Integrand size = 21, antiderivative size = 772 \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d+e x^2\right )^3} \, dx=-\frac {b c e x \left (1-c^2 x^2\right )}{8 d^2 \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )}+\frac {a+b \text {arccosh}(c x)}{4 d \left (d+e x^2\right )^2}+\frac {a+b \text {arccosh}(c x)}{2 d^2 \left (d+e x^2\right )}+\frac {(a+b \text {arccosh}(c x))^2}{b d^3}-\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^{5/2} \sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c \left (2 c^2 d+e\right ) \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 )}{8 d^{5/2} \left (c^2 d+e\right )^{3/2} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{d^3}-\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^3}-\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^3}-\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^3}-\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^3}-\frac {b \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )}{2 d^3}-\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^3}-\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^3}-\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^3}-\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^3} \]
1/4*(a+b*arccosh(c*x))/d/(e*x^2+d)^2+1/2*(a+b*arccosh(c*x))/d^2/(e*x^2+d)+ (a+b*arccosh(c*x))^2/b/d^3+(a+b*arccosh(c*x))*ln(1+1/(c*x+(c*x-1)^(1/2)*(c *x+1)^(1/2))^2)/d^3-1/2*(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^3-1/2*(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^3-1/2*(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^3-1/2*(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^3-1/2*b*polylog(2,-1/(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)/d^3-1/2*b*poly log(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*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*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*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^ 3-1/8*b*c*e*x*(-c^2*x^2+1)/d^2/(c^2*d+e)/(e*x^2+d)/(c*x-1)^(1/2)/(c*x+1)^( 1/2)-1/8*b*c*(2*c^2*d+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)^(3/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)- 1/2*b*c*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 = 6.07 (sec) , antiderivative size = 1204, normalized size of antiderivative = 1.56 \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d+e x^2\right )^3} \, dx=\frac {a}{4 d \left (d+e x^2\right )^2}+\frac {a}{2 d^2 \left (d+e x^2\right )}+\frac {a \log (x)}{d^3}-\frac {a \log \left (d+e x^2\right )}{2 d^3}+b \left (-\frac {5 i \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 )}{16 d^{5/2}}-\frac {5 i \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 )}{16 d^{5/2}}+\frac {\sqrt {e} \left (\frac {c \sqrt {-1+c x} \sqrt {1+c x}}{\left (c^2 d+e\right ) \left (-i \sqrt {d}+\sqrt {e} x\right )}-\frac {\text {arccosh}(c x)}{\sqrt {e} \left (-i \sqrt {d}+\sqrt {e} x\right )^2}+\frac {c^3 \sqrt {d} \left (\log (4)+\log \left (\frac {e \sqrt {c^2 d+e} \left (-i \sqrt {e}-c^2 \sqrt {d} x+\sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}\right )}{c^3 \left (d+i \sqrt {d} \sqrt {e} x\right )}\right )\right )}{\sqrt {e} \left (c^2 d+e\right )^{3/2}}\right )}{16 d^2}+\frac {\sqrt {e} \left (\frac {c \sqrt {-1+c x} \sqrt {1+c x}}{\left (c^2 d+e\right ) \left (i \sqrt {d}+\sqrt {e} x\right )}-\frac {\text {arccosh}(c x)}{\sqrt {e} \left (i \sqrt {d}+\sqrt {e} x\right )^2}-\frac {c^3 \sqrt {d} \left (\log (4)+\log \left (\frac {e \sqrt {c^2 d+e} \left (-i \sqrt {e}+c^2 \sqrt {d} x+\sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}\right )}{c^3 \left (d-i \sqrt {d} \sqrt {e} x\right )}\right )\right )}{\sqrt {e} \left (c^2 d+e\right )^{3/2}}\right )}{16 d^2}+\frac {\text {arccosh}(c x) \left (\text {arccosh}(c x)+2 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )-\operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )}{2 d^3}-\frac {\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 )}{4 d^3}-\frac {\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 )}{4 d^3}\right ) \]
a/(4*d*(d + e*x^2)^2) + a/(2*d^2*(d + e*x^2)) + (a*Log[x])/d^3 - (a*Log[d + e*x^2])/(2*d^3) + b*((((-5*I)/16)*(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))])/S qrt[-(c^2*d) - e]))/d^(5/2) - (((5*I)/16)*(-(ArcCosh[c*x]/(I*Sqrt[d] + Sqr t[e]*x)) - (c*Log[(2*e*(-Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[-(c^2*d) - e]*Sq rt[-1 + c*x]*Sqrt[1 + c*x]))/(c*Sqrt[-(c^2*d) - e]*(I*Sqrt[d] + Sqrt[e]*x) )])/Sqrt[-(c^2*d) - e]))/d^(5/2) + (Sqrt[e]*((c*Sqrt[-1 + c*x]*Sqrt[1 + c* x])/((c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x)) - ArcCosh[c*x]/(Sqrt[e]*((-I) *Sqrt[d] + Sqrt[e]*x)^2) + (c^3*Sqrt[d]*(Log[4] + Log[(e*Sqrt[c^2*d + e]*( (-I)*Sqrt[e] - c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x ]))/(c^3*(d + I*Sqrt[d]*Sqrt[e]*x))]))/(Sqrt[e]*(c^2*d + e)^(3/2))))/(16*d ^2) + (Sqrt[e]*((c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/((c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x)) - ArcCosh[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) - (c^3*Sqr t[d]*(Log[4] + Log[(e*Sqrt[c^2*d + e]*((-I)*Sqrt[e] + c^2*Sqrt[d]*x + Sqrt [c^2*d + e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c^3*(d - I*Sqrt[d]*Sqrt[e]*x)) ]))/(Sqrt[e]*(c^2*d + e)^(3/2))))/(16*d^2) + (ArcCosh[c*x]*(ArcCosh[c*x] + 2*Log[1 + E^(-2*ArcCosh[c*x])]) - PolyLog[2, -E^(-2*ArcCosh[c*x])])/(2*d^ 3) - (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...
Time = 1.71 (sec) , antiderivative size = 772, 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.
| \(\displaystyle \int \frac {a+b \text {arccosh}(c x)}{x \left (d+e x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 6374 |
| \(\displaystyle \int \left (-\frac {e x (a+b \text {arccosh}(c x))}{d^3 \left (d+e x^2\right )}+\frac {a+b \text {arccosh}(c x)}{d^3 x}-\frac {e x (a+b \text {arccosh}(c x))}{d^2 \left (d+e x^2\right )^2}-\frac {e x (a+b \text {arccosh}(c x))}{d \left (d+e x^2\right )^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{2 d^3}-\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}+1\right )}{2 d^3}-\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{2 d^3}-\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}+1\right )}{2 d^3}+\frac {(a+b \text {arccosh}(c x))^2}{b d^3}+\frac {\log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))}{d^3}+\frac {a+b \text {arccosh}(c x)}{2 d^2 \left (d+e x^2\right )}+\frac {a+b \text {arccosh}(c x)}{4 d \left (d+e x^2\right )^2}-\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 d^3}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 d^3}-\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 d^3}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 d^3}-\frac {b \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )}{2 d^3}-\frac {b c \sqrt {c^2 x^2-1} \left (2 c^2 d+e\right ) \text {arctanh}\left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{8 d^{5/2} \sqrt {c x-1} \sqrt {c x+1} \left (c^2 d+e\right )^{3/2}}-\frac {b c \sqrt {c^2 x^2-1} \text {arctanh}\left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{2 d^{5/2} \sqrt {c x-1} \sqrt {c x+1} \sqrt {c^2 d+e}}-\frac {b c e x \left (1-c^2 x^2\right )}{8 d^2 \sqrt {c x-1} \sqrt {c x+1} \left (c^2 d+e\right ) \left (d+e x^2\right )}\) |
-1/8*(b*c*e*x*(1 - c^2*x^2))/(d^2*(c^2*d + e)*Sqrt[-1 + c*x]*Sqrt[1 + c*x] *(d + e*x^2)) + (a + b*ArcCosh[c*x])/(4*d*(d + e*x^2)^2) + (a + b*ArcCosh[ c*x])/(2*d^2*(d + e*x^2)) + (a + b*ArcCosh[c*x])^2/(b*d^3) - (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 ^(5/2)*Sqrt[c^2*d + e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*(2*c^2*d + e)* Sqrt[-1 + c^2*x^2]*ArcTanh[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[-1 + c^2*x^2] )])/(8*d^(5/2)*(c^2*d + e)^(3/2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + ((a + b*A rcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])])/d^3 - ((a + b*ArcCosh[c*x])*Log [1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*d^3) - ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqr t[-(c^2*d) - e])])/(2*d^3) - ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcC osh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*d^3) - ((a + b*ArcCosh[c* x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/( 2*d^3) - (b*PolyLog[2, -E^(-2*ArcCosh[c*x])])/(2*d^3) - (b*PolyLog[2, -((S qrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e]))])/(2*d^3) - (b*P olyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2* d^3) - (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d ) - e]))])/(2*d^3) - (b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*d^3)
3.6.9.3.1 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 = 2.44 (sec) , antiderivative size = 1225, normalized size of antiderivative = 1.59
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1225\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1278\) |
| default | \(\text {Expression too large to display}\) | \(1278\) |
1/4*a/d/(e*x^2+d)^2-1/2*a/d^3*ln(e*x^2+d)+1/2*a/d^2/(e*x^2+d)+a/d^3*ln(x)+ b*(1/8*c^2*(6*c^4*d^2*arccosh(c*x)+4*arccosh(c*x)*c^4*d*e*x^2+(c*x-1)^(1/2 )*(c*x+1)^(1/2)*c^3*d*e*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)*e^2*c^3*x^3-c^4*d^2- 2*c^4*d*e*x^2-c^4*e^2*x^4+6*c^2*d*e*arccosh(c*x)+4*arccosh(c*x)*e^2*c^2*x^ 2)/d^2/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2+5/8*(d*c^2*(c^2*d+e))^(1/2)/(c^2*d+e) ^2/d^3*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))+3/4*(d*c^2*(c^2*d+e))^(1/2)/(c^2*d+e)^2/d^2*c^2* 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/(c^2*d+e)/d^3*e*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))+1/(c^2*d+e)/d^3*e*arccosh(c*x)*ln(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1) ^(1/2)))+1/(c^2*d+e)/d^3*e*arccosh(c*x)*ln(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^ (1/2)))+1/(c^2*d+e)/d^3*e*dilog(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))+1/( c^2*d+e)/d^3*e*dilog(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))-1/4/(c^2*d+e)/ d^3*e^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)+dilog((_R1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R 1)),_R1=RootOf(e*_Z^4+(4*c^2*d+2*e)*_Z^2+e))-1/4/(c^2*d+e)/d^2*c^2*sum((_R 1^2*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)...
\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d+e x^2\right )^3} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3} x} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d+e x^2\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d+e x^2\right )^3} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3} x} \,d x } \]
1/4*a*((2*e*x^2 + 3*d)/(d^2*e^2*x^4 + 2*d^3*e*x^2 + d^4) - 2*log(e*x^2 + d )/d^3 + 4*log(x)/d^3) + b*integrate(log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) /(e^3*x^7 + 3*d*e^2*x^5 + 3*d^2*e*x^3 + d^3*x), x)
\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d+e x^2\right )^3} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3} x} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d+e x^2\right )^3} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x\,{\left (e\,x^2+d\right )}^3} \,d x \]