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 = 0.03, antiderivative size = 27, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.07, 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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fricas [A] time = 0.53, size = 30, normalized size = 1.11 \[ \frac {\sqrt {a x^{2} + a} {\left (x - 1\right )} e^{\operatorname {arccot}\relax (x)}}{2 \, {\left (a^{2} x^{2} + a^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\operatorname {arccot}\relax (x)}}{{\left (a x^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 23, normalized size = 0.85 \[ \frac {\left (x^{2}+1\right ) \left (x -1\right ) {\mathrm e}^{\mathrm {arccot}\relax (x )}}{2 \left (a \,x^{2}+a \right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\operatorname {arccot}\relax (x)}}{{\left (a x^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.68, size = 25, normalized size = 0.93 \[ \frac {{\mathrm {e}}^{\mathrm {acot}\relax (x)}\,\left (\frac {x}{2\,a}-\frac {1}{2\,a}\right )}{\sqrt {a\,x^2+a}} \]
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 \[ \int \frac {e^{\operatorname {acot}{\relax (x )}}}{\left (a \left (x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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