Optimal. Leaf size=9 \[ -\frac{e^{\cot ^{-1}(x)}}{a} \]
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Rubi [A] time = 0.0204457, antiderivative size = 9, 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 = {5113} \[ -\frac{e^{\cot ^{-1}(x)}}{a} \]
Antiderivative was successfully verified.
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Rule 5113
Rubi steps
\begin{align*} \int \frac{e^{\cot ^{-1}(x)}}{a+a x^2} \, dx &=-\frac{e^{\cot ^{-1}(x)}}{a}\\ \end{align*}
Mathematica [A] time = 0.0319867, size = 9, normalized size = 1. \[ -\frac{e^{\cot ^{-1}(x)}}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 9, normalized size = 1. \begin{align*} -{\frac{{{\rm e}^{{\rm arccot} \left (x\right )}}}{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50966, size = 12, normalized size = 1.33 \begin{align*} -\frac{e^{\left (\arctan \left (1, x\right )\right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.53757, size = 22, normalized size = 2.44 \begin{align*} -\frac{e^{\operatorname{arccot}\left (x\right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.21094, size = 7, normalized size = 0.78 \begin{align*} - \frac{e^{\operatorname{acot}{\left (x \right )}}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11449, size = 14, normalized size = 1.56 \begin{align*} -\frac{e^{\arctan \left (\frac{1}{x}\right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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