3.12 \(\int x^5 \cot ^{-1}(a x)^2 \, dx\)

Optimal. Leaf size=104 \[ \frac{x^4}{60 a^2}-\frac{4 x^2}{45 a^4}+\frac{23 \log \left (a^2 x^2+1\right )}{90 a^6}-\frac{x^3 \cot ^{-1}(a x)}{9 a^3}+\frac{x \cot ^{-1}(a x)}{3 a^5}+\frac{\cot ^{-1}(a x)^2}{6 a^6}+\frac{1}{6} x^6 \cot ^{-1}(a x)^2+\frac{x^5 \cot ^{-1}(a x)}{15 a} \]

[Out]

(-4*x^2)/(45*a^4) + x^4/(60*a^2) + (x*ArcCot[a*x])/(3*a^5) - (x^3*ArcCot[a*x])/(9*a^3) + (x^5*ArcCot[a*x])/(15
*a) + ArcCot[a*x]^2/(6*a^6) + (x^6*ArcCot[a*x]^2)/6 + (23*Log[1 + a^2*x^2])/(90*a^6)

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Rubi [A]  time = 0.221475, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {4853, 4917, 266, 43, 4847, 260, 4885} \[ \frac{x^4}{60 a^2}-\frac{4 x^2}{45 a^4}+\frac{23 \log \left (a^2 x^2+1\right )}{90 a^6}-\frac{x^3 \cot ^{-1}(a x)}{9 a^3}+\frac{x \cot ^{-1}(a x)}{3 a^5}+\frac{\cot ^{-1}(a x)^2}{6 a^6}+\frac{1}{6} x^6 \cot ^{-1}(a x)^2+\frac{x^5 \cot ^{-1}(a x)}{15 a} \]

Antiderivative was successfully verified.

[In]

Int[x^5*ArcCot[a*x]^2,x]

[Out]

(-4*x^2)/(45*a^4) + x^4/(60*a^2) + (x*ArcCot[a*x])/(3*a^5) - (x^3*ArcCot[a*x])/(9*a^3) + (x^5*ArcCot[a*x])/(15
*a) + ArcCot[a*x]^2/(6*a^6) + (x^6*ArcCot[a*x]^2)/6 + (23*Log[1 + a^2*x^2])/(90*a^6)

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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 4847

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

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 4885

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

Rubi steps

\begin{align*} \int x^5 \cot ^{-1}(a x)^2 \, dx &=\frac{1}{6} x^6 \cot ^{-1}(a x)^2+\frac{1}{3} a \int \frac{x^6 \cot ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac{1}{6} x^6 \cot ^{-1}(a x)^2+\frac{\int x^4 \cot ^{-1}(a x) \, dx}{3 a}-\frac{\int \frac{x^4 \cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{3 a}\\ &=\frac{x^5 \cot ^{-1}(a x)}{15 a}+\frac{1}{6} x^6 \cot ^{-1}(a x)^2+\frac{1}{15} \int \frac{x^5}{1+a^2 x^2} \, dx-\frac{\int x^2 \cot ^{-1}(a x) \, dx}{3 a^3}+\frac{\int \frac{x^2 \cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{3 a^3}\\ &=-\frac{x^3 \cot ^{-1}(a x)}{9 a^3}+\frac{x^5 \cot ^{-1}(a x)}{15 a}+\frac{1}{6} x^6 \cot ^{-1}(a x)^2+\frac{1}{30} \operatorname{Subst}\left (\int \frac{x^2}{1+a^2 x} \, dx,x,x^2\right )+\frac{\int \cot ^{-1}(a x) \, dx}{3 a^5}-\frac{\int \frac{\cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{3 a^5}-\frac{\int \frac{x^3}{1+a^2 x^2} \, dx}{9 a^2}\\ &=\frac{x \cot ^{-1}(a x)}{3 a^5}-\frac{x^3 \cot ^{-1}(a x)}{9 a^3}+\frac{x^5 \cot ^{-1}(a x)}{15 a}+\frac{\cot ^{-1}(a x)^2}{6 a^6}+\frac{1}{6} x^6 \cot ^{-1}(a x)^2+\frac{1}{30} \operatorname{Subst}\left (\int \left (-\frac{1}{a^4}+\frac{x}{a^2}+\frac{1}{a^4 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )+\frac{\int \frac{x}{1+a^2 x^2} \, dx}{3 a^4}-\frac{\operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^2\right )}{18 a^2}\\ &=-\frac{x^2}{30 a^4}+\frac{x^4}{60 a^2}+\frac{x \cot ^{-1}(a x)}{3 a^5}-\frac{x^3 \cot ^{-1}(a x)}{9 a^3}+\frac{x^5 \cot ^{-1}(a x)}{15 a}+\frac{\cot ^{-1}(a x)^2}{6 a^6}+\frac{1}{6} x^6 \cot ^{-1}(a x)^2+\frac{\log \left (1+a^2 x^2\right )}{5 a^6}-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a^2}-\frac{1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )}{18 a^2}\\ &=-\frac{4 x^2}{45 a^4}+\frac{x^4}{60 a^2}+\frac{x \cot ^{-1}(a x)}{3 a^5}-\frac{x^3 \cot ^{-1}(a x)}{9 a^3}+\frac{x^5 \cot ^{-1}(a x)}{15 a}+\frac{\cot ^{-1}(a x)^2}{6 a^6}+\frac{1}{6} x^6 \cot ^{-1}(a x)^2+\frac{23 \log \left (1+a^2 x^2\right )}{90 a^6}\\ \end{align*}

Mathematica [A]  time = 0.0228788, size = 79, normalized size = 0.76 \[ \frac{3 a^4 x^4-16 a^2 x^2+46 \log \left (a^2 x^2+1\right )+4 a x \left (3 a^4 x^4-5 a^2 x^2+15\right ) \cot ^{-1}(a x)+30 \left (a^6 x^6+1\right ) \cot ^{-1}(a x)^2}{180 a^6} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*ArcCot[a*x]^2,x]

[Out]

(-16*a^2*x^2 + 3*a^4*x^4 + 4*a*x*(15 - 5*a^2*x^2 + 3*a^4*x^4)*ArcCot[a*x] + 30*(1 + a^6*x^6)*ArcCot[a*x]^2 + 4
6*Log[1 + a^2*x^2])/(180*a^6)

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Maple [A]  time = 0.049, size = 102, normalized size = 1. \begin{align*}{\frac{{x}^{6} \left ({\rm arccot} \left (ax\right ) \right ) ^{2}}{6}}+{\frac{{x}^{5}{\rm arccot} \left (ax\right )}{15\,a}}-{\frac{{x}^{3}{\rm arccot} \left (ax\right )}{9\,{a}^{3}}}+{\frac{x{\rm arccot} \left (ax\right )}{3\,{a}^{5}}}-{\frac{{\rm arccot} \left (ax\right )\arctan \left ( ax \right ) }{3\,{a}^{6}}}+{\frac{{x}^{4}}{60\,{a}^{2}}}-{\frac{4\,{x}^{2}}{45\,{a}^{4}}}+{\frac{23\,\ln \left ({a}^{2}{x}^{2}+1 \right ) }{90\,{a}^{6}}}-{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{2}}{6\,{a}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*arccot(a*x)^2,x)

[Out]

1/6*x^6*arccot(a*x)^2+1/15*x^5*arccot(a*x)/a-1/9*x^3*arccot(a*x)/a^3+1/3*x*arccot(a*x)/a^5-1/3/a^6*arccot(a*x)
*arctan(a*x)+1/60*x^4/a^2-4/45*x^2/a^4+23/90*ln(a^2*x^2+1)/a^6-1/6/a^6*arctan(a*x)^2

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Maxima [A]  time = 1.52281, size = 128, normalized size = 1.23 \begin{align*} \frac{1}{6} \, x^{6} \operatorname{arccot}\left (a x\right )^{2} + \frac{1}{45} \, a{\left (\frac{3 \, a^{4} x^{5} - 5 \, a^{2} x^{3} + 15 \, x}{a^{6}} - \frac{15 \, \arctan \left (a x\right )}{a^{7}}\right )} \operatorname{arccot}\left (a x\right ) + \frac{3 \, a^{4} x^{4} - 16 \, a^{2} x^{2} - 30 \, \arctan \left (a x\right )^{2} + 46 \, \log \left (a^{2} x^{2} + 1\right )}{180 \, a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*arccot(a*x)^2,x, algorithm="maxima")

[Out]

1/6*x^6*arccot(a*x)^2 + 1/45*a*((3*a^4*x^5 - 5*a^2*x^3 + 15*x)/a^6 - 15*arctan(a*x)/a^7)*arccot(a*x) + 1/180*(
3*a^4*x^4 - 16*a^2*x^2 - 30*arctan(a*x)^2 + 46*log(a^2*x^2 + 1))/a^6

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Fricas [A]  time = 1.98936, size = 189, normalized size = 1.82 \begin{align*} \frac{3 \, a^{4} x^{4} - 16 \, a^{2} x^{2} + 30 \,{\left (a^{6} x^{6} + 1\right )} \operatorname{arccot}\left (a x\right )^{2} + 4 \,{\left (3 \, a^{5} x^{5} - 5 \, a^{3} x^{3} + 15 \, a x\right )} \operatorname{arccot}\left (a x\right ) + 46 \, \log \left (a^{2} x^{2} + 1\right )}{180 \, a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*arccot(a*x)^2,x, algorithm="fricas")

[Out]

1/180*(3*a^4*x^4 - 16*a^2*x^2 + 30*(a^6*x^6 + 1)*arccot(a*x)^2 + 4*(3*a^5*x^5 - 5*a^3*x^3 + 15*a*x)*arccot(a*x
) + 46*log(a^2*x^2 + 1))/a^6

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Sympy [A]  time = 2.48021, size = 104, normalized size = 1. \begin{align*} \begin{cases} \frac{x^{6} \operatorname{acot}^{2}{\left (a x \right )}}{6} + \frac{x^{5} \operatorname{acot}{\left (a x \right )}}{15 a} + \frac{x^{4}}{60 a^{2}} - \frac{x^{3} \operatorname{acot}{\left (a x \right )}}{9 a^{3}} - \frac{4 x^{2}}{45 a^{4}} + \frac{x \operatorname{acot}{\left (a x \right )}}{3 a^{5}} + \frac{23 \log{\left (a^{2} x^{2} + 1 \right )}}{90 a^{6}} + \frac{\operatorname{acot}^{2}{\left (a x \right )}}{6 a^{6}} & \text{for}\: a \neq 0 \\\frac{\pi ^{2} x^{6}}{24} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*acot(a*x)**2,x)

[Out]

Piecewise((x**6*acot(a*x)**2/6 + x**5*acot(a*x)/(15*a) + x**4/(60*a**2) - x**3*acot(a*x)/(9*a**3) - 4*x**2/(45
*a**4) + x*acot(a*x)/(3*a**5) + 23*log(a**2*x**2 + 1)/(90*a**6) + acot(a*x)**2/(6*a**6), Ne(a, 0)), (pi**2*x**
6/24, True))

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Giac [A]  time = 1.11885, size = 190, normalized size = 1.83 \begin{align*} \frac{1}{6} \, x^{6} \arctan \left (\frac{1}{a x}\right )^{2} + \frac{12 \, a^{5} i x^{5} \log \left (\frac{a x - i}{a x + i}\right ) + 6 \, a^{4} x^{4} - 20 \, a^{3} i x^{3} \log \left (\frac{a x - i}{a x + i}\right ) - 32 \, a^{2} x^{2} + 60 \, a i x \log \left (\frac{a x - i}{a x + i}\right ) - 15 \, \log \left (\frac{a x - i}{a x + i}\right )^{2} + 92 \, \log \left (a^{2} x^{2} + 1\right )}{360 \, a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*arccot(a*x)^2,x, algorithm="giac")

[Out]

1/6*x^6*arctan(1/(a*x))^2 + 1/360*(12*a^5*i*x^5*log((a*x - i)/(a*x + i)) + 6*a^4*x^4 - 20*a^3*i*x^3*log((a*x -
i)/(a*x + i)) - 32*a^2*x^2 + 60*a*i*x*log((a*x - i)/(a*x + i)) - 15*log((a*x - i)/(a*x + i))^2 + 92*log(a^2*x
^2 + 1))/a^6