Optimal. Leaf size=51 \[ -\frac{\left (\frac{1}{5}-\frac{2 i}{5}\right ) 2^{1+i} c (1-i a x)^{2-i} \, _2F_1\left (-1-i,2-i;3-i;\frac{1}{2} (1-i a x)\right )}{a} \]
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Rubi [A] time = 0.0258661, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {5073, 69} \[ -\frac{\left (\frac{1}{5}-\frac{2 i}{5}\right ) 2^{1+i} c (1-i a x)^{2-i} \, _2F_1\left (-1-i,2-i;3-i;\frac{1}{2} (1-i a x)\right )}{a} \]
Antiderivative was successfully verified.
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Rule 5073
Rule 69
Rubi steps
\begin{align*} \int e^{-2 \tan ^{-1}(a x)} \left (c+a^2 c x^2\right ) \, dx &=c \int (1-i a x)^{1-i} (1+i a x)^{1+i} \, dx\\ &=-\frac{\left (\frac{1}{5}-\frac{2 i}{5}\right ) 2^{1+i} c (1-i a x)^{2-i} \, _2F_1\left (-1-i,2-i;3-i;\frac{1}{2} (1-i a x)\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.0118533, size = 51, normalized size = 1. \[ -\frac{\left (\frac{1}{5}-\frac{2 i}{5}\right ) 2^{1+i} c (1-i a x)^{2-i} \, _2F_1\left (-1-i,2-i;3-i;\frac{1}{2} (1-i a x)\right )}{a} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.059, size = 0, normalized size = 0. \begin{align*} \int{\frac{{a}^{2}c{x}^{2}+c}{{{\rm e}^{2\,\arctan \left ( ax \right ) }}}}\, dx \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{\left (a^{2} c x^{2} + c\right )} e^{\left (-2 \, \arctan \left (a x\right )\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{2} c x^{2} + c\right )} e^{\left (-2 \, \arctan \left (a x\right )\right )}, x\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{\left (a^{2} c x^{2} + c\right )} e^{\left (-2 \, \arctan \left (a x\right )\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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