\(\int \frac {\text {arccosh}(c x)^2}{(d+e x)^3} \, dx\) [13]

Optimal result

Integrand size = 14, antiderivative size = 352 \[ \int \frac {\text {arccosh}(c x)^2}{(d+e x)^3} \, dx=-\frac {c \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \text {arccosh}(c x)}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\text {arccosh}(c x)^2}{2 e (d+e x)^2}+\frac {c^3 d \text {arccosh}(c x) \log \left (1+\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {c^3 d \text {arccosh}(c x) \log \left (1+\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}+\frac {c^2 \log (d+e x)}{e \left (c^2 d^2-e^2\right )}+\frac {c^3 d \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {c^3 d \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}} \] Output:

-c*(-(-c*x+1)/(c*x+1))^(1/2)*(c*x+1)*arccosh(c*x)/(c^2*d^2-e^2)/(e*x+d)-1/ 
2*arccosh(c*x)^2/e/(e*x+d)^2+c^3*d*arccosh(c*x)*ln(1+e*(c*x+(c*x-1)^(1/2)* 
(c*x+1)^(1/2))/(c*d-(c^2*d^2-e^2)^(1/2)))/e/(c^2*d^2-e^2)^(3/2)-c^3*d*arcc 
osh(c*x)*ln(1+e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2) 
))/e/(c^2*d^2-e^2)^(3/2)+c^2*ln(e*x+d)/e/(c^2*d^2-e^2)+c^3*d*polylog(2,-e* 
(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*d-(c^2*d^2-e^2)^(1/2)))/e/(c^2*d^2-e^ 
2)^(3/2)-c^3*d*polylog(2,-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*d+(c^2*d^ 
2-e^2)^(1/2)))/e/(c^2*d^2-e^2)^(3/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 3.02 (sec) , antiderivative size = 936, normalized size of antiderivative = 2.66 \[ \int \frac {\text {arccosh}(c x)^2}{(d+e x)^3} \, dx =\text {Too large to display} \] Input:

Integrate[ArcCosh[c*x]^2/(d + e*x)^3,x]
 

Output:

c^2*(-((Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x])/((c*d - e)*(c*d 
 + e)*(c*d + c*e*x))) - ArcCosh[c*x]^2/(2*e*(c*d + c*e*x)^2) + Log[1 + (e* 
x)/d]/(c^2*d^2*e - e^3) + (c*d*(2*ArcCosh[c*x]*ArcTan[((c*d + e)*Coth[ArcC 
osh[c*x]/2])/Sqrt[-(c^2*d^2) + e^2]] - (2*I)*ArcCos[-((c*d)/e)]*ArcTan[((- 
(c*d) + e)*Tanh[ArcCosh[c*x]/2])/Sqrt[-(c^2*d^2) + e^2]] + (ArcCos[-((c*d) 
/e)] + 2*(ArcTan[((c*d + e)*Coth[ArcCosh[c*x]/2])/Sqrt[-(c^2*d^2) + e^2]] 
+ ArcTan[((-(c*d) + e)*Tanh[ArcCosh[c*x]/2])/Sqrt[-(c^2*d^2) + e^2]]))*Log 
[Sqrt[-(c^2*d^2) + e^2]/(Sqrt[2]*Sqrt[e]*E^(ArcCosh[c*x]/2)*Sqrt[c*d + c*e 
*x])] + (ArcCos[-((c*d)/e)] - 2*(ArcTan[((c*d + e)*Coth[ArcCosh[c*x]/2])/S 
qrt[-(c^2*d^2) + e^2]] + ArcTan[((-(c*d) + e)*Tanh[ArcCosh[c*x]/2])/Sqrt[- 
(c^2*d^2) + e^2]]))*Log[(Sqrt[-(c^2*d^2) + e^2]*E^(ArcCosh[c*x]/2))/(Sqrt[ 
2]*Sqrt[e]*Sqrt[c*d + c*e*x])] - (ArcCos[-((c*d)/e)] + 2*ArcTan[((-(c*d) + 
 e)*Tanh[ArcCosh[c*x]/2])/Sqrt[-(c^2*d^2) + e^2]])*Log[((c*d + e)*(c*d - e 
 + I*Sqrt[-(c^2*d^2) + e^2])*(-1 + Tanh[ArcCosh[c*x]/2]))/(e*(c*d + e + I* 
Sqrt[-(c^2*d^2) + e^2]*Tanh[ArcCosh[c*x]/2]))] - (ArcCos[-((c*d)/e)] - 2*A 
rcTan[((-(c*d) + e)*Tanh[ArcCosh[c*x]/2])/Sqrt[-(c^2*d^2) + e^2]])*Log[((c 
*d + e)*(-(c*d) + e + I*Sqrt[-(c^2*d^2) + e^2])*(1 + Tanh[ArcCosh[c*x]/2]) 
)/(e*(c*d + e + I*Sqrt[-(c^2*d^2) + e^2]*Tanh[ArcCosh[c*x]/2]))] + I*(Poly 
Log[2, ((c*d - I*Sqrt[-(c^2*d^2) + e^2])*(c*d + e - I*Sqrt[-(c^2*d^2) + e^ 
2]*Tanh[ArcCosh[c*x]/2]))/(e*(c*d + e + I*Sqrt[-(c^2*d^2) + e^2]*Tanh[A...
 

Rubi [A] (warning: unable to verify)

Time = 1.41 (sec) , antiderivative size = 338, normalized size of antiderivative = 0.96, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.071, Rules used = {6378, 6395, 3042, 3805, 26, 3042, 26, 3147, 16, 3801, 2694, 27, 2620, 2715, 2838}

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.

\(\displaystyle \int \frac {\text {arccosh}(c x)^2}{(d+e x)^3} \, dx\)

\(\Big \downarrow \) 6378

\(\displaystyle \frac {c \int \frac {\text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1} (d+e x)^2}dx}{e}-\frac {\text {arccosh}(c x)^2}{2 e (d+e x)^2}\)

\(\Big \downarrow \) 6395

\(\displaystyle \frac {c^2 \int \frac {\text {arccosh}(c x)}{(c d+c e x)^2}d\text {arccosh}(c x)}{e}-\frac {\text {arccosh}(c x)^2}{2 e (d+e x)^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\text {arccosh}(c x)^2}{2 e (d+e x)^2}+\frac {c^2 \int \frac {\text {arccosh}(c x)}{\left (c d+e \sin \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )\right )^2}d\text {arccosh}(c x)}{e}\)

\(\Big \downarrow \) 3805

\(\displaystyle -\frac {\text {arccosh}(c x)^2}{2 e (d+e x)^2}+\frac {c^2 \left (\frac {i e \int -\frac {i \sqrt {\frac {c x-1}{c x+1}} (c x+1)}{c d+c e x}d\text {arccosh}(c x)}{c^2 d^2-e^2}+\frac {c d \int \frac {\text {arccosh}(c x)}{c d+c e x}d\text {arccosh}(c x)}{c^2 d^2-e^2}-\frac {e \sqrt {\frac {c x-1}{c x+1}} (c x+1) \text {arccosh}(c x)}{\left (c^2 d^2-e^2\right ) (c d+c e x)}\right )}{e}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {c^2 \left (\frac {e \int \frac {\sqrt {\frac {c x-1}{c x+1}} (c x+1)}{c d+c e x}d\text {arccosh}(c x)}{c^2 d^2-e^2}+\frac {c d \int \frac {\text {arccosh}(c x)}{c d+c e x}d\text {arccosh}(c x)}{c^2 d^2-e^2}-\frac {e \sqrt {\frac {c x-1}{c x+1}} (c x+1) \text {arccosh}(c x)}{\left (c^2 d^2-e^2\right ) (c d+c e x)}\right )}{e}-\frac {\text {arccosh}(c x)^2}{2 e (d+e x)^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\text {arccosh}(c x)^2}{2 e (d+e x)^2}+\frac {c^2 \left (\frac {c d \int \frac {\text {arccosh}(c x)}{c d+e \sin \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )}d\text {arccosh}(c x)}{c^2 d^2-e^2}+\frac {e \int -\frac {i \cos \left (i \text {arccosh}(c x)-\frac {\pi }{2}\right )}{c d-e \sin \left (i \text {arccosh}(c x)-\frac {\pi }{2}\right )}d\text {arccosh}(c x)}{c^2 d^2-e^2}-\frac {e \sqrt {\frac {c x-1}{c x+1}} (c x+1) \text {arccosh}(c x)}{\left (c^2 d^2-e^2\right ) (c d+c e x)}\right )}{e}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {\text {arccosh}(c x)^2}{2 e (d+e x)^2}+\frac {c^2 \left (\frac {c d \int \frac {\text {arccosh}(c x)}{c d+e \sin \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )}d\text {arccosh}(c x)}{c^2 d^2-e^2}-\frac {i e \int \frac {\cos \left (i \text {arccosh}(c x)-\frac {\pi }{2}\right )}{c d-e \sin \left (i \text {arccosh}(c x)-\frac {\pi }{2}\right )}d\text {arccosh}(c x)}{c^2 d^2-e^2}-\frac {e \sqrt {\frac {c x-1}{c x+1}} (c x+1) \text {arccosh}(c x)}{\left (c^2 d^2-e^2\right ) (c d+c e x)}\right )}{e}\)

\(\Big \downarrow \) 3147

\(\displaystyle -\frac {\text {arccosh}(c x)^2}{2 e (d+e x)^2}+\frac {c^2 \left (\frac {c d \int \frac {\text {arccosh}(c x)}{c d+e \sin \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )}d\text {arccosh}(c x)}{c^2 d^2-e^2}+\frac {\int \frac {1}{c d+c e x}d(c e x)}{c^2 d^2-e^2}-\frac {e \sqrt {\frac {c x-1}{c x+1}} (c x+1) \text {arccosh}(c x)}{\left (c^2 d^2-e^2\right ) (c d+c e x)}\right )}{e}\)

\(\Big \downarrow \) 16

\(\displaystyle -\frac {\text {arccosh}(c x)^2}{2 e (d+e x)^2}+\frac {c^2 \left (\frac {c d \int \frac {\text {arccosh}(c x)}{c d+e \sin \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )}d\text {arccosh}(c x)}{c^2 d^2-e^2}-\frac {e \sqrt {\frac {c x-1}{c x+1}} (c x+1) \text {arccosh}(c x)}{\left (c^2 d^2-e^2\right ) (c d+c e x)}+\frac {\log (c d+c e x)}{c^2 d^2-e^2}\right )}{e}\)

\(\Big \downarrow \) 3801

\(\displaystyle \frac {c^2 \left (\frac {2 c d \int \frac {e^{\text {arccosh}(c x)} \text {arccosh}(c x)}{2 c e^{\text {arccosh}(c x)} d+e e^{2 \text {arccosh}(c x)}+e}d\text {arccosh}(c x)}{c^2 d^2-e^2}-\frac {e \sqrt {\frac {c x-1}{c x+1}} (c x+1) \text {arccosh}(c x)}{\left (c^2 d^2-e^2\right ) (c d+c e x)}+\frac {\log (c d+c e x)}{c^2 d^2-e^2}\right )}{e}-\frac {\text {arccosh}(c x)^2}{2 e (d+e x)^2}\)

\(\Big \downarrow \) 2694

\(\displaystyle \frac {c^2 \left (\frac {2 c d \left (\frac {e \int \frac {e^{\text {arccosh}(c x)} \text {arccosh}(c x)}{2 \left (c d+e e^{\text {arccosh}(c x)}-\sqrt {c^2 d^2-e^2}\right )}d\text {arccosh}(c x)}{\sqrt {c^2 d^2-e^2}}-\frac {e \int \frac {e^{\text {arccosh}(c x)} \text {arccosh}(c x)}{2 \left (c d+e e^{\text {arccosh}(c x)}+\sqrt {c^2 d^2-e^2}\right )}d\text {arccosh}(c x)}{\sqrt {c^2 d^2-e^2}}\right )}{c^2 d^2-e^2}-\frac {e \sqrt {\frac {c x-1}{c x+1}} (c x+1) \text {arccosh}(c x)}{\left (c^2 d^2-e^2\right ) (c d+c e x)}+\frac {\log (c d+c e x)}{c^2 d^2-e^2}\right )}{e}-\frac {\text {arccosh}(c x)^2}{2 e (d+e x)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {c^2 \left (\frac {2 c d \left (\frac {e \int \frac {e^{\text {arccosh}(c x)} \text {arccosh}(c x)}{c d+e e^{\text {arccosh}(c x)}-\sqrt {c^2 d^2-e^2}}d\text {arccosh}(c x)}{2 \sqrt {c^2 d^2-e^2}}-\frac {e \int \frac {e^{\text {arccosh}(c x)} \text {arccosh}(c x)}{c d+e e^{\text {arccosh}(c x)}+\sqrt {c^2 d^2-e^2}}d\text {arccosh}(c x)}{2 \sqrt {c^2 d^2-e^2}}\right )}{c^2 d^2-e^2}-\frac {e \sqrt {\frac {c x-1}{c x+1}} (c x+1) \text {arccosh}(c x)}{\left (c^2 d^2-e^2\right ) (c d+c e x)}+\frac {\log (c d+c e x)}{c^2 d^2-e^2}\right )}{e}-\frac {\text {arccosh}(c x)^2}{2 e (d+e x)^2}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {c^2 \left (\frac {2 c d \left (\frac {e \left (\frac {\text {arccosh}(c x) \log \left (\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}+1\right )}{e}-\frac {\int \log \left (\frac {e^{\text {arccosh}(c x)} e}{c d-\sqrt {c^2 d^2-e^2}}+1\right )d\text {arccosh}(c x)}{e}\right )}{2 \sqrt {c^2 d^2-e^2}}-\frac {e \left (\frac {\text {arccosh}(c x) \log \left (\frac {e e^{\text {arccosh}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}+1\right )}{e}-\frac {\int \log \left (\frac {e^{\text {arccosh}(c x)} e}{c d+\sqrt {c^2 d^2-e^2}}+1\right )d\text {arccosh}(c x)}{e}\right )}{2 \sqrt {c^2 d^2-e^2}}\right )}{c^2 d^2-e^2}-\frac {e \sqrt {\frac {c x-1}{c x+1}} (c x+1) \text {arccosh}(c x)}{\left (c^2 d^2-e^2\right ) (c d+c e x)}+\frac {\log (c d+c e x)}{c^2 d^2-e^2}\right )}{e}-\frac {\text {arccosh}(c x)^2}{2 e (d+e x)^2}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {c^2 \left (\frac {2 c d \left (\frac {e \left (\frac {\text {arccosh}(c x) \log \left (\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}+1\right )}{e}-\frac {\int e^{-\text {arccosh}(c x)} \log \left (\frac {e^{\text {arccosh}(c x)} e}{c d-\sqrt {c^2 d^2-e^2}}+1\right )de^{\text {arccosh}(c x)}}{e}\right )}{2 \sqrt {c^2 d^2-e^2}}-\frac {e \left (\frac {\text {arccosh}(c x) \log \left (\frac {e e^{\text {arccosh}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}+1\right )}{e}-\frac {\int e^{-\text {arccosh}(c x)} \log \left (\frac {e^{\text {arccosh}(c x)} e}{c d+\sqrt {c^2 d^2-e^2}}+1\right )de^{\text {arccosh}(c x)}}{e}\right )}{2 \sqrt {c^2 d^2-e^2}}\right )}{c^2 d^2-e^2}-\frac {e \sqrt {\frac {c x-1}{c x+1}} (c x+1) \text {arccosh}(c x)}{\left (c^2 d^2-e^2\right ) (c d+c e x)}+\frac {\log (c d+c e x)}{c^2 d^2-e^2}\right )}{e}-\frac {\text {arccosh}(c x)^2}{2 e (d+e x)^2}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {c^2 \left (\frac {2 c d \left (\frac {e \left (\frac {\operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\text {arccosh}(c x) \log \left (\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}+1\right )}{e}\right )}{2 \sqrt {c^2 d^2-e^2}}-\frac {e \left (\frac {\operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\text {arccosh}(c x) \log \left (\frac {e e^{\text {arccosh}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}+1\right )}{e}\right )}{2 \sqrt {c^2 d^2-e^2}}\right )}{c^2 d^2-e^2}-\frac {e \sqrt {\frac {c x-1}{c x+1}} (c x+1) \text {arccosh}(c x)}{\left (c^2 d^2-e^2\right ) (c d+c e x)}+\frac {\log (c d+c e x)}{c^2 d^2-e^2}\right )}{e}-\frac {\text {arccosh}(c x)^2}{2 e (d+e x)^2}\)

Input:

Int[ArcCosh[c*x]^2/(d + e*x)^3,x]
 

Output:

-1/2*ArcCosh[c*x]^2/(e*(d + e*x)^2) + (c^2*(-((e*Sqrt[(-1 + c*x)/(1 + c*x) 
]*(1 + c*x)*ArcCosh[c*x])/((c^2*d^2 - e^2)*(c*d + c*e*x))) + Log[c*d + c*e 
*x]/(c^2*d^2 - e^2) + (2*c*d*((e*((ArcCosh[c*x]*Log[1 + (e*E^ArcCosh[c*x]) 
/(c*d - Sqrt[c^2*d^2 - e^2])])/e + PolyLog[2, -((e*E^ArcCosh[c*x])/(c*d - 
Sqrt[c^2*d^2 - e^2]))]/e))/(2*Sqrt[c^2*d^2 - e^2]) - (e*((ArcCosh[c*x]*Log 
[1 + (e*E^ArcCosh[c*x])/(c*d + Sqrt[c^2*d^2 - e^2])])/e + PolyLog[2, -((e* 
E^ArcCosh[c*x])/(c*d + Sqrt[c^2*d^2 - e^2]))]/e))/(2*Sqrt[c^2*d^2 - e^2])) 
)/(c^2*d^2 - e^2)))/e
 

Defintions of rubi rules used

rule 16

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

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2620

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[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 2694

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]}, Simp[2*(c/q)   Int 
[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[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 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[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 2838

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 3042

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

rule 3147

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m 
_.), x_Symbol] :> Simp[1/(b^p*f)   Subst[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] && IntegerQ[(p 
 - 1)/2] && NeQ[a^2 - b^2, 0]
 

rule 3801

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + Pi*(k_.) + (Comple 
x[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Simp[2   Int[((c + d*x)^m*(E^((-I)*e 
+ f*fz*x)/(b + (2*a*E^((-I)*e + f*fz*x))/E^(I*Pi*(k - 1/2)) - (b*E^(2*((-I) 
*e + f*fz*x)))/E^(2*I*k*Pi))))/E^(I*Pi*(k - 1/2)), x], x] /; FreeQ[{a, b, c 
, d, e, f, fz}, x] && IntegerQ[2*k] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
 

rule 3805

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] + (Simp[a/(a^2 - b^2)   Int[(c + d*x)^m/(a + b*Sin[e + f*x]), x] 
, x] - Simp[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 6378

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

rule 6395

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/( 
Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[1/( 
c^(m + 1)*Sqrt[(-d1)*d2])   Subst[Int[(a + b*x)^n*(c*f + g*Cosh[x])^m, x], 
x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g, n}, x] && EqQ 
[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[m] && GtQ[d1, 0] && LtQ[d2, 
 0] && (GtQ[m, 0] || IGtQ[n, 0])
 

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 605, normalized size of antiderivative = 1.72

method	result	size
derivativedivides	\(\frac {-\frac {c^{3} \operatorname {arccosh}\left (c x \right ) \left (2 c d e \sqrt {c x -1}\, \sqrt {c x +1}+2 c \,e^{2} x \sqrt {c x -1}\, \sqrt {c x +1}+c^{2} d^{2} \operatorname {arccosh}\left (c x \right )-2 c^{2} d^{2}-4 c^{2} d e x -2 c^{2} e^{2} x^{2}-e^{2} \operatorname {arccosh}\left (c x \right )\right )}{2 e \left (e c x +c d \right )^{2} \left (c^{2} d^{2}-e^{2}\right )}+\frac {c^{4} d \,\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {-c d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{\left (c^{2} d^{2}-e^{2}\right )^{\frac {3}{2}} e}-\frac {c^{4} d \,\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {c d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{\left (c^{2} d^{2}-e^{2}\right )^{\frac {3}{2}} e}+\frac {c^{4} d \operatorname {dilog}\left (\frac {-c d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{\left (c^{2} d^{2}-e^{2}\right )^{\frac {3}{2}} e}-\frac {c^{4} d \operatorname {dilog}\left (\frac {c d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{\left (c^{2} d^{2}-e^{2}\right )^{\frac {3}{2}} e}-\frac {2 c^{3} \ln \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{\left (c^{2} d^{2}-e^{2}\right ) e}+\frac {c^{3} \ln \left (2 d c \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}+e \right )}{\left (c^{2} d^{2}-e^{2}\right ) e}}{c}\)	\(605\)
default	\(\frac {-\frac {c^{3} \operatorname {arccosh}\left (c x \right ) \left (2 c d e \sqrt {c x -1}\, \sqrt {c x +1}+2 c \,e^{2} x \sqrt {c x -1}\, \sqrt {c x +1}+c^{2} d^{2} \operatorname {arccosh}\left (c x \right )-2 c^{2} d^{2}-4 c^{2} d e x -2 c^{2} e^{2} x^{2}-e^{2} \operatorname {arccosh}\left (c x \right )\right )}{2 e \left (e c x +c d \right )^{2} \left (c^{2} d^{2}-e^{2}\right )}+\frac {c^{4} d \,\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {-c d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{\left (c^{2} d^{2}-e^{2}\right )^{\frac {3}{2}} e}-\frac {c^{4} d \,\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {c d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{\left (c^{2} d^{2}-e^{2}\right )^{\frac {3}{2}} e}+\frac {c^{4} d \operatorname {dilog}\left (\frac {-c d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{\left (c^{2} d^{2}-e^{2}\right )^{\frac {3}{2}} e}-\frac {c^{4} d \operatorname {dilog}\left (\frac {c d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{\left (c^{2} d^{2}-e^{2}\right )^{\frac {3}{2}} e}-\frac {2 c^{3} \ln \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{\left (c^{2} d^{2}-e^{2}\right ) e}+\frac {c^{3} \ln \left (2 d c \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}+e \right )}{\left (c^{2} d^{2}-e^{2}\right ) e}}{c}\)	\(605\)

Input:

int(arccosh(c*x)^2/(e*x+d)^3,x,method=_RETURNVERBOSE)
 

Output:

1/c*(-1/2*c^3*arccosh(c*x)*(2*c*d*e*(c*x-1)^(1/2)*(c*x+1)^(1/2)+2*c*e^2*x* 
(c*x-1)^(1/2)*(c*x+1)^(1/2)+c^2*d^2*arccosh(c*x)-2*c^2*d^2-4*c^2*d*e*x-2*c 
^2*e^2*x^2-e^2*arccosh(c*x))/e/(c*e*x+c*d)^2/(c^2*d^2-e^2)+1/(c^2*d^2-e^2) 
^(3/2)/e*c^4*d*arccosh(c*x)*ln((-c*d-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+( 
c^2*d^2-e^2)^(1/2))/(-c*d+(c^2*d^2-e^2)^(1/2)))-1/(c^2*d^2-e^2)^(3/2)/e*c^ 
4*d*arccosh(c*x)*ln((c*d+e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+(c^2*d^2-e^2) 
^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2)))+1/(c^2*d^2-e^2)^(3/2)/e*c^4*d*dilog((-c 
*d-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+(c^2*d^2-e^2)^(1/2))/(-c*d+(c^2*d^2 
-e^2)^(1/2)))-1/(c^2*d^2-e^2)^(3/2)/e*c^4*d*dilog((c*d+e*(c*x+(c*x-1)^(1/2 
)*(c*x+1)^(1/2))+(c^2*d^2-e^2)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2)))-2/(c^2*d^ 
2-e^2)/e*c^3*ln(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+1/(c^2*d^2-e^2)/e*c^3*ln( 
2*d*c*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2) 
)^2+e))
 

Fricas [F]

\[ \int \frac {\text {arccosh}(c x)^2}{(d+e x)^3} \, dx=\int { \frac {\operatorname {arcosh}\left (c x\right )^{2}}{{\left (e x + d\right )}^{3}} \,d x } \] Input:

integrate(arccosh(c*x)^2/(e*x+d)^3,x, algorithm="fricas")
 

Output:

integral(arccosh(c*x)^2/(e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3), x)
 

Sympy [F]

\[ \int \frac {\text {arccosh}(c x)^2}{(d+e x)^3} \, dx=\int \frac {\operatorname {acosh}^{2}{\left (c x \right )}}{\left (d + e x\right )^{3}}\, dx \] Input:

integrate(acosh(c*x)**2/(e*x+d)**3,x)
 

Output:

Integral(acosh(c*x)**2/(d + e*x)**3, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\text {arccosh}(c x)^2}{(d+e x)^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(arccosh(c*x)^2/(e*x+d)^3,x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume((e-c*d)*(e+c*d)>0)', see `assume 
?` for mor
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\text {arccosh}(c x)^2}{(d+e x)^3} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(arccosh(c*x)^2/(e*x+d)^3,x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Limit: Max order reached or unable 
to make series expansion Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arccosh}(c x)^2}{(d+e x)^3} \, dx=\int \frac {{\mathrm {acosh}\left (c\,x\right )}^2}{{\left (d+e\,x\right )}^3} \,d x \] Input:

int(acosh(c*x)^2/(d + e*x)^3,x)
 

Output:

int(acosh(c*x)^2/(d + e*x)^3, x)
 

Reduce [F]

\[ \int \frac {\text {arccosh}(c x)^2}{(d+e x)^3} \, dx=\int \frac {\mathit {acosh} \left (c x \right )^{2}}{e^{3} x^{3}+3 d \,e^{2} x^{2}+3 d^{2} e x +d^{3}}d x \] Input:

int(acosh(c*x)^2/(e*x+d)^3,x)
                                                                                    
                                                                                    
 

Output:

int(acosh(c*x)**2/(d**3 + 3*d**2*e*x + 3*d*e**2*x**2 + e**3*x**3),x)