\(\int \frac {\csc ^{-1}(a x^n)}{x} \, dx\) [16]

Optimal result

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

1/2*I*arccsc(a*x^n)^2/n-arccsc(a*x^n)*ln(1-(I/a/(x^n)+(1-1/a^2/(x^n)^2)^(1 
/2))^2)/n+1/2*I*polylog(2,(I/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 = 63, normalized size of antiderivative = 0.91 \[ \int \frac {\csc ^{-1}\left (a x^n\right )}{x} \, dx=-\frac {x^{-n} \, _3F_2\left (\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};\frac {x^{-2 n}}{a^2}\right )}{a n}+\left (\csc ^{-1}\left (a x^n\right )-\arcsin \left (\frac {x^{-n}}{a}\right )\right ) \log (x) \] Input:

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

Rubi [A] (warning: unable to verify)

Time = 0.49 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.17, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {7282, 5742, 5136, 3042, 25, 4200, 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.

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

\(\Big \downarrow \) 7282

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

\(\Big \downarrow \) 5742

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

\(\Big \downarrow \) 5136

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4200

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

-(((-1/2*I)*x^(2*n) - (2*I)*((I/2)*ArcSin[1/(a*x^n)]*Log[1 - E^((2*I)*ArcS 
in[1/(a*x^n)])] + PolyLog[2, E^((2*I)*ArcSin[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 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 4200

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))), x] 
, x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]
 

rule 5136

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

rule 5742

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> -Subst[Int[(a + b 
*ArcSin[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 = 1.05 (sec) , antiderivative size = 155, normalized size of antiderivative = 2.25

method	result	size
derivativedivides	\(\frac {\frac {i \operatorname {arccsc}\left (a \,x^{n}\right )^{2}}{2}-\operatorname {arccsc}\left (a \,x^{n}\right ) \ln \left (1+\frac {i x^{-n}}{a}+\sqrt {1-\frac {x^{-2 n}}{a^{2}}}\right )+i \operatorname {polylog}\left (2, -\frac {i x^{-n}}{a}-\sqrt {1-\frac {x^{-2 n}}{a^{2}}}\right )-\operatorname {arccsc}\left (a \,x^{n}\right ) \ln \left (1-\frac {i x^{-n}}{a}-\sqrt {1-\frac {x^{-2 n}}{a^{2}}}\right )+i \operatorname {polylog}\left (2, \frac {i x^{-n}}{a}+\sqrt {1-\frac {x^{-2 n}}{a^{2}}}\right )}{n}\)	\(155\)
default	\(\frac {\frac {i \operatorname {arccsc}\left (a \,x^{n}\right )^{2}}{2}-\operatorname {arccsc}\left (a \,x^{n}\right ) \ln \left (1+\frac {i x^{-n}}{a}+\sqrt {1-\frac {x^{-2 n}}{a^{2}}}\right )+i \operatorname {polylog}\left (2, -\frac {i x^{-n}}{a}-\sqrt {1-\frac {x^{-2 n}}{a^{2}}}\right )-\operatorname {arccsc}\left (a \,x^{n}\right ) \ln \left (1-\frac {i x^{-n}}{a}-\sqrt {1-\frac {x^{-2 n}}{a^{2}}}\right )+i \operatorname {polylog}\left (2, \frac {i x^{-n}}{a}+\sqrt {1-\frac {x^{-2 n}}{a^{2}}}\right )}{n}\)	\(155\)

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(arccsc(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) + arctan2(1, sqrt(a*x^n + 1)*sqrt(a*x^n - 1))*log(x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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