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

Optimal. Leaf size=31 \[ \frac{\log \left (a^2 x^4+1\right )}{4 a}+\frac{1}{2} x^2 \cot ^{-1}\left (a x^2\right ) \]

[Out]

(x^2*ArcCot[a*x^2])/2 + Log[1 + a^2*x^4]/(4*a)

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Rubi [A]  time = 0.0091751, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5034, 260} \[ \frac{\log \left (a^2 x^4+1\right )}{4 a}+\frac{1}{2} x^2 \cot ^{-1}\left (a x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*ArcCot[a*x^2])/2 + Log[1 + a^2*x^4]/(4*a)

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 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]

Rubi steps

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

Mathematica [A]  time = 0.006245, size = 31, normalized size = 1. \[ \frac{\log \left (a^2 x^4+1\right )}{4 a}+\frac{1}{2} x^2 \cot ^{-1}\left (a x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*ArcCot[a*x^2])/2 + Log[1 + a^2*x^4]/(4*a)

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Maple [A]  time = 0.039, size = 28, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}{\rm arccot} \left (a{x}^{2}\right )}{2}}+{\frac{\ln \left ({a}^{2}{x}^{4}+1 \right ) }{4\,a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2*arccot(a*x^2)+1/4*ln(a^2*x^4+1)/a

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Maxima [A]  time = 0.993607, size = 38, normalized size = 1.23 \begin{align*} \frac{2 \, a x^{2} \operatorname{arccot}\left (a x^{2}\right ) + \log \left (a^{2} x^{4} + 1\right )}{4 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.21044, size = 68, normalized size = 2.19 \begin{align*} \frac{2 \, a x^{2} \operatorname{arccot}\left (a x^{2}\right ) + \log \left (a^{2} x^{4} + 1\right )}{4 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.839381, size = 31, normalized size = 1. \begin{align*} \begin{cases} \frac{x^{2} \operatorname{acot}{\left (a x^{2} \right )}}{2} + \frac{\log{\left (a^{2} x^{4} + 1 \right )}}{4 a} & \text{for}\: a \neq 0 \\\frac{\pi x^{2}}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x**2*acot(a*x**2)/2 + log(a**2*x**4 + 1)/(4*a), Ne(a, 0)), (pi*x**2/4, True))

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Giac [A]  time = 1.09056, size = 39, normalized size = 1.26 \begin{align*} \frac{1}{2} \, x^{2} \arctan \left (\frac{1}{a x^{2}}\right ) + \frac{\log \left (a^{2} x^{4} + 1\right )}{4 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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