3.1.19 \(\int \frac {\cot ^{-1}(a x)^2}{x^2} \, dx\) [19]

Optimal. Leaf size=66 \[ -i a \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{x}-2 a \cot ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )-i a \text {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right ) \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4947, 5045, 4989, 2497} \begin {gather*} -i a \text {Li}_2\left (\frac {2}{1-i a x}-1\right )-i a \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{x}-2 a \log \left (2-\frac {2}{1-i a x}\right ) \cot ^{-1}(a x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcCot[a*x]^2/x^2,x]

[Out]

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

Rule 2497

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/D[u, x])]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rule 4947

Int[((a_.) + ArcCot[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcCot[c*x^
n])^p/(m + 1)), x] + Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcCot[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))),
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] && IntegerQ[m])) && NeQ[m, -1]

Rule 4989

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Simp[(a + b*ArcCot[c*x])
^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] + Dist[b*c*(p/d), Int[(a + b*ArcCot[c*x])^(p - 1)*(Log[2 - 2/(1 + e*(x/d))
]/(1 + c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]

Rule 5045

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[I*((a + b*ArcCot[
c*x])^(p + 1)/(b*d*(p + 1))), x] + Dist[I/d, Int[(a + b*ArcCot[c*x])^p/(x*(I + c*x)), x], x] /; FreeQ[{a, b, c
, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 64, normalized size = 0.97 \begin {gather*} a \left (i \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{a x}-2 \cot ^{-1}(a x) \log \left (1+e^{2 i \cot ^{-1}(a x)}\right )+i \text {PolyLog}\left (2,-e^{2 i \cot ^{-1}(a x)}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[a*x]^2/x^2,x]

[Out]

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

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 224 vs. \(2 (62 ) = 124\).
time = 0.32, size = 225, normalized size = 3.41

method	result	size
derivativedivides	\(a \left (-\frac {\mathrm {arccot}\left (a x \right )^{2}}{a x}-2 \ln \left (a x \right ) \mathrm {arccot}\left (a x \right )+\mathrm {arccot}\left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {i \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )}{2}+\frac {i \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )}{2}+\frac {i \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )}{2}+\frac {i \ln \left (a x -i\right )^{2}}{4}+\frac {i \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )}{2}-\frac {i \dilog \left (\frac {i \left (a x -i\right )}{2}\right )}{2}-\frac {i \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )}{2}-\frac {i \ln \left (a x +i\right )^{2}}{4}+i \ln \left (a x \right ) \ln \left (i a x +1\right )-i \ln \left (a x \right ) \ln \left (-i a x +1\right )+i \dilog \left (i a x +1\right )-i \dilog \left (-i a x +1\right )\right )\)	\(225\)
default	\(a \left (-\frac {\mathrm {arccot}\left (a x \right )^{2}}{a x}-2 \ln \left (a x \right ) \mathrm {arccot}\left (a x \right )+\mathrm {arccot}\left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {i \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )}{2}+\frac {i \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )}{2}+\frac {i \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )}{2}+\frac {i \ln \left (a x -i\right )^{2}}{4}+\frac {i \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )}{2}-\frac {i \dilog \left (\frac {i \left (a x -i\right )}{2}\right )}{2}-\frac {i \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )}{2}-\frac {i \ln \left (a x +i\right )^{2}}{4}+i \ln \left (a x \right ) \ln \left (i a x +1\right )-i \ln \left (a x \right ) \ln \left (-i a x +1\right )+i \dilog \left (i a x +1\right )-i \dilog \left (-i a x +1\right )\right )\)	\(225\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccot(a*x)^2/x^2,x,method=_RETURNVERBOSE)

[Out]

a*(-arccot(a*x)^2/a/x-2*ln(a*x)*arccot(a*x)+arccot(a*x)*ln(a^2*x^2+1)-1/2*I*ln(a*x-I)*ln(a^2*x^2+1)+1/2*I*dilo
g(-1/2*I*(I+a*x))+1/2*I*ln(a*x-I)*ln(-1/2*I*(I+a*x))+1/4*I*ln(a*x-I)^2+1/2*I*ln(I+a*x)*ln(a^2*x^2+1)-1/2*I*dil
og(1/2*I*(a*x-I))-1/2*I*ln(I+a*x)*ln(1/2*I*(a*x-I))-1/4*I*ln(I+a*x)^2+I*ln(a*x)*ln(1+I*a*x)-I*ln(a*x)*ln(1-I*a
*x)+I*dilog(1+I*a*x)-I*dilog(1-I*a*x))

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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)^2/x^2,x, algorithm="maxima")

[Out]

1/16*(4*(3*a*arctan(a*x)*arctan(1/(a*x))^2 + (arctan(a*x)^3/a + 3*arctan(a*x)^2*arctan(1/(a*x))/a)*a^2 + 4*a^2
*integrate(1/16*x^2*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) - 16*a^2*integrate(1/16*x^2*log(a^2*x^2 + 1)/(a^2*x
^4 + x^2), x) - 32*a*integrate(1/16*x*arctan(1/(a*x))/(a^2*x^4 + x^2), x) + 48*integrate(1/16*arctan(1/(a*x))^
2/(a^2*x^4 + x^2), x) + 4*integrate(1/16*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x))*x - 4*arctan2(1, a*x)^2 + log
(a^2*x^2 + 1)^2)/x

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Fricas [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)^2/x^2,x, algorithm="fricas")

[Out]

integral(arccot(a*x)^2/x^2, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {acot}^{2}{\left (a x \right )}}{x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acot(a*x)**2/x**2,x)

[Out]

Integral(acot(a*x)**2/x**2, x)

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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)^2/x^2,x, algorithm="giac")

[Out]

integrate(arccot(a*x)^2/x^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\mathrm {acot}\left (a\,x\right )}^2}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acot(a*x)^2/x^2,x)

[Out]

int(acot(a*x)^2/x^2, x)

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