\(\int \frac {a+b \text {sech}^{-1}(c x)}{x (d+e x^2)} \, dx\) [112]

Optimal result

Integrand size = 21, antiderivative size = 417 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x \left (d+e x^2\right )} \, dx=\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{2 b d}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d}-\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d}-\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d}-\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d}-\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d} \] Output:

1/2*(a+b*arcsech(c*x))^2/b/d-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-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-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-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-1/2*b*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-1/2*b*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-1/2*b*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-1/2*b*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
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.35 (sec) , antiderivative size = 841, normalized size of antiderivative = 2.02 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x \left (d+e x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/2*(b*ArcSech[c*x]^2 + (4*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]] + (4*I)*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcT 
anh[((I*c*Sqrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]/2])/Sqrt[c^2*d + e]] + b*Ar 
cSech[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c* 
x])] - (2*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])] + b*ArcSech[c*x 
]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - ( 
2*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])] + b*ArcSech[c*x]*Log[1 
 - (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (2*I)*b*A 
rcSin[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])] + b*ArcSech[c*x]*Log[1 + (I*(Sq 
rt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (2*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*a*Log[x] + a*Log[d + e*x^2] - b*Poly 
Log[2, ((-I)*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - b 
*PolyLog[2, (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - 
 b*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x] 
)] - b*PolyLog[2, (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[...
 

Rubi [A] (verified)

Time = 1.30 (sec) , antiderivative size = 469, normalized size of antiderivative = 1.12, 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.

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

\(\Big \downarrow \) 6857

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

\(\Big \downarrow \) 6374

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(a + b*ArcCosh[1/(c*x)])^2/(2*b*d) - ((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) - ((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) - ((a + b*ArcCosh[1/(c*x)])*Log[1 - (c*Sqrt[-d]*E^Arc 
Cosh[1/(c*x)])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d) - ((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) - (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] - Sqrt[c 
^2*d + e]))])/(2*d) - (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[ 
e] - Sqrt[c^2*d + e])])/(2*d) - (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCosh[1/(c 
*x)])/(Sqrt[e] + Sqrt[c^2*d + e]))])/(2*d) - (b*PolyLog[2, (c*Sqrt[-d]*E^A 
rcCosh[1/(c*x)])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d)
 

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] (warning: unable to verify)

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

Time = 1.42 (sec) , antiderivative size = 2081, normalized size of antiderivative = 4.99

method	result	size
parts	\(\text {Expression too large to display}\)	\(2081\)
derivativedivides	\(\text {Expression too large to display}\)	\(2108\)
default	\(\text {Expression too large to display}\)	\(2108\)

Input:

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

Output:

-1/2*a/d*ln(e*x^2+d)+a/d*ln(x)+b*(1/2/d*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)/(c^2*d+e)/c^2/d^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)+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)/e/(c^2*d+e)/d*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*( 
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)/c^2/ 
d^2*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)/(c^2*d+e)/c^2/d^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))+1/2*(e*(c^2*d+e)) 
^(1/2)/(c^2*d+e)/d*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)/c^4/d^3*e*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*(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^2*d+e)/d^3/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/(c^2*d+e)/d^3/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))+1/4*(e*(c^2*d+e))^(1/2)/e/(c^2* 
d+e)*c^2*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))-1/4*(-c^2*d*(e*(c^2*d+e))^(1/2)+2...
 

Fricas [F]

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

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

Output:

integral((b*arcsech(c*x) + a)/(e*x^3 + d*x), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-1/2*a*(log(e*x^2 + d)/d - 2*log(x)/d) + b*integrate(log(sqrt(1/(c*x) + 1) 
*sqrt(1/(c*x) - 1) + 1/(c*x))/(e*x^3 + d*x), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \text {sech}^{-1}(c x)}{x \left (d+e x^2\right )} \, dx=\frac {2 \left (\int \frac {\mathit {asech} \left (c x \right )}{e \,x^{3}+d x}d x \right ) b d -\mathrm {log}\left (e \,x^{2}+d \right ) a +2 \,\mathrm {log}\left (x \right ) a}{2 d} \] Input:

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

Output:

(2*int(asech(c*x)/(d*x + e*x**3),x)*b*d - log(d + e*x**2)*a + 2*log(x)*a)/ 
(2*d)