\(\int \frac {\text {arccosh}(a+b x)}{x^4} \, dx\) [8]

Optimal result

Integrand size = 10, antiderivative size = 154 \[ \int \frac {\text {arccosh}(a+b x)}{x^4} \, dx=\frac {b \sqrt {-1+a+b x} \sqrt {1+a+b x}}{6 \left (1-a^2\right ) x^2}+\frac {a b^2 \sqrt {-1+a+b x} \sqrt {1+a+b x}}{2 \left (1-a^2\right )^2 x}-\frac {\text {arccosh}(a+b x)}{3 x^3}-\frac {\left (1+2 a^2\right ) b^3 \arctan \left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {-1+a+b x}}\right )}{3 \left (1-a^2\right )^{5/2}} \] Output:

1/6*b*(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2)/(-a^2+1)/x^2+1/2*a*b^2*(b*x+a-1)^(1/ 
2)*(b*x+a+1)^(1/2)/(-a^2+1)^2/x-1/3*arccosh(b*x+a)/x^3-1/3*(2*a^2+1)*b^3*a 
rctan((1-a)^(1/2)*(b*x+a+1)^(1/2)/(1+a)^(1/2)/(b*x+a-1)^(1/2))/(-a^2+1)^(5 
/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.23 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.05 \[ \int \frac {\text {arccosh}(a+b x)}{x^4} \, dx=\frac {1}{6} \left (\frac {b \sqrt {-1+a+b x} \sqrt {1+a+b x} \left (1-a^2+3 a b x\right )}{\left (-1+a^2\right )^2 x^2}-\frac {2 \text {arccosh}(a+b x)}{x^3}-\frac {i \left (1+2 a^2\right ) b^3 \log \left (\frac {12 \left (1-a^2\right )^{3/2} \left (-i+i a^2+i a b x+\sqrt {1-a^2} \sqrt {-1+a+b x} \sqrt {1+a+b x}\right )}{b^3 \left (x+2 a^2 x\right )}\right )}{\left (1-a^2\right )^{5/2}}\right ) \] Input:

Integrate[ArcCosh[a + b*x]/x^4,x]
 

Output:

((b*Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x]*(1 - a^2 + 3*a*b*x))/((-1 + a^2)^ 
2*x^2) - (2*ArcCosh[a + b*x])/x^3 - (I*(1 + 2*a^2)*b^3*Log[(12*(1 - a^2)^( 
3/2)*(-I + I*a^2 + I*a*b*x + Sqrt[1 - a^2]*Sqrt[-1 + a + b*x]*Sqrt[1 + a + 
 b*x]))/(b^3*(x + 2*a^2*x))])/(1 - a^2)^(5/2))/6
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.14, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6411, 27, 6378, 114, 25, 168, 25, 27, 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.

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

\(\Big \downarrow \) 6411

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

\(\Big \downarrow \) 27

\(\displaystyle b^3 \int \frac {\text {arccosh}(a+b x)}{b^4 x^4}d(a+b x)\)

\(\Big \downarrow \) 6378

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 218

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

Input:

Int[ArcCosh[a + b*x]/x^4,x]
 

Output:

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

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 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 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(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] + S 
imp[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*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[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 = 0.08 (sec) , antiderivative size = 308, normalized size of antiderivative = 2.00

method	result	size
parts	\(-\frac {\operatorname {arccosh}\left (b x +a \right )}{3 x^{3}}-\frac {b \sqrt {b x +a -1}\, \sqrt {b x +a +1}\, \operatorname {csgn}\left (b \right )^{2} \left (2 \sqrt {a^{2}-1}\, \ln \left (\frac {2 a^{2}-2+2 a b x +2 \sqrt {a^{2}-1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}{x}\right ) a^{2} b^{2} x^{2}+\sqrt {a^{2}-1}\, \ln \left (\frac {2 a^{2}-2+2 a b x +2 \sqrt {a^{2}-1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}{x}\right ) b^{2} x^{2}-3 a^{3} b x \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+a^{4} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+3 a b x \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}-2 a^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\right )}{6 x^{2} \left (a^{2}-1\right )^{2} \left (1+a \right ) \left (a -1\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}\)	\(308\)
derivativedivides	\(b^{3} \left (-\frac {\operatorname {arccosh}\left (b x +a \right )}{3 b^{3} x^{3}}-\frac {\sqrt {b x +a +1}\, \sqrt {b x +a -1}\, \left (2 \sqrt {a^{2}-1}\, \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 a \left (b x +a \right )-2}{b x}\right ) a^{4}-4 \sqrt {a^{2}-1}\, \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 a \left (b x +a \right )-2}{b x}\right ) a^{3} \left (b x +a \right )+2 \sqrt {a^{2}-1}\, \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 a \left (b x +a \right )-2}{b x}\right ) a^{2} \left (b x +a \right )^{2}+4 \sqrt {\left (b x +a \right )^{2}-1}\, a^{4}-3 \sqrt {\left (b x +a \right )^{2}-1}\, a^{3} \left (b x +a \right )+\sqrt {a^{2}-1}\, \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 a \left (b x +a \right )-2}{b x}\right ) a^{2}-2 \sqrt {a^{2}-1}\, \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 a \left (b x +a \right )-2}{b x}\right ) a \left (b x +a \right )+\sqrt {a^{2}-1}\, \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 a \left (b x +a \right )-2}{b x}\right ) \left (b x +a \right )^{2}-5 a^{2} \sqrt {\left (b x +a \right )^{2}-1}+3 \left (b x +a \right ) a \sqrt {\left (b x +a \right )^{2}-1}+\sqrt {\left (b x +a \right )^{2}-1}\right )}{6 b^{2} x^{2} \left (a^{2}-1\right )^{2} \left (1+a \right ) \left (a -1\right ) \sqrt {\left (b x +a \right )^{2}-1}}\right )\)	\(467\)
default	\(b^{3} \left (-\frac {\operatorname {arccosh}\left (b x +a \right )}{3 b^{3} x^{3}}-\frac {\sqrt {b x +a +1}\, \sqrt {b x +a -1}\, \left (2 \sqrt {a^{2}-1}\, \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 a \left (b x +a \right )-2}{b x}\right ) a^{4}-4 \sqrt {a^{2}-1}\, \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 a \left (b x +a \right )-2}{b x}\right ) a^{3} \left (b x +a \right )+2 \sqrt {a^{2}-1}\, \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 a \left (b x +a \right )-2}{b x}\right ) a^{2} \left (b x +a \right )^{2}+4 \sqrt {\left (b x +a \right )^{2}-1}\, a^{4}-3 \sqrt {\left (b x +a \right )^{2}-1}\, a^{3} \left (b x +a \right )+\sqrt {a^{2}-1}\, \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 a \left (b x +a \right )-2}{b x}\right ) a^{2}-2 \sqrt {a^{2}-1}\, \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 a \left (b x +a \right )-2}{b x}\right ) a \left (b x +a \right )+\sqrt {a^{2}-1}\, \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 a \left (b x +a \right )-2}{b x}\right ) \left (b x +a \right )^{2}-5 a^{2} \sqrt {\left (b x +a \right )^{2}-1}+3 \left (b x +a \right ) a \sqrt {\left (b x +a \right )^{2}-1}+\sqrt {\left (b x +a \right )^{2}-1}\right )}{6 b^{2} x^{2} \left (a^{2}-1\right )^{2} \left (1+a \right ) \left (a -1\right ) \sqrt {\left (b x +a \right )^{2}-1}}\right )\)	\(467\)

Input:

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

Output:

-1/3*arccosh(b*x+a)/x^3-1/6*b*(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2)*csgn(b)^2*(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^2*b^2*x^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)*b^2*x^2-3*a^3*b*x*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+a 
^4*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+3*a*b*x*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)-2*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^2/(a^2-1 
)^2/(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. 275 vs. \(2 (124) = 248\).

Time = 0.14 (sec) , antiderivative size = 566, normalized size of antiderivative = 3.68 \[ \int \frac {\text {arccosh}(a+b x)}{x^4} \, dx=\left [\frac {{\left (2 \, a^{2} + 1\right )} \sqrt {a^{2} - 1} b^{3} x^{3} \log \left (\frac {a^{2} b x + a^{3} + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (a^{2} - \sqrt {a^{2} - 1} a - 1\right )} - {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1} - a}{x}\right ) + 3 \, {\left (a^{3} - a\right )} b^{3} x^{3} + 2 \, {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} x^{3} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - 2 \, {\left (a^{6} - 3 \, a^{4} - {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} x^{3} + 3 \, a^{2} - 1\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + {\left (3 \, {\left (a^{3} - a\right )} b^{2} x^{2} - {\left (a^{4} - 2 \, a^{2} + 1\right )} b x\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}}{6 \, {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} x^{3}}, \frac {2 \, {\left (2 \, a^{2} + 1\right )} \sqrt {-a^{2} + 1} b^{3} x^{3} \arctan \left (-\frac {\sqrt {-a^{2} + 1} b x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} \sqrt {-a^{2} + 1}}{a^{2} - 1}\right ) + 3 \, {\left (a^{3} - a\right )} b^{3} x^{3} + 2 \, {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} x^{3} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - 2 \, {\left (a^{6} - 3 \, a^{4} - {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} x^{3} + 3 \, a^{2} - 1\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + {\left (3 \, {\left (a^{3} - a\right )} b^{2} x^{2} - {\left (a^{4} - 2 \, a^{2} + 1\right )} b x\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}}{6 \, {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} x^{3}}\right ] \] Input:

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

Output:

[1/6*((2*a^2 + 1)*sqrt(a^2 - 1)*b^3*x^3*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) + 3*(a^3 - a)*b^3*x^3 + 2*(a^6 - 3*a^4 + 3*a^2 - 1)*x^3*l 
og(-b*x - a + sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)) - 2*(a^6 - 3*a^4 - (a^6 - 
 3*a^4 + 3*a^2 - 1)*x^3 + 3*a^2 - 1)*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x 
+ a^2 - 1)) + (3*(a^3 - a)*b^2*x^2 - (a^4 - 2*a^2 + 1)*b*x)*sqrt(b^2*x^2 + 
 2*a*b*x + a^2 - 1))/((a^6 - 3*a^4 + 3*a^2 - 1)*x^3), 1/6*(2*(2*a^2 + 1)*s 
qrt(-a^2 + 1)*b^3*x^3*arctan(-(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)) + 3*(a^3 - a)*b^3*x^3 + 2*(a^6 - 3* 
a^4 + 3*a^2 - 1)*x^3*log(-b*x - a + sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)) - 2 
*(a^6 - 3*a^4 - (a^6 - 3*a^4 + 3*a^2 - 1)*x^3 + 3*a^2 - 1)*log(b*x + a + s 
qrt(b^2*x^2 + 2*a*b*x + a^2 - 1)) + (3*(a^3 - a)*b^2*x^2 - (a^4 - 2*a^2 + 
1)*b*x)*sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1))/((a^6 - 3*a^4 + 3*a^2 - 1)*x^3) 
]
 

Sympy [F]

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

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

Output:

Integral(acosh(a + b*x)/x**4, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\text {arccosh}(a+b x)}{x^4} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(arccosh(b*x+a)/x^4,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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 340 vs. \(2 (124) = 248\).

Time = 0.16 (sec) , antiderivative size = 340, normalized size of antiderivative = 2.21 \[ \int \frac {\text {arccosh}(a+b x)}{x^4} \, dx=\frac {1}{3} \, b {\left (\frac {{\left (2 \, a^{2} b^{2} + b^{2}\right )} \arctan \left (-\frac {x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}}{\sqrt {-a^{2} + 1}}\right )}{{\left (a^{4} - 2 \, a^{2} + 1\right )} \sqrt {-a^{2} + 1}} - \frac {2 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )}^{3} a^{2} b^{2} - 6 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )} a^{4} b^{2} - 4 \, a^{5} b {\left | b \right |} + {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )}^{3} b^{2} + 7 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )} a^{2} b^{2} + 8 \, a^{3} b {\left | b \right |} - {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )} b^{2} - 4 \, a b {\left | b \right |}}{{\left (a^{4} - 2 \, a^{2} + 1\right )} {\left ({\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )}^{2} - a^{2} + 1\right )}^{2}}\right )} - \frac {\log \left (b x + a + \sqrt {{\left (b x + a\right )}^{2} - 1}\right )}{3 \, x^{3}} \] Input:

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

Output:

1/3*b*((2*a^2*b^2 + b^2)*arctan(-(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 
- 1))/sqrt(-a^2 + 1))/((a^4 - 2*a^2 + 1)*sqrt(-a^2 + 1)) - (2*(x*abs(b) - 
sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1))^3*a^2*b^2 - 6*(x*abs(b) - sqrt(b^2*x^2 
+ 2*a*b*x + a^2 - 1))*a^4*b^2 - 4*a^5*b*abs(b) + (x*abs(b) - sqrt(b^2*x^2 
+ 2*a*b*x + a^2 - 1))^3*b^2 + 7*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 - 
 1))*a^2*b^2 + 8*a^3*b*abs(b) - (x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 - 
 1))*b^2 - 4*a*b*abs(b))/((a^4 - 2*a^2 + 1)*((x*abs(b) - sqrt(b^2*x^2 + 2* 
a*b*x + a^2 - 1))^2 - a^2 + 1)^2)) - 1/3*log(b*x + a + sqrt((b*x + a)^2 - 
1))/x^3
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 332, normalized size of antiderivative = 2.16 \[ \int \frac {\text {arccosh}(a+b x)}{x^4} \, dx=\frac {-4 \mathit {acosh} \left (b x +a \right ) a^{7}+12 \mathit {acosh} \left (b x +a \right ) a^{5}-12 \mathit {acosh} \left (b x +a \right ) a^{3}+4 \mathit {acosh} \left (b x +a \right ) a +8 \sqrt {-a^{2}+1}\, \mathit {atan} \left (\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+b x}{\sqrt {-a^{2}+1}}\right ) a^{3} b^{3} x^{3}+4 \sqrt {-a^{2}+1}\, \mathit {atan} \left (\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+b x}{\sqrt {-a^{2}+1}}\right ) a \,b^{3} x^{3}+2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, a^{5} b x -6 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, a^{4} b^{2} x^{2}-4 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, a^{3} b x +6 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, a^{2} b^{2} x^{2}+2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, a b x +4 a^{4} b^{3} x^{3}-5 a^{2} b^{3} x^{3}+b^{3} x^{3}}{12 a \,x^{3} \left (a^{6}-3 a^{4}+3 a^{2}-1\right )} \] Input:

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

Output:

( - 4*acosh(a + b*x)*a**7 + 12*acosh(a + b*x)*a**5 - 12*acosh(a + b*x)*a** 
3 + 4*acosh(a + b*x)*a + 8*sqrt( - a**2 + 1)*atan((sqrt(a**2 + 2*a*b*x + b 
**2*x**2 - 1) + b*x)/sqrt( - a**2 + 1))*a**3*b**3*x**3 + 4*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**3*x**3 + 2*sqrt(a**2 + 2*a*b*x + b**2*x**2 - 1)*a**5*b*x - 6*sqrt(a**2 
+ 2*a*b*x + b**2*x**2 - 1)*a**4*b**2*x**2 - 4*sqrt(a**2 + 2*a*b*x + b**2*x 
**2 - 1)*a**3*b*x + 6*sqrt(a**2 + 2*a*b*x + b**2*x**2 - 1)*a**2*b**2*x**2 
+ 2*sqrt(a**2 + 2*a*b*x + b**2*x**2 - 1)*a*b*x + 4*a**4*b**3*x**3 - 5*a**2 
*b**3*x**3 + b**3*x**3)/(12*a*x**3*(a**6 - 3*a**4 + 3*a**2 - 1))