3.21 \(\int \frac {\cot ^{-1}(a x)^2}{x^4} \, dx\)

Optimal. Leaf size=113 \[ \frac {1}{3} i a^3 \text {Li}_2\left (\frac {2}{1-i a x}-1\right )-\frac {1}{3} a^3 \tan ^{-1}(a x)+\frac {1}{3} i a^3 \cot ^{-1}(a x)^2+\frac {2}{3} a^3 \log \left (2-\frac {2}{1-i a x}\right ) \cot ^{-1}(a x)-\frac {a^2}{3 x}-\frac {\cot ^{-1}(a x)^2}{3 x^3}+\frac {a \cot ^{-1}(a x)}{3 x^2} \]

[Out]

-1/3*a^2/x+1/3*a*arccot(a*x)/x^2+1/3*I*a^3*arccot(a*x)^2-1/3*arccot(a*x)^2/x^3-1/3*a^3*arctan(a*x)+2/3*a^3*arc
cot(a*x)*ln(2-2/(1-I*a*x))+1/3*I*a^3*polylog(2,-1+2/(1-I*a*x))

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Rubi [A]  time = 0.16, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4853, 4919, 325, 203, 4925, 4869, 2447} \[ \frac {1}{3} i a^3 \text {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )-\frac {a^2}{3 x}-\frac {1}{3} a^3 \tan ^{-1}(a x)+\frac {1}{3} i a^3 \cot ^{-1}(a x)^2+\frac {2}{3} a^3 \log \left (2-\frac {2}{1-i a x}\right ) \cot ^{-1}(a x)+\frac {a \cot ^{-1}(a x)}{3 x^2}-\frac {\cot ^{-1}(a x)^2}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[a*x]^2/x^4,x]

[Out]

-a^2/(3*x) + (a*ArcCot[a*x])/(3*x^2) + (I/3)*a^3*ArcCot[a*x]^2 - ArcCot[a*x]^2/(3*x^3) - (a^3*ArcTan[a*x])/3 +
(2*a^3*ArcCot[a*x]*Log[2 - 2/(1 - I*a*x)])/3 + (I/3)*a^3*PolyLog[2, -1 + 2/(1 - I*a*x)]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, 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 4919

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

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^4} \, dx &=-\frac {\cot ^{-1}(a x)^2}{3 x^3}-\frac {1}{3} (2 a) \int \frac {\cot ^{-1}(a x)}{x^3 \left (1+a^2 x^2\right )} \, dx\\ &=-\frac {\cot ^{-1}(a x)^2}{3 x^3}-\frac {1}{3} (2 a) \int \frac {\cot ^{-1}(a x)}{x^3} \, dx+\frac {1}{3} \left (2 a^3\right ) \int \frac {\cot ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx\\ &=\frac {a \cot ^{-1}(a x)}{3 x^2}+\frac {1}{3} i a^3 \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{3 x^3}+\frac {1}{3} a^2 \int \frac {1}{x^2 \left (1+a^2 x^2\right )} \, dx+\frac {1}{3} \left (2 i a^3\right ) \int \frac {\cot ^{-1}(a x)}{x (i+a x)} \, dx\\ &=-\frac {a^2}{3 x}+\frac {a \cot ^{-1}(a x)}{3 x^2}+\frac {1}{3} i a^3 \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{3 x^3}+\frac {2}{3} a^3 \cot ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{3} a^4 \int \frac {1}{1+a^2 x^2} \, dx+\frac {1}{3} \left (2 a^4\right ) \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac {a^2}{3 x}+\frac {a \cot ^{-1}(a x)}{3 x^2}+\frac {1}{3} i a^3 \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{3 x^3}-\frac {1}{3} a^3 \tan ^{-1}(a x)+\frac {2}{3} a^3 \cot ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )+\frac {1}{3} i a^3 \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )\\ \end {align*}

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Mathematica [A]  time = 0.25, size = 96, normalized size = 0.85 \[ \frac {-i a^3 x^3 \text {Li}_2\left (-e^{2 i \cot ^{-1}(a x)}\right )+\left (-1-i a^3 x^3\right ) \cot ^{-1}(a x)^2-a^2 x^2+a x \cot ^{-1}(a x) \left (a^2 x^2+2 a^2 x^2 \log \left (1+e^{2 i \cot ^{-1}(a x)}\right )+1\right )}{3 x^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCot[a*x]^2/x^4,x]

[Out]

(-(a^2*x^2) + (-1 - I*a^3*x^3)*ArcCot[a*x]^2 + a*x*ArcCot[a*x]*(1 + a^2*x^2 + 2*a^2*x^2*Log[1 + E^((2*I)*ArcCo
t[a*x])]) - I*a^3*x^3*PolyLog[2, -E^((2*I)*ArcCot[a*x])])/(3*x^3)

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fricas [F]  time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arccot}\left (a x\right )^{2}}{x^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x)^2/x^4,x, algorithm="fricas")

[Out]

integral(arccot(a*x)^2/x^4, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arccot}\left (a x\right )^{2}}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x)^2/x^4,x, algorithm="giac")

[Out]

integrate(arccot(a*x)^2/x^4, x)

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maple [B]  time = 0.14, size = 290, normalized size = 2.57 \[ -\frac {\mathrm {arccot}\left (a x \right )^{2}}{3 x^{3}}+\frac {a \,\mathrm {arccot}\left (a x \right )}{3 x^{2}}+\frac {2 a^{3} \mathrm {arccot}\left (a x \right ) \ln \left (a x \right )}{3}-\frac {a^{3} \mathrm {arccot}\left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{3}-\frac {i a^{3} \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )}{6}+\frac {i a^{3} \dilog \left (\frac {i \left (a x -i\right )}{2}\right )}{6}-\frac {i a^{3} \ln \left (a x \right ) \ln \left (i a x +1\right )}{3}-\frac {i a^{3} \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )}{6}-\frac {i a^{3} \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )}{6}+\frac {i a^{3} \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )}{6}+\frac {i a^{3} \dilog \left (-i a x +1\right )}{3}-\frac {i a^{3} \ln \left (a x -i\right )^{2}}{12}-\frac {a^{2}}{3 x}-\frac {a^{3} \arctan \left (a x \right )}{3}-\frac {i a^{3} \dilog \left (i a x +1\right )}{3}+\frac {i a^{3} \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )}{6}+\frac {i a^{3} \ln \left (a x +i\right )^{2}}{12}+\frac {i a^{3} \ln \left (a x \right ) \ln \left (-i a x +1\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccot(a*x)^2/x^4,x)

[Out]

-1/3*arccot(a*x)^2/x^3+1/3*a*arccot(a*x)/x^2+2/3*a^3*arccot(a*x)*ln(a*x)-1/3*a^3*arccot(a*x)*ln(a^2*x^2+1)-1/6
*I*a^3*ln(a*x-I)*ln(-1/2*I*(I+a*x))-1/3*I*a^3*ln(a*x)*ln(1+I*a*x)-1/6*I*a^3*ln(I+a*x)*ln(a^2*x^2+1)-1/3*I*a^3*
dilog(1+I*a*x)+1/6*I*a^3*ln(I+a*x)*ln(1/2*I*(a*x-I))+1/3*I*a^3*dilog(1-I*a*x)-1/6*I*a^3*dilog(-1/2*I*(I+a*x))+
1/6*I*a^3*dilog(1/2*I*(a*x-I))-1/3*a^2/x-1/3*a^3*arctan(a*x)-1/12*I*a^3*ln(a*x-I)^2+1/6*I*a^3*ln(a*x-I)*ln(a^2
*x^2+1)+1/12*I*a^3*ln(I+a*x)^2+1/3*I*a^3*ln(a*x)*ln(1-I*a*x)

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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^4,x, algorithm="maxima")

[Out]

Timed out

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {acot}\left (a\,x\right )}^2}{x^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acot(a*x)^2/x^4,x)

[Out]

int(acot(a*x)^2/x^4, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acot}^{2}{\left (a x \right )}}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acot(a*x)**2/x**4,x)

[Out]

Integral(acot(a*x)**2/x**4, x)

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