3.6 \(\int \frac{\csc ^{-1}(\sqrt{x})}{x} \, dx\)

Optimal. Leaf size=56 \[ i \text{PolyLog}\left (2,e^{2 i \csc ^{-1}\left (\sqrt{x}\right )}\right )+i \csc ^{-1}\left (\sqrt{x}\right )^2-2 \csc ^{-1}\left (\sqrt{x}\right ) \log \left (1-e^{2 i \csc ^{-1}\left (\sqrt{x}\right )}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0826394, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {5219, 4625, 3717, 2190, 2279, 2391} \[ i \text{PolyLog}\left (2,e^{2 i \csc ^{-1}\left (\sqrt{x}\right )}\right )+i \csc ^{-1}\left (\sqrt{x}\right )^2-2 \csc ^{-1}\left (\sqrt{x}\right ) \log \left (1-e^{2 i \csc ^{-1}\left (\sqrt{x}\right )}\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcCsc[Sqrt[x]]/x,x]

[Out]

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

Rule 5219

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> -Subst[Int[(a + b*ArcSin[x/c])/x, x], x, 1/x] /; Fre
eQ[{a, b, c}, x]

Rule 4625

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

Rule 3717

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 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2279

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 2391

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]

Rubi steps

\begin{align*} \int \frac{\csc ^{-1}\left (\sqrt{x}\right )}{x} \, dx &=2 \operatorname{Subst}\left (\int \frac{\csc ^{-1}(x)}{x} \, dx,x,\sqrt{x}\right )\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{\sin ^{-1}(x)}{x} \, dx,x,\frac{1}{\sqrt{x}}\right )\right )\\ &=-\left (2 \operatorname{Subst}\left (\int x \cot (x) \, dx,x,\sin ^{-1}\left (\frac{1}{\sqrt{x}}\right )\right )\right )\\ &=i \sin ^{-1}\left (\frac{1}{\sqrt{x}}\right )^2+4 i \operatorname{Subst}\left (\int \frac{e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\sin ^{-1}\left (\frac{1}{\sqrt{x}}\right )\right )\\ &=i \sin ^{-1}\left (\frac{1}{\sqrt{x}}\right )^2-2 \sin ^{-1}\left (\frac{1}{\sqrt{x}}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (\frac{1}{\sqrt{x}}\right )}\right )+2 \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}\left (\frac{1}{\sqrt{x}}\right )\right )\\ &=i \sin ^{-1}\left (\frac{1}{\sqrt{x}}\right )^2-2 \sin ^{-1}\left (\frac{1}{\sqrt{x}}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (\frac{1}{\sqrt{x}}\right )}\right )-i \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}\left (\frac{1}{\sqrt{x}}\right )}\right )\\ &=i \sin ^{-1}\left (\frac{1}{\sqrt{x}}\right )^2-2 \sin ^{-1}\left (\frac{1}{\sqrt{x}}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (\frac{1}{\sqrt{x}}\right )}\right )+i \text{Li}_2\left (e^{2 i \sin ^{-1}\left (\frac{1}{\sqrt{x}}\right )}\right )\\ \end{align*}

Mathematica [A]  time = 0.031124, size = 54, normalized size = 0.96 \[ i \left (\text{PolyLog}\left (2,e^{2 i \csc ^{-1}\left (\sqrt{x}\right )}\right )+\csc ^{-1}\left (\sqrt{x}\right ) \left (\csc ^{-1}\left (\sqrt{x}\right )+2 i \log \left (1-e^{2 i \csc ^{-1}\left (\sqrt{x}\right )}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCsc[Sqrt[x]]/x,x]

[Out]

I*(ArcCsc[Sqrt[x]]*(ArcCsc[Sqrt[x]] + (2*I)*Log[1 - E^((2*I)*ArcCsc[Sqrt[x]])]) + PolyLog[2, E^((2*I)*ArcCsc[S
qrt[x]])])

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Maple [A]  time = 0.168, size = 105, normalized size = 1.9 \begin{align*} i \left ({\rm arccsc} \left (\sqrt{x}\right ) \right ) ^{2}-2\,{\rm arccsc} \left (\sqrt{x}\right )\ln \left ( 1-{\frac{i}{\sqrt{x}}}-\sqrt{1-{x}^{-1}} \right ) -2\,{\rm arccsc} \left (\sqrt{x}\right )\ln \left ( 1+{\frac{i}{\sqrt{x}}}+\sqrt{1-{x}^{-1}} \right ) +2\,i{\it polylog} \left ( 2,{-i{\frac{1}{\sqrt{x}}}}-\sqrt{1-{x}^{-1}} \right ) +2\,i{\it polylog} \left ( 2,{i{\frac{1}{\sqrt{x}}}}+\sqrt{1-{x}^{-1}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccsc(x^(1/2))/x,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arccsc}\left (\sqrt{x}\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccsc(x^(1/2))/x,x, algorithm="maxima")

[Out]

integrate(arccsc(sqrt(x))/x, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{arccsc}\left (\sqrt{x}\right )}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccsc(x^(1/2))/x,x, algorithm="fricas")

[Out]

integral(arccsc(sqrt(x))/x, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acsc}{\left (\sqrt{x} \right )}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acsc(x**(1/2))/x,x)

[Out]

Integral(acsc(sqrt(x))/x, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arccsc}\left (\sqrt{x}\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccsc(x^(1/2))/x,x, algorithm="giac")

[Out]

integrate(arccsc(sqrt(x))/x, x)