3.13 \(\int \frac{(a+b \text{csch}^{-1}(c+d x))^2}{(e+f x)^3} \, dx\)

Optimal. Leaf size=1024 \[ \text{result too large to display} \]

[Out]

-((b*d^2*f*Sqrt[1 + (c + d*x)^(-2)]*(a + b*ArcCsch[c + d*x]))/((d*e - c*f)*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^
2)*(f + (d*e - c*f)/(c + d*x)))) + (d^2*(a + b*ArcCsch[c + d*x])^2)/(2*f*(d*e - c*f)^2) - (a + b*ArcCsch[c + d
*x])^2/(2*f*(e + f*x)^2) + (b*d^2*f^2*(a + b*ArcCsch[c + d*x])*Log[1 + (E^ArcCsch[c + d*x]*(d*e - c*f))/(f - S
qrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2])])/((d*e - c*f)^2*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)^(3/2)) - (2*b
*d^2*(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)^2*Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]) - (b*d^2*f^2*(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)^2*(d^2*e^
2 - 2*c*d*e*f + (1 + c^2)*f^2)^(3/2)) + (2*b*d^2*(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)^2*Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2
]) + (b^2*d^2*f*Log[f + (d*e - c*f)/(c + d*x)])/((d*e - c*f)^2*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)) + (b^2*d
^2*f^2*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)^2*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)^(3/2)) - (2*b^2*d^2*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)^2*Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]
) - (b^2*d^2*f^2*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)^2*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)^(3/2)) + (2*b^2*d^2*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)^2*Sqrt[d^2*e^2 - 2*c*d*e*f + (1 +
c^2)*f^2])

________________________________________________________________________________________

Rubi [A]  time = 2.29602, antiderivative size = 1024, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 11, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.55, Rules used = {6322, 5469, 4191, 3322, 2264, 2190, 2279, 2391, 3324, 2668, 31} \[ \frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2 d^2}{2 f (d e-c f)^2}-\frac{b f \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right ) d^2}{(d e-c f) \left (d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2\right ) \left (f+\frac{d e-c f}{c+d x}\right )}-\frac{2 b \left (a+b \text{csch}^{-1}(c+d x)\right ) \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 ) d^2}{(d e-c f)^2 \sqrt{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}+\frac{b f^2 \left (a+b \text{csch}^{-1}(c+d x)\right ) \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 ) d^2}{(d e-c f)^2 \left (d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2\right )^{3/2}}+\frac{2 b \left (a+b \text{csch}^{-1}(c+d x)\right ) \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 ) d^2}{(d e-c f)^2 \sqrt{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}-\frac{b f^2 \left (a+b \text{csch}^{-1}(c+d x)\right ) \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 ) d^2}{(d e-c f)^2 \left (d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2\right )^{3/2}}+\frac{b^2 f \log \left (f+\frac{d e-c f}{c+d x}\right ) d^2}{(d e-c f)^2 \left (d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2\right )}-\frac{2 b^2 \text{PolyLog}\left (2,-\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}}\right ) d^2}{(d e-c f)^2 \sqrt{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}+\frac{b^2 f^2 \text{PolyLog}\left (2,-\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}}\right ) d^2}{(d e-c f)^2 \left (d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2\right )^{3/2}}+\frac{2 b^2 \text{PolyLog}\left (2,-\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}}\right ) d^2}{(d e-c f)^2 \sqrt{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}-\frac{b^2 f^2 \text{PolyLog}\left (2,-\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}}\right ) d^2}{(d e-c f)^2 \left (d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2\right )^{3/2}}-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2}{2 f (e+f x)^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCsch[c + d*x])^2/(e + f*x)^3,x]

[Out]

-((b*d^2*f*Sqrt[1 + (c + d*x)^(-2)]*(a + b*ArcCsch[c + d*x]))/((d*e - c*f)*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^
2)*(f + (d*e - c*f)/(c + d*x)))) + (d^2*(a + b*ArcCsch[c + d*x])^2)/(2*f*(d*e - c*f)^2) - (a + b*ArcCsch[c + d
*x])^2/(2*f*(e + f*x)^2) + (b*d^2*f^2*(a + b*ArcCsch[c + d*x])*Log[1 + (E^ArcCsch[c + d*x]*(d*e - c*f))/(f - S
qrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2])])/((d*e - c*f)^2*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)^(3/2)) - (2*b
*d^2*(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)^2*Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]) - (b*d^2*f^2*(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)^2*(d^2*e^
2 - 2*c*d*e*f + (1 + c^2)*f^2)^(3/2)) + (2*b*d^2*(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)^2*Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2
]) + (b^2*d^2*f*Log[f + (d*e - c*f)/(c + d*x)])/((d*e - c*f)^2*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)) + (b^2*d
^2*f^2*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)^2*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)^(3/2)) - (2*b^2*d^2*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)^2*Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]
) - (b^2*d^2*f^2*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)^2*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)^(3/2)) + (2*b^2*d^2*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)^2*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]

Rule 3324

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

Rule 2668

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2}{(e+f x)^3} \, dx &=-\left (d^2 \operatorname{Subst}\left (\int \frac{(a+b x)^2 \coth (x) \text{csch}(x)}{(d e-c f+f \text{csch}(x))^3} \, dx,x,\text{csch}^{-1}(c+d x)\right )\right )\\ &=-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2}{2 f (e+f x)^2}+\frac{\left (b d^2\right ) \operatorname{Subst}\left (\int \frac{a+b x}{(d e-c f+f \text{csch}(x))^2} \, dx,x,\text{csch}^{-1}(c+d x)\right )}{f}\\ &=-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2}{2 f (e+f x)^2}+\frac{\left (b d^2\right ) \operatorname{Subst}\left (\int \left (\frac{a+b x}{(d e-c f)^2}+\frac{2 f (a+b x)}{(d e-c f)^2 \left (-f-d e \left (1-\frac{c f}{d e}\right ) \sinh (x)\right )}+\frac{f^2 (a+b x)}{(d e-c f)^2 \left (f+d e \left (1-\frac{c f}{d e}\right ) \sinh (x)\right )^2}\right ) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{f}\\ &=\frac{d^2 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{2 f (d e-c f)^2}-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2}{2 f (e+f x)^2}+\frac{\left (2 b d^2\right ) \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)^2}+\frac{\left (b d^2 f\right ) \operatorname{Subst}\left (\int \frac{a+b x}{\left (f+d e \left (1-\frac{c f}{d e}\right ) \sinh (x)\right )^2} \, dx,x,\text{csch}^{-1}(c+d x)\right )}{(d e-c f)^2}\\ &=-\frac{b d^2 f \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{(d e-c f) \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (f+\frac{d e-c f}{c+d x}\right )}+\frac{d^2 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{2 f (d e-c f)^2}-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2}{2 f (e+f x)^2}+\frac{\left (4 b d^2\right ) \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)^2}+\frac{\left (b d^2 f^2\right ) \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)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac{\left (b^2 d^2 f\right ) \operatorname{Subst}\left (\int \frac{\cosh (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) \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}\\ &=-\frac{b d^2 f \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{(d e-c f) \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (f+\frac{d e-c f}{c+d x}\right )}+\frac{d^2 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{2 f (d e-c f)^2}-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2}{2 f (e+f x)^2}+\frac{\left (b^2 d^2 f\right ) \operatorname{Subst}\left (\int \frac{1}{f+x} \, dx,x,\frac{d e \left (1-\frac{c f}{d e}\right )}{c+d x}\right )}{(d e-c f)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac{\left (2 b d^2 f^2\right ) \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)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac{\left (4 b d^2\right ) \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 )}{(d e-c f) \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac{\left (4 b d^2\right ) \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 )}{(d e-c f) \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\\ &=-\frac{b d^2 f \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{(d e-c f) \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (f+\frac{d e-c f}{c+d x}\right )}+\frac{d^2 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{2 f (d e-c f)^2}-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2}{2 f (e+f x)^2}-\frac{2 b d^2 \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)^2 \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac{2 b d^2 \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)^2 \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac{b^2 d^2 f \log \left (\frac{e+f x}{c+d x}\right )}{(d e-c f)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac{\left (2 b d^2 f^2\right ) \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 )}{(d e-c f) \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^{3/2}}-\frac{\left (2 b d^2 f^2\right ) \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 )}{(d e-c f) \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^{3/2}}-\frac{\left (2 b^2 d^2\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)^2 \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac{\left (2 b^2 d^2\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)^2 \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\\ &=-\frac{b d^2 f \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{(d e-c f) \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (f+\frac{d e-c f}{c+d x}\right )}+\frac{d^2 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{2 f (d e-c f)^2}-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2}{2 f (e+f x)^2}+\frac{b d^2 f^2 \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)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^{3/2}}-\frac{2 b d^2 \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)^2 \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}-\frac{b d^2 f^2 \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)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^{3/2}}+\frac{2 b d^2 \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)^2 \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac{b^2 d^2 f \log \left (\frac{e+f x}{c+d x}\right )}{(d e-c f)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac{\left (b^2 d^2 f^2\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)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^{3/2}}+\frac{\left (b^2 d^2 f^2\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)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^{3/2}}-\frac{\left (2 b^2 d^2\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)^2 \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac{\left (2 b^2 d^2\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)^2 \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\\ &=-\frac{b d^2 f \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{(d e-c f) \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (f+\frac{d e-c f}{c+d x}\right )}+\frac{d^2 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{2 f (d e-c f)^2}-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2}{2 f (e+f x)^2}+\frac{b d^2 f^2 \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)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^{3/2}}-\frac{2 b d^2 \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)^2 \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}-\frac{b d^2 f^2 \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)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^{3/2}}+\frac{2 b d^2 \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)^2 \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac{b^2 d^2 f \log \left (\frac{e+f x}{c+d x}\right )}{(d e-c f)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac{2 b^2 d^2 \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)^2 \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac{2 b^2 d^2 \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)^2 \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}-\frac{\left (b^2 d^2 f^2\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)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^{3/2}}+\frac{\left (b^2 d^2 f^2\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)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^{3/2}}\\ &=-\frac{b d^2 f \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{(d e-c f) \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (f+\frac{d e-c f}{c+d x}\right )}+\frac{d^2 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{2 f (d e-c f)^2}-\frac{\left (a+b \text{csch}^{-1}(c+d x)\right )^2}{2 f (e+f x)^2}+\frac{b d^2 f^2 \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)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^{3/2}}-\frac{2 b d^2 \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)^2 \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}-\frac{b d^2 f^2 \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)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^{3/2}}+\frac{2 b d^2 \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)^2 \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac{b^2 d^2 f \log \left (\frac{e+f x}{c+d x}\right )}{(d e-c f)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac{b^2 d^2 f^2 \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)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^{3/2}}-\frac{2 b^2 d^2 \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)^2 \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}-\frac{b^2 d^2 f^2 \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)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^{3/2}}+\frac{2 b^2 d^2 \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)^2 \sqrt{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\\ \end{align*}

Mathematica [C]  time = 14.1639, size = 8348, normalized size = 8.15 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcCsch[c + d*x])^2/(e + f*x)^3,x]

[Out]

Result too large to show

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Maple [F]  time = 0.53, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\rm arccsch} \left (dx+c\right ) \right ) ^{2}}{ \left ( fx+e \right ) ^{3}}}\, 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)^3,x)

[Out]

int((a+b*arccsch(d*x+c))^2/(f*x+e)^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(d*x+c))^2/(f*x+e)^3,x, algorithm="maxima")

[Out]

-1/2*b^2*log(sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1) + 1)^2/(f^3*x^2 + 2*e*f^2*x + e^2*f) - 1/2*a^2/(f^3*x^2 + 2*e*f
^2*x + e^2*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 + 4*a*b*c*d*f*x + 2*(c^2*f + f)*a*b - 2*(
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 + 2*(c^2*f + f)*a*b + (2*a*b*d^2*f
+ b^2*d^2*f)*x^2 + (4*a*b*c*d*f + (d^2*e + c*d*f)*b^2)*x - 2*(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^4*x^5 + c^2*e^3*f + (3*d^2*e*f^3 + 2*c*d*f^4)*x^4 + e^3*f +
(3*d^2*e^2*f^2 + 6*c*d*e*f^3 + c^2*f^4 + f^4)*x^3 + (d^2*e^3*f + 6*c*d*e^2*f^2 + 3*c^2*e*f^3 + 3*e*f^3)*x^2 +
(2*c*d*e^3*f + 3*c^2*e^2*f^2 + 3*e^2*f^2)*x + (d^2*f^4*x^5 + c^2*e^3*f + (3*d^2*e*f^3 + 2*c*d*f^4)*x^4 + e^3*
f + (3*d^2*e^2*f^2 + 6*c*d*e*f^3 + c^2*f^4 + f^4)*x^3 + (d^2*e^3*f + 6*c*d*e^2*f^2 + 3*c^2*e*f^3 + 3*e*f^3)*x^
2 + (2*c*d*e^3*f + 3*c^2*e^2*f^2 + 3*e^2*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^{3} x^{3} + 3 \, e f^{2} x^{2} + 3 \, e^{2} f x + e^{3}}, 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)^3,x, algorithm="fricas")

[Out]

integral((b^2*arccsch(d*x + c)^2 + 2*a*b*arccsch(d*x + c) + a^2)/(f^3*x^3 + 3*e*f^2*x^2 + 3*e^2*f*x + e^3), 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)**3,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 )}^{3}}\,{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)^3,x, algorithm="giac")

[Out]

integrate((b*arccsch(d*x + c) + a)^2/(f*x + e)^3, x)