3.67 \(\int \frac{\cot ^{-1}(x)}{(a+a x^2)^{3/2}} \, dx\)

Optimal. Leaf size=35 \[ \frac{x \cot ^{-1}(x)}{a \sqrt{a x^2+a}}-\frac{1}{a \sqrt{a x^2+a}} \]

[Out]

-(1/(a*Sqrt[a + a*x^2])) + (x*ArcCot[x])/(a*Sqrt[a + a*x^2])

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Rubi [A]  time = 0.0203738, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {4895} \[ \frac{x \cot ^{-1}(x)}{a \sqrt{a x^2+a}}-\frac{1}{a \sqrt{a x^2+a}} \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[x]/(a + a*x^2)^(3/2),x]

[Out]

-(1/(a*Sqrt[a + a*x^2])) + (x*ArcCot[x])/(a*Sqrt[a + a*x^2])

Rule 4895

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> -Simp[b/(c*d*Sqrt[d + e*x^2])
, x] + Simp[(x*(a + b*ArcCot[c*x]))/(d*Sqrt[d + e*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d]

Rubi steps

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

Mathematica [A]  time = 0.0227613, size = 21, normalized size = 0.6 \[ \frac{x \cot ^{-1}(x)-1}{a \sqrt{a \left (x^2+1\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[x]/(a + a*x^2)^(3/2),x]

[Out]

(-1 + x*ArcCot[x])/(a*Sqrt[a*(1 + x^2)])

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Maple [C]  time = 0.381, size = 68, normalized size = 1.9 \begin{align*}{\frac{ \left ({\rm arccot} \left (x\right )+i \right ) \left ( x+i \right ) }{ \left ( 2\,{x}^{2}+2 \right ){a}^{2}}\sqrt{a \left ( x+i \right ) \left ( x-i \right ) }}+{\frac{ \left ( x-i \right ) \left ({\rm arccot} \left (x\right )-i \right ) }{ \left ( 2\,{x}^{2}+2 \right ){a}^{2}}\sqrt{a \left ( x+i \right ) \left ( x-i \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccot(x)/(a*x^2+a)^(3/2),x)

[Out]

1/2*(arccot(x)+I)*(x+I)*(a*(x+I)*(x-I))^(1/2)/(x^2+1)/a^2+1/2*(a*(x+I)*(x-I))^(1/2)*(x-I)*(arccot(x)-I)/(x^2+1
)/a^2

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Maxima [A]  time = 1.46939, size = 42, normalized size = 1.2 \begin{align*} \frac{x \operatorname{arccot}\left (x\right )}{\sqrt{a x^{2} + a} a} - \frac{1}{\sqrt{a x^{2} + a} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(x)/(a*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

x*arccot(x)/(sqrt(a*x^2 + a)*a) - 1/(sqrt(a*x^2 + a)*a)

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Fricas [A]  time = 2.11805, size = 69, normalized size = 1.97 \begin{align*} \frac{\sqrt{a x^{2} + a}{\left (x \operatorname{arccot}\left (x\right ) - 1\right )}}{a^{2} x^{2} + a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(x)/(a*x^2+a)^(3/2),x, algorithm="fricas")

[Out]

sqrt(a*x^2 + a)*(x*arccot(x) - 1)/(a^2*x^2 + a^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acot}{\left (x \right )}}{\left (a \left (x^{2} + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acot(x)/(a*x**2+a)**(3/2),x)

[Out]

Integral(acot(x)/(a*(x**2 + 1))**(3/2), x)

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Giac [A]  time = 1.14389, size = 45, normalized size = 1.29 \begin{align*} \frac{x \arctan \left (\frac{1}{x}\right )}{\sqrt{a x^{2} + a} a} - \frac{1}{\sqrt{a x^{2} + a} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(x)/(a*x^2+a)^(3/2),x, algorithm="giac")

[Out]

x*arctan(1/x)/(sqrt(a*x^2 + a)*a) - 1/(sqrt(a*x^2 + a)*a)