Integrand size = 21, antiderivative size = 634 \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d+e x^2\right )^2} \, dx=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d^2 x}-\frac {a+b \text {arccosh}(c x)}{2 d^2 x^2}-\frac {e (a+b \text {arccosh}(c x))}{2 d^2 \left (d+e x^2\right )}-\frac {2 e (a+b \text {arccosh}(c x))^2}{b d^3}+\frac {b c e \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{2 d^{5/2} \sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 e (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{d^3}+\frac {e (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{d^3}+\frac {e (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{d^3}+\frac {e (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{d^3}+\frac {e (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{d^3}+\frac {b e \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )}{d^3}+\frac {b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{d^3}+\frac {b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{d^3}+\frac {b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{d^3}+\frac {b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{d^3} \]
1/2*(-a-b*arccosh(c*x))/d^2/x^2-1/2*e*(a+b*arccosh(c*x))/d^2/(e*x^2+d)-2*e *(a+b*arccosh(c*x))^2/b/d^3-2*e*(a+b*arccosh(c*x))*ln(1+1/(c*x+(c*x-1)^(1/ 2)*(c*x+1)^(1/2))^2)/d^3+e*(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)))/d^3+e*(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)))/d^3+e*(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)))/d^3+e*(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)))/d^3 +b*e*polylog(2,-1/(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)/d^3+b*e*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))) /d^3+b*e*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)))/d^3+b*e*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)))/d^3+b*e*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)))/d^3+1/2*b*c*(c* x-1)^(1/2)*(c*x+1)^(1/2)/d^2/x+1/2*b*c*e*arctanh(x*(c^2*d+e)^(1/2)/d^(1/2) /(c^2*x^2-1)^(1/2))*(c^2*x^2-1)^(1/2)/d^(5/2)/(c^2*d+e)^(1/2)/(c*x-1)^(1/2 )/(c*x+1)^(1/2)
Result contains complex when optimal does not.
Time = 1.55 (sec) , antiderivative size = 792, normalized size of antiderivative = 1.25 \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d+e x^2\right )^2} \, dx=\frac {-\frac {2 a d}{x^2}-\frac {2 a d e}{d+e x^2}-8 a e \log (x)+4 a e \log \left (d+e x^2\right )+b \left (\frac {2 d \left (c x \sqrt {-1+c x} \sqrt {1+c x}-\text {arccosh}(c x)\right )}{x^2}-4 e \text {arccosh}(c x)^2-4 e \text {arccosh}(c x) \left (\text {arccosh}(c x)+2 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )+4 e \text {arccosh}(c x) \left (\log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+\log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )+4 e \text {arccosh}(c x) \left (\log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}-\sqrt {-c^2 d-e}}\right )+\log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )+i \sqrt {d} e \left (\frac {\text {arccosh}(c x)}{-i \sqrt {d}+\sqrt {e} x}+\frac {c \log \left (\frac {2 e \left (i \sqrt {e}+c^2 \sqrt {d} x-i \sqrt {-c^2 d-e} \sqrt {-1+c x} \sqrt {1+c x}\right )}{c \sqrt {-c^2 d-e} \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{\sqrt {-c^2 d-e}}\right )+i \sqrt {d} e \left (-\frac {\text {arccosh}(c x)}{i \sqrt {d}+\sqrt {e} x}-\frac {c \log \left (\frac {2 e \left (-\sqrt {e}-i c^2 \sqrt {d} x+\sqrt {-c^2 d-e} \sqrt {-1+c x} \sqrt {1+c x}\right )}{c \sqrt {-c^2 d-e} \left (i \sqrt {d}+\sqrt {e} x\right )}\right )}{\sqrt {-c^2 d-e}}\right )+4 e \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )+4 e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}-\sqrt {-c^2 d-e}}\right )+4 e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+4 e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+4 e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )}{4 d^3} \]
((-2*a*d)/x^2 - (2*a*d*e)/(d + e*x^2) - 8*a*e*Log[x] + 4*a*e*Log[d + e*x^2 ] + b*((2*d*(c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - ArcCosh[c*x]))/x^2 - 4*e*A rcCosh[c*x]^2 - 4*e*ArcCosh[c*x]*(ArcCosh[c*x] + 2*Log[1 + E^(-2*ArcCosh[c *x])]) + 4*e*ArcCosh[c*x]*(Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/((-I)*c*Sqrt[d ] + Sqrt[-(c^2*d) - e])] + Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])]) + 4*e*ArcCosh[c*x]*(Log[1 + (Sqrt[e]*E^ArcCosh[c*x] )/(I*c*Sqrt[d] - Sqrt[-(c^2*d) - e])] + Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/( I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])]) + I*Sqrt[d]*e*(ArcCosh[c*x]/((-I)*Sqrt [d] + Sqrt[e]*x) + (c*Log[(2*e*(I*Sqrt[e] + c^2*Sqrt[d]*x - I*Sqrt[-(c^2*d ) - e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c*Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*S qrt[e]*x))])/Sqrt[-(c^2*d) - e]) + I*Sqrt[d]*e*(-(ArcCosh[c*x]/(I*Sqrt[d] + Sqrt[e]*x)) - (c*Log[(2*e*(-Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[-(c^2*d) - e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c*Sqrt[-(c^2*d) - e]*(I*Sqrt[d] + Sqrt[ e]*x))])/Sqrt[-(c^2*d) - e]) + 4*e*PolyLog[2, -E^(-2*ArcCosh[c*x])] + 4*e* PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] - Sqrt[-(c^2*d) - e])] + 4*e*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e])] + 4*e*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2 *d) - e]))] + 4*e*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[ -(c^2*d) - e])]))/(4*d^3)
Time = 1.61 (sec) , antiderivative size = 634, 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.
| \(\displaystyle \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d+e x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 6374 |
| \(\displaystyle \int \left (\frac {2 e^2 x (a+b \text {arccosh}(c x))}{d^3 \left (d+e x^2\right )}-\frac {2 e (a+b \text {arccosh}(c x))}{d^3 x}+\frac {e^2 x (a+b \text {arccosh}(c x))}{d^2 \left (d+e x^2\right )^2}+\frac {a+b \text {arccosh}(c x)}{d^2 x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{d^3}+\frac {e (a+b \text {arccosh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}+1\right )}{d^3}+\frac {e (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{d^3}+\frac {e (a+b \text {arccosh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}+1\right )}{d^3}-\frac {2 e (a+b \text {arccosh}(c x))^2}{b d^3}-\frac {2 e \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))}{d^3}-\frac {e (a+b \text {arccosh}(c x))}{2 d^2 \left (d+e x^2\right )}-\frac {a+b \text {arccosh}(c x)}{2 d^2 x^2}+\frac {b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{d^3}+\frac {b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{d^3}+\frac {b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{d^3}+\frac {b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{d^3}+\frac {b e \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )}{d^3}+\frac {b c e \sqrt {c^2 x^2-1} \text {arctanh}\left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{2 d^{5/2} \sqrt {c x-1} \sqrt {c x+1} \sqrt {c^2 d+e}}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 d^2 x}\) |
(b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*d^2*x) - (a + b*ArcCosh[c*x])/(2*d^2 *x^2) - (e*(a + b*ArcCosh[c*x]))/(2*d^2*(d + e*x^2)) - (2*e*(a + b*ArcCosh [c*x])^2)/(b*d^3) + (b*c*e*Sqrt[-1 + c^2*x^2]*ArcTanh[(Sqrt[c^2*d + e]*x)/ (Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(2*d^(5/2)*Sqrt[c^2*d + e]*Sqrt[-1 + c*x]*S qrt[1 + c*x]) - (2*e*(a + b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])])/d^ 3 + (e*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/d^3 + (e*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^Ar cCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/d^3 + (e*(a + b*ArcCosh[c* x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/d ^3 + (e*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/d^3 + (b*e*PolyLog[2, -E^(-2*ArcCosh[c*x])])/d^3 + (b*e*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e]))])/d^3 + (b*e*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[- (c^2*d) - e])])/d^3 + (b*e*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[- d] + Sqrt[-(c^2*d) - e]))])/d^3 + (b*e*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x]) /(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/d^3
3.6.1.3.1 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 = 1.61 (sec) , antiderivative size = 635, normalized size of antiderivative = 1.00
| method | result | size |
| parts | \(\frac {a e \ln \left (e \,x^{2}+d \right )}{d^{3}}-\frac {a e}{2 d^{2} \left (e \,x^{2}+d \right )}-\frac {a}{2 d^{2} x^{2}}-\frac {2 a e \ln \left (x \right )}{d^{3}}+b \,c^{2} \left (-\frac {-\sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} d x -\sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} e \,x^{3}+c^{4} d \,x^{2}+c^{4} e \,x^{4}+\operatorname {arccosh}\left (c x \right ) c^{2} d +2 \,\operatorname {arccosh}\left (c x \right ) c^{2} e \,x^{2}}{2 c^{2} x^{2} \left (c^{2} e \,x^{2}+c^{2} d \right ) d^{2}}-\frac {\sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, e \,\operatorname {arctanh}\left (\frac {4 c^{2} d +2 e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}+2 e}{4 \sqrt {c^{4} d^{2}+c^{2} d e}}\right )}{2 d^{3} c^{2} \left (c^{2} d +e \right )}+\frac {e \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}^{2} e +2 c^{2} d +e}\right )}{2 d^{3} c^{2}}-\frac {2 e \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3} c^{2}}-\frac {2 e \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3} c^{2}}-\frac {2 e \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3} c^{2}}-\frac {2 e \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3} c^{2}}+\frac {e^{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 {\left (\textit {\_R1}^{2}+1\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}^{2} e +2 c^{2} d +e}\right )}{2 d^{3} c^{2}}\right )\) | \(635\) |
| derivativedivides | \(c^{2} \left (-\frac {a}{2 d^{2} c^{2} x^{2}}-\frac {2 a e \ln \left (c x \right )}{c^{2} d^{3}}+\frac {a e \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{c^{2} d^{3}}-\frac {a e}{2 d^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}+b \,c^{4} \left (-\frac {-\sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} d x -\sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} e \,x^{3}+c^{4} d \,x^{2}+c^{4} e \,x^{4}+\operatorname {arccosh}\left (c x \right ) c^{2} d +2 \,\operatorname {arccosh}\left (c x \right ) c^{2} e \,x^{2}}{2 \left (c^{2} e \,x^{2}+c^{2} d \right ) d^{2} c^{6} x^{2}}-\frac {\sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, e \,\operatorname {arctanh}\left (\frac {4 c^{2} d +2 e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}+2 e}{4 \sqrt {c^{4} d^{2}+c^{2} d e}}\right )}{2 c^{6} d^{3} \left (c^{2} d +e \right )}-\frac {2 e \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{c^{6} d^{3}}-\frac {2 e \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{c^{6} d^{3}}-\frac {2 e \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{c^{6} d^{3}}-\frac {2 e \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{c^{6} d^{3}}+\frac {e \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}^{2} e +2 c^{2} d +e}\right )}{2 c^{6} d^{3}}+\frac {e^{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 {\left (\textit {\_R1}^{2}+1\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}^{2} e +2 c^{2} d +e}\right )}{2 c^{6} d^{3}}\right )\right )\) | \(664\) |
| default | \(c^{2} \left (-\frac {a}{2 d^{2} c^{2} x^{2}}-\frac {2 a e \ln \left (c x \right )}{c^{2} d^{3}}+\frac {a e \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{c^{2} d^{3}}-\frac {a e}{2 d^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}+b \,c^{4} \left (-\frac {-\sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} d x -\sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} e \,x^{3}+c^{4} d \,x^{2}+c^{4} e \,x^{4}+\operatorname {arccosh}\left (c x \right ) c^{2} d +2 \,\operatorname {arccosh}\left (c x \right ) c^{2} e \,x^{2}}{2 \left (c^{2} e \,x^{2}+c^{2} d \right ) d^{2} c^{6} x^{2}}-\frac {\sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, e \,\operatorname {arctanh}\left (\frac {4 c^{2} d +2 e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}+2 e}{4 \sqrt {c^{4} d^{2}+c^{2} d e}}\right )}{2 c^{6} d^{3} \left (c^{2} d +e \right )}-\frac {2 e \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{c^{6} d^{3}}-\frac {2 e \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{c^{6} d^{3}}-\frac {2 e \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{c^{6} d^{3}}-\frac {2 e \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{c^{6} d^{3}}+\frac {e \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}^{2} e +2 c^{2} d +e}\right )}{2 c^{6} d^{3}}+\frac {e^{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 {\left (\textit {\_R1}^{2}+1\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}^{2} e +2 c^{2} d +e}\right )}{2 c^{6} d^{3}}\right )\right )\) | \(664\) |
a*e/d^3*ln(e*x^2+d)-1/2*a*e/d^2/(e*x^2+d)-1/2*a/d^2/x^2-2*a/d^3*e*ln(x)+b* c^2*(-1/2*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*d*x-(c*x-1)^(1/2)*(c*x+1)^(1/2 )*c^3*e*x^3+c^4*d*x^2+c^4*e*x^4+arccosh(c*x)*c^2*d+2*arccosh(c*x)*c^2*e*x^ 2)/c^2/x^2/(c^2*e*x^2+c^2*d)/d^2-1/2*(d*c^2*(c^2*d+e))^(1/2)/d^3/c^2/(c^2* d+e)*e*arctanh(1/4*(4*c^2*d+2*e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2+2*e)/( c^4*d^2+c^2*d*e)^(1/2))+1/2*e/d^3/c^2*sum((_R1^2*e+4*c^2*d+e)/(_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))-2*e/d^3/c^2*arccosh(c*x)*ln(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2 )))-2*e/d^3/c^2*arccosh(c*x)*ln(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))-2*e /d^3/c^2*dilog(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))-2*e/d^3/c^2*dilog(1- I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))+1/2*e^2/d^3/c^2*sum((_R1^2+1)/(_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)))
\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d+e x^2\right )^2} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x^{3}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d+e x^2\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d+e x^2\right )^2} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x^{3}} \,d x } \]
-1/2*a*((2*e*x^2 + d)/(d^2*e*x^4 + d^3*x^2) - 2*e*log(e*x^2 + d)/d^3 + 4*e *log(x)/d^3) + b*integrate(log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(e^2*x^7 + 2*d*e*x^5 + d^2*x^3), x)
\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d+e x^2\right )^2} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x^{3}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d+e x^2\right )^2} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^3\,{\left (e\,x^2+d\right )}^2} \,d x \]