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 = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.03, size = 21, normalized size = 0.60 \[ \frac {x \cot ^{-1}(x)-1}{a \sqrt {a \left (x^2+1\right )}} \]
Antiderivative was successfully verified.
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fricas [A] time = 3.04, size = 29, normalized size = 0.83 \[ \frac {\sqrt {a x^{2} + a} {\left (x \operatorname {arccot}\relax (x) - 1\right )}}{a^{2} x^{2} + a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 33, normalized size = 0.94 \[ \frac {x \arctan \left (\frac {1}{x}\right )}{\sqrt {a x^{2} + a} a} - \frac {1}{\sqrt {a x^{2} + a} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.70, size = 68, normalized size = 1.94 \[ \frac {\left (i+\mathrm {arccot}\relax (x )\right ) \left (x +i\right ) \sqrt {a \left (x +i\right ) \left (x -i\right )}}{2 \left (x^{2}+1\right ) a^{2}}+\frac {\sqrt {a \left (x +i\right ) \left (x -i\right )}\, \left (x -i\right ) \left (\mathrm {arccot}\relax (x )-i\right )}{2 \left (x^{2}+1\right ) a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 31, normalized size = 0.89 \[ \frac {x \operatorname {arccot}\relax (x)}{\sqrt {a x^{2} + a} a} - \frac {1}{\sqrt {a x^{2} + a} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\mathrm {acot}\relax (x)}{{\left (a\,x^2+a\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acot}{\relax (x )}}{\left (a \left (x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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