\(\int \frac {(a+b \text {sech}^{-1}(c x))^2}{x^3} \, dx\) [39]

Optimal result

Integrand size = 14, antiderivative size = 106 \[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{x^3} \, dx=-\frac {b^2 (1-c x) (1+c x)}{4 x^2}+\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}-\frac {1}{4} c^2 \left (a+b \text {sech}^{-1}(c x)\right )^2-\frac {(1-c x) (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 x^2} \] Output:

-1/4*b^2*(-c*x+1)*(c*x+1)/x^2+1/2*b*((-c*x+1)/(c*x+1))^(1/2)*(c*x+1)*(a+b* 
arcsech(c*x))/x^2-1/4*c^2*(a+b*arcsech(c*x))^2-1/2*(-c*x+1)*(c*x+1)*(a+b*a 
rcsech(c*x))^2/x^2
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.73 \[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{x^3} \, dx=\frac {-2 a^2-b^2+2 a b \sqrt {\frac {1-c x}{1+c x}}+2 a b c x \sqrt {\frac {1-c x}{1+c x}}+2 b \left (-2 a+b \sqrt {\frac {1-c x}{1+c x}} (1+c x)\right ) \text {sech}^{-1}(c x)+b^2 \left (-2+c^2 x^2\right ) \text {sech}^{-1}(c x)^2-2 a b c^2 x^2 \log (x)+2 a b c^2 x^2 \log \left (1+\sqrt {\frac {1-c x}{1+c x}}+c x \sqrt {\frac {1-c x}{1+c x}}\right )}{4 x^2} \] Input:

Integrate[(a + b*ArcSech[c*x])^2/x^3,x]
 

Output:

(-2*a^2 - b^2 + 2*a*b*Sqrt[(1 - c*x)/(1 + c*x)] + 2*a*b*c*x*Sqrt[(1 - c*x) 
/(1 + c*x)] + 2*b*(-2*a + b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))*ArcSech[c 
*x] + b^2*(-2 + c^2*x^2)*ArcSech[c*x]^2 - 2*a*b*c^2*x^2*Log[x] + 2*a*b*c^2 
*x^2*Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]])/( 
4*x^2)
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6839, 5969, 3042, 25, 3791, 17}

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.

\(\displaystyle \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{x^3} \, dx\)

\(\Big \downarrow \) 6839

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

\(\Big \downarrow \) 5969

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3791

\(\displaystyle -c^2 \left (b \left (\frac {1}{2} \int \left (a+b \text {sech}^{-1}(c x)\right )d\text {sech}^{-1}(c x)-\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^2 x^2}+\frac {b (1-c x) (c x+1)}{4 c^2 x^2}\right )+\frac {(1-c x) (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^2 x^2}\right )\)

\(\Big \downarrow \) 17

\(\displaystyle -c^2 \left (\frac {(1-c x) (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^2 x^2}+b \left (-\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^2 x^2}+\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{4 b}+\frac {b (1-c x) (c x+1)}{4 c^2 x^2}\right )\right )\)

Input:

Int[(a + b*ArcSech[c*x])^2/x^3,x]
 

Output:

-(c^2*(((1 - c*x)*(1 + c*x)*(a + b*ArcSech[c*x])^2)/(2*c^2*x^2) + b*((b*(1 
 - c*x)*(1 + c*x))/(4*c^2*x^2) - (Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(a + 
 b*ArcSech[c*x]))/(2*c^2*x^2) + (a + b*ArcSech[c*x])^2/(4*b))))
 

Defintions of rubi rules used

rule 17

Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 
)/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 3042

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

rule 3791

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> 
 Simp[d*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Simp[b*(c + d*x)*Cos[e + f*x 
]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^2*((n - 1)/n)   Int[(c + d* 
x)*(b*Sin[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 
 1]
 

rule 5969

Int[Cosh[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)* 
(x_)]^(n_.), x_Symbol] :> Simp[(c + d*x)^m*(Sinh[a + b*x]^(n + 1)/(b*(n + 1 
))), x] - Simp[d*(m/(b*(n + 1)))   Int[(c + d*x)^(m - 1)*Sinh[a + b*x]^(n + 
 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
 

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 = 0.49 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.48

method	result	size
parts	\(-\frac {a^{2}}{2 x^{2}}+b^{2} c^{2} \left (-\frac {\cosh \left (2 \,\operatorname {arcsech}\left (c x \right )\right ) \operatorname {arcsech}\left (c x \right )^{2}}{4}+\frac {\sinh \left (2 \,\operatorname {arcsech}\left (c x \right )\right ) \operatorname {arcsech}\left (c x \right )}{4}-\frac {\cosh \left (2 \,\operatorname {arcsech}\left (c x \right )\right )}{8}\right )+2 a b \,c^{2} \left (-\frac {\operatorname {arcsech}\left (c x \right )}{2 c^{2} x^{2}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}\right )}{4 c x \sqrt {-c^{2} x^{2}+1}}\right )\)	\(157\)
derivativedivides	\(c^{2} \left (-\frac {a^{2}}{2 c^{2} x^{2}}+b^{2} \left (-\frac {\cosh \left (2 \,\operatorname {arcsech}\left (c x \right )\right ) \operatorname {arcsech}\left (c x \right )^{2}}{4}+\frac {\sinh \left (2 \,\operatorname {arcsech}\left (c x \right )\right ) \operatorname {arcsech}\left (c x \right )}{4}-\frac {\cosh \left (2 \,\operatorname {arcsech}\left (c x \right )\right )}{8}\right )+2 a b \left (-\frac {\operatorname {arcsech}\left (c x \right )}{2 c^{2} x^{2}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}\right )}{4 c x \sqrt {-c^{2} x^{2}+1}}\right )\right )\)	\(158\)
default	\(c^{2} \left (-\frac {a^{2}}{2 c^{2} x^{2}}+b^{2} \left (-\frac {\cosh \left (2 \,\operatorname {arcsech}\left (c x \right )\right ) \operatorname {arcsech}\left (c x \right )^{2}}{4}+\frac {\sinh \left (2 \,\operatorname {arcsech}\left (c x \right )\right ) \operatorname {arcsech}\left (c x \right )}{4}-\frac {\cosh \left (2 \,\operatorname {arcsech}\left (c x \right )\right )}{8}\right )+2 a b \left (-\frac {\operatorname {arcsech}\left (c x \right )}{2 c^{2} x^{2}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}\right )}{4 c x \sqrt {-c^{2} x^{2}+1}}\right )\right )\)	\(158\)

Input:

int((a+b*arcsech(c*x))^2/x^3,x,method=_RETURNVERBOSE)
 

Output:

-1/2*a^2/x^2+b^2*c^2*(-1/4*cosh(2*arcsech(c*x))*arcsech(c*x)^2+1/4*sinh(2* 
arcsech(c*x))*arcsech(c*x)-1/8*cosh(2*arcsech(c*x)))+2*a*b*c^2*(-1/2/c^2/x 
^2*arcsech(c*x)+1/4*(-(c*x-1)/c/x)^(1/2)/c/x*((c*x+1)/c/x)^(1/2)*(arctanh( 
1/(-c^2*x^2+1)^(1/2))*c^2*x^2+(-c^2*x^2+1)^(1/2))/(-c^2*x^2+1)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.56 \[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{x^3} \, dx=\frac {2 \, a b c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + {\left (b^{2} c^{2} x^{2} - 2 \, b^{2}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} - 2 \, a^{2} - b^{2} + 2 \, {\left (a b c^{2} x^{2} + b^{2} c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 2 \, a b\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )}{4 \, x^{2}} \] Input:

integrate((a+b*arcsech(c*x))^2/x^3,x, algorithm="fricas")
 

Output:

1/4*(2*a*b*c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + (b^2*c^2*x^2 - 2*b^2)*log( 
(c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x))^2 - 2*a^2 - b^2 + 2*(a*b*c 
^2*x^2 + b^2*c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 2*a*b)*log((c*x*sqrt(-(c 
^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)))/x^2
 

Sympy [F]

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

integrate((a+b*asech(c*x))**2/x**3,x)
 

Output:

Integral((a + b*asech(c*x))**2/x**3, x)
 

Maxima [F]

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

integrate((a+b*arcsech(c*x))^2/x^3,x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((a+b*arcsech(c*x))^2/x^3,x, algorithm="giac")
 

Output:

integrate((b*arcsech(c*x) + a)^2/x^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{x^3} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^2}{x^3} \,d x \] Input:

int((a + b*acosh(1/(c*x)))^2/x^3,x)
 

Output:

int((a + b*acosh(1/(c*x)))^2/x^3, x)
 

Reduce [F]

\[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{x^3} \, dx=\frac {4 \left (\int \frac {\mathit {asech} \left (c x \right )}{x^{3}}d x \right ) a b \,x^{2}+2 \left (\int \frac {\mathit {asech} \left (c x \right )^{2}}{x^{3}}d x \right ) b^{2} x^{2}-a^{2}}{2 x^{2}} \] Input:

int((a+b*asech(c*x))^2/x^3,x)
 

Output:

(4*int(asech(c*x)/x**3,x)*a*b*x**2 + 2*int(asech(c*x)**2/x**3,x)*b**2*x**2 
 - a**2)/(2*x**2)