\(\int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^2 \, dx\) [23]

Optimal result

Integrand size = 23, antiderivative size = 186 \[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^2 \, dx=\frac {3 b^2 e^3 (c+d x)^2}{32 d}+\frac {b^2 e^3 (c+d x)^4}{32 d}-\frac {3 b e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))}{16 d}-\frac {b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))}{8 d}-\frac {3 e^3 (a+b \text {arccosh}(c+d x))^2}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arccosh}(c+d x))^2}{4 d} \] Output:

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

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.14 \[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^2 \, dx=\frac {e^3 \left (3 b^2 (c+d x)^2+\left (8 a^2+b^2\right ) (c+d x)^4+2 a b \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \left (-3-2 (c+d x)^2\right )+2 b (c+d x) \left (8 a (c+d x)^3-3 b \sqrt {-1+c+d x} \sqrt {1+c+d x}-2 b \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}\right ) \text {arccosh}(c+d x)+b^2 \left (-3+8 (c+d x)^4\right ) \text {arccosh}(c+d x)^2-6 a b \log \left (c+d x+\sqrt {-1+c+d x} \sqrt {1+c+d x}\right )\right )}{32 d} \] Input:

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

Output:

(e^3*(3*b^2*(c + d*x)^2 + (8*a^2 + b^2)*(c + d*x)^4 + 2*a*b*Sqrt[-1 + c + 
d*x]*(c + d*x)*Sqrt[1 + c + d*x]*(-3 - 2*(c + d*x)^2) + 2*b*(c + d*x)*(8*a 
*(c + d*x)^3 - 3*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x] - 2*b*Sqrt[-1 + c 
+ d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])*ArcCosh[c + d*x] + b^2*(-3 + 8*(c + 
d*x)^4)*ArcCosh[c + d*x]^2 - 6*a*b*Log[c + d*x + Sqrt[-1 + c + d*x]*Sqrt[1 
 + c + d*x]]))/(32*d)
 

Rubi [A] (verified)

Time = 1.11 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.89, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {6411, 27, 6298, 6354, 15, 6354, 15, 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.

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

\(\Big \downarrow \) 6411

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 6354

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

\(\Big \downarrow \) 15

\(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arccosh}(c+d x))^2-\frac {1}{2} b \left (\frac {3}{4} \int \frac {(c+d x)^2 (a+b \text {arccosh}(c+d x))}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {1}{4} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3 (a+b \text {arccosh}(c+d x))-\frac {1}{16} b (c+d x)^4\right )\right )}{d}\)

\(\Big \downarrow \) 6354

\(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arccosh}(c+d x))^2-\frac {1}{2} b \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \text {arccosh}(c+d x)}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)-\frac {1}{2} b \int (c+d x)d(c+d x)+\frac {1}{2} \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))\right )+\frac {1}{4} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3 (a+b \text {arccosh}(c+d x))-\frac {1}{16} b (c+d x)^4\right )\right )}{d}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arccosh}(c+d x))^2-\frac {1}{2} b \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \text {arccosh}(c+d x)}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {1}{2} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x) (a+b \text {arccosh}(c+d x))-\frac {1}{4} b (c+d x)^2\right )+\frac {1}{4} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3 (a+b \text {arccosh}(c+d x))-\frac {1}{16} b (c+d x)^4\right )\right )}{d}\)

\(\Big \downarrow \) 6308

\(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arccosh}(c+d x))^2-\frac {1}{2} b \left (\frac {1}{4} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3 (a+b \text {arccosh}(c+d x))+\frac {3}{4} \left (\frac {1}{2} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x) (a+b \text {arccosh}(c+d x))+\frac {(a+b \text {arccosh}(c+d x))^2}{4 b}-\frac {1}{4} b (c+d x)^2\right )-\frac {1}{16} b (c+d x)^4\right )\right )}{d}\)

Input:

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

Output:

(e^3*(((c + d*x)^4*(a + b*ArcCosh[c + d*x])^2)/4 - (b*(-1/16*(b*(c + d*x)^ 
4) + (Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + 
d*x]))/4 + (3*(-1/4*(b*(c + d*x)^2) + (Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 
 + c + d*x]*(a + b*ArcCosh[c + d*x]))/2 + (a + b*ArcCosh[c + d*x])^2/(4*b) 
))/4))/2))/d
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -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 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 = 0.21 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.31

method	result	size
derivativedivides	\(\frac {\frac {e^{3} a^{2} \left (d x +c \right )^{4}}{4}+e^{3} b^{2} \left (\frac {\left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )^{2}}{4}-\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{8}-\frac {3 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )}{16}-\frac {3 \operatorname {arccosh}\left (d x +c \right )^{2}}{32}+\frac {\left (d x +c \right )^{4}}{32}+\frac {3 \left (d x +c \right )^{2}}{32}\right )+2 e^{3} a b \left (\frac {\left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )}{4}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (2 \left (d x +c \right )^{3} \sqrt {\left (d x +c \right )^{2}-1}+3 \left (d x +c \right ) \sqrt {\left (d x +c \right )^{2}-1}+3 \ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )\right )}{32 \sqrt {\left (d x +c \right )^{2}-1}}\right )}{d}\)	\(244\)
default	\(\frac {\frac {e^{3} a^{2} \left (d x +c \right )^{4}}{4}+e^{3} b^{2} \left (\frac {\left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )^{2}}{4}-\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{8}-\frac {3 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )}{16}-\frac {3 \operatorname {arccosh}\left (d x +c \right )^{2}}{32}+\frac {\left (d x +c \right )^{4}}{32}+\frac {3 \left (d x +c \right )^{2}}{32}\right )+2 e^{3} a b \left (\frac {\left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )}{4}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (2 \left (d x +c \right )^{3} \sqrt {\left (d x +c \right )^{2}-1}+3 \left (d x +c \right ) \sqrt {\left (d x +c \right )^{2}-1}+3 \ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )\right )}{32 \sqrt {\left (d x +c \right )^{2}-1}}\right )}{d}\)	\(244\)
parts	\(\frac {e^{3} a^{2} \left (d x +c \right )^{4}}{4 d}+\frac {e^{3} b^{2} \left (\frac {\left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )^{2}}{4}-\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{8}-\frac {3 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )}{16}-\frac {3 \operatorname {arccosh}\left (d x +c \right )^{2}}{32}+\frac {\left (d x +c \right )^{4}}{32}+\frac {3 \left (d x +c \right )^{2}}{32}\right )}{d}+\frac {2 e^{3} a b \left (\frac {\left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )}{4}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (2 \left (d x +c \right )^{3} \sqrt {\left (d x +c \right )^{2}-1}+3 \left (d x +c \right ) \sqrt {\left (d x +c \right )^{2}-1}+3 \ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )\right )}{32 \sqrt {\left (d x +c \right )^{2}-1}}\right )}{d}\)	\(249\)
orering	\(\frac {\left (37 d^{6} x^{6}+222 c \,d^{5} x^{5}+555 c^{2} d^{4} x^{4}+740 c^{3} d^{3} x^{3}+546 c^{4} d^{2} x^{2}+21 d^{4} x^{4}+204 c^{5} d x +84 c \,d^{3} x^{3}+28 c^{6}+99 c^{2} d^{2} x^{2}+30 c^{3} d x +6 c^{4}-60 d^{2} x^{2}-120 c d x -24 c^{2}\right ) \left (d e x +c e \right )^{3} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2}}{64 d \left (d x +c \right )^{5}}-\frac {\left (9 d^{6} x^{6}+54 c \,d^{5} x^{5}+135 c^{2} d^{4} x^{4}+180 c^{3} d^{3} x^{3}+130 c^{4} d^{2} x^{2}+11 d^{4} x^{4}+44 c^{5} d x +44 c \,d^{3} x^{3}+4 c^{6}+51 c^{2} d^{2} x^{2}+14 c^{3} d x +2 c^{4}-24 d^{2} x^{2}-48 c d x -6 c^{2}\right ) \left (3 \left (d e x +c e \right )^{2} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2} d e +\frac {2 \left (d e x +c e \right )^{3} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right ) b d}{\sqrt {d x +c -1}\, \sqrt {d x +c +1}}\right )}{64 d^{2} \left (d x +c \right )^{4}}+\frac {x \left (d^{3} x^{3}+4 c \,d^{2} x^{2}+6 c^{2} d x +4 c^{3}+3 d x +6 c \right ) \left (d x +c -1\right ) \left (d x +c +1\right ) \left (6 \left (d e x +c e \right ) \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2} d^{2} e^{2}+\frac {12 \left (d e x +c e \right )^{2} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right ) d^{2} e b}{\sqrt {d x +c -1}\, \sqrt {d x +c +1}}+\frac {2 \left (d e x +c e \right )^{3} b^{2} d^{2}}{\left (d x +c -1\right ) \left (d x +c +1\right )}-\frac {\left (d e x +c e \right )^{3} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right ) b \,d^{2}}{\left (d x +c -1\right )^{\frac {3}{2}} \sqrt {d x +c +1}}-\frac {\left (d e x +c e \right )^{3} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right ) b \,d^{2}}{\sqrt {d x +c -1}\, \left (d x +c +1\right )^{\frac {3}{2}}}\right )}{64 d^{2} \left (d x +c \right )^{3}}\)	\(608\)

Input:

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

Output:

1/d*(1/4*e^3*a^2*(d*x+c)^4+e^3*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)+2*e^3*a*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. 481 vs. \(2 (166) = 332\).

Time = 0.12 (sec) , antiderivative size = 481, normalized size of antiderivative = 2.59 \[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^2 \, dx=\frac {{\left (8 \, a^{2} + b^{2}\right )} d^{4} e^{3} x^{4} + 4 \, {\left (8 \, a^{2} + b^{2}\right )} c d^{3} e^{3} x^{3} + 3 \, {\left (2 \, {\left (8 \, a^{2} + b^{2}\right )} c^{2} + b^{2}\right )} d^{2} e^{3} x^{2} + 2 \, {\left (2 \, {\left (8 \, a^{2} + b^{2}\right )} c^{3} + 3 \, b^{2} c\right )} d e^{3} x + {\left (8 \, b^{2} d^{4} e^{3} x^{4} + 32 \, b^{2} c d^{3} e^{3} x^{3} + 48 \, b^{2} c^{2} d^{2} e^{3} x^{2} + 32 \, b^{2} c^{3} d e^{3} x + {\left (8 \, b^{2} c^{4} - 3 \, b^{2}\right )} e^{3}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{2} + 2 \, {\left (8 \, a b d^{4} e^{3} x^{4} + 32 \, a b c d^{3} e^{3} x^{3} + 48 \, a b c^{2} d^{2} e^{3} x^{2} + 32 \, a b c^{3} d e^{3} x + {\left (8 \, a b c^{4} - 3 \, a b\right )} e^{3} - {\left (2 \, b^{2} d^{3} e^{3} x^{3} + 6 \, b^{2} c d^{2} e^{3} x^{2} + 3 \, {\left (2 \, b^{2} c^{2} + b^{2}\right )} d e^{3} x + {\left (2 \, b^{2} c^{3} + 3 \, b^{2} c\right )} e^{3}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 2 \, {\left (2 \, a b d^{3} e^{3} x^{3} + 6 \, a b c d^{2} e^{3} x^{2} + 3 \, {\left (2 \, a b c^{2} + a b\right )} d e^{3} x + {\left (2 \, a b c^{3} + 3 \, a b c\right )} e^{3}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{32 \, d} \] Input:

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

Output:

1/32*((8*a^2 + b^2)*d^4*e^3*x^4 + 4*(8*a^2 + b^2)*c*d^3*e^3*x^3 + 3*(2*(8* 
a^2 + b^2)*c^2 + b^2)*d^2*e^3*x^2 + 2*(2*(8*a^2 + b^2)*c^3 + 3*b^2*c)*d*e^ 
3*x + (8*b^2*d^4*e^3*x^4 + 32*b^2*c*d^3*e^3*x^3 + 48*b^2*c^2*d^2*e^3*x^2 + 
 32*b^2*c^3*d*e^3*x + (8*b^2*c^4 - 3*b^2)*e^3)*log(d*x + c + sqrt(d^2*x^2 
+ 2*c*d*x + c^2 - 1))^2 + 2*(8*a*b*d^4*e^3*x^4 + 32*a*b*c*d^3*e^3*x^3 + 48 
*a*b*c^2*d^2*e^3*x^2 + 32*a*b*c^3*d*e^3*x + (8*a*b*c^4 - 3*a*b)*e^3 - (2*b 
^2*d^3*e^3*x^3 + 6*b^2*c*d^2*e^3*x^2 + 3*(2*b^2*c^2 + b^2)*d*e^3*x + (2*b^ 
2*c^3 + 3*b^2*c)*e^3)*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))*log(d*x + c + sqr 
t(d^2*x^2 + 2*c*d*x + c^2 - 1)) - 2*(2*a*b*d^3*e^3*x^3 + 6*a*b*c*d^2*e^3*x 
^2 + 3*(2*a*b*c^2 + a*b)*d*e^3*x + (2*a*b*c^3 + 3*a*b*c)*e^3)*sqrt(d^2*x^2 
 + 2*c*d*x + c^2 - 1))/d
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/4*a^2*d^3*e^3*x^4 + a^2*c*d^2*e^3*x^3 + 3/2*a^2*c^2*d*e^3*x^2 + 3/2*(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*b*c^2*d*e^3 + 1/3*(6*x^3*arccosh( 
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*sqrt 
(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*b*c*d^2* 
e^3 + 1/48*(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*log(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*b*d^3*e^ 
3 + a^2*c^3*e^3*x + 2*((d*x + c)*arccosh(d*x + c) - sqrt((d*x + c)^2 - 1)) 
*a*b*c^3*e^3/d + 1/4*(b^2*d^3*e^3*x^4 + 4*b^2*c*d^2*e^3*x^3 + 6*b^2*c^2...
 

Giac [F]

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

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

Output:

integrate((d*e*x + c*e)^3*(b*arccosh(d*x + c) + a)^2, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^2 \, dx=\frac {e^{3} \left (32 \mathit {acosh} \left (d x +c \right ) a b \,c^{4}+32 \mathit {acosh} \left (d x +c \right ) a b \,c^{3} d x +48 \mathit {acosh} \left (d x +c \right ) a b \,c^{2} d^{2} x^{2}+32 \mathit {acosh} \left (d x +c \right ) a b c \,d^{3} x^{3}+8 \mathit {acosh} \left (d x +c \right ) a b \,d^{4} x^{4}+30 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, a b \,c^{3}-6 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, a b \,c^{2} d x -6 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, a b c \,d^{2} x^{2}-3 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, a b c -2 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, a b \,d^{3} x^{3}-3 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, a b d x -32 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, a b \,c^{3}+16 \left (\int \mathit {acosh} \left (d x +c \right )^{2}d x \right ) b^{2} c^{3} d +16 \left (\int \mathit {acosh} \left (d x +c \right )^{2} x^{3}d x \right ) b^{2} d^{4}+48 \left (\int \mathit {acosh} \left (d x +c \right )^{2} x^{2}d x \right ) b^{2} c \,d^{3}+48 \left (\int \mathit {acosh} \left (d x +c \right )^{2} x d x \right ) b^{2} c^{2} d^{2}-24 \,\mathrm {log}\left (\sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}+c +d x \right ) a b \,c^{4}-3 \,\mathrm {log}\left (\sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}+c +d x \right ) a b +16 a^{2} c^{3} d x +24 a^{2} c^{2} d^{2} x^{2}+16 a^{2} c \,d^{3} x^{3}+4 a^{2} d^{4} x^{4}\right )}{16 d} \] Input:

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

Output:

(e**3*(32*acosh(c + d*x)*a*b*c**4 + 32*acosh(c + d*x)*a*b*c**3*d*x + 48*ac 
osh(c + d*x)*a*b*c**2*d**2*x**2 + 32*acosh(c + d*x)*a*b*c*d**3*x**3 + 8*ac 
osh(c + d*x)*a*b*d**4*x**4 + 30*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*a*b*c 
**3 - 6*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*a*b*c**2*d*x - 6*sqrt(c**2 + 
2*c*d*x + d**2*x**2 - 1)*a*b*c*d**2*x**2 - 3*sqrt(c**2 + 2*c*d*x + d**2*x* 
*2 - 1)*a*b*c - 2*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*a*b*d**3*x**3 - 3*s 
qrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*a*b*d*x - 32*sqrt(c + d*x + 1)*sqrt(c 
+ d*x - 1)*a*b*c**3 + 16*int(acosh(c + d*x)**2,x)*b**2*c**3*d + 16*int(aco 
sh(c + d*x)**2*x**3,x)*b**2*d**4 + 48*int(acosh(c + d*x)**2*x**2,x)*b**2*c 
*d**3 + 48*int(acosh(c + d*x)**2*x,x)*b**2*c**2*d**2 - 24*log(sqrt(c**2 + 
2*c*d*x + d**2*x**2 - 1) + c + d*x)*a*b*c**4 - 3*log(sqrt(c**2 + 2*c*d*x + 
 d**2*x**2 - 1) + c + d*x)*a*b + 16*a**2*c**3*d*x + 24*a**2*c**2*d**2*x**2 
 + 16*a**2*c*d**3*x**3 + 4*a**2*d**4*x**4))/(16*d)