### 3.11 $$\int \frac{(a+b \text{csch}^{-1}(c+d x))^2}{e+f x} \, dx$$

Optimal. Leaf size=475 $\frac{2 b \left (a+b \text{csch}^{-1}(c+d x)\right ) \text{PolyLog}\left (2,-\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{f-\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}\right )}{f}+\frac{2 b \left (a+b \text{csch}^{-1}(c+d x)\right ) \text{PolyLog}\left (2,-\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+f}\right )}{f}-\frac{b \text{PolyLog}\left (2,e^{2 \text{csch}^{-1}(c+d x)}\right ) \left (a+b \text{csch}^{-1}(c+d x)\right )}{f}-\frac{2 b^2 \text{PolyLog}\left (3,-\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{f-\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}\right )}{f}-\frac{2 b^2 \text{PolyLog}\left (3,-\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+f}\right )}{f}+\frac{b^2 \text{PolyLog}\left (3,e^{2 \text{csch}^{-1}(c+d x)}\right )}{2 f}+\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2 \log \left (\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{f-\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}+1\right )}{f}+\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2 \log \left (\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+f}+1\right )}{f}-\frac{\log \left (1-e^{2 \text{csch}^{-1}(c+d x)}\right ) \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f}$

[Out]

-(((a + b*ArcCsch[c + d*x])^2*Log[1 - E^(2*ArcCsch[c + d*x])])/f) + ((a + b*ArcCsch[c + d*x])^2*Log[1 + (E^Arc
Csch[c + d*x]*(d*e - c*f))/(f - Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2])])/f + ((a + b*ArcCsch[c + d*x])^2*L
og[1 + (E^ArcCsch[c + d*x]*(d*e - c*f))/(f + Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2])])/f - (b*(a + b*ArcCsc
h[c + d*x])*PolyLog[2, E^(2*ArcCsch[c + d*x])])/f + (2*b*(a + b*ArcCsch[c + d*x])*PolyLog[2, -((E^ArcCsch[c +
d*x]*(d*e - c*f))/(f - Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]))])/f + (2*b*(a + b*ArcCsch[c + d*x])*PolyLog
[2, -((E^ArcCsch[c + d*x]*(d*e - c*f))/(f + Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]))])/f + (b^2*PolyLog[3,
E^(2*ArcCsch[c + d*x])])/(2*f) - (2*b^2*PolyLog[3, -((E^ArcCsch[c + d*x]*(d*e - c*f))/(f - Sqrt[d^2*e^2 - 2*c*
d*e*f + (1 + c^2)*f^2]))])/f - (2*b^2*PolyLog[3, -((E^ArcCsch[c + d*x]*(d*e - c*f))/(f + Sqrt[d^2*e^2 - 2*c*d*
e*f + (1 + c^2)*f^2]))])/f

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Rubi [A]  time = 1.07282, antiderivative size = 475, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 9, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.45, Rules used = {6322, 5596, 5569, 3716, 2190, 2531, 2282, 6589, 5561} $\frac{2 b \left (a+b \text{csch}^{-1}(c+d x)\right ) \text{PolyLog}\left (2,-\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{f-\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}\right )}{f}+\frac{2 b \left (a+b \text{csch}^{-1}(c+d x)\right ) \text{PolyLog}\left (2,-\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+f}\right )}{f}-\frac{b \text{PolyLog}\left (2,e^{2 \text{csch}^{-1}(c+d x)}\right ) \left (a+b \text{csch}^{-1}(c+d x)\right )}{f}-\frac{2 b^2 \text{PolyLog}\left (3,-\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{f-\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}\right )}{f}-\frac{2 b^2 \text{PolyLog}\left (3,-\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+f}\right )}{f}+\frac{b^2 \text{PolyLog}\left (3,e^{2 \text{csch}^{-1}(c+d x)}\right )}{2 f}+\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2 \log \left (\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{f-\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}+1\right )}{f}+\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2 \log \left (\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+f}+1\right )}{f}-\frac{\log \left (1-e^{2 \text{csch}^{-1}(c+d x)}\right ) \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(((a + b*ArcCsch[c + d*x])^2*Log[1 - E^(2*ArcCsch[c + d*x])])/f) + ((a + b*ArcCsch[c + d*x])^2*Log[1 + (E^Arc
Csch[c + d*x]*(d*e - c*f))/(f - Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2])])/f + ((a + b*ArcCsch[c + d*x])^2*L
og[1 + (E^ArcCsch[c + d*x]*(d*e - c*f))/(f + Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2])])/f - (b*(a + b*ArcCsc
h[c + d*x])*PolyLog[2, E^(2*ArcCsch[c + d*x])])/f + (2*b*(a + b*ArcCsch[c + d*x])*PolyLog[2, -((E^ArcCsch[c +
d*x]*(d*e - c*f))/(f - Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]))])/f + (2*b*(a + b*ArcCsch[c + d*x])*PolyLog
[2, -((E^ArcCsch[c + d*x]*(d*e - c*f))/(f + Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]))])/f + (b^2*PolyLog[3,
E^(2*ArcCsch[c + d*x])])/(2*f) - (2*b^2*PolyLog[3, -((E^ArcCsch[c + d*x]*(d*e - c*f))/(f - Sqrt[d^2*e^2 - 2*c*
d*e*f + (1 + c^2)*f^2]))])/f - (2*b^2*PolyLog[3, -((E^ArcCsch[c + d*x]*(d*e - c*f))/(f + Sqrt[d^2*e^2 - 2*c*d*
e*f + (1 + c^2)*f^2]))])/f

Rule 6322

Int[((a_.) + ArcCsch[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> -Dist[(d^(m + 1)
)^(-1), Subst[Int[(a + b*x)^p*Csch[x]*Coth[x]*(d*e - c*f + f*Csch[x])^m, x], x, ArcCsch[c + d*x]], x] /; FreeQ
[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IntegerQ[m]

Rule 5596

Int[(((e_.) + (f_.)*(x_))^(m_.)*(F_)[(c_.) + (d_.)*(x_)]^(n_.)*(G_)[(c_.) + (d_.)*(x_)]^(p_.))/(Csch[(c_.) + (
d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Int[((e + f*x)^m*Sinh[c + d*x]*F[c + d*x]^n*G[c + d*x]^p)/(b + a*Sinh[c
+ d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && HyperbolicQ[F] && HyperbolicQ[G] && IntegersQ[m, n, p]

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 3716

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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)))/(E^(2*I*k*Pi)*(1 + E^(2*
(-(I*e) + f*fz*x))/E^(2*I*k*Pi))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && 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 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

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]

Rubi steps

\begin{align*} \int \frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2}{e+f x} \, dx &=-\operatorname{Subst}\left (\int \frac{(a+b x)^2 \coth (x) \text{csch}(x)}{d e-c f+f \text{csch}(x)} \, dx,x,\text{csch}^{-1}(c+d x)\right )\\ &=-\operatorname{Subst}\left (\int \frac{(a+b x)^2 \coth (x)}{f+(d e-c f) \sinh (x)} \, dx,x,\text{csch}^{-1}(c+d x)\right )\\ &=-\frac{\operatorname{Subst}\left (\int (a+b x)^2 \coth (x) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{f}+\frac{(d e-c f) \operatorname{Subst}\left (\int \frac{(a+b x)^2 \cosh (x)}{f+(d e-c f) \sinh (x)} \, dx,x,\text{csch}^{-1}(c+d x)\right )}{f}\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)^2}{1-e^{2 x}} \, dx,x,\text{csch}^{-1}(c+d x)\right )}{f}+\frac{(d e-c f) \operatorname{Subst}\left (\int \frac{e^x (a+b x)^2}{f+e^x (d e-c f)-\sqrt{d^2 e^2-2 c d e f+f^2+c^2 f^2}} \, dx,x,\text{csch}^{-1}(c+d x)\right )}{f}+\frac{(d e-c f) \operatorname{Subst}\left (\int \frac{e^x (a+b x)^2}{f+e^x (d e-c f)+\sqrt{d^2 e^2-2 c d e f+f^2+c^2 f^2}} \, dx,x,\text{csch}^{-1}(c+d x)\right )}{f}\\ &=-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2 \log \left (1-e^{2 \text{csch}^{-1}(c+d x)}\right )}{f}+\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2 \log \left (1+\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}+\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2 \log \left (1+\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}+\frac{(2 b) \operatorname{Subst}\left (\int (a+b x) \log \left (1-e^{2 x}\right ) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{f}-\frac{(2 b) \operatorname{Subst}\left (\int (a+b x) \log \left (1+\frac{e^x (d e-c f)}{f-\sqrt{d^2 e^2-2 c d e f+f^2+c^2 f^2}}\right ) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{f}-\frac{(2 b) \operatorname{Subst}\left (\int (a+b x) \log \left (1+\frac{e^x (d e-c f)}{f+\sqrt{d^2 e^2-2 c d e f+f^2+c^2 f^2}}\right ) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{f}\\ &=-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2 \log \left (1-e^{2 \text{csch}^{-1}(c+d x)}\right )}{f}+\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2 \log \left (1+\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}+\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2 \log \left (1+\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}-\frac{b \left (a+b \text{csch}^{-1}(c+d x)\right ) \text{Li}_2\left (e^{2 \text{csch}^{-1}(c+d x)}\right )}{f}+\frac{2 b \left (a+b \text{csch}^{-1}(c+d x)\right ) \text{Li}_2\left (-\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}+\frac{2 b \left (a+b \text{csch}^{-1}(c+d x)\right ) \text{Li}_2\left (-\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}+\frac{b^2 \operatorname{Subst}\left (\int \text{Li}_2\left (e^{2 x}\right ) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{f}-\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{e^x (d e-c f)}{f-\sqrt{d^2 e^2-2 c d e f+f^2+c^2 f^2}}\right ) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{f}-\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{e^x (d e-c f)}{f+\sqrt{d^2 e^2-2 c d e f+f^2+c^2 f^2}}\right ) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{f}\\ &=-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2 \log \left (1-e^{2 \text{csch}^{-1}(c+d x)}\right )}{f}+\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2 \log \left (1+\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}+\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2 \log \left (1+\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}-\frac{b \left (a+b \text{csch}^{-1}(c+d x)\right ) \text{Li}_2\left (e^{2 \text{csch}^{-1}(c+d x)}\right )}{f}+\frac{2 b \left (a+b \text{csch}^{-1}(c+d x)\right ) \text{Li}_2\left (-\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}+\frac{2 b \left (a+b \text{csch}^{-1}(c+d x)\right ) \text{Li}_2\left (-\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 \text{csch}^{-1}(c+d x)}\right )}{2 f}-\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{(d e-c f) x}{-f+\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{x} \, dx,x,e^{\text{csch}^{-1}(c+d x)}\right )}{f}-\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{(d e-c f) x}{f+\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{x} \, dx,x,e^{\text{csch}^{-1}(c+d x)}\right )}{f}\\ &=-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2 \log \left (1-e^{2 \text{csch}^{-1}(c+d x)}\right )}{f}+\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2 \log \left (1+\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}+\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2 \log \left (1+\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}-\frac{b \left (a+b \text{csch}^{-1}(c+d x)\right ) \text{Li}_2\left (e^{2 \text{csch}^{-1}(c+d x)}\right )}{f}+\frac{2 b \left (a+b \text{csch}^{-1}(c+d x)\right ) \text{Li}_2\left (-\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}+\frac{2 b \left (a+b \text{csch}^{-1}(c+d x)\right ) \text{Li}_2\left (-\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}+\frac{b^2 \text{Li}_3\left (e^{2 \text{csch}^{-1}(c+d x)}\right )}{2 f}-\frac{2 b^2 \text{Li}_3\left (-\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}-\frac{2 b^2 \text{Li}_3\left (-\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}\\ \end{align*}

Mathematica [C]  time = 2.78343, size = 1008, normalized size = 2.12 $\frac{6 \log (e+f x) a^2+6 b \left (\frac{1}{4} \left (\pi -2 i \text{csch}^{-1}(c+d x)\right )^2-\text{csch}^{-1}(c+d x)^2-8 \sin ^{-1}\left (\sqrt{\frac{d e-c f+i f}{2 d e-2 c f}}\right ) \tan ^{-1}\left (\frac{(i d e-i c f+f) \cot \left (\frac{1}{4} \left (2 i \text{csch}^{-1}(c+d x)+\pi \right )\right )}{\sqrt{f^2+(d e-c f)^2}}\right )-2 \text{csch}^{-1}(c+d x) \log \left (1-e^{-2 \text{csch}^{-1}(c+d x)}\right )+\left (2 \text{csch}^{-1}(c+d x)+i \left (4 \sin ^{-1}\left (\sqrt{\frac{d e-c f+i f}{2 d e-2 c f}}\right )+\pi \right )\right ) \log \left (\frac{d e-e^{\text{csch}^{-1}(c+d x)} f-c f+e^{\text{csch}^{-1}(c+d x)} \sqrt{f^2+(d e-c f)^2}}{d e-c f}\right )+\left (2 \text{csch}^{-1}(c+d x)+i \left (\pi -4 \sin ^{-1}\left (\sqrt{\frac{d e-c f+i f}{2 d e-2 c f}}\right )\right )\right ) \log \left (-\frac{-d e+e^{\text{csch}^{-1}(c+d x)} f+c f+e^{\text{csch}^{-1}(c+d x)} \sqrt{f^2+(d e-c f)^2}}{d e-c f}\right )+2 \text{csch}^{-1}(c+d x) \log \left (\frac{d (e+f x)}{c+d x}\right )-\left (2 \text{csch}^{-1}(c+d x)+i \pi \right ) \log \left (\frac{d (e+f x)}{c+d x}\right )+\text{PolyLog}\left (2,e^{-2 \text{csch}^{-1}(c+d x)}\right )+2 \text{PolyLog}\left (2,\frac{e^{\text{csch}^{-1}(c+d x)} \left (f-\sqrt{f^2+(d e-c f)^2}\right )}{d e-c f}\right )+2 \text{PolyLog}\left (2,\frac{e^{\text{csch}^{-1}(c+d x)} \left (f+\sqrt{f^2+(d e-c f)^2}\right )}{d e-c f}\right )\right ) a+b^2 \left (-2 \text{csch}^{-1}(c+d x)^3-6 \log \left (1+e^{-\text{csch}^{-1}(c+d x)}\right ) \text{csch}^{-1}(c+d x)^2-6 \log \left (1-e^{\text{csch}^{-1}(c+d x)}\right ) \text{csch}^{-1}(c+d x)^2+6 \log \left (\frac{e^{\text{csch}^{-1}(c+d x)} (c f-d e)}{\sqrt{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}-f}+1\right ) \text{csch}^{-1}(c+d x)^2+6 \log \left (\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}+1\right ) \text{csch}^{-1}(c+d x)^2+12 \text{PolyLog}\left (2,-e^{-\text{csch}^{-1}(c+d x)}\right ) \text{csch}^{-1}(c+d x)-12 \text{PolyLog}\left (2,e^{\text{csch}^{-1}(c+d x)}\right ) \text{csch}^{-1}(c+d x)+12 \text{PolyLog}\left (2,\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{\sqrt{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}-f}\right ) \text{csch}^{-1}(c+d x)+12 \text{PolyLog}\left (2,\frac{e^{\text{csch}^{-1}(c+d x)} (c f-d e)}{f+\sqrt{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}\right ) \text{csch}^{-1}(c+d x)+12 \text{PolyLog}\left (3,-e^{-\text{csch}^{-1}(c+d x)}\right )+12 \text{PolyLog}\left (3,e^{\text{csch}^{-1}(c+d x)}\right )-12 \text{PolyLog}\left (3,\frac{e^{\text{csch}^{-1}(c+d x)} (d e-c f)}{\sqrt{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}-f}\right )-12 \text{PolyLog}\left (3,\frac{e^{\text{csch}^{-1}(c+d x)} (c f-d e)}{f+\sqrt{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}\right )-i \pi ^3\right )}{6 f}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(6*a^2*Log[e + f*x] + 6*a*b*((Pi - (2*I)*ArcCsch[c + d*x])^2/4 - ArcCsch[c + d*x]^2 - 8*ArcSin[Sqrt[(d*e + I*f
- c*f)/(2*d*e - 2*c*f)]]*ArcTan[((I*d*e + f - I*c*f)*Cot[(Pi + (2*I)*ArcCsch[c + d*x])/4])/Sqrt[f^2 + (d*e -
c*f)^2]] - 2*ArcCsch[c + d*x]*Log[1 - E^(-2*ArcCsch[c + d*x])] + (2*ArcCsch[c + d*x] + I*(Pi + 4*ArcSin[Sqrt[(
d*e + I*f - c*f)/(2*d*e - 2*c*f)]]))*Log[(d*e - c*f - E^ArcCsch[c + d*x]*f + E^ArcCsch[c + d*x]*Sqrt[f^2 + (d*
e - c*f)^2])/(d*e - c*f)] + (2*ArcCsch[c + d*x] + I*(Pi - 4*ArcSin[Sqrt[(d*e + I*f - c*f)/(2*d*e - 2*c*f)]]))*
Log[-((-(d*e) + c*f + E^ArcCsch[c + d*x]*f + E^ArcCsch[c + d*x]*Sqrt[f^2 + (d*e - c*f)^2])/(d*e - c*f))] + 2*A
rcCsch[c + d*x]*Log[(d*(e + f*x))/(c + d*x)] - (I*Pi + 2*ArcCsch[c + d*x])*Log[(d*(e + f*x))/(c + d*x)] + Poly
Log[2, E^(-2*ArcCsch[c + d*x])] + 2*PolyLog[2, (E^ArcCsch[c + d*x]*(f - Sqrt[f^2 + (d*e - c*f)^2]))/(d*e - c*f
)] + 2*PolyLog[2, (E^ArcCsch[c + d*x]*(f + Sqrt[f^2 + (d*e - c*f)^2]))/(d*e - c*f)]) + b^2*((-I)*Pi^3 - 2*ArcC
sch[c + d*x]^3 - 6*ArcCsch[c + d*x]^2*Log[1 + E^(-ArcCsch[c + d*x])] - 6*ArcCsch[c + d*x]^2*Log[1 - E^ArcCsch[
c + d*x]] + 6*ArcCsch[c + d*x]^2*Log[1 + (E^ArcCsch[c + d*x]*(-(d*e) + c*f))/(-f + Sqrt[d^2*e^2 - 2*c*d*e*f +
(1 + c^2)*f^2])] + 6*ArcCsch[c + d*x]^2*Log[1 + (E^ArcCsch[c + d*x]*(d*e - c*f))/(f + Sqrt[d^2*e^2 - 2*c*d*e*f
+ (1 + c^2)*f^2])] + 12*ArcCsch[c + d*x]*PolyLog[2, -E^(-ArcCsch[c + d*x])] - 12*ArcCsch[c + d*x]*PolyLog[2,
E^ArcCsch[c + d*x]] + 12*ArcCsch[c + d*x]*PolyLog[2, (E^ArcCsch[c + d*x]*(d*e - c*f))/(-f + Sqrt[d^2*e^2 - 2*c
*d*e*f + (1 + c^2)*f^2])] + 12*ArcCsch[c + d*x]*PolyLog[2, (E^ArcCsch[c + d*x]*(-(d*e) + c*f))/(f + Sqrt[d^2*e
^2 - 2*c*d*e*f + (1 + c^2)*f^2])] + 12*PolyLog[3, -E^(-ArcCsch[c + d*x])] + 12*PolyLog[3, E^ArcCsch[c + d*x]]
- 12*PolyLog[3, (E^ArcCsch[c + d*x]*(d*e - c*f))/(-f + Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2])] - 12*PolyLo
g[3, (E^ArcCsch[c + d*x]*(-(d*e) + c*f))/(f + Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2])]))/(6*f)

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Maple [F]  time = 0.611, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\rm arccsch} \left (dx+c\right ) \right ) ^{2}}{fx+e}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccsch(d*x+c))^2/(f*x+e),x)

[Out]

int((a+b*arccsch(d*x+c))^2/(f*x+e),x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(d*x+c))^2/(f*x+e),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \operatorname{arcsch}\left (d x + c\right )^{2} + 2 \, a b \operatorname{arcsch}\left (d x + c\right ) + a^{2}}{f x + e}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(d*x+c))^2/(f*x+e),x, algorithm="fricas")

[Out]

integral((b^2*arccsch(d*x + c)^2 + 2*a*b*arccsch(d*x + c) + a^2)/(f*x + e), x)

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Sympy [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((a+b*acsch(d*x+c))**2/(f*x+e),x)

[Out]

Timed out

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(d*x+c))^2/(f*x+e),x, algorithm="giac")

[Out]

integrate((b*arccsch(d*x + c) + a)^2/(f*x + e), x)