\(\int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^4} \, dx\) [19]

Optimal result

Integrand size = 21, antiderivative size = 99 \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^4} \, dx=\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{6 d e^4 (c+d x)^2}-\frac {a+b \text {arccosh}(c+d x)}{3 d e^4 (c+d x)^3}+\frac {b \arctan \left (\sqrt {-1+c+d x} \sqrt {1+c+d x}\right )}{6 d e^4} \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.16 \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^4} \, dx=\frac {-\frac {2 a}{(c+d x)^3}+\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{(c+d x)^2}-\frac {2 b \text {arccosh}(c+d x)}{(c+d x)^3}+\frac {b \sqrt {-1+(c+d x)^2} \arctan \left (\sqrt {-1+(c+d x)^2}\right )}{\sqrt {-1+c+d x} \sqrt {1+c+d x}}}{6 d e^4} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 92, 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, 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.

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

\(\Big \downarrow \) 6411

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 103

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

\(\Big \downarrow \) 216

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

Input:

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

Output:

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

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 = 0.14 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.11

method	result	size
derivativedivides	\(\frac {-\frac {a}{3 e^{4} \left (d x +c \right )^{3}}+\frac {b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )}{3 \left (d x +c \right )^{3}}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\arctan \left (\frac {1}{\sqrt {\left (d x +c \right )^{2}-1}}\right ) \left (d x +c \right )^{2}-\sqrt {\left (d x +c \right )^{2}-1}\right )}{6 \left (d x +c \right )^{2} \sqrt {\left (d x +c \right )^{2}-1}}\right )}{e^{4}}}{d}\)	\(110\)
default	\(\frac {-\frac {a}{3 e^{4} \left (d x +c \right )^{3}}+\frac {b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )}{3 \left (d x +c \right )^{3}}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\arctan \left (\frac {1}{\sqrt {\left (d x +c \right )^{2}-1}}\right ) \left (d x +c \right )^{2}-\sqrt {\left (d x +c \right )^{2}-1}\right )}{6 \left (d x +c \right )^{2} \sqrt {\left (d x +c \right )^{2}-1}}\right )}{e^{4}}}{d}\)	\(110\)
parts	\(-\frac {a}{3 e^{4} \left (d x +c \right )^{3} d}+\frac {b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )}{3 \left (d x +c \right )^{3}}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\arctan \left (\frac {1}{\sqrt {\left (d x +c \right )^{2}-1}}\right ) \left (d x +c \right )^{2}-\sqrt {\left (d x +c \right )^{2}-1}\right )}{6 \left (d x +c \right )^{2} \sqrt {\left (d x +c \right )^{2}-1}}\right )}{e^{4} d}\)	\(112\)

Input:

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

Output:

1/d*(-1/3*a/e^4/(d*x+c)^3+b/e^4*(-1/3/(d*x+c)^3*arccosh(d*x+c)-1/6*(d*x+c- 
1)^(1/2)*(d*x+c+1)^(1/2)*(arctan(1/((d*x+c)^2-1)^(1/2))*(d*x+c)^2-((d*x+c) 
^2-1)^(1/2))/(d*x+c)^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. 276 vs. \(2 (85) = 170\).

Time = 0.13 (sec) , antiderivative size = 276, normalized size of antiderivative = 2.79 \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^4} \, dx=-\frac {2 \, a c^{3} - 2 \, {\left (b c^{3} d^{3} x^{3} + 3 \, b c^{4} d^{2} x^{2} + 3 \, b c^{5} d x + b c^{6}\right )} \arctan \left (-d x - c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 2 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 2 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (-d x - c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left (b c^{3} d x + b c^{4}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{6 \, {\left (c^{3} d^{4} e^{4} x^{3} + 3 \, c^{4} d^{3} e^{4} x^{2} + 3 \, c^{5} d^{2} e^{4} x + c^{6} d e^{4}\right )}} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/6*b*((2*d^2*x^2 + 4*c*d*x + 2*c^2 - (d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + 
 c^3)*log(d*x + c + 1) + (d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)*log(d*x 
 + c - 1) - 2*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c))/(d^4*e^4 
*x^3 + 3*c*d^3*e^4*x^2 + 3*c^2*d^2*e^4*x + c^3*d*e^4) - 6*integrate(1/3/(d 
^6*e^4*x^6 + 6*c*d^5*e^4*x^5 + c^6*e^4 - c^4*e^4 + (15*c^2*d^4*e^4 - d^4*e 
^4)*x^4 + 4*(5*c^3*d^3*e^4 - c*d^3*e^4)*x^3 + 3*(5*c^4*d^2*e^4 - 2*c^2*d^2 
*e^4)*x^2 + 2*(3*c^5*d*e^4 - 2*c^3*d*e^4)*x + (d^5*e^4*x^5 + 5*c*d^4*e^4*x 
^4 + c^5*e^4 - c^3*e^4 + (10*c^2*d^3*e^4 - d^3*e^4)*x^3 + (10*c^3*d^2*e^4 
- 3*c*d^2*e^4)*x^2 + (5*c^4*d*e^4 - 3*c^2*d*e^4)*x)*e^(1/2*log(d*x + c + 1 
) + 1/2*log(d*x + c - 1))), x)) - 1/3*a/(d^4*e^4*x^3 + 3*c*d^3*e^4*x^2 + 3 
*c^2*d^2*e^4*x + c^3*d*e^4)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(3*int(acosh(c + d*x)/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x** 
3 + d**4*x**4),x)*b*c**3*d + 9*int(acosh(c + d*x)/(c**4 + 4*c**3*d*x + 6*c 
**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4),x)*b*c**2*d**2*x + 9*int(acosh( 
c + d*x)/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4 
),x)*b*c*d**3*x**2 + 3*int(acosh(c + d*x)/(c**4 + 4*c**3*d*x + 6*c**2*d**2 
*x**2 + 4*c*d**3*x**3 + d**4*x**4),x)*b*d**4*x**3 - a)/(3*d*e**4*(c**3 + 3 
*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3))