\(\int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^6} \, dx\) [21]

Optimal result

Integrand size = 21, antiderivative size = 137 \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^6} \, dx=\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{20 d e^6 (c+d x)^4}+\frac {3 b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{40 d e^6 (c+d x)^2}-\frac {a+b \text {arccosh}(c+d x)}{5 d e^6 (c+d x)^5}+\frac {3 b \arctan \left (\sqrt {-1+c+d x} \sqrt {1+c+d x}\right )}{40 d e^6} \] Output:

1/20*b*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d/e^6/(d*x+c)^4+3/40*b*(d*x+c-1)^(1 
/2)*(d*x+c+1)^(1/2)/d/e^6/(d*x+c)^2-1/5*(a+b*arccosh(d*x+c))/d/e^6/(d*x+c) 
^5+3/40*b*arctan((d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/d/e^6
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.07 \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^6} \, dx=\frac {-\frac {8 a}{(c+d x)^5}+\frac {2 b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{(c+d x)^4}+\frac {3 b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{(c+d x)^2}-\frac {8 b \text {arccosh}(c+d x)}{(c+d x)^5}+\frac {3 b \sqrt {-1+(c+d x)^2} \arctan \left (\sqrt {-1+(c+d x)^2}\right )}{\sqrt {-1+c+d x} \sqrt {1+c+d x}}}{40 d e^6} \] Input:

Integrate[(a + b*ArcCosh[c + d*x])/(c*e + d*e*x)^6,x]
 

Output:

((-8*a)/(c + d*x)^5 + (2*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(c + d*x) 
^4 + (3*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(c + d*x)^2 - (8*b*ArcCosh 
[c + d*x])/(c + d*x)^5 + (3*b*Sqrt[-1 + (c + d*x)^2]*ArcTan[Sqrt[-1 + (c + 
 d*x)^2]])/(Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]))/(40*d*e^6)
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.93, 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, 114, 27, 114, 103, 216}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^6} \, dx\)

\(\Big \downarrow \) 6411

\(\displaystyle \frac {\int \frac {a+b \text {arccosh}(c+d x)}{e^6 (c+d x)^6}d(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {a+b \text {arccosh}(c+d x)}{(c+d x)^6}d(c+d x)}{d e^6}\)

\(\Big \downarrow \) 6298

\(\displaystyle \frac {\frac {1}{5} b \int \frac {1}{\sqrt {c+d x-1} (c+d x)^5 \sqrt {c+d x+1}}d(c+d x)-\frac {a+b \text {arccosh}(c+d x)}{5 (c+d x)^5}}{d e^6}\)

\(\Big \downarrow \) 114

\(\displaystyle \frac {\frac {1}{5} b \left (\frac {1}{4} \int \frac {3}{\sqrt {c+d x-1} (c+d x)^3 \sqrt {c+d x+1}}d(c+d x)+\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{4 (c+d x)^4}\right )-\frac {a+b \text {arccosh}(c+d x)}{5 (c+d x)^5}}{d e^6}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{5} b \left (\frac {3}{4} \int \frac {1}{\sqrt {c+d x-1} (c+d x)^3 \sqrt {c+d x+1}}d(c+d x)+\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{4 (c+d x)^4}\right )-\frac {a+b \text {arccosh}(c+d x)}{5 (c+d x)^5}}{d e^6}\)

\(\Big \downarrow \) 114

\(\displaystyle \frac {\frac {1}{5} b \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1}}d(c+d x)+\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{2 (c+d x)^2}\right )+\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{4 (c+d x)^4}\right )-\frac {a+b \text {arccosh}(c+d x)}{5 (c+d x)^5}}{d e^6}\)

\(\Big \downarrow \) 103

\(\displaystyle \frac {\frac {1}{5} b \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{(c+d x-1) (c+d x+1)+1}d\left (\sqrt {c+d x-1} \sqrt {c+d x+1}\right )+\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{2 (c+d x)^2}\right )+\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{4 (c+d x)^4}\right )-\frac {a+b \text {arccosh}(c+d x)}{5 (c+d x)^5}}{d e^6}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\frac {1}{5} b \left (\frac {3}{4} \left (\frac {1}{2} \arctan \left (\sqrt {c+d x-1} \sqrt {c+d x+1}\right )+\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{2 (c+d x)^2}\right )+\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{4 (c+d x)^4}\right )-\frac {a+b \text {arccosh}(c+d x)}{5 (c+d x)^5}}{d e^6}\)

Input:

Int[(a + b*ArcCosh[c + d*x])/(c*e + d*e*x)^6,x]
 

Output:

(-1/5*(a + b*ArcCosh[c + d*x])/(c + d*x)^5 + (b*((Sqrt[-1 + c + d*x]*Sqrt[ 
1 + c + d*x])/(4*(c + d*x)^4) + (3*((Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]) 
/(2*(c + d*x)^2) + ArcTan[Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]]/2))/4))/5) 
/(d*e^6)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 103

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

rule 114

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 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e 
 - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
 

rule 216

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

rule 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 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 = 0.15 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.96

method	result	size
derivativedivides	\(\frac {-\frac {a}{5 e^{6} \left (d x +c \right )^{5}}+\frac {b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )}{5 \left (d x +c \right )^{5}}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (3 \arctan \left (\frac {1}{\sqrt {\left (d x +c \right )^{2}-1}}\right ) \left (d x +c \right )^{4}-3 \left (d x +c \right )^{2} \sqrt {\left (d x +c \right )^{2}-1}-2 \sqrt {\left (d x +c \right )^{2}-1}\right )}{40 \sqrt {\left (d x +c \right )^{2}-1}\, \left (d x +c \right )^{4}}\right )}{e^{6}}}{d}\)	\(131\)
default	\(\frac {-\frac {a}{5 e^{6} \left (d x +c \right )^{5}}+\frac {b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )}{5 \left (d x +c \right )^{5}}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (3 \arctan \left (\frac {1}{\sqrt {\left (d x +c \right )^{2}-1}}\right ) \left (d x +c \right )^{4}-3 \left (d x +c \right )^{2} \sqrt {\left (d x +c \right )^{2}-1}-2 \sqrt {\left (d x +c \right )^{2}-1}\right )}{40 \sqrt {\left (d x +c \right )^{2}-1}\, \left (d x +c \right )^{4}}\right )}{e^{6}}}{d}\)	\(131\)
parts	\(-\frac {a}{5 e^{6} \left (d x +c \right )^{5} d}+\frac {b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )}{5 \left (d x +c \right )^{5}}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (3 \arctan \left (\frac {1}{\sqrt {\left (d x +c \right )^{2}-1}}\right ) \left (d x +c \right )^{4}-3 \left (d x +c \right )^{2} \sqrt {\left (d x +c \right )^{2}-1}-2 \sqrt {\left (d x +c \right )^{2}-1}\right )}{40 \sqrt {\left (d x +c \right )^{2}-1}\, \left (d x +c \right )^{4}}\right )}{e^{6} d}\)	\(133\)

Input:

int((a+b*arccosh(d*x+c))/(d*e*x+c*e)^6,x,method=_RETURNVERBOSE)
 

Output:

1/d*(-1/5*a/e^6/(d*x+c)^5+b/e^6*(-1/5/(d*x+c)^5*arccosh(d*x+c)-1/40*(d*x+c 
-1)^(1/2)*(d*x+c+1)^(1/2)*(3*arctan(1/((d*x+c)^2-1)^(1/2))*(d*x+c)^4-3*(d* 
x+c)^2*((d*x+c)^2-1)^(1/2)-2*((d*x+c)^2-1)^(1/2))/((d*x+c)^2-1)^(1/2)/(d*x 
+c)^4))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 416 vs. \(2 (117) = 234\).

Time = 0.16 (sec) , antiderivative size = 416, normalized size of antiderivative = 3.04 \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^6} \, dx=-\frac {8 \, a c^{5} - 6 \, {\left (b c^{5} d^{5} x^{5} + 5 \, b c^{6} d^{4} x^{4} + 10 \, b c^{7} d^{3} x^{3} + 10 \, b c^{8} d^{2} x^{2} + 5 \, b c^{9} d x + b c^{10}\right )} \arctan \left (-d x - c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 8 \, {\left (b d^{5} x^{5} + 5 \, b c d^{4} x^{4} + 10 \, b c^{2} d^{3} x^{3} + 10 \, b c^{3} d^{2} x^{2} + 5 \, b c^{4} d x\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 8 \, {\left (b d^{5} x^{5} + 5 \, b c d^{4} x^{4} + 10 \, b c^{2} d^{3} x^{3} + 10 \, b c^{3} d^{2} x^{2} + 5 \, b c^{4} d x + b c^{5}\right )} \log \left (-d x - c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left (3 \, b c^{5} d^{3} x^{3} + 9 \, b c^{6} d^{2} x^{2} + 3 \, b c^{8} + 2 \, b c^{6} + {\left (9 \, b c^{7} + 2 \, b c^{5}\right )} d x\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{40 \, {\left (c^{5} d^{6} e^{6} x^{5} + 5 \, c^{6} d^{5} e^{6} x^{4} + 10 \, c^{7} d^{4} e^{6} x^{3} + 10 \, c^{8} d^{3} e^{6} x^{2} + 5 \, c^{9} d^{2} e^{6} x + c^{10} d e^{6}\right )}} \] Input:

integrate((a+b*arccosh(d*x+c))/(d*e*x+c*e)^6,x, algorithm="fricas")
 

Output:

-1/40*(8*a*c^5 - 6*(b*c^5*d^5*x^5 + 5*b*c^6*d^4*x^4 + 10*b*c^7*d^3*x^3 + 1 
0*b*c^8*d^2*x^2 + 5*b*c^9*d*x + b*c^10)*arctan(-d*x - c + sqrt(d^2*x^2 + 2 
*c*d*x + c^2 - 1)) - 8*(b*d^5*x^5 + 5*b*c*d^4*x^4 + 10*b*c^2*d^3*x^3 + 10* 
b*c^3*d^2*x^2 + 5*b*c^4*d*x)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 
1)) - 8*(b*d^5*x^5 + 5*b*c*d^4*x^4 + 10*b*c^2*d^3*x^3 + 10*b*c^3*d^2*x^2 + 
 5*b*c^4*d*x + b*c^5)*log(-d*x - c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)) - 
(3*b*c^5*d^3*x^3 + 9*b*c^6*d^2*x^2 + 3*b*c^8 + 2*b*c^6 + (9*b*c^7 + 2*b*c^ 
5)*d*x)*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))/(c^5*d^6*e^6*x^5 + 5*c^6*d^5*e^ 
6*x^4 + 10*c^7*d^4*e^6*x^3 + 10*c^8*d^3*e^6*x^2 + 5*c^9*d^2*e^6*x + c^10*d 
*e^6)
 

Sympy [F]

\[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^6} \, dx=\frac {\int \frac {a}{c^{6} + 6 c^{5} d x + 15 c^{4} d^{2} x^{2} + 20 c^{3} d^{3} x^{3} + 15 c^{2} d^{4} x^{4} + 6 c d^{5} x^{5} + d^{6} x^{6}}\, dx + \int \frac {b \operatorname {acosh}{\left (c + d x \right )}}{c^{6} + 6 c^{5} d x + 15 c^{4} d^{2} x^{2} + 20 c^{3} d^{3} x^{3} + 15 c^{2} d^{4} x^{4} + 6 c d^{5} x^{5} + d^{6} x^{6}}\, dx}{e^{6}} \] Input:

integrate((a+b*acosh(d*x+c))/(d*e*x+c*e)**6,x)
 

Output:

(Integral(a/(c**6 + 6*c**5*d*x + 15*c**4*d**2*x**2 + 20*c**3*d**3*x**3 + 1 
5*c**2*d**4*x**4 + 6*c*d**5*x**5 + d**6*x**6), x) + Integral(b*acosh(c + d 
*x)/(c**6 + 6*c**5*d*x + 15*c**4*d**2*x**2 + 20*c**3*d**3*x**3 + 15*c**2*d 
**4*x**4 + 6*c*d**5*x**5 + d**6*x**6), x))/e**6
 

Maxima [F]

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

integrate((a+b*arccosh(d*x+c))/(d*e*x+c*e)^6,x, algorithm="maxima")
 

Output:

1/30*b*((6*d^4*x^4 + 24*c*d^3*x^3 + 6*c^4 + 2*(18*c^2*d^2 + d^2)*x^2 + 2*c 
^2 + 4*(6*c^3*d + c*d)*x - 3*(d^5*x^5 + 5*c*d^4*x^4 + 10*c^2*d^3*x^3 + 10* 
c^3*d^2*x^2 + 5*c^4*d*x + c^5)*log(d*x + c + 1) + 3*(d^5*x^5 + 5*c*d^4*x^4 
 + 10*c^2*d^3*x^3 + 10*c^3*d^2*x^2 + 5*c^4*d*x + c^5)*log(d*x + c - 1) - 6 
*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c))/(d^6*e^6*x^5 + 5*c*d^ 
5*e^6*x^4 + 10*c^2*d^4*e^6*x^3 + 10*c^3*d^3*e^6*x^2 + 5*c^4*d^2*e^6*x + c^ 
5*d*e^6) - 30*integrate(1/5/(d^8*e^6*x^8 + 8*c*d^7*e^6*x^7 + c^8*e^6 - c^6 
*e^6 + (28*c^2*d^6*e^6 - d^6*e^6)*x^6 + 2*(28*c^3*d^5*e^6 - 3*c*d^5*e^6)*x 
^5 + 5*(14*c^4*d^4*e^6 - 3*c^2*d^4*e^6)*x^4 + 4*(14*c^5*d^3*e^6 - 5*c^3*d^ 
3*e^6)*x^3 + (28*c^6*d^2*e^6 - 15*c^4*d^2*e^6)*x^2 + 2*(4*c^7*d*e^6 - 3*c^ 
5*d*e^6)*x + (d^7*e^6*x^7 + 7*c*d^6*e^6*x^6 + c^7*e^6 - c^5*e^6 + (21*c^2* 
d^5*e^6 - d^5*e^6)*x^5 + 5*(7*c^3*d^4*e^6 - c*d^4*e^6)*x^4 + 5*(7*c^4*d^3* 
e^6 - 2*c^2*d^3*e^6)*x^3 + (21*c^5*d^2*e^6 - 10*c^3*d^2*e^6)*x^2 + (7*c^6* 
d*e^6 - 5*c^4*d*e^6)*x)*e^(1/2*log(d*x + c + 1) + 1/2*log(d*x + c - 1))), 
x)) - 1/5*a/(d^6*e^6*x^5 + 5*c*d^5*e^6*x^4 + 10*c^2*d^4*e^6*x^3 + 10*c^3*d 
^3*e^6*x^2 + 5*c^4*d^2*e^6*x + c^5*d*e^6)
 

Giac [F]

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

integrate((a+b*arccosh(d*x+c))/(d*e*x+c*e)^6,x, algorithm="giac")
 

Output:

integrate((b*arccosh(d*x + c) + a)/(d*e*x + c*e)^6, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^6} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^6} \,d x \] Input:

int((a + b*acosh(c + d*x))/(c*e + d*e*x)^6,x)
 

Output:

int((a + b*acosh(c + d*x))/(c*e + d*e*x)^6, x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^6} \, dx=\frac {5 \left (\int \frac {\mathit {acosh} \left (d x +c \right )}{d^{6} x^{6}+6 c \,d^{5} x^{5}+15 c^{2} d^{4} x^{4}+20 c^{3} d^{3} x^{3}+15 c^{4} d^{2} x^{2}+6 c^{5} d x +c^{6}}d x \right ) b \,c^{5} d +25 \left (\int \frac {\mathit {acosh} \left (d x +c \right )}{d^{6} x^{6}+6 c \,d^{5} x^{5}+15 c^{2} d^{4} x^{4}+20 c^{3} d^{3} x^{3}+15 c^{4} d^{2} x^{2}+6 c^{5} d x +c^{6}}d x \right ) b \,c^{4} d^{2} x +50 \left (\int \frac {\mathit {acosh} \left (d x +c \right )}{d^{6} x^{6}+6 c \,d^{5} x^{5}+15 c^{2} d^{4} x^{4}+20 c^{3} d^{3} x^{3}+15 c^{4} d^{2} x^{2}+6 c^{5} d x +c^{6}}d x \right ) b \,c^{3} d^{3} x^{2}+50 \left (\int \frac {\mathit {acosh} \left (d x +c \right )}{d^{6} x^{6}+6 c \,d^{5} x^{5}+15 c^{2} d^{4} x^{4}+20 c^{3} d^{3} x^{3}+15 c^{4} d^{2} x^{2}+6 c^{5} d x +c^{6}}d x \right ) b \,c^{2} d^{4} x^{3}+25 \left (\int \frac {\mathit {acosh} \left (d x +c \right )}{d^{6} x^{6}+6 c \,d^{5} x^{5}+15 c^{2} d^{4} x^{4}+20 c^{3} d^{3} x^{3}+15 c^{4} d^{2} x^{2}+6 c^{5} d x +c^{6}}d x \right ) b c \,d^{5} x^{4}+5 \left (\int \frac {\mathit {acosh} \left (d x +c \right )}{d^{6} x^{6}+6 c \,d^{5} x^{5}+15 c^{2} d^{4} x^{4}+20 c^{3} d^{3} x^{3}+15 c^{4} d^{2} x^{2}+6 c^{5} d x +c^{6}}d x \right ) b \,d^{6} x^{5}-a}{5 d \,e^{6} \left (d^{5} x^{5}+5 c \,d^{4} x^{4}+10 c^{2} d^{3} x^{3}+10 c^{3} d^{2} x^{2}+5 c^{4} d x +c^{5}\right )} \] Input:

int((a+b*acosh(d*x+c))/(d*e*x+c*e)^6,x)
 

Output:

(5*int(acosh(c + d*x)/(c**6 + 6*c**5*d*x + 15*c**4*d**2*x**2 + 20*c**3*d** 
3*x**3 + 15*c**2*d**4*x**4 + 6*c*d**5*x**5 + d**6*x**6),x)*b*c**5*d + 25*i 
nt(acosh(c + d*x)/(c**6 + 6*c**5*d*x + 15*c**4*d**2*x**2 + 20*c**3*d**3*x* 
*3 + 15*c**2*d**4*x**4 + 6*c*d**5*x**5 + d**6*x**6),x)*b*c**4*d**2*x + 50* 
int(acosh(c + d*x)/(c**6 + 6*c**5*d*x + 15*c**4*d**2*x**2 + 20*c**3*d**3*x 
**3 + 15*c**2*d**4*x**4 + 6*c*d**5*x**5 + d**6*x**6),x)*b*c**3*d**3*x**2 + 
 50*int(acosh(c + d*x)/(c**6 + 6*c**5*d*x + 15*c**4*d**2*x**2 + 20*c**3*d* 
*3*x**3 + 15*c**2*d**4*x**4 + 6*c*d**5*x**5 + d**6*x**6),x)*b*c**2*d**4*x* 
*3 + 25*int(acosh(c + d*x)/(c**6 + 6*c**5*d*x + 15*c**4*d**2*x**2 + 20*c** 
3*d**3*x**3 + 15*c**2*d**4*x**4 + 6*c*d**5*x**5 + d**6*x**6),x)*b*c*d**5*x 
**4 + 5*int(acosh(c + d*x)/(c**6 + 6*c**5*d*x + 15*c**4*d**2*x**2 + 20*c** 
3*d**3*x**3 + 15*c**2*d**4*x**4 + 6*c*d**5*x**5 + d**6*x**6),x)*b*d**6*x** 
5 - a)/(5*d*e**6*(c**5 + 5*c**4*d*x + 10*c**3*d**2*x**2 + 10*c**2*d**3*x** 
3 + 5*c*d**4*x**4 + d**5*x**5))