Integrand size = 21, antiderivative size = 741 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx=-\frac {b e \left (c^2-\frac {1}{x^2}\right )}{8 c d^2 \left (c^2 d+e\right ) \left (e+\frac {d}{x^2}\right ) \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}+\frac {e^2 \left (a+b \text {sech}^{-1}(c x)\right )}{4 d^3 \left (e+\frac {d}{x^2}\right )^2}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{d^3 \left (e+\frac {d}{x^2}\right )}+\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{2 b d^3}+\frac {b \sqrt {e} \sqrt {-1+\frac {1}{c^2 x^2}} \text {arctanh}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {-1+\frac {1}{c^2 x^2}} x}\right )}{d^3 \sqrt {c^2 d+e} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}-\frac {b \sqrt {e} \left (c^2 d+2 e\right ) \sqrt {-1+\frac {1}{c^2 x^2}} \text {arctanh}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {-1+\frac {1}{c^2 x^2}} x}\right )}{8 d^3 \left (c^2 d+e\right )^{3/2} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^3}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^3}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^3}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^3}-\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^3}-\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^3}-\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^3}-\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^3} \]
1/4*e^2*(a+b*arcsech(c*x))/d^3/(e+d/x^2)^2-e*(a+b*arcsech(c*x))/d^3/(e+d/x ^2)+1/2*(a+b*arcsech(c*x))^2/b/d^3-1/2*(a+b*arcsech(c*x))*ln(1-c*(1/c/x+(- 1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))*(-d)^(1/2)/(e^(1/2)-(c^2*d+e)^(1/2)))/d^3- 1/2*(a+b*arcsech(c*x))*ln(1+c*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))*(-d )^(1/2)/(e^(1/2)-(c^2*d+e)^(1/2)))/d^3-1/2*(a+b*arcsech(c*x))*ln(1-c*(1/c/ x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))*(-d)^(1/2)/(e^(1/2)+(c^2*d+e)^(1/2)))/ d^3-1/2*(a+b*arcsech(c*x))*ln(1+c*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2)) *(-d)^(1/2)/(e^(1/2)+(c^2*d+e)^(1/2)))/d^3-1/2*b*polylog(2,-c*(1/c/x+(-1+1 /c/x)^(1/2)*(1+1/c/x)^(1/2))*(-d)^(1/2)/(e^(1/2)-(c^2*d+e)^(1/2)))/d^3-1/2 *b*polylog(2,c*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))*(-d)^(1/2)/(e^(1/2 )-(c^2*d+e)^(1/2)))/d^3-1/2*b*polylog(2,-c*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/ x)^(1/2))*(-d)^(1/2)/(e^(1/2)+(c^2*d+e)^(1/2)))/d^3-1/2*b*polylog(2,c*(1/c /x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))*(-d)^(1/2)/(e^(1/2)+(c^2*d+e)^(1/2))) /d^3-1/8*b*e*(c^2-1/x^2)/c/d^2/(c^2*d+e)/(e+d/x^2)/x/(-1+1/c/x)^(1/2)/(1+1 /c/x)^(1/2)-1/8*b*(c^2*d+2*e)*arctanh((c^2*d+e)^(1/2)/c/x/e^(1/2)/(-1+1/c^ 2/x^2)^(1/2))*e^(1/2)*(-1+1/c^2/x^2)^(1/2)/d^3/(c^2*d+e)^(3/2)/(-1+1/c/x)^ (1/2)/(1+1/c/x)^(1/2)+b*arctanh((c^2*d+e)^(1/2)/c/x/e^(1/2)/(-1+1/c^2/x^2) ^(1/2))*e^(1/2)*(-1+1/c^2/x^2)^(1/2)/d^3/(c^2*d+e)^(1/2)/(-1+1/c/x)^(1/2)/ (1+1/c/x)^(1/2)
Result contains complex when optimal does not.
Time = 6.09 (sec) , antiderivative size = 2054, normalized size of antiderivative = 2.77 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx=\text {Result too large to show} \]
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*((Sqrt[e]*(((-I)*Sqrt[e]*Sqrt[(1 - c*x)/(1 + c*x)]*( 1 + c*x))/(Sqrt[d]*(c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x)) - ArcSech[c*x]/ (Sqrt[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2) + Log[x]/(d*Sqrt[e]) - Log[1 + Sqrt [(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]]/(d*Sqrt[e]) + ((2*c ^2*d + e)*Log[(-4*d*Sqrt[e]*Sqrt[c^2*d + e]*(Sqrt[e] - I*c^2*Sqrt[d]*x + S qrt[c^2*d + e]*Sqrt[(1 - c*x)/(1 + c*x)] + c*Sqrt[c^2*d + e]*x*Sqrt[(1 - c *x)/(1 + c*x)]))/((2*c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x))])/(d*(c^2*d + e)^(3/2))))/(16*d^2) + (Sqrt[e]*((I*Sqrt[e]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(Sqrt[d]*(c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x)) - ArcSech[c*x]/(Sqrt[ e]*(I*Sqrt[d] + Sqrt[e]*x)^2) + Log[x]/(d*Sqrt[e]) - Log[1 + Sqrt[(1 - c*x )/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]]/(d*Sqrt[e]) + ((2*c^2*d + e) *Log[(-4*d*Sqrt[e]*Sqrt[c^2*d + e]*(Sqrt[e] + I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[(1 - c*x)/(1 + c*x)] + c*Sqrt[c^2*d + e]*x*Sqrt[(1 - c*x)/(1 + c*x)]))/((2*c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x))])/(d*(c^2*d + e)^(3/2))))/ (16*d^2) - (((5*I)/16)*Sqrt[e]*(-(ArcSech[c*x]/(I*Sqrt[d]*Sqrt[e] + e*x)) + (I*(Log[x]/Sqrt[e] - Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c *x)/(1 + c*x)]]/Sqrt[e] + Log[((2*I)*Sqrt[e]*(Sqrt[d]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) + (Sqrt[d]*Sqrt[e] + I*c^2*d*x)/Sqrt[c^2*d + e]))/(I*Sqrt[ d] + Sqrt[e]*x)]/Sqrt[c^2*d + e]))/Sqrt[d]))/d^(5/2) + (((5*I)/16)*Sqrt...
Time = 1.83 (sec) , antiderivative size = 801, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6857, 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.
| \(\displaystyle \int \frac {a+b \text {sech}^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 6857 |
| \(\displaystyle -\int \frac {a+b \text {arccosh}\left (\frac {1}{c x}\right )}{\left (\frac {d}{x^2}+e\right )^3 x^5}d\frac {1}{x}\) |
| \(\Big \downarrow \) 6374 |
| \(\displaystyle -\int \left (\frac {\left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) e^2}{d^2 \left (\frac {d}{x^2}+e\right )^3 x}-\frac {2 \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) e}{d^2 \left (\frac {d}{x^2}+e\right )^2 x}+\frac {a+b \text {arccosh}\left (\frac {1}{c x}\right )}{d^2 \left (\frac {d}{x^2}+e\right ) x}\right )d\frac {1}{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) e^2}{4 d^3 \left (\frac {d}{x^2}+e\right )^2}-\frac {\left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) e}{d^3 \left (\frac {d}{x^2}+e\right )}-\frac {b \left (c^2-\frac {1}{x^2}\right ) e}{8 c d^2 \left (d c^2+e\right ) \left (\frac {d}{x^2}+e\right ) \sqrt {\frac {1}{c x}-1} \sqrt {1+\frac {1}{c x}} x}-\frac {b \left (d c^2+2 e\right ) \sqrt {\frac {1}{c^2 x^2}-1} \text {arctanh}\left (\frac {\sqrt {d c^2+e}}{c \sqrt {e} \sqrt {\frac {1}{c^2 x^2}-1} x}\right ) \sqrt {e}}{8 d^3 \left (d c^2+e\right )^{3/2} \sqrt {\frac {1}{c x}-1} \sqrt {1+\frac {1}{c x}}}+\frac {b \sqrt {\frac {1}{c^2 x^2}-1} \text {arctanh}\left (\frac {\sqrt {d c^2+e}}{c \sqrt {e} \sqrt {\frac {1}{c^2 x^2}-1} x}\right ) \sqrt {e}}{d^3 \sqrt {d c^2+e} \sqrt {\frac {1}{c x}-1} \sqrt {1+\frac {1}{c x}}}+\frac {\left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )^2}{2 b d^3}-\frac {\left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 d^3}-\frac {\left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) \log \left (\frac {\sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )} c}{\sqrt {e}-\sqrt {d c^2+e}}+1\right )}{2 d^3}-\frac {\left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 d^3}-\frac {\left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) \log \left (\frac {\sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )} c}{\sqrt {e}+\sqrt {d c^2+e}}+1\right )}{2 d^3}-\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 d^3}-\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 d^3}-\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 d^3}-\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 d^3}\) |
-1/8*(b*e*(c^2 - x^(-2)))/(c*d^2*(c^2*d + e)*(e + d/x^2)*Sqrt[-1 + 1/(c*x) ]*Sqrt[1 + 1/(c*x)]*x) + (e^2*(a + b*ArcCosh[1/(c*x)]))/(4*d^3*(e + d/x^2) ^2) - (e*(a + b*ArcCosh[1/(c*x)]))/(d^3*(e + d/x^2)) + (a + b*ArcCosh[1/(c *x)])^2/(2*b*d^3) + (b*Sqrt[e]*Sqrt[-1 + 1/(c^2*x^2)]*ArcTanh[Sqrt[c^2*d + e]/(c*Sqrt[e]*Sqrt[-1 + 1/(c^2*x^2)]*x)])/(d^3*Sqrt[c^2*d + e]*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) - (b*Sqrt[e]*(c^2*d + 2*e)*Sqrt[-1 + 1/(c^2*x^ 2)]*ArcTanh[Sqrt[c^2*d + e]/(c*Sqrt[e]*Sqrt[-1 + 1/(c^2*x^2)]*x)])/(8*d^3* (c^2*d + e)^(3/2)*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) - ((a + b*ArcCosh[ 1/(c*x)])*Log[1 - (c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*d^3) - ((a + b*ArcCosh[1/(c*x)])*Log[1 + (c*Sqrt[-d]*E^ArcCosh[1/ (c*x)])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*d^3) - ((a + b*ArcCosh[1/(c*x)])* Log[1 - (c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d ^3) - ((a + b*ArcCosh[1/(c*x)])*Log[1 + (c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(S qrt[e] + Sqrt[c^2*d + e])])/(2*d^3) - (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCos h[1/(c*x)])/(Sqrt[e] - Sqrt[c^2*d + e]))])/(2*d^3) - (b*PolyLog[2, (c*Sqrt [-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*d^3) - (b*PolyLo g[2, -((c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] + Sqrt[c^2*d + e]))])/(2*d ^3) - (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d^3)
3.2.26.3.1 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]
Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_.) + (e_.)*(x_ )^2)^(p_.), x_Symbol] :> -Subst[Int[(e + d*x^2)^p*((a + b*ArcCosh[x/c])^n/x ^(m + 2*(p + 1))), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0 ] && IntegersQ[m, p]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.33 (sec) , antiderivative size = 3727, normalized size of antiderivative = 5.03
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(3727\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(3801\) |
| default | \(\text {Expression too large to display}\) | \(3801\) |
-1/2*a/d^3*ln(e*x^2+d)+1/4*a/d/(e*x^2+d)^2+1/2*a/d^2/(e*x^2+d)+a/d^3*ln(x) +b*(-2*(-c^2*d*(e*(c^2*d+e))^(1/2)+2*c^2*d*e+2*e^2-2*(e*(c^2*d+e))^(1/2)*e )*e/(c^4*d^2+2*c^2*d*e+e^2)/d^4/c^2*arcsech(c*x)^2+(-c^2*d*(e*(c^2*d+e))^( 1/2)+2*c^2*d*e+2*e^2-2*(e*(c^2*d+e))^(1/2)*e)*e/(c^4*d^2+2*c^2*d*e+e^2)/d^ 4/c^2*polylog(2,d*c^2*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2 *(e*(c^2*d+e))^(1/2)-2*e))-3/4*(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)/d^4/(c^2* d+e)/c^2*e*polylog(2,d*c^2*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^ 2*d-2*(e*(c^2*d+e))^(1/2)-2*e))-(-c^2*d*(e*(c^2*d+e))^(1/2)+2*c^2*d*e+2*e^ 2-2*(e*(c^2*d+e))^(1/2)*e)/(c^4*d^2+2*c^2*d*e+e^2)*e^2/d^5/c^4*arcsech(c*x )^2-1/4*(e*(c^2*d+e))^(1/2)/(c^2*d+e)^2/d/e*c^4*arcsech(c*x)^2+1/8*(e*(c^2 *d+e))^(1/2)/(c^2*d+e)^2/d/e*c^4*polylog(2,d*c^2*(1/c/x+(-1+1/c/x)^(1/2)*( 1+1/c/x)^(1/2))^2/(-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e))+1/2*(-c^2*d*(e*(c^2* d+e))^(1/2)+2*c^2*d*e+2*e^2-2*(e*(c^2*d+e))^(1/2)*e)/(c^4*d^2+2*c^2*d*e+e^ 2)*e^2/d^5/c^4*polylog(2,d*c^2*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/ (-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*e))+1/2*(e*(c^2*d+e))^(1/2)/(c^2*d+e)^2/d^ 3*e*arcsech(c*x)*ln(1-d*c^2*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c ^2*d+2*(e*(c^2*d+e))^(1/2)-2*e))+(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)/d^5/(c^ 2*d+e)/c^4*e^2*arcsech(c*x)^2+3/2*(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)/d^4/(c ^2*d+e)/c^2*e*arcsech(c*x)^2+3/4*(e*(c^2*d+e))^(1/2)/(c^2*d+e)^2/d^2*c^2*a rcsech(c*x)*ln(1-d*c^2*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2...
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3} x} \,d x } \]
Timed out. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3} x} \,d x } \]
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(sqrt(1/(c*x) + 1)*sqrt(1/(c*x) - 1 ) + 1/(c*x))/(e^3*x^7 + 3*d*e^2*x^5 + 3*d^2*e*x^3 + d^3*x), x)
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3} x} \,d x } \]
Timed out. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{x\,{\left (e\,x^2+d\right )}^3} \,d x \]