3.98 \(\int \frac{\cot ^{-1}(a x^5)}{x} \, dx\)

Optimal. Leaf size=37 \[ \frac{1}{10} i \text{PolyLog}\left (2,\frac{i}{a x^5}\right )-\frac{1}{10} i \text{PolyLog}\left (2,-\frac{i}{a x^5}\right ) \]

[Out]

(-I/10)*PolyLog[2, (-I)/(a*x^5)] + (I/10)*PolyLog[2, I/(a*x^5)]

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Rubi [A]  time = 0.0344804, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5032, 4849, 2391} \[ \frac{1}{10} i \text{PolyLog}\left (2,\frac{i}{a x^5}\right )-\frac{1}{10} i \text{PolyLog}\left (2,-\frac{i}{a x^5}\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[a*x^5]/x,x]

[Out]

(-I/10)*PolyLog[2, (-I)/(a*x^5)] + (I/10)*PolyLog[2, I/(a*x^5)]

Rule 5032

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

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}\left (a x^5\right )}{x} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{\cot ^{-1}(a x)}{x} \, dx,x,x^5\right )\\ &=\frac{1}{10} i \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{i}{a x}\right )}{x} \, dx,x,x^5\right )-\frac{1}{10} i \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{i}{a x}\right )}{x} \, dx,x,x^5\right )\\ &=-\frac{1}{10} i \text{Li}_2\left (-\frac{i}{a x^5}\right )+\frac{1}{10} i \text{Li}_2\left (\frac{i}{a x^5}\right )\\ \end{align*}

Mathematica [A]  time = 0.0069422, size = 37, normalized size = 1. \[ \frac{1}{10} i \text{PolyLog}\left (2,\frac{i}{a x^5}\right )-\frac{1}{10} i \text{PolyLog}\left (2,-\frac{i}{a x^5}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[a*x^5]/x,x]

[Out]

(-I/10)*PolyLog[2, (-I)/(a*x^5)] + (I/10)*PolyLog[2, I/(a*x^5)]

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Maple [C]  time = 0.116, size = 57, normalized size = 1.5 \begin{align*} \ln \left ( x \right ){\rm arccot} \left (a{x}^{5}\right )+{\frac{1}{2\,a}\sum _{{\it \_R1}={\it RootOf} \left ({{\it \_Z}}^{10}{a}^{2}+1 \right ) }{\frac{1}{{{\it \_R1}}^{5}} \left ( \ln \left ( x \right ) \ln \left ({\frac{{\it \_R1}-x}{{\it \_R1}}} \right ) +{\it dilog} \left ({\frac{{\it \_R1}-x}{{\it \_R1}}} \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccot(a*x^5)/x,x)

[Out]

ln(x)*arccot(a*x^5)+1/2/a*sum(1/_R1^5*(ln(x)*ln((_R1-x)/_R1)+dilog((_R1-x)/_R1)),_R1=RootOf(_Z^10*a^2+1))

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Maxima [B]  time = 1.62922, size = 108, normalized size = 2.92 \begin{align*} -\frac{1}{5} i \, \arctan \left (a x^{5}\right ) \arctan \left (0, a\right ) + \frac{1}{20} \, \pi \log \left (a^{2} x^{10} + 1\right ) - \frac{1}{5} \, \arctan \left (a x^{5}\right ) \log \left (x^{5}{\left | a \right |}\right ) + \operatorname{arccot}\left (a x^{5}\right ) \log \left (x\right ) + \arctan \left (a x^{5}\right ) \log \left (x\right ) + \frac{1}{10} i \,{\rm Li}_2\left (i \, a x^{5} + 1\right ) - \frac{1}{10} i \,{\rm Li}_2\left (-i \, a x^{5} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x^5)/x,x, algorithm="maxima")

[Out]

-1/5*I*arctan(a*x^5)*arctan2(0, a) + 1/20*pi*log(a^2*x^10 + 1) - 1/5*arctan(a*x^5)*log(x^5*abs(a)) + arccot(a*
x^5)*log(x) + arctan(a*x^5)*log(x) + 1/10*I*dilog(I*a*x^5 + 1) - 1/10*I*dilog(-I*a*x^5 + 1)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{arccot}\left (a x^{5}\right )}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x^5)/x,x, algorithm="fricas")

[Out]

integral(arccot(a*x^5)/x, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acot(a*x**5)/x,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arccot}\left (a x^{5}\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x^5)/x,x, algorithm="giac")

[Out]

integrate(arccot(a*x^5)/x, x)