\(\int \frac {(a+b \text {arccosh}(c+d x))^3}{(c e+d e x)^4} \, dx\) [39]

Optimal result

Integrand size = 23, antiderivative size = 297 \[ \int \frac {(a+b \text {arccosh}(c+d x))^3}{(c e+d e x)^4} \, dx=\frac {b^2 (a+b \text {arccosh}(c+d x))}{d e^4 (c+d x)}+\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^2}{2 d e^4 (c+d x)^2}-\frac {(a+b \text {arccosh}(c+d x))^3}{3 d e^4 (c+d x)^3}+\frac {b (a+b \text {arccosh}(c+d x))^2 \arctan \left (e^{\text {arccosh}(c+d x)}\right )}{d e^4}-\frac {b^3 \arctan \left (\sqrt {-1+c+d x} \sqrt {1+c+d x}\right )}{d e^4}-\frac {i b^2 (a+b \text {arccosh}(c+d x)) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c+d x)}\right )}{d e^4}+\frac {i b^2 (a+b \text {arccosh}(c+d x)) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c+d x)}\right )}{d e^4}+\frac {i b^3 \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c+d x)}\right )}{d e^4}-\frac {i b^3 \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c+d x)}\right )}{d e^4} \] Output:

b^2*(a+b*arccosh(d*x+c))/d/e^4/(d*x+c)+1/2*b*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/ 
2)*(a+b*arccosh(d*x+c))^2/d/e^4/(d*x+c)^2-1/3*(a+b*arccosh(d*x+c))^3/d/e^4 
/(d*x+c)^3+b*(a+b*arccosh(d*x+c))^2*arctan(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1) 
^(1/2))/d/e^4-b^3*arctan((d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/d/e^4-I*b^2*(a+b 
*arccosh(d*x+c))*polylog(2,-I*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)))/d/e 
^4+I*b^2*(a+b*arccosh(d*x+c))*polylog(2,I*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1) 
^(1/2)))/d/e^4+I*b^3*polylog(3,-I*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))) 
/d/e^4-I*b^3*polylog(3,I*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)))/d/e^4
 

Mathematica [A] (warning: unable to verify)

Time = 1.72 (sec) , antiderivative size = 489, normalized size of antiderivative = 1.65 \[ \int \frac {(a+b \text {arccosh}(c+d x))^3}{(c e+d e x)^4} \, dx=\frac {-\frac {2 a^3}{(c+d x)^3}+\frac {3 a^2 b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{(c+d x)^2}-\frac {6 a^2 b \text {arccosh}(c+d x)}{(c+d x)^3}-3 a^2 b \arctan \left (\frac {1}{\sqrt {-1+c+d x} \sqrt {1+c+d x}}\right )+6 a b^2 \left (\frac {1}{c+d x}+\frac {\sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x) \text {arccosh}(c+d x)}{(c+d x)^2}-\frac {\text {arccosh}(c+d x)^2}{(c+d x)^3}-i \text {arccosh}(c+d x) \log \left (1-i e^{-\text {arccosh}(c+d x)}\right )+i \text {arccosh}(c+d x) \log \left (1+i e^{-\text {arccosh}(c+d x)}\right )-i \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c+d x)}\right )+i \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c+d x)}\right )\right )+b^3 \left (\frac {6 \text {arccosh}(c+d x)}{c+d x}+\frac {3 \sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x) \text {arccosh}(c+d x)^2}{(c+d x)^2}-\frac {2 \text {arccosh}(c+d x)^3}{(c+d x)^3}+3 i \left (4 i \arctan \left (e^{\text {arccosh}(c+d x)}\right )+\text {arccosh}(c+d x)^2 \log \left (1-i e^{\text {arccosh}(c+d x)}\right )-\text {arccosh}(c+d x)^2 \log \left (1+i e^{\text {arccosh}(c+d x)}\right )-2 \text {arccosh}(c+d x) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c+d x)}\right )+2 \text {arccosh}(c+d x) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c+d x)}\right )+2 \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c+d x)}\right )-2 \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c+d x)}\right )\right )\right )}{6 d e^4} \] Input:

Integrate[(a + b*ArcCosh[c + d*x])^3/(c*e + d*e*x)^4,x]
 

Output:

((-2*a^3)/(c + d*x)^3 + (3*a^2*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(c 
+ d*x)^2 - (6*a^2*b*ArcCosh[c + d*x])/(c + d*x)^3 - 3*a^2*b*ArcTan[1/(Sqrt 
[-1 + c + d*x]*Sqrt[1 + c + d*x])] + 6*a*b^2*((c + d*x)^(-1) + (Sqrt[(-1 + 
 c + d*x)/(1 + c + d*x)]*(1 + c + d*x)*ArcCosh[c + d*x])/(c + d*x)^2 - Arc 
Cosh[c + d*x]^2/(c + d*x)^3 - I*ArcCosh[c + d*x]*Log[1 - I/E^ArcCosh[c + d 
*x]] + I*ArcCosh[c + d*x]*Log[1 + I/E^ArcCosh[c + d*x]] - I*PolyLog[2, (-I 
)/E^ArcCosh[c + d*x]] + I*PolyLog[2, I/E^ArcCosh[c + d*x]]) + b^3*((6*ArcC 
osh[c + d*x])/(c + d*x) + (3*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d 
*x)*ArcCosh[c + d*x]^2)/(c + d*x)^2 - (2*ArcCosh[c + d*x]^3)/(c + d*x)^3 + 
 (3*I)*((4*I)*ArcTan[E^ArcCosh[c + d*x]] + ArcCosh[c + d*x]^2*Log[1 - I*E^ 
ArcCosh[c + d*x]] - ArcCosh[c + d*x]^2*Log[1 + I*E^ArcCosh[c + d*x]] - 2*A 
rcCosh[c + d*x]*PolyLog[2, (-I)*E^ArcCosh[c + d*x]] + 2*ArcCosh[c + d*x]*P 
olyLog[2, I*E^ArcCosh[c + d*x]] + 2*PolyLog[3, (-I)*E^ArcCosh[c + d*x]] - 
2*PolyLog[3, I*E^ArcCosh[c + d*x]])))/(6*d*e^4)
 

Rubi [A] (verified)

Time = 2.04 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.84, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {6411, 27, 6298, 6348, 6298, 103, 216, 6362, 3042, 4668, 3011, 2720, 7143}

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.

\(\displaystyle \int \frac {(a+b \text {arccosh}(c+d x))^3}{(c e+d e x)^4} \, dx\)

\(\Big \downarrow \) 6411

\(\displaystyle \frac {\int \frac {(a+b \text {arccosh}(c+d x))^3}{e^4 (c+d x)^4}d(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(a+b \text {arccosh}(c+d x))^3}{(c+d x)^4}d(c+d x)}{d e^4}\)

\(\Big \downarrow \) 6298

\(\displaystyle \frac {b \int \frac {(a+b \text {arccosh}(c+d x))^2}{\sqrt {c+d x-1} (c+d x)^3 \sqrt {c+d x+1}}d(c+d x)-\frac {(a+b \text {arccosh}(c+d x))^3}{3 (c+d x)^3}}{d e^4}\)

\(\Big \downarrow \) 6348

\(\displaystyle \frac {b \left (-b \int \frac {a+b \text {arccosh}(c+d x)}{(c+d x)^2}d(c+d x)+\frac {1}{2} \int \frac {(a+b \text {arccosh}(c+d x))^2}{\sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1}}d(c+d x)+\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^2}{2 (c+d x)^2}\right )-\frac {(a+b \text {arccosh}(c+d x))^3}{3 (c+d x)^3}}{d e^4}\)

\(\Big \downarrow \) 6298

\(\displaystyle \frac {b \left (-b \left (b \int \frac {1}{\sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1}}d(c+d x)-\frac {a+b \text {arccosh}(c+d x)}{c+d x}\right )+\frac {1}{2} \int \frac {(a+b \text {arccosh}(c+d x))^2}{\sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1}}d(c+d x)+\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^2}{2 (c+d x)^2}\right )-\frac {(a+b \text {arccosh}(c+d x))^3}{3 (c+d x)^3}}{d e^4}\)

\(\Big \downarrow \) 103

\(\displaystyle \frac {b \left (-b \left (b \int \frac {1}{(c+d x-1) (c+d x+1)+1}d\left (\sqrt {c+d x-1} \sqrt {c+d x+1}\right )-\frac {a+b \text {arccosh}(c+d x)}{c+d x}\right )+\frac {1}{2} \int \frac {(a+b \text {arccosh}(c+d x))^2}{\sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1}}d(c+d x)+\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^2}{2 (c+d x)^2}\right )-\frac {(a+b \text {arccosh}(c+d x))^3}{3 (c+d x)^3}}{d e^4}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {b \left (\frac {1}{2} \int \frac {(a+b \text {arccosh}(c+d x))^2}{\sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1}}d(c+d x)-b \left (b \arctan \left (\sqrt {c+d x-1} \sqrt {c+d x+1}\right )-\frac {a+b \text {arccosh}(c+d x)}{c+d x}\right )+\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^2}{2 (c+d x)^2}\right )-\frac {(a+b \text {arccosh}(c+d x))^3}{3 (c+d x)^3}}{d e^4}\)

\(\Big \downarrow \) 6362

\(\displaystyle \frac {b \left (\frac {1}{2} \int \frac {(a+b \text {arccosh}(c+d x))^2}{c+d x}d\text {arccosh}(c+d x)-b \left (b \arctan \left (\sqrt {c+d x-1} \sqrt {c+d x+1}\right )-\frac {a+b \text {arccosh}(c+d x)}{c+d x}\right )+\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^2}{2 (c+d x)^2}\right )-\frac {(a+b \text {arccosh}(c+d x))^3}{3 (c+d x)^3}}{d e^4}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {(a+b \text {arccosh}(c+d x))^3}{3 (c+d x)^3}+b \left (\frac {1}{2} \int (a+b \text {arccosh}(c+d x))^2 \csc \left (i \text {arccosh}(c+d x)+\frac {\pi }{2}\right )d\text {arccosh}(c+d x)-b \left (b \arctan \left (\sqrt {c+d x-1} \sqrt {c+d x+1}\right )-\frac {a+b \text {arccosh}(c+d x)}{c+d x}\right )+\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^2}{2 (c+d x)^2}\right )}{d e^4}\)

\(\Big \downarrow \) 4668

\(\displaystyle \frac {-\frac {(a+b \text {arccosh}(c+d x))^3}{3 (c+d x)^3}+b \left (\frac {1}{2} \left (-2 i b \int (a+b \text {arccosh}(c+d x)) \log \left (1-i e^{\text {arccosh}(c+d x)}\right )d\text {arccosh}(c+d x)+2 i b \int (a+b \text {arccosh}(c+d x)) \log \left (1+i e^{\text {arccosh}(c+d x)}\right )d\text {arccosh}(c+d x)+2 \arctan \left (e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))^2\right )-b \left (b \arctan \left (\sqrt {c+d x-1} \sqrt {c+d x+1}\right )-\frac {a+b \text {arccosh}(c+d x)}{c+d x}\right )+\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^2}{2 (c+d x)^2}\right )}{d e^4}\)

\(\Big \downarrow \) 3011

\(\displaystyle \frac {-\frac {(a+b \text {arccosh}(c+d x))^3}{3 (c+d x)^3}+b \left (\frac {1}{2} \left (2 i b \left (b \int \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c+d x)}\right )d\text {arccosh}(c+d x)-\operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))\right )-2 i b \left (b \int \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c+d x)}\right )d\text {arccosh}(c+d x)-\operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))\right )+2 \arctan \left (e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))^2\right )-b \left (b \arctan \left (\sqrt {c+d x-1} \sqrt {c+d x+1}\right )-\frac {a+b \text {arccosh}(c+d x)}{c+d x}\right )+\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^2}{2 (c+d x)^2}\right )}{d e^4}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {-\frac {(a+b \text {arccosh}(c+d x))^3}{3 (c+d x)^3}+b \left (\frac {1}{2} \left (2 i b \left (b \int e^{-\text {arccosh}(c+d x)} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c+d x)}\right )de^{\text {arccosh}(c+d x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))\right )-2 i b \left (b \int e^{-\text {arccosh}(c+d x)} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c+d x)}\right )de^{\text {arccosh}(c+d x)}-\operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))\right )+2 \arctan \left (e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))^2\right )-b \left (b \arctan \left (\sqrt {c+d x-1} \sqrt {c+d x+1}\right )-\frac {a+b \text {arccosh}(c+d x)}{c+d x}\right )+\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^2}{2 (c+d x)^2}\right )}{d e^4}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {-\frac {(a+b \text {arccosh}(c+d x))^3}{3 (c+d x)^3}+b \left (\frac {1}{2} \left (2 \arctan \left (e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))^2+2 i b \left (b \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c+d x)}\right )-\operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))\right )-2 i b \left (b \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c+d x)}\right )-\operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))\right )\right )-b \left (b \arctan \left (\sqrt {c+d x-1} \sqrt {c+d x+1}\right )-\frac {a+b \text {arccosh}(c+d x)}{c+d x}\right )+\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^2}{2 (c+d x)^2}\right )}{d e^4}\)

Input:

Int[(a + b*ArcCosh[c + d*x])^3/(c*e + d*e*x)^4,x]
 

Output:

(-1/3*(a + b*ArcCosh[c + d*x])^3/(c + d*x)^3 + b*((Sqrt[-1 + c + d*x]*Sqrt 
[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^2)/(2*(c + d*x)^2) - b*(-((a + b*Ar 
cCosh[c + d*x])/(c + d*x)) + b*ArcTan[Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x] 
]) + (2*(a + b*ArcCosh[c + d*x])^2*ArcTan[E^ArcCosh[c + d*x]] + (2*I)*b*(- 
((a + b*ArcCosh[c + d*x])*PolyLog[2, (-I)*E^ArcCosh[c + d*x]]) + b*PolyLog 
[3, (-I)*E^ArcCosh[c + d*x]]) - (2*I)*b*(-((a + b*ArcCosh[c + d*x])*PolyLo 
g[2, I*E^ArcCosh[c + d*x]]) + b*PolyLog[3, I*E^ArcCosh[c + d*x]]))/2))/(d* 
e^4)
 

Defintions of rubi rules used

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 103

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_ 
))), x_] :> Simp[b*f   Subst[Int[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sq 
rt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d 
*e - f*(b*c + a*d), 0]
 

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{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 3011

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] + Simp[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 3042

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

rule 4668

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

rule 6298

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

rule 6348

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e 
1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[(f*x)^(m + 1) 
*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(d1*d2*f*( 
m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1)))   Int[(f*x)^(m + 2)* 
(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x], x] + Simp[b*c*(n/(f 
*(m + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p] 
   Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCos 
h[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, p}, x] && Eq 
Q[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && ILtQ[m, -1]
 

rule 6362

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

rule 6411

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

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> 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]
 

Maple [F]

\[\int \frac {\left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{3}}{\left (d e x +c e \right )^{4}}d x\]

Input:

int((a+b*arccosh(d*x+c))^3/(d*e*x+c*e)^4,x)
 

Output:

int((a+b*arccosh(d*x+c))^3/(d*e*x+c*e)^4,x)
 

Fricas [F]

\[ \int \frac {(a+b \text {arccosh}(c+d x))^3}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{4}} \,d x } \] Input:

integrate((a+b*arccosh(d*x+c))^3/(d*e*x+c*e)^4,x, algorithm="fricas")
 

Output:

integral((b^3*arccosh(d*x + c)^3 + 3*a*b^2*arccosh(d*x + c)^2 + 3*a^2*b*ar 
ccosh(d*x + c) + a^3)/(d^4*e^4*x^4 + 4*c*d^3*e^4*x^3 + 6*c^2*d^2*e^4*x^2 + 
 4*c^3*d*e^4*x + c^4*e^4), x)
 

Sympy [F]

\[ \int \frac {(a+b \text {arccosh}(c+d x))^3}{(c e+d e x)^4} \, dx=\frac {\int \frac {a^{3}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {3 a b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {3 a^{2} b \operatorname {acosh}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx}{e^{4}} \] Input:

integrate((a+b*acosh(d*x+c))**3/(d*e*x+c*e)**4,x)
 

Output:

(Integral(a**3/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d** 
4*x**4), x) + Integral(b**3*acosh(c + d*x)**3/(c**4 + 4*c**3*d*x + 6*c**2* 
d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4), x) + Integral(3*a*b**2*acosh(c + d 
*x)**2/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4), 
 x) + Integral(3*a**2*b*acosh(c + d*x)/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x* 
*2 + 4*c*d**3*x**3 + d**4*x**4), x))/e**4
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {(a+b \text {arccosh}(c+d x))^3}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{4}} \,d x } \] Input:

integrate((a+b*arccosh(d*x+c))^3/(d*e*x+c*e)^4,x, algorithm="maxima")
 

Output:

-1/3*b^3*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^3/(d^4*e^4*x^3 
 + 3*c*d^3*e^4*x^2 + 3*c^2*d^2*e^4*x + c^3*d*e^4) - 1/3*a^3/(d^4*e^4*x^3 + 
 3*c*d^3*e^4*x^2 + 3*c^2*d^2*e^4*x + c^3*d*e^4) + integrate(((3*(c^3 - c)* 
a*b^2 + (c^3 - c)*b^3 + (3*a*b^2*d^3 + b^3*d^3)*x^3 + 3*(3*a*b^2*c*d^2 + b 
^3*c*d^2)*x^2 + (b^3*c^2 + 3*(c^2 - 1)*a*b^2 + (3*a*b^2*d^2 + b^3*d^2)*x^2 
 + 2*(3*a*b^2*c*d + b^3*c*d)*x)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + (3*( 
3*c^2*d - d)*a*b^2 + (3*c^2*d - d)*b^3)*x)*log(d*x + sqrt(d*x + c + 1)*sqr 
t(d*x + c - 1) + c)^2 + 3*(a^2*b*d^3*x^3 + 3*a^2*b*c*d^2*x^2 + (3*c^2*d - 
d)*a^2*b*x + (c^3 - c)*a^2*b + (a^2*b*d^2*x^2 + 2*a^2*b*c*d*x + (c^2 - 1)* 
a^2*b)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1))*log(d*x + sqrt(d*x + c + 1)*sq 
rt(d*x + c - 1) + c))/(d^7*e^4*x^7 + 7*c*d^6*e^4*x^6 + c^7*e^4 - c^5*e^4 + 
 (21*c^2*d^5*e^4 - d^5*e^4)*x^5 + 5*(7*c^3*d^4*e^4 - c*d^4*e^4)*x^4 + 5*(7 
*c^4*d^3*e^4 - 2*c^2*d^3*e^4)*x^3 + (21*c^5*d^2*e^4 - 10*c^3*d^2*e^4)*x^2 
+ (d^6*e^4*x^6 + 6*c*d^5*e^4*x^5 + c^6*e^4 - c^4*e^4 + (15*c^2*d^4*e^4 - d 
^4*e^4)*x^4 + 4*(5*c^3*d^3*e^4 - c*d^3*e^4)*x^3 + 3*(5*c^4*d^2*e^4 - 2*c^2 
*d^2*e^4)*x^2 + 2*(3*c^5*d*e^4 - 2*c^3*d*e^4)*x)*sqrt(d*x + c + 1)*sqrt(d* 
x + c - 1) + (7*c^6*d*e^4 - 5*c^4*d*e^4)*x), x)
 

Giac [F]

\[ \int \frac {(a+b \text {arccosh}(c+d x))^3}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{4}} \,d x } \] Input:

integrate((a+b*arccosh(d*x+c))^3/(d*e*x+c*e)^4,x, algorithm="giac")
 

Output:

integrate((b*arccosh(d*x + c) + a)^3/(d*e*x + c*e)^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arccosh}(c+d x))^3}{(c e+d e x)^4} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^3}{{\left (c\,e+d\,e\,x\right )}^4} \,d x \] Input:

int((a + b*acosh(c + d*x))^3/(c*e + d*e*x)^4,x)
 

Output:

int((a + b*acosh(c + d*x))^3/(c*e + d*e*x)^4, x)
 

Reduce [F]

\[ \int \frac {(a+b \text {arccosh}(c+d x))^3}{(c e+d e x)^4} \, dx =\text {Too large to display} \] Input:

int((a+b*acosh(d*x+c))^3/(d*e*x+c*e)^4,x)
 

Output:

(9*int(acosh(c + d*x)/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x** 
3 + d**4*x**4),x)*a**2*b*c**3*d + 27*int(acosh(c + d*x)/(c**4 + 4*c**3*d*x 
 + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4),x)*a**2*b*c**2*d**2*x + 2 
7*int(acosh(c + d*x)/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 
 + d**4*x**4),x)*a**2*b*c*d**3*x**2 + 9*int(acosh(c + d*x)/(c**4 + 4*c**3* 
d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4),x)*a**2*b*d**4*x**3 + 
3*int(acosh(c + d*x)**3/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x 
**3 + d**4*x**4),x)*b**3*c**3*d + 9*int(acosh(c + d*x)**3/(c**4 + 4*c**3*d 
*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4),x)*b**3*c**2*d**2*x + 9 
*int(acosh(c + d*x)**3/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x* 
*3 + d**4*x**4),x)*b**3*c*d**3*x**2 + 3*int(acosh(c + d*x)**3/(c**4 + 4*c* 
*3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4),x)*b**3*d**4*x**3 + 
 9*int(acosh(c + d*x)**2/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3* 
x**3 + d**4*x**4),x)*a*b**2*c**3*d + 27*int(acosh(c + d*x)**2/(c**4 + 4*c* 
*3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4),x)*a*b**2*c**2*d**2 
*x + 27*int(acosh(c + d*x)**2/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c* 
d**3*x**3 + d**4*x**4),x)*a*b**2*c*d**3*x**2 + 9*int(acosh(c + d*x)**2/(c* 
*4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4),x)*a*b**2* 
d**4*x**3 - a**3)/(3*d*e**4*(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3 
))