\(\int (c e+d e x) (a+b \text {arccosh}(c+d x))^3 \, dx\) [34]

Optimal result

Integrand size = 21, antiderivative size = 175 \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^3 \, dx=-\frac {3 b^3 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{8 d}-\frac {3 b^3 e \text {arccosh}(c+d x)}{8 d}+\frac {3 b^2 e (c+d x)^2 (a+b \text {arccosh}(c+d x))}{4 d}-\frac {3 b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^2}{4 d}-\frac {e (a+b \text {arccosh}(c+d x))^3}{4 d}+\frac {e (c+d x)^2 (a+b \text {arccosh}(c+d x))^3}{2 d} \] Output:

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

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.39 \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^3 \, dx=\frac {e \left (2 a \left (2 a^2+3 b^2\right ) (c+d x)^2-3 b \left (2 a^2+b^2\right ) \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}-6 b (c+d x) \left (-2 a^2 (c+d x)-b^2 (c+d x)+2 a b \sqrt {-1+c+d x} \sqrt {1+c+d x}\right ) \text {arccosh}(c+d x)+6 b^2 \left (-a+2 a (c+d x)^2-b \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}\right ) \text {arccosh}(c+d x)^2+2 b^3 \left (-1+2 (c+d x)^2\right ) \text {arccosh}(c+d x)^3-3 b \left (2 a^2+b^2\right ) \log \left (c+d x+\sqrt {-1+c+d x} \sqrt {1+c+d x}\right )\right )}{8 d} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.21 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.94, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {6411, 27, 6298, 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.

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

\(\Big \downarrow \) 6411

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 6354

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

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 101

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

\(\Big \downarrow \) 43

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

\(\Big \downarrow \) 6308

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

Input:

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

Output:

(e*(((c + d*x)^2*(a + b*ArcCosh[c + d*x])^3)/2 - (3*b*((Sqrt[-1 + c + d*x] 
*(c + d*x)*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^2)/2 + (a + b*ArcCos 
h[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)))/2))/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 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 = 0.25 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.33

method	result	size
derivativedivides	\(\frac {\frac {e \,a^{3} \left (d x +c \right )^{2}}{2}+e \,b^{3} \left (\frac {\cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )^{3}}{4}-\frac {3 \sinh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )^{2}}{8}+\frac {3 \cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )}{8}-\frac {3 \sinh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right )}{16}\right )+3 e a \,b^{2} \left (\frac {\cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )^{2}}{4}-\frac {\sinh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )}{4}+\frac {\cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right )}{8}\right )+3 e \,a^{2} b \left (\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )}{2}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right ) \sqrt {\left (d x +c \right )^{2}-1}+\ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )\right )}{4 \sqrt {\left (d x +c \right )^{2}-1}}\right )}{d}\)	\(233\)
default	\(\frac {\frac {e \,a^{3} \left (d x +c \right )^{2}}{2}+e \,b^{3} \left (\frac {\cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )^{3}}{4}-\frac {3 \sinh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )^{2}}{8}+\frac {3 \cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )}{8}-\frac {3 \sinh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right )}{16}\right )+3 e a \,b^{2} \left (\frac {\cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )^{2}}{4}-\frac {\sinh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )}{4}+\frac {\cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right )}{8}\right )+3 e \,a^{2} b \left (\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )}{2}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right ) \sqrt {\left (d x +c \right )^{2}-1}+\ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )\right )}{4 \sqrt {\left (d x +c \right )^{2}-1}}\right )}{d}\)	\(233\)
parts	\(e \,a^{3} \left (\frac {1}{2} d \,x^{2}+c x \right )+\frac {e \,b^{3} \left (\frac {\cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )^{3}}{4}-\frac {3 \sinh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )^{2}}{8}+\frac {3 \cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )}{8}-\frac {3 \sinh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right )}{16}\right )}{d}+\frac {3 e a \,b^{2} \left (\frac {\cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )^{2}}{4}-\frac {\sinh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )}{4}+\frac {\cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right )}{8}\right )}{d}+\frac {3 e \,a^{2} b \left (\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )}{2}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right ) \sqrt {\left (d x +c \right )^{2}-1}+\ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )\right )}{4 \sqrt {\left (d x +c \right )^{2}-1}}\right )}{d}\)	\(240\)
orering	\(\frac {\left (15 d^{4} x^{4}+60 c \,d^{3} x^{3}+90 c^{2} d^{2} x^{2}+60 c^{3} d x +15 c^{4}-20 d^{2} x^{2}-40 c d x -20 c^{2}+8\right ) \left (d e x +c e \right ) \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{3}}{16 d \left (d x +c \right )^{3}}-\frac {\left (7 d^{4} x^{4}+28 c \,d^{3} x^{3}+42 c^{2} d^{2} x^{2}+28 c^{3} d x +7 c^{4}-16 d^{2} x^{2}-32 c d x -16 c^{2}+8\right ) \left (d e \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{3}+\frac {3 \left (d e x +c e \right ) \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2} b d}{\sqrt {d x +c -1}\, \sqrt {d x +c +1}}\right )}{16 \left (d x +c \right )^{2} d^{2}}+\frac {\left (d x +c -1\right ) \left (d x +c +1\right ) \left (d^{2} x^{2}+2 c d x +c^{2}-2\right ) \left (\frac {6 d^{2} e \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2} b}{\sqrt {d x +c -1}\, \sqrt {d x +c +1}}+\frac {6 \left (d e x +c e \right ) \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right ) b^{2} d^{2}}{\left (d x +c -1\right ) \left (d x +c +1\right )}-\frac {3 \left (d e x +c e \right ) \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2} b \,d^{2}}{2 \left (d x +c -1\right )^{\frac {3}{2}} \sqrt {d x +c +1}}-\frac {3 \left (d e x +c e \right ) \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2} b \,d^{2}}{2 \sqrt {d x +c -1}\, \left (d x +c +1\right )^{\frac {3}{2}}}\right )}{8 \left (d x +c \right ) d^{3}}-\frac {\left (d x +c -1\right )^{2} \left (d x +c +1\right )^{2} \left (\frac {18 d^{3} e \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right ) b^{2}}{\left (d x +c -1\right ) \left (d x +c +1\right )}-\frac {9 d^{3} e \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2} b}{2 \left (d x +c -1\right )^{\frac {3}{2}} \sqrt {d x +c +1}}-\frac {9 d^{3} e \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2} b}{2 \sqrt {d x +c -1}\, \left (d x +c +1\right )^{\frac {3}{2}}}+\frac {6 \left (d e x +c e \right ) b^{3} d^{3}}{\left (d x +c -1\right )^{\frac {3}{2}} \left (d x +c +1\right )^{\frac {3}{2}}}-\frac {9 \left (d e x +c e \right ) \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right ) b^{2} d^{3}}{\left (d x +c -1\right )^{2} \left (d x +c +1\right )}-\frac {9 \left (d e x +c e \right ) \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right ) b^{2} d^{3}}{\left (d x +c -1\right ) \left (d x +c +1\right )^{2}}+\frac {9 \left (d e x +c e \right ) \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2} b \,d^{3}}{4 \left (d x +c -1\right )^{\frac {5}{2}} \sqrt {d x +c +1}}+\frac {3 \left (d e x +c e \right ) \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2} b \,d^{3}}{2 \left (d x +c -1\right )^{\frac {3}{2}} \left (d x +c +1\right )^{\frac {3}{2}}}+\frac {9 \left (d e x +c e \right ) \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2} b \,d^{3}}{4 \sqrt {d x +c -1}\, \left (d x +c +1\right )^{\frac {5}{2}}}\right )}{16 d^{4}}\)	\(794\)

Input:

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

Output:

1/d*(1/2*e*a^3*(d*x+c)^2+e*b^3*(1/4*cosh(2*arccosh(d*x+c))*arccosh(d*x+c)^ 
3-3/8*sinh(2*arccosh(d*x+c))*arccosh(d*x+c)^2+3/8*cosh(2*arccosh(d*x+c))*a 
rccosh(d*x+c)-3/16*sinh(2*arccosh(d*x+c)))+3*e*a*b^2*(1/4*cosh(2*arccosh(d 
*x+c))*arccosh(d*x+c)^2-1/4*sinh(2*arccosh(d*x+c))*arccosh(d*x+c)+1/8*cosh 
(2*arccosh(d*x+c)))+3*e*a^2*b*(1/2*(d*x+c)^2*arccosh(d*x+c)-1/4*(d*x+c-1)^ 
(1/2)*(d*x+c+1)^(1/2)*((d*x+c)*((d*x+c)^2-1)^(1/2)+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. 395 vs. \(2 (155) = 310\).

Time = 0.11 (sec) , antiderivative size = 395, normalized size of antiderivative = 2.26 \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^3 \, dx=\frac {2 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} d^{2} e x^{2} + 4 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} c d e x + 2 \, {\left (2 \, b^{3} d^{2} e x^{2} + 4 \, b^{3} c d e x + {\left (2 \, b^{3} c^{2} - b^{3}\right )} e\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{3} + 6 \, {\left (2 \, a b^{2} d^{2} e x^{2} + 4 \, a b^{2} c d e x + {\left (2 \, a b^{2} c^{2} - a b^{2}\right )} e - {\left (b^{3} d e x + b^{3} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{2} + 3 \, {\left (2 \, {\left (2 \, a^{2} b + b^{3}\right )} d^{2} e x^{2} + 4 \, {\left (2 \, a^{2} b + b^{3}\right )} c d e x - {\left (2 \, a^{2} b + b^{3} - 2 \, {\left (2 \, a^{2} b + b^{3}\right )} c^{2}\right )} e - 4 \, {\left (a b^{2} d e x + a b^{2} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 3 \, {\left ({\left (2 \, a^{2} b + b^{3}\right )} d e x + {\left (2 \, a^{2} b + b^{3}\right )} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{8 \, d} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

e*(Integral(a**3*c, x) + Integral(a**3*d*x, x) + Integral(b**3*c*acosh(c + 
 d*x)**3, x) + Integral(3*a*b**2*c*acosh(c + d*x)**2, x) + Integral(3*a**2 
*b*c*acosh(c + d*x), x) + Integral(b**3*d*x*acosh(c + d*x)**3, x) + Integr 
al(3*a*b**2*d*x*acosh(c + d*x)**2, x) + Integral(3*a**2*b*d*x*acosh(c + d* 
x), x))
 

Maxima [F]

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

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

Output:

1/2*a^3*d*e*x^2 + 3/4*(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*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*c/d^3))*a^2*b*d*e 
 + a^3*c*e*x + 3*((d*x + c)*arccosh(d*x + c) - sqrt((d*x + c)^2 - 1))*a^2* 
b*c*e/d + 1/2*(b^3*d*e*x^2 + 2*b^3*c*e*x)*log(d*x + sqrt(d*x + c + 1)*sqrt 
(d*x + c - 1) + c)^3 + integrate(3/2*((2*a*b^2*d^4*e - b^3*d^4*e)*x^4 + 2* 
(c^4*e - c^2*e)*a*b^2 + 4*(2*a*b^2*c*d^3*e - b^3*c*d^3*e)*x^3 + (2*(6*c^2* 
d^2*e - d^2*e)*a*b^2 - (5*c^2*d^2*e - d^2*e)*b^3)*x^2 + (2*(c^3*e - c*e)*a 
*b^2 + (2*a*b^2*d^3*e - b^3*d^3*e)*x^3 + 3*(2*a*b^2*c*d^2*e - b^3*c*d^2*e) 
*x^2 - 2*(b^3*c^2*d*e - (3*c^2*d*e - d*e)*a*b^2)*x)*sqrt(d*x + c + 1)*sqrt 
(d*x + c - 1) + 2*(2*(2*c^3*d*e - c*d*e)*a*b^2 - (c^3*d*e - c*d*e)*b^3)*x) 
*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^2/(d^3*x^3 + 3*c*d^2*x 
^2 + c^3 + (d^2*x^2 + 2*c*d*x + c^2 - 1)*sqrt(d*x + c + 1)*sqrt(d*x + c - 
1) + (3*c^2*d - d)*x - c), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^3 \, dx=\frac {e \left (12 \mathit {acosh} \left (d x +c \right ) a^{2} b \,c^{2}+12 \mathit {acosh} \left (d x +c \right ) a^{2} b c d x +6 \mathit {acosh} \left (d x +c \right ) a^{2} b \,d^{2} x^{2}+9 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, a^{2} b c -3 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, a^{2} b d x -12 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, a^{2} b c +4 \left (\int \mathit {acosh} \left (d x +c \right )^{3}d x \right ) b^{3} c d +12 \left (\int \mathit {acosh} \left (d x +c \right )^{2}d x \right ) a \,b^{2} c d +4 \left (\int \mathit {acosh} \left (d x +c \right )^{3} x d x \right ) b^{3} d^{2}+12 \left (\int \mathit {acosh} \left (d x +c \right )^{2} x d x \right ) a \,b^{2} d^{2}-6 \,\mathrm {log}\left (\sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}+c +d x \right ) a^{2} b \,c^{2}-3 \,\mathrm {log}\left (\sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}+c +d x \right ) a^{2} b +4 a^{3} c d x +2 a^{3} d^{2} x^{2}\right )}{4 d} \] Input:

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

Output:

(e*(12*acosh(c + d*x)*a**2*b*c**2 + 12*acosh(c + d*x)*a**2*b*c*d*x + 6*aco 
sh(c + d*x)*a**2*b*d**2*x**2 + 9*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*a**2 
*b*c - 3*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*a**2*b*d*x - 12*sqrt(c + d*x 
 + 1)*sqrt(c + d*x - 1)*a**2*b*c + 4*int(acosh(c + d*x)**3,x)*b**3*c*d + 1 
2*int(acosh(c + d*x)**2,x)*a*b**2*c*d + 4*int(acosh(c + d*x)**3*x,x)*b**3* 
d**2 + 12*int(acosh(c + d*x)**2*x,x)*a*b**2*d**2 - 6*log(sqrt(c**2 + 2*c*d 
*x + d**2*x**2 - 1) + c + d*x)*a**2*b*c**2 - 3*log(sqrt(c**2 + 2*c*d*x + d 
**2*x**2 - 1) + c + d*x)*a**2*b + 4*a**3*c*d*x + 2*a**3*d**2*x**2))/(4*d)