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

Optimal. Leaf size=41 \[ -\frac{\log \left (a^2 x^4+1\right )}{12 a^3}+\frac{x^4}{12 a}+\frac{1}{6} x^6 \cot ^{-1}\left (a x^2\right ) \]

[Out]

x^4/(12*a) + (x^6*ArcCot[a*x^2])/6 - Log[1 + a^2*x^4]/(12*a^3)

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Rubi [A]  time = 0.0253619, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5034, 266, 43} \[ -\frac{\log \left (a^2 x^4+1\right )}{12 a^3}+\frac{x^4}{12 a}+\frac{1}{6} x^6 \cot ^{-1}\left (a x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^4/(12*a) + (x^6*ArcCot[a*x^2])/6 - Log[1 + a^2*x^4]/(12*a^3)

Rule 5034

Int[((a_.) + ArcCot[(c_.)*(x_)^(n_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcCot
[c*x^n]))/(d*(m + 1)), x] + Dist[(b*c*n)/(d*(m + 1)), Int[(x^(n - 1)*(d*x)^(m + 1))/(1 + c^2*x^(2*n)), x], x]
/; FreeQ[{a, b, c, d, m, n}, x] && NeQ[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])

Rubi steps

\begin{align*} \int x^5 \cot ^{-1}\left (a x^2\right ) \, dx &=\frac{1}{6} x^6 \cot ^{-1}\left (a x^2\right )+\frac{1}{3} a \int \frac{x^7}{1+a^2 x^4} \, dx\\ &=\frac{1}{6} x^6 \cot ^{-1}\left (a x^2\right )+\frac{1}{12} a \operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^4\right )\\ &=\frac{1}{6} x^6 \cot ^{-1}\left (a x^2\right )+\frac{1}{12} a \operatorname{Subst}\left (\int \left (\frac{1}{a^2}-\frac{1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^4\right )\\ &=\frac{x^4}{12 a}+\frac{1}{6} x^6 \cot ^{-1}\left (a x^2\right )-\frac{\log \left (1+a^2 x^4\right )}{12 a^3}\\ \end{align*}

Mathematica [A]  time = 0.0144449, size = 41, normalized size = 1. \[ -\frac{\log \left (a^2 x^4+1\right )}{12 a^3}+\frac{x^4}{12 a}+\frac{1}{6} x^6 \cot ^{-1}\left (a x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^4/(12*a) + (x^6*ArcCot[a*x^2])/6 - Log[1 + a^2*x^4]/(12*a^3)

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Maple [A]  time = 0.045, size = 36, normalized size = 0.9 \begin{align*}{\frac{{x}^{4}}{12\,a}}+{\frac{{x}^{6}{\rm arccot} \left (a{x}^{2}\right )}{6}}-{\frac{\ln \left ({a}^{2}{x}^{4}+1 \right ) }{12\,{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*x^4/a+1/6*x^6*arccot(a*x^2)-1/12*ln(a^2*x^4+1)/a^3

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Maxima [A]  time = 0.974629, size = 51, normalized size = 1.24 \begin{align*} \frac{1}{6} \, x^{6} \operatorname{arccot}\left (a x^{2}\right ) + \frac{1}{12} \,{\left (\frac{x^{4}}{a^{2}} - \frac{\log \left (a^{2} x^{4} + 1\right )}{a^{4}}\right )} a \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) + 1/12*(x^4/a^2 - log(a^2*x^4 + 1)/a^4)*a

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Fricas [A]  time = 2.1604, size = 88, normalized size = 2.15 \begin{align*} \frac{2 \, a^{3} x^{6} \operatorname{arccot}\left (a x^{2}\right ) + a^{2} x^{4} - \log \left (a^{2} x^{4} + 1\right )}{12 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(2*a^3*x^6*arccot(a*x^2) + a^2*x^4 - log(a^2*x^4 + 1))/a^3

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Sympy [A]  time = 2.93952, size = 39, normalized size = 0.95 \begin{align*} \begin{cases} \frac{x^{6} \operatorname{acot}{\left (a x^{2} \right )}}{6} + \frac{x^{4}}{12 a} - \frac{\log{\left (a^{2} x^{4} + 1 \right )}}{12 a^{3}} & \text{for}\: a \neq 0 \\\frac{\pi x^{6}}{12} & \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**4/(12*a) - log(a**2*x**4 + 1)/(12*a**3), Ne(a, 0)), (pi*x**6/12, True))

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Giac [A]  time = 1.10586, size = 54, normalized size = 1.32 \begin{align*} \frac{1}{6} \, x^{6} \arctan \left (\frac{1}{a x^{2}}\right ) + \frac{1}{12} \,{\left (\frac{x^{4}}{a^{2}} - \frac{\log \left (a^{2} x^{4} + 1\right )}{a^{4}}\right )} a \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)) + 1/12*(x^4/a^2 - log(a^2*x^4 + 1)/a^4)*a