\(\int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^5} \, dx\) [20]

Optimal result

Integrand size = 21, antiderivative size = 104 \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^5} \, dx=\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{12 d e^5 (c+d x)^3}+\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{6 d e^5 (c+d x)}-\frac {a+b \text {arccosh}(c+d x)}{4 d e^5 (c+d x)^4} \] Output:

1/12*b*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d/e^5/(d*x+c)^3+1/6*b*(d*x+c-1)^(1/ 
2)*(d*x+c+1)^(1/2)/d/e^5/(d*x+c)-1/4*(a+b*arccosh(d*x+c))/d/e^5/(d*x+c)^4
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.83 \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^5} \, dx=\frac {-3 a+b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (c+2 c^3+d x+6 c^2 d x+6 c d^2 x^2+2 d^3 x^3\right )-3 b \text {arccosh}(c+d x)}{12 d e^5 (c+d x)^4} \] Input:

Integrate[(a + b*ArcCosh[c + d*x])/(c*e + d*e*x)^5,x]
 

Output:

(-3*a + b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(c + 2*c^3 + d*x + 6*c^2*d* 
x + 6*c*d^2*x^2 + 2*d^3*x^3) - 3*b*ArcCosh[c + d*x])/(12*d*e^5*(c + d*x)^4 
)
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.93, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6411, 27, 6298, 114, 27, 106}

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.

\(\displaystyle \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^5} \, dx\)

\(\Big \downarrow \) 6411

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 106

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

Input:

Int[(a + b*ArcCosh[c + d*x])/(c*e + d*e*x)^5,x]
 

Output:

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

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 106

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] /; FreeQ[{a, b, c, d, e, f, m, n, 
p}, x] && EqQ[Simplify[m + n + p + 3], 0] && EqQ[a*d*f*(m + 1) + b*c*f*(n + 
 1) + b*d*e*(p + 1), 0] && NeQ[m, -1]
 

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 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 = 0.14 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.73

method	result	size
derivativedivides	\(\frac {-\frac {a}{4 e^{5} \left (d x +c \right )^{4}}+\frac {b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )}{4 \left (d x +c \right )^{4}}+\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (2 \left (d x +c \right )^{2}+1\right )}{12 \left (d x +c \right )^{3}}\right )}{e^{5}}}{d}\)	\(76\)
default	\(\frac {-\frac {a}{4 e^{5} \left (d x +c \right )^{4}}+\frac {b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )}{4 \left (d x +c \right )^{4}}+\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (2 \left (d x +c \right )^{2}+1\right )}{12 \left (d x +c \right )^{3}}\right )}{e^{5}}}{d}\)	\(76\)
parts	\(-\frac {a}{4 e^{5} \left (d x +c \right )^{4} d}+\frac {b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )}{4 \left (d x +c \right )^{4}}+\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (2 \left (d x +c \right )^{2}+1\right )}{12 \left (d x +c \right )^{3}}\right )}{e^{5} d}\)	\(78\)
orering	\(\frac {\left (d x +c \right ) \left (10 d^{4} x^{4}+40 c \,d^{3} x^{3}+60 c^{2} d^{2} x^{2}+40 c^{3} d x +10 c^{4}-5 d^{2} x^{2}-10 c d x -5 c^{2}-8\right ) \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}{12 d \left (d e x +c e \right )^{5}}+\frac {\left (2 d^{2} x^{2}+4 c d x +2 c^{2}+1\right ) \left (d x +c -1\right ) \left (d x +c +1\right ) \left (d x +c \right )^{2} \left (\frac {b d}{\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d e x +c e \right )^{5}}-\frac {5 \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right ) d e}{\left (d e x +c e \right )^{6}}\right )}{12 d^{2}}\)	\(190\)

Input:

int((a+b*arccosh(d*x+c))/(d*e*x+c*e)^5,x,method=_RETURNVERBOSE)
 

Output:

1/d*(-1/4*a/e^5/(d*x+c)^4+b/e^5*(-1/4/(d*x+c)^4*arccosh(d*x+c)+1/12*(d*x+c 
-1)^(1/2)*(d*x+c+1)^(1/2)*(2*(d*x+c)^2+1)/(d*x+c)^3))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 208 vs. \(2 (90) = 180\).

Time = 0.11 (sec) , antiderivative size = 208, normalized size of antiderivative = 2.00 \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^5} \, dx=\frac {3 \, a d^{4} x^{4} + 12 \, a c d^{3} x^{3} + 18 \, a c^{2} d^{2} x^{2} + 12 \, a c^{3} d x - 3 \, b c^{4} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) + {\left (2 \, b c^{4} d^{3} x^{3} + 6 \, b c^{5} d^{2} x^{2} + 2 \, b c^{7} + b c^{5} + {\left (6 \, b c^{6} + b c^{4}\right )} d x\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{12 \, {\left (c^{4} d^{5} e^{5} x^{4} + 4 \, c^{5} d^{4} e^{5} x^{3} + 6 \, c^{6} d^{3} e^{5} x^{2} + 4 \, c^{7} d^{2} e^{5} x + c^{8} d e^{5}\right )}} \] Input:

integrate((a+b*arccosh(d*x+c))/(d*e*x+c*e)^5,x, algorithm="fricas")
 

Output:

1/12*(3*a*d^4*x^4 + 12*a*c*d^3*x^3 + 18*a*c^2*d^2*x^2 + 12*a*c^3*d*x - 3*b 
*c^4*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)) + (2*b*c^4*d^3*x^3 + 
 6*b*c^5*d^2*x^2 + 2*b*c^7 + b*c^5 + (6*b*c^6 + b*c^4)*d*x)*sqrt(d^2*x^2 + 
 2*c*d*x + c^2 - 1))/(c^4*d^5*e^5*x^4 + 4*c^5*d^4*e^5*x^3 + 6*c^6*d^3*e^5* 
x^2 + 4*c^7*d^2*e^5*x + c^8*d*e^5)
 

Sympy [F]

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

integrate((a+b*acosh(d*x+c))/(d*e*x+c*e)**5,x)
 

Output:

(Integral(a/(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), x) + Integral(b*acosh(c + d*x)/(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), x 
))/e**5
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 260 vs. \(2 (90) = 180\).

Time = 0.06 (sec) , antiderivative size = 260, normalized size of antiderivative = 2.50 \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^5} \, dx=\frac {1}{12} \, b {\left (\frac {{\left (2 \, d^{4} x^{4} + 8 \, c d^{3} x^{3} + 2 \, c^{4} + {\left (12 \, c^{2} d^{2} - d^{2}\right )} x^{2} - c^{2} + 2 \, {\left (4 \, c^{3} d - c d\right )} x - 1\right )} d}{{\left (d^{5} e^{5} x^{3} + 3 \, c d^{4} e^{5} x^{2} + 3 \, c^{2} d^{3} e^{5} x + c^{3} d^{2} e^{5}\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1}} - \frac {3 \, \operatorname {arcosh}\left (d x + c\right )}{d^{5} e^{5} x^{4} + 4 \, c d^{4} e^{5} x^{3} + 6 \, c^{2} d^{3} e^{5} x^{2} + 4 \, c^{3} d^{2} e^{5} x + c^{4} d e^{5}}\right )} - \frac {a}{4 \, {\left (d^{5} e^{5} x^{4} + 4 \, c d^{4} e^{5} x^{3} + 6 \, c^{2} d^{3} e^{5} x^{2} + 4 \, c^{3} d^{2} e^{5} x + c^{4} d e^{5}\right )}} \] Input:

integrate((a+b*arccosh(d*x+c))/(d*e*x+c*e)^5,x, algorithm="maxima")
 

Output:

1/12*b*((2*d^4*x^4 + 8*c*d^3*x^3 + 2*c^4 + (12*c^2*d^2 - d^2)*x^2 - c^2 + 
2*(4*c^3*d - c*d)*x - 1)*d/((d^5*e^5*x^3 + 3*c*d^4*e^5*x^2 + 3*c^2*d^3*e^5 
*x + c^3*d^2*e^5)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1)) - 3*arccosh(d*x + c 
)/(d^5*e^5*x^4 + 4*c*d^4*e^5*x^3 + 6*c^2*d^3*e^5*x^2 + 4*c^3*d^2*e^5*x + c 
^4*d*e^5)) - 1/4*a/(d^5*e^5*x^4 + 4*c*d^4*e^5*x^3 + 6*c^2*d^3*e^5*x^2 + 4* 
c^3*d^2*e^5*x + c^4*d*e^5)
                                                                                    
                                                                                    
 

Giac [F(-2)]

Exception generated. \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^5} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a+b*arccosh(d*x+c))/(d*e*x+c*e)^5,x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^5} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^5} \,d x \] Input:

int((a + b*acosh(c + d*x))/(c*e + d*e*x)^5,x)
 

Output:

int((a + b*acosh(c + d*x))/(c*e + d*e*x)^5, x)
 

Reduce [F]

\[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^5} \, dx=\frac {4 \left (\int \frac {\mathit {acosh} \left (d x +c \right )}{d^{5} x^{5}+5 c \,d^{4} x^{4}+10 c^{2} d^{3} x^{3}+10 c^{3} d^{2} x^{2}+5 c^{4} d x +c^{5}}d x \right ) b \,c^{4} d +16 \left (\int \frac {\mathit {acosh} \left (d x +c \right )}{d^{5} x^{5}+5 c \,d^{4} x^{4}+10 c^{2} d^{3} x^{3}+10 c^{3} d^{2} x^{2}+5 c^{4} d x +c^{5}}d x \right ) b \,c^{3} d^{2} x +24 \left (\int \frac {\mathit {acosh} \left (d x +c \right )}{d^{5} x^{5}+5 c \,d^{4} x^{4}+10 c^{2} d^{3} x^{3}+10 c^{3} d^{2} x^{2}+5 c^{4} d x +c^{5}}d x \right ) b \,c^{2} d^{3} x^{2}+16 \left (\int \frac {\mathit {acosh} \left (d x +c \right )}{d^{5} x^{5}+5 c \,d^{4} x^{4}+10 c^{2} d^{3} x^{3}+10 c^{3} d^{2} x^{2}+5 c^{4} d x +c^{5}}d x \right ) b c \,d^{4} x^{3}+4 \left (\int \frac {\mathit {acosh} \left (d x +c \right )}{d^{5} x^{5}+5 c \,d^{4} x^{4}+10 c^{2} d^{3} x^{3}+10 c^{3} d^{2} x^{2}+5 c^{4} d x +c^{5}}d x \right ) b \,d^{5} x^{4}-a}{4 d \,e^{5} \left (d^{4} x^{4}+4 c \,d^{3} x^{3}+6 c^{2} d^{2} x^{2}+4 c^{3} d x +c^{4}\right )} \] Input:

int((a+b*acosh(d*x+c))/(d*e*x+c*e)^5,x)
 

Output:

(4*int(acosh(c + d*x)/(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),x)*b*c**4*d + 16*int(acosh(c + d*x)/(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),x)*b*c**3*d**2*x + 24*int(acosh(c + d*x)/(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),x)*b*c* 
*2*d**3*x**2 + 16*int(acosh(c + d*x)/(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),x)*b*c*d**4*x**3 + 4*in 
t(acosh(c + d*x)/(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),x)*b*d**5*x**4 - a)/(4*d*e**5*(c**4 + 4*c** 
3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4))