\(\int \frac {\sec ^{-1}(a x^n)}{x} \, dx\) [17]

Optimal result

Integrand size = 10, antiderivative size = 69 \[ \int \frac {\sec ^{-1}\left (a x^n\right )}{x} \, dx=\frac {i \sec ^{-1}\left (a x^n\right )^2}{2 n}-\frac {\sec ^{-1}\left (a x^n\right ) \log \left (1+e^{2 i \sec ^{-1}\left (a x^n\right )}\right )}{n}+\frac {i \operatorname {PolyLog}\left (2,-e^{2 i \sec ^{-1}\left (a x^n\right )}\right )}{2 n} \] Output:

1/2*I*arcsec(a*x^n)^2/n-arcsec(a*x^n)*ln(1+(1/a/(x^n)+I*(1-1/a^2/(x^n)^2)^ 
(1/2))^2)/n+1/2*I*polylog(2,-(1/a/(x^n)+I*(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 = 60, normalized size of antiderivative = 0.87 \[ \int \frac {\sec ^{-1}\left (a x^n\right )}{x} \, dx=\frac {x^{-n} \, _3F_2\left (\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};\frac {x^{-2 n}}{a^2}\right )}{a n}+\left (\sec ^{-1}\left (a x^n\right )+\arcsin \left (\frac {x^{-n}}{a}\right )\right ) \log (x) \] Input:

Integrate[ArcSec[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) 
+ (ArcSec[a*x^n] + ArcSin[1/(a*x^n)])*Log[x]
 

Rubi [A] (warning: unable to verify)

Time = 0.49 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.16, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {7282, 5741, 5137, 3042, 4202, 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.

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

\(\Big \downarrow \) 7282

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

\(\Big \downarrow \) 5741

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

\(\Big \downarrow \) 5137

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \arccos \left (\frac {x^{-n}}{a}\right ) \tan \left (\arccos \left (\frac {x^{-n}}{a}\right )\right )d\arccos \left (\frac {x^{-n}}{a}\right )}{n}\)

\(\Big \downarrow \) 4202

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

((I/2)*x^(2*n) - (2*I)*((-1/2*I)*ArcCos[1/(a*x^n)]*Log[1 + E^((2*I)*ArcCos 
[1/(a*x^n)])] - PolyLog[2, -E^((2*I)*ArcCos[1/(a*x^n)])]/4))/n
 

Defintions of rubi rules used

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 4202

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (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*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt 
Q[m, 0]
 

rule 5137

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

rule 5741

Int[((a_.) + ArcSec[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> -Subst[Int[(a + b 
*ArcCos[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 [A] (verified)

Time = 0.80 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.35

method	result	size
derivativedivides	\(\frac {\frac {i \operatorname {arcsec}\left (a \,x^{n}\right )^{2}}{2}-\operatorname {arcsec}\left (a \,x^{n}\right ) \ln \left (1+\left (\frac {x^{-n}}{a}+i \sqrt {1-\frac {x^{-2 n}}{a^{2}}}\right )^{2}\right )+\frac {i \operatorname {polylog}\left (2, -\left (\frac {x^{-n}}{a}+i \sqrt {1-\frac {x^{-2 n}}{a^{2}}}\right )^{2}\right )}{2}}{n}\)	\(93\)
default	\(\frac {\frac {i \operatorname {arcsec}\left (a \,x^{n}\right )^{2}}{2}-\operatorname {arcsec}\left (a \,x^{n}\right ) \ln \left (1+\left (\frac {x^{-n}}{a}+i \sqrt {1-\frac {x^{-2 n}}{a^{2}}}\right )^{2}\right )+\frac {i \operatorname {polylog}\left (2, -\left (\frac {x^{-n}}{a}+i \sqrt {1-\frac {x^{-2 n}}{a^{2}}}\right )^{2}\right )}{2}}{n}\)	\(93\)

Input:

int(arcsec(a*x^n)/x,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F(-2)]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-a^2*n*integrate(sqrt(a*x^n + 1)*sqrt(a*x^n - 1)*log(x)/(a^4*x*x^(2*n) - a 
^2*x), x) + arctan(sqrt(a*x^n + 1)*sqrt(a*x^n - 1))*log(x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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