\(\int \frac {x^4 (a+b \text {arccosh}(c x))}{d+e x^2} \, dx\) [387]

Optimal result

Integrand size = 21, antiderivative size = 627 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{d+e x^2} \, dx=-\frac {a d x}{e^2}+\frac {b d \sqrt {-1+c x} \sqrt {1+c x}}{c e^2}-\frac {2 b \sqrt {-1+c x} \sqrt {1+c x}}{9 c^3 e}-\frac {b x^2 \sqrt {-1+c x} \sqrt {1+c x}}{9 c e}-\frac {b d x \text {arccosh}(c x)}{e^2}+\frac {x^3 (a+b \text {arccosh}(c x))}{3 e}+\frac {(-d)^{3/2} (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^{5/2}}-\frac {(-d)^{3/2} (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^{5/2}}+\frac {(-d)^{3/2} (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^{5/2}}-\frac {(-d)^{3/2} (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^{5/2}}-\frac {b (-d)^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^{5/2}}+\frac {b (-d)^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^{5/2}}-\frac {b (-d)^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^{5/2}}+\frac {b (-d)^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^{5/2}} \] Output:

-a*d*x/e^2+b*d*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c/e^2-2/9*b*(c*x-1)^(1/2)*(c*x+ 
1)^(1/2)/c^3/e-1/9*b*x^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c/e-b*d*x*arccosh(c*x 
)/e^2+1/3*x^3*(a+b*arccosh(c*x))/e+1/2*(-d)^(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))) 
/e^(5/2)-1/2*(-d)^(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)))/e^(5/2)+1/2*(-d)^(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)))/e^(5/2)-1/2*(-d)^(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)))/e^( 
5/2)-1/2*b*(-d)^(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)))/e^(5/2)+1/2*b*(-d)^(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)))/e^(5 
/2)-1/2*b*(-d)^(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)))/e^(5/2)+1/2*b*(-d)^(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)))/e^(5/ 
2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.79 (sec) , antiderivative size = 524, normalized size of antiderivative = 0.84 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{d+e x^2} \, dx=-\frac {a d x}{e^2}+\frac {a x^3}{3 e}+\frac {a d^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{5/2}}+\frac {b \left (\frac {4 d \sqrt {e} \left (\sqrt {\frac {-1+c x}{1+c x}} (1+c x)-c x \text {arccosh}(c x)\right )}{c}-\frac {4 e^{3/2} \left (\sqrt {-1+c x} \sqrt {1+c x} \left (2+c^2 x^2\right )-3 c^3 x^3 \text {arccosh}(c x)\right )}{9 c^3}-i d^{3/2} \left (\text {arccosh}(c x) \left (-\text {arccosh}(c x)+2 \left (\log \left (1+\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+\log \left (1+\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )+i d^{3/2} \left (\text {arccosh}(c x) \left (-\text {arccosh}(c x)+2 \left (\log \left (1+\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+\log \left (1-\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )}{4 e^{5/2}} \] Input:

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

Output:

-((a*d*x)/e^2) + (a*x^3)/(3*e) + (a*d^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/e 
^(5/2) + (b*((4*d*Sqrt[e]*(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) - c*x*ArcC 
osh[c*x]))/c - (4*e^(3/2)*(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(2 + c^2*x^2) - 3* 
c^3*x^3*ArcCosh[c*x]))/(9*c^3) - I*d^(3/2)*(ArcCosh[c*x]*(-ArcCosh[c*x] + 
2*(Log[1 + (I*Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[d] - Sqrt[c^2*d + e])] + Log 
[1 + (I*Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyL 
og[2, (I*Sqrt[e]*E^ArcCosh[c*x])/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + 2*Pol 
yLog[2, ((-I)*Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[d] + Sqrt[c^2*d + e])]) + I* 
d^(3/2)*(ArcCosh[c*x]*(-ArcCosh[c*x] + 2*(Log[1 + (I*Sqrt[e]*E^ArcCosh[c*x 
])/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + Log[1 - (I*Sqrt[e]*E^ArcCosh[c*x])/ 
(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog[2, (I*Sqrt[e]*E^ArcCosh[c*x]) 
/(c*Sqrt[d] - Sqrt[c^2*d + e])] + 2*PolyLog[2, (I*Sqrt[e]*E^ArcCosh[c*x])/ 
(c*Sqrt[d] + Sqrt[c^2*d + e])])))/(4*e^(5/2))
 

Rubi [A] (verified)

Time = 1.48 (sec) , antiderivative size = 627, 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.

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

\(\Big \downarrow \) 6374

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(-d)^{3/2} (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{2 e^{5/2}}-\frac {(-d)^{3/2} (a+b \text {arccosh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}+1\right )}{2 e^{5/2}}+\frac {(-d)^{3/2} (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{2 e^{5/2}}-\frac {(-d)^{3/2} (a+b \text {arccosh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}+1\right )}{2 e^{5/2}}+\frac {x^3 (a+b \text {arccosh}(c x))}{3 e}-\frac {a d x}{e^2}-\frac {b (-d)^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 e^{5/2}}+\frac {b (-d)^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 e^{5/2}}-\frac {b (-d)^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 e^{5/2}}+\frac {b (-d)^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 e^{5/2}}-\frac {b d x \text {arccosh}(c x)}{e^2}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1}}{9 c^3 e}+\frac {b d \sqrt {c x-1} \sqrt {c x+1}}{c e^2}-\frac {b x^2 \sqrt {c x-1} \sqrt {c x+1}}{9 c e}\)

Input:

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

Output:

-((a*d*x)/e^2) + (b*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(c*e^2) - (2*b*Sqrt[-1 
 + c*x]*Sqrt[1 + c*x])/(9*c^3*e) - (b*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(9 
*c*e) - (b*d*x*ArcCosh[c*x])/e^2 + (x^3*(a + b*ArcCosh[c*x]))/(3*e) + ((-d 
)^(3/2)*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] 
- Sqrt[-(c^2*d) - e])])/(2*e^(5/2)) - ((-d)^(3/2)*(a + b*ArcCosh[c*x])*Log 
[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*e^(5/ 
2)) + ((-d)^(3/2)*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c 
*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*e^(5/2)) - ((-d)^(3/2)*(a + b*ArcCosh 
[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])] 
)/(2*e^(5/2)) - (b*(-d)^(3/2)*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqr 
t[-d] - Sqrt[-(c^2*d) - e]))])/(2*e^(5/2)) + (b*(-d)^(3/2)*PolyLog[2, (Sqr 
t[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*e^(5/2)) - (b* 
(-d)^(3/2)*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2* 
d) - e]))])/(2*e^(5/2)) + (b*(-d)^(3/2)*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x] 
)/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*e^(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] (verified)

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

Time = 23.09 (sec) , antiderivative size = 364, normalized size of antiderivative = 0.58

method	result	size
parts	\(\frac {a \,x^{3}}{3 e}-\frac {a d x}{e^{2}}+\frac {a \,d^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{e^{2} \sqrt {d e}}+\frac {b c \,d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{2 e^{2}}-\frac {b c \,d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right )}{2 e^{2}}+\frac {b \,\operatorname {arccosh}\left (c x \right ) x^{3}}{3 e}-\frac {2 b \sqrt {c x -1}\, \sqrt {c x +1}}{9 c^{3} e}-\frac {b \,x^{2} \sqrt {c x -1}\, \sqrt {c x +1}}{9 c e}-\frac {b d x \,\operatorname {arccosh}\left (c x \right )}{e^{2}}+\frac {b d \sqrt {c x -1}\, \sqrt {c x +1}}{c \,e^{2}}\)	\(364\)
derivativedivides	\(\frac {-\frac {a \,c^{5} d x}{e^{2}}+\frac {a \,c^{5} x^{3}}{3 e}+\frac {a \,c^{5} d^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{e^{2} \sqrt {d e}}-\frac {b \,c^{5} \operatorname {arccosh}\left (c x \right ) d x}{e^{2}}+\frac {b \,c^{4} \sqrt {c x +1}\, \sqrt {c x -1}\, d}{e^{2}}+\frac {b \,c^{5} \operatorname {arccosh}\left (c x \right ) x^{3}}{3 e}-\frac {2 b \,c^{2} \sqrt {c x -1}\, \sqrt {c x +1}}{9 e}+\frac {b \,c^{6} d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{2 e^{2}}-\frac {b \,c^{6} d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right )}{2 e^{2}}-\frac {b \,c^{4} \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2}}{9 e}}{c^{5}}\)	\(387\)
default	\(\frac {-\frac {a \,c^{5} d x}{e^{2}}+\frac {a \,c^{5} x^{3}}{3 e}+\frac {a \,c^{5} d^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{e^{2} \sqrt {d e}}-\frac {b \,c^{5} \operatorname {arccosh}\left (c x \right ) d x}{e^{2}}+\frac {b \,c^{4} \sqrt {c x +1}\, \sqrt {c x -1}\, d}{e^{2}}+\frac {b \,c^{5} \operatorname {arccosh}\left (c x \right ) x^{3}}{3 e}-\frac {2 b \,c^{2} \sqrt {c x -1}\, \sqrt {c x +1}}{9 e}+\frac {b \,c^{6} d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{2 e^{2}}-\frac {b \,c^{6} d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right )}{2 e^{2}}-\frac {b \,c^{4} \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2}}{9 e}}{c^{5}}\)	\(387\)

Input:

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

Output:

1/3*a/e*x^3-a*d*x/e^2+a*d^2/e^2/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))+1/2*b* 
c/e^2*d^2*sum(_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))-1/2*b*c/e^2*d^2*sum(1/_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)+d 
ilog((_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))+1/3*b/e*arccosh(c*x)*x^3-2/9*b*(c*x-1)^(1/2)*(c*x+1)^(1/2) 
/c^3/e-1/9*b*x^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c/e-b*d*x*arccosh(c*x)/e^2+b* 
d*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c/e^2
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x^4*(a+b*arccosh(c*x))/(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]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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