Optimal. Leaf size=58 \[ -\frac{(1-4 x) e^{\cot ^{-1}(x)}}{17 a^3 \left (x^2+1\right )^2}-\frac{12 (1-2 x) e^{\cot ^{-1}(x)}}{85 a^3 \left (x^2+1\right )}-\frac{24 e^{\cot ^{-1}(x)}}{85 a^3} \]
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Rubi [A] time = 0.0668886, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {5115, 5113} \[ -\frac{(1-4 x) e^{\cot ^{-1}(x)}}{17 a^3 \left (x^2+1\right )^2}-\frac{12 (1-2 x) e^{\cot ^{-1}(x)}}{85 a^3 \left (x^2+1\right )}-\frac{24 e^{\cot ^{-1}(x)}}{85 a^3} \]
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 )^3} \, dx &=-\frac{e^{\cot ^{-1}(x)} (1-4 x)}{17 a^3 \left (1+x^2\right )^2}+\frac{12 \int \frac{e^{\cot ^{-1}(x)}}{\left (a+a x^2\right )^2} \, dx}{17 a}\\ &=-\frac{e^{\cot ^{-1}(x)} (1-4 x)}{17 a^3 \left (1+x^2\right )^2}-\frac{12 e^{\cot ^{-1}(x)} (1-2 x)}{85 a^3 \left (1+x^2\right )}+\frac{24 \int \frac{e^{\cot ^{-1}(x)}}{a+a x^2} \, dx}{85 a^2}\\ &=-\frac{24 e^{\cot ^{-1}(x)}}{85 a^3}-\frac{e^{\cot ^{-1}(x)} (1-4 x)}{17 a^3 \left (1+x^2\right )^2}-\frac{12 e^{\cot ^{-1}(x)} (1-2 x)}{85 a^3 \left (1+x^2\right )}\\ \end{align*}
Mathematica [A] time = 0.144909, size = 38, normalized size = 0.66 \[ -\frac{\left (24 x^4-24 x^3+60 x^2-44 x+41\right ) e^{\cot ^{-1}(x)}}{85 a^3 \left (x^2+1\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 36, normalized size = 0.6 \begin{align*} -{\frac{{{\rm e}^{{\rm arccot} \left (x\right )}} \left ( 24\,{x}^{4}-24\,{x}^{3}+60\,{x}^{2}-44\,x+41 \right ) }{85\, \left ({x}^{2}+1 \right ) ^{2}{a}^{3}}} \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 )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.49881, size = 116, normalized size = 2. \begin{align*} -\frac{{\left (24 \, x^{4} - 24 \, x^{3} + 60 \, x^{2} - 44 \, x + 41\right )} e^{\operatorname{arccot}\left (x\right )}}{85 \,{\left (a^{3} x^{4} + 2 \, a^{3} x^{2} + a^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 16.5687, size = 155, normalized size = 2.67 \begin{align*} - \frac{24 x^{4} e^{\operatorname{acot}{\left (x \right )}}}{85 a^{3} x^{4} + 170 a^{3} x^{2} + 85 a^{3}} + \frac{24 x^{3} e^{\operatorname{acot}{\left (x \right )}}}{85 a^{3} x^{4} + 170 a^{3} x^{2} + 85 a^{3}} - \frac{60 x^{2} e^{\operatorname{acot}{\left (x \right )}}}{85 a^{3} x^{4} + 170 a^{3} x^{2} + 85 a^{3}} + \frac{44 x e^{\operatorname{acot}{\left (x \right )}}}{85 a^{3} x^{4} + 170 a^{3} x^{2} + 85 a^{3}} - \frac{41 e^{\operatorname{acot}{\left (x \right )}}}{85 a^{3} x^{4} + 170 a^{3} x^{2} + 85 a^{3}} \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 )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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