Integrand size = 21, antiderivative size = 542 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x \left (d+e x^2\right )^2} \, dx=-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{2 d^2 \left (e+\frac {d}{x^2}\right )}+\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{2 b d^2}+\frac {b \sqrt {e} \sqrt {-1+\frac {1}{c^2 x^2}} \text {arctanh}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {-1+\frac {1}{c^2 x^2}} x}\right )}{2 d^2 \sqrt {c^2 d+e} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^2} \]
-1/2*e*(a+b*arcsech(c*x))/d^2/(e+d/x^2)+1/2*(a+b*arcsech(c*x))^2/b/d^2-1/2 *(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^2-1/2*(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^2 -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^2-1/2*(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^2-1/2*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^2-1/2*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^2-1/2*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^2-1/2*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^2+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^2/(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 = 1.99 (sec) , antiderivative size = 1189, normalized size of antiderivative = 2.19 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x \left (d+e x^2\right )^2} \, dx=\frac {\frac {2 a d}{d+e x^2}+\frac {b \sqrt {d} \text {sech}^{-1}(c x)}{\sqrt {d}-i \sqrt {e} x}+\frac {b \sqrt {d} \text {sech}^{-1}(c x)}{\sqrt {d}+i \sqrt {e} x}-2 b \text {sech}^{-1}(c x)^2-8 i b \arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \text {arctanh}\left (\frac {\left (-i c \sqrt {d}+\sqrt {e}\right ) \tanh \left (\frac {1}{2} \text {sech}^{-1}(c x)\right )}{\sqrt {c^2 d+e}}\right )-8 i b \arcsin \left (\frac {\sqrt {1+\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \text {arctanh}\left (\frac {\left (i c \sqrt {d}+\sqrt {e}\right ) \tanh \left (\frac {1}{2} \text {sech}^{-1}(c x)\right )}{\sqrt {c^2 d+e}}\right )-2 b \text {sech}^{-1}(c x) \log \left (1+\frac {i \left (\sqrt {e}-\sqrt {c^2 d+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )+4 i b \arcsin \left (\frac {\sqrt {1+\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {i \left (\sqrt {e}-\sqrt {c^2 d+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )-2 b \text {sech}^{-1}(c x) \log \left (1+\frac {i \left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )+4 i b \arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {i \left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )-2 b \text {sech}^{-1}(c x) \log \left (1-\frac {i \left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )-4 i b \arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1-\frac {i \left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )-2 b \text {sech}^{-1}(c x) \log \left (1+\frac {i \left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )-4 i b \arcsin \left (\frac {\sqrt {1+\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {i \left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )+4 a \log (x)+2 b \log (x)-2 a \log \left (d+e x^2\right )-2 b \log \left (1+\sqrt {\frac {1-c x}{1+c x}}+c x \sqrt {\frac {1-c x}{1+c x}}\right )+\frac {b \sqrt {e} \log \left (\frac {2 i \sqrt {e} \left (\sqrt {d} \sqrt {\frac {1-c x}{1+c x}} (1+c x)+\frac {\sqrt {d} \sqrt {e}+i c^2 d x}{\sqrt {c^2 d+e}}\right )}{i \sqrt {d}+\sqrt {e} x}\right )}{\sqrt {c^2 d+e}}+\frac {b \sqrt {e} \log \left (\frac {2 \sqrt {e} \left (i \sqrt {d} \sqrt {\frac {1-c x}{1+c x}} (1+c x)+\frac {i \sqrt {d} \sqrt {e}+c^2 d x}{\sqrt {c^2 d+e}}\right )}{-i \sqrt {d}+\sqrt {e} x}\right )}{\sqrt {c^2 d+e}}+2 b \operatorname {PolyLog}\left (2,-\frac {i \left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )+2 b \operatorname {PolyLog}\left (2,\frac {i \left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )+2 b \operatorname {PolyLog}\left (2,-\frac {i \left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )+2 b \operatorname {PolyLog}\left (2,\frac {i \left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )}{4 d^2} \]
((2*a*d)/(d + e*x^2) + (b*Sqrt[d]*ArcSech[c*x])/(Sqrt[d] - I*Sqrt[e]*x) + (b*Sqrt[d]*ArcSech[c*x])/(Sqrt[d] + I*Sqrt[e]*x) - 2*b*ArcSech[c*x]^2 - (8 *I)*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[(((-I)*c*S qrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]/2])/Sqrt[c^2*d + e]] - (8*I)*b*ArcSin[ Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[((I*c*Sqrt[d] + Sqrt[e] )*Tanh[ArcSech[c*x]/2])/Sqrt[c^2*d + e]] - 2*b*ArcSech[c*x]*Log[1 + (I*(Sq rt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (4*I)*b*ArcSin[Sqr t[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - 2*b*ArcSech[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (4*I)*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e])) /(c*Sqrt[d]*E^ArcSech[c*x])] - 2*b*ArcSech[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt [c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (4*I)*b*ArcSin[Sqrt[1 - (I*Sqr t[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqr t[d]*E^ArcSech[c*x])] - 2*b*ArcSech[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (4*I)*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/( c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ ArcSech[c*x])] + 4*a*Log[x] + 2*b*Log[x] - 2*a*Log[d + e*x^2] - 2*b*Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]] + (b*Sqrt[e]* Log[((2*I)*Sqrt[e]*(Sqrt[d]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) + (Sqrt...
Time = 1.52 (sec) , antiderivative size = 598, normalized size of antiderivative = 1.10, 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.
| \(\displaystyle \int \frac {a+b \text {sech}^{-1}(c x)}{x \left (d+e x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 6857 |
| \(\displaystyle -\int \frac {a+b \text {arccosh}\left (\frac {1}{c x}\right )}{\left (\frac {d}{x^2}+e\right )^2 x^3}d\frac {1}{x}\) |
| \(\Big \downarrow \) 6374 |
| \(\displaystyle -\int \left (\frac {a+b \text {arccosh}\left (\frac {1}{c x}\right )}{d \left (\frac {d}{x^2}+e\right ) x}-\frac {e \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{d \left (\frac {d}{x^2}+e\right )^2 x}\right )d\frac {1}{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) \log \left (\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {c^2 d+e}}+1\right )}{2 d^2}-\frac {\left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{2 d^2}-\frac {\left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) \log \left (\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {c^2 d+e}+\sqrt {e}}+1\right )}{2 d^2}-\frac {e \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{2 d^2 \left (\frac {d}{x^2}+e\right )}+\frac {\left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )^2}{2 b d^2}-\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 d^2}-\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 d^2}-\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 d^2}-\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 d^2}+\frac {b \sqrt {e} \sqrt {\frac {1}{c^2 x^2}-1} \text {arctanh}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} x \sqrt {\frac {1}{c^2 x^2}-1}}\right )}{2 d^2 \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1} \sqrt {c^2 d+e}}\) |
-1/2*(e*(a + b*ArcCosh[1/(c*x)]))/(d^2*(e + d/x^2)) + (a + b*ArcCosh[1/(c* x)])^2/(2*b*d^2) + (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)])/(2*d^2*Sqrt[c^2*d + e]*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^2) - ((a + b*Arc Cosh[1/(c*x)])*Log[1 + (c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] - Sqrt[c^2 *d + e])])/(2*d^2) - ((a + b*ArcCosh[1/(c*x)])*Log[1 - (c*Sqrt[-d]*E^ArcCo sh[1/(c*x)])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d^2) - ((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^2) - (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] - Sqrt [c^2*d + e]))])/(2*d^2) - (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(S qrt[e] - Sqrt[c^2*d + e])])/(2*d^2) - (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCos h[1/(c*x)])/(Sqrt[e] + Sqrt[c^2*d + e]))])/(2*d^2) - (b*PolyLog[2, (c*Sqrt [-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d^2)
3.2.18.3.1 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 = 7.00 (sec) , antiderivative size = 2226, normalized size of antiderivative = 4.11
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(2226\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(2275\) |
| default | \(\text {Expression too large to display}\) | \(2275\) |
-1/2*a/d^2*ln(e*x^2+d)+1/2*a/d/(e*x^2+d)+a/d^2*ln(x)+b*(-(-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^2*d+e)/d^3/c^2*arcs ech(c*x)^2+1/8*(-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)/d^2/e/(c^2*d+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))-1/2*(c^2*d-2*(e*(c^2*d+e))^ (1/2)+2*e)/d^4/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))*e-1/2*(c^2*d-2*(e*(c^2*d+e))^(1/2) +2*e)/d^3/c^2*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))*arcsech(c*x)+(c^2*d-2*(e*(c^2*d+e))^(1/2)+2* e)/d^4/c^4*arcsech(c*x)^2*e-1/4*(-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)/d^2/e/(c^2*d+e)*arcsech(c*x)^2+1/2*(e*(c^2*d+e) )^(1/2)/d^2/(c^2*d+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))+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^2*d+e)/d^3/c^2*polylo g(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)*e/d^4/(c^2*d+e)/c^4*arcsech(c*x)^2+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)*e/d^4/(c^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))+1/8*(e*(c^2*d+e))^(1/2)/d/e/(c^2*d+e)*c^2*polylog(2,d*c^2*(1...
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{x \left (d+e x^2\right )^2} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x} \,d x } \]
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{x \left (d+e x^2\right )^2} \, dx=\int \frac {a + b \operatorname {asech}{\left (c x \right )}}{x \left (d + e x^{2}\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x \left (d+e x^2\right )^2} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{x \left (d+e x^2\right )^2} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x} \,d x } \]
Timed out. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x \left (d+e x^2\right )^2} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{x\,{\left (e\,x^2+d\right )}^2} \,d x \]