\(\int \frac {\text {csch}^{-1}(a x^n)}{x} \, dx\) [23]

Optimal result

Integrand size = 10, antiderivative size = 61 \[ \int \frac {\text {csch}^{-1}\left (a x^n\right )}{x} \, dx=\frac {\text {csch}^{-1}\left (a x^n\right )^2}{2 n}-\frac {\text {csch}^{-1}\left (a x^n\right ) \log \left (1-e^{2 \text {csch}^{-1}\left (a x^n\right )}\right )}{n}-\frac {\operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}\left (a x^n\right )}\right )}{2 n} \] Output:

1/2*arccsch(a*x^n)^2/n-arccsch(a*x^n)*ln(1-(1/a/(x^n)+(1+1/a^2/(x^n)^2)^(1 
/2))^2)/n-1/2*polylog(2,(1/a/(x^n)+(1+1/a^2/(x^n)^2)^(1/2))^2)/n
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.06 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.05 \[ \int \frac {\text {csch}^{-1}\left (a x^n\right )}{x} \, dx=-\frac {x^{-n} \, _3F_2\left (\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};-\frac {x^{-2 n}}{a^2}\right )}{a n}+\left (\text {csch}^{-1}\left (a x^n\right )-\text {arcsinh}\left (\frac {x^{-n}}{a}\right )\right ) \log (x) \] Input:

Integrate[ArcCsch[a*x^n]/x,x]
 

Output:

-(HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, -(1/(a^2*x^(2*n)))]/(a*n* 
x^n)) + (ArcCsch[a*x^n] - ArcSinh[1/(a*x^n)])*Log[x]
 

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.49 (sec) , antiderivative size = 77, 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 = {7282, 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.

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

\(\Big \downarrow \) 7282

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

\(\Big \downarrow \) 6836

\(\displaystyle -\frac {\int x^{-n} \text {arcsinh}\left (\frac {x^{-n}}{a}\right )dx^{-n}}{n}\)

\(\Big \downarrow \) 6190

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4199

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

Int[ArcCsch[a*x^n]/x,x]
 

Output:

(I*((-1/2*I)*x^(2*n) - (2*I)*(-1/2*(ArcSinh[1/(a*x^n)]*Log[1 - E^(2*ArcSin 
h[1/(a*x^n)])]) - PolyLog[2, E^(2*ArcSinh[1/(a*x^n)])]/4)))/n
 

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 7282

Int[(u_)/(x_), x_Symbol] :> With[{lst = PowerVariableExpn[u, 0, x]}, Simp[1 
/lst[[2]]   Subst[Int[NormalizeIntegrand[Simplify[lst[[1]]/x], x], x], x, ( 
lst[[3]]*x)^lst[[2]]], x] /;  !FalseQ[lst] && NeQ[lst[[2]], 0]] /; NonsumQ[ 
u] &&  !RationalFunctionQ[u, x]
 

Maple [F]

\[\int \frac {\operatorname {arccsch}\left (a \,x^{n}\right )}{x}d x\]

Input:

int(arccsch(a*x^n)/x,x)
 

Output:

int(arccsch(a*x^n)/x,x)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {\text {csch}^{-1}\left (a x^n\right )}{x} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(arccsch(a*x^n)/x,x, algorithm="fricas")
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

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

integrate(acsch(a*x**n)/x,x)
 

Output:

Integral(acsch(a*x**n)/x, x)
 

Maxima [F]

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

integrate(arccsch(a*x^n)/x,x, algorithm="maxima")
 

Output:

a^2*n*integrate(x^(2*n)*log(x)/(a^2*x*x^(2*n) + (a^2*x*x^(2*n) + x)*sqrt(a 
^2*x^(2*n) + 1) + x), x) + n*integrate(log(x)/(a^2*x*x^(2*n) + x), x) - lo 
g(a)*log(x) - log(x)*log(x^n) + log(x)*log(sqrt(a^2*x^(2*n) + 1) + 1)
 

Giac [F]

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

integrate(arccsch(a*x^n)/x,x, algorithm="giac")
 

Output:

integrate(arccsch(a*x^n)/x, x)
 

Mupad [F(-1)]

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

int(asinh(1/(a*x^n))/x,x)
                                                                                    
                                                                                    
 

Output:

int(asinh(1/(a*x^n))/x, x)
 

Reduce [F]

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

int(acsch(a*x^n)/x,x)
 

Output:

int(acsch(x**n*a)/x,x)