Integrand size = 21, antiderivative size = 792 \[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx=\frac {a+b \text {arccosh}(c x)}{4 e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {a+b \text {arccosh}(c x)}{4 e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {-1+c x}}\right )}{2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{3/2}}+\frac {b c \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {-1+c x}}\right )}{2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{3/2}}+\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{4 \sqrt {-d} e^{3/2}} \]
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)))/e^(3/2)/(-d)^(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) ))/e^(3/2)/(-d)^(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)))/e^(3/2)/(-d)^(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)))/e^(3/2)/(-d)^(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)))/e^(3/2)/(-d) ^(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)))/e^(3/2)/(-d)^(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)))/e^(3/2)/(-d)^ (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)))/e^(3/2)/(-d)^(1/2)+1/4*(a+b*arccosh(c*x))/e^(3/2)/( (-d)^(1/2)-x*e^(1/2))+1/4*(-a-b*arccosh(c*x))/e^(3/2)/((-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))/e^(3/2)/(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))/e^(3/2)/(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 = 6.08 (sec) , antiderivative size = 719, normalized size of antiderivative = 0.91 \[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx=\frac {-\frac {4 a \sqrt {e} x}{d+e x^2}+\frac {4 a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}+b \left (-\frac {2 \text {arccosh}(c x)}{i \sqrt {d}+\sqrt {e} x}-2 \left (\frac {\text {arccosh}(c x)}{-i \sqrt {d}+\sqrt {e} x}+\frac {c \log \left (\frac {2 e \left (i \sqrt {e}+c^2 \sqrt {d} x-i \sqrt {-c^2 d-e} \sqrt {-1+c x} \sqrt {1+c x}\right )}{c \sqrt {-c^2 d-e} \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{\sqrt {-c^2 d-e}}\right )-\frac {2 c \log \left (\frac {2 e \left (-\sqrt {e}-i c^2 \sqrt {d} x+\sqrt {-c^2 d-e} \sqrt {-1+c x} \sqrt {1+c x}\right )}{c \sqrt {-c^2 d-e} \left (i \sqrt {d}+\sqrt {e} x\right )}\right )}{\sqrt {-c^2 d-e}}+\frac {i \left (\text {arccosh}(c x) \left (-\text {arccosh}(c x)+2 \left (\log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}-\sqrt {-c^2 d-e}}\right )+\log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )}{\sqrt {d}}+\frac {i \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)-2 \left (\log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+\log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )\right )-2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )-2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )}{\sqrt {d}}\right )}{8 e^{3/2}} \]
((-4*a*Sqrt[e]*x)/(d + e*x^2) + (4*a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[d] + b*((-2*ArcCosh[c*x])/(I*Sqrt[d] + Sqrt[e]*x) - 2*(ArcCosh[c*x]/((-I)*Sqr t[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*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*(ArcCosh[c*x]*(-Arc Cosh[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] + Sqr t[-(c^2*d) - e]))]))/Sqrt[d] + (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*PolyLo g[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])]) )/Sqrt[d]))/(8*e^(3/2))
Time = 2.38 (sec) , antiderivative size = 792, 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.
| \(\displaystyle \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 6374 |
| \(\displaystyle \int \left (\frac {a+b \text {arccosh}(c x)}{e \left (d+e x^2\right )}-\frac {d (a+b \text {arccosh}(c x))}{e \left (d+e x^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}+1\right )}{4 \sqrt {-d} e^{3/2}}+\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}+1\right )}{4 \sqrt {-d} e^{3/2}}+\frac {a+b \text {arccosh}(c x)}{4 e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {a+b \text {arccosh}(c x)}{4 e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {b c \text {arctanh}\left (\frac {\sqrt {c x+1} \sqrt {c \sqrt {-d}-\sqrt {e}}}{\sqrt {c x-1} \sqrt {c \sqrt {-d}+\sqrt {e}}}\right )}{2 e^{3/2} \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}}}+\frac {b c \text {arctanh}\left (\frac {\sqrt {c x+1} \sqrt {c \sqrt {-d}+\sqrt {e}}}{\sqrt {c x-1} \sqrt {c \sqrt {-d}-\sqrt {e}}}\right )}{2 e^{3/2} \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}}}\) |
(a + b*ArcCosh[c*x])/(4*e^(3/2)*(Sqrt[-d] - Sqrt[e]*x)) - (a + b*ArcCosh[c *x])/(4*e^(3/2)*(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*S qrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[c*Sqrt[-d] + Sqrt[e]]*e^(3/2)) + (b*c*ArcTa nh[(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] - Sqrt[e]]* Sqrt[-1 + c*x])])/(2*Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[c*Sqrt[-d] + Sqrt[e]] *e^(3/2)) + ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt [-d] - Sqrt[-(c^2*d) - e])])/(4*Sqrt[-d]*e^(3/2)) - ((a + b*ArcCosh[c*x])* Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(4*Sq rt[-d]*e^(3/2)) + ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/( c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(4*Sqrt[-d]*e^(3/2)) - ((a + b*ArcCosh[ c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])]) /(4*Sqrt[-d]*e^(3/2)) - (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[- d] - Sqrt[-(c^2*d) - e]))])/(4*Sqrt[-d]*e^(3/2)) + (b*PolyLog[2, (Sqrt[e]* E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(4*Sqrt[-d]*e^(3/2)) - (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e] ))])/(4*Sqrt[-d]*e^(3/2)) + (b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt [-d] + Sqrt[-(c^2*d) - e])])/(4*Sqrt[-d]*e^(3/2))
3.6.3.3.1 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 = 24.16 (sec) , antiderivative size = 813, normalized size of antiderivative = 1.03
| method | result | size |
| derivativedivides | \(\frac {-\frac {a \,c^{5} x}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {a \,c^{3} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 e \sqrt {d e}}+b \,c^{4} \left (-\frac {\operatorname {arccosh}\left (c x \right ) c x}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}-\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 e^{4} \left (c^{2} d +e \right )}+\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 e^{4}}-\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 e^{4} \left (c^{2} d +e \right )}+\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 e^{4}}+\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 e}-\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 e}\right )}{c^{3}}\) | \(813\) |
| default | \(\frac {-\frac {a \,c^{5} x}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {a \,c^{3} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 e \sqrt {d e}}+b \,c^{4} \left (-\frac {\operatorname {arccosh}\left (c x \right ) c x}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}-\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 e^{4} \left (c^{2} d +e \right )}+\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 e^{4}}-\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 e^{4} \left (c^{2} d +e \right )}+\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 e^{4}}+\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 e}-\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 e}\right )}{c^{3}}\) | \(813\) |
| parts | \(-\frac {a x}{2 e \left (e \,x^{2}+d \right )}+\frac {a \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 e \sqrt {d e}}+\frac {b \left (-\frac {c^{5} \operatorname {arccosh}\left (c x \right ) x}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}-\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^{4} \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 e^{4} \left (c^{2} d +e \right )}+\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^{4}}{2 e^{4}}-\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^{4} \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 e^{4} \left (c^{2} d +e \right )}+\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^{4}}{2 e^{4}}+\frac {c^{4} \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 e}-\frac {c^{4} \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 e}\right )}{c^{3}}\) | \(816\) |
1/c^3*(-1/2*a*c^5/e*x/(c^2*e*x^2+c^2*d)+1/2*a*c^3/e/(d*e)^(1/2)*arctan(e*x /(d*e)^(1/2))+b*c^4*(-1/2*arccosh(c*x)/e*c*x/(c^2*e*x^2+c^2*d)-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)*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))/e^4/(c^2 *d+e)+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))/e^4-1/2*(-(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)*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))/e^4/(c^2*d+e)+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))/e^4+1/4/e*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))-1/4/e*s um(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)),_R1=RootO f(e*_Z^4+(4*c^2*d+2*e)*_Z^2+e))))
\[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
\[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx=\int \frac {x^{2} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]