Integrand size = 14, antiderivative size = 774 \[ \int \frac {\text {arccosh}(a x)}{\left (c+d x^2\right )^2} \, dx=-\frac {\text {arccosh}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\text {arccosh}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}+\sqrt {d} x\right )}+\frac {a \text {arctanh}\left (\frac {\sqrt {a \sqrt {-c}-\sqrt {d}} \sqrt {1+a x}}{\sqrt {a \sqrt {-c}+\sqrt {d}} \sqrt {-1+a x}}\right )}{2 c \sqrt {a \sqrt {-c}-\sqrt {d}} \sqrt {a \sqrt {-c}+\sqrt {d}} \sqrt {d}}-\frac {a \text {arctanh}\left (\frac {\sqrt {a \sqrt {-c}+\sqrt {d}} \sqrt {1+a x}}{\sqrt {a \sqrt {-c}-\sqrt {d}} \sqrt {-1+a x}}\right )}{2 c \sqrt {a \sqrt {-c}-\sqrt {d}} \sqrt {a \sqrt {-c}+\sqrt {d}} \sqrt {d}}-\frac {\text {arccosh}(a x) \log \left (1-\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\text {arccosh}(a x) \log \left (1+\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\text {arccosh}(a x) \log \left (1-\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\text {arccosh}(a x) \log \left (1+\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt {d}} \]
-1/4*arccosh(a*x)*ln(1-(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))*d^(1/2)/(a*(-c)^( 1/2)-(-a^2*c-d)^(1/2)))/(-c)^(3/2)/d^(1/2)+1/4*arccosh(a*x)*ln(1+(a*x+(a*x -1)^(1/2)*(a*x+1)^(1/2))*d^(1/2)/(a*(-c)^(1/2)-(-a^2*c-d)^(1/2)))/(-c)^(3/ 2)/d^(1/2)-1/4*arccosh(a*x)*ln(1-(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))*d^(1/2) /(a*(-c)^(1/2)+(-a^2*c-d)^(1/2)))/(-c)^(3/2)/d^(1/2)+1/4*arccosh(a*x)*ln(1 +(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))*d^(1/2)/(a*(-c)^(1/2)+(-a^2*c-d)^(1/2)) )/(-c)^(3/2)/d^(1/2)+1/4*polylog(2,-(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))*d^(1 /2)/(a*(-c)^(1/2)-(-a^2*c-d)^(1/2)))/(-c)^(3/2)/d^(1/2)-1/4*polylog(2,(a*x +(a*x-1)^(1/2)*(a*x+1)^(1/2))*d^(1/2)/(a*(-c)^(1/2)-(-a^2*c-d)^(1/2)))/(-c )^(3/2)/d^(1/2)+1/4*polylog(2,-(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))*d^(1/2)/( a*(-c)^(1/2)+(-a^2*c-d)^(1/2)))/(-c)^(3/2)/d^(1/2)-1/4*polylog(2,(a*x+(a*x -1)^(1/2)*(a*x+1)^(1/2))*d^(1/2)/(a*(-c)^(1/2)+(-a^2*c-d)^(1/2)))/(-c)^(3/ 2)/d^(1/2)-1/4*arccosh(a*x)/c/d^(1/2)/((-c)^(1/2)-x*d^(1/2))+1/4*arccosh(a *x)/c/d^(1/2)/((-c)^(1/2)+x*d^(1/2))+1/2*a*arctanh((a*x+1)^(1/2)*(a*(-c)^( 1/2)-d^(1/2))^(1/2)/(a*x-1)^(1/2)/(a*(-c)^(1/2)+d^(1/2))^(1/2))/c/d^(1/2)/ (a*(-c)^(1/2)-d^(1/2))^(1/2)/(a*(-c)^(1/2)+d^(1/2))^(1/2)-1/2*a*arctanh((a *x+1)^(1/2)*(a*(-c)^(1/2)+d^(1/2))^(1/2)/(a*x-1)^(1/2)/(a*(-c)^(1/2)-d^(1/ 2))^(1/2))/c/d^(1/2)/(a*(-c)^(1/2)-d^(1/2))^(1/2)/(a*(-c)^(1/2)+d^(1/2))^( 1/2)
Result contains complex when optimal does not.
Time = 5.85 (sec) , antiderivative size = 687, normalized size of antiderivative = 0.89 \[ \int \frac {\text {arccosh}(a x)}{\left (c+d x^2\right )^2} \, dx=\frac {2 \sqrt {c} \left (\frac {\text {arccosh}(a x)}{-i \sqrt {c}+\sqrt {d} x}+\frac {a \log \left (\frac {2 d \left (i \sqrt {d}+a^2 \sqrt {c} x-i \sqrt {-a^2 c-d} \sqrt {-1+a x} \sqrt {1+a x}\right )}{a \sqrt {-a^2 c-d} \left (\sqrt {c}+i \sqrt {d} x\right )}\right )}{\sqrt {-a^2 c-d}}\right )-2 \sqrt {c} \left (-\frac {\text {arccosh}(a x)}{i \sqrt {c}+\sqrt {d} x}-\frac {a \log \left (\frac {2 d \left (-\sqrt {d}-i a^2 \sqrt {c} x+\sqrt {-a^2 c-d} \sqrt {-1+a x} \sqrt {1+a x}\right )}{a \sqrt {-a^2 c-d} \left (i \sqrt {c}+\sqrt {d} x\right )}\right )}{\sqrt {-a^2 c-d}}\right )+i \left (\text {arccosh}(a x) \left (-\text {arccosh}(a x)+2 \left (\log \left (1+\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{i a \sqrt {c}-\sqrt {-a^2 c-d}}\right )+\log \left (1+\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{i a \sqrt {c}+\sqrt {-a^2 c-d}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{-i a \sqrt {c}+\sqrt {-a^2 c-d}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{i a \sqrt {c}+\sqrt {-a^2 c-d}}\right )\right )-i \left (\text {arccosh}(a x) \left (-\text {arccosh}(a x)+2 \left (\log \left (1+\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{-i a \sqrt {c}+\sqrt {-a^2 c-d}}\right )+\log \left (1-\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{i a \sqrt {c}+\sqrt {-a^2 c-d}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{-i a \sqrt {c}+\sqrt {-a^2 c-d}}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{i a \sqrt {c}+\sqrt {-a^2 c-d}}\right )\right )}{8 c^{3/2} \sqrt {d}} \]
(2*Sqrt[c]*(ArcCosh[a*x]/((-I)*Sqrt[c] + Sqrt[d]*x) + (a*Log[(2*d*(I*Sqrt[ d] + a^2*Sqrt[c]*x - I*Sqrt[-(a^2*c) - d]*Sqrt[-1 + a*x]*Sqrt[1 + a*x]))/( a*Sqrt[-(a^2*c) - d]*(Sqrt[c] + I*Sqrt[d]*x))])/Sqrt[-(a^2*c) - d]) - 2*Sq rt[c]*(-(ArcCosh[a*x]/(I*Sqrt[c] + Sqrt[d]*x)) - (a*Log[(2*d*(-Sqrt[d] - I *a^2*Sqrt[c]*x + Sqrt[-(a^2*c) - d]*Sqrt[-1 + a*x]*Sqrt[1 + a*x]))/(a*Sqrt [-(a^2*c) - d]*(I*Sqrt[c] + Sqrt[d]*x))])/Sqrt[-(a^2*c) - d]) + I*(ArcCosh [a*x]*(-ArcCosh[a*x] + 2*(Log[1 + (Sqrt[d]*E^ArcCosh[a*x])/(I*a*Sqrt[c] - Sqrt[-(a^2*c) - d])] + Log[1 + (Sqrt[d]*E^ArcCosh[a*x])/(I*a*Sqrt[c] + Sqr t[-(a^2*c) - d])])) + 2*PolyLog[2, (Sqrt[d]*E^ArcCosh[a*x])/((-I)*a*Sqrt[c ] + Sqrt[-(a^2*c) - d])] + 2*PolyLog[2, -((Sqrt[d]*E^ArcCosh[a*x])/(I*a*Sq rt[c] + Sqrt[-(a^2*c) - d]))]) - I*(ArcCosh[a*x]*(-ArcCosh[a*x] + 2*(Log[1 + (Sqrt[d]*E^ArcCosh[a*x])/((-I)*a*Sqrt[c] + Sqrt[-(a^2*c) - d])] + Log[1 - (Sqrt[d]*E^ArcCosh[a*x])/(I*a*Sqrt[c] + Sqrt[-(a^2*c) - d])])) + 2*Poly Log[2, -((Sqrt[d]*E^ArcCosh[a*x])/((-I)*a*Sqrt[c] + Sqrt[-(a^2*c) - d]))] + 2*PolyLog[2, (Sqrt[d]*E^ArcCosh[a*x])/(I*a*Sqrt[c] + Sqrt[-(a^2*c) - d]) ]))/(8*c^(3/2)*Sqrt[d])
Time = 1.75 (sec) , antiderivative size = 774, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6324, 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.
| \(\displaystyle \int \frac {\text {arccosh}(a x)}{\left (c+d x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 6324 |
| \(\displaystyle \int \left (-\frac {d \text {arccosh}(a x)}{2 c \left (-c d-d^2 x^2\right )}-\frac {d \text {arccosh}(a x)}{4 c \left (\sqrt {-c} \sqrt {d}-d x\right )^2}-\frac {d \text {arccosh}(a x)}{4 c \left (\sqrt {-c} \sqrt {d}+d x\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}-\sqrt {-c a^2-d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}-\sqrt {-c a^2-d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{\sqrt {-c} a+\sqrt {-c a^2-d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{\sqrt {-c} a+\sqrt {-c a^2-d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\text {arccosh}(a x) \log \left (1-\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}-\sqrt {a^2 (-c)-d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\text {arccosh}(a x) \log \left (\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}-\sqrt {a^2 (-c)-d}}+1\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\text {arccosh}(a x) \log \left (1-\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{\sqrt {a^2 (-c)-d}+a \sqrt {-c}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\text {arccosh}(a x) \log \left (\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{\sqrt {a^2 (-c)-d}+a \sqrt {-c}}+1\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\text {arccosh}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\text {arccosh}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}+\sqrt {d} x\right )}+\frac {a \text {arctanh}\left (\frac {\sqrt {a x+1} \sqrt {a \sqrt {-c}-\sqrt {d}}}{\sqrt {a x-1} \sqrt {a \sqrt {-c}+\sqrt {d}}}\right )}{2 c \sqrt {d} \sqrt {a \sqrt {-c}-\sqrt {d}} \sqrt {a \sqrt {-c}+\sqrt {d}}}-\frac {a \text {arctanh}\left (\frac {\sqrt {a x+1} \sqrt {a \sqrt {-c}+\sqrt {d}}}{\sqrt {a x-1} \sqrt {a \sqrt {-c}-\sqrt {d}}}\right )}{2 c \sqrt {d} \sqrt {a \sqrt {-c}-\sqrt {d}} \sqrt {a \sqrt {-c}+\sqrt {d}}}\) |
-1/4*ArcCosh[a*x]/(c*Sqrt[d]*(Sqrt[-c] - Sqrt[d]*x)) + ArcCosh[a*x]/(4*c*S qrt[d]*(Sqrt[-c] + Sqrt[d]*x)) + (a*ArcTanh[(Sqrt[a*Sqrt[-c] - Sqrt[d]]*Sq rt[1 + a*x])/(Sqrt[a*Sqrt[-c] + Sqrt[d]]*Sqrt[-1 + a*x])])/(2*c*Sqrt[a*Sqr t[-c] - Sqrt[d]]*Sqrt[a*Sqrt[-c] + Sqrt[d]]*Sqrt[d]) - (a*ArcTanh[(Sqrt[a* Sqrt[-c] + Sqrt[d]]*Sqrt[1 + a*x])/(Sqrt[a*Sqrt[-c] - Sqrt[d]]*Sqrt[-1 + a *x])])/(2*c*Sqrt[a*Sqrt[-c] - Sqrt[d]]*Sqrt[a*Sqrt[-c] + Sqrt[d]]*Sqrt[d]) - (ArcCosh[a*x]*Log[1 - (Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] - Sqrt[-(a^2 *c) - d])])/(4*(-c)^(3/2)*Sqrt[d]) + (ArcCosh[a*x]*Log[1 + (Sqrt[d]*E^ArcC osh[a*x])/(a*Sqrt[-c] - Sqrt[-(a^2*c) - d])])/(4*(-c)^(3/2)*Sqrt[d]) - (Ar cCosh[a*x]*Log[1 - (Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] + Sqrt[-(a^2*c) - d])])/(4*(-c)^(3/2)*Sqrt[d]) + (ArcCosh[a*x]*Log[1 + (Sqrt[d]*E^ArcCosh[a* x])/(a*Sqrt[-c] + Sqrt[-(a^2*c) - d])])/(4*(-c)^(3/2)*Sqrt[d]) + PolyLog[2 , -((Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] - Sqrt[-(a^2*c) - d]))]/(4*(-c)^( 3/2)*Sqrt[d]) - PolyLog[2, (Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] - Sqrt[-(a ^2*c) - d])]/(4*(-c)^(3/2)*Sqrt[d]) + PolyLog[2, -((Sqrt[d]*E^ArcCosh[a*x] )/(a*Sqrt[-c] + Sqrt[-(a^2*c) - d]))]/(4*(-c)^(3/2)*Sqrt[d]) - PolyLog[2, (Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] + Sqrt[-(a^2*c) - d])]/(4*(-c)^(3/2)* Sqrt[d])
3.1.46.3.1 Defintions of rubi rules used
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcCosh[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (p > 0 || IGtQ[n, 0])
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 27.27 (sec) , antiderivative size = 790, normalized size of antiderivative = 1.02
| method | result | size |
| derivativedivides | \(\frac {\frac {\operatorname {arccosh}\left (a x \right ) a^{3} x}{2 c \left (a^{2} d \,x^{2}+c \,a^{2}\right )}+\frac {a^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}+\left (4 c \,a^{2}+2 d \right ) \textit {\_Z}^{2}+d \right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {arccosh}\left (a x \right ) \ln \left (\frac {\textit {\_R1} -a x -\sqrt {a x -1}\, \sqrt {a x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -a x -\sqrt {a x -1}\, \sqrt {a x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} d +2 c \,a^{2}+d}\right )}{4 c}+\frac {\sqrt {\left (2 c \,a^{2}+2 \sqrt {a^{2} c \left (c \,a^{2}+d \right )}+d \right ) d}\, \left (-2 \sqrt {a^{2} c \left (c \,a^{2}+d \right )}\, a^{2} c +2 a^{4} c^{2}+2 a^{2} c d -\sqrt {a^{2} c \left (c \,a^{2}+d \right )}\, d \right ) a^{2} \arctan \left (\frac {d \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{\sqrt {\left (2 c \,a^{2}+2 \sqrt {a^{2} c \left (c \,a^{2}+d \right )}+d \right ) d}}\right )}{2 c \left (c \,a^{2}+d \right ) d^{3}}-\frac {\sqrt {\left (2 c \,a^{2}+2 \sqrt {a^{2} c \left (c \,a^{2}+d \right )}+d \right ) d}\, \left (2 c \,a^{2}-2 \sqrt {a^{2} c \left (c \,a^{2}+d \right )}+d \right ) \arctan \left (\frac {d \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{\sqrt {\left (2 c \,a^{2}+2 \sqrt {a^{2} c \left (c \,a^{2}+d \right )}+d \right ) d}}\right ) a^{2}}{2 c \,d^{3}}+\frac {\sqrt {-\left (2 c \,a^{2}-2 \sqrt {a^{2} c \left (c \,a^{2}+d \right )}+d \right ) d}\, \left (2 \sqrt {a^{2} c \left (c \,a^{2}+d \right )}\, a^{2} c +2 a^{4} c^{2}+2 a^{2} c d +\sqrt {a^{2} c \left (c \,a^{2}+d \right )}\, d \right ) a^{2} \operatorname {arctanh}\left (\frac {d \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{\sqrt {\left (-2 c \,a^{2}+2 \sqrt {a^{2} c \left (c \,a^{2}+d \right )}-d \right ) d}}\right )}{2 c \left (c \,a^{2}+d \right ) d^{3}}-\frac {\sqrt {-\left (2 c \,a^{2}-2 \sqrt {a^{2} c \left (c \,a^{2}+d \right )}+d \right ) d}\, \left (2 c \,a^{2}+2 \sqrt {a^{2} c \left (c \,a^{2}+d \right )}+d \right ) \operatorname {arctanh}\left (\frac {d \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{\sqrt {\left (-2 c \,a^{2}+2 \sqrt {a^{2} c \left (c \,a^{2}+d \right )}-d \right ) d}}\right ) a^{2}}{2 c \,d^{3}}-\frac {a^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}+\left (4 c \,a^{2}+2 d \right ) \textit {\_Z}^{2}+d \right )}{\sum }\frac {\operatorname {arccosh}\left (a x \right ) \ln \left (\frac {\textit {\_R1} -a x -\sqrt {a x -1}\, \sqrt {a x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -a x -\sqrt {a x -1}\, \sqrt {a x +1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} d +2 c \,a^{2}+d \right )}\right )}{4 c}}{a}\) | \(790\) |
| default | \(\frac {\frac {\operatorname {arccosh}\left (a x \right ) a^{3} x}{2 c \left (a^{2} d \,x^{2}+c \,a^{2}\right )}+\frac {a^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}+\left (4 c \,a^{2}+2 d \right ) \textit {\_Z}^{2}+d \right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {arccosh}\left (a x \right ) \ln \left (\frac {\textit {\_R1} -a x -\sqrt {a x -1}\, \sqrt {a x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -a x -\sqrt {a x -1}\, \sqrt {a x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} d +2 c \,a^{2}+d}\right )}{4 c}+\frac {\sqrt {\left (2 c \,a^{2}+2 \sqrt {a^{2} c \left (c \,a^{2}+d \right )}+d \right ) d}\, \left (-2 \sqrt {a^{2} c \left (c \,a^{2}+d \right )}\, a^{2} c +2 a^{4} c^{2}+2 a^{2} c d -\sqrt {a^{2} c \left (c \,a^{2}+d \right )}\, d \right ) a^{2} \arctan \left (\frac {d \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{\sqrt {\left (2 c \,a^{2}+2 \sqrt {a^{2} c \left (c \,a^{2}+d \right )}+d \right ) d}}\right )}{2 c \left (c \,a^{2}+d \right ) d^{3}}-\frac {\sqrt {\left (2 c \,a^{2}+2 \sqrt {a^{2} c \left (c \,a^{2}+d \right )}+d \right ) d}\, \left (2 c \,a^{2}-2 \sqrt {a^{2} c \left (c \,a^{2}+d \right )}+d \right ) \arctan \left (\frac {d \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{\sqrt {\left (2 c \,a^{2}+2 \sqrt {a^{2} c \left (c \,a^{2}+d \right )}+d \right ) d}}\right ) a^{2}}{2 c \,d^{3}}+\frac {\sqrt {-\left (2 c \,a^{2}-2 \sqrt {a^{2} c \left (c \,a^{2}+d \right )}+d \right ) d}\, \left (2 \sqrt {a^{2} c \left (c \,a^{2}+d \right )}\, a^{2} c +2 a^{4} c^{2}+2 a^{2} c d +\sqrt {a^{2} c \left (c \,a^{2}+d \right )}\, d \right ) a^{2} \operatorname {arctanh}\left (\frac {d \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{\sqrt {\left (-2 c \,a^{2}+2 \sqrt {a^{2} c \left (c \,a^{2}+d \right )}-d \right ) d}}\right )}{2 c \left (c \,a^{2}+d \right ) d^{3}}-\frac {\sqrt {-\left (2 c \,a^{2}-2 \sqrt {a^{2} c \left (c \,a^{2}+d \right )}+d \right ) d}\, \left (2 c \,a^{2}+2 \sqrt {a^{2} c \left (c \,a^{2}+d \right )}+d \right ) \operatorname {arctanh}\left (\frac {d \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{\sqrt {\left (-2 c \,a^{2}+2 \sqrt {a^{2} c \left (c \,a^{2}+d \right )}-d \right ) d}}\right ) a^{2}}{2 c \,d^{3}}-\frac {a^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}+\left (4 c \,a^{2}+2 d \right ) \textit {\_Z}^{2}+d \right )}{\sum }\frac {\operatorname {arccosh}\left (a x \right ) \ln \left (\frac {\textit {\_R1} -a x -\sqrt {a x -1}\, \sqrt {a x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -a x -\sqrt {a x -1}\, \sqrt {a x +1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} d +2 c \,a^{2}+d \right )}\right )}{4 c}}{a}\) | \(790\) |
1/a*(1/2*arccosh(a*x)*a^3*x/c/(a^2*d*x^2+a^2*c)+1/4/c*a^2*sum(_R1/(_R1^2*d +2*a^2*c+d)*(arccosh(a*x)*ln((_R1-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))/_R1)+di log((_R1-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))/_R1)),_R1=RootOf(d*_Z^4+(4*a^2*c +2*d)*_Z^2+d))+1/2*((2*c*a^2+2*(a^2*c*(a^2*c+d))^(1/2)+d)*d)^(1/2)*(-2*(a^ 2*c*(a^2*c+d))^(1/2)*a^2*c+2*a^4*c^2+2*a^2*c*d-(a^2*c*(a^2*c+d))^(1/2)*d)* a^2*arctan(d*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/((2*c*a^2+2*(a^2*c*(a^2*c+d ))^(1/2)+d)*d)^(1/2))/c/(a^2*c+d)/d^3-1/2*((2*c*a^2+2*(a^2*c*(a^2*c+d))^(1 /2)+d)*d)^(1/2)*(2*c*a^2-2*(a^2*c*(a^2*c+d))^(1/2)+d)*arctan(d*(a*x+(a*x-1 )^(1/2)*(a*x+1)^(1/2))/((2*c*a^2+2*(a^2*c*(a^2*c+d))^(1/2)+d)*d)^(1/2))*a^ 2/c/d^3+1/2*(-(2*c*a^2-2*(a^2*c*(a^2*c+d))^(1/2)+d)*d)^(1/2)*(2*(a^2*c*(a^ 2*c+d))^(1/2)*a^2*c+2*a^4*c^2+2*a^2*c*d+(a^2*c*(a^2*c+d))^(1/2)*d)*a^2*arc tanh(d*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/((-2*c*a^2+2*(a^2*c*(a^2*c+d))^(1 /2)-d)*d)^(1/2))/c/(a^2*c+d)/d^3-1/2*(-(2*c*a^2-2*(a^2*c*(a^2*c+d))^(1/2)+ d)*d)^(1/2)*(2*c*a^2+2*(a^2*c*(a^2*c+d))^(1/2)+d)*arctanh(d*(a*x+(a*x-1)^( 1/2)*(a*x+1)^(1/2))/((-2*c*a^2+2*(a^2*c*(a^2*c+d))^(1/2)-d)*d)^(1/2))*a^2/ c/d^3-1/4/c*a^2*sum(1/_R1/(_R1^2*d+2*a^2*c+d)*(arccosh(a*x)*ln((_R1-a*x-(a *x-1)^(1/2)*(a*x+1)^(1/2))/_R1)+dilog((_R1-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2) )/_R1)),_R1=RootOf(d*_Z^4+(4*a^2*c+2*d)*_Z^2+d)))
\[ \int \frac {\text {arccosh}(a x)}{\left (c+d x^2\right )^2} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )}{{\left (d x^{2} + c\right )}^{2}} \,d x } \]
\[ \int \frac {\text {arccosh}(a x)}{\left (c+d x^2\right )^2} \, dx=\int \frac {\operatorname {acosh}{\left (a x \right )}}{\left (c + d x^{2}\right )^{2}}\, dx \]
\[ \int \frac {\text {arccosh}(a x)}{\left (c+d x^2\right )^2} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )}{{\left (d x^{2} + c\right )}^{2}} \,d x } \]
\[ \int \frac {\text {arccosh}(a x)}{\left (c+d x^2\right )^2} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )}{{\left (d x^{2} + c\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\text {arccosh}(a x)}{\left (c+d x^2\right )^2} \, dx=\int \frac {\mathrm {acosh}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^2} \,d x \]