Integrand size = 21, antiderivative size = 581 \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d+e x^2\right )^2} \, dx=\frac {a+b \text {arccosh}(c x)}{2 d \left (d+e x^2\right )}-\frac {b c \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{2 d^{3/2} \sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^2}-\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^2}-\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^2}-\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^2}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+e^{2 \text {arccosh}(c x)}\right )}{d^2}-\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^2}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^2}-\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^2}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^2}+\frac {b \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )}{2 d^2} \] Output:
1/2*(a+b*arccosh(c*x))/d/(e*x^2+d)-1/2*b*c*(c^2*x^2-1)^(1/2)*arctanh((c^2* d+e)^(1/2)*x/d^(1/2)/(c^2*x^2-1)^(1/2))/d^(3/2)/(c^2*d+e)^(1/2)/(c*x-1)^(1 /2)/(c*x+1)^(1/2)-1/2*(a+b*arccosh(c*x))*ln(1-e^(1/2)*(c*x+(c*x-1)^(1/2)*( c*x+1)^(1/2))/(c*(-d)^(1/2)-(-c^2*d-e)^(1/2)))/d^2-1/2*(a+b*arccosh(c*x))* ln(1+e^(1/2)*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*(-d)^(1/2)-(-c^2*d-e)^(1 /2)))/d^2-1/2*(a+b*arccosh(c*x))*ln(1-e^(1/2)*(c*x+(c*x-1)^(1/2)*(c*x+1)^( 1/2))/(c*(-d)^(1/2)+(-c^2*d-e)^(1/2)))/d^2-1/2*(a+b*arccosh(c*x))*ln(1+e^( 1/2)*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*(-d)^(1/2)+(-c^2*d-e)^(1/2)))/d^ 2+(a+b*arccosh(c*x))*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)/d^2-1/2*b*p olylog(2,-e^(1/2)*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*(-d)^(1/2)-(-c^2*d- e)^(1/2)))/d^2-1/2*b*polylog(2,e^(1/2)*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/( c*(-d)^(1/2)-(-c^2*d-e)^(1/2)))/d^2-1/2*b*polylog(2,-e^(1/2)*(c*x+(c*x-1)^ (1/2)*(c*x+1)^(1/2))/(c*(-d)^(1/2)+(-c^2*d-e)^(1/2)))/d^2-1/2*b*polylog(2, e^(1/2)*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*(-d)^(1/2)+(-c^2*d-e)^(1/2))) /d^2+1/2*b*polylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)/d^2
Result contains complex when optimal does not.
Time = 1.06 (sec) , antiderivative size = 754, normalized size of antiderivative = 1.30 \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d+e x^2\right )^2} \, dx=\frac {a}{2 d^2+2 d e x^2}+\frac {a \log (x)}{d^2}-\frac {a \log \left (d+e x^2\right )}{2 d^2}-\frac {b \left (-\frac {\sqrt {d} \text {arccosh}(c x)}{\sqrt {d}-i \sqrt {e} x}-\frac {\sqrt {d} \text {arccosh}(c x)}{\sqrt {d}+i \sqrt {e} x}-4 \text {arccosh}(c x)^2-4 \text {arccosh}(c x) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )+2 \text {arccosh}(c x) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}-\sqrt {-c^2 d-e}}\right )+2 \text {arccosh}(c x) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+2 \text {arccosh}(c x) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+2 \text {arccosh}(c x) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+\frac {i c \sqrt {d} \log \left (\frac {2 e \left (i \sqrt {e}+c^2 \sqrt {d} x-i \sqrt {-c^2 d-e} \sqrt {-1+c x} \sqrt {1+c x}\right )}{c \sqrt {-c^2 d-e} \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{\sqrt {-c^2 d-e}}-\frac {i c \sqrt {d} \log \left (\frac {2 e \left (-\sqrt {e}-i c^2 \sqrt {d} x+\sqrt {-c^2 d-e} \sqrt {-1+c x} \sqrt {1+c x}\right )}{c \sqrt {-c^2 d-e} \left (i \sqrt {d}+\sqrt {e} x\right )}\right )}{\sqrt {-c^2 d-e}}+2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}-\sqrt {-c^2 d-e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )}{4 d^2} \] Input:
Integrate[(a + b*ArcCosh[c*x])/(x*(d + e*x^2)^2),x]
Output:
a/(2*d^2 + 2*d*e*x^2) + (a*Log[x])/d^2 - (a*Log[d + e*x^2])/(2*d^2) - (b*( -((Sqrt[d]*ArcCosh[c*x])/(Sqrt[d] - I*Sqrt[e]*x)) - (Sqrt[d]*ArcCosh[c*x]) /(Sqrt[d] + I*Sqrt[e]*x) - 4*ArcCosh[c*x]^2 - 4*ArcCosh[c*x]*Log[1 + E^(-2 *ArcCosh[c*x])] + 2*ArcCosh[c*x]*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqr t[d] - Sqrt[-(c^2*d) - e])] + 2*ArcCosh[c*x]*Log[1 + (Sqrt[e]*E^ArcCosh[c* x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e])] + 2*ArcCosh[c*x]*Log[1 - (Sqrt[ e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])] + 2*ArcCosh[c*x]*Lo g[1 + (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])] + (I*c* Sqrt[d]*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))])/S qrt[-(c^2*d) - e] - (I*c*Sqrt[d]*Log[(2*e*(-Sqrt[e] - I*c^2*Sqrt[d]*x + Sq rt[-(c^2*d) - e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c*Sqrt[-(c^2*d) - e]*(I*S qrt[d] + Sqrt[e]*x))])/Sqrt[-(c^2*d) - e] + 2*PolyLog[2, -E^(-2*ArcCosh[c* x])] + 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])] + 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] + Sq rt[-(c^2*d) - e])]))/(4*d^2)
Time = 1.85 (sec) , antiderivative size = 598, normalized size of antiderivative = 1.03, 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.
| \(\displaystyle \int \frac {a+b \text {arccosh}(c x)}{x \left (d+e x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 6374 |
| \(\displaystyle \int \left (-\frac {e x (a+b \text {arccosh}(c x))}{d^2 \left (d+e x^2\right )}+\frac {a+b \text {arccosh}(c x)}{d^2 x}-\frac {e x (a+b \text {arccosh}(c x))}{d \left (d+e x^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{2 d^2}-\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}+1\right )}{2 d^2}-\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{2 d^2}-\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}+1\right )}{2 d^2}+\frac {(a+b \text {arccosh}(c x))^2}{b d^2}+\frac {\log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))}{d^2}+\frac {a+b \text {arccosh}(c x)}{2 d \left (d+e x^2\right )}-\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 d^2}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 d^2}-\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 d^2}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 d^2}-\frac {b \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )}{2 d^2}-\frac {b c \sqrt {c^2 x^2-1} \text {arctanh}\left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{2 d^{3/2} \sqrt {c x-1} \sqrt {c x+1} \sqrt {c^2 d+e}}\) |
Input:
Int[(a + b*ArcCosh[c*x])/(x*(d + e*x^2)^2),x]
Output:
(a + b*ArcCosh[c*x])/(2*d*(d + e*x^2)) + (a + b*ArcCosh[c*x])^2/(b*d^2) - (b*c*Sqrt[-1 + c^2*x^2]*ArcTanh[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[-1 + c^2 *x^2])])/(2*d^(3/2)*Sqrt[c^2*d + e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + ((a + b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])])/d^2 - ((a + b*ArcCosh[c*x])* Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*d^ 2) - ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*d^2) - ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^A rcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*d^2) - ((a + b*ArcCosh [c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])] )/(2*d^2) - (b*PolyLog[2, -E^(-2*ArcCosh[c*x])])/(2*d^2) - (b*PolyLog[2, - ((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e]))])/(2*d^2) - ( b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/ (2*d^2) - (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^ 2*d) - e]))])/(2*d^2) - (b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*d^2)
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 = 0.63 (sec) , antiderivative size = 505, normalized size of antiderivative = 0.87
| method | result | size |
| parts | \(\frac {a}{2 d \left (e \,x^{2}+d \right )}-\frac {a \ln \left (e \,x^{2}+d \right )}{2 d^{2}}+\frac {a \ln \left (x \right )}{d^{2}}+b \left (\frac {c^{2} \operatorname {arccosh}\left (c x \right )}{2 d \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {\sqrt {c^{2} d \left (c^{2} d +e \right )}\, \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^{2} \left (c^{2} d +e \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 {\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}}{4 d^{2}}+\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{2}}+\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{2}}+\frac {\operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{2}}+\frac {\operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{2}}-\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}+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 )}{4 d^{2}}\right )\) | \(505\) |
| derivativedivides | \(\frac {a \ln \left (c x \right )}{d^{2}}+\frac {a \,c^{2}}{2 d \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {a \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{2 d^{2}}+\frac {b \,c^{2} \operatorname {arccosh}\left (c x \right )}{2 d \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {b \sqrt {c^{2} d \left (c^{2} d +e \right )}\, \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^{2} \left (c^{2} d +e \right )}-\frac {b \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 )}{4 d^{2}}+\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{2}}+\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{2}}+\frac {b \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{2}}+\frac {b \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{2}}-\frac {b 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}+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 )}{4 d^{2}}\) | \(529\) |
| default | \(\frac {a \ln \left (c x \right )}{d^{2}}+\frac {a \,c^{2}}{2 d \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {a \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{2 d^{2}}+\frac {b \,c^{2} \operatorname {arccosh}\left (c x \right )}{2 d \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {b \sqrt {c^{2} d \left (c^{2} d +e \right )}\, \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^{2} \left (c^{2} d +e \right )}-\frac {b \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 )}{4 d^{2}}+\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{2}}+\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{2}}+\frac {b \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{2}}+\frac {b \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{2}}-\frac {b 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}+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 )}{4 d^{2}}\) | \(529\) |
Input:
int((a+b*arccosh(c*x))/x/(e*x^2+d)^2,x,method=_RETURNVERBOSE)
Output:
1/2*a/d/(e*x^2+d)-1/2*a/d^2*ln(e*x^2+d)+a/d^2*ln(x)+b*(1/2*c^2*arccosh(c*x )/d/(c^2*e*x^2+c^2*d)+1/2*(c^2*d*(c^2*d+e))^(1/2)/d^2/(c^2*d+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))-1/4/d^2*sum((_R1^2*e+4*c^2*d+e)/(_R1^2*e+2*c^2*d+e)*(arccosh(c*x)*l n((_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/d^2*arccos h(c*x)*ln(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))+1/d^2*arccosh(c*x)*ln(1-I *(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))+1/d^2*dilog(1+I*(c*x+(c*x-1)^(1/2)*(c* x+1)^(1/2)))+1/d^2*dilog(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))-1/4/d^2*e* 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)+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)))
\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d+e x^2\right )^2} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x} \,d x } \] Input:
integrate((a+b*arccosh(c*x))/x/(e*x^2+d)^2,x, algorithm="fricas")
Output:
integral((b*arccosh(c*x) + a)/(e^2*x^5 + 2*d*e*x^3 + d^2*x), x)
\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d+e x^2\right )^2} \, dx=\int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{x \left (d + e x^{2}\right )^{2}}\, dx \] Input:
integrate((a+b*acosh(c*x))/x/(e*x**2+d)**2,x)
Output:
Integral((a + b*acosh(c*x))/(x*(d + e*x**2)**2), x)
\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d+e x^2\right )^2} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x} \,d x } \] Input:
integrate((a+b*arccosh(c*x))/x/(e*x^2+d)^2,x, algorithm="maxima")
Output:
1/2*a*(1/(d*e*x^2 + d^2) - log(e*x^2 + d)/d^2 + 2*log(x)/d^2) + b*integrat e(log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(e^2*x^5 + 2*d*e*x^3 + d^2*x), x)
\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d+e x^2\right )^2} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x} \,d x } \] Input:
integrate((a+b*arccosh(c*x))/x/(e*x^2+d)^2,x, algorithm="giac")
Output:
integrate((b*arccosh(c*x) + a)/((e*x^2 + d)^2*x), x)
Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d+e x^2\right )^2} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x\,{\left (e\,x^2+d\right )}^2} \,d x \] Input:
int((a + b*acosh(c*x))/(x*(d + e*x^2)^2),x)
Output:
int((a + b*acosh(c*x))/(x*(d + e*x^2)^2), x)
\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d+e x^2\right )^2} \, dx=\frac {2 \left (\int \frac {\mathit {acosh} \left (c x \right )}{e^{2} x^{5}+2 d e \,x^{3}+d^{2} x}d x \right ) b \,d^{3}+2 \left (\int \frac {\mathit {acosh} \left (c x \right )}{e^{2} x^{5}+2 d e \,x^{3}+d^{2} x}d x \right ) b \,d^{2} e \,x^{2}-\mathrm {log}\left (e \,x^{2}+d \right ) a d -\mathrm {log}\left (e \,x^{2}+d \right ) a e \,x^{2}+2 \,\mathrm {log}\left (x \right ) a d +2 \,\mathrm {log}\left (x \right ) a e \,x^{2}-a e \,x^{2}}{2 d^{2} \left (e \,x^{2}+d \right )} \] Input:
int((a+b*acosh(c*x))/x/(e*x^2+d)^2,x)
Output:
(2*int(acosh(c*x)/(d**2*x + 2*d*e*x**3 + e**2*x**5),x)*b*d**3 + 2*int(acos h(c*x)/(d**2*x + 2*d*e*x**3 + e**2*x**5),x)*b*d**2*e*x**2 - log(d + e*x**2 )*a*d - log(d + e*x**2)*a*e*x**2 + 2*log(x)*a*d + 2*log(x)*a*e*x**2 - a*e* x**2)/(2*d**2*(d + e*x**2))