Optimal. Leaf size=27 \[ -\frac{(1-x) e^{\cot ^{-1}(x)}}{2 a \sqrt{a x^2+a}} \]
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Rubi [A] time = 0.028173, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {5114} \[ -\frac{(1-x) e^{\cot ^{-1}(x)}}{2 a \sqrt{a x^2+a}} \]
Antiderivative was successfully verified.
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Rule 5114
Rubi steps
\begin{align*} \int \frac{e^{\cot ^{-1}(x)}}{\left (a+a x^2\right )^{3/2}} \, dx &=-\frac{e^{\cot ^{-1}(x)} (1-x)}{2 a \sqrt{a+a x^2}}\\ \end{align*}
Mathematica [A] time = 0.0673215, size = 25, normalized size = 0.93 \[ \frac{(x-1) e^{\cot ^{-1}(x)}}{2 a \sqrt{a \left (x^2+1\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 23, normalized size = 0.9 \begin{align*}{\frac{ \left ({x}^{2}+1 \right ) \left ( x-1 \right ){{\rm e}^{{\rm arccot} \left (x\right )}}}{2} \left ( a{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\operatorname{arccot}\left (x\right )}}{{\left (a x^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.44181, size = 77, normalized size = 2.85 \begin{align*} \frac{\sqrt{a x^{2} + a}{\left (x - 1\right )} e^{\operatorname{arccot}\left (x\right )}}{2 \,{\left (a^{2} x^{2} + a^{2}\right )}} \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{e^{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\operatorname{arccot}\left (x\right )}}{{\left (a x^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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