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

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5373, 4721, 3798, 2221, 2317, 2438} \[ \int \frac {\csc ^{-1}\left (\frac {a}{x}\right )}{x} \, dx=-\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin \left (\frac {x}{a}\right )}\right )-\frac {1}{2} i \arcsin \left (\frac {x}{a}\right )^2+\arcsin \left (\frac {x}{a}\right ) \log \left (1-e^{2 i \arcsin \left (\frac {x}{a}\right )}\right ) \]

[In]

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

[Out]

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

Rule 2221

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]
- Dist[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 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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 2438

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 3798

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[I*((c + d*x)^(m + 1)/(d*(
m + 1))), x] - Dist[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 4721

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 5373

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]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\arcsin \left (\frac {x}{a}\right )}{x} \, dx \\ & = \text {Subst}\left (\int x \cot (x) \, dx,x,\arcsin \left (\frac {x}{a}\right )\right ) \\ & = -\frac {1}{2} i \arcsin \left (\frac {x}{a}\right )^2-2 i \text {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\arcsin \left (\frac {x}{a}\right )\right ) \\ & = -\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 )-\text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\arcsin \left (\frac {x}{a}\right )\right ) \\ & = -\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 \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \arcsin \left (\frac {x}{a}\right )}\right ) \\ & = -\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 ) \\ \end{align*}

Mathematica [A] (verified)

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

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.61 (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	\(-\frac {i \operatorname {arccsc}\left (\frac {a}{x}\right )^{2}}{2}+\operatorname {arccsc}\left (\frac {a}{x}\right ) \ln \left (1+\frac {i x}{a}+\sqrt {1-\frac {x^{2}}{a^{2}}}\right )-i \operatorname {polylog}\left (2, -\frac {i x}{a}-\sqrt {1-\frac {x^{2}}{a^{2}}}\right )+\operatorname {arccsc}\left (\frac {a}{x}\right ) \ln \left (1-\frac {i x}{a}-\sqrt {1-\frac {x^{2}}{a^{2}}}\right )-i \operatorname {polylog}\left (2, \frac {i x}{a}+\sqrt {1-\frac {x^{2}}{a^{2}}}\right )\)	\(125\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(arccsc(a/x)/x, x)

Sympy [F]

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

[In]

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

[Out]

Integral(acsc(a/x)/x, x)

Maxima [F]

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

[In]

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

[Out]

integrate(arccsc(a/x)/x, x)

Giac [F]

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

[In]

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

[Out]

integrate(arccsc(a/x)/x, x)

Mupad [B] (verification not implemented)

Time = 0.77 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.83 \[ \int \frac {\csc ^{-1}\left (\frac {a}{x}\right )}{x} \, dx=-\frac {\mathrm {polylog}\left (2,{\mathrm {e}}^{\mathrm {asin}\left (\frac {x}{a}\right )\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}}{2}+\ln \left (1-{\mathrm {e}}^{\mathrm {asin}\left (\frac {x}{a}\right )\,2{}\mathrm {i}}\right )\,\mathrm {asin}\left (\frac {x}{a}\right )-\frac {{\mathrm {asin}\left (\frac {x}{a}\right )}^2\,1{}\mathrm {i}}{2} \]

[In]

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

[Out]

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