3.1.26 \(\int \frac {a+b \text {sech}^{-1}(c x)}{x} \, dx\) [26]

3.1.26.1 Optimal result

Integrand size = 12, antiderivative size = 56 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x} \, dx=-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{2 b}-\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+e^{-2 \text {sech}^{-1}(c x)}\right )+\frac {1}{2} b \operatorname {PolyLog}\left (2,-e^{-2 \text {sech}^{-1}(c x)}\right ) \]

output

-1/2*(a+b*arcsech(c*x))^2/b-(a+b*arcsech(c*x))*ln(1+1/(1/c/x+(-1+1/c/x)^(1 
/2)*(1+1/c/x)^(1/2))^2)+1/2*b*polylog(2,-1/(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/ 
x)^(1/2))^2)
 

3.1.26.2 Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.84 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x} \, dx=a \log (x)+\frac {1}{2} b \left (-\text {sech}^{-1}(c x) \left (\text {sech}^{-1}(c x)+2 \log \left (1+e^{-2 \text {sech}^{-1}(c x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-2 \text {sech}^{-1}(c x)}\right )\right ) \]

input

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

output

a*Log[x] + (b*(-(ArcSech[c*x]*(ArcSech[c*x] + 2*Log[1 + E^(-2*ArcSech[c*x] 
)])) + PolyLog[2, -E^(-2*ArcSech[c*x])]))/2
 

3.1.26.3 Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.50 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.39, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6835, 6297, 25, 3042, 26, 4201, 2620, 2715, 2838}

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.

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

\(\Big \downarrow \) 6835

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

\(\Big \downarrow \) 6297

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {i \left (2 i \left (-\frac {1}{4} b^2 \int x \log \left (1+e^{-2 \text {arccosh}\left (\frac {1}{c x}\right )}\right )de^{-2 \text {arccosh}\left (\frac {1}{c x}\right )}-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}\left (\frac {1}{c x}\right )}+1\right ) \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )\right )-\frac {i}{2 x^2}\right )}{b}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}\left (2,-a-b \text {arccosh}\left (\frac {1}{c x}\right )\right )-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}\left (\frac {1}{c x}\right )}+1\right ) \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )\right )-\frac {i}{2 x^2}\right )}{b}\)

input

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

output

((-I)*((-1/2*I)/x^2 + (2*I)*(-1/2*(b*(a + b*ArcCosh[1/(c*x)])*Log[1 + E^(- 
2*ArcCosh[1/(c*x)])]) + (b^2*PolyLog[2, -a - b*ArcCosh[1/(c*x)]])/4)))/b
 

3.1.26.3.1 Defintions of rubi rules used

rule 25

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

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 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

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

rule 4201

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x 
_Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I   Int[ 
(c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] 
 /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
 

rule 6297

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Simp[1/b 
 Subst[Int[x^n*Tanh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a 
, b, c}, x] && IGtQ[n, 0]
 

rule 6835

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> -Subst[Int[(a + 
b*ArcCosh[x/c])/x, x], x, 1/x] /; FreeQ[{a, b, c}, x]
 

3.1.26.4 Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.75

method	result	size
parts	\(a \ln \left (x \right )+b \left (\frac {\operatorname {arcsech}\left (c x \right )^{2}}{2}-\operatorname {arcsech}\left (c x \right ) \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{2}\right )\)	\(98\)
derivativedivides	\(a \ln \left (c x \right )+b \left (\frac {\operatorname {arcsech}\left (c x \right )^{2}}{2}-\operatorname {arcsech}\left (c x \right ) \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{2}\right )\)	\(100\)
default	\(a \ln \left (c x \right )+b \left (\frac {\operatorname {arcsech}\left (c x \right )^{2}}{2}-\operatorname {arcsech}\left (c x \right ) \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{2}\right )\)	\(100\)

input

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

output

a*ln(x)+b*(1/2*arcsech(c*x)^2-arcsech(c*x)*ln(1+(1/c/x+(-1+1/c/x)^(1/2)*(1 
+1/c/x)^(1/2))^2)-1/2*polylog(2,-(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^ 
2))
 

3.1.26.5 Fricas [F]

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

input

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

output

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

3.1.26.6 Sympy [F]

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

input

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

output

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

3.1.26.7 Maxima [F]

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

input

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

output

b*integrate(log(sqrt(1/(c*x) + 1)*sqrt(1/(c*x) - 1) + 1/(c*x))/x, x) + a*l 
og(x)
 

3.1.26.8 Giac [F]

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

input

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

output

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

3.1.26.9 Mupad [F(-1)]

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

input

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

output

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