\(\int \frac {1}{x^2 (a+b \text {sech}^{-1}(c x))} \, dx\) [54]

Optimal result

Integrand size = 14, antiderivative size = 46 \[ \int \frac {1}{x^2 \left (a+b \text {sech}^{-1}(c x)\right )} \, dx=\frac {c \text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )}{b}-\frac {c \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{b} \] Output:

c*Chi(a/b+arcsech(c*x))*sinh(a/b)/b-c*cosh(a/b)*Shi(a/b+arcsech(c*x))/b
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x^2 \left (a+b \text {sech}^{-1}(c x)\right )} \, dx=\frac {c \left (\text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )-\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )\right )}{b} \] Input:

Integrate[1/(x^2*(a + b*ArcSech[c*x])),x]
 

Output:

(c*(CoshIntegral[a/b + ArcSech[c*x]]*Sinh[a/b] - Cosh[a/b]*SinhIntegral[a/ 
b + ArcSech[c*x]]))/b
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.46 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.17, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6839, 3042, 26, 3784, 26, 3042, 26, 3779, 3782}

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.

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

\(\Big \downarrow \) 6839

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

\(\Big \downarrow \) 3042

\(\displaystyle -c \int -\frac {i \sin \left (i \text {sech}^{-1}(c x)\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)\)

\(\Big \downarrow \) 26

\(\displaystyle i c \int \frac {\sin \left (i \text {sech}^{-1}(c x)\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)\)

\(\Big \downarrow \) 3784

\(\displaystyle i c \left (\cosh \left (\frac {a}{b}\right ) \int \frac {i \sinh \left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)-i \sinh \left (\frac {a}{b}\right ) \int \frac {\cosh \left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)\right )\)

\(\Big \downarrow \) 26

\(\displaystyle i c \left (i \cosh \left (\frac {a}{b}\right ) \int \frac {\sinh \left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)-i \sinh \left (\frac {a}{b}\right ) \int \frac {\cosh \left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle i c \left (i \cosh \left (\frac {a}{b}\right ) \int -\frac {i \sin \left (\frac {i a}{b}+i \text {sech}^{-1}(c x)\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)-i \sinh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i a}{b}+i \text {sech}^{-1}(c x)+\frac {\pi }{2}\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)\right )\)

\(\Big \downarrow \) 26

\(\displaystyle i c \left (\cosh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i a}{b}+i \text {sech}^{-1}(c x)\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)-i \sinh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i a}{b}+i \text {sech}^{-1}(c x)+\frac {\pi }{2}\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)\right )\)

\(\Big \downarrow \) 3779

\(\displaystyle i c \left (\frac {i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{b}-i \sinh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i a}{b}+i \text {sech}^{-1}(c x)+\frac {\pi }{2}\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)\right )\)

\(\Big \downarrow \) 3782

\(\displaystyle i c \left (\frac {i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{b}-\frac {i \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{b}\right )\)

Input:

Int[1/(x^2*(a + b*ArcSech[c*x])),x]
 

Output:

I*c*(((-I)*CoshIntegral[a/b + ArcSech[c*x]]*Sinh[a/b])/b + (I*Cosh[a/b]*Si 
nhIntegral[a/b + ArcSech[c*x]])/b)
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3779

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> Simp[I*(SinhIntegral[c*f*(fz/d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f 
, fz}, x] && EqQ[d*e - c*f*fz*I, 0]
 

rule 3782

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> Simp[CoshIntegral[c*f*(fz/d) + f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz 
}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]
 

rule 3784

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* 
e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* 
f)/d]   Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] 
&& NeQ[d*e - c*f, 0]
 

rule 6839

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 
-(c^(m + 1))^(-1)   Subst[Int[(a + b*x)^n*Sech[x]^(m + 1)*Tanh[x], x], x, A 
rcSech[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (G 
tQ[n, 0] || LtQ[m, -1])
 

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.17

method	result	size
derivativedivides	\(c \left (-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\frac {a}{b}+\operatorname {arcsech}\left (c x \right )\right )}{2 b}+\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arcsech}\left (c x \right )-\frac {a}{b}\right )}{2 b}\right )\)	\(54\)
default	\(c \left (-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\frac {a}{b}+\operatorname {arcsech}\left (c x \right )\right )}{2 b}+\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arcsech}\left (c x \right )-\frac {a}{b}\right )}{2 b}\right )\)	\(54\)

Input:

int(1/x^2/(a+b*arcsech(c*x)),x,method=_RETURNVERBOSE)
 

Output:

c*(-1/2/b*exp(1/b*a)*Ei(1,1/b*a+arcsech(c*x))+1/2/b*exp(-1/b*a)*Ei(1,-arcs 
ech(c*x)-1/b*a))
 

Fricas [F]

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

integrate(1/x^2/(a+b*arcsech(c*x)),x, algorithm="fricas")
                                                                                    
                                                                                    
 

Output:

integral(1/(b*x^2*arcsech(c*x) + a*x^2), x)
 

Sympy [F]

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

integrate(1/x**2/(a+b*asech(c*x)),x)
 

Output:

Integral(1/(x**2*(a + b*asech(c*x))), x)
 

Maxima [F]

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

integrate(1/x^2/(a+b*arcsech(c*x)),x, algorithm="maxima")
 

Output:

integrate(1/((b*arcsech(c*x) + a)*x^2), x)
 

Giac [F]

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

integrate(1/x^2/(a+b*arcsech(c*x)),x, algorithm="giac")
 

Output:

integrate(1/((b*arcsech(c*x) + a)*x^2), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int(1/x^2/(a+b*asech(c*x)),x)
 

Output:

int(1/(asech(c*x)*b*x**2 + a*x**2),x)