\(\int \frac {\cot ^{-1}(a x)}{x} \, dx\) [7]

Optimal result

Integrand size = 8, antiderivative size = 37 \[ \int \frac {\cot ^{-1}(a x)}{x} \, dx=-\frac {1}{2} i \operatorname {PolyLog}\left (2,-\frac {i}{a x}\right )+\frac {1}{2} i \operatorname {PolyLog}\left (2,\frac {i}{a x}\right ) \]

[Out]

-1/2*I*polylog(2,-I/a/x)+1/2*I*polylog(2,I/a/x)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4941, 2438} \[ \int \frac {\cot ^{-1}(a x)}{x} \, dx=\frac {1}{2} i \operatorname {PolyLog}\left (2,\frac {i}{a x}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,-\frac {i}{a x}\right ) \]

[In]

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

[Out]

(-1/2*I)*PolyLog[2, (-I)/(a*x)] + (I/2)*PolyLog[2, I/(a*x)]

Rule 2438

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 4941

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]

Rubi steps \begin{align*} \text {integral}& = \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 \operatorname {PolyLog}\left (2,-\frac {i}{a x}\right )+\frac {1}{2} i \operatorname {PolyLog}\left (2,\frac {i}{a x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00 \[ \int \frac {\cot ^{-1}(a x)}{x} \, dx=-\frac {1}{2} i \operatorname {PolyLog}\left (2,-\frac {i}{a x}\right )+\frac {1}{2} i \operatorname {PolyLog}\left (2,\frac {i}{a x}\right ) \]

[In]

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

[Out]

(-1/2*I)*PolyLog[2, (-I)/(a*x)] + (I/2)*PolyLog[2, I/(a*x)]

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.89

method	result	size
risch	\(\frac {\pi \ln \left (-i a x \right )}{2}+\frac {i \operatorname {dilog}\left (-i a x +1\right )}{2}-\frac {i \operatorname {dilog}\left (i a x +1\right )}{2}\)	\(33\)
derivativedivides	\(\ln \left (a x \right ) \operatorname {arccot}\left (a x \right )-\frac {i \ln \left (a x \right ) \ln \left (i a x +1\right )}{2}+\frac {i \ln \left (a x \right ) \ln \left (-i a x +1\right )}{2}-\frac {i \operatorname {dilog}\left (i a x +1\right )}{2}+\frac {i \operatorname {dilog}\left (-i a x +1\right )}{2}\)	\(63\)
default	\(\ln \left (a x \right ) \operatorname {arccot}\left (a x \right )-\frac {i \ln \left (a x \right ) \ln \left (i a x +1\right )}{2}+\frac {i \ln \left (a x \right ) \ln \left (-i a x +1\right )}{2}-\frac {i \operatorname {dilog}\left (i a x +1\right )}{2}+\frac {i \operatorname {dilog}\left (-i a x +1\right )}{2}\)	\(63\)
parts	\(\ln \left (x \right ) \operatorname {arccot}\left (a x \right )+a \left (-\frac {i \ln \left (x \right ) \left (-\ln \left (-i a x +1\right )+\ln \left (i a x +1\right )\right )}{2 a}-\frac {i \left (\operatorname {dilog}\left (i a x +1\right )-\operatorname {dilog}\left (-i a x +1\right )\right )}{2 a}\right )\)	\(64\)

[In]

int(arccot(a*x)/x,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

\[ \int \frac {\cot ^{-1}(a x)}{x} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )}{x} \,d x } \]

[In]

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

[Out]

integral(arccot(a*x)/x, x)

Sympy [F]

\[ \int \frac {\cot ^{-1}(a x)}{x} \, dx=\int \frac {\operatorname {acot}{\left (a x \right )}}{x}\, dx \]

[In]

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

[Out]

Integral(acot(a*x)/x, x)

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (23) = 46\).

Time = 0.30 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.51 \[ \int \frac {\cot ^{-1}(a x)}{x} \, dx=\frac {1}{4} \, \pi \log \left (a^{2} x^{2} + 1\right ) - \arctan \left (a x\right ) \log \left (a x\right ) + \operatorname {arccot}\left (a x\right ) \log \left (x\right ) + \arctan \left (a x\right ) \log \left (x\right ) + \frac {1}{2} i \, {\rm Li}_2\left (i \, a x + 1\right ) - \frac {1}{2} i \, {\rm Li}_2\left (-i \, a x + 1\right ) \]

[In]

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

[Out]

1/4*pi*log(a^2*x^2 + 1) - arctan(a*x)*log(a*x) + arccot(a*x)*log(x) + arctan(a*x)*log(x) + 1/2*I*dilog(I*a*x +
1) - 1/2*I*dilog(-I*a*x + 1)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.03 \[ \int \frac {\cot ^{-1}(a x)}{x} \, dx=-\frac {1}{2} \, {\left (\frac {x^{2} \arctan \left (\frac {1}{a x}\right )}{a} + \frac {x}{a^{2}} + \frac {\arctan \left (\frac {1}{a x}\right )}{a^{3}}\right )} a^{2} \]

[In]

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

[Out]

-1/2*(x^2*arctan(1/(a*x))/a + x/a^2 + arctan(1/(a*x))/a^3)*a^2

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^{-1}(a x)}{x} \, dx=\int \frac {\mathrm {acot}\left (a\,x\right )}{x} \,d x \]

[In]

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

[Out]

int(acot(a*x)/x, x)