### 3.12 $$\int \frac{(a+b \text{csch}^{-1}(c+d x))^2}{(e+f x)^2} \, dx$$

Optimal. Leaf size=448 $-\frac{2 b^2 d \text{PolyLog}\left (2,-\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{f-\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}\right )}{(d e-c f) \sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}+\frac{2 b^2 d \text{PolyLog}\left (2,-\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+f}\right )}{(d e-c f) \sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}-\frac{2 b d \left (a+b \text{csch}^{-1}(c+d x)\right ) \log \left (\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{f-\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}+1\right )}{(d e-c f) \sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}+\frac{2 b d \left (a+b \text{csch}^{-1}(c+d x)\right ) \log \left (\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+f}+1\right )}{(d e-c f) \sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}+\frac{d \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f (d e-c f)}-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f (e+f x)}$

[Out]

(d*(a + b*ArcCsch[c + d*x])^2)/(f*(d*e - c*f)) - (a + b*ArcCsch[c + d*x])^2/(f*(e + f*x)) - (2*b*d*(a + b*ArcC
sch[c + d*x])*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])])/((d*e
- c*f)*Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]) + (2*b*d*(a + b*ArcCsch[c + d*x])*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])])/((d*e - c*f)*Sqrt[d^2*e^2 - 2*c*d*e*f + (1 +
c^2)*f^2]) - (2*b^2*d*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]))])/((d*e - c*f)*Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]) + (2*b^2*d*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]))])/((d*e - c*f)*Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c
^2)*f^2])

________________________________________________________________________________________

Rubi [A]  time = 1.11005, antiderivative size = 448, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.4, Rules used = {6322, 5469, 4191, 3322, 2264, 2190, 2279, 2391} $-\frac{2 b^2 d \text{PolyLog}\left (2,-\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{f-\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}\right )}{(d e-c f) \sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}+\frac{2 b^2 d \text{PolyLog}\left (2,-\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+f}\right )}{(d e-c f) \sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}-\frac{2 b d \left (a+b \text{csch}^{-1}(c+d x)\right ) \log \left (\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{f-\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}+1\right )}{(d e-c f) \sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}+\frac{2 b d \left (a+b \text{csch}^{-1}(c+d x)\right ) \log \left (\frac{(d e-c f) e^{\text{csch}^{-1}(c+d x)}}{\sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+f}+1\right )}{(d e-c f) \sqrt{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}+\frac{d \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f (d e-c f)}-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f (e+f x)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(d*(a + b*ArcCsch[c + d*x])^2)/(f*(d*e - c*f)) - (a + b*ArcCsch[c + d*x])^2/(f*(e + f*x)) - (2*b*d*(a + b*ArcC
sch[c + d*x])*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])])/((d*e
- c*f)*Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]) + (2*b*d*(a + b*ArcCsch[c + d*x])*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])])/((d*e - c*f)*Sqrt[d^2*e^2 - 2*c*d*e*f + (1 +
c^2)*f^2]) - (2*b^2*d*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]))])/((d*e - c*f)*Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]) + (2*b^2*d*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]))])/((d*e - c*f)*Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c
^2)*f^2])

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 5469

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

Rule 4191

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, 1/(Sin[e + f*x]^n/(b + a*Sin[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 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]

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]

Rubi steps

\begin{align*} \int \frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx &=-\left (d \operatorname{Subst}\left (\int \frac{(a+b x)^2 \coth (x) \text{csch}(x)}{(d e-c f+f \text{csch}(x))^2} \, dx,x,\text{csch}^{-1}(c+d x)\right )\right )\\ &=-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f (e+f x)}+\frac{(2 b d) \operatorname{Subst}\left (\int \frac{a+b x}{d e-c f+f \text{csch}(x)} \, dx,x,\text{csch}^{-1}(c+d x)\right )}{f}\\ &=-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f (e+f x)}+\frac{(2 b d) \operatorname{Subst}\left (\int \left (\frac{a+b x}{d e-c f}+\frac{f (a+b x)}{(-d e+c f) \left (f+d e \left (1-\frac{c f}{d e}\right ) \sinh (x)\right )}\right ) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{f}\\ &=\frac{d \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f (d e-c f)}-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f (e+f x)}-\frac{(2 b d) \operatorname{Subst}\left (\int \frac{a+b x}{f+d e \left (1-\frac{c f}{d e}\right ) \sinh (x)} \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d e-c f}\\ &=\frac{d \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f (d e-c f)}-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f (e+f x)}-\frac{(4 b d) \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{2 e^x f-d e \left (1-\frac{c f}{d e}\right )+d e e^{2 x} \left (1-\frac{c f}{d e}\right )} \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d e-c f}\\ &=\frac{d \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f (d e-c f)}-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f (e+f x)}-\frac{(4 b d) \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{2 f+2 d e e^x \left (1-\frac{c f}{d e}\right )-2 \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 )}{\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac{(4 b d) \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{2 f+2 d e e^x \left (1-\frac{c f}{d e}\right )+2 \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 )}{\sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\\ &=\frac{d \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f (d e-c f)}-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f (e+f x)}-\frac{2 b d \left (a+b \text{csch}^{-1}(c+d x)\right ) \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 )}{(d e-c f) \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac{2 b d \left (a+b \text{csch}^{-1}(c+d x)\right ) \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 )}{(d e-c f) \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac{\left (2 b^2 d\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{2 d e e^x \left (1-\frac{c f}{d e}\right )}{2 f-2 \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 )}{(d e-c f) \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}-\frac{\left (2 b^2 d\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{2 d e e^x \left (1-\frac{c f}{d e}\right )}{2 f+2 \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 )}{(d e-c f) \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\\ &=\frac{d \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f (d e-c f)}-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f (e+f x)}-\frac{2 b d \left (a+b \text{csch}^{-1}(c+d x)\right ) \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 )}{(d e-c f) \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac{2 b d \left (a+b \text{csch}^{-1}(c+d x)\right ) \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 )}{(d e-c f) \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac{\left (2 b^2 d\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 d e \left (1-\frac{c f}{d e}\right ) x}{2 f-2 \sqrt{d^2 e^2-2 c d e f+f^2+c^2 f^2}}\right )}{x} \, dx,x,e^{\text{csch}^{-1}(c+d x)}\right )}{(d e-c f) \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}-\frac{\left (2 b^2 d\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 d e \left (1-\frac{c f}{d e}\right ) x}{2 f+2 \sqrt{d^2 e^2-2 c d e f+f^2+c^2 f^2}}\right )}{x} \, dx,x,e^{\text{csch}^{-1}(c+d x)}\right )}{(d e-c f) \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\\ &=\frac{d \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f (d e-c f)}-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2}{f (e+f x)}-\frac{2 b d \left (a+b \text{csch}^{-1}(c+d x)\right ) \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 )}{(d e-c f) \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac{2 b d \left (a+b \text{csch}^{-1}(c+d x)\right ) \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 )}{(d e-c f) \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}-\frac{2 b^2 d \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 )}{(d e-c f) \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac{2 b^2 d \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 )}{(d e-c f) \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\\ \end{align*}

Mathematica [C]  time = 12.8638, size = 2061, normalized size = 4.6 $\text{Result too large to show}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(a^2/(f*(e + f*x))) - (2*a*b*(c + d*x)^2*(f + (d*e - c*f)/(c + d*x))^2*(ArcCsch[c + d*x]/(f + (d*e)/(c + d*x)
- (c*f)/(c + d*x)) - (2*ArcTan[(d*e - c*f - f*Tanh[ArcCsch[c + d*x]/2])/Sqrt[-(d^2*e^2) + 2*c*d*e*f - (1 + c^
2)*f^2]])/Sqrt[-(d^2*e^2) + 2*c*d*e*f - (1 + c^2)*f^2]))/(d*(-(d*e) + c*f)*(e + f*x)^2) - (b^2*(c + d*x)^2*(f
+ (d*e - c*f)/(c + d*x))^2*(ArcCsch[c + d*x]^2/((-(d*e) + c*f)*(f + (d*e)/(c + d*x) - (c*f)/(c + d*x))) + (2*(
((-I)*Pi*ArcTanh[(-(d*e) + c*f + f*Tanh[ArcCsch[c + d*x]/2])/Sqrt[f^2 + (d*e - c*f)^2]])/Sqrt[f^2 + (d*e - c*f
)^2] - (2*(Pi/2 - I*ArcCsch[c + d*x])*ArcTanh[((f - I*(d*e - c*f))*Cot[(Pi/2 - I*ArcCsch[c + d*x])/2])/Sqrt[-(
d^2*e^2) + 2*c*d*e*f - f^2 - c^2*f^2]] - 2*ArcCos[((-I)*f)/(d*e - c*f)]*ArcTanh[((-f - I*(d*e - c*f))*Tan[(Pi/
2 - I*ArcCsch[c + d*x])/2])/Sqrt[-(d^2*e^2) + 2*c*d*e*f - f^2 - c^2*f^2]] + (ArcCos[((-I)*f)/(d*e - c*f)] - (2
*I)*(ArcTanh[((f - I*(d*e - c*f))*Cot[(Pi/2 - I*ArcCsch[c + d*x])/2])/Sqrt[-(d^2*e^2) + 2*c*d*e*f - f^2 - c^2*
f^2]] - ArcTanh[((-f - I*(d*e - c*f))*Tan[(Pi/2 - I*ArcCsch[c + d*x])/2])/Sqrt[-(d^2*e^2) + 2*c*d*e*f - f^2 -
c^2*f^2]]))*Log[Sqrt[-(d^2*e^2) + 2*c*d*e*f - f^2 - c^2*f^2]/(Sqrt[2]*E^((I/2)*(Pi/2 - I*ArcCsch[c + d*x]))*Sq
rt[(-I)*(d*e - c*f)]*Sqrt[f + (d*e - c*f)/(c + d*x)])] + (ArcCos[((-I)*f)/(d*e - c*f)] + (2*I)*(ArcTanh[((f -
I*(d*e - c*f))*Cot[(Pi/2 - I*ArcCsch[c + d*x])/2])/Sqrt[-(d^2*e^2) + 2*c*d*e*f - f^2 - c^2*f^2]] - ArcTanh[((-
f - I*(d*e - c*f))*Tan[(Pi/2 - I*ArcCsch[c + d*x])/2])/Sqrt[-(d^2*e^2) + 2*c*d*e*f - f^2 - c^2*f^2]]))*Log[(E^
((I/2)*(Pi/2 - I*ArcCsch[c + d*x]))*Sqrt[-(d^2*e^2) + 2*c*d*e*f - f^2 - c^2*f^2])/(Sqrt[2]*Sqrt[(-I)*(d*e - c*
f)]*Sqrt[f + (d*e - c*f)/(c + d*x)])] - (ArcCos[((-I)*f)/(d*e - c*f)] + (2*I)*ArcTanh[((-f - I*(d*e - c*f))*Ta
n[(Pi/2 - I*ArcCsch[c + d*x])/2])/Sqrt[-(d^2*e^2) + 2*c*d*e*f - f^2 - c^2*f^2]])*Log[1 - (I*(f - I*Sqrt[-(d^2*
e^2) + 2*c*d*e*f - f^2 - c^2*f^2])*(f - I*(d*e - c*f) - Sqrt[-(d^2*e^2) + 2*c*d*e*f - f^2 - c^2*f^2]*Tan[(Pi/2
- I*ArcCsch[c + d*x])/2]))/((d*e - c*f)*(f - I*(d*e - c*f) + Sqrt[-(d^2*e^2) + 2*c*d*e*f - f^2 - c^2*f^2]*Tan
[(Pi/2 - I*ArcCsch[c + d*x])/2]))] + (-ArcCos[((-I)*f)/(d*e - c*f)] + (2*I)*ArcTanh[((-f - I*(d*e - c*f))*Tan[
(Pi/2 - I*ArcCsch[c + d*x])/2])/Sqrt[-(d^2*e^2) + 2*c*d*e*f - f^2 - c^2*f^2]])*Log[1 - (I*(f + I*Sqrt[-(d^2*e^
2) + 2*c*d*e*f - f^2 - c^2*f^2])*(f - I*(d*e - c*f) - Sqrt[-(d^2*e^2) + 2*c*d*e*f - f^2 - c^2*f^2]*Tan[(Pi/2 -
I*ArcCsch[c + d*x])/2]))/((d*e - c*f)*(f - I*(d*e - c*f) + Sqrt[-(d^2*e^2) + 2*c*d*e*f - f^2 - c^2*f^2]*Tan[(
Pi/2 - I*ArcCsch[c + d*x])/2]))] + I*(PolyLog[2, (I*(f - I*Sqrt[-(d^2*e^2) + 2*c*d*e*f - f^2 - c^2*f^2])*(f -
I*(d*e - c*f) - Sqrt[-(d^2*e^2) + 2*c*d*e*f - f^2 - c^2*f^2]*Tan[(Pi/2 - I*ArcCsch[c + d*x])/2]))/((d*e - c*f)
*(f - I*(d*e - c*f) + Sqrt[-(d^2*e^2) + 2*c*d*e*f - f^2 - c^2*f^2]*Tan[(Pi/2 - I*ArcCsch[c + d*x])/2]))] - Pol
yLog[2, (I*(f + I*Sqrt[-(d^2*e^2) + 2*c*d*e*f - f^2 - c^2*f^2])*(f - I*(d*e - c*f) - Sqrt[-(d^2*e^2) + 2*c*d*e
*f - f^2 - c^2*f^2]*Tan[(Pi/2 - I*ArcCsch[c + d*x])/2]))/((d*e - c*f)*(f - I*(d*e - c*f) + Sqrt[-(d^2*e^2) + 2
*c*d*e*f - f^2 - c^2*f^2]*Tan[(Pi/2 - I*ArcCsch[c + d*x])/2]))]))/Sqrt[-(d^2*e^2) + 2*c*d*e*f - f^2 - c^2*f^2]
))/(d*e - c*f)))/(d*(e + f*x)^2)

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Maple [F]  time = 0.543, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\rm arccsch} \left (dx+c\right ) \right ) ^{2}}{ \left ( fx+e \right ) ^{2}}}\, 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)^2,x)

[Out]

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

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

[Out]

-b^2*log(sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1) + 1)^2/(f^2*x + e*f) - a^2/(f^2*x + e*f) - integrate(-((b^2*d^2*f*x
^2 + 2*b^2*c*d*f*x + (c^2*f + f)*b^2)*log(d*x + c)^2 - 2*(a*b*d^2*f*x^2 + 2*a*b*c*d*f*x + (c^2*f + f)*a*b)*log
(d*x + c) + 2*(a*b*d^2*f*x^2 + 2*a*b*c*d*f*x + (c^2*f + f)*a*b - (b^2*d^2*f*x^2 + 2*b^2*c*d*f*x + (c^2*f + f)*
b^2)*log(d*x + c) + (b^2*c*d*e + (c^2*f + f)*a*b + (a*b*d^2*f + b^2*d^2*f)*x^2 + (2*a*b*c*d*f + (d^2*e + c*d*f
)*b^2)*x - (b^2*d^2*f*x^2 + 2*b^2*c*d*f*x + (c^2*f + f)*b^2)*log(d*x + c))*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))*
log(sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1) + 1) + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*((b^2*d^2*f*x^2 + 2*b^2*c*d*f*x
+ (c^2*f + f)*b^2)*log(d*x + c)^2 - 2*(a*b*d^2*f*x^2 + 2*a*b*c*d*f*x + (c^2*f + f)*a*b)*log(d*x + c)))/(d^2*f
^3*x^4 + c^2*e^2*f + 2*(d^2*e*f^2 + c*d*f^3)*x^3 + e^2*f + (d^2*e^2*f + 4*c*d*e*f^2 + c^2*f^3 + f^3)*x^2 + 2*(
c*d*e^2*f + c^2*e*f^2 + e*f^2)*x + (d^2*f^3*x^4 + c^2*e^2*f + 2*(d^2*e*f^2 + c*d*f^3)*x^3 + e^2*f + (d^2*e^2*f
+ 4*c*d*e*f^2 + c^2*f^3 + f^3)*x^2 + 2*(c*d*e^2*f + c^2*e*f^2 + e*f^2)*x)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)),
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^{2} x^{2} + 2 \, e f x + e^{2}}, 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)^2,x, algorithm="fricas")

[Out]

integral((b^2*arccsch(d*x + c)^2 + 2*a*b*arccsch(d*x + c) + a^2)/(f^2*x^2 + 2*e*f*x + e^2), 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)**2,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}}{{\left (f x + e\right )}^{2}}\,{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)^2,x, algorithm="giac")

[Out]

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