3.157 \(\int \frac {\tan ^{-1}(\sqrt {x})}{x} \, dx\)

Optimal. Leaf size=31 \[ i \text {Li}_2\left (-i \sqrt {x}\right )-i \text {Li}_2\left (i \sqrt {x}\right ) \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5031, 4848, 2391} \[ i \text {PolyLog}\left (2,-i \sqrt {x}\right )-i \text {PolyLog}\left (2,i \sqrt {x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcTan[Sqrt[x]]/x,x]

[Out]

I*PolyLog[2, (-I)*Sqrt[x]] - I*PolyLog[2, I*Sqrt[x]]

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]

Rule 4848

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Dist[(I*b)/2, Int[Log[1 - I*c*x
]/x, x], x] - Dist[(I*b)/2, Int[Log[1 + I*c*x]/x, x], x]) /; FreeQ[{a, b, c}, x]

Rule 5031

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/n, Subst[Int[(a + b*ArcTan[c*x])^p
/x, x], x, x^n], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 31, normalized size = 1.00 \[ i \text {Li}_2\left (-i \sqrt {x}\right )-i \text {Li}_2\left (i \sqrt {x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTan[Sqrt[x]]/x,x]

[Out]

I*PolyLog[2, (-I)*Sqrt[x]] - I*PolyLog[2, I*Sqrt[x]]

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fricas [F]  time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\arctan \left (\sqrt {x}\right )}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arctan(sqrt(x))/x, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\arctan \left (\sqrt {x}\right )}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arctan(sqrt(x))/x, x)

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maple [B]  time = 0.04, size = 61, normalized size = 1.97 \[ \ln \relax (x ) \arctan \left (\sqrt {x}\right )+\frac {i \ln \relax (x ) \ln \left (1+i \sqrt {x}\right )}{2}-\frac {i \ln \relax (x ) \ln \left (1-i \sqrt {x}\right )}{2}+i \dilog \left (1+i \sqrt {x}\right )-i \dilog \left (1-i \sqrt {x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(x)*arctan(x^(1/2))+1/2*I*ln(x)*ln(1+I*x^(1/2))-1/2*I*ln(x)*ln(1-I*x^(1/2))+I*dilog(1+I*x^(1/2))-I*dilog(1-I
*x^(1/2))

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maxima [B]  time = 0.42, size = 35, normalized size = 1.13 \[ -\frac {1}{2} \, \pi \log \left (x + 1\right ) + \arctan \left (\sqrt {x}\right ) \log \relax (x) - i \, {\rm Li}_2\left (i \, \sqrt {x} + 1\right ) + i \, {\rm Li}_2\left (-i \, \sqrt {x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*pi*log(x + 1) + arctan(sqrt(x))*log(x) - I*dilog(I*sqrt(x) + 1) + I*dilog(-I*sqrt(x) + 1)

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mupad [B]  time = 0.30, size = 24, normalized size = 0.77 \[ -{\mathrm {Li}}_{\mathrm {2}}\left (1-\sqrt {x}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}+\mathrm {polylog}\left (2,-\sqrt {x}\,1{}\mathrm {i}\right )\,1{}\mathrm {i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

polylog(2, -x^(1/2)*1i)*1i - dilog(1 - x^(1/2)*1i)*1i

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {atan}{\left (\sqrt {x} \right )}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(atan(sqrt(x))/x, x)

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