3.1.12 \(\int x^5 \cot ^{-1}(a x)^2 \, dx\) [12]

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

[Out]

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

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Rubi [A]
time = 0.15, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4947, 5037, 272, 45, 4931, 266, 5005} \begin {gather*} \frac {\cot ^{-1}(a x)^2}{6 a^6}+\frac {x \cot ^{-1}(a x)}{3 a^5}-\frac {4 x^2}{45 a^4}-\frac {x^3 \cot ^{-1}(a x)}{9 a^3}+\frac {x^4}{60 a^2}+\frac {23 \log \left (a^2 x^2+1\right )}{90 a^6}+\frac {1}{6} x^6 \cot ^{-1}(a x)^2+\frac {x^5 \cot ^{-1}(a x)}{15 a} \end {gather*}

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 45

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 266

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 272

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 4931

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

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 5005

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]

Rule 5037

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]

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} \text {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} \text {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 {\text {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 {\text {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*}

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Mathematica [A]
time = 0.02, size = 79, normalized size = 0.76 \begin {gather*} \frac {-16 a^2 x^2+3 a^4 x^4+4 a x \left (15-5 a^2 x^2+3 a^4 x^4\right ) \cot ^{-1}(a x)+30 \left (1+a^6 x^6\right ) \cot ^{-1}(a x)^2+46 \log \left (1+a^2 x^2\right )}{180 a^6} \end {gather*}

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 = 0.11, size = 98, normalized size = 0.94

method	result	size
derivativedivides	\(\frac {\frac {a^{6} x^{6} \mathrm {arccot}\left (a x \right )^{2}}{6}+\frac {a^{5} x^{5} \mathrm {arccot}\left (a x \right )}{15}-\frac {a^{3} x^{3} \mathrm {arccot}\left (a x \right )}{9}+\frac {a x \,\mathrm {arccot}\left (a x \right )}{3}-\frac {\mathrm {arccot}\left (a x \right ) \arctan \left (a x \right )}{3}+\frac {a^{4} x^{4}}{60}-\frac {4 a^{2} x^{2}}{45}+\frac {23 \ln \left (a^{2} x^{2}+1\right )}{90}-\frac {\arctan \left (a x \right )^{2}}{6}}{a^{6}}\)	\(98\)
default	\(\frac {\frac {a^{6} x^{6} \mathrm {arccot}\left (a x \right )^{2}}{6}+\frac {a^{5} x^{5} \mathrm {arccot}\left (a x \right )}{15}-\frac {a^{3} x^{3} \mathrm {arccot}\left (a x \right )}{9}+\frac {a x \,\mathrm {arccot}\left (a x \right )}{3}-\frac {\mathrm {arccot}\left (a x \right ) \arctan \left (a x \right )}{3}+\frac {a^{4} x^{4}}{60}-\frac {4 a^{2} x^{2}}{45}+\frac {23 \ln \left (a^{2} x^{2}+1\right )}{90}-\frac {\arctan \left (a x \right )^{2}}{6}}{a^{6}}\)	\(98\)
risch	\(-\frac {\left (x^{6} a^{6}+1\right ) \ln \left (i a x +1\right )^{2}}{24 a^{6}}+\frac {\left (15 i \pi \,a^{6} x^{6}+15 x^{6} \ln \left (-i a x +1\right ) a^{6}+6 i a^{5} x^{5}-10 i a^{3} x^{3}+30 i a x +15 \ln \left (-i a x +1\right )\right ) \ln \left (i a x +1\right )}{180 a^{6}}-\frac {i x \ln \left (-i a x +1\right )}{6 a^{5}}+\frac {x^{6} \pi ^{2}}{24}-\frac {x^{6} \ln \left (-i a x +1\right )^{2}}{24}-\frac {i x^{5} \ln \left (-i a x +1\right )}{30 a}+\frac {\pi \,x^{5}}{30 a}+\frac {i x^{3} \ln \left (-i a x +1\right )}{18 a^{3}}+\frac {x^{4}}{60 a^{2}}-\frac {\pi \,x^{3}}{18 a^{3}}-\frac {i \pi \,x^{6} \ln \left (-i a x +1\right )}{12}-\frac {4 x^{2}}{45 a^{4}}+\frac {\pi x}{6 a^{5}}-\frac {\pi \arctan \left (a x \right )}{6 a^{6}}-\frac {\ln \left (-i a x +1\right )^{2}}{24 a^{6}}+\frac {23 \ln \left (a^{2} x^{2}+1\right )}{90 a^{6}}\)	\(267\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*arccot(a*x)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.49, size = 95, normalized size = 0.91 \begin {gather*} \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 {gather*}

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 = 3.10, size = 78, normalized size = 0.75 \begin {gather*} \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 {gather*}

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 = 0.40, size = 104, normalized size = 1.00 \begin {gather*} \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 {gather*}

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

[Out]

integrate(x^5*arccot(a*x)^2, x)

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Mupad [B]
time = 0.81, size = 85, normalized size = 0.82 \begin {gather*} \frac {x^6\,{\mathrm {acot}\left (a\,x\right )}^2}{6}+\frac {\frac {23\,\ln \left (a^2\,x^2+1\right )}{90}-\frac {4\,a^2\,x^2}{45}+\frac {a^4\,x^4}{60}+\frac {{\mathrm {acot}\left (a\,x\right )}^2}{6}-\frac {a^3\,x^3\,\mathrm {acot}\left (a\,x\right )}{9}+\frac {a^5\,x^5\,\mathrm {acot}\left (a\,x\right )}{15}+\frac {a\,x\,\mathrm {acot}\left (a\,x\right )}{3}}{a^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^6*acot(a*x)^2)/6 + ((23*log(a^2*x^2 + 1))/90 - (4*a^2*x^2)/45 + (a^4*x^4)/60 + acot(a*x)^2/6 - (a^3*x^3*aco
t(a*x))/9 + (a^5*x^5*acot(a*x))/15 + (a*x*acot(a*x))/3)/a^6

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