3.7 \(\int \frac{\cot ^{-1}(a x)}{x} \, dx\)

Optimal. Leaf size=37 \[ \frac{1}{2} i \text{PolyLog}\left (2,\frac{i}{a x}\right )-\frac{1}{2} i \text{PolyLog}\left (2,-\frac{i}{a x}\right ) \]

[Out]

(-I/2)*PolyLog[2, (-I)/(a*x)] + (I/2)*PolyLog[2, I/(a*x)]

________________________________________________________________________________________

Rubi [A]  time = 0.0252205, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4849, 2391} \[ \frac{1}{2} i \text{PolyLog}\left (2,\frac{i}{a x}\right )-\frac{1}{2} i \text{PolyLog}\left (2,-\frac{i}{a x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[a*x]/x,x]

[Out]

(-I/2)*PolyLog[2, (-I)/(a*x)] + (I/2)*PolyLog[2, I/(a*x)]

Rule 4849

Int[((a_.) + ArcCot[(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 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{\cot ^{-1}(a x)}{x} \, dx &=\frac{1}{2} i \int \frac{\log \left (1-\frac{i}{a x}\right )}{x} \, dx-\frac{1}{2} i \int \frac{\log \left (1+\frac{i}{a x}\right )}{x} \, dx\\ &=-\frac{1}{2} i \text{Li}_2\left (-\frac{i}{a x}\right )+\frac{1}{2} i \text{Li}_2\left (\frac{i}{a x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0026943, size = 37, normalized size = 1. \[ \frac{1}{2} i \text{PolyLog}\left (2,\frac{i}{a x}\right )-\frac{1}{2} i \text{PolyLog}\left (2,-\frac{i}{a x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[a*x]/x,x]

[Out]

(-I/2)*PolyLog[2, (-I)/(a*x)] + (I/2)*PolyLog[2, I/(a*x)]

________________________________________________________________________________________

Maple [B]  time = 0.049, size = 63, normalized size = 1.7 \begin{align*} \ln \left ( ax \right ){\rm arccot} \left (ax\right )-{\frac{i}{2}}\ln \left ( ax \right ) \ln \left ( 1+iax \right ) +{\frac{i}{2}}\ln \left ( ax \right ) \ln \left ( 1-iax \right ) -{\frac{i}{2}}{\it dilog} \left ( 1+iax \right ) +{\frac{i}{2}}{\it dilog} \left ( 1-iax \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccot(a*x)/x,x)

[Out]

ln(a*x)*arccot(a*x)-1/2*I*ln(a*x)*ln(1+I*a*x)+1/2*I*ln(a*x)*ln(1-I*a*x)-1/2*I*dilog(1+I*a*x)+1/2*I*dilog(1-I*a
*x)

________________________________________________________________________________________

Maxima [B]  time = 1.62585, size = 89, normalized size = 2.41 \begin{align*} -i \, \arctan \left (a x\right ) \arctan \left (0, a\right ) + \frac{1}{4} \, \pi \log \left (a^{2} x^{2} + 1\right ) - \arctan \left (a x\right ) \log \left (x{\left | a \right |}\right ) + \operatorname{arccot}\left (a x\right ) \log \left (x\right ) + \arctan \left (a x\right ) \log \left (x\right ) + \frac{1}{2} i \,{\rm Li}_2\left (i \, a x + 1\right ) - \frac{1}{2} i \,{\rm Li}_2\left (-i \, a x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x)/x,x, algorithm="maxima")

[Out]

-I*arctan(a*x)*arctan2(0, a) + 1/4*pi*log(a^2*x^2 + 1) - arctan(a*x)*log(x*abs(a)) + arccot(a*x)*log(x) + arct
an(a*x)*log(x) + 1/2*I*dilog(I*a*x + 1) - 1/2*I*dilog(-I*a*x + 1)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x)/x,x, algorithm="fricas")

[Out]

integral(arccot(a*x)/x, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acot(a*x)/x,x)

[Out]

Integral(acot(a*x)/x, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x)/x,x, algorithm="giac")

[Out]

integrate(arccot(a*x)/x, x)