3.1.60 \(\int \frac {1}{x^2 (a+b \text {sech}^{-1}(c x))^2} \, dx\) [60]

3.1.60.1 Optimal result

Integrand size = 14, antiderivative size = 86 \[ \int \frac {1}{x^2 \left (a+b \text {sech}^{-1}(c x)\right )^2} \, dx=\frac {\sqrt {\frac {1-c x}{1+c x}} (1+c x)}{b x \left (a+b \text {sech}^{-1}(c x)\right )}-\frac {c \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{b^2}+\frac {c \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{b^2} \]

output

-c*Chi(a/b+arcsech(c*x))*cosh(a/b)/b^2+c*Shi(a/b+arcsech(c*x))*sinh(a/b)/b 
^2+(c*x+1)*((-c*x+1)/(c*x+1))^(1/2)/b/x/(a+b*arcsech(c*x))
 

3.1.60.2 Mathematica [A] (verified)

Time = 0.46 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x^2 \left (a+b \text {sech}^{-1}(c x)\right )^2} \, dx=\frac {\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{x \left (a+b \text {sech}^{-1}(c x)\right )}-c \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )+c \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{b^2} \]

input

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

output

((b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(x*(a + b*ArcSech[c*x])) - c*Cosh 
[a/b]*CoshIntegral[a/b + ArcSech[c*x]] + c*Sinh[a/b]*SinhIntegral[a/b + Ar 
cSech[c*x]])/b^2
 

3.1.60.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.56 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.20, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {6839, 3042, 26, 3778, 3042, 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.

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

\(\Big \downarrow \) 6839

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3784

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3779

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

\(\Big \downarrow \) 3782

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

input

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

output

I*c*(((-I)*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(b*c*x*(a + b*ArcSech[c*x] 
)) + (I*((Cosh[a/b]*CoshIntegral[a/b + ArcSech[c*x]])/b - (Sinh[a/b]*SinhI 
ntegral[a/b + ArcSech[c*x]])/b))/b)
 

3.1.60.3.1 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 3778

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c 
 + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m + 1))), x] - Simp[f/(d*(m + 1))   Int[( 
c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[m, - 
1]
 

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])
 

3.1.60.4 Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.91

method	result	size
derivativedivides	\(c \left (\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}-1}{2 c x b \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )}+\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\frac {a}{b}+\operatorname {arcsech}\left (c x \right )\right )}{2 b^{2}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}+1}{2 b c x \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )}+\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsech}\left (c x \right )-\frac {a}{b}\right )}{2 b^{2}}\right )\)	\(164\)
default	\(c \left (\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}-1}{2 c x b \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )}+\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\frac {a}{b}+\operatorname {arcsech}\left (c x \right )\right )}{2 b^{2}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}+1}{2 b c x \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )}+\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsech}\left (c x \right )-\frac {a}{b}\right )}{2 b^{2}}\right )\)	\(164\)

input

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

output

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

3.1.60.5 Fricas [F]

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

input

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

output

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

3.1.60.6 Sympy [F]

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

input

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

output

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

3.1.60.7 Maxima [F]

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

input

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

output

-(c^2*x^3 + (c^2*x^3 - x)*sqrt(c*x + 1)*sqrt(-c*x + 1) - x)/((b^2*c^2*x^2 
- b^2)*x^2*log(x) + ((b^2*c^2*log(c) - a*b*c^2)*x^2 - b^2*log(c) + a*b)*x^ 
2 - (b^2*x^2*log(x) + (b^2*log(c) - a*b)*x^2)*sqrt(c*x + 1)*sqrt(-c*x + 1) 
 + (sqrt(c*x + 1)*sqrt(-c*x + 1)*b^2*x^2 - (b^2*c^2*x^2 - b^2)*x^2)*log(sq 
rt(c*x + 1)*sqrt(-c*x + 1) + 1)) + integrate(-(c^4*x^4 - 2*c^2*x^2 - (c^2* 
x^2 + 1)*(c*x + 1)*(c*x - 1) - (c^2*x^2 - 2)*sqrt(c*x + 1)*sqrt(-c*x + 1) 
+ 1)/((b^2*c^4*x^4 - 2*b^2*c^2*x^2 + b^2)*x^2*log(x) - (b^2*x^2*log(x) + ( 
b^2*log(c) - a*b)*x^2)*(c*x + 1)*(c*x - 1) + ((b^2*c^4*log(c) - a*b*c^4)*x 
^4 - 2*(b^2*c^2*log(c) - a*b*c^2)*x^2 + b^2*log(c) - a*b)*x^2 - 2*((b^2*c^ 
2*x^2 - b^2)*x^2*log(x) + ((b^2*c^2*log(c) - a*b*c^2)*x^2 - b^2*log(c) + a 
*b)*x^2)*sqrt(c*x + 1)*sqrt(-c*x + 1) + ((c*x + 1)*(c*x - 1)*b^2*x^2 + 2*( 
b^2*c^2*x^2 - b^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)*x^2 - (b^2*c^4*x^4 - 2*b^2 
*c^2*x^2 + b^2)*x^2)*log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 1)), x)
 

3.1.60.8 Giac [F]

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

input

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

output

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

3.1.60.9 Mupad [F(-1)]

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

input

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

output

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