\(\int \frac {\text {csch}^{-1}(\sqrt {x})}{x} \, dx\) [18]

Optimal result

Integrand size = 10, antiderivative size = 46 \[ \int \frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{x} \, dx=\text {csch}^{-1}\left (\sqrt {x}\right )^2-2 \text {csch}^{-1}\left (\sqrt {x}\right ) \log \left (1-e^{2 \text {csch}^{-1}\left (\sqrt {x}\right )}\right )-\operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}\left (\sqrt {x}\right )}\right ) \] Output:

arccsch(x^(1/2))^2-2*arccsch(x^(1/2))*ln(1-(1/x^(1/2)+(1+1/x)^(1/2))^2)-po 
lylog(2,(1/x^(1/2)+(1+1/x)^(1/2))^2)
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00 \[ \int \frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{x} \, dx=\text {csch}^{-1}\left (\sqrt {x}\right ) \left (\text {csch}^{-1}\left (\sqrt {x}\right )-2 \log \left (1-e^{2 \text {csch}^{-1}\left (\sqrt {x}\right )}\right )\right )-\operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}\left (\sqrt {x}\right )}\right ) \] Input:

Integrate[ArcCsch[Sqrt[x]]/x,x]
 

Output:

ArcCsch[Sqrt[x]]*(ArcCsch[Sqrt[x]] - 2*Log[1 - E^(2*ArcCsch[Sqrt[x]])]) - 
PolyLog[2, E^(2*ArcCsch[Sqrt[x]])]
 

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.46 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.26, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {7267, 6836, 6190, 3042, 26, 4199, 25, 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.

\(\displaystyle \int \frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{x} \, dx\)

\(\Big \downarrow \) 7267

\(\displaystyle 2 \int \frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{\sqrt {x}}d\sqrt {x}\)

\(\Big \downarrow \) 6836

\(\displaystyle -2 \int \frac {\text {arcsinh}\left (\frac {1}{\sqrt {x}}\right )}{\sqrt {x}}d\frac {1}{\sqrt {x}}\)

\(\Big \downarrow \) 6190

\(\displaystyle -2 \int \sqrt {1+\frac {1}{x}} \sqrt {x} \text {arcsinh}\left (\frac {1}{\sqrt {x}}\right )d\text {arcsinh}\left (\frac {1}{\sqrt {x}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -2 \int -i \text {arcsinh}\left (\frac {1}{\sqrt {x}}\right ) \tan \left (i \text {arcsinh}\left (\frac {1}{\sqrt {x}}\right )+\frac {\pi }{2}\right )d\text {arcsinh}\left (\frac {1}{\sqrt {x}}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle 2 i \int \text {arcsinh}\left (\frac {1}{\sqrt {x}}\right ) \tan \left (i \text {arcsinh}\left (\frac {1}{\sqrt {x}}\right )+\frac {\pi }{2}\right )d\text {arcsinh}\left (\frac {1}{\sqrt {x}}\right )\)

\(\Big \downarrow \) 4199

\(\displaystyle 2 i \left (2 i \int -\frac {e^{2 \text {arcsinh}\left (\frac {1}{\sqrt {x}}\right )} \text {arcsinh}\left (\frac {1}{\sqrt {x}}\right )}{1-e^{2 \text {arcsinh}\left (\frac {1}{\sqrt {x}}\right )}}d\text {arcsinh}\left (\frac {1}{\sqrt {x}}\right )-\frac {i x}{2}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle 2 i \left (-2 i \int \frac {e^{2 \text {arcsinh}\left (\frac {1}{\sqrt {x}}\right )} \text {arcsinh}\left (\frac {1}{\sqrt {x}}\right )}{1-e^{2 \text {arcsinh}\left (\frac {1}{\sqrt {x}}\right )}}d\text {arcsinh}\left (\frac {1}{\sqrt {x}}\right )-\frac {i x}{2}\right )\)

\(\Big \downarrow \) 2620

\(\displaystyle 2 i \left (-2 i \left (\frac {1}{2} \int \log \left (1-e^{2 \text {arcsinh}\left (\frac {1}{\sqrt {x}}\right )}\right )d\text {arcsinh}\left (\frac {1}{\sqrt {x}}\right )-\frac {1}{2} \text {arcsinh}\left (\frac {1}{\sqrt {x}}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {1}{\sqrt {x}}\right )}\right )\right )-\frac {i x}{2}\right )\)

\(\Big \downarrow \) 2715

\(\displaystyle 2 i \left (-2 i \left (\frac {1}{4} \int e^{2 \text {arcsinh}\left (\frac {1}{\sqrt {x}}\right )} \log \left (1-e^{2 \text {arcsinh}\left (\frac {1}{\sqrt {x}}\right )}\right )de^{2 \text {arcsinh}\left (\frac {1}{\sqrt {x}}\right )}-\frac {1}{2} \text {arcsinh}\left (\frac {1}{\sqrt {x}}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {1}{\sqrt {x}}\right )}\right )\right )-\frac {i x}{2}\right )\)

\(\Big \downarrow \) 2838

\(\displaystyle 2 i \left (-2 i \left (-\frac {1}{4} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (\frac {1}{\sqrt {x}}\right )}\right )-\frac {1}{2} \text {arcsinh}\left (\frac {1}{\sqrt {x}}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {1}{\sqrt {x}}\right )}\right )\right )-\frac {i x}{2}\right )\)

Input:

Int[ArcCsch[Sqrt[x]]/x,x]
 

Output:

(2*I)*((-1/2*I)*x - (2*I)*(-1/2*(ArcSinh[1/Sqrt[x]]*Log[1 - E^(2*ArcSinh[1 
/Sqrt[x]])]) - PolyLog[2, E^(2*ArcSinh[1/Sqrt[x]])]/4))
 

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 4199

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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 
))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && In 
tegerQ[4*k] && IGtQ[m, 0]
 

rule 6190

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

rule 6836

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

rule 7267

Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si 
mp[lst[[2]]*lst[[4]]   Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x 
] /;  !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
 

Maple [F]

\[\int \frac {\operatorname {arccsch}\left (\sqrt {x}\right )}{x}d x\]

Input:

int(arccsch(x^(1/2))/x,x)
 

Output:

int(arccsch(x^(1/2))/x,x)
 

Fricas [F]

\[ \int \frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{x} \, dx=\int { \frac {\operatorname {arcsch}\left (\sqrt {x}\right )}{x} \,d x } \] Input:

integrate(arccsch(x^(1/2))/x,x, algorithm="fricas")
 

Output:

integral(arccsch(sqrt(x))/x, x)
 

Sympy [F]

\[ \int \frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{x} \, dx=\int \frac {\operatorname {acsch}{\left (\sqrt {x} \right )}}{x}\, dx \] Input:

integrate(acsch(x**(1/2))/x,x)
 

Output:

Integral(acsch(sqrt(x))/x, x)
 

Maxima [F]

\[ \int \frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{x} \, dx=\int { \frac {\operatorname {arcsch}\left (\sqrt {x}\right )}{x} \,d x } \] Input:

integrate(arccsch(x^(1/2))/x,x, algorithm="maxima")
 

Output:

integrate(arccsch(sqrt(x))/x, x)
 

Giac [F]

\[ \int \frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{x} \, dx=\int { \frac {\operatorname {arcsch}\left (\sqrt {x}\right )}{x} \,d x } \] Input:

integrate(arccsch(x^(1/2))/x,x, algorithm="giac")
 

Output:

integrate(arccsch(sqrt(x))/x, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{x} \, dx=\int \frac {\mathrm {asinh}\left (\frac {1}{\sqrt {x}}\right )}{x} \,d x \] Input:

int(asinh(1/x^(1/2))/x,x)
                                                                                    
                                                                                    
 

Output:

int(asinh(1/x^(1/2))/x, x)
 

Reduce [F]

\[ \int \frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{x} \, dx=\int \frac {\mathit {acsch} \left (\sqrt {x}\right )}{x}d x \] Input:

int(acsch(x^(1/2))/x,x)
 

Output:

int(acsch(sqrt(x))/x,x)