Optimal. Leaf size=35 \[ \frac{x \cot ^{-1}(x)}{a \sqrt{a x^2+a}}-\frac{1}{a \sqrt{a x^2+a}} \]
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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.
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Rule 4895
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.
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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.
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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.
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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.
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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.
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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.
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