Integrand size = 21, antiderivative size = 844 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^2 \left (d+e x^2\right )^2} \, dx=\frac {b c \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{d^2}-\frac {a}{d^2 x}-\frac {b \text {sech}^{-1}(c x)}{d^2 x}+\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b e \arctan \left (\frac {\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{2 d^2 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}}}-\frac {b e \arctan \left (\frac {\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{2 d^2 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}}}-\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}-\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}+\frac {3 b \sqrt {e} \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}-\frac {3 b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac {3 b \sqrt {e} \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}-\frac {3 b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}} \]
-a/d^2/x-b*arcsech(c*x)/d^2/x-3/4*(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)))*e^(1/2)/( -d)^(5/2)+3/4*(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)))*e^(1/2)/(-d)^(5/2)-3/4*(a+b*a rcsech(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)))*e^(1/2)/(-d)^(5/2)+3/4*(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)))*e^(1/2)/(-d)^(5/2)+3/4*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)))*e^(1/2)/(-d)^(5/2)-3/4*b*p olylog(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)))*e^(1/2)/(-d)^(5/2)+3/4*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)))*e^(1/2)/(-d)^(5 /2)-3/4*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)))*e^(1/2)/(-d)^(5/2)+1/4*e*(a+b*arcsech(c*x))/d^2 /(-d/x+(-d)^(1/2)*e^(1/2))-1/4*e*(a+b*arcsech(c*x))/d^2/(d/x+(-d)^(1/2)*e^ (1/2))+b*c*(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2)/d^2-1/2*b*e*arctan((1+1/c/x)^( 1/2)*(c*d-(-d)^(1/2)*e^(1/2))^(1/2)/(-1+1/c/x)^(1/2)/(c*d+(-d)^(1/2)*e^(1/ 2))^(1/2))/d^2/(c*d-(-d)^(1/2)*e^(1/2))^(1/2)/(c*d+(-d)^(1/2)*e^(1/2))^(1/ 2)-1/2*b*e*arctan((1+1/c/x)^(1/2)*(c*d+(-d)^(1/2)*e^(1/2))^(1/2)/(-1+1/c/x )^(1/2)/(c*d-(-d)^(1/2)*e^(1/2))^(1/2))/d^2/(c*d-(-d)^(1/2)*e^(1/2))^(1...
Result contains complex when optimal does not.
Time = 1.35 (sec) , antiderivative size = 1305, normalized size of antiderivative = 1.55 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^2 \left (d+e x^2\right )^2} \, dx =\text {Too large to display} \]
((-4*a*Sqrt[d])/x + 4*b*c*Sqrt[d]*Sqrt[(1 - c*x)/(1 + c*x)] + (4*b*Sqrt[d] *Sqrt[(1 - c*x)/(1 + c*x)])/x - (2*a*Sqrt[d]*e*x)/(d + e*x^2) - (4*b*Sqrt[ d]*ArcSech[c*x])/x - (b*Sqrt[d]*e*ArcSech[c*x])/((-I)*Sqrt[d]*Sqrt[e] + e* x) - (b*Sqrt[d]*e*ArcSech[c*x])/(I*Sqrt[d]*Sqrt[e] + e*x) - 6*a*Sqrt[e]*Ar cTan[(Sqrt[e]*x)/Sqrt[d]] + 12*b*Sqrt[e]*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sq rt[d])]/Sqrt[2]]*ArcTanh[(((-I)*c*Sqrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]/2]) /Sqrt[c^2*d + e]] - 12*b*Sqrt[e]*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]] + (3*I)*b*Sqrt[e]*ArcSech[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e ]))/(c*Sqrt[d]*E^ArcSech[c*x])] + 6*b*Sqrt[e]*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])] - (3*I)*b*Sqrt[e]*ArcSech[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt [c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - 6*b*Sqrt[e]*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])] - (3*I)*b*Sqrt[e]*ArcSech[c*x]*Log[1 - (I*(Sqrt [e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + 6*b*Sqrt[e]*ArcSin[S qrt[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])] + (3*I)*b*Sqrt[e]*ArcSech[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - 6*b*Sqrt[e] *ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] ...
Time = 2.82 (sec) , antiderivative size = 904, normalized size of antiderivative = 1.07, 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.
| \(\displaystyle \int \frac {a+b \text {sech}^{-1}(c x)}{x^2 \left (d+e x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 6857 |
| \(\displaystyle -\int \frac {a+b \text {arccosh}\left (\frac {1}{c x}\right )}{\left (\frac {d}{x^2}+e\right )^2 x^4}d\frac {1}{x}\) |
| \(\Big \downarrow \) 6374 |
| \(\displaystyle -\int \left (\frac {\left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) e^2}{d^2 \left (\frac {d}{x^2}+e\right )^2}-\frac {2 \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) e}{d^2 \left (\frac {d}{x^2}+e\right )}+\frac {a+b \text {arccosh}\left (\frac {1}{c x}\right )}{d^2}\right )d\frac {1}{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a}{d^2 x}-\frac {b \text {arccosh}\left (\frac {1}{c x}\right )}{d^2 x}+\frac {e \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {e \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{4 d^2 \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )}-\frac {b e \arctan \left (\frac {\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {\frac {1}{c x}-1}}\right )}{2 d^2 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}}}-\frac {b e \arctan \left (\frac {\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {\frac {1}{c x}-1}}\right )}{2 d^2 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}}}-\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}-\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}+\frac {3 b \sqrt {e} \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 )}{4 (-d)^{5/2}}-\frac {3 b \sqrt {e} \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 )}{4 (-d)^{5/2}}+\frac {3 b \sqrt {e} \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 )}{4 (-d)^{5/2}}-\frac {3 b \sqrt {e} \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 )}{4 (-d)^{5/2}}+\frac {b c \sqrt {\frac {1}{c x}-1} \sqrt {1+\frac {1}{c x}}}{d^2}\) |
(b*c*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)])/d^2 - a/(d^2*x) - (b*ArcCosh[1/ (c*x)])/(d^2*x) + (e*(a + b*ArcCosh[1/(c*x)]))/(4*d^2*(Sqrt[-d]*Sqrt[e] - d/x)) - (e*(a + b*ArcCosh[1/(c*x)]))/(4*d^2*(Sqrt[-d]*Sqrt[e] + d/x)) - (b *e*ArcTan[(Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[1 + 1/(c*x)])/(Sqrt[c*d + Sqr t[-d]*Sqrt[e]]*Sqrt[-1 + 1/(c*x)])])/(2*d^2*Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*S qrt[c*d + Sqrt[-d]*Sqrt[e]]) - (b*e*ArcTan[(Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*S qrt[1 + 1/(c*x)])/(Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[-1 + 1/(c*x)])])/(2*d ^2*Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[c*d + Sqrt[-d]*Sqrt[e]]) - (3*Sqrt[e] *(a + b*ArcCosh[1/(c*x)])*Log[1 - (c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*(-d)^(5/2)) + (3*Sqrt[e]*(a + b*ArcCosh[1/(c*x)]) *Log[1 + (c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] - Sqrt[c^2*d + e])])/(4* (-d)^(5/2)) - (3*Sqrt[e]*(a + b*ArcCosh[1/(c*x)])*Log[1 - (c*Sqrt[-d]*E^Ar cCosh[1/(c*x)])/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*(-d)^(5/2)) + (3*Sqrt[e]* (a + b*ArcCosh[1/(c*x)])*Log[1 + (c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*(-d)^(5/2)) + (3*b*Sqrt[e]*PolyLog[2, -((c*Sqrt[-d ]*E^ArcCosh[1/(c*x)])/(Sqrt[e] - Sqrt[c^2*d + e]))])/(4*(-d)^(5/2)) - (3*b *Sqrt[e]*PolyLog[2, (c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*(-d)^(5/2)) + (3*b*Sqrt[e]*PolyLog[2, -((c*Sqrt[-d]*E^ArcCosh[1 /(c*x)])/(Sqrt[e] + Sqrt[c^2*d + e]))])/(4*(-d)^(5/2)) - (3*b*Sqrt[e]*Poly Log[2, (c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] + Sqrt[c^2*d + e])])/(4...
3.2.22.3.1 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 = 109.90 (sec) , antiderivative size = 1007, normalized size of antiderivative = 1.19
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1007\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1034\) |
| default | \(\text {Expression too large to display}\) | \(1034\) |
a*(-1/d^2*e*(1/2*x/(e*x^2+d)+3/2/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2)))-1/d^ 2/x)+b*c*(-1/2*(-1+arcsech(c*x))/d^2*((-(c*x-1)/c/x)^(1/2)*c*x*((c*x+1)/c/ x)^(1/2)+1)/x/c+1/2*((-(c*x-1)/c/x)^(1/2)*c*x*((c*x+1)/c/x)^(1/2)-1)*(1+ar csech(c*x))/d^2/x/c-1/2*arcsech(c*x)/d^2*e*x*c/(c^2*e*x^2+c^2*d)+1/2*(-(c^ 2*d-2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*(c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)* e*arctanh(c*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/((-c^2*d+2*(e*(c^2* d+e))^(1/2)-2*e)*d)^(1/2))/d^5/c^5-1/2*(-(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e) *d)^(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*arctanh(c*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/((-c^2*d+2*(e*( c^2*d+e))^(1/2)-2*e)*d)^(1/2))/d^5/c^5/(c^2*d+e)+1/2*((c^2*d+2*(e*(c^2*d+e ))^(1/2)+2*e)*d)^(1/2)*(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)*e*arctan(c*d*(1/c /x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d) ^(1/2))/d^5/c^5-1/2*((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(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*arctan(c*d*(1 /c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)* d)^(1/2))/d^5/c^5/(c^2*d+e)-3/4/d^2*e*sum(1/_R1/(_R1^2*c^2*d+c^2*d+2*e)*(a rcsech(c*x)*ln((_R1-1/c/x-(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/_R1)+dilog((_R 1-1/c/x-(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/_R1)),_R1=RootOf(c^2*d*_Z^4+(2*c ^2*d+4*e)*_Z^2+c^2*d))+3/4/d^2*e*sum(_R1/(_R1^2*c^2*d+c^2*d+2*e)*(arcsech( c*x)*ln((_R1-1/c/x-(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/_R1)+dilog((_R1-1/...
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^2 \left (d+e x^2\right )^2} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x^{2}} \,d x } \]
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^2 \left (d+e x^2\right )^2} \, dx=\int \frac {a + b \operatorname {asech}{\left (c x \right )}}{x^{2} \left (d + e x^{2}\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^2 \left (d+e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^2 \left (d+e x^2\right )^2} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x^{2}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^2 \left (d+e x^2\right )^2} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{x^2\,{\left (e\,x^2+d\right )}^2} \,d x \]