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

Optimal result

Integrand size = 10, antiderivative size = 62 \[ \int \frac {\sec ^{-1}\left (a x^5\right )}{x} \, dx=\frac {1}{10} i \sec ^{-1}\left (a x^5\right )^2-\frac {1}{5} \sec ^{-1}\left (a x^5\right ) \log \left (1+e^{2 i \sec ^{-1}\left (a x^5\right )}\right )+\frac {1}{10} i \operatorname {PolyLog}\left (2,-e^{2 i \sec ^{-1}\left (a x^5\right )}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.90 \[ \int \frac {\sec ^{-1}\left (a x^5\right )}{x} \, dx=\frac {1}{10} i \left (\sec ^{-1}\left (a x^5\right ) \left (\sec ^{-1}\left (a x^5\right )+2 i \log \left (1+e^{2 i \sec ^{-1}\left (a x^5\right )}\right )\right )+\operatorname {PolyLog}\left (2,-e^{2 i \sec ^{-1}\left (a x^5\right )}\right )\right ) \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.46 (sec) , antiderivative size = 72, 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.

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

\(\Big \downarrow \) 7282

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

\(\Big \downarrow \) 5741

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

\(\Big \downarrow \) 5137

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4202

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

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 [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-5*a^2*integrate(sqrt(a*x^5 + 1)*sqrt(a*x^5 - 1)*log(x)/(a^4*x^11 - a^2*x) 
, x) - 5*I*a^2*integrate(log(x)/(a^4*x^11 - a^2*x), x) + arctan(sqrt(a*x^5 
 + 1)*sqrt(a*x^5 - 1))*log(x) - 1/2*I*log(a^2*x^10)*log(x) + 1/2*I*log(a*x 
^5 + 1)*log(x) + 1/2*I*log(-a*x^5 + 1)*log(x) + I*log(a)*log(x) + 5/2*I*lo 
g(x)^2 + 1/10*I*dilog(a*x^5) + 1/10*I*dilog(-a*x^5)
                                                                                    
                                                                                    
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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