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

Optimal. Leaf size=39 \[ -\frac {\log \left (a^2 x^2+1\right )}{6 a^3}+\frac {1}{3} x^3 \cot ^{-1}(a x)+\frac {x^2}{6 a} \]

[Out]

1/6*x^2/a+1/3*x^3*arccot(a*x)-1/6*ln(a^2*x^2+1)/a^3

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Rubi [A]  time = 0.03, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4853, 266, 43} \[ -\frac {\log \left (a^2 x^2+1\right )}{6 a^3}+\frac {x^2}{6 a}+\frac {1}{3} x^3 \cot ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^2/(6*a) + (x^3*ArcCot[a*x])/3 - Log[1 + a^2*x^2]/(6*a^3)

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 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 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]

Rubi steps

\begin {align*} \int x^2 \cot ^{-1}(a x) \, dx &=\frac {1}{3} x^3 \cot ^{-1}(a x)+\frac {1}{3} a \int \frac {x^3}{1+a^2 x^2} \, dx\\ &=\frac {1}{3} x^3 \cot ^{-1}(a x)+\frac {1}{6} a \operatorname {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )\\ &=\frac {1}{3} x^3 \cot ^{-1}(a x)+\frac {1}{6} a \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 )\\ &=\frac {x^2}{6 a}+\frac {1}{3} x^3 \cot ^{-1}(a x)-\frac {\log \left (1+a^2 x^2\right )}{6 a^3}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 39, normalized size = 1.00 \[ -\frac {\log \left (a^2 x^2+1\right )}{6 a^3}+\frac {1}{3} x^3 \cot ^{-1}(a x)+\frac {x^2}{6 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^2/(6*a) + (x^3*ArcCot[a*x])/3 - Log[1 + a^2*x^2]/(6*a^3)

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fricas [A]  time = 0.73, size = 37, normalized size = 0.95 \[ \frac {2 \, a^{3} x^{3} \operatorname {arccot}\left (a x\right ) + a^{2} x^{2} - \log \left (a^{2} x^{2} + 1\right )}{6 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*a^3*x^3*arccot(a*x) + a^2*x^2 - log(a^2*x^2 + 1))/a^3

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giac [A]  time = 0.12, size = 64, normalized size = 1.64 \[ \frac {1}{6} \, {\left (\frac {2 \, x^{3} \arctan \left (\frac {1}{a x}\right )}{a} - \frac {x^{2} {\left (\frac {1}{a^{2} x^{2}} - 1\right )}}{a^{2}} - \frac {\log \left (\frac {1}{a^{2} x^{2}} + 1\right )}{a^{4}} + \frac {\log \left (\frac {1}{a^{2} x^{2}}\right )}{a^{4}}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*x^3*arctan(1/(a*x))/a - x^2*(1/(a^2*x^2) - 1)/a^2 - log(1/(a^2*x^2) + 1)/a^4 + log(1/(a^2*x^2))/a^4)*a

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maple [A]  time = 0.04, size = 34, normalized size = 0.87 \[ \frac {x^{2}}{6 a}+\frac {x^{3} \mathrm {arccot}\left (a x \right )}{3}-\frac {\ln \left (a^{2} x^{2}+1\right )}{6 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x^2/a+1/3*x^3*arccot(a*x)-1/6*ln(a^2*x^2+1)/a^3

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maxima [A]  time = 0.31, size = 36, normalized size = 0.92 \[ \frac {1}{3} \, x^{3} \operatorname {arccot}\left (a x\right ) + \frac {1}{6} \, a {\left (\frac {x^{2}}{a^{2}} - \frac {\log \left (a^{2} x^{2} + 1\right )}{a^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*arccot(a*x) + 1/6*a*(x^2/a^2 - log(a^2*x^2 + 1)/a^4)

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mupad [B]  time = 0.69, size = 49, normalized size = 1.26 \[ \left \{\begin {array}{cl} \frac {\pi \,x^3}{6} & \text {\ if\ \ }a=0\\ \frac {\frac {x^2}{2}-\frac {\ln \left (a^2\,x^2+1\right )}{2\,a^2}}{3\,a}+\frac {x^3\,\mathrm {acot}\left (a\,x\right )}{3} & \text {\ if\ \ }a\neq 0 \end {array}\right . \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

piecewise(a == 0, (x^3*pi)/6, a ~= 0, (x^2/2 - log(a^2*x^2 + 1)/(2*a^2))/(3*a) + (x^3*acot(a*x))/3)

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sympy [A]  time = 0.48, size = 37, normalized size = 0.95 \[ \begin {cases} \frac {x^{3} \operatorname {acot}{\left (a x \right )}}{3} + \frac {x^{2}}{6 a} - \frac {\log {\left (a^{2} x^{2} + 1 \right )}}{6 a^{3}} & \text {for}\: a \neq 0 \\\frac {\pi x^{3}}{6} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x**3*acot(a*x)/3 + x**2/(6*a) - log(a**2*x**2 + 1)/(6*a**3), Ne(a, 0)), (pi*x**3/6, True))

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