3.19 \(\int \frac {\cot ^{-1}(a x)^2}{x^2} \, dx\)

Optimal. Leaf size=66 \[ -i a \text {Li}_2\left (\frac {2}{1-i a x}-1\right )-i a \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{x}-2 a \log \left (2-\frac {2}{1-i a x}\right ) \cot ^{-1}(a x) \]

[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 = 0.11, 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 = {4853, 4925, 4869, 2447} \[ -i a \text {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )-i a \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{x}-2 a \log \left (2-\frac {2}{1-i a x}\right ) \cot ^{-1}(a x) \]

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 2447

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 4853

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

Rule 4869

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 4925

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 = 0.05, size = 64, normalized size = 0.97 \[ a \left (i \text {Li}_2\left (-e^{2 i \cot ^{-1}(a x)}\right )-\frac {\cot ^{-1}(a x)^2}{a x}+i \cot ^{-1}(a x)^2-2 \cot ^{-1}(a x) \log \left (1+e^{2 i \cot ^{-1}(a x)}\right )\right ) \]

Warning: Unable to verify antiderivative.

[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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fricas [F]  time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arccot}\left (a x\right )^{2}}{x^{2}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arccot}\left (a x\right )^{2}}{x^{2}}\,{d x} \]

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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maple [B]  time = 0.14, size = 234, normalized size = 3.55 \[ -\frac {\mathrm {arccot}\left (a x \right )^{2}}{x}-2 a \,\mathrm {arccot}\left (a x \right ) \ln \left (a x \right )+a \,\mathrm {arccot}\left (a x \right ) \ln \left (a^{2} x^{2}+1\right )+i a \ln \left (a x \right ) \ln \left (i a x +1\right )-i a \ln \left (a x \right ) \ln \left (-i a x +1\right )+i a \dilog \left (i a x +1\right )-i a \dilog \left (-i a x +1\right )-\frac {i a \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )}{2}+\frac {i a \ln \left (a x -i\right )^{2}}{4}+\frac {i a \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )}{2}+\frac {i a \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )}{2}+\frac {i a \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )}{2}-\frac {i a \ln \left (a x +i\right )^{2}}{4}-\frac {i a \dilog \left (\frac {i \left (a x -i\right )}{2}\right )}{2}-\frac {i a \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x)^2/x^2,x, algorithm="maxima")

[Out]

Timed out

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

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

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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