Optimal. Leaf size=87 \[ -\frac{\left (\frac{2}{5}-\frac{i}{5}\right ) 2^{\frac{1}{2}+i} (1-i a x)^{\frac{1}{2}-i} \sqrt{a^2 x^2+1} \, _2F_1\left (\frac{1}{2}-i,\frac{1}{2}-i;\frac{3}{2}-i;\frac{1}{2} (1-i a x)\right )}{a \sqrt{a^2 c x^2+c}} \]
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Rubi [A] time = 0.0746697, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {5076, 5073, 69} \[ -\frac{\left (\frac{2}{5}-\frac{i}{5}\right ) 2^{\frac{1}{2}+i} (1-i a x)^{\frac{1}{2}-i} \sqrt{a^2 x^2+1} \, _2F_1\left (\frac{1}{2}-i,\frac{1}{2}-i;\frac{3}{2}-i;\frac{1}{2} (1-i a x)\right )}{a \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 5076
Rule 5073
Rule 69
Rubi steps
\begin{align*} \int \frac{e^{-2 \tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx &=\frac{\sqrt{1+a^2 x^2} \int \frac{e^{-2 \tan ^{-1}(a x)}}{\sqrt{1+a^2 x^2}} \, dx}{\sqrt{c+a^2 c x^2}}\\ &=\frac{\sqrt{1+a^2 x^2} \int (1-i a x)^{-\frac{1}{2}-i} (1+i a x)^{-\frac{1}{2}+i} \, dx}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{\left (\frac{2}{5}-\frac{i}{5}\right ) 2^{\frac{1}{2}+i} (1-i a x)^{\frac{1}{2}-i} \sqrt{1+a^2 x^2} \, _2F_1\left (\frac{1}{2}-i,\frac{1}{2}-i;\frac{3}{2}-i;\frac{1}{2} (1-i a x)\right )}{a \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0208826, size = 87, normalized size = 1. \[ -\frac{\left (\frac{2}{5}-\frac{i}{5}\right ) 2^{\frac{1}{2}+i} (1-i a x)^{\frac{1}{2}-i} \sqrt{a^2 x^2+1} \, _2F_1\left (\frac{1}{2}-i,\frac{1}{2}-i;\frac{3}{2}-i;\frac{1}{2} (1-i a x)\right )}{a \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.285, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{{\rm e}^{2\,\arctan \left ( ax \right ) }}}{\frac{1}{\sqrt{{a}^{2}c{x}^{2}+c}}}}\, 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 \frac{e^{\left (-2 \, \arctan \left (a x\right )\right )}}{\sqrt{a^{2} c x^{2} + c}}\,{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 (\frac{e^{\left (-2 \, \arctan \left (a x\right )\right )}}{\sqrt{a^{2} c x^{2} + c}}, 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 \frac{e^{\left (-2 \, \arctan \left (a x\right )\right )}}{\sqrt{a^{2} c x^{2} + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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