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

Optimal. Leaf size=111 \[ -\frac{i \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{3 a^3}+\frac{x}{3 a^2}-\frac{\tan ^{-1}(a x)}{3 a^3}-\frac{i \cot ^{-1}(a x)^2}{3 a^3}+\frac{2 \log \left (\frac{2}{1+i a x}\right ) \cot ^{-1}(a x)}{3 a^3}+\frac{1}{3} x^3 \cot ^{-1}(a x)^2+\frac{x^2 \cot ^{-1}(a x)}{3 a} \]

[Out]

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

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Rubi [A]  time = 0.140106, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8, Rules used = {4853, 4917, 321, 203, 4921, 4855, 2402, 2315} \[ -\frac{i \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{3 a^3}+\frac{x}{3 a^2}-\frac{\tan ^{-1}(a x)}{3 a^3}-\frac{i \cot ^{-1}(a x)^2}{3 a^3}+\frac{2 \log \left (\frac{2}{1+i a x}\right ) \cot ^{-1}(a x)}{3 a^3}+\frac{1}{3} x^3 \cot ^{-1}(a x)^2+\frac{x^2 \cot ^{-1}(a x)}{3 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 4917

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

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, 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 4921

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

Rule 4855

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcCot[c*x])^p*Lo
g[2/(1 + (e*x)/d)])/e, x] - Dist[(b*c*p)/e, Int[((a + b*ArcCot[c*x])^(p - 1)*Log[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 2402

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> -Dist[e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rubi steps

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

Mathematica [A]  time = 0.262658, size = 76, normalized size = 0.68 \[ \frac{-i \text{PolyLog}\left (2,e^{2 i \cot ^{-1}(a x)}\right )+\left (a^3 x^3-i\right ) \cot ^{-1}(a x)^2+\cot ^{-1}(a x) \left (a^2 x^2+2 \log \left (1-e^{2 i \cot ^{-1}(a x)}\right )+1\right )+a x}{3 a^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B]  time = 0.112, size = 213, normalized size = 1.9 \begin{align*}{\frac{{x}^{3} \left ({\rm arccot} \left (ax\right ) \right ) ^{2}}{3}}+{\frac{{x}^{2}{\rm arccot} \left (ax\right )}{3\,a}}-{\frac{{\rm arccot} \left (ax\right )\ln \left ({a}^{2}{x}^{2}+1 \right ) }{3\,{a}^{3}}}+{\frac{x}{3\,{a}^{2}}}-{\frac{\arctan \left ( ax \right ) }{3\,{a}^{3}}}-{\frac{{\frac{i}{12}} \left ( \ln \left ( ax-i \right ) \right ) ^{2}}{{a}^{3}}}-{\frac{{\frac{i}{6}}\ln \left ( ax-i \right ) \ln \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{a}^{3}}}+{\frac{{\frac{i}{6}}\ln \left ( ax-i \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{{a}^{3}}}-{\frac{{\frac{i}{6}}{\it dilog} \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{a}^{3}}}+{\frac{{\frac{i}{12}} \left ( \ln \left ( ax+i \right ) \right ) ^{2}}{{a}^{3}}}+{\frac{{\frac{i}{6}}\ln \left ( ax+i \right ) \ln \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{a}^{3}}}-{\frac{{\frac{i}{6}}\ln \left ( ax+i \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{{a}^{3}}}+{\frac{{\frac{i}{6}}{\it dilog} \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{12} \, x^{3} \arctan \left (1, a x\right )^{2} - \frac{1}{48} \, x^{3} \log \left (a^{2} x^{2} + 1\right )^{2} + \int \frac{36 \, a^{2} x^{4} \arctan \left (1, a x\right )^{2} + 4 \, a^{2} x^{4} \log \left (a^{2} x^{2} + 1\right ) + 8 \, a x^{3} \arctan \left (1, a x\right ) + 36 \, x^{2} \arctan \left (1, a x\right )^{2} + 3 \,{\left (a^{2} x^{4} + x^{2}\right )} \log \left (a^{2} x^{2} + 1\right )^{2}}{48 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*x^3*arctan2(1, a*x)^2 - 1/48*x^3*log(a^2*x^2 + 1)^2 + integrate(1/48*(36*a^2*x^4*arctan2(1, a*x)^2 + 4*a^
2*x^4*log(a^2*x^2 + 1) + 8*a*x^3*arctan2(1, a*x) + 36*x^2*arctan2(1, a*x)^2 + 3*(a^2*x^4 + x^2)*log(a^2*x^2 +
1)^2)/(a^2*x^2 + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{2} \operatorname{arccot}\left (a x\right )^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^2*arccot(a*x)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*acot(a*x)**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \operatorname{arccot}\left (a x\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2*arccot(a*x)^2, x)