\(\int \frac {\text {csch}^{-1}(a x^5)}{x} \, dx\) [24]

Optimal result

Integrand size = 10, antiderivative size = 54 \[ \int \frac {\text {csch}^{-1}\left (a x^5\right )}{x} \, dx=\frac {1}{10} \text {csch}^{-1}\left (a x^5\right )^2-\frac {1}{5} \text {csch}^{-1}\left (a x^5\right ) \log \left (1-e^{2 \text {csch}^{-1}\left (a x^5\right )}\right )-\frac {1}{10} \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}\left (a x^5\right )}\right ) \] Output:

1/10*arccsch(a*x^5)^2-1/5*arccsch(a*x^5)*ln(1-(1/a/x^5+(1+1/a^2/x^10)^(1/2 
))^2)-1/10*polylog(2,(1/a/x^5+(1+1/a^2/x^10)^(1/2))^2)
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.93 \[ \int \frac {\text {csch}^{-1}\left (a x^5\right )}{x} \, dx=\frac {1}{10} \left (\text {csch}^{-1}\left (a x^5\right )^2-2 \text {csch}^{-1}\left (a x^5\right ) \log \left (1-e^{2 \text {csch}^{-1}\left (a x^5\right )}\right )-\operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}\left (a x^5\right )}\right )\right ) \] Input:

Integrate[ArcCsch[a*x^5]/x,x]
 

Output:

(ArcCsch[a*x^5]^2 - 2*ArcCsch[a*x^5]*Log[1 - E^(2*ArcCsch[a*x^5])] - PolyL 
og[2, E^(2*ArcCsch[a*x^5])])/10
 

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.50 (sec) , antiderivative size = 68, 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.

\(\displaystyle \int \frac {\text {csch}^{-1}\left (a x^5\right )}{x} \, dx\)

\(\Big \downarrow \) 7282

\(\displaystyle \frac {1}{5} \int \frac {\text {csch}^{-1}\left (a x^5\right )}{x^5}dx^5\)

\(\Big \downarrow \) 6836

\(\displaystyle -\frac {1}{5} \int \frac {\text {arcsinh}\left (\frac {1}{a x^5}\right )}{x^5}d\frac {1}{x^5}\)

\(\Big \downarrow \) 6190

\(\displaystyle -\frac {1}{5} \int a \sqrt {1+\frac {1}{x^{10} a^2}} x^5 \text {arcsinh}\left (\frac {1}{a x^5}\right )d\text {arcsinh}\left (\frac {1}{a x^5}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{5} \int -i \text {arcsinh}\left (\frac {1}{a x^5}\right ) \tan \left (i \text {arcsinh}\left (\frac {1}{a x^5}\right )+\frac {\pi }{2}\right )d\text {arcsinh}\left (\frac {1}{a x^5}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {1}{5} i \int \text {arcsinh}\left (\frac {1}{a x^5}\right ) \tan \left (i \text {arcsinh}\left (\frac {1}{a x^5}\right )+\frac {\pi }{2}\right )d\text {arcsinh}\left (\frac {1}{a x^5}\right )\)

\(\Big \downarrow \) 4199

\(\displaystyle \frac {1}{5} i \left (2 i \int -\frac {e^{2 \text {arcsinh}\left (\frac {1}{a x^5}\right )} \text {arcsinh}\left (\frac {1}{a x^5}\right )}{1-e^{2 \text {arcsinh}\left (\frac {1}{a x^5}\right )}}d\text {arcsinh}\left (\frac {1}{a x^5}\right )-\frac {i x^{10}}{2}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{5} i \left (-2 i \int \frac {e^{2 \text {arcsinh}\left (\frac {1}{a x^5}\right )} \text {arcsinh}\left (\frac {1}{a x^5}\right )}{1-e^{2 \text {arcsinh}\left (\frac {1}{a x^5}\right )}}d\text {arcsinh}\left (\frac {1}{a x^5}\right )-\frac {i x^{10}}{2}\right )\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {1}{5} i \left (-2 i \left (\frac {1}{2} \int \log \left (1-e^{2 \text {arcsinh}\left (\frac {1}{a x^5}\right )}\right )d\text {arcsinh}\left (\frac {1}{a x^5}\right )-\frac {1}{2} \text {arcsinh}\left (\frac {1}{a x^5}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {1}{a x^5}\right )}\right )\right )-\frac {i x^{10}}{2}\right )\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {1}{5} i \left (-2 i \left (\frac {1}{4} \int e^{2 \text {arcsinh}\left (\frac {1}{a x^5}\right )} \log \left (1-e^{2 \text {arcsinh}\left (\frac {1}{a x^5}\right )}\right )de^{2 \text {arcsinh}\left (\frac {1}{a x^5}\right )}-\frac {1}{2} \text {arcsinh}\left (\frac {1}{a x^5}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {1}{a x^5}\right )}\right )\right )-\frac {i x^{10}}{2}\right )\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {1}{5} i \left (-2 i \left (-\frac {1}{4} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (\frac {1}{a x^5}\right )}\right )-\frac {1}{2} \text {arcsinh}\left (\frac {1}{a x^5}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {1}{a x^5}\right )}\right )\right )-\frac {i x^{10}}{2}\right )\)

Input:

Int[ArcCsch[a*x^5]/x,x]
 

Output:

(I/5)*((-1/2*I)*x^10 - (2*I)*(-1/2*(ArcSinh[1/(a*x^5)]*Log[1 - E^(2*ArcSin 
h[1/(a*x^5)])]) - PolyLog[2, E^(2*ArcSinh[1/(a*x^5)])]/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 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]

\[\int \frac {\operatorname {arccsch}\left (a \,x^{5}\right )}{x}d x\]

Input:

int(arccsch(a*x^5)/x,x)
 

Output:

int(arccsch(a*x^5)/x,x)
 

Fricas [F]

\[ \int \frac {\text {csch}^{-1}\left (a x^5\right )}{x} \, dx=\int { \frac {\operatorname {arcsch}\left (a x^{5}\right )}{x} \,d x } \] Input:

integrate(arccsch(a*x^5)/x,x, algorithm="fricas")
 

Output:

integral(arccsch(a*x^5)/x, x)
 

Sympy [F]

\[ \int \frac {\text {csch}^{-1}\left (a x^5\right )}{x} \, dx=\int \frac {\operatorname {acsch}{\left (a x^{5} \right )}}{x}\, dx \] Input:

integrate(acsch(a*x**5)/x,x)
 

Output:

Integral(acsch(a*x**5)/x, x)
 

Maxima [F]

\[ \int \frac {\text {csch}^{-1}\left (a x^5\right )}{x} \, dx=\int { \frac {\operatorname {arcsch}\left (a x^{5}\right )}{x} \,d x } \] Input:

integrate(arccsch(a*x^5)/x,x, algorithm="maxima")
 

Output:

5*a^2*integrate(x^9*log(x)/(a^2*x^10 + (a^2*x^10 + 1)^(3/2) + 1), x) - 1/2 
*log(a^2*x^10 + 1)*log(x) - log(a)*log(x) - 5/2*log(x)^2 + log(x)*log(sqrt 
(a^2*x^10 + 1) + 1) - 1/20*dilog(-a^2*x^10)
 

Giac [F]

\[ \int \frac {\text {csch}^{-1}\left (a x^5\right )}{x} \, dx=\int { \frac {\operatorname {arcsch}\left (a x^{5}\right )}{x} \,d x } \] Input:

integrate(arccsch(a*x^5)/x,x, algorithm="giac")
 

Output:

integrate(arccsch(a*x^5)/x, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {csch}^{-1}\left (a x^5\right )}{x} \, dx=\int \frac {\mathrm {asinh}\left (\frac {1}{a\,x^5}\right )}{x} \,d x \] Input:

int(asinh(1/(a*x^5))/x,x)
                                                                                    
                                                                                    
 

Output:

int(asinh(1/(a*x^5))/x, x)
 

Reduce [F]

\[ \int \frac {\text {csch}^{-1}\left (a x^5\right )}{x} \, dx=\int \frac {\mathit {acsch} \left (a \,x^{5}\right )}{x}d x \] Input:

int(acsch(a*x^5)/x,x)
 

Output:

int(acsch(a*x**5)/x,x)