3.1.13 \(\int \frac {(a+b \text {csch}^{-1}(c+d x))^2}{(e+f x)^3} \, dx\) [13]

3.1.13.1 Optimal result

Integrand size = 20, antiderivative size = 1024 \[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{(e+f x)^3} \, dx=-\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 (f+\frac {d e-c f}{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 \operatorname {PolyLog}\left (2,-\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 \operatorname {PolyLog}\left (2,-\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 \operatorname {PolyLog}\left (2,-\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 \operatorname {PolyLog}\left (2,-\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}} \]

output

1/2*d^2*(a+b*arccsch(d*x+c))^2/f/(-c*f+d*e)^2-1/2*(a+b*arccsch(d*x+c))^2/f 
/(f*x+e)^2+b^2*d^2*f*ln(f+(-c*f+d*e)/(d*x+c))/(-c*f+d*e)^2/(d^2*e^2-2*c*d* 
e*f+(c^2+1)*f^2)+b*d^2*f^2*(a+b*arccsch(d*x+c))*ln(1+(1/(d*x+c)+(1+1/(d*x+ 
c)^2)^(1/2))*(-c*f+d*e)/(f-(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)^(1/2)))/(-c*f+d 
*e)^2/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)^(3/2)-b*d^2*f^2*(a+b*arccsch(d*x+c)) 
*ln(1+(1/(d*x+c)+(1+1/(d*x+c)^2)^(1/2))*(-c*f+d*e)/(f+(d^2*e^2-2*c*d*e*f+( 
c^2+1)*f^2)^(1/2)))/(-c*f+d*e)^2/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)^(3/2)+b^2 
*d^2*f^2*polylog(2,-(1/(d*x+c)+(1+1/(d*x+c)^2)^(1/2))*(-c*f+d*e)/(f-(d^2*e 
^2-2*c*d*e*f+(c^2+1)*f^2)^(1/2)))/(-c*f+d*e)^2/(d^2*e^2-2*c*d*e*f+(c^2+1)* 
f^2)^(3/2)-b^2*d^2*f^2*polylog(2,-(1/(d*x+c)+(1+1/(d*x+c)^2)^(1/2))*(-c*f+ 
d*e)/(f+(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)^(1/2)))/(-c*f+d*e)^2/(d^2*e^2-2*c* 
d*e*f+(c^2+1)*f^2)^(3/2)-2*b*d^2*(a+b*arccsch(d*x+c))*ln(1+(1/(d*x+c)+(1+1 
/(d*x+c)^2)^(1/2))*(-c*f+d*e)/(f-(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)^(1/2)))/( 
-c*f+d*e)^2/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)^(1/2)+2*b*d^2*(a+b*arccsch(d*x 
+c))*ln(1+(1/(d*x+c)+(1+1/(d*x+c)^2)^(1/2))*(-c*f+d*e)/(f+(d^2*e^2-2*c*d*e 
*f+(c^2+1)*f^2)^(1/2)))/(-c*f+d*e)^2/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)^(1/2) 
-2*b^2*d^2*polylog(2,-(1/(d*x+c)+(1+1/(d*x+c)^2)^(1/2))*(-c*f+d*e)/(f-(d^2 
*e^2-2*c*d*e*f+(c^2+1)*f^2)^(1/2)))/(-c*f+d*e)^2/(d^2*e^2-2*c*d*e*f+(c^2+1 
)*f^2)^(1/2)+2*b^2*d^2*polylog(2,-(1/(d*x+c)+(1+1/(d*x+c)^2)^(1/2))*(-c*f+ 
d*e)/(f+(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)^(1/2)))/(-c*f+d*e)^2/(d^2*e^2-2...
 

3.1.13.2 Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 13.37 (sec) , antiderivative size = 8350, normalized size of antiderivative = 8.15 \[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{(e+f x)^3} \, dx=\text {Result too large to show} \]

input

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

output

Result too large to show
 

3.1.13.3 Rubi [A] (verified)

Time = 2.43 (sec) , antiderivative size = 1006, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6876, 5992, 3042, 4679, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 6876

\(\displaystyle -d^2 \int \frac {(c+d x)^2 \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{(d e-c f+f (c+d x))^3}d\text {csch}^{-1}(c+d x)\)

\(\Big \downarrow \) 5992

\(\displaystyle -d^2 \left (\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 f (f (c+d x)-c f+d e)^2}-\frac {b \int \frac {a+b \text {csch}^{-1}(c+d x)}{(d e-c f+f (c+d x))^2}d\text {csch}^{-1}(c+d x)}{f}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -d^2 \left (\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 f (f (c+d x)-c f+d e)^2}-\frac {b \int \frac {a+b \text {csch}^{-1}(c+d x)}{\left (d e-c f+i f \csc \left (i \text {csch}^{-1}(c+d x)\right )\right )^2}d\text {csch}^{-1}(c+d x)}{f}\right )\)

\(\Big \downarrow \) 4679

\(\displaystyle -d^2 \left (\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 f (f (c+d x)-c f+d e)^2}-\frac {b \int \left (\frac {\left (a+b \text {csch}^{-1}(c+d x)\right ) f^2}{(d e-c f)^2 \left (f+\frac {d e \left (1-\frac {c f}{d e}\right )}{c+d x}\right )^2}+\frac {2 \left (a+b \text {csch}^{-1}(c+d x)\right ) f}{(d e-c f)^2 \left (-f-\frac {d e \left (1-\frac {c f}{d e}\right )}{c+d x}\right )}+\frac {a+b \text {csch}^{-1}(c+d x)}{(d e-c f)^2}\right )d\text {csch}^{-1}(c+d x)}{f}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle -d^2 \left (\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 f (d e-c f+f (c+d x))^2}-\frac {b \left (\frac {\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 ) f^3}{(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 ) \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 ) f^3}{(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 \operatorname {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 ) f^3}{(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 \operatorname {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 ) f^3}{(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 {\sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right ) f^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 {b \log \left (f+\frac {d e-c f}{c+d x}\right ) f^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 \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 ) f}{(d e-c f)^2 \sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}+\frac {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 ) f}{(d e-c f)^2 \sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}-\frac {2 b \operatorname {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 ) f}{(d e-c f)^2 \sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}+\frac {2 b \operatorname {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 ) f}{(d e-c f)^2 \sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}+\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 b (d e-c f)^2}\right )}{f}\right )\)

input

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

output

-(d^2*((a + b*ArcCsch[c + d*x])^2/(2*f*(d*e - c*f + f*(c + d*x))^2) - (b*( 
-((f^2*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)))) + (a + b* 
ArcCsch[c + d*x])^2/(2*b*(d*e - c*f)^2) + (f^3*(a + b*ArcCsch[c + d*x])*Lo 
g[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*f*(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]) - (f^3*(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* 
f*(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*Sqrt[d^2*e^2 - 
2*c*d*e*f + (1 + c^2)*f^2]) + (b*f^2*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*f^3*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* 
f*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*f^3*PolyLog[2, -((E^ArcCsch[c + d*x]*(d*e - c*f))/(f + Sq...
 

3.1.13.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4679

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*Si 
n[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] && IGt 
Q[m, 0]
 

rule 5992

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] + Simp[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 6876

Int[((a_.) + ArcCsch[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^( 
m_.), x_Symbol] :> Simp[-(d^(m + 1))^(-1)   Subst[Int[(a + b*x)^p*Csch[x]*C 
oth[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]
 

3.1.13.4 Maple [F]

\[\int \frac {\left (a +b \,\operatorname {arccsch}\left (d x +c \right )\right )^{2}}{\left (f x +e \right )^{3}}d x\]

input

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

output

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

3.1.13.5 Fricas [F]

\[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{(e+f x)^3} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (d x + c\right ) + a\right )}^{2}}{{\left (f x + e\right )}^{3}} \,d x } \]

input

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

output

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)
 

3.1.13.6 Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{(e+f x)^3} \, dx=\text {Timed out} \]

input

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

output

Timed out
 

3.1.13.7 Maxima [F]

\[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{(e+f x)^3} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (d x + c\right ) + a\right )}^{2}}{{\left (f x + e\right )}^{3}} \,d x } \]

input

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

output

-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)
 

3.1.13.8 Giac [F]

\[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{(e+f x)^3} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (d x + c\right ) + a\right )}^{2}}{{\left (f x + e\right )}^{3}} \,d x } \]

input

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

output

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

3.1.13.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{(e+f x)^3} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c+d\,x}\right )\right )}^2}{{\left (e+f\,x\right )}^3} \,d x \]

input

int((a + b*asinh(1/(c + d*x)))^2/(e + f*x)^3,x)
 

output

int((a + b*asinh(1/(c + d*x)))^2/(e + f*x)^3, x)