\(\int \frac {\text {arccosh}(a+b x)}{x^3} \, dx\) [7]

Optimal result

Integrand size = 10, antiderivative size = 106 \[ \int \frac {\text {arccosh}(a+b x)}{x^3} \, dx=\frac {b \sqrt {-1+a+b x} \sqrt {1+a+b x}}{2 \left (1-a^2\right ) x}-\frac {\text {arccosh}(a+b x)}{2 x^2}-\frac {a b^2 \arctan \left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {-1+a+b x}}\right )}{\left (1-a^2\right )^{3/2}} \] Output:

1/2*b*(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2)/(-a^2+1)/x-1/2*arccosh(b*x+a)/x^2-a* 
b^2*arctan((1-a)^(1/2)*(b*x+a+1)^(1/2)/(1+a)^(1/2)/(b*x+a-1)^(1/2))/(-a^2+ 
1)^(3/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.20 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.28 \[ \int \frac {\text {arccosh}(a+b x)}{x^3} \, dx=\frac {-\text {arccosh}(a+b x)+\frac {b x \left (-\sqrt {-1+a+b x} \sqrt {1+a+b x}+\frac {i a b x \log \left (\frac {4 i \sqrt {1-a^2} \left (-1+a^2+a b x-i \sqrt {1-a^2} \sqrt {-1+a+b x} \sqrt {1+a+b x}\right )}{a b^2 x}\right )}{\sqrt {1-a^2}}\right )}{-1+a^2}}{2 x^2} \] Input:

Integrate[ArcCosh[a + b*x]/x^3,x]
 

Output:

(-ArcCosh[a + b*x] + (b*x*(-(Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x]) + (I*a* 
b*x*Log[((4*I)*Sqrt[1 - a^2]*(-1 + a^2 + a*b*x - I*Sqrt[1 - a^2]*Sqrt[-1 + 
 a + b*x]*Sqrt[1 + a + b*x]))/(a*b^2*x)])/Sqrt[1 - a^2]))/(-1 + a^2))/(2*x 
^2)
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {6411, 25, 27, 6378, 107, 104, 218}

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.

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

\(\Big \downarrow \) 6411

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

\(\displaystyle -b^2 \int -\frac {\text {arccosh}(a+b x)}{b^3 x^3}d(a+b x)\)

\(\Big \downarrow \) 6378

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

\(\Big \downarrow \) 107

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 218

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

Input:

Int[ArcCosh[a + b*x]/x^3,x]
 

Output:

-(b^2*(ArcCosh[a + b*x]/(2*b^2*x^2) + (-((Sqrt[-1 + a + b*x]*Sqrt[1 + a + 
b*x])/((1 - a^2)*b*x)) + (2*a*ArcTan[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt 
[1 + a]*Sqrt[-1 + a + b*x])])/(1 - a^2)^(3/2))/2))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 107

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

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 6378

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x 
_Symbol] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(e*(m + 1))), x] 
 - Simp[b*c*(n/(e*(m + 1)))   Int[(d + e*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, e, 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 [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.07 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.62

method	result	size
parts	\(-\frac {\operatorname {arccosh}\left (b x +a \right )}{2 x^{2}}+\frac {b \sqrt {b x +a -1}\, \sqrt {b x +a +1}\, \operatorname {csgn}\left (b \right )^{2} \left (\sqrt {a^{2}-1}\, \ln \left (\frac {2 a^{2}-2+2 a b x +2 \sqrt {a^{2}-1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}{x}\right ) a b x -a^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\right )}{2 x \left (a^{2}-1\right ) \left (1+a \right ) \left (a -1\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}\)	\(172\)
derivativedivides	\(b^{2} \left (-\frac {\operatorname {arccosh}\left (b x +a \right )}{2 b^{2} x^{2}}-\frac {\sqrt {b x +a +1}\, \sqrt {b x +a -1}\, \left (\sqrt {a^{2}-1}\, \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 a \left (b x +a \right )-2}{b x}\right ) a^{2}-\sqrt {a^{2}-1}\, \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 a \left (b x +a \right )-2}{b x}\right ) a \left (b x +a \right )+a^{2} \sqrt {\left (b x +a \right )^{2}-1}-\sqrt {\left (b x +a \right )^{2}-1}\right )}{2 b x \left (a^{2}-1\right ) \left (1+a \right ) \left (a -1\right ) \sqrt {\left (b x +a \right )^{2}-1}}\right )\)	\(201\)
default	\(b^{2} \left (-\frac {\operatorname {arccosh}\left (b x +a \right )}{2 b^{2} x^{2}}-\frac {\sqrt {b x +a +1}\, \sqrt {b x +a -1}\, \left (\sqrt {a^{2}-1}\, \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 a \left (b x +a \right )-2}{b x}\right ) a^{2}-\sqrt {a^{2}-1}\, \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 a \left (b x +a \right )-2}{b x}\right ) a \left (b x +a \right )+a^{2} \sqrt {\left (b x +a \right )^{2}-1}-\sqrt {\left (b x +a \right )^{2}-1}\right )}{2 b x \left (a^{2}-1\right ) \left (1+a \right ) \left (a -1\right ) \sqrt {\left (b x +a \right )^{2}-1}}\right )\)	\(201\)

Input:

int(arccosh(b*x+a)/x^3,x,method=_RETURNVERBOSE)
 

Output:

-1/2*arccosh(b*x+a)/x^2+1/2*b*(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2)*csgn(b)^2*(( 
a^2-1)^(1/2)*ln(2*(a*b*x+(a^2-1)^(1/2)*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+a^2-1 
)/x)*a*b*x-a^2*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+(b^2*x^2+2*a*b*x+a^2-1)^(1/2) 
)/x/(a^2-1)/(1+a)/(a-1)/(b^2*x^2+2*a*b*x+a^2-1)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 222 vs. \(2 (86) = 172\).

Time = 0.12 (sec) , antiderivative size = 460, normalized size of antiderivative = 4.34 \[ \int \frac {\text {arccosh}(a+b x)}{x^3} \, dx=\left [\frac {\sqrt {a^{2} - 1} a b^{2} x^{2} \log \left (\frac {a^{2} b x + a^{3} + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (a^{2} + \sqrt {a^{2} - 1} a - 1\right )} + {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1} - a}{x}\right ) - {\left (a^{2} - 1\right )} b^{2} x^{2} + {\left (a^{4} - 2 \, a^{2} + 1\right )} x^{2} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (a^{2} - 1\right )} b x - {\left (a^{4} - {\left (a^{4} - 2 \, a^{2} + 1\right )} x^{2} - 2 \, a^{2} + 1\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )}{2 \, {\left (a^{4} - 2 \, a^{2} + 1\right )} x^{2}}, -\frac {2 \, \sqrt {-a^{2} + 1} a b^{2} x^{2} \arctan \left (-\frac {\sqrt {-a^{2} + 1} b x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} \sqrt {-a^{2} + 1}}{a^{2} - 1}\right ) + {\left (a^{2} - 1\right )} b^{2} x^{2} - {\left (a^{4} - 2 \, a^{2} + 1\right )} x^{2} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (a^{2} - 1\right )} b x + {\left (a^{4} - {\left (a^{4} - 2 \, a^{2} + 1\right )} x^{2} - 2 \, a^{2} + 1\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )}{2 \, {\left (a^{4} - 2 \, a^{2} + 1\right )} x^{2}}\right ] \] Input:

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

Output:

[1/2*(sqrt(a^2 - 1)*a*b^2*x^2*log((a^2*b*x + a^3 + sqrt(b^2*x^2 + 2*a*b*x 
+ a^2 - 1)*(a^2 + sqrt(a^2 - 1)*a - 1) + (a*b*x + a^2 - 1)*sqrt(a^2 - 1) - 
 a)/x) - (a^2 - 1)*b^2*x^2 + (a^4 - 2*a^2 + 1)*x^2*log(-b*x - a + sqrt(b^2 
*x^2 + 2*a*b*x + a^2 - 1)) - sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*(a^2 - 1)*b 
*x - (a^4 - (a^4 - 2*a^2 + 1)*x^2 - 2*a^2 + 1)*log(b*x + a + sqrt(b^2*x^2 
+ 2*a*b*x + a^2 - 1)))/((a^4 - 2*a^2 + 1)*x^2), -1/2*(2*sqrt(-a^2 + 1)*a*b 
^2*x^2*arctan(-(sqrt(-a^2 + 1)*b*x - sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*sqr 
t(-a^2 + 1))/(a^2 - 1)) + (a^2 - 1)*b^2*x^2 - (a^4 - 2*a^2 + 1)*x^2*log(-b 
*x - a + sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)) + sqrt(b^2*x^2 + 2*a*b*x + a^2 
 - 1)*(a^2 - 1)*b*x + (a^4 - (a^4 - 2*a^2 + 1)*x^2 - 2*a^2 + 1)*log(b*x + 
a + sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)))/((a^4 - 2*a^2 + 1)*x^2)]
 

Sympy [F]

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

integrate(acosh(b*x+a)/x**3,x)
 

Output:

Integral(acosh(a + b*x)/x**3, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\text {arccosh}(a+b x)}{x^3} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(a-1>0)', see `assume?` for more 
details)Is
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.60 \[ \int \frac {\text {arccosh}(a+b x)}{x^3} \, dx=-{\left (\frac {a b \arctan \left (-\frac {x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}}{\sqrt {-a^{2} + 1}}\right )}{{\left (a^{2} - 1\right )} \sqrt {-a^{2} + 1}} - \frac {{\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )} a b + a^{2} {\left | b \right |} - {\left | b \right |}}{{\left ({\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )}^{2} - a^{2} + 1\right )} {\left (a^{2} - 1\right )}}\right )} b - \frac {\log \left (b x + a + \sqrt {{\left (b x + a\right )}^{2} - 1}\right )}{2 \, x^{2}} \] Input:

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

Output:

-(a*b*arctan(-(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1))/sqrt(-a^2 + 1 
))/((a^2 - 1)*sqrt(-a^2 + 1)) - ((x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 
- 1))*a*b + a^2*abs(b) - abs(b))/(((x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^ 
2 - 1))^2 - a^2 + 1)*(a^2 - 1)))*b - 1/2*log(b*x + a + sqrt((b*x + a)^2 - 
1))/x^2
 

Mupad [F(-1)]

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

int(acosh(a + b*x)/x^3,x)
 

Output:

int(acosh(a + b*x)/x^3, x)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.37 \[ \int \frac {\text {arccosh}(a+b x)}{x^3} \, dx=\frac {-\mathit {acosh} \left (b x +a \right ) a^{4}+2 \mathit {acosh} \left (b x +a \right ) a^{2}-\mathit {acosh} \left (b x +a \right )-2 \sqrt {-a^{2}+1}\, \mathit {atan} \left (\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+b x}{\sqrt {-a^{2}+1}}\right ) a \,b^{2} x^{2}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, a^{2} b x -\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, b x}{2 x^{2} \left (a^{4}-2 a^{2}+1\right )} \] Input:

int(acosh(b*x+a)/x^3,x)
 

Output:

( - acosh(a + b*x)*a**4 + 2*acosh(a + b*x)*a**2 - acosh(a + b*x) - 2*sqrt( 
 - a**2 + 1)*atan((sqrt(a**2 + 2*a*b*x + b**2*x**2 - 1) + b*x)/sqrt( - a** 
2 + 1))*a*b**2*x**2 + sqrt(a**2 + 2*a*b*x + b**2*x**2 - 1)*a**2*b*x - sqrt 
(a**2 + 2*a*b*x + b**2*x**2 - 1)*b*x)/(2*x**2*(a**4 - 2*a**2 + 1))