\(\int (c e+d e x) (a+b \text {arccosh}(c+d x))^4 \, dx\) [42]

Optimal result

Integrand size = 21, antiderivative size = 209 \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^4 \, dx=\frac {3 b^4 e (c+d x)^2}{4 d}-\frac {3 b^3 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))}{2 d}-\frac {3 b^2 e (a+b \text {arccosh}(c+d x))^2}{4 d}+\frac {3 b^2 e (c+d x)^2 (a+b \text {arccosh}(c+d x))^2}{2 d}-\frac {b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^3}{d}-\frac {e (a+b \text {arccosh}(c+d x))^4}{4 d}+\frac {e (c+d x)^2 (a+b \text {arccosh}(c+d x))^4}{2 d} \] Output:

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

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.72 \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^4 \, dx=\frac {e \left (\left (2 a^4+6 a^2 b^2+3 b^4\right ) (c+d x)^2-2 a b \left (2 a^2+3 b^2\right ) \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}-2 b (c+d x) \left (-4 a^3 (c+d x)-6 a b^2 (c+d x)+6 a^2 b \sqrt {-1+c+d x} \sqrt {1+c+d x}+3 b^3 \sqrt {-1+c+d x} \sqrt {1+c+d x}\right ) \text {arccosh}(c+d x)+3 b^2 \left (-2 a^2-b^2+4 a^2 (c+d x)^2+2 b^2 (c+d x)^2-4 a b \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}\right ) \text {arccosh}(c+d x)^2+4 b^3 \left (-a+2 a (c+d x)^2-b \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}\right ) \text {arccosh}(c+d x)^3+b^4 \left (-1+2 (c+d x)^2\right ) \text {arccosh}(c+d x)^4-2 a b \left (2 a^2+3 b^2\right ) \log \left (c+d x+\sqrt {-1+c+d x} \sqrt {1+c+d x}\right )\right )}{4 d} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.37 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.94, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6411, 27, 6298, 6354, 6298, 6308, 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.

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

\(\Big \downarrow \) 6411

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 6354

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

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 6308

\(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arccosh}(c+d x))^4-2 b \left (-\frac {3}{2} b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arccosh}(c+d x))^2-b \int \frac {(c+d x)^2 (a+b \text {arccosh}(c+d x))}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)\right )+\frac {(a+b \text {arccosh}(c+d x))^4}{8 b}+\frac {1}{2} \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3\right )\right )}{d}\)

\(\Big \downarrow \) 6354

\(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arccosh}(c+d x))^4-2 b \left (-\frac {3}{2} b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arccosh}(c+d x))^2-b \left (\frac {1}{2} \int \frac {a+b \text {arccosh}(c+d x)}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)-\frac {1}{2} b \int (c+d x)d(c+d x)+\frac {1}{2} \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))\right )\right )+\frac {(a+b \text {arccosh}(c+d x))^4}{8 b}+\frac {1}{2} \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3\right )\right )}{d}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arccosh}(c+d x))^4-2 b \left (-\frac {3}{2} b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arccosh}(c+d x))^2-b \left (\frac {1}{2} \int \frac {a+b \text {arccosh}(c+d x)}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {1}{2} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x) (a+b \text {arccosh}(c+d x))-\frac {1}{4} b (c+d x)^2\right )\right )+\frac {(a+b \text {arccosh}(c+d x))^4}{8 b}+\frac {1}{2} \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3\right )\right )}{d}\)

\(\Big \downarrow \) 6308

\(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arccosh}(c+d x))^4-2 b \left (\frac {(a+b \text {arccosh}(c+d x))^4}{8 b}+\frac {1}{2} \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3-\frac {3}{2} b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arccosh}(c+d x))^2-b \left (\frac {1}{2} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x) (a+b \text {arccosh}(c+d x))+\frac {(a+b \text {arccosh}(c+d x))^2}{4 b}-\frac {1}{4} b (c+d x)^2\right )\right )\right )\right )}{d}\)

Input:

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

Output:

(e*(((c + d*x)^2*(a + b*ArcCosh[c + d*x])^4)/2 - 2*b*((Sqrt[-1 + c + d*x]* 
(c + d*x)*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^3)/2 + (a + b*ArcCosh 
[c + d*x])^4/(8*b) - (3*b*(((c + d*x)^2*(a + b*ArcCosh[c + d*x])^2)/2 - b* 
(-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))))/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 = 0.36 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.57

method	result	size
derivativedivides	\(\frac {\frac {e \,a^{4} \left (d x +c \right )^{2}}{2}+e \,b^{4} \left (\frac {\cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )^{4}}{4}-\frac {\sinh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )^{3}}{2}+\frac {3 \cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )^{2}}{4}-\frac {3 \sinh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )}{4}+\frac {3 \cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right )}{8}\right )+4 e a \,b^{3} \left (\frac {\cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )^{3}}{4}-\frac {3 \sinh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )^{2}}{8}+\frac {3 \cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )}{8}-\frac {3 \sinh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right )}{16}\right )+6 e \,a^{2} b^{2} \left (\frac {\cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )^{2}}{4}-\frac {\sinh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )}{4}+\frac {\cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right )}{8}\right )+4 e \,a^{3} b \left (\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )}{2}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right ) \sqrt {\left (d x +c \right )^{2}-1}+\ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )\right )}{4 \sqrt {\left (d x +c \right )^{2}-1}}\right )}{d}\)	\(328\)
default	\(\frac {\frac {e \,a^{4} \left (d x +c \right )^{2}}{2}+e \,b^{4} \left (\frac {\cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )^{4}}{4}-\frac {\sinh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )^{3}}{2}+\frac {3 \cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )^{2}}{4}-\frac {3 \sinh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )}{4}+\frac {3 \cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right )}{8}\right )+4 e a \,b^{3} \left (\frac {\cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )^{3}}{4}-\frac {3 \sinh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )^{2}}{8}+\frac {3 \cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )}{8}-\frac {3 \sinh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right )}{16}\right )+6 e \,a^{2} b^{2} \left (\frac {\cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )^{2}}{4}-\frac {\sinh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )}{4}+\frac {\cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right )}{8}\right )+4 e \,a^{3} b \left (\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )}{2}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right ) \sqrt {\left (d x +c \right )^{2}-1}+\ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )\right )}{4 \sqrt {\left (d x +c \right )^{2}-1}}\right )}{d}\)	\(328\)
parts	\(e \,a^{4} \left (\frac {1}{2} d \,x^{2}+c x \right )+\frac {e \,b^{4} \left (\frac {\cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )^{4}}{4}-\frac {\sinh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )^{3}}{2}+\frac {3 \cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )^{2}}{4}-\frac {3 \sinh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )}{4}+\frac {3 \cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right )}{8}\right )}{d}+\frac {4 e a \,b^{3} \left (\frac {\cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )^{3}}{4}-\frac {3 \sinh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )^{2}}{8}+\frac {3 \cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )}{8}-\frac {3 \sinh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right )}{16}\right )}{d}+\frac {6 e \,a^{2} b^{2} \left (\frac {\cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )^{2}}{4}-\frac {\sinh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )}{4}+\frac {\cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right )}{8}\right )}{d}+\frac {4 e \,a^{3} b \left (\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )}{2}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right ) \sqrt {\left (d x +c \right )^{2}-1}+\ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )\right )}{4 \sqrt {\left (d x +c \right )^{2}-1}}\right )}{d}\)	\(338\)
orering	\(\text {Expression too large to display}\)	\(1735\)

Input:

int((d*e*x+c*e)*(a+b*arccosh(d*x+c))^4,x,method=_RETURNVERBOSE)
 

Output:

1/d*(1/2*e*a^4*(d*x+c)^2+e*b^4*(1/4*cosh(2*arccosh(d*x+c))*arccosh(d*x+c)^ 
4-1/2*sinh(2*arccosh(d*x+c))*arccosh(d*x+c)^3+3/4*cosh(2*arccosh(d*x+c))*a 
rccosh(d*x+c)^2-3/4*sinh(2*arccosh(d*x+c))*arccosh(d*x+c)+3/8*cosh(2*arcco 
sh(d*x+c)))+4*e*a*b^3*(1/4*cosh(2*arccosh(d*x+c))*arccosh(d*x+c)^3-3/8*sin 
h(2*arccosh(d*x+c))*arccosh(d*x+c)^2+3/8*cosh(2*arccosh(d*x+c))*arccosh(d* 
x+c)-3/16*sinh(2*arccosh(d*x+c)))+6*e*a^2*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*arcc 
osh(d*x+c)))+4*e*a^3*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. 579 vs. \(2 (189) = 378\).

Time = 0.11 (sec) , antiderivative size = 579, normalized size of antiderivative = 2.77 \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^4 \, dx=\frac {{\left (2 \, a^{4} + 6 \, a^{2} b^{2} + 3 \, b^{4}\right )} d^{2} e x^{2} + 2 \, {\left (2 \, a^{4} + 6 \, a^{2} b^{2} + 3 \, b^{4}\right )} c d e x + {\left (2 \, b^{4} d^{2} e x^{2} + 4 \, b^{4} c d e x + {\left (2 \, b^{4} c^{2} - b^{4}\right )} e\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{4} + 4 \, {\left (2 \, a b^{3} d^{2} e x^{2} + 4 \, a b^{3} c d e x + {\left (2 \, a b^{3} c^{2} - a b^{3}\right )} e - {\left (b^{4} d e x + b^{4} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{3} + 3 \, {\left (2 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} d^{2} e x^{2} + 4 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} c d e x - {\left (2 \, a^{2} b^{2} + b^{4} - 2 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} c^{2}\right )} e - 4 \, {\left (a b^{3} d e x + a b^{3} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{2} + 2 \, {\left (2 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} d^{2} e x^{2} + 4 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} c d e x - {\left (2 \, a^{3} b + 3 \, a b^{3} - 2 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} c^{2}\right )} e - 3 \, {\left ({\left (2 \, a^{2} b^{2} + b^{4}\right )} d e x + {\left (2 \, a^{2} b^{2} + b^{4}\right )} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 2 \, {\left ({\left (2 \, a^{3} b + 3 \, a b^{3}\right )} d e x + {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{4 \, d} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^4 \, dx=\frac {e \left (8 \mathit {acosh} \left (d x +c \right ) a^{3} b \,c^{2}+8 \mathit {acosh} \left (d x +c \right ) a^{3} b c d x +4 \mathit {acosh} \left (d x +c \right ) a^{3} b \,d^{2} x^{2}+6 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, a^{3} b c -2 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, a^{3} b d x -8 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, a^{3} b c +2 \left (\int \mathit {acosh} \left (d x +c \right )^{4}d x \right ) b^{4} c d +8 \left (\int \mathit {acosh} \left (d x +c \right )^{3}d x \right ) a \,b^{3} c d +12 \left (\int \mathit {acosh} \left (d x +c \right )^{2}d x \right ) a^{2} b^{2} c d +2 \left (\int \mathit {acosh} \left (d x +c \right )^{4} x d x \right ) b^{4} d^{2}+8 \left (\int \mathit {acosh} \left (d x +c \right )^{3} x d x \right ) a \,b^{3} d^{2}+12 \left (\int \mathit {acosh} \left (d x +c \right )^{2} x d x \right ) a^{2} b^{2} d^{2}-4 \,\mathrm {log}\left (\sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}+c +d x \right ) a^{3} b \,c^{2}-2 \,\mathrm {log}\left (\sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}+c +d x \right ) a^{3} b +2 a^{4} c d x +a^{4} d^{2} x^{2}\right )}{2 d} \] Input:

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

Output:

(e*(8*acosh(c + d*x)*a**3*b*c**2 + 8*acosh(c + d*x)*a**3*b*c*d*x + 4*acosh 
(c + d*x)*a**3*b*d**2*x**2 + 6*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*a**3*b 
*c - 2*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*a**3*b*d*x - 8*sqrt(c + d*x + 
1)*sqrt(c + d*x - 1)*a**3*b*c + 2*int(acosh(c + d*x)**4,x)*b**4*c*d + 8*in 
t(acosh(c + d*x)**3,x)*a*b**3*c*d + 12*int(acosh(c + d*x)**2,x)*a**2*b**2* 
c*d + 2*int(acosh(c + d*x)**4*x,x)*b**4*d**2 + 8*int(acosh(c + d*x)**3*x,x 
)*a*b**3*d**2 + 12*int(acosh(c + d*x)**2*x,x)*a**2*b**2*d**2 - 4*log(sqrt( 
c**2 + 2*c*d*x + d**2*x**2 - 1) + c + d*x)*a**3*b*c**2 - 2*log(sqrt(c**2 + 
 2*c*d*x + d**2*x**2 - 1) + c + d*x)*a**3*b + 2*a**4*c*d*x + a**4*d**2*x** 
2))/(2*d)