Integrand size = 18, antiderivative size = 804 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^2} \, dx=-\frac {a+b \text {arccosh}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \text {arccosh}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {-1+c x}}\right )}{2 d \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {e}}-\frac {b c \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {-1+c x}}\right )}{2 d \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {e}}-\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{4 (-d)^{3/2} \sqrt {e}} \]
-1/4*(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)^(3/2)/e^(1/2)+1/4*(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)^(3/2)/e^(1/2)-1/4*(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)^(3/2)/e^(1/2)+1/4 *(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)^(3/2)/e^(1/2)+1/4*b*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)^(3/2)/ e^(1/2)-1/4*b*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)^(3/2)/e^(1/2)+1/4*b*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)^(3/2)/e ^(1/2)-1/4*b*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)^(3/2)/e^(1/2)+1/4*(-a-b*arccosh(c*x))/d/e^(1/ 2)/((-d)^(1/2)-x*e^(1/2))+1/4*(a+b*arccosh(c*x))/d/e^(1/2)/((-d)^(1/2)+x*e ^(1/2))+1/2*b*c*arctanh((c*x+1)^(1/2)*(c*(-d)^(1/2)-e^(1/2))^(1/2)/(c*x-1) ^(1/2)/(c*(-d)^(1/2)+e^(1/2))^(1/2))/d/e^(1/2)/(c*(-d)^(1/2)-e^(1/2))^(1/2 )/(c*(-d)^(1/2)+e^(1/2))^(1/2)-1/2*b*c*arctanh((c*x+1)^(1/2)*(c*(-d)^(1/2) +e^(1/2))^(1/2)/(c*x-1)^(1/2)/(c*(-d)^(1/2)-e^(1/2))^(1/2))/d/e^(1/2)/(c*( -d)^(1/2)-e^(1/2))^(1/2)/(c*(-d)^(1/2)+e^(1/2))^(1/2)
Result contains complex when optimal does not.
Time = 6.39 (sec) , antiderivative size = 734, normalized size of antiderivative = 0.91 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^2} \, dx=\frac {1}{2} \left (\frac {a x}{d^2+d e x^2}+\frac {a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2} \sqrt {e}}+\frac {b \left (2 \sqrt {d} \left (\frac {\text {arccosh}(c x)}{-i \sqrt {d}+\sqrt {e} x}+\frac {c \log \left (\frac {2 e \left (i \sqrt {e}+c^2 \sqrt {d} x-i \sqrt {-c^2 d-e} \sqrt {-1+c x} \sqrt {1+c x}\right )}{c \sqrt {-c^2 d-e} \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{\sqrt {-c^2 d-e}}\right )-2 \sqrt {d} \left (-\frac {\text {arccosh}(c x)}{i \sqrt {d}+\sqrt {e} x}-\frac {c \log \left (\frac {2 e \left (-\sqrt {e}-i c^2 \sqrt {d} x+\sqrt {-c^2 d-e} \sqrt {-1+c x} \sqrt {1+c x}\right )}{c \sqrt {-c^2 d-e} \left (i \sqrt {d}+\sqrt {e} x\right )}\right )}{\sqrt {-c^2 d-e}}\right )+i \left (\text {arccosh}(c x) \left (-\text {arccosh}(c x)+2 \left (\log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}-\sqrt {-c^2 d-e}}\right )+\log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )-i \left (\text {arccosh}(c x) \left (-\text {arccosh}(c x)+2 \left (\log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+\log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}-\sqrt {-c^2 d-e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )\right )}{4 d^{3/2} \sqrt {e}}\right ) \]
((a*x)/(d^2 + d*e*x^2) + (a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(d^(3/2)*Sqrt[e]) + (b*(2*Sqrt[d]*(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*Sqrt[e]*x))])/Sqrt[-(c^2*d) - e]) - 2*Sqrt[d]*(-(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]) + I*(A rcCosh[c*x]*(-ArcCosh[c*x] + 2*(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])])) + 2*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/((-I)*c* Sqrt[d] + Sqrt[-(c^2*d) - e])] + 2*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/( I*c*Sqrt[d] + Sqrt[-(c^2*d) - e]))]) - I*(ArcCosh[c*x]*(-ArcCosh[c*x] + 2* (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])])) + 2*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] - Sqrt[-(c^2*d) - e])] + 2*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e]) ])))/(4*d^(3/2)*Sqrt[e]))/2
Time = 1.52 (sec) , antiderivative size = 804, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, 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.
| \(\displaystyle \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 6324 |
| \(\displaystyle \int \left (-\frac {e (a+b \text {arccosh}(c x))}{2 d \left (-d e-e^2 x^2\right )}-\frac {e (a+b \text {arccosh}(c x))}{4 d \left (\sqrt {-d} \sqrt {e}-e x\right )^2}-\frac {e (a+b \text {arccosh}(c x))}{4 d \left (\sqrt {-d} \sqrt {e}+e x\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right ) (a+b \text {arccosh}(c x))}{4 (-d)^{3/2} \sqrt {e}}+\frac {\log \left (\frac {e^{\text {arccosh}(c x)} \sqrt {e}}{c \sqrt {-d}-\sqrt {-d c^2-e}}+1\right ) (a+b \text {arccosh}(c x))}{4 (-d)^{3/2} \sqrt {e}}-\frac {\log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right ) (a+b \text {arccosh}(c x))}{4 (-d)^{3/2} \sqrt {e}}+\frac {\log \left (\frac {e^{\text {arccosh}(c x)} \sqrt {e}}{\sqrt {-d} c+\sqrt {-d c^2-e}}+1\right ) (a+b \text {arccosh}(c x))}{4 (-d)^{3/2} \sqrt {e}}-\frac {a+b \text {arccosh}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \text {arccosh}(c x)}{4 d \sqrt {e} \left (\sqrt {e} x+\sqrt {-d}\right )}+\frac {b c \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c x+1}}{\sqrt {\sqrt {-d} c+\sqrt {e}} \sqrt {c x-1}}\right )}{2 d \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {\sqrt {-d} c+\sqrt {e}} \sqrt {e}}-\frac {b c \text {arctanh}\left (\frac {\sqrt {\sqrt {-d} c+\sqrt {e}} \sqrt {c x+1}}{\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c x-1}}\right )}{2 d \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {\sqrt {-d} c+\sqrt {e}} \sqrt {e}}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{4 (-d)^{3/2} \sqrt {e}}\) |
-1/4*(a + b*ArcCosh[c*x])/(d*Sqrt[e]*(Sqrt[-d] - Sqrt[e]*x)) + (a + b*ArcC osh[c*x])/(4*d*Sqrt[e]*(Sqrt[-d] + Sqrt[e]*x)) + (b*c*ArcTanh[(Sqrt[c*Sqrt [-d] - Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[-1 + c*x]) ])/(2*d*Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[e]) - ( b*c*ArcTanh[(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[-1 + c*x])])/(2*d*Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[e]) - ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c* x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(4*(-d)^(3/2)*Sqrt[e]) + ((a + b*A rcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(4*(-d)^(3/2)*Sqrt[e]) - ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(4*(-d)^(3/2)*Sqrt[e]) + ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt [-(c^2*d) - e])])/(4*(-d)^(3/2)*Sqrt[e]) + (b*PolyLog[2, -((Sqrt[e]*E^ArcC osh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e]))])/(4*(-d)^(3/2)*Sqrt[e]) - (b *PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/( 4*(-d)^(3/2)*Sqrt[e]) + (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[- d] + Sqrt[-(c^2*d) - e]))])/(4*(-d)^(3/2)*Sqrt[e]) - (b*PolyLog[2, (Sqrt[e ]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(4*(-d)^(3/2)*Sqrt[e ])
3.6.4.3.1 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 = 20.61 (sec) , antiderivative size = 828, normalized size of antiderivative = 1.03
| method | result | size |
| parts | \(\frac {a x}{2 d \left (e \,x^{2}+d \right )}+\frac {a \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 d \sqrt {d e}}+\frac {b \left (\frac {c^{3} \operatorname {arccosh}\left (c x \right ) x}{2 d \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {c^{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 )}{4 d}+\frac {\sqrt {-\left (2 c^{2} d -2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) e}\, \left (2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, c^{2} d +2 c^{4} d^{2}+2 c^{2} d e +\sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, e \right ) c^{2} \operatorname {arctanh}\left (\frac {e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}-e \right ) e}}\right )}{2 d \left (c^{2} d +e \right ) e^{3}}-\frac {\sqrt {-\left (2 c^{2} d -2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) e}\, \left (2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) \operatorname {arctanh}\left (\frac {e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}-e \right ) e}}\right ) c^{2}}{2 d \,e^{3}}+\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) e}\, \left (-2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, c^{2} d +2 c^{4} d^{2}+2 c^{2} d e -\sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, e \right ) c^{2} \arctan \left (\frac {e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) e}}\right )}{2 d \left (c^{2} d +e \right ) e^{3}}-\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) e}\, \left (2 c^{2} d -2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) \arctan \left (\frac {e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) e}}\right ) c^{2}}{2 d \,e^{3}}-\frac {c^{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 )}{4 d}\right )}{c}\) | \(828\) |
| derivativedivides | \(\frac {\frac {a \,c^{3} x}{2 d \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {a c \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 d \sqrt {d e}}+b \,c^{4} \left (\frac {\operatorname {arccosh}\left (c x \right ) x}{2 c d \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {\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}}{4 d \,c^{2}}+\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) e}\, \left (-2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, c^{2} d +2 c^{4} d^{2}+2 c^{2} d e -\sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, e \right ) \arctan \left (\frac {e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) e}}\right )}{2 d \,c^{2} \left (c^{2} d +e \right ) e^{3}}-\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) e}\, \left (2 c^{2} d -2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) \arctan \left (\frac {e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) e}}\right )}{2 c^{2} d \,e^{3}}+\frac {\sqrt {-\left (2 c^{2} d -2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) e}\, \left (2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, c^{2} d +2 c^{4} d^{2}+2 c^{2} d e +\sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, e \right ) \operatorname {arctanh}\left (\frac {e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}-e \right ) e}}\right )}{2 d \,c^{2} \left (c^{2} d +e \right ) e^{3}}-\frac {\sqrt {-\left (2 c^{2} d -2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) e}\, \left (2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) \operatorname {arctanh}\left (\frac {e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}-e \right ) e}}\right )}{2 c^{2} d \,e^{3}}-\frac {\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 )}}{4 d \,c^{2}}\right )}{c}\) | \(843\) |
| default | \(\frac {\frac {a \,c^{3} x}{2 d \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {a c \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 d \sqrt {d e}}+b \,c^{4} \left (\frac {\operatorname {arccosh}\left (c x \right ) x}{2 c d \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {\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}}{4 d \,c^{2}}+\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) e}\, \left (-2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, c^{2} d +2 c^{4} d^{2}+2 c^{2} d e -\sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, e \right ) \arctan \left (\frac {e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) e}}\right )}{2 d \,c^{2} \left (c^{2} d +e \right ) e^{3}}-\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) e}\, \left (2 c^{2} d -2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) \arctan \left (\frac {e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) e}}\right )}{2 c^{2} d \,e^{3}}+\frac {\sqrt {-\left (2 c^{2} d -2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) e}\, \left (2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, c^{2} d +2 c^{4} d^{2}+2 c^{2} d e +\sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, e \right ) \operatorname {arctanh}\left (\frac {e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}-e \right ) e}}\right )}{2 d \,c^{2} \left (c^{2} d +e \right ) e^{3}}-\frac {\sqrt {-\left (2 c^{2} d -2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) e}\, \left (2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) \operatorname {arctanh}\left (\frac {e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}-e \right ) e}}\right )}{2 c^{2} d \,e^{3}}-\frac {\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 )}}{4 d \,c^{2}}\right )}{c}\) | \(843\) |
1/2*a*x/d/(e*x^2+d)+1/2*a/d/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))+b/c*(1/2*c ^3*arccosh(c*x)*x/d/(c^2*e*x^2+c^2*d)+1/4/d*c^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*(-(2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)+e)*e)^(1/2)*(2*(d*c^2*(c^2*d +e))^(1/2)*c^2*d+2*c^4*d^2+2*c^2*d*e+(d*c^2*(c^2*d+e))^(1/2)*e)*c^2*arctan h(e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/((-2*c^2*d+2*(d*c^2*(c^2*d+e))^(1/2) -e)*e)^(1/2))/d/(c^2*d+e)/e^3-1/2*(-(2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)+e)* e)^(1/2)*(2*c^2*d+2*(d*c^2*(c^2*d+e))^(1/2)+e)*arctanh(e*(c*x+(c*x-1)^(1/2 )*(c*x+1)^(1/2))/((-2*c^2*d+2*(d*c^2*(c^2*d+e))^(1/2)-e)*e)^(1/2))*c^2/d/e ^3+1/2*((2*c^2*d+2*(d*c^2*(c^2*d+e))^(1/2)+e)*e)^(1/2)*(-2*(d*c^2*(c^2*d+e ))^(1/2)*c^2*d+2*c^4*d^2+2*c^2*d*e-(d*c^2*(c^2*d+e))^(1/2)*e)*c^2*arctan(e *(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/((2*c^2*d+2*(d*c^2*(c^2*d+e))^(1/2)+e)* e)^(1/2))/d/(c^2*d+e)/e^3-1/2*((2*c^2*d+2*(d*c^2*(c^2*d+e))^(1/2)+e)*e)^(1 /2)*(2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)+e)*arctan(e*(c*x+(c*x-1)^(1/2)*(c*x +1)^(1/2))/((2*c^2*d+2*(d*c^2*(c^2*d+e))^(1/2)+e)*e)^(1/2))*c^2/d/e^3-1/4/ d*c^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))/_R1)),_R 1=RootOf(e*_Z^4+(4*c^2*d+2*e)*_Z^2+e)))
\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^2} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^2} \, dx=\int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{\left (d + e x^{2}\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^2} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^2} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]