\(\int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^3 \, dx\) [32]

Optimal result

Integrand size = 23, antiderivative size = 307 \[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^3 \, dx=-\frac {45 b^3 e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{256 d}-\frac {3 b^3 e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{128 d}-\frac {45 b^3 e^3 \text {arccosh}(c+d x)}{256 d}+\frac {9 b^2 e^3 (c+d x)^2 (a+b \text {arccosh}(c+d x))}{32 d}+\frac {3 b^2 e^3 (c+d x)^4 (a+b \text {arccosh}(c+d x))}{32 d}-\frac {9 b e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^2}{32 d}-\frac {3 b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^2}{16 d}-\frac {3 e^3 (a+b \text {arccosh}(c+d x))^3}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arccosh}(c+d x))^3}{4 d} \] Output:

-45/256*b^3*e^3*(d*x+c-1)^(1/2)*(d*x+c)*(d*x+c+1)^(1/2)/d-3/128*b^3*e^3*(d 
*x+c-1)^(1/2)*(d*x+c)^3*(d*x+c+1)^(1/2)/d-45/256*b^3*e^3*arccosh(d*x+c)/d+ 
9/32*b^2*e^3*(d*x+c)^2*(a+b*arccosh(d*x+c))/d+3/32*b^2*e^3*(d*x+c)^4*(a+b* 
arccosh(d*x+c))/d-9/32*b*e^3*(d*x+c-1)^(1/2)*(d*x+c)*(d*x+c+1)^(1/2)*(a+b* 
arccosh(d*x+c))^2/d-3/16*b*e^3*(d*x+c-1)^(1/2)*(d*x+c)^3*(d*x+c+1)^(1/2)*( 
a+b*arccosh(d*x+c))^2/d-3/32*e^3*(a+b*arccosh(d*x+c))^3/d+1/4*e^3*(d*x+c)^ 
4*(a+b*arccosh(d*x+c))^3/d
 

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.17 \[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^3 \, dx=\frac {e^3 \left (72 a b^2 (c+d x)^2+8 a \left (8 a^2+3 b^2\right ) (c+d x)^4+3 b \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \left (-3 \left (8 a^2+5 b^2\right )-2 \left (8 a^2+b^2\right ) (c+d x)^2\right )-24 b (c+d x) \left (-3 b^2 (c+d x)-8 a^2 (c+d x)^3-b^2 (c+d x)^3+6 a b \sqrt {-1+c+d x} \sqrt {1+c+d x}+4 a b \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}\right ) \text {arccosh}(c+d x)+24 b^2 \left (-3 a+8 a (c+d x)^4-3 b \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}-2 b \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}\right ) \text {arccosh}(c+d x)^2+8 b^3 \left (-3+8 (c+d x)^4\right ) \text {arccosh}(c+d x)^3-9 b \left (8 a^2+5 b^2\right ) \log \left (c+d x+\sqrt {-1+c+d x} \sqrt {1+c+d x}\right )\right )}{256 d} \] Input:

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

Output:

(e^3*(72*a*b^2*(c + d*x)^2 + 8*a*(8*a^2 + 3*b^2)*(c + d*x)^4 + 3*b*Sqrt[-1 
 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x]*(-3*(8*a^2 + 5*b^2) - 2*(8*a^2 + b 
^2)*(c + d*x)^2) - 24*b*(c + d*x)*(-3*b^2*(c + d*x) - 8*a^2*(c + d*x)^3 - 
b^2*(c + d*x)^3 + 6*a*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x] + 4*a*b*Sqrt[ 
-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])*ArcCosh[c + d*x] + 24*b^2*(-3 
*a + 8*a*(c + d*x)^4 - 3*b*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x] 
- 2*b*Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[1 + c + d*x])*ArcCosh[c + d*x]^2 
 + 8*b^3*(-3 + 8*(c + d*x)^4)*ArcCosh[c + d*x]^3 - 9*b*(8*a^2 + 5*b^2)*Log 
[c + d*x + Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]]))/(256*d)
 

Rubi [A] (verified)

Time = 1.64 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.05, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {6411, 27, 6298, 6354, 6298, 111, 27, 101, 43, 6354, 6298, 101, 43, 6308}

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.

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

\(\Big \downarrow \) 6411

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 6354

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

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 111

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 101

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

\(\Big \downarrow \) 43

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

\(\Big \downarrow \) 6354

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

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 101

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

\(\Big \downarrow \) 43

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

\(\Big \downarrow \) 6308

\(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arccosh}(c+d x))^3-\frac {3}{4} b \left (\frac {1}{4} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3 (a+b \text {arccosh}(c+d x))^2-\frac {1}{2} b \left (\frac {1}{4} (c+d x)^4 (a+b \text {arccosh}(c+d x))-\frac {1}{4} b \left (\frac {3}{4} \left (\frac {1}{2} \text {arccosh}(c+d x)+\frac {1}{2} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)\right )+\frac {1}{4} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3\right )\right )+\frac {3}{4} \left (\frac {(a+b \text {arccosh}(c+d x))^3}{6 b}+\frac {1}{2} \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^2-b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arccosh}(c+d x))-\frac {1}{2} b \left (\frac {1}{2} \text {arccosh}(c+d x)+\frac {1}{2} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)\right )\right )\right )\right )\right )}{d}\)

Input:

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

Output:

(e^3*(((c + d*x)^4*(a + b*ArcCosh[c + d*x])^3)/4 - (3*b*((Sqrt[-1 + c + d* 
x]*(c + d*x)^3*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^2)/4 - (b*(-1/4* 
(b*((Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[1 + c + d*x])/4 + (3*((Sqrt[-1 + 
c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/2 + ArcCosh[c + d*x]/2))/4)) + ((c + 
 d*x)^4*(a + b*ArcCosh[c + d*x]))/4))/2 + (3*((Sqrt[-1 + c + d*x]*(c + d*x 
)*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^2)/2 + (a + b*ArcCosh[c + d*x 
])^3/(6*b) - b*(-1/2*(b*((Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/ 
2 + ArcCosh[c + d*x]/2)) + ((c + d*x)^2*(a + b*ArcCosh[c + d*x]))/2)))/4)) 
/4))/d
 

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 43

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
ArcCosh[b*(x/a)]/(b*Sqrt[d/b]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a 
*d, 0] && GtQ[a, 0] && GtQ[d/b, 0]
 

rule 101

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + 
 p + 3))), x] + Simp[1/(d*f*(n + p + 3))   Int[(c + d*x)^n*(e + f*x)^p*Simp 
[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f 
*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, 
 c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
 

rule 111

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1))   Int[(a + b*x) 
^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m 
 - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m 
 + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & 
& GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]
 

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 6308

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sq 
rt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + 
 c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*ArcCosh[ 
c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1 
] && EqQ[e2, (-c)*d2] && NeQ[n, -1]
 

rule 6354

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

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]
 

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.41

method	result	size
derivativedivides	\(\frac {\frac {e^{3} a^{3} \left (d x +c \right )^{4}}{4}+e^{3} b^{3} \left (\frac {\left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )^{3}}{4}-\frac {3 \left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{16}-\frac {9 \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )}{32}-\frac {3 \operatorname {arccosh}\left (d x +c \right )^{3}}{32}+\frac {3 \left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )}{32}-\frac {3 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{3}}{128}-\frac {45 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )}{256}-\frac {45 \,\operatorname {arccosh}\left (d x +c \right )}{256}+\frac {9 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )}{32}\right )+3 e^{3} a \,b^{2} \left (\frac {\left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )^{2}}{4}-\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{8}-\frac {3 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )}{16}-\frac {3 \operatorname {arccosh}\left (d x +c \right )^{2}}{32}+\frac {\left (d x +c \right )^{4}}{32}+\frac {3 \left (d x +c \right )^{2}}{32}\right )+3 e^{3} a^{2} b \left (\frac {\left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )}{4}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (2 \left (d x +c \right )^{3} \sqrt {\left (d x +c \right )^{2}-1}+3 \left (d x +c \right ) \sqrt {\left (d x +c \right )^{2}-1}+3 \ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )\right )}{32 \sqrt {\left (d x +c \right )^{2}-1}}\right )}{d}\)	\(433\)
default	\(\frac {\frac {e^{3} a^{3} \left (d x +c \right )^{4}}{4}+e^{3} b^{3} \left (\frac {\left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )^{3}}{4}-\frac {3 \left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{16}-\frac {9 \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )}{32}-\frac {3 \operatorname {arccosh}\left (d x +c \right )^{3}}{32}+\frac {3 \left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )}{32}-\frac {3 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{3}}{128}-\frac {45 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )}{256}-\frac {45 \,\operatorname {arccosh}\left (d x +c \right )}{256}+\frac {9 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )}{32}\right )+3 e^{3} a \,b^{2} \left (\frac {\left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )^{2}}{4}-\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{8}-\frac {3 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )}{16}-\frac {3 \operatorname {arccosh}\left (d x +c \right )^{2}}{32}+\frac {\left (d x +c \right )^{4}}{32}+\frac {3 \left (d x +c \right )^{2}}{32}\right )+3 e^{3} a^{2} b \left (\frac {\left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )}{4}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (2 \left (d x +c \right )^{3} \sqrt {\left (d x +c \right )^{2}-1}+3 \left (d x +c \right ) \sqrt {\left (d x +c \right )^{2}-1}+3 \ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )\right )}{32 \sqrt {\left (d x +c \right )^{2}-1}}\right )}{d}\)	\(433\)
parts	\(\frac {e^{3} a^{3} \left (d x +c \right )^{4}}{4 d}+\frac {e^{3} b^{3} \left (\frac {\left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )^{3}}{4}-\frac {3 \left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{16}-\frac {9 \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )}{32}-\frac {3 \operatorname {arccosh}\left (d x +c \right )^{3}}{32}+\frac {3 \left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )}{32}-\frac {3 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{3}}{128}-\frac {45 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )}{256}-\frac {45 \,\operatorname {arccosh}\left (d x +c \right )}{256}+\frac {9 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )}{32}\right )}{d}+\frac {3 e^{3} a \,b^{2} \left (\frac {\left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )^{2}}{4}-\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{8}-\frac {3 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )}{16}-\frac {3 \operatorname {arccosh}\left (d x +c \right )^{2}}{32}+\frac {\left (d x +c \right )^{4}}{32}+\frac {3 \left (d x +c \right )^{2}}{32}\right )}{d}+\frac {3 e^{3} a^{2} b \left (\frac {\left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )}{4}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (2 \left (d x +c \right )^{3} \sqrt {\left (d x +c \right )^{2}-1}+3 \left (d x +c \right ) \sqrt {\left (d x +c \right )^{2}-1}+3 \ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )\right )}{32 \sqrt {\left (d x +c \right )^{2}-1}}\right )}{d}\)	\(441\)
orering	\(\text {Expression too large to display}\)	\(1154\)

Input:

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

Output:

1/d*(1/4*e^3*a^3*(d*x+c)^4+e^3*b^3*(1/4*(d*x+c)^4*arccosh(d*x+c)^3-3/16*(d 
*x+c)^3*arccosh(d*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)-9/32*arccosh(d*x+ 
c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*(d*x+c)-3/32*arccosh(d*x+c)^3+3/32*(d 
*x+c)^4*arccosh(d*x+c)-3/128*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*(d*x+c)^3-45/ 
256*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*(d*x+c)-45/256*arccosh(d*x+c)+9/32*(d* 
x+c)^2*arccosh(d*x+c))+3*e^3*a*b^2*(1/4*(d*x+c)^4*arccosh(d*x+c)^2-1/8*(d* 
x+c)^3*arccosh(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)-3/16*arccosh(d*x+c)* 
(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*(d*x+c)-3/32*arccosh(d*x+c)^2+1/32*(d*x+c) 
^4+3/32*(d*x+c)^2)+3*e^3*a^2*b*(1/4*(d*x+c)^4*arccosh(d*x+c)-1/32*(d*x+c-1 
)^(1/2)*(d*x+c+1)^(1/2)*(2*(d*x+c)^3*((d*x+c)^2-1)^(1/2)+3*(d*x+c)*((d*x+c 
)^2-1)^(1/2)+3*ln(d*x+c+((d*x+c)^2-1)^(1/2)))/((d*x+c)^2-1)^(1/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 828 vs. \(2 (273) = 546\).

Time = 0.12 (sec) , antiderivative size = 828, normalized size of antiderivative = 2.70 \[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^3 \, dx =\text {Too large to display} \] Input:

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

Output:

1/256*(8*(8*a^3 + 3*a*b^2)*d^4*e^3*x^4 + 32*(8*a^3 + 3*a*b^2)*c*d^3*e^3*x^ 
3 + 24*(3*a*b^2 + 2*(8*a^3 + 3*a*b^2)*c^2)*d^2*e^3*x^2 + 16*(9*a*b^2*c + 2 
*(8*a^3 + 3*a*b^2)*c^3)*d*e^3*x + 8*(8*b^3*d^4*e^3*x^4 + 32*b^3*c*d^3*e^3* 
x^3 + 48*b^3*c^2*d^2*e^3*x^2 + 32*b^3*c^3*d*e^3*x + (8*b^3*c^4 - 3*b^3)*e^ 
3)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))^3 + 24*(8*a*b^2*d^4*e^ 
3*x^4 + 32*a*b^2*c*d^3*e^3*x^3 + 48*a*b^2*c^2*d^2*e^3*x^2 + 32*a*b^2*c^3*d 
*e^3*x + (8*a*b^2*c^4 - 3*a*b^2)*e^3 - (2*b^3*d^3*e^3*x^3 + 6*b^3*c*d^2*e^ 
3*x^2 + 3*(2*b^3*c^2 + b^3)*d*e^3*x + (2*b^3*c^3 + 3*b^3*c)*e^3)*sqrt(d^2* 
x^2 + 2*c*d*x + c^2 - 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)) 
^2 + 3*(8*(8*a^2*b + b^3)*d^4*e^3*x^4 + 32*(8*a^2*b + b^3)*c*d^3*e^3*x^3 + 
 24*(b^3 + 2*(8*a^2*b + b^3)*c^2)*d^2*e^3*x^2 + 16*(3*b^3*c + 2*(8*a^2*b + 
 b^3)*c^3)*d*e^3*x + (24*b^3*c^2 + 8*(8*a^2*b + b^3)*c^4 - 24*a^2*b - 15*b 
^3)*e^3 - 16*(2*a*b^2*d^3*e^3*x^3 + 6*a*b^2*c*d^2*e^3*x^2 + 3*(2*a*b^2*c^2 
 + a*b^2)*d*e^3*x + (2*a*b^2*c^3 + 3*a*b^2*c)*e^3)*sqrt(d^2*x^2 + 2*c*d*x 
+ c^2 - 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)) - 3*(2*(8*a^2 
*b + b^3)*d^3*e^3*x^3 + 6*(8*a^2*b + b^3)*c*d^2*e^3*x^2 + 3*(8*a^2*b + 5*b 
^3 + 2*(8*a^2*b + b^3)*c^2)*d*e^3*x + (2*(8*a^2*b + b^3)*c^3 + 3*(8*a^2*b 
+ 5*b^3)*c)*e^3)*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))/d
 

Sympy [F]

\[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^3 \, dx=e^{3} \left (\int a^{3} c^{3}\, dx + \int a^{3} d^{3} x^{3}\, dx + \int b^{3} c^{3} \operatorname {acosh}^{3}{\left (c + d x \right )}\, dx + \int 3 a b^{2} c^{3} \operatorname {acosh}^{2}{\left (c + d x \right )}\, dx + \int 3 a^{2} b c^{3} \operatorname {acosh}{\left (c + d x \right )}\, dx + \int 3 a^{3} c d^{2} x^{2}\, dx + \int 3 a^{3} c^{2} d x\, dx + \int b^{3} d^{3} x^{3} \operatorname {acosh}^{3}{\left (c + d x \right )}\, dx + \int 3 a b^{2} d^{3} x^{3} \operatorname {acosh}^{2}{\left (c + d x \right )}\, dx + \int 3 a^{2} b d^{3} x^{3} \operatorname {acosh}{\left (c + d x \right )}\, dx + \int 3 b^{3} c d^{2} x^{2} \operatorname {acosh}^{3}{\left (c + d x \right )}\, dx + \int 3 b^{3} c^{2} d x \operatorname {acosh}^{3}{\left (c + d x \right )}\, dx + \int 9 a b^{2} c d^{2} x^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}\, dx + \int 9 a b^{2} c^{2} d x \operatorname {acosh}^{2}{\left (c + d x \right )}\, dx + \int 9 a^{2} b c d^{2} x^{2} \operatorname {acosh}{\left (c + d x \right )}\, dx + \int 9 a^{2} b c^{2} d x \operatorname {acosh}{\left (c + d x \right )}\, dx\right ) \] Input:

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

Output:

e**3*(Integral(a**3*c**3, x) + Integral(a**3*d**3*x**3, x) + Integral(b**3 
*c**3*acosh(c + d*x)**3, x) + Integral(3*a*b**2*c**3*acosh(c + d*x)**2, x) 
 + Integral(3*a**2*b*c**3*acosh(c + d*x), x) + Integral(3*a**3*c*d**2*x**2 
, x) + Integral(3*a**3*c**2*d*x, x) + Integral(b**3*d**3*x**3*acosh(c + d* 
x)**3, x) + Integral(3*a*b**2*d**3*x**3*acosh(c + d*x)**2, x) + Integral(3 
*a**2*b*d**3*x**3*acosh(c + d*x), x) + Integral(3*b**3*c*d**2*x**2*acosh(c 
 + d*x)**3, x) + Integral(3*b**3*c**2*d*x*acosh(c + d*x)**3, x) + Integral 
(9*a*b**2*c*d**2*x**2*acosh(c + d*x)**2, x) + Integral(9*a*b**2*c**2*d*x*a 
cosh(c + d*x)**2, x) + Integral(9*a**2*b*c*d**2*x**2*acosh(c + d*x), x) + 
Integral(9*a**2*b*c**2*d*x*acosh(c + d*x), x))
 

Maxima [F]

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

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

Output:

1/4*a^3*d^3*e^3*x^4 + a^3*c*d^2*e^3*x^3 + 3/2*a^3*c^2*d*e^3*x^2 + 9/4*(2*x 
^2*arccosh(d*x + c) - d*(3*c^2*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c* 
d*x + c^2 - 1)*d)/d^3 + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*x/d^2 - (c^2 - 1 
)*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^3 - 3*sqr 
t(d^2*x^2 + 2*c*d*x + c^2 - 1)*c/d^3))*a^2*b*c^2*d*e^3 + 1/2*(6*x^3*arccos 
h(d*x + c) - d*(2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*x^2/d^2 - 15*c^3*log(2 
*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^4 - 5*sqrt(d^2*x 
^2 + 2*c*d*x + c^2 - 1)*c*x/d^3 + 9*(c^2 - 1)*c*log(2*d^2*x + 2*c*d + 2*sq 
rt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^4 + 15*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 
 1)*c^2/d^4 - 4*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*(c^2 - 1)/d^4))*a^2*b*c* 
d^2*e^3 + 1/32*(24*x^4*arccosh(d*x + c) - (6*sqrt(d^2*x^2 + 2*c*d*x + c^2 
- 1)*x^3/d^2 - 14*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*c*x^2/d^3 + 105*c^4*lo 
g(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^5 + 35*sqrt(d 
^2*x^2 + 2*c*d*x + c^2 - 1)*c^2*x/d^4 - 90*(c^2 - 1)*c^2*log(2*d^2*x + 2*c 
*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^5 - 105*sqrt(d^2*x^2 + 2*c*d 
*x + c^2 - 1)*c^3/d^5 - 9*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*(c^2 - 1)*x/d^ 
4 + 9*(c^2 - 1)^2*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1 
)*d)/d^5 + 55*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*(c^2 - 1)*c/d^5)*d)*a^2*b* 
d^3*e^3 + a^3*c^3*e^3*x + 3*((d*x + c)*arccosh(d*x + c) - sqrt((d*x + c)^2 
 - 1))*a^2*b*c^3*e^3/d + 1/4*(b^3*d^3*e^3*x^4 + 4*b^3*c*d^2*e^3*x^3 + 6...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^3 \, dx=\frac {e^{3} \left (-9 \,\mathrm {log}\left (\sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}+c +d x \right ) a^{2} b +8 a^{3} d^{4} x^{4}+24 \mathit {acosh} \left (d x +c \right ) a^{2} b \,d^{4} x^{4}-6 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, a^{2} b \,d^{3} x^{3}-9 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, a^{2} b d x -96 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, a^{2} b \,c^{3}+96 \left (\int \mathit {acosh} \left (d x +c \right )^{3} x^{2}d x \right ) b^{3} c \,d^{3}+96 \left (\int \mathit {acosh} \left (d x +c \right )^{3} x d x \right ) b^{3} c^{2} d^{2}+96 \left (\int \mathit {acosh} \left (d x +c \right )^{2} x^{3}d x \right ) a \,b^{2} d^{4}+96 \mathit {acosh} \left (d x +c \right ) a^{2} b \,c^{4}+90 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, a^{2} b \,c^{3}-9 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, a^{2} b c -72 \,\mathrm {log}\left (\sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}+c +d x \right ) a^{2} b \,c^{4}+32 a^{3} c^{3} d x +48 a^{3} c^{2} d^{2} x^{2}+32 a^{3} c \,d^{3} x^{3}+32 \left (\int \mathit {acosh} \left (d x +c \right )^{3}d x \right ) b^{3} c^{3} d +32 \left (\int \mathit {acosh} \left (d x +c \right )^{3} x^{3}d x \right ) b^{3} d^{4}+288 \left (\int \mathit {acosh} \left (d x +c \right )^{2} x^{2}d x \right ) a \,b^{2} c \,d^{3}+288 \left (\int \mathit {acosh} \left (d x +c \right )^{2} x d x \right ) a \,b^{2} c^{2} d^{2}+96 \mathit {acosh} \left (d x +c \right ) a^{2} b \,c^{3} d x +144 \mathit {acosh} \left (d x +c \right ) a^{2} b \,c^{2} d^{2} x^{2}+96 \mathit {acosh} \left (d x +c \right ) a^{2} b c \,d^{3} x^{3}-18 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, a^{2} b \,c^{2} d x -18 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, a^{2} b c \,d^{2} x^{2}+96 \left (\int \mathit {acosh} \left (d x +c \right )^{2}d x \right ) a \,b^{2} c^{3} d \right )}{32 d} \] Input:

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

Output:

(e**3*(96*acosh(c + d*x)*a**2*b*c**4 + 96*acosh(c + d*x)*a**2*b*c**3*d*x + 
 144*acosh(c + d*x)*a**2*b*c**2*d**2*x**2 + 96*acosh(c + d*x)*a**2*b*c*d** 
3*x**3 + 24*acosh(c + d*x)*a**2*b*d**4*x**4 + 90*sqrt(c**2 + 2*c*d*x + d** 
2*x**2 - 1)*a**2*b*c**3 - 18*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*a**2*b*c 
**2*d*x - 18*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*a**2*b*c*d**2*x**2 - 9*s 
qrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*a**2*b*c - 6*sqrt(c**2 + 2*c*d*x + d** 
2*x**2 - 1)*a**2*b*d**3*x**3 - 9*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*a**2 
*b*d*x - 96*sqrt(c + d*x + 1)*sqrt(c + d*x - 1)*a**2*b*c**3 + 32*int(acosh 
(c + d*x)**3,x)*b**3*c**3*d + 96*int(acosh(c + d*x)**2,x)*a*b**2*c**3*d + 
32*int(acosh(c + d*x)**3*x**3,x)*b**3*d**4 + 96*int(acosh(c + d*x)**3*x**2 
,x)*b**3*c*d**3 + 96*int(acosh(c + d*x)**3*x,x)*b**3*c**2*d**2 + 96*int(ac 
osh(c + d*x)**2*x**3,x)*a*b**2*d**4 + 288*int(acosh(c + d*x)**2*x**2,x)*a* 
b**2*c*d**3 + 288*int(acosh(c + d*x)**2*x,x)*a*b**2*c**2*d**2 - 72*log(sqr 
t(c**2 + 2*c*d*x + d**2*x**2 - 1) + c + d*x)*a**2*b*c**4 - 9*log(sqrt(c**2 
 + 2*c*d*x + d**2*x**2 - 1) + c + d*x)*a**2*b + 32*a**3*c**3*d*x + 48*a**3 
*c**2*d**2*x**2 + 32*a**3*c*d**3*x**3 + 8*a**3*d**4*x**4))/(32*d)