Optimal. Leaf size=35 \[ -\frac{(1-2 x) e^{\cot ^{-1}(x)}}{5 a^2 \left (x^2+1\right )}-\frac{2 e^{\cot ^{-1}(x)}}{5 a^2} \]
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Rubi [A] time = 0.0431212, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {5115, 5113} \[ -\frac{(1-2 x) e^{\cot ^{-1}(x)}}{5 a^2 \left (x^2+1\right )}-\frac{2 e^{\cot ^{-1}(x)}}{5 a^2} \]
Antiderivative was successfully verified.
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Rule 5115
Rule 5113
Rubi steps
\begin{align*} \int \frac{e^{\cot ^{-1}(x)}}{\left (a+a x^2\right )^2} \, dx &=-\frac{e^{\cot ^{-1}(x)} (1-2 x)}{5 a^2 \left (1+x^2\right )}+\frac{2 \int \frac{e^{\cot ^{-1}(x)}}{a+a x^2} \, dx}{5 a}\\ &=-\frac{2 e^{\cot ^{-1}(x)}}{5 a^2}-\frac{e^{\cot ^{-1}(x)} (1-2 x)}{5 a^2 \left (1+x^2\right )}\\ \end{align*}
Mathematica [A] time = 0.106613, size = 28, normalized size = 0.8 \[ -\frac{\left (2 x^2-2 x+3\right ) e^{\cot ^{-1}(x)}}{5 a^2 \left (x^2+1\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 26, normalized size = 0.7 \begin{align*} -{\frac{{{\rm e}^{{\rm arccot} \left (x\right )}} \left ( 2\,{x}^{2}-2\,x+3 \right ) }{ \left ( 5\,{x}^{2}+5 \right ){a}^{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 )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.37982, size = 70, normalized size = 2. \begin{align*} -\frac{{\left (2 \, x^{2} - 2 \, x + 3\right )} e^{\operatorname{arccot}\left (x\right )}}{5 \,{\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 [B] time = 5.07343, size = 65, normalized size = 1.86 \begin{align*} - \frac{2 x^{2} e^{\operatorname{acot}{\left (x \right )}}}{5 a^{2} x^{2} + 5 a^{2}} + \frac{2 x e^{\operatorname{acot}{\left (x \right )}}}{5 a^{2} x^{2} + 5 a^{2}} - \frac{3 e^{\operatorname{acot}{\left (x \right )}}}{5 a^{2} x^{2} + 5 a^{2}} \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 )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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