Optimal. Leaf size=9 \[ -\frac {e^{\cot ^{-1}(x)}}{a} \]
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Rubi [A] time = 0.02, antiderivative size = 9, 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 = {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*}
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Mathematica [A] time = 0.04, size = 9, normalized size = 1.00 \[ -\frac {e^{\cot ^{-1}(x)}}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 8, normalized size = 0.89 \[ -\frac {e^{\operatorname {arccot}\relax (x)}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 10, normalized size = 1.11 \[ -\frac {e^{\arctan \left (\frac {1}{x}\right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 9, normalized size = 1.00 \[ -\frac {{\mathrm e}^{\mathrm {arccot}\relax (x )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 9, normalized size = 1.00 \[ -\frac {e^{\left (\arctan \left (1, x\right )\right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 8, normalized size = 0.89 \[ -\frac {{\mathrm {e}}^{\mathrm {acot}\relax (x)}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.75, size = 7, normalized size = 0.78 \[ - \frac {e^{\operatorname {acot}{\relax (x )}}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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