\(\int \frac {a+b \text {sech}^{-1}(c x)}{x^2 (d+e x^2)} \, dx\) [113]

Optimal result

Integrand size = 21, antiderivative size = 523 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^2 \left (d+e x^2\right )} \, dx=\frac {b c \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{d}-\frac {a}{d x}-\frac {b \text {sech}^{-1}(c x)}{d x}+\frac {\sqrt {e} \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{3/2}}-\frac {\sqrt {e} \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{3/2}}+\frac {\sqrt {e} \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{3/2}}-\frac {\sqrt {e} \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{3/2}}-\frac {b \sqrt {e} \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{3/2}}+\frac {b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{3/2}}-\frac {b \sqrt {e} \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{3/2}}+\frac {b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{3/2}} \] Output:

b*c*(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2)/d-a/d/x-b*arcsech(c*x)/d/x+1/2*e^(1/2 
)*(a+b*arcsech(c*x))*ln(1-c*(-d)^(1/2)*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^( 
1/2))/(e^(1/2)-(c^2*d+e)^(1/2)))/(-d)^(3/2)-1/2*e^(1/2)*(a+b*arcsech(c*x)) 
*ln(1+c*(-d)^(1/2)*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/(e^(1/2)-(c^2* 
d+e)^(1/2)))/(-d)^(3/2)+1/2*e^(1/2)*(a+b*arcsech(c*x))*ln(1-c*(-d)^(1/2)*( 
1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/(e^(1/2)+(c^2*d+e)^(1/2)))/(-d)^(3 
/2)-1/2*e^(1/2)*(a+b*arcsech(c*x))*ln(1+c*(-d)^(1/2)*(1/c/x+(-1+1/c/x)^(1/ 
2)*(1+1/c/x)^(1/2))/(e^(1/2)+(c^2*d+e)^(1/2)))/(-d)^(3/2)-1/2*b*e^(1/2)*po 
lylog(2,-c*(-d)^(1/2)*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/(e^(1/2)-(c 
^2*d+e)^(1/2)))/(-d)^(3/2)+1/2*b*e^(1/2)*polylog(2,c*(-d)^(1/2)*(1/c/x+(-1 
+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/(e^(1/2)-(c^2*d+e)^(1/2)))/(-d)^(3/2)-1/2*b 
*e^(1/2)*polylog(2,-c*(-d)^(1/2)*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/ 
(e^(1/2)+(c^2*d+e)^(1/2)))/(-d)^(3/2)+1/2*b*e^(1/2)*polylog(2,c*(-d)^(1/2) 
*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/(e^(1/2)+(c^2*d+e)^(1/2)))/(-d)^ 
(3/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 2.03 (sec) , antiderivative size = 933, normalized size of antiderivative = 1.78 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^2 \left (d+e x^2\right )} \, dx =\text {Too large to display} \] Input:

Integrate[(a + b*ArcSech[c*x])/(x^2*(d + e*x^2)),x]
 

Output:

(-4*a*Sqrt[d] - 4*a*Sqrt[e]*x*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + b*(4*Sqrt[d]*S 
qrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) - 4*Sqrt[d]*ArcSech[c*x] - (2*I)*Sqrt[e 
]*x*((-4*I)*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]] + ArcSech[c*x] 
*Log[1 + E^(-2*ArcSech[c*x])] - ArcSech[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^ 
2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (2*I)*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])] - ArcSech[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c 
*Sqrt[d]*E^ArcSech[c*x])] - (2*I)*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 
])] + PolyLog[2, (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x 
])] + PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c 
*x])]) + (2*I)*Sqrt[e]*x*((-4*I)*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]] + ArcSech[c*x]*Log[1 + E^(-2*ArcSech[c*x])] - ArcSech[c*x]*Log[1 
 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (2*I)*Ar 
cSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqr 
t[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - ArcSech[c*x]*Log[1 - (I*(Sqrt 
[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (2*I)*ArcSin[Sqrt[1 
- (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + ...
 

Rubi [A] (verified)

Time = 1.43 (sec) , antiderivative size = 575, 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.

\(\displaystyle \int \frac {a+b \text {sech}^{-1}(c x)}{x^2 \left (d+e x^2\right )} \, dx\)

\(\Big \downarrow \) 6857

\(\displaystyle -\int \frac {a+b \text {arccosh}\left (\frac {1}{c x}\right )}{\left (\frac {d}{x^2}+e\right ) x^2}d\frac {1}{x}\)

\(\Big \downarrow \) 6374

\(\displaystyle -\int \left (\frac {a+b \text {arccosh}\left (\frac {1}{c x}\right )}{d}-\frac {e \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{d \left (\frac {d}{x^2}+e\right )}\right )d\frac {1}{x}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {e} \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{3/2}}-\frac {\sqrt {e} \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) \log \left (\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {c^2 d+e}}+1\right )}{2 (-d)^{3/2}}+\frac {\sqrt {e} \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{2 (-d)^{3/2}}-\frac {\sqrt {e} \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) \log \left (\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {c^2 d+e}+\sqrt {e}}+1\right )}{2 (-d)^{3/2}}-\frac {a}{d x}-\frac {b \sqrt {e} \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 (-d)^{3/2}}+\frac {b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 (-d)^{3/2}}-\frac {b \sqrt {e} \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 (-d)^{3/2}}+\frac {b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 (-d)^{3/2}}-\frac {b \text {arccosh}\left (\frac {1}{c x}\right )}{d x}+\frac {b c \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}{d}\)

Input:

Int[(a + b*ArcSech[c*x])/(x^2*(d + e*x^2)),x]
 

Output:

(b*c*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)])/d - a/(d*x) - (b*ArcCosh[1/(c*x 
)])/(d*x) + (Sqrt[e]*(a + b*ArcCosh[1/(c*x)])*Log[1 - (c*Sqrt[-d]*E^ArcCos 
h[1/(c*x)])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*(-d)^(3/2)) - (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])])/(2*(-d)^(3/2)) + (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])])/(2*(-d)^(3/2 
)) - (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])])/(2*(-d)^(3/2)) - (b*Sqrt[e]*PolyLog[2, 
-((c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] - Sqrt[c^2*d + e]))])/(2*(-d)^( 
3/2)) + (b*Sqrt[e]*PolyLog[2, (c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] - S 
qrt[c^2*d + e])])/(2*(-d)^(3/2)) - (b*Sqrt[e]*PolyLog[2, -((c*Sqrt[-d]*E^A 
rcCosh[1/(c*x)])/(Sqrt[e] + Sqrt[c^2*d + e]))])/(2*(-d)^(3/2)) + (b*Sqrt[e 
]*PolyLog[2, (c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] + Sqrt[c^2*d + e])]) 
/(2*(-d)^(3/2))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6374

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]
 

rule 6857

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]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 30.19 (sec) , antiderivative size = 372, normalized size of antiderivative = 0.71

method	result	size
parts	\(-\frac {a e \arctan \left (\frac {x e}{\sqrt {d e}}\right )}{d \sqrt {d e}}-\frac {a}{d x}+\frac {c b \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{d}-\frac {b \,\operatorname {arcsech}\left (c x \right )}{d x}+\frac {c b e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {arcsech}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e}\right )}{2 d}-\frac {c b e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\operatorname {arcsech}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e \right )}\right )}{2 d}\)	\(372\)
derivativedivides	\(c \left (-\frac {a e \arctan \left (\frac {x e}{\sqrt {d e}}\right )}{c d \sqrt {d e}}-\frac {a}{d c x}+\frac {b \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{d}-\frac {b \,\operatorname {arcsech}\left (c x \right )}{d c x}+\frac {b e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {arcsech}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e}\right )}{2 d}-\frac {b e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\operatorname {arcsech}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e \right )}\right )}{2 d}\right )\)	\(380\)
default	\(c \left (-\frac {a e \arctan \left (\frac {x e}{\sqrt {d e}}\right )}{c d \sqrt {d e}}-\frac {a}{d c x}+\frac {b \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{d}-\frac {b \,\operatorname {arcsech}\left (c x \right )}{d c x}+\frac {b e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {arcsech}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e}\right )}{2 d}-\frac {b e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\operatorname {arcsech}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e \right )}\right )}{2 d}\right )\)	\(380\)

Input:

int((a+b*arcsech(c*x))/x^2/(e*x^2+d),x,method=_RETURNVERBOSE)
 

Output:

-a*e/d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))-a/d/x+c*b/d*(-(c*x-1)/c/x)^(1/2 
)*((c*x+1)/c/x)^(1/2)-b*arcsech(c*x)/d/x+1/2*c*b*e/d*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/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))-1/2*c*b*e/d*sum(1/_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/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))
 

Fricas [F]

\[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^2 \left (d+e x^2\right )} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{{\left (e x^{2} + d\right )} x^{2}} \,d x } \] Input:

integrate((a+b*arcsech(c*x))/x^2/(e*x^2+d),x, algorithm="fricas")
 

Output:

integral((b*arcsech(c*x) + a)/(e*x^4 + d*x^2), x)
 

Sympy [F]

\[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^2 \left (d+e x^2\right )} \, dx=\int \frac {a + b \operatorname {asech}{\left (c x \right )}}{x^{2} \left (d + e x^{2}\right )}\, dx \] Input:

integrate((a+b*asech(c*x))/x**2/(e*x**2+d),x)
 

Output:

Integral((a + b*asech(c*x))/(x**2*(d + e*x**2)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^2 \left (d+e x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a+b*arcsech(c*x))/x^2/(e*x^2+d),x, algorithm="maxima")
 

Output:

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
 

Giac [F]

\[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^2 \left (d+e x^2\right )} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{{\left (e x^{2} + d\right )} x^{2}} \,d x } \] Input:

integrate((a+b*arcsech(c*x))/x^2/(e*x^2+d),x, algorithm="giac")
 

Output:

integrate((b*arcsech(c*x) + a)/((e*x^2 + d)*x^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^2 \left (d+e x^2\right )} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{x^2\,\left (e\,x^2+d\right )} \,d x \] Input:

int((a + b*acosh(1/(c*x)))/(x^2*(d + e*x^2)),x)
 

Output:

int((a + b*acosh(1/(c*x)))/(x^2*(d + e*x^2)), x)
 

Reduce [F]

\[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^2 \left (d+e x^2\right )} \, dx=\frac {-\sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a x +\left (\int \frac {\mathit {asech} \left (c x \right )}{e \,x^{4}+d \,x^{2}}d x \right ) b \,d^{2} x -a d}{d^{2} x} \] Input:

int((a+b*asech(c*x))/x^2/(e*x^2+d),x)
                                                                                    
                                                                                    
 

Output:

( - sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*x + int(asech(c*x)/(d* 
x**2 + e*x**4),x)*b*d**2*x - a*d)/(d**2*x)