3.364 \(\int \frac{\sin ^{-1}(\sqrt{x})}{x} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0610309, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4830, 3717, 2190, 2279, 2391} \[ -i \text{PolyLog}\left (2,e^{2 i \sin ^{-1}\left (\sqrt{x}\right )}\right )-i \sin ^{-1}\left (\sqrt{x}\right )^2+2 \sin ^{-1}\left (\sqrt{x}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (\sqrt{x}\right )}\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcSin[Sqrt[x]]/x,x]

[Out]

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

Rule 4830

Int[ArcSin[(a_.)*(x_)^(p_)]^(n_.)/(x_), x_Symbol] :> Dist[1/p, Subst[Int[x^n*Cot[x], x], x, ArcSin[a*x^p]], x]
/; FreeQ[{a, p}, 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{\sin ^{-1}\left (\sqrt{x}\right )}{x} \, dx &=2 \operatorname{Subst}\left (\int x \cot (x) \, dx,x,\sin ^{-1}\left (\sqrt{x}\right )\right )\\ &=-i \sin ^{-1}\left (\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 (\sqrt{x}\right )\right )\\ &=-i \sin ^{-1}\left (\sqrt{x}\right )^2+2 \sin ^{-1}\left (\sqrt{x}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (\sqrt{x}\right )}\right )-2 \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}\left (\sqrt{x}\right )\right )\\ &=-i \sin ^{-1}\left (\sqrt{x}\right )^2+2 \sin ^{-1}\left (\sqrt{x}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (\sqrt{x}\right )}\right )+i \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}\left (\sqrt{x}\right )}\right )\\ &=-i \sin ^{-1}\left (\sqrt{x}\right )^2+2 \sin ^{-1}\left (\sqrt{x}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (\sqrt{x}\right )}\right )-i \text{Li}_2\left (e^{2 i \sin ^{-1}\left (\sqrt{x}\right )}\right )\\ \end{align*}

Mathematica [A]  time = 0.0306429, size = 53, normalized size = 0.95 \[ 2 \sin ^{-1}\left (\sqrt{x}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (\sqrt{x}\right )}\right )-i \left (\sin ^{-1}\left (\sqrt{x}\right )^2+\text{PolyLog}\left (2,e^{2 i \sin ^{-1}\left (\sqrt{x}\right )}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSin[Sqrt[x]]/x,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.038, size = 97, normalized size = 1.7 \begin{align*} -i \left ( \arcsin \left ( \sqrt{x} \right ) \right ) ^{2}+2\,\arcsin \left ( \sqrt{x} \right ) \ln \left ( 1+i\sqrt{x}+\sqrt{1-x} \right ) +2\,\arcsin \left ( \sqrt{x} \right ) \ln \left ( 1-i\sqrt{x}-\sqrt{1-x} \right ) -2\,i{\it polylog} \left ( 2,-i\sqrt{x}-\sqrt{1-x} \right ) -2\,i{\it polylog} \left ( 2,i\sqrt{x}+\sqrt{1-x} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arcsin(sqrt(x))/x, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arcsin(sqrt(x))/x, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(asin(sqrt(x))/x, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arcsin(sqrt(x))/x, x)