3.1.97 \(\int \frac {\cot ^{-1}(a x^n)}{x} \, dx\) [97]

Optimal. Leaf size=47 \[ -\frac {i \text {PolyLog}\left (2,-\frac {i x^{-n}}{a}\right )}{2 n}+\frac {i \text {PolyLog}\left (2,\frac {i x^{-n}}{a}\right )}{2 n} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 47, 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 = {4945, 4941, 2438} \begin {gather*} \frac {i \text {Li}_2\left (\frac {i x^{-n}}{a}\right )}{2 n}-\frac {i \text {Li}_2\left (-\frac {i x^{-n}}{a}\right )}{2 n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

Rule 4945

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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 40, normalized size = 0.85 \begin {gather*} -\frac {i \left (\text {PolyLog}\left (2,-\frac {i x^{-n}}{a}\right )-\text {PolyLog}\left (2,\frac {i x^{-n}}{a}\right )\right )}{2 n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (39 ) = 78\).
time = 0.15, size = 83, normalized size = 1.77

method	result	size
derivativedivides	\(\frac {\ln \left (a \,x^{n}\right ) \mathrm {arccot}\left (a \,x^{n}\right )-\frac {i \ln \left (a \,x^{n}\right ) \ln \left (1+i x^{n} a \right )}{2}+\frac {i \ln \left (a \,x^{n}\right ) \ln \left (1-i x^{n} a \right )}{2}-\frac {i \dilog \left (1+i x^{n} a \right )}{2}+\frac {i \dilog \left (1-i x^{n} a \right )}{2}}{n}\)	\(83\)
default	\(\frac {\ln \left (a \,x^{n}\right ) \mathrm {arccot}\left (a \,x^{n}\right )-\frac {i \ln \left (a \,x^{n}\right ) \ln \left (1+i x^{n} a \right )}{2}+\frac {i \ln \left (a \,x^{n}\right ) \ln \left (1-i x^{n} a \right )}{2}-\frac {i \dilog \left (1+i x^{n} a \right )}{2}+\frac {i \dilog \left (1-i x^{n} a \right )}{2}}{n}\)	\(83\)
risch	\(\frac {i \ln \left (x \right ) \ln \left (1+i x^{n} a \right )}{2}+\frac {\pi \ln \left (x \right )}{2}+\frac {i \dilog \left (1-i x^{n} a \right )}{2 n}-\frac {i \ln \left (-i \left (-a \,x^{n}+i\right )\right ) \ln \left (x \right )}{2}+\frac {i \ln \left (-i \left (-a \,x^{n}+i\right )\right ) \ln \left (-i x^{n} a \right )}{2 n}+\frac {i \dilog \left (-i x^{n} a \right )}{2 n}\)	\(97\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*n*integrate(x^n*log(x)/(a^2*x*x^(2*n) + x), x) + arctan(1/(a*x^n))*log(x)

________________________________________________________________________________________

Fricas [A]
time = 2.55, size = 63, normalized size = 1.34 \begin {gather*} \frac {2 \, n \operatorname {arccot}\left (a x^{n}\right ) \log \left (x\right ) - i \, n \log \left (i \, a x^{n} + 1\right ) \log \left (x\right ) + i \, n \log \left (-i \, a x^{n} + 1\right ) \log \left (x\right ) + i \, {\rm Li}_2\left (i \, a x^{n}\right ) - i \, {\rm Li}_2\left (-i \, a x^{n}\right )}{2 \, n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*n*arccot(a*x^n)*log(x) - I*n*log(I*a*x^n + 1)*log(x) + I*n*log(-I*a*x^n + 1)*log(x) + I*dilog(I*a*x^n)
- I*dilog(-I*a*x^n))/n

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {acot}{\left (a x^{n} \right )}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acot(a*x**n)/x, x)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccot(a*x^n)/x, x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\mathrm {acot}\left (a\,x^n\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(acot(a*x^n)/x, x)

________________________________________________________________________________________