\(\int \frac {a+b \text {arccosh}(c x)}{x^4 (d+e x^2)} \, dx\) [395]

Optimal result

Integrand size = 21, antiderivative size = 624 \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d+e x^2\right )} \, dx=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d x^2}-\frac {a+b \text {arccosh}(c x)}{3 d x^3}+\frac {e (a+b \text {arccosh}(c x))}{d^2 x}+\frac {b c^3 \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{6 d}-\frac {b c e \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d^2}+\frac {e^{3/2} (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 (-d)^{5/2}}-\frac {e^{3/2} (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 (-d)^{5/2}}+\frac {e^{3/2} (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 (-d)^{5/2}}-\frac {e^{3/2} (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 (-d)^{5/2}}-\frac {b e^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 (-d)^{5/2}}+\frac {b e^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 (-d)^{5/2}}-\frac {b e^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 (-d)^{5/2}}+\frac {b e^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 (-d)^{5/2}} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 1.03 (sec) , antiderivative size = 657, normalized size of antiderivative = 1.05 \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d+e x^2\right )} \, dx=\frac {1}{6} \left (-\frac {2 a}{d x^3}+\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{d x^2}-\frac {2 b \text {arccosh}(c x)}{d x^3}+\frac {6 e (a+b \text {arccosh}(c x))}{d^2 x}+\frac {b c^3 \sqrt {-1+c^2 x^2} \arctan \left (\sqrt {-1+c^2 x^2}\right )}{d \sqrt {-1+c x} \sqrt {1+c x}}-\frac {6 b c e \sqrt {-1+c^2 x^2} \arctan \left (\sqrt {-1+c^2 x^2}\right )}{d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 e^{3/2} (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{(-d)^{5/2}}+\frac {3 e^{3/2} (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{(-d)^{5/2}}+\frac {3 e^{3/2} (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{(-d)^{5/2}}-\frac {3 e^{3/2} (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{(-d)^{5/2}}+\frac {3 b e^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{(-d)^{5/2}}-\frac {3 b e^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{(-d)^{5/2}}-\frac {3 b e^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{(-d)^{5/2}}+\frac {3 b e^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{(-d)^{5/2}}\right ) \] Input:

Integrate[(a + b*ArcCosh[c*x])/(x^4*(d + e*x^2)),x]
 

Output:

((-2*a)/(d*x^3) + (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(d*x^2) - (2*b*ArcCos 
h[c*x])/(d*x^3) + (6*e*(a + b*ArcCosh[c*x]))/(d^2*x) + (b*c^3*Sqrt[-1 + c^ 
2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(d*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (6*b 
*c*e*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(d^2*Sqrt[-1 + c*x]*Sq 
rt[1 + c*x]) - (3*e^(3/2)*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[ 
c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(-d)^(5/2) + (3*e^(3/2)*(a + b*A 
rcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(-(c*Sqrt[-d]) + Sqrt[-(c^2* 
d) - e])])/(-d)^(5/2) + (3*e^(3/2)*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E 
^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(-d)^(5/2) - (3*e^(3/2) 
*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[ 
-(c^2*d) - e])])/(-d)^(5/2) + (3*b*e^(3/2)*PolyLog[2, (Sqrt[e]*E^ArcCosh[c 
*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(-d)^(5/2) - (3*b*e^(3/2)*PolyLog 
[2, (Sqrt[e]*E^ArcCosh[c*x])/(-(c*Sqrt[-d]) + Sqrt[-(c^2*d) - e])])/(-d)^( 
5/2) - (3*b*e^(3/2)*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sq 
rt[-(c^2*d) - e]))])/(-d)^(5/2) + (3*b*e^(3/2)*PolyLog[2, (Sqrt[e]*E^ArcCo 
sh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(-d)^(5/2))/6
 

Rubi [A] (verified)

Time = 1.83 (sec) , antiderivative size = 624, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {6374, 2009}

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.

\(\displaystyle \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d+e x^2\right )} \, dx\)

\(\Big \downarrow \) 6374

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

\(\Big \downarrow \) 2009

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

Input:

Int[(a + b*ArcCosh[c*x])/(x^4*(d + e*x^2)),x]
 

Output:

(b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*d*x^2) - (a + b*ArcCosh[c*x])/(3*d*x 
^3) + (e*(a + b*ArcCosh[c*x]))/(d^2*x) + (b*c^3*ArcTan[Sqrt[-1 + c*x]*Sqrt 
[1 + c*x]])/(6*d) - (b*c*e*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/d^2 + (e^ 
(3/2)*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - 
Sqrt[-(c^2*d) - e])])/(2*(-d)^(5/2)) - (e^(3/2)*(a + b*ArcCosh[c*x])*Log[1 
 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*(-d)^(5 
/2)) + (e^(3/2)*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*S 
qrt[-d] + Sqrt[-(c^2*d) - e])])/(2*(-d)^(5/2)) - (e^(3/2)*(a + b*ArcCosh[c 
*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/ 
(2*(-d)^(5/2)) - (b*e^(3/2)*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[ 
-d] - Sqrt[-(c^2*d) - e]))])/(2*(-d)^(5/2)) + (b*e^(3/2)*PolyLog[2, (Sqrt[ 
e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*(-d)^(5/2)) - (b 
*e^(3/2)*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) 
 - e]))])/(2*(-d)^(5/2)) + (b*e^(3/2)*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/ 
(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*(-d)^(5/2))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6374

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e 
_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcCosh[c*x])^n, 
 (f*x)^m*(d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[c^2*d 
 + e, 0] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 12.45 (sec) , antiderivative size = 424, normalized size of antiderivative = 0.68

method	result	size
parts	\(a \left (\frac {e^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{d^{2} \sqrt {d e}}-\frac {1}{3 d \,x^{3}}+\frac {e}{d^{2} x}\right )+\frac {b \left (8 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{7} d^{2} x^{3}+4 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} d^{2} x -48 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{5} d e \,x^{3}-8 c^{4} d^{2} \operatorname {arccosh}\left (c x \right )+24 \,\operatorname {arccosh}\left (c x \right ) c^{4} d e \,x^{2}+3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (4 \textit {\_R1}^{2} c^{2} d +\textit {\_R1}^{2} e +e \right ) \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right ) e^{2} c^{3} x^{3}-3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right ) e^{2} c^{3} x^{3}\right )}{24 c^{4} d^{3} x^{3}}\)	\(424\)
derivativedivides	\(c^{3} \left (-\frac {a}{3 d \,c^{3} x^{3}}+\frac {a e}{c^{3} d^{2} x}+\frac {a \,e^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{c^{3} d^{2} \sqrt {d e}}+\frac {b \left (8 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{7} d^{2} x^{3}+4 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} d^{2} x -48 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{5} d e \,x^{3}-8 c^{4} d^{2} \operatorname {arccosh}\left (c x \right )+24 \,\operatorname {arccosh}\left (c x \right ) c^{4} d e \,x^{2}+3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (4 \textit {\_R1}^{2} c^{2} d +\textit {\_R1}^{2} e +e \right ) \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right ) e^{2} c^{3} x^{3}-3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right ) e^{2} c^{3} x^{3}\right )}{24 c^{7} d^{3} x^{3}}\right )\)	\(437\)
default	\(c^{3} \left (-\frac {a}{3 d \,c^{3} x^{3}}+\frac {a e}{c^{3} d^{2} x}+\frac {a \,e^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{c^{3} d^{2} \sqrt {d e}}+\frac {b \left (8 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{7} d^{2} x^{3}+4 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} d^{2} x -48 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{5} d e \,x^{3}-8 c^{4} d^{2} \operatorname {arccosh}\left (c x \right )+24 \,\operatorname {arccosh}\left (c x \right ) c^{4} d e \,x^{2}+3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (4 \textit {\_R1}^{2} c^{2} d +\textit {\_R1}^{2} e +e \right ) \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right ) e^{2} c^{3} x^{3}-3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right ) e^{2} c^{3} x^{3}\right )}{24 c^{7} d^{3} x^{3}}\right )\)	\(437\)

Input:

int((a+b*arccosh(c*x))/x^4/(e*x^2+d),x,method=_RETURNVERBOSE)
 

Output:

a*(e^2/d^2/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))-1/3/d/x^3+e/d^2/x)+1/24*b/c 
^4*(8*arctan(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*c^7*d^2*x^3+4*(c*x+1)^(1/2)* 
(c*x-1)^(1/2)*c^5*d^2*x-48*arctan(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*c^5*d*e 
*x^3-8*c^4*d^2*arccosh(c*x)+24*arccosh(c*x)*c^4*d*e*x^2+3*sum((4*_R1^2*c^2 
*d+_R1^2*e+e)/_R1/(_R1^2*e+2*c^2*d+e)*(arccosh(c*x)*ln((_R1-c*x-(c*x-1)^(1 
/2)*(c*x+1)^(1/2))/_R1)+dilog((_R1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R1)), 
_R1=RootOf(e*_Z^4+(4*c^2*d+2*e)*_Z^2+e))*e^2*c^3*x^3-3*sum((_R1^2*e+4*c^2* 
d+e)/_R1/(_R1^2*e+2*c^2*d+e)*(arccosh(c*x)*ln((_R1-c*x-(c*x-1)^(1/2)*(c*x+ 
1)^(1/2))/_R1)+dilog((_R1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R1)),_R1=RootO 
f(e*_Z^4+(4*c^2*d+2*e)*_Z^2+e))*e^2*c^3*x^3)/d^3/x^3
 

Fricas [F]

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

integrate((a+b*arccosh(c*x))/x^4/(e*x^2+d),x, algorithm="fricas")
 

Output:

integral((b*arccosh(c*x) + a)/(e*x^6 + d*x^4), x)
 

Sympy [F]

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

integrate((a+b*acosh(c*x))/x**4/(e*x**2+d),x)
 

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d+e x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a+b*arccosh(c*x))/x^4/(e*x^2+d),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(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F]

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

integrate((a+b*arccosh(c*x))/x^4/(e*x^2+d),x, algorithm="giac")
 

Output:

integrate((b*arccosh(c*x) + a)/((e*x^2 + d)*x^4), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

int((a + b*acosh(c*x))/(x^4*(d + e*x^2)),x)
 

Output:

int((a + b*acosh(c*x))/(x^4*(d + e*x^2)), x)
 

Reduce [F]

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

int((a+b*acosh(c*x))/x^4/(e*x^2+d),x)
 

Output:

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