{b d^2}+\frac{a (e+f x) \text{sech}(c+d x)}{b^2 d}+\frac{a^2 (e+f x) \tanh (c+d x)}{b^3 d}-\frac{(e+f x) \tanh (c+d x)}{b d}-\frac{a^3 \int \left (a (e+f x) \text{sech}^2(c+d x)-b (e+f x) \text{sech}(c+d x) \tanh (c+d x)\right ) \, dx}{b^3 \left (a^2+b^2\right )}-\frac{\left (2 a^3\right ) \int \frac{e^{c+d x} (e+f x)}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{b \left (a^2+b^2\right )}\\ &=\frac{e x}{b}+\frac{f x^2}{2 b}-\frac{a f \tan ^{-1}(\sinh (c+d x))}{b^2 d^2}-\frac{a^2 f \log (\cosh (c+d x))}{b^3 d^2}+\frac{f \log (\cosh (c+d x))}{b d^2}+\frac{a (e+f x) \text{sech}(c+d x)}{b^2 d}+\frac{a^2 (e+f x) \tanh (c+d x)}{b^3 d}-\frac{(e+f x) \tanh (c+d x)}{b d}-\frac{\left (2 a^3\right ) \int \frac{e^{c+d x} (e+f x)}{2 a-2 \sqrt{a^2+b^2}+2 b e^{c+d x}} \, dx}{\left (a^2+b^2\right )^{3/2}}+\frac{\left (2 a^3\right ) \int \frac{e^{c+d x} (e+f x)}{2 a+2 \sqrt{a^2+b^2}+2 b e^{c+d x}} \, dx}{\left (a^2+b^2\right )^{3/2}}-\frac{a^4 \int (e+f x) \text{sech}^2(c+d x) \, dx}{b^3 \left (a^2+b^2\right )}+\frac{a^3 \int (e+f x) \text{sech}(c+d x) \tanh (c+d x) \, dx}{b^2 \left (a^2+b^2\right )}\\ &=\frac{e x}{b}+\frac{f x^2}{2 b}-\frac{a f \tan ^{-1}(\sinh (c+d x))}{b^2 d^2}-\frac{a^3 (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 )^{3/2} d}+\frac{a^3 (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 )^{3/2} d}-\frac{a^2 f \log (\cosh (c+d x))}{b^3 d^2}+\frac{f \log (\cosh (c+d x))}{b d^2}+\frac{a (e+f x) \text{sech}(c+d x)}{b^2 d}-\frac{a^3 (e+f x) \text{sech}(c+d x)}{b^2 \left (a^2+b^2\right ) d}+\frac{a^2 (e+f x) \tanh (c+d x)}{b^3 d}-\frac{(e+f x) \tanh (c+d x)}{b d}-\frac{a^4 (e+f x) \tanh (c+d x)}{b^3 \left (a^2+b^2\right ) d}+\frac{\left (a^3 f\right ) \int \log \left (1+\frac{2 b e^{c+d x}}{2 a-2 \sqrt{a^2+b^2}}\right ) \, dx}{b \left (a^2+b^2\right )^{3/2} d}-\frac{\left (a^3 f\right ) \int \log \left (1+\frac{2 b e^{c+d x}}{2 a+2 \sqrt{a^2+b^2}}\right ) \, dx}{b \left (a^2+b^2\right )^{3/2} d}+\frac{\left (a^4 f\right ) \int \tanh (c+d x) \, dx}{b^3 \left (a^2+b^2\right ) d}+\frac{\left (a^3 f\right ) \int \text{sech}(c+d x) \, dx}{b^2 \left (a^2+b^2\right ) d}\\ &=\frac{e x}{b}+\frac{f x^2}{2 b}-\frac{a f \tan ^{-1}(\sinh (c+d x))}{b^2 d^2}+\frac{a^3 f \tan ^{-1}(\sinh (c+d x))}{b^2 \left (a^2+b^2\right ) d^2}-\frac{a^3 (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 )^{3/2} d}+\frac{a^3 (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 )^{3/2} d}-\frac{a^2 f \log (\cosh (c+d x))}{b^3 d^2}+\frac{f \log (\cosh (c+d x))}{b d^2}+\frac{a^4 f \log (\cosh (c+d x))}{b^3 \left (a^2+b^2\right ) d^2}+\frac{a (e+f x) \text{sech}(c+d x)}{b^2 d}-\frac{a^3 (e+f x) \text{sech}(c+d x)}{b^2 \left (a^2+b^2\right ) d}+\frac{a^2 (e+f x) \tanh (c+d x)}{b^3 d}-\frac{(e+f x) \tanh (c+d x)}{b d}-\frac{a^4 (e+f x) \tanh (c+d x)}{b^3 \left (a^2+b^2\right ) d}+\frac{\left (a^3 f\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 b x}{2 a-2 \sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}-\frac{\left (a^3 f\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 b x}{2 a+2 \sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}\\ &=\frac{e x}{b}+\frac{f x^2}{2 b}-\frac{a f \tan ^{-1}(\sinh (c+d x))}{b^2 d^2}+\frac{a^3 f \tan ^{-1}(\sinh (c+d x))}{b^2 \left (a^2+b^2\right ) d^2}-\frac{a^3 (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 )^{3/2} d}+\frac{a^3 (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 )^{3/2} d}-\frac{a^2 f \log (\cosh (c+d x))}{b^3 d^2}+\frac{f \log (\cosh (c+d x))}{b d^2}+\frac{a^4 f \log (\cosh (c+d x))}{b^3 \left (a^2+b^2\right ) d^2}-\frac{a^3 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 )^{3/2} d^2}+\frac{a^3 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 )^{3/2} d^2}+\frac{a (e+f x) \text{sech}(c+d x)}{b^2 d}-\frac{a^3 (e+f x) \text{sech}(c+d x)}{b^2 \left (a^2+b^2\right ) d}+\frac{a^2 (e+f x) \tanh (c+d x)}{b^3 d}-\frac{(e+f x) \tanh (c+d x)}{b d}-\frac{a^4 (e+f x) \tanh (c+d x)}{b^3 \left (a^2+b^2\right ) d}\\ \end{align*}

Mathematica [A]  time = 4.54347, size = 317, normalized size = 0.7 \[ \frac{\frac{2 a^3 \left (-f \text{PolyLog}\left (2,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )+f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )+2 d e \tanh ^{-1}\left (\frac{a+b e^{c+d x}}{\sqrt{a^2+b^2}}\right )-f (c+d x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )+f (c+d x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )-2 c f \tanh ^{-1}\left (\frac{a+b e^{c+d x}}{\sqrt{a^2+b^2}}\right )\right )}{b \left (a^2+b^2\right )^{3/2}}+\frac{2 d (e+f x) \text{sech}(c+d x) (a-b \sinh (c+d x))}{a^2+b^2}-\frac{4 a f \tan ^{-1}\left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{a^2+b^2}+\frac{2 b f \log (\cosh (c+d x))}{a^2+b^2}-\frac{(c+d x) (c f-d (2 e+f x))}{b}}{2 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(((c + d*x)*(c*f - d*(2*e + f*x)))/b) - (4*a*f*ArcTan[Tanh[(c + d*x)/2]])/(a^2 + b^2) + (2*b*f*Log[Cosh[c +
d*x]])/(a^2 + b^2) + (2*a^3*(2*d*e*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] - 2*c*f*ArcTanh[(a + b*E^(c +
d*x))/Sqrt[a^2 + b^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])] - f*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + f*PolyLog[2, -((b*
E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(b*(a^2 + b^2)^(3/2)) + (2*d*(e + f*x)*Sech[c + d*x]*(a - b*Sinh[c + d*
x]))/(a^2 + b^2))/(2*d^2)

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Maple [B]  time = 0.214, size = 1897, normalized size = 4.2 \begin{align*} \text{result 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)^2/(a+b*sinh(d*x+c)),x)

[Out]

1/2*f*x^2/b+1/(a^2+b^2)^2/d^2*f*b*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)*a^2+2/(a^2+b^2)/d^2*f*b^3/(2*a^2+2*b^2
)*ln(1+exp(2*d*x+2*c))-1/(a^2+b^2)/d^2*f*b^3/(2*a^2+2*b^2)*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)-4/(a^2+b^2)/d
^2*a^3*f/(2*a^2+2*b^2)*arctan(exp(d*x+c))-2/(a^2+b^2)^(5/2)/d^2*f*b*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)
^(1/2))*a^3-2/(a^2+b^2)^(5/2)/d^2*f*b^3*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))*a-4/(a^2+b^2)/d^2*f*
b^2/(2*a^2+2*b^2)*a*arctan(exp(d*x+c))+2/(a^2+b^2)/d^2*a^2*b*f/(2*a^2+2*b^2)*ln(1+exp(2*d*x+2*c))-2/(a^2+b^2)/
d^2*a^2*b*f/(2*a^2+2*b^2)*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)+e*x/b+2/(a^2+b^2)^(3/2)/b/d^2*a^5*f/(2*a^2+2*b
^2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c-2/(a^2+b^2)^(3/2)/b/d^2*a^5*f*c/(2*a^2+2*b^2)*a
rctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-2/(a^2+b^2)^(3/2)/d^2*a^3*b*f*c/(2*a^2+2*b^2)*arctanh(1/2*(2*
b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-2/(a^2+b^2)^(3/2)/b/d*a^5*f/(2*a^2+2*b^2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)
-a)/(-a+(a^2+b^2)^(1/2)))*x+2/(a^2+b^2)^(3/2)/b/d*a^5*f/(2*a^2+2*b^2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(
a^2+b^2)^(1/2)))*x-2/(a^2+b^2)^(3/2)/d^2*a^3*b*f/(2*a^2+2*b^2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b
^2)^(1/2)))*c+2/(a^2+b^2)^(3/2)/d^2*a^3*b*f/(2*a^2+2*b^2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/
2)))*c+2/(a^2+b^2)^(3/2)/d*a^3*b*f/(2*a^2+2*b^2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x-2/
(a^2+b^2)^(3/2)/d*a^3*b*f/(2*a^2+2*b^2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x-2/(a^2+b^
2)^(3/2)/b/d^2*a^5*f/(2*a^2+2*b^2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c-2/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/(a^2+b^2)^(3/2)/d*a^3*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/(a^2+b^2)^(3/2)/d^2*a^3*b*f/(2*a^2+2*b^2)*di
log((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))-2/(a^2+b^2)^(3/2)/d^2*a^3*b*f/(2*a^2+2*b^2)*dilog((-
b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))+2/(a^2+b^2)^(1/2)/d^2*a*b*f/(2*a^2+2*b^2)*arctanh(1/2*(2
*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+2/(a^2+b^2)^(3/2)/d^2*a*b^3*f/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*
a)/(a^2+b^2)^(1/2))+2/(a^2+b^2)^(3/2)/d^2*a^3*b*f/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/
2))+2/(a^2+b^2)^(3/2)/b/d*a^5*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/(a^2+b^2)^(3
/2)/b/d^2*a^5*f/(2*a^2+2*b^2)*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))-2/(a^2+b^2)^(3/2)/b/
d^2*a^5*f/(2*a^2+2*b^2)*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))-2/(a^2+b^2)/d^2*b*f*ln(e
xp(d*x+c))+1/2/(a^2+b^2)^2/d^2*f*b^3*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)+2*(f*x+e)*(a*exp(d*x+c)+b)/d/(a^2+b
^2)/(1+exp(2*d*x+2*c))+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))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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)^2/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.01406, size = 3753, normalized size = 8.27 \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)^2/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

1/2*((a^4 + 2*a^2*b^2 + b^4)*d^2*f*x^2 + 2*(a^4 + 2*a^2*b^2 + b^4)*d^2*e*x + 4*(a^2*b^2 + b^4)*d*e + ((a^4 + 2
*a^2*b^2 + b^4)*d^2*f*x^2 + 2*((a^4 + 2*a^2*b^2 + b^4)*d^2*e - 2*(a^2*b^2 + b^4)*d*f)*x)*cosh(d*x + c)^2 + ((a
^4 + 2*a^2*b^2 + b^4)*d^2*f*x^2 + 2*((a^4 + 2*a^2*b^2 + b^4)*d^2*e - 2*(a^2*b^2 + b^4)*d*f)*x)*sinh(d*x + c)^2
- 2*(a^3*b*f*cosh(d*x + c)^2 + 2*a^3*b*f*cosh(d*x + c)*sinh(d*x + c) + a^3*b*f*sinh(d*x + c)^2 + a^3*b*f)*sqr
t((a^2 + b^2)/b^2)*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*(a^3*b*f*cosh(d*x + c)^2 + 2*a^3*b*f*cosh(d*x + c)*sinh(d*x + c) + a^3*b*f*sinh(d*x
+ c)^2 + a^3*b*f)*sqrt((a^2 + b^2)/b^2)*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*(a^3*b*d*e - a^3*b*c*f + (a^3*b*d*e - a^3*b*c*f)*cosh(d*x + c)^
2 + 2*(a^3*b*d*e - a^3*b*c*f)*cosh(d*x + c)*sinh(d*x + c) + (a^3*b*d*e - a^3*b*c*f)*sinh(d*x + c)^2)*sqrt((a^2
+ b^2)/b^2)*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^3*b*d*e - a^3
*b*c*f + (a^3*b*d*e - a^3*b*c*f)*cosh(d*x + c)^2 + 2*(a^3*b*d*e - a^3*b*c*f)*cosh(d*x + c)*sinh(d*x + c) + (a^
3*b*d*e - a^3*b*c*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sq
rt((a^2 + b^2)/b^2) + 2*a) - 2*(a^3*b*d*f*x + a^3*b*c*f + (a^3*b*d*f*x + a^3*b*c*f)*cosh(d*x + c)^2 + 2*(a^3*b
*d*f*x + a^3*b*c*f)*cosh(d*x + c)*sinh(d*x + c) + (a^3*b*d*f*x + a^3*b*c*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/
b^2)*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^3*b*d*f*x + a^3*b*c*f + (a^3*b*d*f*x + a^3*b*c*f)*cosh(d*x + c)^2 + 2*(a^3*b*d*f*x + a^3*b*c*f)*cosh
(d*x + c)*sinh(d*x + c) + (a^3*b*d*f*x + a^3*b*c*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*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) - 4*((a^3*b + a*b^3)*
f*cosh(d*x + c)^2 + 2*(a^3*b + a*b^3)*f*cosh(d*x + c)*sinh(d*x + c) + (a^3*b + a*b^3)*f*sinh(d*x + c)^2 + (a^3
*b + a*b^3)*f)*arctan(cosh(d*x + c) + sinh(d*x + c)) + 4*((a^3*b + a*b^3)*d*f*x + (a^3*b + a*b^3)*d*e)*cosh(d*
x + c) + 2*((a^2*b^2 + b^4)*f*cosh(d*x + c)^2 + 2*(a^2*b^2 + b^4)*f*cosh(d*x + c)*sinh(d*x + c) + (a^2*b^2 + b
^4)*f*sinh(d*x + c)^2 + (a^2*b^2 + b^4)*f)*log(2*cosh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) + 2*(2*(a^3*b
+ a*b^3)*d*f*x + 2*(a^3*b + a*b^3)*d*e + ((a^4 + 2*a^2*b^2 + b^4)*d^2*f*x^2 + 2*((a^4 + 2*a^2*b^2 + b^4)*d^2*e
- 2*(a^2*b^2 + b^4)*d*f)*x)*cosh(d*x + c))*sinh(d*x + c))/((a^4*b + 2*a^2*b^3 + b^5)*d^2*cosh(d*x + c)^2 + 2*
(a^4*b + 2*a^2*b^3 + b^5)*d^2*cosh(d*x + c)*sinh(d*x + c) + (a^4*b + 2*a^2*b^3 + b^5)*d^2*sinh(d*x + c)^2 + (a
^4*b + 2*a^2*b^3 + b^5)*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 ^{2}{\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)**2/(a+b*sinh(d*x+c)),x)

[Out]

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

[Out]

Timed out