\(\int \frac {\csc ^{-1}(\frac {a}{x})}{x} \, dx\) [12]

Optimal result

Integrand size = 10, antiderivative size = 59 \[ \int \frac {\csc ^{-1}\left (\frac {a}{x}\right )}{x} \, dx=-\frac {1}{2} i \arcsin \left (\frac {x}{a}\right )^2+\arcsin \left (\frac {x}{a}\right ) \log \left (1-e^{2 i \arcsin \left (\frac {x}{a}\right )}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin \left (\frac {x}{a}\right )}\right ) \] Output:

-1/2*I*arcsin(x/a)^2+arcsin(x/a)*ln(1-(I*x/a+(1-x^2/a^2)^(1/2))^2)-1/2*I*p 
olylog(2,(I*x/a+(1-x^2/a^2)^(1/2))^2)
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.92 \[ \int \frac {\csc ^{-1}\left (\frac {a}{x}\right )}{x} \, dx=\csc ^{-1}\left (\frac {a}{x}\right ) \log \left (1-e^{2 i \csc ^{-1}\left (\frac {a}{x}\right )}\right )-\frac {1}{2} i \left (\csc ^{-1}\left (\frac {a}{x}\right )^2+\operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}\left (\frac {a}{x}\right )}\right )\right ) \] Input:

Integrate[ArcCsc[a/x]/x,x]
 

Output:

ArcCsc[a/x]*Log[1 - E^((2*I)*ArcCsc[a/x])] - (I/2)*(ArcCsc[a/x]^2 + PolyLo 
g[2, E^((2*I)*ArcCsc[a/x])])
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.14, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {5788, 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.

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

\(\Big \downarrow \) 5788

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

\(\Big \downarrow \) 5136

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4200

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

Int[ArcCsc[a/x]/x,x]
 

Output:

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

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 5788

Int[ArcCsc[(c_.)/((a_.) + (b_.)*(x_)^(n_.))]^(m_.)*(u_.), x_Symbol] :> Int[ 
u*ArcSin[a/c + b*(x^n/c)]^m, x] /; FreeQ[{a, b, c, n, m}, x]
 

Maple [A] (verified)

Time = 1.34 (sec) , antiderivative size = 125, normalized size of antiderivative = 2.12

method	result	size
derivativedivides	\(-\frac {i \operatorname {arccsc}\left (\frac {a}{x}\right )^{2}}{2}+\operatorname {arccsc}\left (\frac {a}{x}\right ) \ln \left (1+\frac {i x}{a}+\sqrt {1-\frac {x^{2}}{a^{2}}}\right )-i \operatorname {polylog}\left (2, -\frac {i x}{a}-\sqrt {1-\frac {x^{2}}{a^{2}}}\right )+\operatorname {arccsc}\left (\frac {a}{x}\right ) \ln \left (1-\frac {i x}{a}-\sqrt {1-\frac {x^{2}}{a^{2}}}\right )-i \operatorname {polylog}\left (2, \frac {i x}{a}+\sqrt {1-\frac {x^{2}}{a^{2}}}\right )\)	\(125\)
default	\(-\frac {i \operatorname {arccsc}\left (\frac {a}{x}\right )^{2}}{2}+\operatorname {arccsc}\left (\frac {a}{x}\right ) \ln \left (1+\frac {i x}{a}+\sqrt {1-\frac {x^{2}}{a^{2}}}\right )-i \operatorname {polylog}\left (2, -\frac {i x}{a}-\sqrt {1-\frac {x^{2}}{a^{2}}}\right )+\operatorname {arccsc}\left (\frac {a}{x}\right ) \ln \left (1-\frac {i x}{a}-\sqrt {1-\frac {x^{2}}{a^{2}}}\right )-i \operatorname {polylog}\left (2, \frac {i x}{a}+\sqrt {1-\frac {x^{2}}{a^{2}}}\right )\)	\(125\)

Input:

int(arccsc(a/x)/x,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F]

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

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

Output:

integral(arccsc(a/x)/x, x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(arccsc(a/x)/x,x, algorithm="maxima")
 

Output:

integrate(arccsc(a/x)/x, x)
 

Giac [F]

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

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

Output:

integrate(arccsc(a/x)/x, x)
 

Mupad [B] (verification not implemented)

Time = 0.85 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.83 \[ \int \frac {\csc ^{-1}\left (\frac {a}{x}\right )}{x} \, dx=-\frac {\mathrm {polylog}\left (2,{\mathrm {e}}^{\mathrm {asin}\left (\frac {x}{a}\right )\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}}{2}+\ln \left (1-{\mathrm {e}}^{\mathrm {asin}\left (\frac {x}{a}\right )\,2{}\mathrm {i}}\right )\,\mathrm {asin}\left (\frac {x}{a}\right )-\frac {{\mathrm {asin}\left (\frac {x}{a}\right )}^2\,1{}\mathrm {i}}{2} \] Input:

int(asin(x/a)/x,x)
 

Output:

log(1 - exp(asin(x/a)*2i))*asin(x/a) - (polylog(2, exp(asin(x/a)*2i))*1i)/ 
2 - (asin(x/a)^2*1i)/2
 

Reduce [F]

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

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

Output:

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