3.63 \(\int \frac{\cos ^{-1}(\sqrt{x})}{x} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0558409, 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 = {4831, 3719, 2190, 2279, 2391} \[ -i \text{PolyLog}\left (2,-e^{2 i \cos ^{-1}\left (\sqrt{x}\right )}\right )-i \cos ^{-1}\left (\sqrt{x}\right )^2+2 \cos ^{-1}\left (\sqrt{x}\right ) \log \left (1+e^{2 i \cos ^{-1}\left (\sqrt{x}\right )}\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcCos[Sqrt[x]]/x,x]

[Out]

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

Rule 4831

Int[ArcCos[(a_.)*(x_)^(p_)]^(n_.)/(x_), x_Symbol] :> -Dist[p^(-1), Subst[Int[x^n*Tan[x], x], x, ArcCos[a*x^p]]
, x] /; FreeQ[{a, p}, x] && IGtQ[n, 0]

Rule 3719

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (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*(e + f*x)))/(1 + E^(2*I*(e + f*x))), x], x] /; FreeQ[{c, d, e, f}, x] &&
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{\cos ^{-1}\left (\sqrt{x}\right )}{x} \, dx &=-\left (2 \operatorname{Subst}\left (\int x \tan (x) \, dx,x,\cos ^{-1}\left (\sqrt{x}\right )\right )\right )\\ &=-i \cos ^{-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,\cos ^{-1}\left (\sqrt{x}\right )\right )\\ &=-i \cos ^{-1}\left (\sqrt{x}\right )^2+2 \cos ^{-1}\left (\sqrt{x}\right ) \log \left (1+e^{2 i \cos ^{-1}\left (\sqrt{x}\right )}\right )-2 \operatorname{Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\cos ^{-1}\left (\sqrt{x}\right )\right )\\ &=-i \cos ^{-1}\left (\sqrt{x}\right )^2+2 \cos ^{-1}\left (\sqrt{x}\right ) \log \left (1+e^{2 i \cos ^{-1}\left (\sqrt{x}\right )}\right )+i \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i \cos ^{-1}\left (\sqrt{x}\right )}\right )\\ &=-i \cos ^{-1}\left (\sqrt{x}\right )^2+2 \cos ^{-1}\left (\sqrt{x}\right ) \log \left (1+e^{2 i \cos ^{-1}\left (\sqrt{x}\right )}\right )-i \text{Li}_2\left (-e^{2 i \cos ^{-1}\left (\sqrt{x}\right )}\right )\\ \end{align*}

Mathematica [A]  time = 0.0260489, size = 54, normalized size = 0.96 \[ -i \left (\text{PolyLog}\left (2,-e^{2 i \cos ^{-1}\left (\sqrt{x}\right )}\right )+\cos ^{-1}\left (\sqrt{x}\right ) \left (\cos ^{-1}\left (\sqrt{x}\right )+2 i \log \left (1+e^{2 i \cos ^{-1}\left (\sqrt{x}\right )}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCos[Sqrt[x]]/x,x]

[Out]

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

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Maple [A]  time = 0.002, size = 59, normalized size = 1.1 \begin{align*} -i \left ( \arccos \left ( \sqrt{x} \right ) \right ) ^{2}+2\,\arccos \left ( \sqrt{x} \right ) \ln \left ( 1+ \left ( \sqrt{x}+i\sqrt{1-x} \right ) ^{2} \right ) -i{\it polylog} \left ( 2,- \left ( \sqrt{x}+i\sqrt{1-x} \right ) ^{2} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccos(sqrt(x))/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arccos(sqrt(x))/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acos(sqrt(x))/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccos(sqrt(x))/x, x)