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 = 0.04, antiderivative size = 67, normalized size of antiderivative = 1.00, 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 37
Rule 45
Rule 5073
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.50, size = 51, normalized size = 0.76 \[ \frac {a^{2} x^{2} - 2 i \, a x + \sqrt {a^{2} x^{2} + 1} {\left (a x - 2 i\right )} - 1}{3 \, a^{3} x^{2} - 6 i \, a^{2} x - 3 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 66, normalized size = 0.99 \[ \frac {2 \, {\left (3 \, a i {\left (\sqrt {a^{2} + \frac {1}{x^{2}}} - \frac {1}{x}\right )} + 2 \, a^{2} - 3 \, {\left (\sqrt {a^{2} + \frac {1}{x^{2}}} - \frac {1}{x}\right )}^{2}\right )}}{3 \, {\left (a i - \sqrt {a^{2} + \frac {1}{x^{2}}} + \frac {1}{x}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 93, normalized size = 1.39 \[ -\frac {\frac {i \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{3 a \left (x -\frac {i}{a}\right )^{2}}-\frac {\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{3 \left (x -\frac {i}{a}\right )}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 59, normalized size = 0.88 \[ -\frac {i \, \sqrt {a^{2} x^{2} + 1}}{3 \, a^{3} x^{2} - 6 i \, a^{2} x - 3 \, a} + \frac {i \, \sqrt {a^{2} x^{2} + 1}}{3 i \, a^{2} x + 3 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 31, normalized size = 0.46 \[ -\frac {\sqrt {a^2\,x^2+1}\,\left (a\,x-2{}\mathrm {i}\right )}{3\,a\,{\left (1+a\,x\,1{}\mathrm {i}\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {1}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} - 2 i a x \sqrt {a^{2} x^{2} + 1} - \sqrt {a^{2} x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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