Optimal. Leaf size=67 \[ \frac{i \sqrt{1-i a x}}{3 a \sqrt{1+i a x}}+\frac{i \sqrt{1-i a x}}{3 a (1+i a x)^{3/2}} \]
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Rubi [A] time = 0.0403993, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {5073, 45, 37} \[ \frac{i \sqrt{1-i a x}}{3 a \sqrt{1+i a x}}+\frac{i \sqrt{1-i a x}}{3 a (1+i a x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 5073
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{e^{-2 i \tan ^{-1}(a x)}}{\left (1+a^2 x^2\right )^{3/2}} \, dx &=\int \frac{1}{\sqrt{1-i a x} (1+i a x)^{5/2}} \, dx\\ &=\frac{i \sqrt{1-i a x}}{3 a (1+i a x)^{3/2}}+\frac{1}{3} \int \frac{1}{\sqrt{1-i a x} (1+i a x)^{3/2}} \, dx\\ &=\frac{i \sqrt{1-i a x}}{3 a (1+i a x)^{3/2}}+\frac{i \sqrt{1-i a x}}{3 a \sqrt{1+i a x}}\\ \end{align*}
Mathematica [A] time = 0.0142586, size = 48, normalized size = 0.72 \[ \frac{\sqrt{1-i a x} (2+i a x)}{3 a \sqrt{1+i a x} (a x-i)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.067, size = 93, normalized size = 1.4 \begin{align*} -{\frac{1}{{a}^{2}} \left ({\frac{{\frac{i}{3}}}{a}\sqrt{{a}^{2} \left ( x-{\frac{i}{a}} \right ) ^{2}+2\,ia \left ( x-{\frac{i}{a}} \right ) } \left ( x-{\frac{i}{a}} \right ) ^{-2}}-{\frac{1}{3}\sqrt{{a}^{2} \left ( x-{\frac{i}{a}} \right ) ^{2}+2\,ia \left ( x-{\frac{i}{a}} \right ) } \left ( x-{\frac{i}{a}} \right ) ^{-1}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47495, size = 80, normalized size = 1.19 \begin{align*} -\frac{i \, \sqrt{a^{2} x^{2} + 1}}{3 \, a^{3} x^{2} - 6 i \, a^{2} x - 3 \, a} + \frac{i \, \sqrt{a^{2} x^{2} + 1}}{3 i \, a^{2} x + 3 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.89626, size = 117, normalized size = 1.75 \begin{align*} \frac{a^{2} x^{2} - 2 i \, a x + \sqrt{a^{2} x^{2} + 1}{\left (a x - 2 i\right )} - 1}{3 \, a^{3} x^{2} - 6 i \, a^{2} x - 3 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a^{2} x^{2} + 1} \left (i a x + 1\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21691, size = 93, normalized size = 1.39 \begin{align*} -\frac{2 \,{\left (3 \, a i{\left (\sqrt{a^{2} + \frac{1}{x^{2}}} - \frac{1}{x}\right )} + 2 \, a^{2} - 3 \,{\left (\sqrt{a^{2} + \frac{1}{x^{2}}} - \frac{1}{x}\right )}^{2}\right )} i^{2}}{3 \,{\left (a i - \sqrt{a^{2} + \frac{1}{x^{2}}} + \frac{1}{x}\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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