Integrand size = 21, antiderivative size = 834 \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d+e x^2\right )^3} \, dx=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d^3 x}+\frac {b c e^2 x \left (1-c^2 x^2\right )}{8 d^3 \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)}{2 d^3 x^2}-\frac {e (a+b \text {arccosh}(c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {e (a+b \text {arccosh}(c x))}{d^3 \left (d+e x^2\right )}-\frac {3 e (a+b \text {arccosh}(c x))^2}{b d^4}+\frac {b c e \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 )}{d^{7/2} \sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c e \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^{7/2} \left (c^2 d+e\right )^{3/2} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 e (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{d^4}+\frac {3 e (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^4}+\frac {3 e (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^4}+\frac {3 e (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^4}+\frac {3 e (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^4}+\frac {3 b e \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )}{2 d^4}+\frac {3 b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^4} \]
1/2*(-a-b*arccosh(c*x))/d^3/x^2-1/4*e*(a+b*arccosh(c*x))/d^2/(e*x^2+d)^2-e *(a+b*arccosh(c*x))/d^3/(e*x^2+d)-3*e*(a+b*arccosh(c*x))^2/b/d^4-3*e*(a+b* arccosh(c*x))*ln(1+1/(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)/d^4+3/2*e*(a+b*a rccosh(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^4+3/2*e*(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^4+3/2*e*(a+b*arcco sh(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^4+3/2*e*(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^4+3/2*b*e*polylog(2,-1 /(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)/d^4+3/2*b*e*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^4+3/2*b*e* 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^4+3/2*b*e*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^4+3/2*b*e*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^4+1/8*b*c*e^2* x*(-c^2*x^2+1)/d^3/(c^2*d+e)/(e*x^2+d)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/2*b*c *(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/x+1/8*b*c*e*(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^(7/2)/(c^2*d+e)^(3 /2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+b*c*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^(7/2)/(c^2*d+e)^(1/2)/(c*x-1)^(1/2...
Result contains complex when optimal does not.
Time = 6.09 (sec) , antiderivative size = 1261, normalized size of antiderivative = 1.51 \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d+e x^2\right )^3} \, dx=-\frac {a}{2 d^3 x^2}-\frac {a e}{4 d^2 \left (d+e x^2\right )^2}-\frac {a e}{d^3 \left (d+e x^2\right )}-\frac {3 a e \log (x)}{d^4}+\frac {3 a e \log \left (d+e x^2\right )}{2 d^4}+b \left (\frac {c x \sqrt {-1+c x} \sqrt {1+c x}-\text {arccosh}(c x)}{2 d^3 x^2}+\frac {9 i e \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^{7/2}}+\frac {9 i e \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^{7/2}}-\frac {e^{3/2} \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^3}-\frac {e^{3/2} \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^3}-\frac {3 e \left (\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 )\right )}{2 d^4}+\frac {3 e \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 )}{4 d^4}+\frac {3 e \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 )}{4 d^4}\right ) \]
-1/2*a/(d^3*x^2) - (a*e)/(4*d^2*(d + e*x^2)^2) - (a*e)/(d^3*(d + e*x^2)) - (3*a*e*Log[x])/d^4 + (3*a*e*Log[d + e*x^2])/(2*d^4) + b*((c*x*Sqrt[-1 + c *x]*Sqrt[1 + c*x] - ArcCosh[c*x])/(2*d^3*x^2) + (((9*I)/16)*e*(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]*(S qrt[d] + I*Sqrt[e]*x))])/Sqrt[-(c^2*d) - e]))/d^(7/2) + (((9*I)/16)*e*(-(A rcCosh[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]))/d^(7/2) - (e^(3/2)*(( 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^3) - (e^(3/2)*((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)*Sq rt[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^3) - ( 3*e*(ArcCosh[c*x]*(ArcCosh[c*x] + 2*Log[1 + E^(-2*ArcCosh[c*x])]) - PolyLo g[2, -E^(-2*ArcCosh[c*x])]))/(2*d^4) + (3*e*(ArcCosh[c*x]*(-ArcCosh[c*x...
Time = 1.83 (sec) , antiderivative size = 834, 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.
| \(\displaystyle \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d+e x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 6374 |
| \(\displaystyle \int \left (\frac {3 e^2 x (a+b \text {arccosh}(c x))}{d^4 \left (d+e x^2\right )}-\frac {3 e (a+b \text {arccosh}(c x))}{d^4 x}+\frac {2 e^2 x (a+b \text {arccosh}(c x))}{d^3 \left (d+e x^2\right )^2}+\frac {a+b \text {arccosh}(c x)}{d^3 x^3}+\frac {e^2 x (a+b \text {arccosh}(c x))}{d^2 \left (d+e x^2\right )^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b c x \left (1-c^2 x^2\right ) e^2}{8 d^3 \left (d c^2+e\right ) \sqrt {c x-1} \sqrt {c x+1} \left (e x^2+d\right )}-\frac {3 (a+b \text {arccosh}(c x))^2 e}{b d^4}-\frac {(a+b \text {arccosh}(c x)) e}{d^3 \left (e x^2+d\right )}-\frac {(a+b \text {arccosh}(c x)) e}{4 d^2 \left (e x^2+d\right )^2}+\frac {b c \left (2 d c^2+e\right ) \sqrt {c^2 x^2-1} \text {arctanh}\left (\frac {\sqrt {d c^2+e} x}{\sqrt {d} \sqrt {c^2 x^2-1}}\right ) e}{8 d^{7/2} \left (d c^2+e\right )^{3/2} \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c \sqrt {c^2 x^2-1} \text {arctanh}\left (\frac {\sqrt {d c^2+e} x}{\sqrt {d} \sqrt {c^2 x^2-1}}\right ) e}{d^{7/2} \sqrt {d c^2+e} \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right ) e}{d^4}+\frac {3 (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right ) e}{2 d^4}+\frac {3 (a+b \text {arccosh}(c x)) \log \left (\frac {e^{\text {arccosh}(c x)} \sqrt {e}}{c \sqrt {-d}-\sqrt {-d c^2-e}}+1\right ) e}{2 d^4}+\frac {3 (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right ) e}{2 d^4}+\frac {3 (a+b \text {arccosh}(c x)) \log \left (\frac {e^{\text {arccosh}(c x)} \sqrt {e}}{\sqrt {-d} c+\sqrt {-d c^2-e}}+1\right ) e}{2 d^4}+\frac {3 b \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right ) e}{2 d^4}+\frac {3 b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right ) e}{2 d^4}+\frac {3 b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right ) e}{2 d^4}+\frac {3 b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right ) e}{2 d^4}+\frac {3 b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right ) e}{2 d^4}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 d^3 x}\) |
(b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*d^3*x) + (b*c*e^2*x*(1 - c^2*x^2))/( 8*d^3*(c^2*d + e)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x^2)) - (a + b*ArcCo sh[c*x])/(2*d^3*x^2) - (e*(a + b*ArcCosh[c*x]))/(4*d^2*(d + e*x^2)^2) - (e *(a + b*ArcCosh[c*x]))/(d^3*(d + e*x^2)) - (3*e*(a + b*ArcCosh[c*x])^2)/(b *d^4) + (b*c*e*Sqrt[-1 + c^2*x^2]*ArcTanh[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqr t[-1 + c^2*x^2])])/(d^(7/2)*Sqrt[c^2*d + e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c*e*(2*c^2*d + e)*Sqrt[-1 + c^2*x^2]*ArcTanh[(Sqrt[c^2*d + e]*x)/(Sqr t[d]*Sqrt[-1 + c^2*x^2])])/(8*d^(7/2)*(c^2*d + e)^(3/2)*Sqrt[-1 + c*x]*Sqr t[1 + c*x]) - (3*e*(a + b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])])/d^4 + (3*e*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*d^4) + (3*e*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e ]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*d^4) + (3*e*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2* d) - e])])/(2*d^4) + (3*e*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[ c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*d^4) + (3*b*e*PolyLog[2, -E^( -2*ArcCosh[c*x])])/(2*d^4) + (3*b*e*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/ (c*Sqrt[-d] - Sqrt[-(c^2*d) - e]))])/(2*d^4) + (3*b*e*PolyLog[2, (Sqrt[e]* E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*d^4) + (3*b*e*PolyL og[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e]))])/(2*d ^4) + (3*b*e*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c...
3.6.10.3.1 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 = 2.69 (sec) , antiderivative size = 1455, normalized size of antiderivative = 1.74
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1455\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1515\) |
| default | \(\text {Expression too large to display}\) | \(1515\) |
3/2*a*e/d^4*ln(e*x^2+d)-1/4*a*e/d^2/(e*x^2+d)^2-a*e/d^3/(e*x^2+d)-1/2*a/d^ 3/x^2-3*a/d^4*e*ln(x)+b*c^2*(-1/8*(-4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^7*d^3* x-8*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^7*d^2*e*x^3-4*(c*x-1)^(1/2)*(c*x+1)^(1/2 )*c^7*d*e^2*x^5+4*c^8*d^3*x^2+8*c^8*d^2*e*x^4+4*c^8*d*e^2*x^6+4*c^6*d^3*ar ccosh(c*x)+18*arccosh(c*x)*c^6*d^2*e*x^2+12*arccosh(c*x)*c^6*d*e^2*x^4-4*( c*x-1)^(1/2)*(c*x+1)^(1/2)*c^5*d^2*e*x-7*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^5*d *e^2*x^3-3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*e^3*c^5*x^5+3*c^6*d^2*e*x^2+6*c^6*d *e^2*x^4+3*c^6*e^3*x^6+4*c^4*d^2*e*arccosh(c*x)+18*arccosh(c*x)*c^4*d*e^2* x^2+12*arccosh(c*x)*e^3*c^4*x^4)/c^2/x^2/d^3/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2 -9/8*(d*c^2*(c^2*d+e))^(1/2)/(c^2*d+e)^2/d^4/c^2*e^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))-5/4* (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/(c^2*d+e)*e ^2/d^4/c^2*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))-3/(c^2*d+e)*e^2/ d^4/c^2*arccosh(c*x)*ln(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))-3/(c^2*d+e) *e^2/d^4/c^2*arccosh(c*x)*ln(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))-3/(c^2 *d+e)*e^2/d^4/c^2*dilog(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))-3/(c^2*d+e) *e^2/d^4/c^2*dilog(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))+3/4/(c^2*d+e)...
\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d+e x^2\right )^3} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3} x^{3}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d+e x^2\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d+e x^2\right )^3} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3} x^{3}} \,d x } \]
-1/4*a*((6*e^2*x^4 + 9*d*e*x^2 + 2*d^2)/(d^3*e^2*x^6 + 2*d^4*e*x^4 + d^5*x ^2) - 6*e*log(e*x^2 + d)/d^4 + 12*e*log(x)/d^4) + b*integrate(log(c*x + sq rt(c*x + 1)*sqrt(c*x - 1))/(e^3*x^9 + 3*d*e^2*x^7 + 3*d^2*e*x^5 + d^3*x^3) , x)
\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d+e x^2\right )^3} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3} x^{3}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d+e x^2\right )^3} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^3\,{\left (e\,x^2+d\right )}^3} \,d x \]