Optimal. Leaf size=89 \[ -\frac{(1-2 a x) e^{-2 \tan ^{-1}(a x)}}{10 a c^3 \left (a^2 x^2+1\right )^2}-\frac{3 (1-a x) e^{-2 \tan ^{-1}(a x)}}{20 a c^3 \left (a^2 x^2+1\right )}-\frac{3 e^{-2 \tan ^{-1}(a x)}}{40 a c^3} \]
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Rubi [A] time = 0.0875743, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {5070, 5071} \[ -\frac{(1-2 a x) e^{-2 \tan ^{-1}(a x)}}{10 a c^3 \left (a^2 x^2+1\right )^2}-\frac{3 (1-a x) e^{-2 \tan ^{-1}(a x)}}{20 a c^3 \left (a^2 x^2+1\right )}-\frac{3 e^{-2 \tan ^{-1}(a x)}}{40 a c^3} \]
Antiderivative was successfully verified.
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Rule 5070
Rule 5071
Rubi steps
\begin{align*} \int \frac{e^{-2 \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^3} \, dx &=-\frac{e^{-2 \tan ^{-1}(a x)} (1-2 a x)}{10 a c^3 \left (1+a^2 x^2\right )^2}+\frac{3 \int \frac{e^{-2 \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{5 c}\\ &=-\frac{e^{-2 \tan ^{-1}(a x)} (1-2 a x)}{10 a c^3 \left (1+a^2 x^2\right )^2}-\frac{3 e^{-2 \tan ^{-1}(a x)} (1-a x)}{20 a c^3 \left (1+a^2 x^2\right )}+\frac{3 \int \frac{e^{-2 \tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx}{20 c^2}\\ &=-\frac{3 e^{-2 \tan ^{-1}(a x)}}{40 a c^3}-\frac{e^{-2 \tan ^{-1}(a x)} (1-2 a x)}{10 a c^3 \left (1+a^2 x^2\right )^2}-\frac{3 e^{-2 \tan ^{-1}(a x)} (1-a x)}{20 a c^3 \left (1+a^2 x^2\right )}\\ \end{align*}
Mathematica [C] time = 0.14501, size = 85, normalized size = 0.96 \[ \frac{(8 a x-4) e^{-2 \tan ^{-1}(a x)}-3 (1-i a x)^{-i} (1+i a x)^i \left (a^2 x^2+1\right ) \left (a^2 x^2-2 a x+3\right )}{40 a c^3 \left (a^2 x^2+1\right )^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.038, size = 59, normalized size = 0.7 \begin{align*} -{\frac{3\,{a}^{4}{x}^{4}-6\,{a}^{3}{x}^{3}+12\,{a}^{2}{x}^{2}-14\,ax+13}{40\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}{{\rm e}^{2\,\arctan \left ( ax \right ) }}a}} \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^{\left (-2 \, \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.0834, size = 155, normalized size = 1.74 \begin{align*} -\frac{{\left (3 \, a^{4} x^{4} - 6 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 14 \, a x + 13\right )} e^{\left (-2 \, \arctan \left (a x\right )\right )}}{40 \,{\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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^{\left (-2 \, \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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