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

Optimal. Leaf size=294 \[ \frac{f \sqrt{a^2+b^2} \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 d^2}-\frac{f \sqrt{a^2+b^2} \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^2 d^2}+\frac{b f \text{PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^2}-\frac{b f \text{PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^2}+\frac{\sqrt{a^2+b^2} (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a^2 d}-\frac{\sqrt{a^2+b^2} (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a^2 d}+\frac{2 b (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d}+\frac{f \log (\sinh (c+d x))}{a d^2}-\frac{(e+f x) \coth (c+d x)}{a d} \]

[Out]

(2*b*(e + f*x)*ArcTanh[E^(c + d*x)])/(a^2*d) - ((e + f*x)*Coth[c + d*x])/(a*d) + (Sqrt[a^2 + b^2]*(e + f*x)*Lo
g[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^2*d) - (Sqrt[a^2 + b^2]*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a
+ Sqrt[a^2 + b^2])])/(a^2*d) + (f*Log[Sinh[c + d*x]])/(a*d^2) + (b*f*PolyLog[2, -E^(c + d*x)])/(a^2*d^2) - (b*
f*PolyLog[2, E^(c + d*x)])/(a^2*d^2) + (Sqrt[a^2 + b^2]*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))]
)/(a^2*d^2) - (Sqrt[a^2 + b^2]*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*d^2)

________________________________________________________________________________________

Rubi [A]  time = 0.688467, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 14, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {5569, 3720, 3475, 5585, 5450, 3296, 2637, 4182, 2279, 2391, 5565, 3322, 2264, 2190} \[ \frac{f \sqrt{a^2+b^2} \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 d^2}-\frac{f \sqrt{a^2+b^2} \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^2 d^2}+\frac{b f \text{PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^2}-\frac{b f \text{PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^2}+\frac{\sqrt{a^2+b^2} (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a^2 d}-\frac{\sqrt{a^2+b^2} (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a^2 d}+\frac{2 b (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d}+\frac{f \log (\sinh (c+d x))}{a d^2}-\frac{(e+f x) \coth (c+d x)}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b*(e + f*x)*ArcTanh[E^(c + d*x)])/(a^2*d) - ((e + f*x)*Coth[c + d*x])/(a*d) + (Sqrt[a^2 + b^2]*(e + f*x)*Lo
g[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^2*d) - (Sqrt[a^2 + b^2]*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a
+ Sqrt[a^2 + b^2])])/(a^2*d) + (f*Log[Sinh[c + d*x]])/(a*d^2) + (b*f*PolyLog[2, -E^(c + d*x)])/(a^2*d^2) - (b*
f*PolyLog[2, E^(c + d*x)])/(a^2*d^2) + (Sqrt[a^2 + b^2]*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))]
)/(a^2*d^2) - (Sqrt[a^2 + b^2]*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*d^2)

Rule 5569

Int[(Coth[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[1/a, Int[(e + f*x)^m*Coth[c + d*x]^n, x], x] - Dist[b/a, Int[((e + f*x)^m*Cosh[c + d*x]*Coth[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 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 5585

Int[(Cosh[(c_.) + (d_.)*(x_)]^(p_.)*Coth[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[(e + f*x)^m*Cosh[c + d*x]^p*Coth[c + d*x]^n, x], x] - Dis
t[b/a, Int[((e + f*x)^m*Cosh[c + d*x]^(p + 1)*Coth[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 5450

Int[Cosh[(a_.) + (b_.)*(x_)]^(n_.)*Coth[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int
[(c + d*x)^m*Cosh[a + b*x]^n*Coth[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Cosh[a + b*x]^(n - 2)*Coth[a + b*x]^p
, x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]

Rule 3296

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

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 4182

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*Ar
cTanh[E^(-(I*e) + f*fz*x)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 - E^(-(I*e) + f*
fz*x)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e) + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, 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 5565

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

Rubi steps

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

Mathematica [A]  time = 3.66865, size = 364, normalized size = 1.24 \[ \frac{2 \sqrt{a^2+b^2} \left (f \text{PolyLog}\left (2,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )-f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )-2 d e \tanh ^{-1}\left (\frac{a+b e^{c+d x}}{\sqrt{a^2+b^2}}\right )+f (c+d x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )-f (c+d x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )+2 c f \tanh ^{-1}\left (\frac{a+b e^{c+d x}}{\sqrt{a^2+b^2}}\right )\right )+2 b f \left (-\text{PolyLog}\left (2,-e^{-c-d x}\right )+\text{PolyLog}\left (2,e^{-c-d x}\right )-(c+d x) \left (\log \left (1-e^{-c-d x}\right )-\log \left (e^{-c-d x}+1\right )\right )\right )-a d (e+f x) \tanh \left (\frac{1}{2} (c+d x)\right )-a d (e+f x) \coth \left (\frac{1}{2} (c+d x)\right )+2 a f \log (\sinh (c+d x))-2 b d e \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )+2 b c f \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{2 a^2 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.213, size = 1017, normalized size = 3.5 \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)*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x)

[Out]

-2/d*e/(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^2*f/(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/d^2*f/(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/a^2/d*b^2*e/(a^2+b^2)^(1/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+1/a^2/d
^2*b*f*c*ln(exp(d*x+c)-1)+1/a^2/d^2*b^2*f/(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/a^2/d^2*b^2*f/(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/a^2/d*
b*f*ln(exp(d*x+c)+1)*x-1/a^2/d*b^2*f/(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-1/a^2/d^2*b^2*f/(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/a^2/d^2*b^2*f*c
/(a^2+b^2)^(1/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+1/a^2/d*b^2*f/(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+1/a^2/d^2*b^2*f/(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/d*(f*x+e)/a/(exp(2*d*x+2*c)-1)-1/a^2/d*b*e*ln(exp(d*x+c)-1)+1/a^2/d*b*e*ln(ex
p(d*x+c)+1)+1/a^2/d^2*b*f*dilog(exp(d*x+c))+1/a^2/d^2*b*f*dilog(exp(d*x+c)+1)-2/d^2/a*f*ln(exp(d*x+c))+1/d^2/a
*f*ln(exp(d*x+c)-1)+1/d^2/a*f*ln(exp(d*x+c)+1)+1/d*f/(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+1/d^2*f/(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-1/d*f
/(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-1/d^2*f/(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/d^2*f*c/(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)*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.35579, size = 3430, normalized size = 11.67 \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)*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e + f x\right ) \coth ^{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)*coth(d*x+c)**2/(a+b*sinh(d*x+c)),x)

[Out]

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

[Out]

Timed out