Integrand size = 21, antiderivative size = 627 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{d+e x^2} \, dx=-\frac {a d x}{e^2}+\frac {b d \sqrt {-1+c x} \sqrt {1+c x}}{c e^2}-\frac {2 b \sqrt {-1+c x} \sqrt {1+c x}}{9 c^3 e}-\frac {b x^2 \sqrt {-1+c x} \sqrt {1+c x}}{9 c e}-\frac {b d x \text {arccosh}(c x)}{e^2}+\frac {x^3 (a+b \text {arccosh}(c x))}{3 e}+\frac {(-d)^{3/2} (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^{5/2}}-\frac {(-d)^{3/2} (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^{5/2}}+\frac {(-d)^{3/2} (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^{5/2}}-\frac {(-d)^{3/2} (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^{5/2}}-\frac {b (-d)^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^{5/2}}+\frac {b (-d)^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^{5/2}}-\frac {b (-d)^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^{5/2}}+\frac {b (-d)^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^{5/2}} \]
-a*d*x/e^2-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-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e^(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+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e^(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-(c*x+(c*x-1)^(1/2)*(c*x+ 1)^(1/2))*e^(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+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^ (1/2)+(-c^2*d-e)^(1/2)))/e^(5/2)-1/2*b*(-d)^(3/2)*polylog(2,-(c*x+(c*x-1)^ (1/2)*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)-(-c^2*d-e)^(1/2)))/e^(5/2)+1/2* b*(-d)^(3/2)*polylog(2,(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^( 1/2)-(-c^2*d-e)^(1/2)))/e^(5/2)-1/2*b*(-d)^(3/2)*polylog(2,-(c*x+(c*x-1)^( 1/2)*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)+(-c^2*d-e)^(1/2)))/e^(5/2)+1/2*b *(-d)^(3/2)*polylog(2,(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1 /2)+(-c^2*d-e)^(1/2)))/e^(5/2)+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
Result contains complex when optimal does not.
Time = 0.86 (sec) , antiderivative size = 524, normalized size of antiderivative = 0.84 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{d+e x^2} \, dx=-\frac {a d x}{e^2}+\frac {a x^3}{3 e}+\frac {a d^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{5/2}}+\frac {b \left (\frac {4 d \sqrt {e} \left (\sqrt {\frac {-1+c x}{1+c x}} (1+c x)-c x \text {arccosh}(c x)\right )}{c}-\frac {4 e^{3/2} \left (\sqrt {-1+c x} \sqrt {1+c x} \left (2+c^2 x^2\right )-3 c^3 x^3 \text {arccosh}(c x)\right )}{9 c^3}-i d^{3/2} \left (\text {arccosh}(c x) \left (-\text {arccosh}(c x)+2 \left (\log \left (1+\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+\log \left (1+\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )+i d^{3/2} \left (\text {arccosh}(c x) \left (-\text {arccosh}(c x)+2 \left (\log \left (1+\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+\log \left (1-\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )}{4 e^{5/2}} \]
-((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))
Time = 1.45 (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 |
| \(\displaystyle \frac {(-d)^{3/2} (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{2 e^{5/2}}-\frac {(-d)^{3/2} (a+b \text {arccosh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}+1\right )}{2 e^{5/2}}+\frac {(-d)^{3/2} (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{2 e^{5/2}}-\frac {(-d)^{3/2} (a+b \text {arccosh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}+1\right )}{2 e^{5/2}}+\frac {x^3 (a+b \text {arccosh}(c x))}{3 e}-\frac {a d x}{e^2}-\frac {b (-d)^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 e^{5/2}}+\frac {b (-d)^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 e^{5/2}}-\frac {b (-d)^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 e^{5/2}}+\frac {b (-d)^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 e^{5/2}}-\frac {b d x \text {arccosh}(c x)}{e^2}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1}}{9 c^3 e}+\frac {b d \sqrt {c x-1} \sqrt {c x+1}}{c e^2}-\frac {b x^2 \sqrt {c x-1} \sqrt {c x+1}}{9 c e}\) |
-((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))
3.5.89.3.1 Defintions of rubi rules used
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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 45.91 (sec) , antiderivative size = 364, normalized size of antiderivative = 0.58
| method | result | size |
| parts | \(\frac {a \,x^{3}}{3 e}-\frac {a d x}{e^{2}}+\frac {a \,d^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{e^{2} \sqrt {d e}}-\frac {b \,x^{2} \sqrt {c x -1}\, \sqrt {c x +1}}{9 c e}-\frac {b d x \,\operatorname {arccosh}\left (c x \right )}{e^{2}}+\frac {b d \sqrt {c x -1}\, \sqrt {c x +1}}{c \,e^{2}}+\frac {b c \,d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{2 e^{2}}-\frac {b c \,d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right )}{2 e^{2}}+\frac {b \,\operatorname {arccosh}\left (c x \right ) x^{3}}{3 e}-\frac {2 b \sqrt {c x -1}\, \sqrt {c x +1}}{9 c^{3} e}\) | \(364\) |
| derivativedivides | \(\frac {-\frac {a \,c^{5} d x}{e^{2}}+\frac {a \,c^{5} x^{3}}{3 e}+\frac {a \,c^{5} d^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{e^{2} \sqrt {d e}}+\frac {b \,c^{4} \sqrt {c x +1}\, \sqrt {c x -1}\, d}{e^{2}}-\frac {b \,c^{5} \operatorname {arccosh}\left (c x \right ) d x}{e^{2}}+\frac {b \,c^{6} d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{2 e^{2}}-\frac {b \,c^{6} d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right )}{2 e^{2}}-\frac {b \,c^{4} \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2}}{9 e}+\frac {b \,c^{5} \operatorname {arccosh}\left (c x \right ) x^{3}}{3 e}-\frac {2 b \,c^{2} \sqrt {c x -1}\, \sqrt {c x +1}}{9 e}}{c^{5}}\) | \(387\) |
| default | \(\frac {-\frac {a \,c^{5} d x}{e^{2}}+\frac {a \,c^{5} x^{3}}{3 e}+\frac {a \,c^{5} d^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{e^{2} \sqrt {d e}}+\frac {b \,c^{4} \sqrt {c x +1}\, \sqrt {c x -1}\, d}{e^{2}}-\frac {b \,c^{5} \operatorname {arccosh}\left (c x \right ) d x}{e^{2}}+\frac {b \,c^{6} d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{2 e^{2}}-\frac {b \,c^{6} d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right )}{2 e^{2}}-\frac {b \,c^{4} \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2}}{9 e}+\frac {b \,c^{5} \operatorname {arccosh}\left (c x \right ) x^{3}}{3 e}-\frac {2 b \,c^{2} \sqrt {c x -1}\, \sqrt {c x +1}}{9 e}}{c^{5}}\) | \(387\) |
1/3*a*x^3/e-a*d*x/e^2+a*d^2/e^2/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))-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+1/2*b*c/e^2*d^2*sum(_R1/(_R1^2*e+2*c^2*d+e)*(arccos h(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)+dilog((_R1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R 1)),_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
\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{d+e x^2} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}}{e x^{2} + d} \,d x } \]
\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{d+e x^2} \, dx=\int \frac {x^{4} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{d + e x^{2}}\, dx \]
Exception generated. \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{d+e x^2} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{d+e x^2} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}}{e x^{2} + d} \,d x } \]
Timed out. \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{d+e x^2} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{e\,x^2+d} \,d x \]