\(\int \frac {\text {sech}^{-1}(a x^5)}{x} \, dx\) [30]

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.91 \[ \int \frac {\text {sech}^{-1}\left (a x^5\right )}{x} \, dx=\frac {1}{10} \left (-\text {sech}^{-1}\left (a x^5\right ) \left (\text {sech}^{-1}\left (a x^5\right )+2 \log \left (1+e^{-2 \text {sech}^{-1}\left (a x^5\right )}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-2 \text {sech}^{-1}\left (a x^5\right )}\right )\right ) \] Input:

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

Output:

(-(ArcSech[a*x^5]*(ArcSech[a*x^5] + 2*Log[1 + E^(-2*ArcSech[a*x^5])])) + P 
olyLog[2, -E^(-2*ArcSech[a*x^5])])/10
 

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.49 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.26, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {7282, 6835, 6297, 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.

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

\(\Big \downarrow \) 7282

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

\(\Big \downarrow \) 6835

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

\(\Big \downarrow \) 6297

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

\(\displaystyle \frac {1}{5} i \int \text {arccosh}\left (\frac {1}{a x^5}\right ) \tan \left (i \text {arccosh}\left (\frac {1}{a x^5}\right )\right )d\text {arccosh}\left (\frac {1}{a x^5}\right )\)

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

(I/5)*((-1/2*I)*x^10 + (2*I)*((ArcCosh[1/(a*x^5)]*Log[1 + E^(2*ArcCosh[1/( 
a*x^5)])])/2 + PolyLog[2, -E^(2*ArcCosh[1/(a*x^5)])]/4))
 

Defintions of rubi rules used

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]
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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